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0704.1527 | Study of exotic hadrons in s-wave chiral dynamics | Study of exotic hadrons in s-wave chiral dynamics
Tetsuo Hyodo1,∗), Daisuke Jido1 and Atsushi Hosaka2
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502,
Japan
2Research Center for Nuclear Physics (RCNP), Ibaraki, 567-0047 Japan
We study the exotic hadrons in s-wave scattering of the Nambu-Goldstone boson with a
target hadron based on chiral dynamics. Utilizing the low energy theorem of chiral symme-
try, we show that the s-wave interaction is not strong enough to generate bound states in
exotic channels in flavor SU(3) symmetric limit, although the interaction is responsible for
generating some nonexotic hadron resonances dynamically. We discuss the renormalization
condition adopted in this analysis.
§1. Introduction
One of the nontrivial issues in hadron physics is almost complete absence of
flavor exotic hadrons. Experimentally, we have been observing more than hundred
of hadrons,1) whose flavor quantum numbers can be expressed by minimal valence
quark contents of q̄q or qqq. The only one exception is the exotic baryon Θ+ with
S = +1,2) which is composed of at lease five valence quarks. In this way, the exotic
hadrons are indeed “exotic” as an experimental fact. On the other hand, there is
no clear theoretical explanation for the nonobservation of the exotic hadrons. Our
current knowledge does not forbid to construct four or five quark states in QCD and
in effective models. Moreover, the multiquark components in nonexotic hadrons are
evident, as seen in the antiquark distribution (or pion cloud) in nucleon and successful
descriptions of some excited hadrons as resonances in two-hadron scatterings. In view
of these facts, it is fair to say that the nonobservation of the exotic hadrons is not
fully understood theoretically.
§2. Exotic hadrons in s-wave chiral dynamics
In chiral coupled-channel dynamics, some hadron resonances have been success-
fully described in s-wave scattering of a hadron and the Nambu-Goldstone (NG)
boson,3), 4), 5), 6) along the same line with the old studies with phenomenological vec-
tor meson exchange interaction.7), 8) It was found that the generated resonances
turned into bound states in flavor SU(3) symmetric limit.9) We therefore conjecture
that the bound states in the SU(3) limit are the origin of a certain class of physical
resonances, and we examine the possible existence of exotic hadrons as hadron-NG
boson bound states.10), 11)
The low energy interaction of the NG boson (Ad) with a target hadron (T ) in
∗) e-mail address: [email protected]
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1527v1
2 T. Hyodo, D. Jido and A. Hosaka
s-wave is given by
Vα = −
Cα,T , (2.1)
with the decay constant (f) and the energy (ω) of the NG boson. The factor Cα,T is
determined by specifying the flavor representations of the target T and the scattering
system α ∈ T⊗Ad:
Cα,T = −〈2FT · FAd〉α = C2(T )− C2(α) + 3, (2.2)
where C2(R) is the quadratic Casimir of SU(3) for the representation R. Eq. (2.1)
is the model-independent consequence of chiral symmetry, known as the Weinberg-
Tomozawa theorem.12), 13)
We have written down the general expression of the coupling strengths (2.2)
for arbitrary representations of target hadrons in SU(3). In order to specify the
exotic channels, we introduced the exoticness quantum number, as the number of
valence antiquarks to construct the given flavor multiplet for the states with positive
baryon number. Then we find that the Weinberg-Tomozawa interaction in the exotic
channels is repulsive in most cases, and that possible strength of the attractive
interaction is given by a universal value
Cexotic = 1, (2.3)
with α = [p− 1, 2] for T = [p, 0] and p ≥ 3B.10), 11)
Next we construct the scattering amplitude with unitarity condition using the
N/D method.5) The unitarized amplitude is given by
1− Vα(
as a function of the center-of-mass energy
s. The loop function G(
s) is regularized
by the once subtraction as
s) = −ã(s0)−
ρ(s′)
s′ − s
− ρ(s
s′ − s0
, (2.4)
where the phase space integrand is ρ(s) = 2MT
(s− s+)(s− s−)/(8πs), s± =
(m±MT )2, and m and MT are the masses of the target hadron and the NG boson.
In order to determine the subtraction constant ã(s0) and the subtraction point
s0, we adopt the renormalization condition given in Refs. 14), 6),
G(µ) = 0, µ = MT , (2.5)
which is equivalent to tα(µ) = Vα(µ) at this scale. We will discuss the implication
of this prescription in section 3. With the condition (2.5), we show that the bound
state can be obtained if the coupling strength (2.2) is larger than the critical value
Ccrit =
−G(MT +m)
Exotic hadrons in s-wave chiral dynamics 3
Varying the parameters m, MT , and f in the physically allowed region, we show
that the attraction in the exotic channels (2.3) is always smaller than the critical
value Ccrit. Thus, it is not possible to generate bound states in exotic channels in
the SU(3) symmetric limit.
We would like to emphasize that this conclusion is model independent in the
SU(3) limit, as far as we respect chiral symmetry. In this study, we only consider
the exotic hadrons composed of the NG boson and a hadron, so the existence of
exotic states generated by quark dynamics or rotational excitations of chiral solitons
is not excluded. In practice, one should bear in mind that the SU(3) breaking
effect and higher order terms in the chiral Lagrangian would play a substantial role.
Nevertheless, the study of exotic hadrons in a simple extension of a successful model
of hadron resonances as we have done in the present work can partly explain difficulty
of observation of the exotic hadrons.
§3. Interpretation of the renormalization condition
Here we discuss the renormalization condition (2.5) in this analysis. The inter-
action kernel Vα(
s) is constructed from chiral perturbation theory so as to satisfy
the low energy theorem. The low energy theorem also constrains the behavior of the
full unitarized amplitude tα(
s) at a scale
s = µm where the chiral expansion is
valid. Therefore we can match the unitarized amplitude tα(
s) with the tree level
one Vα(
s) at the scale µm:
tα(µm) = Vα(µm) + Vα(µm)G(µm)Vα(µm) + · · · = Vα(µm), (3.1)
This condition determines the subtraction constant such that the loop function
G(µm) vanishes. This is only possible within the region
MT −m ≤ µm ≤ MT +m, (3.2)
since the loop function has an imaginary part outside this region and the subtraction
constant is a real number. We consider that by employing this renormalization
condition, a natural unitarization of the kernel interaction based on chiral symmetry
is realized. Interestingly, if we apply this prescription for the case of the octet baryon
target, the subtraction constant turns out to be “natural size” which was found in
the comparison with three-momentum cutoff,5) and the experimental observables in
the S = −1 meson-baryon channel are successfully reproduced.
We take µm = MT in the present study. The dependence on µm within the
region (3.2) is found that the binding energy of the bound state increases if we
shift the matching scale µm to the lower energy region. This is discussed also in
Refs. 17),18) by varying the subtraction constant. The region µm ≤ MT corresponds
to the u-channel scattering. Thus µm = MT is the most favorable to generate a bound
state within the s-channel regime.
The unitarized amplitudes in the above prescription do not always reproduce ex-
perimental data. In such a case, the subtraction constants ã(s0) should be adjusted
in order to satisfy experimental data. The subtraction constants determined in this
4 T. Hyodo, D. Jido and A. Hosaka
way supplement the role of the higher order chiral Lagrangians, which is lacking in
the kernel interaction. As shown for the ρ meson effect in the meson-meson scatter-
ing,16) the higher order terms may contain the effect of the resonances. Therefore,
if the natural condition (3.1) is badly violated, one may speculate that the seeds
of resonances in the higher order Lagrangian, which are possibly the genuine quark
states, appear in the unitarized amplitude, as in the study of Ref. 19).
In summary, we have argued the following issues.
• In order for the unitarized amplitude tα(
s) to satisfy the low energy theorem,
the loop function should vanish in the region (3.2).
• This requirement can be regarded as a natural unitarization, without introduc-
ing the effect of resonances in the higher order Lagrangian.
• Lower matching scale µm is more favorable to generate a bound state.
Turning to the problem of exotic hadrons, what we have shown is the nonexistence
of the exotic bound states with the most favorable condition to generate a bound
state, without introducing the seed of genuine quark state.
Acknowledgements
T. H. thanks the Japan Society for the Promotion of Science (JSPS) for finan-
cial support. This work is supported in part by the Grant for Scientific Research
(No. 17959600, No. 18042001, and No. 16540252) and by Grant-in-Aid for the 21st
Century COE ”Center for Diversity and Universality in Physics” from the Ministry
of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
References
1) Particle Data Group, W.M. Yao et al., J. of Phys. G33 (2006), 1.
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4) E. Oset and A. Ramos, Nucl. Phys. A 635 (1998), 99
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7) R.H. Dalitz and S.F. Tuan, Ann. of Phys. 10 (1960), 307
8) J.H.W. Wyld, Phys. Rev. 155 (1967), 1649
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10) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. Lett. 97 (2006), 192002
11) T. Hyodo, D. Jido, and A. Hosaka, Phys. Rev. D 75 (2007), 034002
12) S. Weinberg, Phys. Rev. Lett. 17 (2966), 616
13) Y. Tomozawa, Nuovo Cim. 46A (1966), 707
14) K. Igi and K.-i. Hikasa, Phys. Rev. D 59 (1999), 034005
15) T. Hyodo, D. Jido and A. Hosaka, hep-ph/0612333
16) G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989), 311
17) T. Hyodo, S.I. Nam, D. Jido and A. Hosaka, Phys. Rev. C 68 (2003), 018201
18) T. Hyodo, S.I. Nam, D. Jido, and A. Hosaka, Prog. Theor. Phys. 112 (2004), 73
19) J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. D 59 (1999), 074001, Erratum-ibid. D 60
(1999) 099906.
http://arxiv.org/abs/hep-ph/0612333
Introduction
Exotic hadrons in s-wave chiral dynamics
Interpretation of the renormalization condition
|
0704.1528 | Extremely strong-coupling superconductivity and anomalous lattice
properties in the beta-pyrochlore oxide KOs2O6 | Microsoft Word - K-SC6.doc
April 2, 2007
I. INTRODUCTION
The study of non-Cu-based oxide superconductors has been
extended during the last decade, aiming at understanding the role of
electron correlations in the mechanism of superconductivity or
searching for a novel pairing mechanism, hopefully to reach a
higher Tc. An interesting example recently found is a family of
pyrochlore oxide superconductors. The first discovered is
α-pyrochlore rhenate Cd2Re2O7 with Tc = 1.0 K
1-3 and the second
β-pyrochlore osmate AOs2O6 with Tc = 3.3, 6.3, and 9.6 K for A =
Cs,4 Rb,5-7 and K,8 respectively. They crystallize in the cubic
pyrochlore structure of space group Fd-3m and commonly possess
a 3D skeleton made of ReO6 or OsO6 octahedra.
9 The "pyrochlore"
sublattice occupied by the transition metal ions is comprised of
corner-sharing tetrahedra that are known to be highly frustrating for
a localized spin system with antiferromagnetic nearest-neighbor
interactions.
A unique structural feature for the β-pyrochlores is that a
relatively small A ion is located in an oversized atomic cage made
of OsO6 octahedra, Fig. 1. Due to this large size mismatch, the A
atom can rattle in the cage.10 The rattling has been recognized
recently as an interesting phenomenon for a class of compounds
like filled skutterudites11 and Ge/Si clathrates12 and attracted many
researchers, because it may suppress thermal conductivity leading
to an enhanced thermoelectric efficiency. On the other hand, the
rattling is also intriguing from the viewpoint of lattice dynamics:
it gives an almost localized mode even in a crystalline material and
often exhibits unusual anharmonicity.11 Hence, it is considered that
the rattling is a new type of low-lying excitations that potentially
affects various properties in a crystal at low temperature.
In the β pyrochlores, specific heat experiments found
low-energy contributions that could be described approximately by
the Einstein model and determined the Einstein temperature TE to
be 70 K, 60 K, and 40 or 31 K for A = Cs, Rb, and K,
respectively.13, 14 This tendency illustrates uniqueness of the
rattling, because, to the contrary, one expects a higher frequency
for a lighter atom in the case of conventional phonons. Moreover,
it was demonstrated that the specific heat shows an unusual T5
dependence at low temperature below 7 K for A = Cs and Rb,
instead of a usual T3 dependence from a Debye-type phonon.13
On one hand, structural refinements revealed large atomic
displacement parameters at room temperature of 100Uiso = 2.48,
4.26, and 7.35 Å2 for A = Cs, Rb, and K, respectively,10 and 3.41
Å2 for Rb.15 Particularly, the value for K is enormous and may be
the largest among rattlers so far known in related compounds.
This trend over the β-pyrochlore series is ascribed to the fact that
an available space for the A ion to move in a rather rigid cage
increases with decreasing the ionic radius of the A ion.10 Kuneš et
al. calculated an energy potential for each A ion and found in fact a
large anharmonicity, that is, a deviation from a quadratic form
expected for the harmonic oscillator approximation.16 Especially
for the smallest K ion, they found 4 shallow potential minima
locating away from the center (8b site) along the <111> directions
pointing to the nearest K ions, as schematically shown in Fig. 1.
The potential minima are so shallow that the K ion may not stop at
one of them even at very low temperature.
The electronic structures of α-Cd2Re2O7 and β-AOs2O6 have
been calculated by first-principle density-functional methods,
which reveal that a metallic conduction occurs in the (Re, Os)-O
network:16-20 electronic states near the Fermi level originate from
transition metal 5d and O 2p orbitals. Although the overall shape
of the density of states (DOS) is similar for the two pyrochlores, a
difference in band filling may result in different properties; Re5+
for α-Cd2Re2O7 has two 5d electrons, while Os
5.5+ for β-AOs2O6
has two and a half. Moreover, a related α-pyrochlore Cd2Os2O7
with Os5+ (5d3) exhibits a metal-to-insulator transition at 230 K.21,
Various experiments have been carried out on the pyrochlore
oxide superconductors to elucidate the mechanism of the
superconductivity. Most of the results obtained for α-Cd2Re2O7
indicate that it is a weak-coupling BCS-type superconductor.23, 24
In contrast, results on the β-pyrochlores are somewhat
controversial. Although the Tc increases smoothly from Cs to K,
the jump in specific heat at Tc, the upper critical field, and the
Sommerfeld coefficient all exhibit a large enhancement toward K.25
Thus, the K compound is always distinguished from the others.
Pressure dependence of Tc was also studied for the two pyrochlores,
showing a common feature: as pressure increases, Tc first increases,
exhibits a broad maximum and goes to zero above a critical
pressure that depends on the system, for example, about 6 GPa for
the K compound.26-29
On the symmetry of the superconducting gap for the
β-pyrochlores, Rb-NMR and µSR experiments gave evidence for
s-wave superconductivity for RbOs2O6.
30-32 In contrast, Arai et al.
carried out K-NMR experiments and found no coherence peak in
the relaxation rate below Tc for KOs2O6, which seemed to indicate
unconventional superconductivity.32 However, their recent
interpretation is that the absence of a coherence peak does not
necessarily mean non-s pairing, because the relaxation rate probed
by the K nuclei can be affected dominantly by strongly
overdamped phonons.33 Kasahara et al. measured thermal
conductivity using a KOs2O6 single crystal and concluded a full
Extremely strong-coupling superconductivity and anomalous lattice properties in
the β-pyrochlore oxide KOs2O6
Z. Hiroi, S. Yonezawa, Y. Nagao and J. Yamaura
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Superconducting and normal-state properties of the β-pyrochlore oxide KOs2O6 are studied by means of
thermodynamic and transport measurements. It is shown that the superconductivity is of conventional s-wave
type and lies in the extremely strong-coupling regime. Specific heat and resistivity measurements reveal that
there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong
electron-phonon interactions. The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion
confined in an oversized cage made of OsO6 octahedra. It is suggested that this electron-rattler coupling
mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity.
gap from the insensitivity of thermal conductivity to magnetic
fields.34 Moreover, very recent photoemission spectroscopy
(PES) experiments revealed the opening of a large isotropic gap
below Tc.
35 On one hand, a µSR experiment claimed that the gap
of KOs2O6 is anisotropic, or otherwise, there are two gaps.
Therefore, the pairing symmetry of the β-AOs2O6 superconductors
may be of the conventional s wave, aside from minor aspects such
as anisotropy or multi gaps, which means that the fundamental
pairing mechanism is ascribed to phonons. Then, an important
question is what kind of phonons are relevant for the occurrence.
To find out the reason of the observed singular appearance toward
K in the series must be the key to understand interesting physics
involved in this system.
Previous studies have suffered poor quality of samples,
because only polycrystalline samples were available. The
Sommerfeld coefficient γ was estimated from specific heat by
extracting contributions from Os metal impurity to be 40 mJ K-2
mol-1 for both Cs and Rb,13 and 34 mJ K-2 mol-1 for Rb.7 Recently,
Brühwiler et al. obtained large values of γ = 76-110 mJ K-2 mol-1
for KOs2O6 by collecting five dozen of tiny crystals.
14 However,
there is an ambiguity in their values, because of uncertainty in their
extrapolation method. They also reported strong-coupling
superconductivity with a coupling constant λep = 1.0-1.6.
Recently, we successfully prepared a large single crystal of 1 mm
size for KOs2O6 and reported two intriguing phenomena: one is a
sharp and huge peak in specific heat at Tp = 7.5-7.6 K below Tc,
indicative of a first-order structural transition,37, 38 and the other is
anisotropic flux pinning at low magnetic fields around 2 T.39 It
was suggested that the former is associated with the rattling
freedom of the K ion. Moreover, anomalous concave-downward
resistivity was observed down to Tc, suggesting a peculiar
scattering mechanism of carriers.
In this paper, we present specific heat, magnetization and
resistivity measurements on the same high-quality single crystal of
KOs2O6. Reliable data on the superconducting and normal-state
properties are obtained, which provide evidence for an extremely
strong-coupling superconductivity realized in this compound. We
discuss the role of rattling vibrations of the K ion on the
mechanism of the superconductivity.
II. EXPERIMENTAL
A. Sample preparation
A high-quality single crystal was prepared and used for all the
measurements in the present study, which was named KOs-729
after the date of July 29, 2005 when the first experiment was
performed on this crystal. It was grown from a pellet containing
an equimolar mixture of KOsO4 and Os metal in a sealed quartz
tube at 723 K for 24 h. Additional oxygen was supplied by using
the thermal decomposition of AgO placed away from the pellet in
the tube. The KOsO4 powder had been prepared in advance from
KO2 and Os metal in the presence of excess oxygen. It was
necessary to pay attention to avoid the formation of OsO4 in the
course of preparation, which is volatile even at room temperature
and highly toxic to eyes or nose. After the reaction, several tiny
crystals had grown on the surface of the pellet. Although the
mechanism of the crystal growth has not yet been understood
clearly, probably it occurs through partial melting and the
following reaction with a vapor phase.
The KOs-729 crystal possesses a truncated octahedral shape
with a large (111) facet as shown in Fig. 2 and is approximately 1.0
× 0.7 × 0.3 mm3 in size and 1.302 mg in weight. The high quality
of the crystal has been demonstrated by a sharp peak at Tp in
specific heat,38 which was absent in the previous polycrystalline
samples or appeared as broad humps in our previous aggregate of
tiny crystals37 or in five dozen of tiny crystals by Brühwiler et al.14
Moreover, a dramatic angle dependence of flux-flow resistance was
observed on this crystal, indicating that flux pinning is enhanced in
magnetic fields along certain crystallographic directions such as
[110], [001], and [112].39 This evidences the absence of domains
in this relatively large crystal.
A special care has been taken to keep the crystal always in a
dry atmosphere, because it readily undergoes hydration in air, as
reported in isostructural compounds such as KNbWO6.
40, 41 Once
partial hydration takes place in KOs2O6, the second sharp anomaly
in specific heat tends to collapse. In contrast, the superconducting
transition was robust, just slightly broadened after hydration. The
hydration must be relatively slow in a single crystal compared with
the case of polycrystalline samples, which may be the reason for
the broad anomaly or the absence in previous samples.
B. Physical-property measurements
Both specific heat and electrical resistivity were measured in a
temperature range between 300 K and 0.4 K and in magnetic fields
of up to 14 T in a Quantum Design Physical Property Measurement
System (PPMS) equipped with a 3He refrigerator. The magnetic
fields had been calibrated by measuring the magnetization of a
standard Pd specimen and also by measuring the voltage of a Hall
device (F.W. BELL, BHA-921). Specific heat measurements
FIG. 1. (Color online) Crystal structure of the β-pyrochlore
oxide KOs2O6. The K ion (big ball) is located in an oversized
atomic cage made of OsO6 octahedra and can move along the 4
<111> directions pointing to the neighboring K ions in adjacent
cages.
FIG. 2. (Color online) Photograph of the KOs-729 crystal used
in the present study. It possesses a truncated octahedral shape
with 111 facets. The approximate size is 1.0 × 0.7 × 0.3 mm3.
Resistivity measurements were carried out with a current flow
along the [-110] direction.
were performed by the heat-relaxation method. The KOs-729
crystal was attached to an alumina platform by a small amount of
Apiezon N grease. In each measurement, heat capacity was
obtained by fitting a heat relaxation curve recorded after a heat
pulse giving a temperature rise of approximately 2%. The heat
capacity of an addendum had been measured in a separate run
without a sample, and was subtracted from the data. The
measurements were done three times at each temperature with a
scatter less than 0.3% at most.
Resistivity measurements were carried out by the four-probe
method with a current flow along the [-110] direction and magnetic
fields along the [111], [110], [001] or [112] direction of the cubic
crystal structure. All the measurements were done at a current
density of 1.5 A cm-2. Magnetization was measured in magnetic
fields up to 7 T in a Quantum Design magnetic property
measurement system and also up to 14 T in PPMS. The magnetic
fields were applied approximately along the [111] or [-110]
direction.
III. RESULTS
A. Superconducting properties
1. Specific heat
First of all, we analyze specific heat data in order to obtain a
reliable value of the Sommerfeld coefficient γ. There are two
obstacles: one is the large upper critical field Hc2 that is
approximately 2 times greater than our experimental limit of 14 T.
Brühwiler et al. reported γ = 76 (110) mJ K-2 mol-1 assuming µ0Hc2
= 24 (35) T by an extrapolation method.14 Their values should be
modified to γ ~ 100 mJ K-2 mol-1, because recent high magnetic
field experiments revealed µ0Hc2 = 30.6 T or 33 T.
42, 43
Nevertheless, there are still large ambiguity in their extrapolation
method using specific heat data obtained only at H/Hc2 < 0.5. The
other difficulty comes from unusual lattice contributions in specific
heat at low temperature and the existence of a sharp peak at Tp.
Thus, it is not easy to extract the lattice contribution in a standard
way used so far. Here we carefully analyze specific heat data and
reasonably divide them into electronic and lattice parts, from which
a reliable value of γ is determined, and information on the
superconducting gap is attained.
Figure 3 shows the temperature dependence of specific heat of
the KOs-729 crystal measured on cooling at zero field and in a
magnetic field of 14 T applied along the [111] direction. A
superconducting transition at zero field takes place with a large
jump, followed by a huge peak due to the second phase transition at
Tp = 7.5 K. The entropy-conserving construction shown in the inset
gives Tc = 9.60 K, ΔC/Tc = 201.2 mJ K
-2 mol-1, which is close to
the values previously reported,14, 37 and a transition width (ΔTc) of
0.3 K. Tc is reduced to 5.2 K at 14 T, which is evident as a bump
in the 14-T data shown in Fig. 3. In contrast, Brühwiler et al.
reported Tc = 6.2 K at 14 T, though the transition was not clearly
observed in their specific heat data. The C/T at zero field rapidly
decreases to zero as T approaches absolute zero. The absence of a
residual T-linear contribution in specific heat indicates the high
quality of the crystal.
The specific heat of a crystal (C) is the sum of an electronic
contribution (Ce) and an H-independent lattice contribution (Cl).
The former becomes Cen for the normal state above Tc, which is
taken as γT, and Ces for the superconducting state below Tc. The γ
is assumed to be T-independent, though it can not be the case for
compounds with strong electron-phonon couplings.44 In the case
of KOs2O6, Cl is large relative to Ce: for example, Cl is as large as
~90% of the total C at just above Tc, as shown later. In order to
determine the value of γ, it is crucial to know the low-temperature
form of Cl. Since the minimum Tc attained at 14 T is 5.2 K, one
has to estimate the Cl from the T dependence of the total C above
~5.5 K. Two terms in the harmonic-lattice approximation are
often required for an adequate fit; Cl = β3T
3 + β5T
5. The first term
comes from a Debye-type acoustic phonon, and thus is dominant at
low temperature, while the second term expresses a deviation at
high temperature. Actually, this approximation is valid for
α-Cd2Re2O7, where β3 = 0.222 mJ K
-4 mol-1 and β5 = 2.70 × 10
mJ K-6 mol-1 are obtained by a fit to the data below 10 K.1 The
Debye temperature ΘD is 458 K from the β3 value. In strong
contrast, it was found for two members of β-AOs2O6 that the T
term prevails in a wide temperature range; β5 = 14.2 × 10
-3 mJ K-6
mol-1 below 5 K for CsOs2O6 and β5 = 30.2 × 10
-3 mJ K-6 mol-1
below 7 K for RbOs2O6.
The C/T at H = 0 below 7 K shown in Fig. 3 is again plotted in
two ways as functions of T2 and T4 in Fig. 4. It is apparent from
the T2 plot that possible T3 terms expected for ΘD = 458 K and 300
K are negligibly small compared with the whole magnitude of
specific heat, just as observed in other members. On the other
hand, in the T4 plot, there is distinct linear behavior at low
temperature below 4 K, indicating that the C approaches
asymptotically to T5 behavior as T → 0 with a large slope of
0.3481(6) mJ K-6 mol-1. It is reasonable to ascribe this T5
contribution to the lattice, because Ces should decrease quickly as T
→ 0. Note that the value of the β5 for KOs2O6 is more than one
order larger than those in other members. At high temperatures
above 4 K, a downward deviation from the initial T5 behavior is
observed in Fig. 4b. The temperature dependence of the 14-T
data above 5.5 K, which is taken as γT + Cl, is also close to T
5, but
with a smaller slope, which means that a single T5 term is not
appropriate to describe the Cl in such a wide temperature range and
also that an inclusion of higher order term of Tn is not helpful.
Therefore, we adopt expediently an alternative empirical form to
express this strange lattice contribution; Cl = β5T
5f(T), where f(T) =
[1 + exp(1 - pT-q)]-1. Since the f(T) is almost unity below a certain
temperature and decreases gradually with increasing T, this Cl can
reproduce T5 behavior at low temperatures and a weaker T
FIG. 3. (Color online) Specific heat divided by
temperature measured at zero field (circle) and a magnetic
field of 14 T (triangle) applied along the [111] direction.
The inset shows an enlargement of the superconducting
transition with an entropy-conserving construction.
dependence at high temperatures. As shown in Fig. 4b, the 14-T
data in the 5.5-7 K range can be fitted well by the function for a
value of β5 fixed to the initial slope of 0.3481 mJ K
-6 mol-1 and a
given value of γ; for example, p = 6.96(3) and q = 1.09(1) for γ =
70 mJ K-2 mol-1.
In order to determine the value of γ univocally, the entropy
conservation is taken into account for the 14-T data, as shown in
Fig. 5: since the normal-state specific heat expected for the case of
Tc = 0 is given by (Cen + Cl) (dotted line in Fig. 5), the integration
of [Cen + Cl - C(14 T)]/T should become zero due to entropy
balance. It is shown in the inset to Fig. 5 that the integrated value
changes almost linearly with γ and vanishes around γ = 70 mJ K-2
mol-1. Hence, one can determine the value of γ unambiguously.
A certain ambiguity may arise from the assumed lattice function.
However, since the temperature dependence of Cl is substantially
weak in the T range of interest, a possible correction on the γ value
must be minimal, say, less than 1 mJ K-2 mol-1.
Next we determine Ces at zero field by subtracting the Cl
estimated above. The temperature dependence of Ces does have
the BCS form, aexp(-Δ/kBTc), as shown in Fig. 6. The energy gap
Δ obtained by fitting is 22.5 K, which corresponds to 2Δ/kBTc =
4.69, much larger than the BCS value of 3.53. The above Ces at
low temperature below 7 K is again plotted in Fig. 7 together with
high-temperature Ces above 5.5 K, which is obtained by subtracting
the 14-T data from the zero field data as Ces = C(0) - C(14 T).
The two data sets obtained independently overlap well in the 5.5-7
K range, assuring the validity of the above analyses. Because of
the existence of the second peak and its small shifts under magnetic
fields, the data between 7 K and 8.3 K is to be excluded in the
following discussion. Taking γ = 70 mJ K-2 mol-1, the jump in
specific heat at Tc, ΔC/γTc, reaches 2.87, much larger than 1.43
expected for a weak-coupling superconductor, indicating that
KOs2O6 lies in the strong-coupling regime. Comparisons to other
typical strong-coupling superconductors are made in section IV-B.
Here we analyze the data based on the α model that was
developed to provide a semi-empirical approximation to the
thermodynamic properties of strong-coupled superconductors in a
wide range of coupling strengths with a single adjustable parameter,
α = Δ0/kBTc.
45 Recently, it was generalized to a multi-gap
superconductor and successfully applied to the analyses on MgB2
or Nb3Sn.
46, 47 Using the α model, the data in the vicinity of Tc is
well reproduced, as shown in Fig. 7, and we obtain α = 2.50
(2Δ0/kBTc = 5.00), slightly larger than the value obtained above
from the temperature dependence of Ces. One interesting point to
be noted is that there is a significant deviation between the data and
the fitting curve at intermediate temperatures, suggesting the
existence of an additional structure in the gap. Presumably, this
enhancement would be explained if one assumes the coexistence of
another smaller gap (not so small as in MgB2, but intermediate).
This possibility has been already pointed out in the previous µSR
experiment.36 However, ambiguity associated with the second
peak in the present data prevents us from further analyzing the data.
This important issue will be revisited in future work, where the
FIG. 6. (Color online) Temperature dependence of electronic
specific heat measured at H = 0 for the superconducting state
showing an exponential decrease at low temperature. A magnitude
of the gap obtained is 2Δ/kBTc = 4.69.
FIG. 4. (Color online) Low-temperature specific heat below 7 K
plotted as functions of T2 (a) and T4 (b). The broken and dotted
lines in (a) show calculated contributions from Debye T3 phonons of
ΘD = 460 K and 300 K, respectively, which are much smaller than
the experimental values. The broken line in (b) is a linear fit to the
zero-field data as T → 0, which gives a coefficient of the T5 term, β5
= 0.3481(6) mJ K-6 mol-1. The solid and dotted line is a fit to Cl =
β5T5f(T). See text for detail.
FIG. 5. (Color online) Specific heat data same as shown in Fig. 3.
The dotted line shows the estimated contribution of Cen + Cl, and the
broken line represents Cl in the case of γ = 70 mJ K-2 mol-1. The
inset shows a change of entropy balance as a function of γ, from
which the value of γ is decided to be 70 mJ K-2 mol-1.
|
0704.1529 | Analysis of low energy pion spectra | Analysis of Low Energy Pion Spectra
Suk Choi and Kang Seog Lee∗
Department of Physics, Chonnam National University, Gwangju 500-757, Korea
(Dated: Apr. 11, 2007)
The transverse mass spectra and the rapidity distributions of π+ and π− in Au-Au collisions at
2, 4, 6, and 8 GeV·A by E895 collaboration are fitted using an elliptically expanding fireball model
with the contribution from the resonance decays and the final state Coulomb interaction. The
ratio of the total number of produced π− and π+ is used to fit the data. The resulting freeze-out
temperature is rather low(Tf < 60 MeV) with large transverse flow and thus resonance contribution
is very small. The difference in the shape of mt spectra of the oppositely charged pions are found
to be due to the Coulomb interaction of the pions.
PACS numbers: 24.10.Pa,25.75.-q
Pion production just above the threshold energy is
quite different from that at very high energies such as
RHIC energy since the ratio of π− to π+ at very high en-
ergies is one which is not the case at low energies. At just
above the threshold energy, pions are produced through
the production of ∆ resonances and counting all the pos-
sible channels of ∆ decay the difference in the compo-
sition of isospins in the colliding nuclei appears as the
difference in the numbers of the two oppositely charged
pions[1, 2, 3], whereas at high energies many channels
producing pions are open and small asymmetry in the
initial isospin does not matter.
Other features of the pion spectra at low energies
are[1, 2, 3, 4, 5, 6, 7]: (1)The transverse momentum
spectra both of the π− and π+ seem to have two temper-
atures. Usually the low temperature component in the
low momentum region is attributed to the pions decayed
from resonances, especially the delta resonance, while the
higher temperature component in the mid-momentum re-
gion is the thermal ones. (2) Transverse momentum spec-
tra of π− and π+ at very small momentum are different
in the sense that the π+ spectra is convex in its shape
while the π− spectra does not show this behavoir. This
difference in low momentum region is due to the Coulomb
effect. The hadronic matter formed during the collision
has charge which comes from the initially colliding two
nuclei and thus the thermal pions escaping from the sys-
tem experience the Coulomb interaction. The Coulomb
interaction of π− may bend the spectrum in the low mo-
mentum region upward and thus it is hard to disentan-
gle the contribution from the delta resonance and the
Coulomb interaction in the low momentum region. (3)
Width of the rapidity spectra of π− and π+ are much
wider than those from the isotropically expanding ther-
mal model. The wide width may either come from partly
transparent nature of the collision dynamics or the ellip-
soidal expansion geometry. In order to fit large rapidity
width using expanding fireball model one usually needs
large longitudinal expansion velocity.
∗[email protected]
Even though all those features mentioned above are not
new, calculations with all those features put in together
comparing each contributions in detail is hard to find.
There are claims that the properties of ∆ resonance are
modified inside the hadronic matter formed even at this
low energy[5, 6]. In order to draw any conclusion, one
should have a model which can explain all of the features
above mentioned. Lacking one or two features in the
calculations, the result may not be conclusive.
In this paper, we analyze the pion spectra in Au+Au
collisions at 2, 4, 6, and 8 A·GeV measured by the E895
collaboration[4] using the expanding fireball[8, 9, 10]
with the resonance contribution[10] and the final state
Coulomb interaction[1, 11, 12, 13]. The geometry of
the expansion used is ellipsoidal[8] and can be varied to
sphere and cylinder, by taking the transverse size R as a
function of the longitudinal coordinate, z.
At just above the threshold energy, pions are produced
through the production of ∆ resonances and their sub-
sequent decays. The ratio of π− and π+ is given from
the initial isospin conservation as π
5Z2+NZ
∼ 1.94
for Au+Au collisions[1, 2, 3] . Hence it is expected that
at low beam energies near 2 GeV·A in Au+Au collisions,
the ratio is near 1.94 and then as the beam energy is in-
creased the ratio will decrease eventually to one. In the
present calculation, the ratio of normalization constants
for π− and π+, R is taken as a fit parameter in order to
investigate the beam energy dependence of R.
We assume that once the pions are produced they
rescatter among themselves and thermalize before they
decouple from the system. Hence we assume thermaliza-
tion of the pionic matter. However, the total number of
negatively and positively charged pions are not in chem-
ical equilibrium and the ratio is governed by the isospin
asymmetry of the initially colliding nuclei. We keep the
ratio as a fitting parameter and want to compare with the
expected value of 1.94 near threshold of the pion produc-
tion.
After the formation of a pionic fireball, it expands and
cools down until freeze-out when the particle production
is described from the formalism of Cooper-Frye[14]. For
the equilibrium distribution function we use the Lorentz-
boosted Boltzmann distribution function, where uµ is the
http://arxiv.org/abs/0704.1529v1
TABLE I: Fitted values for each parameters.
Ebeam V ηm ρ0 T Pc π
−/π+ χ2 /n
(GeV) (×105) GeV GeV/c
2 1.41 1.12 0.88 46 25 1.96 1.3
4 0.93 1.32 0.92 57 24 1.95 2.9
6 1.44 1.50 1.11 54 18 1.40 2.4
8 1.62 1.58 1.12 55 15 1.38 1.8
expansion velocity the space-time of the system.
(2π)3
pµdσµ(x)f(x, p) (1)
where
f(x, p) = exp(−
pνuν(x)− µ
) (2)
For the geometry of the expanding fireball[8], we as-
sume azimuthal symmetry and further assume boost-
invariance collective dynamics along the longitudinal
direction[15]. In this case it is convenient to use as the
coordinates the longitudinal proper time τ =
t2 − z2,
space-time rapidity η = tanh−1(z/t) and the trans-
verse coordinate r⊥. Then the 4-velocity of expansion
can be expressed as uµ(x) = γ(1, v⊥(x)er , vz(x)) where
vz(τ, r⊥, η) = tanh η, which is the result of the longitudi-
nal boost-invariance. In the transverse direction we take
a linear flow rapidity profile, tanh−1 v⊥ = ρ(η)(r⊥/R0)
where R0 is the transverse radius at midrapidity. Here
one takes ρ(η) = ρ0
1− (η2/η2max) for the elliptic ge-
ometry and a constant value of ρ, i.e. ρ(η) = ρ0 for the
cylindrical geometry. As is the same for the SPS energy
by Dobbler et. al., the elliptic case fits the pion spectra
a little better.
As pions escape from the system at freeze-out, they ex-
perience the Coulomb interaction with the charge of the
system which are mainly due to the initial protons in the
colliding nuclei. The Coulomb effect on the particle spec-
tra are studied in detail in refs.[11, 12, 13] for the static
and dynamical cases. Here due to the low beam energy
we restrict ourselves only to the static case. The system
is expanding rapidly in the longitudinal direction and the
change in the longitudinal momentum is negligible. Only
the transverse momentum of the charged particles will be
shifted by an average amount[11]
pc = ∆p⊥ ∼ 2e2
dN ch
, (3)
where Rf is the transverse radius of the system at freeze-
out. Due to the lack of detailed knowledge, pc is taken
as a fit parameter in the present calculation. And in this
way the beam energy dependence of pc can be studied.
Since the transverse momentum of the escaping ther-
mal pions are shifted by the amount pc, i.e. pt = pt,0±pc.
the invariant cross section can be written in terms of the
unshifted momentum (pt,0, y0).
-1.5 -1 -0.5 0 0.5 1 1.5
E895, Au+Au at 2A·GeV
π- fit
π+ fit
FIG. 1: Rapidity distribution of π− and π+ in Au+Au col-
lision at 2 A·GeV by E895 collaboration[4].
). (4)
where (E d
)0 is the unshifted invariant cross section.
Integrating over the rapidity y one gets the equation for
the transverse momentum spectra,
ptdpt
pt,0dpt,0dy
. (5)
and the rapidity spectra is obtained by integrating over
the transverse mass.
mt,0dmt,0(
pt,0dpt,0
. (6)
Finally one has to add the contribution from resonance
decay[10] to both the transverse spectrum and rapidity
distribution. Here we assume that the resonances decay
far outside the system and the Coulomb interaction of
the pions decayed from the resonances is neglected.
The fitted values for the parameters are tabulated in
the Tab. 1 and the results of the fitting are shown in
Figs.1-2 for 2 A·GeV. The fitted value for the ratio R =
π−/π+ is close to 1.94 at 2 and 4 A·GeV as expected and
decreases to 1.38 at 8 A·GeV, which eventually becomes 1
at higher energies such as RHIC energies. In other words,
at this very low energy pion isospin is not in chemical
equilibrium.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
mt - m0
E895, Au+Au at 2A·GeV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
mt - m0
E895, Au+Au at 2A·GeV
FIG. 2: π− transverse mass spectrum for each rapidity
bin(∆y = 1.0) measured by E895 collaboration[4]in Au+Au
collisions at 2 A·GeV. Data on the top line is for the ra-
pidity bin −0.65 < y < −0.55 and the next one is for
−0.55 < y < −0.45 scaled by 0.1, etc.
The freeze-out temperature is rather small, Tf < 60
MeV and the expansion velocities in both the longitudi-
nal and transverse direction are quite large. The large
longitudinal expansion velocity(> 0.8c) is needed to fit
the large width of the rapidity distribution. The low
freeze-out temperature together with the large transverse
expansion fits the transverse spectra of pions quite nicely.
If not for the large expansion velocity, one usually gets
much larger freeze-out temperature(Tf > 80 MeV.
Since the freeze-out temperature is small, there are
very few resonances at freeze-out, especially ∆, and thus
the resonance contribution is negligible. This is reason-
able since at this low energy near the pion threshold en-
ergy, production of particles with mass larger than pions
is rare and their contribution to the pion spectrum is
negligible.
The pion transverse momentum spectra looks like that
there are two slopes; one for small momentum region and
another for the higher momentum region near the pion
mass. The smaller slope at lower transverse momentum
is usually attributed to the pions from the resonance de-
cay, especially from the ∆ decay[3, 6]. However, present
calculation shows that this is not the case at low beam
energies.
The shape of the transverse mass spectra of π− and
π+ are different especially in the small mass region. As
the transverse mass mt decreases, the mt spectrum of π
increases sharply while that of π+ saturates showing the
convex shape. This difference is due to the Coulomb in-
teraction of pions leaving the system which has the charge
from the initially bombarding nucleons. The change in
the transverse momentum due to the Coulomb interac-
tion decreases from 25 GeV/c at 2 A·GeV to 15 GeV/c at
8 A·GeV. This behavior can be understood from the in-
crease of the screening effect since the number of charged
pions increase at higher energies. At very high energies
such as RHIC energies, the momentum change from the
Coulomb interaction will be small.
The emerging picture of pion production at low en-
ergy is that the pions are produced through the inter-
mediate ∆ formation and thus they are not in chemical
equilibrium in isospin. They make collisions and ther-
malize to form a fireball which expands and cools until
the freeze-out. Since the fireball has charge which is from
the initially colliding nucleus, the pions leaving the sys-
tem experiences the Coulomb interaction which makes
the difference of the mt spectra of the two oppositely
charged pions.
Acknowledgments
This work is financially supported by Chonnam Na-
tional University and the post-BK21 program. We wish
to acknowledge U.W. Heinz for providing the program
and useful discussions.
[1] A. Wagner et. al., Phys. Lett. B420, 20(1998).
[2] B.J. VerWest and R.A. Arndt, Phys. Rev. C25,
1979(1982)
[3] Phys. Rep. 135, 259(1986).
[4] J.L. Klay, et. al., Phys. Rev. C68, 054905(2003).
[5] B. Pin-zhen and J. Rafelski, nucl-th/0507037.
[6] B.S. Hong, J. Korean Phys. Soc. 46, 1083(2005).
[7] B. Hong et. al., Phys. Lett. B407, 115(1999); ibid., J.
Korean Phys. Soc. 46, 1083(2005).
[8] H. Dobbler, J. Sollfrank and U. Heinz, Phys. Lett.
B457,353(1999) .
[9] Kang S. Lee, U. Heinz, and E. Schnedermann, Z. Phys.
C 48,525 (1990).
[10] E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev.
C 48, 2462 (1993).
[11] H.W. Barz, J.P. Bondorf,J.J. Gaardhoje and H. Heisel-
berg, Phys. Rev. C57, 2536(1998).
[12] M. Gyulassy and S. K. Kaufmann, Nucl. Phys. A362,
503(1981).
[13] A. Ayala and J. Kapusta, Phys. Rev. C56, 407(1997).
[14] F. Cooper and G. Frye, Phys. Rev. D10, 186(1974)
[15] J.D. Bjorken, Phys. Rev. D27, 140(1983)
http://arxiv.org/abs/nucl-th/0507037
|
0704.1530 | A study of the $p d \to p d \eta$ reaction | A study of the p d → p d η reaction
N.J.Upadhyay,1, ∗ K.P.Khemchandani,1, 2, † B.K.Jain,1, ‡ and N.G.Kelkar3, §
1Department of Physics, University of Mumbai,
Vidyanagari, Mumbai - 400 098, India
2Departamento de F́ısica Teórica and IFIC,
Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna,
Aptd. 22085, 46071 Valencia, Spain
3Departamento de Fisica, Universidad de los Andes,
Cra.1E No. 18A-10, Santafe de Bogota, Colombia
(Dated: October 25, 2018)
http://arxiv.org/abs/0704.1530v1
Abstract
A study of the p d → p d η reaction in the energy range where the recent data from Uppsala
are available, is done in the two-step model of η production including the final state interaction.
The η − d final state interaction is incorporated through the solution of the Lippmann Schwinger
equation using an elastic scattering matrix element, Tη d→ η d, which is required to be half off-shell.
It is written in a factorized form, with an off-shell form factor multiplying an on-shell part given
by an effective range expansion up to the fourth power in momentum. The parameters of this
expansion have been taken from an existing recent relativistic Faddeev equation solution for the
ηNN system corresponding to different η −N scattering amplitudes. Calculations have also been
done using few body equations within a finite rank approximation (FRA) to generate Tη d→ η d.
The p − d final state interaction is included in the spirit of the Watson-Migdal prescription by
multiplying the matrix element by the inverse of the Jost function. The η − d interaction is found
to be dominant in the region of small invariant η − d mass, Mηd. The p − d interaction enhances
the cross section in the whole region of Mηd, but is larger for large Mηd. We find nearly isotropic
angular distributions of the proton and the deuteron in the final state. All the above observations
are in agreement with data. The production mechanism for the entire range of the existing data
on the p d → p d η reaction seems to be dominated by the two-step model of η production.
PACS numbers: 25.10.+s, 25.40.Ve, 24.10.Eq
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
§Electronic address: [email protected]
mailto:[email protected]
mailto:[email protected]
mailto:[email protected]
mailto:[email protected]
I. INTRODUCTION
The current great interest in the η-nucleus interaction exists because of the attractive
nature of the η − N interaction in the s-wave [1], and the consequent possibility of the
existence of quasi-bound, virtual or resonant η-nucleus states [2]. The exact nature of
these states, of course, depends upon the precise knowledge of the η −N scattering matrix
at low energies. As the η is a highly unstable meson (lifetime ∼ 10−18s), this precise
information is difficult to obtain directly. It can only be obtained from the eta producing
reactions through the final state interaction. With this motivation, starting with the early
experiments near threshold at Saclay on the p d → 3He η and the p d → p d η reactions,
measurements have been carried out near threshold and beyond at Jülich and Uppsala using
the COSY and Celsius rings respectively. In this series of experiments, the recent data
on the p d → p d η reaction using the Wasa/Promice setup at the Celsius storage ring of
the Svedberg laboratory, Uppsala are thematically complete and cover the excess energy,
Q, (Q =
s − mη − mp − md) ranging from around threshold to 107 MeV. The data
[3] (integrated over other variables) include the invariant mass distribution over the whole
excess energy range for the η− d, η− p and p− d systems and angular distributions for the
proton, deuteron and the eta meson. Like the p d → 3He η reaction, the (inclusive) η − d
invariant mass distribution exhibits a large enhancement near threshold and hence appears
promising to study the η − d interaction. The η − p and p− d invariant mass distributions
do not show any such enhancement. All observed angular distributions are nearly isotropic.
Like in our earlier studies on the p d → 3He η reaction our primary aim in this paper is
to investigate the above mentioned data on the η − d invariant mass distribution to obtain
a better understanding of the η − N as well as the η − d interaction. We speculate, from
our experience on the study of the p d → 3He η reaction [4], that in the region of low η− d
relative energy this set of data will be mainly determined by the η − d interaction, though
the three-body nature of the final state may introduce some uncertainty in this conclusion.
We present a study of the p d → p d η reaction which includes the effect of the final state
interaction. We have investigated two possible diagrams for the production mechanism:
the direct mechanism and the two step process of η production. The direct mechanism
proceeds via an intermediate p n → d η reaction with one of the nucleons in the deuteron
as a spectator. The η meson in the two step model is produced in two steps, namely, p p →
d π+ and π+N → η N , hence involving the participation and sharing of the transferred
momentum by three nucleons. The two step model for η production was first used in [5] and
the data on the p d → 3He η reaction was well explained. The vertices at the two steps have
been described by the corresponding off-shell T -matrices. The T -matrix for π+N → η N is
taken from a coupled channel calculation [1], and that for p p → d π+ is obtained from the
SAID program provided by the authors of Ref. [6].
The final state interaction between the η and the deuteron is explicitly incorporated
through an η − d T -matrix, Tηd. This T -matrix, which is required to be half-off-shell, is
described in two ways. One choice involves taking a “factorized form” which is given by an
off-shell form factor multiplied by an on-shell part given by an effective range expansion up to
the fourth power in momentum. The parameters of this expansion have been taken from an
existing recent relativistic Faddeev equation solution for the ηNN system [7] corresponding
to different η − N scattering amplitudes. The off-shell form factor will be described in the
next sections and is chosen to have a form without any adjustable parameters. The second
prescription involves solving few body equations within the finite rank approximation (FRA)
to obtain Tηd. This approach has been used in literature for the η − d, −3He and −4He
systems [8]. We perform calculations for both the prescriptions using different models of the
elementary coupled channel η-nucleon T -matrix which characterize them.
The interaction between the η meson and the proton in the final state, to a certain
extent is contained implicitly in our calculations. This is due to the fact that we describe
the π+N → η N vertex by a T -matrix, which has been modeled to include the η − N
interaction. This off-shell T -matrix treats the π N , η N and π∆ channels in a coupled
channel formalism [1] and reproduces the experimental data on this reaction very well.
The effect of p−d final state interaction (FSI) is incorporated in the spirit of the Watson-
Migdal FSI prescription [9], in which our model p d → p d η production amplitude is mul-
tiplied by a factor which incorporates the FSI between the proton and the deuteron. This
factor is taken to be the frequently used [10, 11, 12, 13] inverse Jost function, [J(p)]−1, where
p is the relative p− d momentum. The assumption implicit in this approximation that the
mechanism for the primary reaction be short ranged is very well fulfilled in the η-production
reactions. The momentum transfer in these reactions near threshold is around 700 MeV/c.
We include FSI for both doublet (2S1/2) and quadruplet (
4S3/2) p− d states.
The η-nucleon T -matrix, which characterizes our calculations, is not precisely known.
Recent theoretical works on the n p → d η reaction [14] conclude that the data on this
reaction can be reproduced with the strength of the real part of the η-nucleon scattering
length ranging between 0.42 and 0.72 fm. In our earlier work on the p d → 3He η reaction
[4], we found a good agreement with data, with the real part of the scattering length taken
to be around 0.75 fm. This value was also found to be in agreement with the n p → d η
data in a K-matrix calculation of the final state η− d interaction in [15]. The same authors
as in [15], recently performed a fit to a wide variety of data which includes the π N → π N ,
π N → η N , γ N → π N and γ N → η N reactions and gave their best fit value of
the η-nucleon scattering length, aηN to be (0.91, 0.27) fm [16]. The η − d effective range
parameters are given in [7] for aηN up to (1.07 , 0.26) fm. Hence, in the present work we
perform calculations with different models of the η − N interaction, which correspond to
three different values of the η − N scattering length, ranging from aηN = (0.42 , 0.34) fm
to (1.07 , 0.26) fm.
We find that the cross sections calculated using the two-step model and the above in-
puts for the final state interaction reproduce most of the features of the experimental data
reasonably well.
A theoretical effort to understand the Uppsala data [3] was made earlier by Tengblad
et. al. [17]. In [17] the contribution of three different diagrams, namely, the pick-up (a
direct one-step mechanism of η production), the impulse approximation and the two-step
mechanism (here the η meson is produced in two steps via the p p → π+ d and π+N → η N
reactions) to the cross section for the p d → p d η reaction is determined. The authors in
[17] conclude that the impulse approximation is in general negligible as compared to the
other two diagrams, the two-step mechanism is dominant in the near threshold region and
the contribution of the pick-up diagram (referred to as the direct mechanism in the present
work) increases with energy and matches the two-step contribution at an excess energy of
Q = 95 MeV. The latter conclusions regarding the contributions of the two step and pick up
diagrams are in contrast to the findings of the present work as well as to existing literature on
similar kind of reactions. We note here that the authors in [17] do not include the final state
interaction in their calculations in any way. They treat the kinematics and the dependence
of the pion propagator (appearing in the two step model) on the Fermi momenta in an
approximate way. The T -matrices which enter as an input to the two step model are simply
extracted from experimental cross sections and are hence not proper off-shell T -matrices.
As a result of the above approximations, the authors in [17] do not reproduce the observed
enhancement in the η−d invariant mass distribution near threshold, and unlike the observed
isotropic distributions, find anisotropy in their calculated angular distributions.
The contribution from the direct mechanism (or the so-called pick-up diagram of [17]) to
the total cross sections is found to be about four orders of magnitude smaller than the two-
step contribution at threshold in the present work. The one-step contribution does increase
with energy (as also found in [17]), however, even at the highest energy for which data is
available (Tp = 1096 MeV) it remains two orders of magnitude smaller than that due to the
two-step model. This is in contrast to the observations in [17], where the two processes give
comparable contributions at high energies. The difference of orders of magnitude between
the two processes can be understood as a result of the large momentum transfer, q, in
the one-step process. This q, which is very large in the threshold region (∼ 840 MeV/c)
continues to be large even at high energies. For example, it is ∼ 600 MeV/c even at the
highest beam energy of 1096 MeV. This finding of ours is very similar to the previous studies
of the reactions involving high momentum transfer. For example, as mentioned above too,
in [5], for the pd →3 Heη reaction up to 2.5 GeV beam energy, the authors comment that
the one-step cross sections underestimate the data by more than two orders of magnitude.
In yet another calculation [18] of the cross section for the pd →3He X reaction (where
X = η, η′, ω, φ) the two-step model was found to describe the data on these reactions up
to 3 GeV quite well. In [19], in connection with the p d → 3HΛK+ reaction, the authors
claim that for a beam energy of 1 - 3 GeV, the one-step mechanism predicts 2 to 3 orders of
magnitude smaller cross sections as compared to the two-step mechanism. The cross sections
obtained from the one-step model, in Ref. [17] are, however, reported to be only one order
of magnitude less than those due to the two-step model at threshold and comparable to the
two-step ones at high energies.
In the next section, we describe the details of the formalism. In the subsequent sections
we present and discuss the results and finally the conclusions.
II. THE FORMALISM
The differential cross section for the p d → p d η reaction, in the center of mass, can be
written as,
2 (2 π)5 s |~kp|
dΩp′ | ~kp′| dMη d |~kη d| dΩη d
〈 |T |2 〉 (1)
where
s is the total energy in the center of mass and ~kp and ~kp′ are the proton momenta in
the initial and final states respectively. Mη d denotes the invariant mass of the η − d system
and ~kη d and Ωη d denote, in the η − d center of mass, the η momentum and its solid angle,
respectively. Ωp′ represents the solid angle of the outgoing proton. Angular brackets around
|T |2 in Eq. (1) represent the sum over the final and initial spins.
The T -matrix, which includes the interaction between the η and the deuteron is given by
T = 〈ψηd( ~kηd), ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd (= − ~kp); mp, md 〉 (2)
where the spin projections for the proton and the deuteron in the initial and final states have
been labeled as mp, md, mp′, and md′ respectively. Tp d→ p d η is the production operator.
The wave function of the interacting η − d in the final state has been represented as
ψηd( ~kηd). In terms of the elastic η − d scattering T -matrix, Tηd, it is written as
〈ψ−ηd | = 〈 ~kηd |+
(2π)3
〈 ~kηd | Tηd | ~q 〉
E(kηd) − E(q) + iǫ
〈 ~q | (3)
The second term here represents the scattered wave. It has two parts originating from
the principal-value and the delta-function part of the propagator in the intermediate state.
Physically they represent the off-shell and the on-shell scattering between the η and the
deuteron. The on-shell part can be shown to be roughly proportional to the η−d momentum
and hence dominant at higher energies. The relative contribution of these terms in our case
would be determined after we substitute the above expression for ψηd( ~kηd) in Eq. (2). We
then get
T = 〈 ~kηd, ~kp′; mp′, md′ | Tpd→pdη | ~kp, ~kd(= −~kp); mp, md 〉 (4)
(2π)3
〈 ~kηd;md′ | Tηd | ~q;m2′ 〉
E(kηd)−E(q) + iǫ
〈 ~q , ~kp′;m2′ , mp′ | Tpd→pdη | ~kp, ~kd; mp, md〉
It can be seen that the Tηd here appears as a half-off-shell T -matrix.
= - k
p kd/ 2 - P
FIG. 1: The two step process production mechanism for the p d → p d η reaction.
A. The production mechanism
For evaluating the η production T -matrix, 〈 | Tpd→pdη | 〉, we assume a two-step mechanism
as shown in Fig. 1. In this model, the incident proton produces a pion in the first step
on interacting with one of the nucleons of the target deuteron. In the second step this
pion produces an η meson on interacting with the other nucleon. Both these nucleons
are off-shell and have a momentum distribution given by the deuteron bound state wave
function. To write the production matrix, we resort to certain standard approximations
used in literature [20] (in particular for the triangle diagram appearing in Fig. 1). The
amplitude for the pN → πd process, which in principal is off-shell, is taken at an on-shell
energy. Considering the high proton beam energy, off-shell effects are not expected to be
significant. The π N → η N process is included via an off-shell T -matrix.
The production matrix is written as [4, 5],
〈 | Tpd→ p d η | 〉 =
(2π)3
〈 p n | d 〉 〈 | Tpp→ π+ d | 〉
k2π − m2π + iǫ
〈 | Tπ+ n→ η p | 〉 (5)
where, the squared four momentum of the intermediate pion, k2π = E
π−~k2π, with the energy,
Eπ, calculated at zero fermi momentum and ~kπ = ~kη + ~kp′ − ~kd/2 + ~P . The summation is
over internal spin projections and the matrix element 〈 p n | d 〉 represents the deuteron wave
function in momentum space, which has been written using the Paris parametrization [21].
The factor 3/2 is a result of summing the diagrams with an intermediate π0 and π+.
The integral over the pion momentum in above includes the contribution from the pole
as well as the principal value term. For the pion propagator itself, as we see, we have taken
= - k
| = - ( kη + kd| )
-1/2 k
FIG. 2: The direct process production mechanism for the p d → p d η reaction.
the plane wave propagator. This thus excludes any effect in our results due to medium mod-
ification of this propagator due to other nucleons. This aspect may be worth investigating
in future.
The T -matrix for the intermediate p p → π+ d process has been taken from an energy
dependent partial wave analysis of the π+ d → p p reaction from threshold to 500 MeV [6].
The various observables in [6] are given in terms of amplitudes which are parametrized to fit
the existing database. We refer the reader to [6] and the references therein for the relevant
expressions of the helicity and partial wave amplitudes and the notation followed by the
authors in [6].
For the π+ n → η p sub-process, different forms of T -matrices are available. We use
the T -matrix from [1] which treats the π N , η N , and π∆ channels in a coupled channel
formalism. This T -matrix consists of the meson - N∗ vertices and the N∗ propagator as given
below:
Tπ+ n→ η p(k
′ , k ; z) =
gN∗ β
(k′ 2 + β2)
τN∗(z)
gN∗ β
(k2 + β2)
with,
τN∗(z) = ( z − M0 − Σπ(z) − Ση(z) + i ǫ)−1
where Σα(z) (α = π , η) are the self energies from the πN and η N loops. The parameters
of this model are, gN∗ = 0.616, β = 2.36 fm
−1 and M0 = 1608.1 MeV. This T -matrix
reproduces the data on the π+n→ ηp reaction very well.
Although the contribution of the direct mechanism (Fig. 2) is known to be small (owing
to the large momentum transfer involved in the process) [5, 18, 19], for completeness, we
calculate its contribution to the total cross section. The T -matrix for this mechanism can
be written as
〈|Tp d→ p d η|2〉 =
〈|Tpn→ d η(
sη d)|2〉 × | φd(q)|2, (7)
where φd represents the deuteron wave function in the initial state. The spin summed
〈| Tpn→ d η |2〉 is given in terms of the total cross section for the p n → d η reaction by
σT (p n → d η) =
2mpmnmd
| ~pf |
|~pi|
〈| Tpn→ d η|2 〉, (8)
where ~pi and ~pf are the initial and final momenta c.m. system. The momentum transfer ~q,
as shown in Fig. 2, is defined as,
~kp + ~kp′ . (9)
The total cross section, σT , for p n → d η reaction is taken from the experiments [22].
B. Final state interaction
1. η − d interaction
This is incorporated through a half-off-shell η− d T -matrix. We construct this T -matrix
using the following two prescriptions:
1. Factorized form of Tηd
In one ansatz we obtain it by multiplying the on-shell η − d T -matrix by an off-shell
extrapolation factor g(k′ , k). Requiring that this T -matrix goes to its on-shell value
in the case of on-shell momenta, we write
Tη−d(k , E(k0) , k
′) = g(k , k0) Tη−d(E(k0)) g(k
′ , k0), (10)
with g(p , q) → 1 as p → q. For a half-off-shell case, this obviously is the ratio of the
half-off-shell to the on-shell scattering amplitude.
For the on-shell η − d T -matrix we use the effective range expansion of the scattering
amplitude up to the fourth power in momentum,
F (k) =
Rk2 + Sk4 − ik
, (11)
where F is related to T by
Tηd(k, k
′) = − 1
(2π)2µηd
Fηd(k, E(k), k
′). (12)
The effective range expansion parameters (A,R,S) are taken from a recent relativistic
Faddeev equation (RFE) calculation of [7]. This calculation uses the relativistic version
of the Faddeev equations for a three particle mNN system, where m is a meson and it
can be an η, π or a σ meson. These particles interact pairwise, and these interactions
are represented with separable potentials. The parameters of the ηN − πN − σN
potentials are fitted to the S11 resonant amplitude and the π
−p → ηn cross sections.
The η − d effective range parameters obtained from these calculations are listed in [7]
for different sets of the meson-nucleon potentials. Each of these sets gives a specific
value of the η −N scattering length, which is also listed in [7].
Since the half-off-shell extrapolation factor g(k′ , k0) is not known with any certainty,
we choose the following two forms for it.
(i) Following the method in [15] for the final state interaction in the η− d system, we
express the off-shell form factor in terms of the deuteron form factor,
g(k′ , k0) =
d~rj0(rk
′/2)φ2d(r)j0(rk0/2) (13)
where for the deuteron wave function, φd(r), we take the Paris parametrization.
(ii) As a second choice, the form factor is taken to be the ratio of the off-shell η − d
T -matrix to its on-shell value, where both of them are calculated using the three
body equations within FRA. The input to these calculations is the elementary η −N
scattering matrix, the details of which are given in the next Section.
2. Few body equations within the finite rank approximation
The other prescription of η−d FSI involves the use of the half-off-shell η−d T -matrix
obtained by solving few body equations within the finite rank approximation (FRA).
For the details of this formalism and the expression for the η-nucleus T -matrix, we refer
the reader to our earlier works [4]. To mention briefly, the FRA involves restricting
the spectral decomposition of the nuclear Hamiltonian in the intermediate state to
the ground state, neglecting thereby all excited and break-up channels of the nucleus.
This is justified in the η − 4He and possibly in the η − 3He case, but in η−deuteron
collisions, where the break-up energy is just 2.225 MeV, the applicability of the FRA
may be limited. However, it should be noted that a comparative study [23] of the η−d
scattering lengths calculated using the FRA and the exact Alt-Grassberger-Sandhas
(AGS) [24] equations (which include these intermediate excitations) shows that they
are not very different if the real part of the η−N scattering length is restricted up to
about 0.5 fm.
2. p− d interaction
We incorporate the p− d FSI in our calculations by multiplying our model T -matrix by
the inverse Jost function, [J(p)]−1. We include the FSI in both the 1/2 and 3/2 spin states
of p − d and restrict it to the s-wave. Since the p and d are charged we also include the
Coulomb effects. Following standard procedure, we write the Jost function in terms of phase
shifts and use the effective range expansion for the later.
The complete expression for the s-wave inverse Jost function squared is written as,
[Jo (kpd)]
−2 = [Jo (kpd)]
Q + [(1 +
)Jo (kpd)]
D . (14)
Here, to include the effect of the existence of one bound state, namely, the spin 1/2 state
(3He), the doublet Jost function is multiplied by a factor (1 +
), where |EB| is the
separation energy of 3He into p− d. Its value is taken to be 5.48 MeV.
The expressions for spin quadruplet (Q) and doublet (D) [Jo (kpd)]
−2 are given by
[Jo (kpd)]
(k2pd + α
2)2 (bcQ)
3C2o k
1 + cot2 δQ
[Jo (kpd)]
(k2pd + α
2)2 (bcD)
3C2o k
1 + cot2 δD
where,
2 bcµ
and acµ and b
µ are defined as
− 2 γ kpdHγ
bcµ =
where µ stands for either Q or D. The factor C2o in above has its origin in the Coulomb
interaction. The phase shifts δQ ,D are obtained from an effective-range expansion [25, 26],
C2o kpd cot δµ = −
pd − 2 γ kpdHγ (20)
αmred
~ kpd
C2o =
2 π γ
e2 π γ − 1
n(n2 + γ2)
− ln (γ) − 0.57722 (23)
Here mred is the reduced mass in the p− d system, γ the Coulomb parameter and α is the
usual electromagnetic coupling constant. The values of the expansion coefficients aµ, bµ in
Eq. (20) are taken as aQ = 11.88 fm, bQ = 2.63 fm, aD = 2.73 fm, and bD = 2.27 fm. They
have been determined from a fit to the p − d elastic-scattering phase shifts in the relative
p− d momentum range up to around 200 MeV/c [27].
The above expression for the Jost function has the required property that for large p,
J0(p) → 1.
III. RESULTS AND DISCUSSION
Before we discuss the results of the present work, in order to highlight the FSI effects
in the experimental η − d invariant mass distribution we remove the phase space from the
experimental dσ/dMηd and plot in Fig. 3 the |f |2, which is then given by,
|f |2 =
phase space
, (24)
where,
phase space =
12 (2 π)5 s |~kp|
dΩp′ | ~kp′| |~kη d| dΩη d (25)
as a function of the excess energy, Qηd = Mηd − mη − md, where Mηd is the invariant mass
of the η − d system. In this figure we also show the plane wave result (i.e. Tp d→ p d η does
not include any FSI). The cross section, dσ/dMη d in Eq. (24), is evaluated for each Mηd
by performing an integral over the p − d centre of mass momenta, kpd. The range of the
allowed values of kpd at each Mηd is shown by the hashed region. One clearly sees a large
0 10 20 30 40 50 60 70
Qηd ( = Mηd − mη − md ) (MeV)
= 1032 MeV
Bilger et. al.
Plane wave
FIG. 3: The ratio of experimental differential cross sections [3] to the phase space (Eq. (25)) as a
function of the excess energy, Qηd, along with range of p−d relative momenta, kpd (hashed region),
contributing to |f |2 at each Qηd.
enhancement in the experimental |f |2 near small values of Qη d, which, most likely is due to
the η − d FSI. We also observe a rise at large values of Qηd. Examining the range of p− d
relative momenta which contribute to |f |2 at each Qηd, one can see that this rise occurs at
small values of kpd, indicating thereby the possibility of a large effect of p − d FSI in this
region.
In Fig. 4, we show two sets of the calculated |f |2 along with the experimental results
for a beam energy of 1032 MeV. These results include only η − d FSI. We limit the range
of Qηd up to about 10 MeV, where, this effect is large. In Fig. 4(a) we show results for
the factorized prescription with the off-shell factor generated from the deuteron form factor
and the on-shell part arising from the relativistic Faddeev equation (RFE) calculation of [7].
The results are shown for three different sets of interaction parameters in the RFE. Since
these sets give uniquely different values of the η − N scattering lengths aηN , we identify
them by their corresponding aηN values. For the results presented here, these values are
0.42 + i0.34 fm, 0.75 + i0.27 fm and 1.07 + i0.26 fm. We see that our results reproduce the
enhancement seen in the experimental |f |2 at small values of Qηd. The absolute magnitude
depends upon the choice of the RFE parameters. It increases with aηN , which designate
these parameter sets. The set corresponding to aηN = 1.07 + i0.26 fm, gives results closest
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
Bilger et. al.
Plane wave
ηd FSI (aηN = 1.07 + i0.26 fm)
ηd FSI (aηN = 0.75 + i0.27 fm)
ηd FSI (aηN = 0.42 + i0.34 fm)
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
Bilger et. al.
Plane wave
ηd FSI (aηN = 0.92 + i0.27 fm)
ηd FSI (aηN = 0.77 + i0.25 fm)
ηd FSI (aηN = 0.4 + i0.3 fm)
FIG. 4: The calculated |f |2 along with the experimental results for a beam energy of 1032 MeV.
(a) The results correspond to the factorized form of Tηd with the off-shell factor generated from the
deuteron form factor. (b) The results correspond to Tηd obtained from few body equations within
the FRA. The data is the same as in Fig. 3.
to the experimental values.
In Fig. 4(b) we show |f |2 calculated using few body equations within the FRA, for η− d
FSI. These results are shown for three different inputs of the η − N T -matrix taken from
[16]. The choice of these T -matrices is such that their scattering length values are close to
those used in Fig. 4(a). Though this model has the limitation of retaining the intermediate
nucleus in its ground state in the η− nucleus elastic scattering, the off-shell re-scattering
effects have been properly included. If we compare Fig. 4(a) and 4(b), the two sets of results
are similar.
In order to check the sensitivity of the results to the off-shell form factor used in the
factorized η − d T -matrix, in Fig. 5(a), we show the |f |2 calculated using two different
off-shell form factors. The on-shell Tηd is obtained from RFE and the off-shell part is either
treated with a deuteron form factor (solid line) or a few body FRA form factor (dash dotted
line) as explained in section II B. The elementary η −N T -matrix parameters required for
the calculation of the FRA form factor are taken from the parametrization of Green and
Wycech [16]. Even though the results (as shown in Fig. 4(a)) corresponding to the aηN =
1.07 + i0.26 fm seem to be the closest to the data, to compare the effect of using different
0 2 4 6 8 10 12 14 16
Qηd ( = Mηd − mη − md ) (MeV)
= 1032 MeV
Bilger et. al.
Plane wave
Deuteron form factor
FRA form factor
2 4 6 8 10 12 14
k (fm
= 51 MeV/c
Deuteron form factor
FRA form factor
FIG. 5: Comparison of the two form factors for aη N = 0.75 + i0.27 fm. (a) Effect of using two
different off-shell extrapolation factors for η − d FSI on |f |2. (b) Two form factors as function of
off-shell momentum (k′).
off-shell form factor, we choose the results corresponding to aη N = 0.75 + i0.27 fm. We
make this choice such that we can compare the two calculations for the inputs corresponding
to a similar η − N scattering length. It should be expected then, that the off-shell form
factors obtained from two different methods should not differ much. This is seen explicitly
in Fig. 5(b) where the two form factors are shown as a function of off-shell momentum (k′)
for an on-shell value, k0 near the low energy peak in the η − d invariant mass distribution
(to be discussed in Fig. 9 later).
Next, we include in our calculations the effect of the p−d FSI. This is done by multiplying
the pd → pdη squared T -matrix (Eq. (4)) used above by the inverse Jost function squared
in Eq. (15) and Eq. (16), and integrating it over the allowed range (as shown in Fig. 3)
of p − d momenta, kpd for each Qη d. We show these results in Fig. 6 for the RFE (with
deuteron form factor) model of η − d FSI, for the parameter set corresponding to aηN =
1.07 + i0.26 fm. We find, that the p− d FSI affects the results in the whole region of Qη d,
while the effect of η − d FSI is confined to small value of Qηd. The large effect of p− d FSI
in the region of small Qηd, however, may not be taken with confidence as the value of kpd
in this region is large (as shown in Fig. 3), where the s-wave effective range expansion (Eq.
(20)) for the calculation of Jost function might not be sufficient. In any case, it appears
0 10 20 30 40 50 60 70
Qηd ( = Mηd − mη − md ) (MeV)
= 1032 MeV
Bilger et. al.
Plane wave
ηd FSI
ηd & pd FSI
FIG. 6: The proton-deuteron final state interaction effects on the p d → p d η reaction at the beam
energy of 1032 MeV. The dashed line shows the plane wave results and the dashed dot (solid) line
shows the effect of η − d (η − d & p− d) FSI for aηN = 1.07 + i0.26 fm.
that the effects of both the η− d and the p− d FSI on the η− d invariant mass distribution
are significant. If we disregard the calculated p− d effect for small Qηd, the η− d and p− d
FSI dominate in regions well separated from each other.
Apart from the FSI, another important ingredient of our calculations is the two-step
description of the production vertex. Because of the large momentum transfer, we believe,
as has also been stressed in Ref. [17], that the angular distribution of the outgoing particles
is probably more sensitive to the description of the production vertex. Inclusive angular
distributions have been measured for all the three outgoing particles in the p d → p d η
reaction. In Fig. 7, we show the calculated angular distributions for all the three outgoing
particles along with the measured distributions. We show results without any FSI, with
η− d FSI and with η− d and p− d FSI both included. As each angle has contribution from
a range of Qη d as well as kpd, the calculated results include integration of the cross section
over these variables. We find that the observed nearly isotropic nature of the experimental
angular distributions for the proton, deuteron and eta already gets reproduced by the plane
wave calculations. The effect of both η − d and p − d FSI is large and persists over all
the angles. Their inclusion brings the magnitudes of the proton and deuteron angular
distributions near to experiments. The magnitude of the eta distribution, however, does not
seem to be affected much with the FSI.
Experimental data also exist on the total cross section. In Fig. 8, we compare the total
Bilger et al
Plane wave
ηd FSI
ηd & pd FSI
-0.8 -0.4 0 0.4 0.8
cos θ
=1032 MeV
FIG. 7: The calculated angular distributions of (a) the deuteron, (b) the proton and (c) the η,
along with the measured cross sections for aηN = 1.07 + i0.26 fm [3].
950 1000 1050 1100
Beam Energy (MeV)
Bilger et al. (PRC 69, 014003 (2004))
Hibou et al. (EPJA 7, 537 (2002))
Two step mechanism
Two step planewave
The direct mechanism
0 20 40 60 80 100
Excess Energy (MeV)
FIG. 8: A comparison of the total cross section for the p d → p d η reaction calculated with the
description of the production vertex as a two step mechanism and direct mechanism, along with
the measured cross sections for aη N = 1.07 + i0.26 fm [3, 28].
2430 2440 2450 2460 2470 2480 2490
dη (MeV)
Total contribution
Off shell contribution
On shell contribution
Plane wave
FIG. 9: Contributions from the off-shell and the on-shell η − d scattering in the final state. The
results are for aη N = 1.07 + i0.26 fm with the inclusion of only the η − d FSI.
cross sections calculated including both the η − d and p − d FSI with the measured cross
sections. The results are shown with the factorized form of η − d FSI with deuteron form
factor for the set corresponding to η −N scattering length equal to 1.07 + i0.26 fm. As we
see, the calculated cross sections are in good agreement with the experimental data.
In Fig. 8 we also give the cross sections calculated for the one-step direct mechanism
(Fig. 2) mentioned in the previous section. Near threshold, these cross sections are about
four orders of magnitude below those obtained from the two-step model and two orders
of magnitude smaller in the high energy range. As mentioned in the Introduction, this
observation is similar to that in other works involving large momentum transfer reactions
[5, 18, 19], and is understandable because the momentum transfer continues to be large
(∼ 600 MeV/c) in the p d → p d η reaction even at an excess energy as large as 100 MeV.
Now we make an observation about the importance of off-shell scattering in treating
η − d FSI near threshold. The scattering part of the η − d wave function (Eq. (3)), gets
contributions from the off-shell as well as the on-shell scattering in the nucleus. To see
quantitatively the relative importance of these two contributions to the cross section for the
p d → p d η reaction, in Fig. 9 we show their contributions separately in the η− d invariant
mass distribution. These results include only the η − d FSI generated from the factorized
prescription using RFE and the deuteron form factor for the η − d T -matrix. We find that
near threshold the off-shell scattering completely dominates the threshold enhancement. At
2430 2440 2450 2460 2470 2480 2490
dη (MeV)
Bilger et. al.
Total contribution
Plane wave
FIG. 10: A comparison of the calculated results including both the η − d and p − d FSI with the
experimental results. The results are for aηN = 1.07 + i0.26 fm.
higher excess energy, as expected, the on-shell contribution takes over.
Finally we show the nature of agreement of our calculated results with the invariant η−d
mass distribution. In Fig. 10, we compare the calculated results including both the η − d
and p − d FSI with the experimental results. The results are for aηN = 1.07 + i0.26 fm
calculated with the factorized prescription using RFE and the off-shell factor generated from
the deuteron form factor. As we see the overall agreement is reasonably good.
IV. SUMMARY
The invariant η − d mass distribution in the p d → p d η reaction has been studied
by describing the production mechanism in terms of a two step model with a pion being
produced in the intermediate state. The η−d final state interaction (FSI) has been included
in (a) a factorized form involving an on-shell Tηd and two types of off-shell form factors and
(b) by solving few body equations within the FRA. The p−d FSI is included through a Jost
function. The conclusions of this investigation can be summarized as:
1. Experimentally observed large enhancement in the cross section near small η−d excess
energy, Qηd is reproduced by the η − d FSI. The rise in the cross section at large Qηd
(which corresponds to a range of small momenta, kpd) can be accounted for by the
p− d FSI.
2. Quantitative reproduction of the large enhancement requires η− d FSI corresponding
to large values of aηN . In the present calculation it is around 1.07 + i0.26 fm.
3. The calculations successfully reproduce the observed isotropic angular distribution
of the proton and the deuteron in the final state. The total cross sections for the
pd→ pdη reaction are also well reproduced.
4. The off-shell part of the η − d scattering dominates near threshold.
5. The results for two different choices of the off-shell extrapolation factor in the factorized
form of the η − d FSI are similar.
V. ACKNOWLEDGMENTS
The authors wish to thank R. A. Arndt for providing the computer codes for evaluating
the p p → π+ d T -matrix. This work is done under a research grant by the Department of
Science and Technology, Government of India. The authors (NJU, KPK and BKJ) gratefully
acknowledge the same.
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http://arxiv.org/abs/nucl-th/9707044
http://arxiv.org/abs/nucl-th/0506024
http://arxiv.org/abs/nucl-th/9510010
http://arxiv.org/abs/nucl-th/9804050
Introduction
The Formalism
The production mechanism
Final state interaction
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p-d interaction
Results and Discussion
Summary
Acknowledgments
References
|
0704.1531 | On the exact formula for neutrino oscillation probability by Kimura,
Takamura and Yokomakura | On the exact formula for neutrino oscillation probability by
Kimura, Takamura and Yokomakura
Osamu Yasuda∗
Department of Physics, Tokyo Metropolitan University,
Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Abstract
The exact formula for the neutrino oscillation probability in matter with constant density, which
was discovered by Kimura, Takamura and Yokomakura, has been applied mostly to the standard
case with three flavor neutrino so far. In this paper applications of their formula to more general
cases are discussed. It is shown that this formalism can be generalized to various cases where the
matter potential have off-diagonal components, and the two non-trivial examples are given: the
case with magnetic moments and a magnetic field and the case with non-standard interactions. It
is pointed out that their formalism can be applied also to the case in the long baseline limit with
matter whose density varies adiabatically as in the case of solar neutrino.
PACS numbers: 14.60.Pq, 14.60.St
∗Electronic address: yasuda˙at˙phys.metro-u.ac.jp
Typeset by REVTEX 1
http://arxiv.org/abs/0704.1531v2
mailto:yasuda_at_phys.metro-u.ac.jp
I. INTRODUCTION
Neutrino oscillations in matter (See, e.g., Ref. [1] for review.) have been discussed by
many people in the past because the oscillation probability has non-trivial behaviors in
matter and due to the matter effect it may exhibit non-trivial enhancement which could
be physically important. Unfortunately, it is not easy to get an analytical formula for the
oscillation probability in the three flavor neutrino scheme in matter, and investigation of
its behaviors has been a difficult but important problem in the phenomenology of neutrino
oscillations. In 2002 Kimura, Takamura and Yokomakura derived a nice compact formula [2,
3] for the neutrino oscillation probability in matter with constant density. Basically what
they showed is that the quantity Ũ∗αjŨβj , which is a factor crucial to express the oscillation
probability analytically, can be expressed as a linear combination of U∗αjUβj, where Ũαj and
Uαj stand for the matrix element of the MNS matrix in matter and in vacuum, respectively.
However, their formula is only applicable to the standard three flavor case. In this pa-
per we show that their result can be generalized to various cases. We also show that their
formalism can be applied also to the case with slowly varying matter density in the limit
of the long neutrino path. In Sect. II, we review briefly some aspects of the oscillation
probabilities, including a simple derivation for the formula by Kimura, Takamura and Yoko-
makura which was given in Ref. [4], because these are used in the following sections. Their
formalism is generalized to the various cases where the matter potential has off-diagonal
components, and we will discuss the case with large magnetic moments and a magnetic field
(Sect. III) and the case with non-standard interactions (Sect. IV). In Sect. V we summarize
our conclusions.
II. GENERALITIES ABOUT OSCILLATION PROBABILITIES
A. The case of constant density
It has been known [5] (See also earlier works [6, 7, 8].) that after eliminating the nega-
tive energy states by a Tani-Foldy-Wouthusen-type transformation, the Dirac equation for
neutrinos propagating in matter is reduced to the familiar form:
UEU−1 +A(t)
Ψ, (1)
where
E ≡ diag (E1, E2, E3) ,
A(t) ≡
2GFdiag (Ne(t)−Nn(t)/2,−Nn(t)/2,−Nn(t)/2) ,
ΨT ≡ (νe, νµ, ντ ) is the flavor eigenstate, U is the Maki-Nakagawa-Sakata (MNS) matrix,
m2j + ~p
(j = 1, 2, 3) is the energy eigenvalue of each mass eigenstate, and the matter
effect A(t) at time (or position ) t is characterized by the density Ne(t) of electrons and the
one Nn(t) of neutrons, respectively. Throughout this paper we assume for simplicity that
the density of matter is either constant or slowly varying so that its derivative is negligible.
The 3× 3 matrix on the right hand side of Eq. (1) can be formally diagonalized as:
UEU−1 +A(t) = Ũ(t)Ẽ(t)Ũ−1(t), (2)
where
Ẽ(t) ≡ diag
Ẽ1(t), Ẽ2(t), Ẽ3(t)
is a diagonal matrix with the energy eigenvalues Ẽj(t) in the presence of the matter effect.
First of all, let us assume that the matter density A(t) is constant. Then all the t
dependence disappears and Eq. (1) can be easily solved, resulting the flavor eigenstate at
the distance L:
Ψ(L) = Ũ exp
−iẼL
Ũ−1Ψ(0). (3)
Thus the oscillation probability P (να → νβ) is given by
P (να → νβ) =
Ũ exp (−iEL) Ũ−1
= δαβ − 4
∆ẼjkL
∆ẼjkL
, (4)
where we have defined
j ≡ ŨαjŨ∗βj ,
∆Ẽjk ≡ Ẽj − Ẽk,
and throughout this paper the indices α, β = (e, µ, τ) and j, k = (1, 2, 3) stand for those
of the flavor and mass eigenstates, respectively. Once we know the eigenvalues Ẽj and the
quantity X̃
j , the oscillation probability can be expressed analytically.
B. The case of adiabatically varying density
Secondly, let us consider the case where the density of the matter varies adiabatically as
in the case of the solar neutrino deficit phenomena. In this case, instead of Eq. (3), we get
Ψ(L) = Ũ(L) exp
Ẽ(t) dt
Ũ(0)−1Ψ(0),
where Ũ(0) and Ũ(L) stand for the effective mixing matrices at the origin t = 0 and at the
end point t = L. The oscillation probability is given by
P (να → νβ) =
Ũ(L) exp
Ẽ(t) dt
Ũ(0)−1
Ũ(L)βjŨ(L)
βkŨ(0)
αjŨ(0)αk exp
∆Ẽ(t)jk dt
. (5)
1 In the standard case with three flavors of neutrinos in matter, the energy eigenvalues Ẽj can be analytically
obtained by the root formula for a cubic equation [9]. So the only non-trivial problem in the standard
case is to obtain the expression for X̃
j , and this was done by Kimura, Takamura and Yokomakura [2, 3].
In general cases, however, the analytic expression for Ẽj is very difficult or impossible to obtain, and we
will discuss below only examples in which the analytic expression for Ẽj is known.
Eq. (5) requires in general the quantity like Ũ(t)βjŨ
∗(t)βk which has the same flavor index
β but different mass eigenstate indices j, k, and it turns out that the analytical expression
for Ũ(t)βjŨ
∗(t)βk is hard to obtain. However, if the length L of the neutrino path is very
large and if |
0 ∆Ẽ(t)jk dt| ≫ 1 is satisfied for j 6= k, as in the case of the solar neutrino
deficit phenomena, after averaging over rapid oscillations Eq. (5) is reduced to
P (να → νβ) =
j (L)X̃
j (0),
where we have defined
X̃ααj (t) ≡
∣Ũ(t)αj
In the case of the solar neutrinos deficit process νe → νe during the daylight, X̃ββj (L) at the
end point t = L and X̃ααj (0) at the origin t = 0 correspond to X
j in vacuum and [X̃
at the center of the Sun, respectively, where
j ≡ UαjU∗βj
ŨαjŨ
are bilinear products of the elements of the mixing matrices in vacuum and at the center of
the Sun, respectively. Thus we obtain
P (νe → νe) =
X̃eej
Hence we see that evaluation of the quantity X̃ααj in the presence of the matter effect is
important not only in the case of constant matter density but also in the case of adiabatically
varying density.
C. Another derivation of the formula by Kimura, Takamura and Yokomakura
In this subsection a systematic derivation of their formula is given because such a deriva-
tion will be crucial for the generalizations in the following sections.2 The arguments are
based on the trivial identities. From the unitarity condition of the matrix Ũ , we have
δαβ =
Ũ Ũ−1
ŨαjŨ
j . (6)
Next we take the (α, β) component of the both hand sides in Eq. (2):
UEU−1 +A
Ũ ẼŨ−1
ŨαjẼjŨ
ẼjX̃
j (7)
2 The argument here is the same as that in Ref. [4]. Since this derivation does not seem to be widely
known, it is reviewed here.
Furthermore, we take the (α, β) component of the square of Eq. (2):
UEU−1 +A
Ũ Ẽ2Ũ−1
ŨαjẼ
Ẽ2j X̃
j (8)
Putting Eqs. (6)–(8) together, we have
1 1 1
Ẽ1 Ẽ2 Ẽ3
Ẽ21 Ẽ
[UEU−1 +A]αβ
(UEU−1 +A)2
which can be easily solved by inverting the Vandermonde matrix:
∆Ẽ21∆Ẽ31
(Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), 1)
∆Ẽ21∆Ẽ32
(Ẽ3Ẽ1, −(Ẽ3 + Ẽ1), 1)
∆Ẽ31∆Ẽ32
(Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), 1)
[UEU−1 +A]αβ
(UEU−1 +A)2
. (9)
[(UEU−1 +A)j]αβ (j = 1, 2) on the right hand side are given by the known quantities:
UEU−1 +A
j + Aδαeδβe
UEU−1 +A
j + A
j + δβeX
+ A2 δαeδβe.
It can be shown that Eq. (9) coincides with the original results by Kimura, Takamura and
Yokomakura [2, 3].
A remark is in order on Eq. (9). Addition of a matrix c1 to Eq. (2) where c is a constant
and 1 is the identity matrix, or in other words, the shift
Ej → Ej + c (j = 1, 2, 3), (10)
should give the same result for X̃
j (j = 1, 2, 3), since Eq. (10) only affects the overall
phase of the oscillation amplitude and the phase has to disappear in the probability. It is
easy to show that the shift (10) indeed gives the same result as Eq. (9). The proof is given
in Appendix A. In practical calculations below, we will always put c = −E1, i.e., we will
consider the mass matrix U(E −E11)U−1 +A instead of the original one UEU−1 +A, since
all the diagonal elements (E − E11)jj = ∆Ej1 = ∆m2j1/2E are expressed in terms of the
relevant variables ∆m2j1, and therefore calculations become simpler. To save space, however,
we will use the matrix UEU−1 +A in most of the following discussions.
D. The case with arbitrary number of neutrinos
It is straightforward to generalize the discussions in sect. II C to the case with arbitrary
number of neutrinos where the matter potential is diagonal in the flavor eigenstate. The
scheme with number of sterile neutrinos is one of the example of these cases [4, 10]. The
time evolution of such a scheme with N neutrino flavors is described by
UNENU−1N +AN
where ΨTN ≡ (να1 , να2 , · · · , ναN ) is the flavor eigenstate,
EN ≡ diag (E1, E2, · · · , EN) (11)
is the energy matrix of the mass eigenstate,
AN ≡ diag (A1, A2, · · · , AN) ,
is the potential matrix for the flavor eigenstate, and UN is the N × N MNS matrix. As in
the previous sect., by taking the α, β components, we get
Ẽmj X̃
UNENU−1N +AN
for m = 0, · · · , N − 1,
which leads to the simultaneous equation
1 1 · · · 1
Ẽ1 Ẽ2 · · · ẼN
ẼN−11 Ẽ
2 · · · ẼN−1N
UNEU−1N +AN
UNEU−1N +AN
. (12)
Eq. (12) can be solved by inverting the N ×N Vandermonde matrix VN :
= V −1N
UNENU−1N +AN
UNENU−1N +AN
. (13)
The determinant of VN is the Vandermonde determinant
j<k ∆Ẽjk, and therefore V
can be analytically obtained as long as we know the value of Ẽj. The factors [(UNENU−1N +
AN)j]αβ on the right hand side of Eq. (13) can be expressed as functions of the energy Ej , the
quantity X
j in vacuum and the matter potential Aγ , since the matrix (UNENU−1N +AN)j is a
sum of products of the matrices [(UNENU−1N )ℓ]γδ =
k (0 ≤ ℓ ≤ j) and [(AN)m]ǫη =
mδǫη (0 ≤ m ≤ j). From Eq. (13) it is clear that enhancement of the oscillation
probability due to the matter effect occurs only when some of ∆Ẽjk becomes small.
III. THE CASE WITH LARGE MAGNETIC MOMENTS AND A MAGNETIC
FIELD
So far we have assumed that the potential term is diagonal in the flavor basis. We can
generalize the present result to the cases where we have off-diagonal potential terms. One of
such examples is the case where there are only three active neutrinos with magnetic moments
and the magnetic field (See, e.g., Ref. [1] for review.). The hermitian matrix3
UEU−1 B
B† U∗E(U∗)−1
B ≡ B µαβ
is the mass matrix for neutrinos and anti-neutrinos without the matter effect where neutrinos
have the magnetic moments µαβ in the magnetic field B. Here we assume the magnetic
interaction of Majorana type
µαβ ν̄α Fλκσ
λκ νcβ + h.c., (15)
and in this case the magnetic moments µαβ are real and anti-symmetric in flavor indices:
µαβ = −µβα.
If the magnetic field is constant, then the oscillation probability can be written as
P (νA → νB) = δAB − 4
X̃ABJ X̃
∆ẼJKL
X̃ABJ X̃
∆ẼJKL
, (16)
where A,B run e, µ, τ, ē, µ̄, τ̄ , and J,K run 1, · · ·, 6, respectively, and X̃ABJ ≡ UAJU∗BJ . ẼJ
(J = 1, · · · , 6) are the eigenvalues of the 6×6 matrix M. On the other hand, if the magnetic
field varies very slowly and if the length L of the baseline is so long that |∆ẼJKL| ≫ 1 is
satisfied for J 6= K, then the oscillation probability is given by
P (νA → νB) =
X̃BBJ (L)X̃
J (0). (17)
Following the same arguments as before, the quantity X̃ABJ is given by inverting the 6 × 6
Vandermonde matrix V6:
X̃AB1
X̃AB2
X̃AB6
= V −16
[M]AB
. (18)
3 See [5] for derivation of Eq. (14) from the Dirac Eq.
As in the previous sections, [(M)J ]AB (J = 0, · · · , 5) on the right hand side of Eq. (18) can
be expressed in terms of the known quantities XABK and BCD, and Eqs. (16) and (18) are
useful only when we know the eigenvalues ẼJ .
To demonstrate the usefulness of these formulae, let us consider the case where the
magnetic field is large at origin but is zero at the end point and the magnetic field varies
adiabatically. For simplicity we assume that θ13 and all the CP phases vanish.
4 In this
case the 6 × 6 matrix M in Eq. (14) becomes real, and we obtain the following oscillation
probabilities:
P (να → νβ) = P (ν̄α → ν̄β) =
(Uβj)
2[Re Ũ(0)αj]
P (να → ν̄β) = P (ν̄α → νβ) =
(Uβj)
2[Im Ũ(0)αj]
2, (19)
where Ũ(0) the 3× 3 unitary matrix which diagonalizes the 3× 3 matrix UEU−1 + iB(0) at
the origin:
UEU−1 + iB(0) = Ũ(0)Ẽ(0)Ũ−1(0).
In this example the energy eigenvalues are degenerate, i.e., the 6 × 6 energy matrix be-
comes diag(Ẽ , Ẽ), and the oscillation probability differs from Eq. (17) because the condition
|∆ẼJKL| ≫ 1(J 6= K) is not satisfied (e.g., ∆ẼJK = 0 not only for J = K = 1 but also for
J = 1, K = 4). Each probability in Eqs. (19) itself is not expressed in terms of X̃ααj (0), but
we find that the following relation holds:
P (να → νβ) + P (ν̄α → νβ) =
(Uβj)
2|Ũ(0)αj|2 =
j (0). (20)
Eq. (20) is a new result and without the present formalism it would be hard to derive it.
The details of derivation of Eq. (19) and explicit forms of X̃ααj (0) are given in Appendix B.
Eq. (20) may be applicable to the case where high energy astrophysical neutrinos, which are
produced in a relatively large magnetic field, are observed on the Earth, on the assumption
that the fluxes of neutrinos and anti-neutrinos are almost equal.
IV. THE CASE WITH NON-STANDARD INTERACTIONS
Another interesting application is the oscillation probability in the presence of new physics
in propagation [11, 12]. In this case the mass matrix is given by
UEU−1 +ANP (21)
4 In the presence of the magnetic interaction (15) of Majorana type, the two CP phases, which are absorbed
by redefinition of the charged lepton fields in the standard case, cannot be absorbed and therefore become
physical. Here, however, we will assume for simplicity that these CP phases vanish.
where
ANP ≡
2GFNe
1 + ǫee ǫeµ ǫeτ
ǫ∗eµ ǫµµ ǫµτ
ǫ∗eτ ǫ
µτ ǫττ
The dimensionless quantities ǫαβ stand for possible deviation from the standard matter effect.
Also in this case the oscillation probability is given by Eqs. (4) and (9), where the standard
potential matrix A has to be replaced by ANP . The extra complication compared to the
standard case is calculations of the eigenvalues Ẽj and the elements [(UEU−1 + ANP )m]αβ
(m = 1, 2).
Again to demonstrate the usefulness of the formalism, here we will discuss for simplicity
the case in which the eigenvalues are the roots of a quadratic equation. It is known [13] that
the constraints on the three parameters ǫee, ǫeτ , ǫττ from various experimental data are weak
and they could be as large as O(1). In Ref. [14] it was found that large values (∼ O(1)) of
the parameters ǫee, ǫeτ , ǫττ are consistent with all the experimental data including those of
the atmospheric neutrino data, provided that one of the eigenvalues of the matrix (21) at
high energy limit, i.e., ANP , becomes zero. Simplifying even further, here we will neglect the
parameters ǫeµ, ǫµµ, ǫµτ which are smaller than O(10−2) and we will consider the potential
matrix
ANP = A
1 + ǫee 0 ǫeτ
0 0 0
ǫ∗eτ 0 ǫττ
, (22)
where A ≡
2GFNe, the three parameters ǫee, ǫeτ , ǫττ are constrained in such a way that
two of the three eigenvalues become zero. We will assume that Ne is constant, and we
will take the limit ∆m221 → 0. The oscillation probability P (νµ → νe) in this case can be
analytically expressed and is given by
P (νµ → νe) = −4Re
− 4Re
(Λ+ − Λ−L)L
8A(∆E31)
Λ+Λ−(Λ+ − Λ−)
|ǫeτXeµ3 X
3 | sin(arg(ǫeµ) + δ)
× sin
(Λ+ − Λ−)L
. (23)
Eq. (23) is another new result and it would be difficult to obtain it without using the
present formalism. The details of derivation of Eq. (23), explanation of the notations and
the explicit forms of all the variables in Eq. (23) are described in Appendix C.
V. CONCLUSIONS
The essence of the exact formula for the neutrino oscillation probability in constant
matter which was discovered by Kimura, Takamura and Yokomakura lies in the fact that
the combination X̃
j ≡ ŨαjŨβj∗ of the mixing matrix elements in matter can be expressed
as polynomials in the same quantity X
j ≡ UαjUβj∗ in vacuum. In this paper we have
discussed applications of their formalism to more general cases. We have pointed out that
their formalism can be useful for the cases in matter not only with constant density but
also with density which varies adiabatically as in the case of the solar neutrino problem,
after taking the limit of the long neutrino path. We have shown that their formalism can be
generalized to the cases where the matter potential has off-diagonal components. As concrete
non-trivial examples, we discussed the case with magnetic moments and a magnetic field,
and the case with non-standard interactions. The application of the present formalism to the
case with unitarity violation has been discussed elsewhere [15]. The formalism by Kimura,
Takamura and Yokomakura is quite general and can be applicable to many problems in
neutrino oscillation phenomenology.
APPENDIX A: PROOF THAT EQ. (10) GIVES THE SAME (9)
In this appendix we show that Eq. (10) gives the same result for X̃
j (j = 1, 2, 3). The
value of X̃
j (j = 1, 2, 3) for
Ẽ + c1
Ũ−1 = UEU−1 +A+ c1
becomes at most quadratic5 in c, and all one has to do is to show that the coefficients of the
terms linear and quadratic in c vanish. Let us introduce the notation
1 1 1
Ẽ1 + c Ẽ2 + c Ẽ3 + c
(Ẽ1 + c)
2 (Ẽ2 + c)
2 (Ẽ3 + c)
≡ (V −1)(0) + c(V −1)(1) + c2(V −1)(2)
[UEU−1 +A+ c1]αβ
(UEU−1 +A+ c1)2
≡ ~B(0) + c ~B(1) + c2 ~B(2),
where V (k) is the coefficient of the inverted Vandermonde matrix which is k-th order in c,
and B
j is the coefficient of the vector (UEU−1 +A+ c1)
which is k-th order in c. Then
the terms linear in c are given by
(V −1)(1) ~B(0) + (V −1)(0) ~B(1)
∆Ẽ21∆Ẽ31
(Ẽ2 + Ẽ3, −2, 0)
∆Ẽ21∆Ẽ32
(−(Ẽ3 + Ẽ1), +2, 0)
∆Ẽ31∆Ẽ32
(Ẽ1 + Ẽ2, −2, 0)
[UEU−1 +A]αβ
(UEU−1 +A)2
5 Notice that all the factors ∆Ẽjk are invariant under the shift (10), and the only change by this shift
comes either from the terms ẼjẼk or from Ẽj + Ẽk in the inverse of the Vandermonde matrix (cf. Eq.
(9)). Hence the difference by Eq. (10) is at most quadratic in c.
∆Ẽ21∆Ẽ31
(+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1)
∆Ẽ21∆Ẽ32
(−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1)
∆Ẽ31∆Ẽ32
(+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)
2 [UEU−1 +A]αβ
= 0,
and the terms quadratic in c are given by
(V −1)(2) ~B(0) + (V −1)(1) ~B(1) + (V −1)(0) ~B(2)
∆Ẽ21∆Ẽ31
(+1, 0, 0)
∆Ẽ21∆Ẽ32
(−1, 0, 0)
∆Ẽ31∆Ẽ32
(+1, 0, 0)
[UEU−1 +A]αβ
(UEU−1 +A)2
∆Ẽ21∆Ẽ31
(Ẽ2 + Ẽ3, −2, 0)
∆Ẽ21∆Ẽ32
(−(Ẽ3 + Ẽ1), +2, 0)
∆Ẽ31∆Ẽ32
(Ẽ1 + Ẽ2, −2, 0)
2 [UEU−1 +A]αβ
∆Ẽ21∆Ẽ31
(+Ẽ2Ẽ3, −(Ẽ2 + Ẽ3), +1)
∆Ẽ21∆Ẽ32
(−Ẽ3Ẽ1, +(Ẽ3 + Ẽ1), −1)
∆Ẽ31∆Ẽ32
(+Ẽ1Ẽ2, −(Ẽ1 + Ẽ2), +1)
= 0.
Thus X̃
j (j = 1, 2, 3) is independent of c, as is claimed.
APPENDIX B: DERIVATION OF EQ. (19)
The matrix (14) can be rewritten as
M = 1
UEU−1 + iB 0
0 UEU−1 − iB
1 −i1
−i1 1
so the problem of diagonalizing the 6 × 6 matrix (14) is reduced to diagonalizing the 3× 3
matrices UEU−1 ± iB. Since we are assuming that θ13 and all the CP phases vanish, all
the matrix elements Uαj and Bαβ = −Bβα are real, UEU−1 ± iB can be diagonalized by a
unitary matrix and its complex conjugate:
UEU−1 + iB = Ũ ẼŨ−1
UEU−1 − iB = Ũ∗Ẽ(Ũ∗)−1.
Therefore, we can diagonalize M by a 6× 6 unitary matrix Ũ as
M = Ũ
Ũ−1,
where
Ũ = 1√
1 −i1
−i1 1
0 Ũ∗
Ũ − iŨ∗
−iŨ Ũ∗
We note in passing that the reason why diagonalization of the 6 × 6 matrix is reduced to
that of the 3× 3 matrix is because the two matrices UEU−1 and B are real.
On the other hand, without a magnetic field the 6× 6 unitary matrix U is given by
where the CP phase δ has dropped out because θ13 = 0. From these we can integrate the
equation of motion and we get the fields at the end point:
Ψc(L)
= Ũ(L)
e−iΦ 0
0 e−iΦ
Ũ(0)−1
Ψc(0)
Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1 −i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1)
i(Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1) Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1
Ψc(0)
where
Ẽ(t) dt,
and we have assumed that a large magnetic field exists at the origin whereas there is no
magnetic field at the end point. Thus the oscillation probabilities for the adiabatic transition
are give by:
P (να → νβ) = P (ν̄α → ν̄β) = lim
Ue−iΦŨ−1 + U∗e−iΦ(Ũ∗)−1
|Uβj|2
Re(Ũαj)
P (ν̄α → νβ) = P (να → ν̄β) = lim
Ue−iΦŨ−1 − U∗e−iΦ(Ũ∗)−1
|Uβj|2
Im(Ũαj)
Hence we obtain the following relation:
P (να → νβ) + P (ν̄α → νβ) = P (να → νβ) + P (να → ν̄β) =
|Uβj |2|Ũαj |2.
To get |Ũαj |2, we need the explicit expression for the eigenvalues and the quantity X̃ααj in the
presence of a magnetic field. In the following we will subtract E11 from the energy matrix E
because it will only change the phase of the oscillation amplitude. For simplicity we will put
θ13 = 0, θ23 = π/4, and we will consider the limit ∆m
21 → 0. Defining ∆Ejk ≡ ∆m2jk/2E
Bαβ = Bµαβ ≡
0 −p −q
p 0 −r
q r 0
we have the eigenvalue equation
0 = |λ1− U(E − E11)U−1 − iB|
= λ3 −∆E31λ2 − (p2 + q2 + r2)λ+
(p− q)2. (B1)
The three roots of the cubic equation (B1) are given by
λ1 = 2R cosϕ+
, λ2 = 2R cos(ϕ+
, λ3 = 2R cos(ϕ−
where
R ≡ [(∆E31/3)2 + (p2 + q2 + r2)/3]3/2,
ϕ ≡ (1/3) cos−1
{(∆E31/3)3 +∆E31(p2 + q2 + r2)/6−∆E31(p− q)2/4}/R
The quantity X̃ααj in the presence of a magnetic field is given by
X̃αα1
X̃αα2
X̃αα3
∆λ21∆λ31
(λ2λ3, −(λ2 + λ3), 1)
∆λ21∆λ32
(λ3λ1, −(λ3 + λ1), 1)
∆λ31∆λ32
(λ1λ2, −(λ1 + λ2), 1)
Y αα2
Y αα3
, (B2)
where
Y αα2 =
U(E −E11)U−1 + iB
= ∆E31X
0 (α = e)
∆E31/2 (α = µ, τ)
Y αα3 =
U(E −E11)U−1 + iB
= (∆E31)
2Xαα3 − (B2)αα
q2 + r2 (α = e)
r2 + p2 + (∆E31)
2/2 (α = µ)
p2 + q2 + (∆E31)
2/2 (α = τ)
. (B4)
In evaluating Y ααj , we have used the facts θ13 = 0, θ23 = π/4, ∆E21 = 0, Bαβ = −Bβα, and
that U(E −E11)U−1 is a symmetric matrix. Using all these results, it is straightforward to
obtain the explicit form for P (να → νβ) +P (ν̄α → νβ) by plugging the results of Eqs. (B2),
(B3), (B4) into the following (although calculations are tedious):
P (να → νe) + P (ν̄α → νe) = c212X̃αα1 + s212X̃αα2
P (να → νβ) + P (ν̄α → νβ) =
X̃αα1 +
X̃αα2 +
X̃αα3 (β = µ, τ),
where s12 ≡ sin θ12, c12 ≡ cos θ12.
APPENDIX C: DERIVATION OF EQ. (23)
The oscillation probability (23) is obtained in two steps. First we will obtain the eigenval-
ues of the matrix (21) with Eq. (22) and then we will plug the expressions for the eigenvalues
into Eq. (9) with A replaced by ANP given in Eq. (22).
Let us introduce notations for 3× 3 hermitian matrices:
0 −i 0
i 0 0
0 0 0
, λ5 ≡
0 0 −i
0 0 0
i 0 0
, λ7 ≡
0 0 0
0 0 −i
0 i 0
1 0 0
0 0 0
0 0 1
, λ9 ≡
1 0 0
0 0 0
0 0 −1
where λ2, λ5 and λ7 are the standard Gell-Mann matrices whereas λ0 and λ9 are the notations
which are defined only in this paper. Simple calculations show that the matrix ANP in Eq.
(22) can be rewritten as
ANP = Aeiγλ9e−iβλ5
1 + ǫee + ǫττ
1 + ǫee − ǫττ
+ |ǫµτ |2
eiβλ5e−iγλ9 , (C1)
where
β ≡ 1
tan−1
2|ǫeτ |2
1 + ǫee − ǫττ
γ ≡ 1
arg (ǫeµ).
From Eq. (C1) we see that the two potentially non-zero eigenvalues λe′ and λτ ′ of the matrix
(22) are given by
1 + ǫee + ǫττ
1 + ǫee − ǫττ
+ |ǫµτ |2
In order for this scheme to be consistent with the atmospheric neutrino data particularly at
high energy, which are perfectly described by vacuum oscillations, λτ ′ has to vanish [14]. In
this case, we have
tanβ =
|ǫeτ |
1 + ǫee
ǫττ =
|ǫeτ |2
1 + ǫee
λe′ = A(1 + ǫee)
|ǫeτ |2
(1 + ǫee)2
A(1 + ǫee)
cos2 β
Thus we have
ANP = Aeiγλ9e−iβλ5diag (λe′ , 0, 0) eiβλ5e−iγλ9 . (C2)
If we did not have β and γ, Eq. (C2) would be the same as the standard three flavor scheme
in matter, which was analytically worked out in Ref. [16] in the limit of ∆m221 → 0. It turns
out that, by redefining the parametrization of the MNS matrix Eq. (C2) can be also treated
analytically in the limit of ∆m221 → 0 as was done in Ref. [16]. The mass matrix can be
written as
UEU−1 +ANP = eiγλ9e−iβλ5
eiβλ5e−iγλ9UEU−1eiγλ9e−iβλ5 + diag (λe′, 0, 0)
eiβλ5e−iγλ9 .
Here we introduce the following two unitary matrices:
U ′ ≡ eiβλ5e−iγλ9 U
≡ diag(1, 1, eiargU ′τ3)U ′′ diag(eiargU ′e1 , eiargU ′e2 , 1),
where U is the 3× 3 MNS matrix in the standard parametrization [17] and U ′′ was defined
in the second line in such a way that the elements U ′′e1, U
e2, U
τ3 be real to be consistent with
the standard parametrization in Ref. [17] 6. Then we have
UEU−1 +ANP = eiγλ9e−iβλ5diag(1, 1, eiargU
U ′′EU ′′−1 + diag (λe′ , 0, 0)
×diag(1, 1, e−iargU ′τ3) eiβλ5e−iγλ9 . (C3)
Before proceeding further, let us obtain the expression for the three mixing angles θ′′jk and
the Dirac phase δ′′ in U ′′. Since
U ′ =
−iγUe1 + sβe
iγUτ1 cβe
−iγUe2 + sβe
iγUτ2 cβe
−iγUe3 + sβe
iγUτ3
Uµ1 Uµ2 Uµ3
−iγUτ1 − sβeiγUe1 cβe−iγUτ2 − sβeiγUe2 cβe−iγUτ3 − sβeiγUe3
where cβ ≡ cos β, sβ ≡ sin β, we get
θ′′13 = sin
−1 |U ′′e3| = sin−1 |cβe−iγUe3 + sβeiγUτ3|
θ′′12 = tan
−1(U ′′e2/U
e1) = tan
|cβe−iγUe2 + sβeiγUτ2|/|cβe−iγUe1 + sβeiγUτ1|
θ′′23 = tan
−1(U ′′µ3/U
τ3) = tan
Uµ3/|cβe−iγUτ3 − sβeiγUe3|
δ′′ = −argU ′′e3 = −arg (cβe−iγUe3 + sβeiγUτ3).
As was shown in Ref. [16], in the limit ∆m221 → 0, the matrix on the right hand side of Eq.
(C3) can be diagonalized as follows:
U ′′EU ′′−1 + diag (λe′ , 0, 0)−E11
= eiθ
λ7Γδ′′e
λ5Γ−1δ′′ e
λ2diag (0, 0,∆E31) e
−iθ′′
λ2Γδ′′e
−iθ′′
λ5Γ−1δ′′ e
−iθ′′
λ7 + diag (λe′, 0, 0)
= eiθ
λ7Γδ′′
λ5diag (0, 0,∆E31) + diag (λe′ , 0, 0)
Γ−1δ′′ e
−iθ′′
= eiθ
λ7Γδ′′e
iθ̃′′
λ5diag (Λ−, 0,Λ+) e
−iθ̃′′
λ5Γ−1δ′′ e
where Γδ′′ ≡ diag(1, 1, e−iδ
), ∆E31 ≡ ∆m231/2E, we have used the standard parametriza-
tion [17] U ′′ ≡ eiθ′′23λ7Γδ′′eiθ
λ5Γ−1δ′′ e
λ2 , and the eigenvalues Λ± are defined by
(∆E31 + λe′)±
(∆E31 cos 2θ
13 − λe′)
+ (∆E31 sin 2θ
6 The element U ′′τ2 has to be also real, but it is already satisfied because U
τ2 = Uτ2.
Having obtained the eigenvalues, by plugging these into Eq. (9) with A → ANP , Ẽ1 →
Λ−, Ẽ2 → 0, Ẽ3 → Λ+, we obtain X̃µe:
Λ−(Λ+ − Λ−)
(0, −Λ+, 1)
(−Λ+Λ−, −(Λ+ + Λ−), 1)
Λ+(Λ+ − Λ−)
(0, −Λ−, 1)
−Y µe3 + Λ+Y
Λ−(Λ+ − Λ−)
3 − (Λ+ + Λ−)Y
3 − Λ−Y
Λ+(Λ+ − Λ−)
where Y
j are defined by
UEU−1 +ANP
and are given by
2 = ∆E31 X
3 = [(∆E31)
2 + A(1 + ǫee)∆E31]X
3 + A∆E31ǫ
Furthermore, by introducing the notations
ξ ≡ [(∆E31)2 + A(1 + ǫee)∆E31]Uµ3|Ue3|
η ≡ A∆E31|ǫeτ |Uµ3Uτ3
ζ ≡ ∆E31Uµ3|Ue3|,
we can rewrite Y
2 = ζe
iδ and Y
3 = ξe
iδ + ηe−2iγ , where δ is the Dirac CP phase of the
MNS matrix U , so we have
Λ−(Λ+ − Λ−)
[ξ + ηe−i(2γ+δ) − Λ+ζ ]
[ξ + ηe−i(2γ+δ) − (Λ+ + Λ−)ζ ]
Λ+(Λ+ − Λ−)
[ξ + ηe−i(2γ+δ) − Λ−ζ ].
Notice that the phase factor eiδ in front of each X̃
j drops out in the oscillation probability
P (νµ → νe) because P (νµ → νe) is expressed in terms of X̃µej X̃
k , and the oscillation
probability (23) depends only on the combination 2γ + δ = arg (ǫeµ) + δ.
In the present case, the matrix Ũ is unitary and because of this three flavor unitarity all
the T violating terms are proportional to one factor:
∆ẼjkL
= 2 Im
∆Ẽ12L
− sin
∆Ẽ13L
+ sin
∆Ẽ23L
= −8 Im
∆Ẽ21L
∆Ẽ31L
∆Ẽ32L
This modified Jarlskog factor Im(X̃
2 ) in matter can be rewritten as
Im(X̃
2 ) =
Λ+Λ−(Λ+ − Λ−)
2 ) = −
ηζ sin(2γ + δ)
Λ+Λ−(Λ+ − Λ−)
= − A(∆E31)
Λ+Λ−(Λ+ − Λ−)
|ǫeτXeµ3 X
3 | sin(arg(ǫeµ) + δ).
This completes derivation of Eq. (23).
ACKNOWLEDGMENTS
The author would like to thank Alexei Smirnov for bringing my attention to Refs. [5,
6, 7, 8]. He would also like to thank He Zhang for calling my attention to Refs. [4, 10]
which were missed in the first version of this paper. This research was supported in part by
a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture,
#19340062.
[1] J. N. Bahcall, R. Davis, P. Parker, A. Smirnov and R. Ulrich, Reading, USA: Addison-Wesley
(1995) 440 p. (Frontiers in physics. 92)
[2] K. Kimura, A. Takamura and H. Yokomakura, Phys. Lett. B 537, 86 (2002)
[arXiv:hep-ph/0203099].
[3] K. Kimura, A. Takamura and H. Yokomakura, Phys. Rev. D 66, 073005 (2002)
[arXiv:hep-ph/0205295].
[4] Z. z. Xing and H. Zhang, Phys. Lett. B 618, 131 (2005) [arXiv:hep-ph/0503118].
[5] W. Grimus and T. Scharnagl, Mod. Phys. Lett. A 8, 1943 (1993).
[6] A. Halprin, Phys. Rev. D 34, 3462 (1986).
[7] P. D. Mannheim, Phys. Rev. D 37, 1935 (1988).
[8] R. F. Sawyer, Phys. Rev. D 42, 3908 (1990).
[9] V. D. Barger, K. Whisnant, S. Pakvasa and R. J. N. Phillips, Phys. Rev. D 22, 2718 (1980).
[10] H. Zhang, arXiv:hep-ph/0606040.
[11] M. M. Guzzo, A. Masiero and S. T. Petcov, Phys. Lett. B 260, 154 (1991);
[12] E. Roulet, Phys. Rev. D 44, 935 (1991).
[13] S. Davidson, C. Pena-Garay, N. Rius and A. Santamaria, JHEP 0303, 011 (2003)
[arXiv:hep-ph/0302093].
[14] A. Friedland and C. Lunardini, Phys. Rev. D 72, 053009 (2005) [arXiv:hep-ph/0506143].
[15] E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon and O. Yasuda, arXiv:hep-ph/0703098.
[16] O. Yasuda, Proceedings of Symposium on New Era in Neutrino Physics (Universal
Academy Press, Inc., Tokyo, eds. H. Minakata and O. Yasuda), p 165 – 177 (1999)
[arXiv:hep-ph/9809205].
[17] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592, 1 (2004).
http://arxiv.org/abs/hep-ph/0203099
http://arxiv.org/abs/hep-ph/0205295
http://arxiv.org/abs/hep-ph/0503118
http://arxiv.org/abs/hep-ph/0606040
http://arxiv.org/abs/hep-ph/0302093
http://arxiv.org/abs/hep-ph/0506143
http://arxiv.org/abs/hep-ph/0703098
http://arxiv.org/abs/hep-ph/9809205
introduction
generalities about oscillation probabilities
The case of constant density
The case of adiabatically varying density
Another derivation of the formula by Kimura, Takamura and Yokomakura
The case with arbitrary number of neutrinos
the case with large magnetic moments and a magnetic field
the case with non-standard interactions
conclusions
proof that Eq. (??) gives the same (??)
Derivation of Eq. (??)
Derivation of Eq. (??)
Acknowledgments
References
|
0704.1532 | Absolute measurement of the nitrogen fluorescence yield in air between
300 and 430 nm | Microsoft Word - fluo_manuscript_rev_S.Lantz.doc
Absolute measurement of the nitrogen fluorescence
yield in air between 300 and 430 nm.
G. Lefeuvrea1, P. Gorodetzkya2, J. Dolbeaua, T. Patzaka, P. Salina
a APC - AstroParticule et Cosmologie, CNRS : UMR7164 – CEA – IN2P3 – Observatoire de Paris, Université
Denis Diderot - Paris VII, 10, rue Alice Domon et Léonie Duquet 75205 Paris Cedex 13, France
Abstract
The nitrogen fluorescence induced in air is used to detect ultra-high energy cosmic rays and to measure their
energy. The precise knowledge of the absolute fluorescence yield is the key quantity to improve the accuracy on
the cosmic ray energy. The total yield has been measured in dry air using a 90Sr source and a [300-430 nm] filter.
The fluorescence yield in air is
4.23 ± 0.20 photons per meter
when normalized to 760 mmHg, 15°C and with an electron energy of 0.85 MeV. This result is consistent with
previous experiments made at various energies, but with an accuracy improved by a factor of about 3. For the
first time, the absolute continuous spectrum of nitrogen excited by 90Sr electrons has also been measured with a
spectrometer. Details of this experiment are given in one of the author's PhD thesis [32].
Keywords: nitrogen fluorescence, air fluorescence, extensive air showers, ultra-high energy cosmic rays.
PACS: 29.30.Dn ; 98.70.Sa
1 Actual address: Department of Physics, Syracuse University, Syracuse, NY 13244, USA
2 Corresponding author: E-mail address: [email protected]
Introduction
One of the current challenges in high energy particle physics is to find the origin and nature of
Ultra-High Energy Cosmic Rays (UHECR, E > 1018 eV). The measurement of their energy spectrum
could confirm their interaction with the cosmological background (GZK theory). But this is a difficult
task: the UHECR are not detectable themselves, but only the air shower they induce while going
through the atmosphere.
Experiments which have tried to solve this problem still have non consistent results for
E > 5•1019 eV. They use different detecting techniques. On one hand, AGASA [1, 2] has an array of
ground detectors separated by about 0.7 km. Muons and electrons of the showers arriving at the
ground are sampled. The energy reconstruction is difficult because of the lack of knowledge on
hadronic cross sections. This method relays on Monte-Carlo particle distributions in the shower
approximation to find the energy of the incoming particle. On the other hand, HiRes [3] uses
fluorescence telescopes to detect the continuous development of the shower. The accuracy of this
method can still be improved if the uncertainties on the fluorescence itself are lowered. Today, the
Auger experiment uses both methods but still calibrates energy extracted from the ground detectors by
the fluorescence (also on the ground, [4]). In the future, space-based experiments like JEM-EUSO will
look for the fluorescence signal from above [5].
Since 1964, several authors [6, 7, 8] have measured the fluorescence yield of each emission
band of nitrogen (see the spectrum in fig. 5). The fluorescence yield is defined as the number of
photons produced when an electron goes through one meter of air. In 2002, the cosmic ray community
[9, 10, 11] decided to start an active campaign of fluorescence measurements for cosmic ray physics.
Some were made at high energy in electron accelerators, the energy ranging from 80 MeV to 28 GeV
[12, 13]. Others were made at low energy, around 1 MeV with a 90Sr radioactive source, or much
lower with electron guns ([14, 15, 18, 21]). These experiments confirmed the hypothesis that the
fluorescence yield, up to some tens of MeV, is proportional to the energy deposit of the particle,
dE/dX. The difference between energy loss and energy deposit is of no importance from 0.1 to
10 MeV [16, 17], which includes our range. Therefore, the absolute scale of the yield can be set with a
measurement performed with electrons from a 90Sr source.
Now, the relative variation of the fluorescence yield with altitude is generally considered as
quite well known [18]. The parameterization of the yield as a function of altitude is deduced from the
relationship between pressure and altitude in an atmospheric model [19, 20]. These parameterizations
are fairly consistent with one another, but can only give the variation of the yield with altitude and not
the absolute value of the yield. Moreover, measurements of the absolute fluorescence yield have never
reached a precision better than 13 to 20 % [13, 21].
The experiment presented in this paper has been realized with a double purpose:
• to improve significantly the precision on the fluorescence yield: it is now 5 % ;
• to achieve a first continuous measurement of the fluorescence spectrum in the range from 300 to
430 nm, which had never been done before with a radioactive source.
1. Experiment
1.1. Principle
Two measurements have been carried out, named “integral” and “spectral” measurements.
Electrons from a strontium source are used to excite nitrogen. Three detectors are needed, all of them
based on Photonis photomultiplier tubes (PMT). One detects the electrons (electron-PMT), the other
two, the photons (photon-PMTs).
On one hand, the nitrogen emits fluorescence light through bands emitting lines from 300 to
430 nm. Thus cosmic ray experiments like HiRes and Auger use band pass filters of that range in front
of their detectors. This is what is done here in the integral measurement.
On the other hand, fluorescence yield experiments generally use narrow filters to separate the
lines. But the overlap between some spectral filter bandwidth and the close proximity of some
nitrogen bands make the separation difficult in reality. For this reason, the spectral measurement has
been performed with an optical grating spectrometer, allowing a continuous check and a good
separation of the lines.
1.2. Description of the setup
This setup has been conceived with the aim of improving the measurement accuracy. The
fluorescence chamber is a cross-shaped stainless steel chamber (a 6-ports accelerator pipe cross). It is
schematically shown in fig. 1. Electrons are emitted by a strong 90Sr source (370 MBq) which is
placed on the top part of the cross. On the bottom part, a plastic scintillator (truncated cone, 20 and
28 mm diameters, 20 mm thick and its angle is the same that the one made by the electron trajectories
at the cone periphery). The electrons path is hence defined as the inside of a cone (100 mm height),
with the source at its top. To homogenize the light between the scintillator and the PMT, a
kaleidoscopic hexagonal light guide is used to carry the light. The phototube is a Photonis XP2262
[22] equipped with an active voltage divider (the rate is high, of the order of 2.5 MHz, and very
stable).
Figure 1. Schematic view of the fluorescence chamber. The strontium source and the scintillator are inside the
lead shield. The internal structure of the lead is shown in fig. 2 (side view).
The consequence of the high counting rate is a high X-ray noise level, due to the interacting
electrons with the surrounding matter, including the PMT. To shield to a maximum these X-rays, a
10 cm diameter lead cylinder has been placed inside the chamber with the 90Sr source at its centre. A
vertical cone is dug in the lead so the electrons can reach the scintillator. This cone is 30 mm diameter
at its base, slightly wider than the electrons cone defined by the scintillator. Such a cone minimises the
number of interactions between electrons and the lead: either they are totally stopped (around the
source), or they touch in a grazing way the cone in their way to the scintillator. A GEANT Monte-
Carlo simulation shows that half of the useful electrons do reach the scintillator without an interaction
with the lead, the other half touching only once.
Figure 2. Internal structure of the lead shield. Inside the hollow cone is the smaller cone of the path of the useful
electrons defined by the source and the scintillator (thick dotted line).
An horizontal cylinder (40 mm diameter) is also dug in the lead, centred on the optical axis, to
let the photons go to the photon-PMTs. This structure is represented in fig. 2.
The effective fluorescence volume is thus, on a first approximation, the intersection of the
vertical cone with the horizontal cylinder. In a more precise way, all geometrical efficiencies have
been determined with Monte-Carlo simulations. They take into account all possible interactions
between electrons and their surroundings and give the exact photon solid angles as well as the exact
useful electron path length (the length used in the yield expressed in photons per meter).
Two kinds of interactions can occur: with the gas and with the lead. First, part of the energy of
the source electrons is lost by ionization, i.e. creation of secondary electrons. If the secondaries go too
far away from the useful fluorescence volume, they can either reach the lead or enter the horizontal
cylinder. Both modify the detection efficiencies (geometry wise) of the photon-PMTs calculated. But
99 % of these secondaries have an energy smaller than 5 keV. Their range in air at atmospheric
pressure is less than 2 mm. This has been checked through a geometrical simulation not to be enough
to have an influence on the solid angles. The effect of these δ rays on the yield expressed in photons
per deposited energy is explained later in paragraph 3.1
The second type of interaction concerns the electrons from the source that do not reach
directly the scintillator. Some of them are scattered by the lead and then bounce back to the
scintillator. This has three consequences, studied by a GEANT Monte-Carlo simulation. The first is
the increase in the electron counting rate, due to the larger solid angle available for the electrons: it is
doubled by this effect. The second is a modification in the electron energy spectrum, which contains
more low energy electrons. This has been taken into account in the calculation of the electrons average
energies. The third is a slight change in the useful electron path length, also taken into account.
Some electrons can also bounce from the scintillator back into the fluorescence volume. This
has been simulated to be a 10-9 effect on the photon detection, hence totally negligible.
Figure 3. Schematic view of the optical components for the spectral measurement.
For the integral measurement, fluorescence photons are detected and counted by a Photonis
XP2020Q (2 in. diameter) placed on the left hand-side of the setup [23]. This PMT will be named
integral-PMT and it is equipped with a fused silica window. As the fluorescence efficiency is known
to be low (around 4 photons per meter), it has to work in single photoelectron mode, meaning at a very
high gain. It is polarized positively (photocathode at ground) in order to reduce its own noise level to
around 300 Hz (instead of 3 kHz with a negative polarity). This effectively removes the tiny
discharges in the silica between the photocathode and the outside world. The optical filter on the
entrance window is a Schott-Desag BG3 (2 mm thick, 34 mm diameter) [24]. It is the same as JEM-
EUSO intends to use, and very similar to those used by Auger. It is glued with Epotec 301-2 [25]
which has the same refraction index than both the window and the filter.
Figure 4. Absolute efficiency of the integral-PMT (dashed line) and spectral-PMT (bold line) as a function of the
size of the effective detecting surface size. A 20 mm diameter diaphragm is used to limit the effective detecting
area of the integral-PMT to the flattest region of the photocathode. Using a diaphragm of 20 mm diameter (314
mm2), the efficiency is 18.7%. The spectral-PMT is used in its central region (~ 10×18 mm) where the efficiency
including the filter is 19.3%. The uncertainty is 1.7% of the efficiency values. Note that the zero suppression on
the vertical axis.
For the spectral measurement, photons are analyzed by an optical grating spectrometer (Jobin-
Yvon H25) and counted by another Photonis XP2020Q, attached to the spectrometer. This PMT will
be named spectral-PMT. The light rays incoming the spectrometer have to be inside its numerical
aperture (f = 250 mm, NA = f/4), so a silica converging lens of 150 mm focal length and 46 mm
diameter [26] is inserted at the right place between the fluorescence volume and the entrance slit of the
spectrometer. The optical image of this volume is thus on the latter. The optical magnification is 1/7.
Entrance and exit slits of the spectrometer have the same dimensions: 2 x 7 mm, the maximum size
available, inducing a resolution (measured with lasers) of 6 nm FWHM. This part of the setup is
shown in fig. 3.
The main uncertainty in this kind of experiments, where very few photons are counted, arises from the
absolute efficiency of the photon-PMTs themselves. The manufacturers provides this value with an
uncertainty of 15 to 20 % (at 1σ), which is not precise enough. For this reason, we designed a new
way to measure the absolute efficiency of the photon-PMTs in the single photoelectron mode very
accurately, based on the comparison with a NIST photodiode.
The relative efficiency map of the photocathodes of the PMTs working in single photoelectron mode
was measured with a 377 nm LED every 3 mm in X and Y with a relative precision better than 0.5%.
This precision was needed to be better than the point to point variations in efficiency (about 2%), in
order to control these variations. A measurement of the absolute efficiency of one of the map points
with an accuracy of 1.7%, transforms that relative efficiency map in an absolute efficiency map. A
dedicated paper, following a patent, will present this measurement. This result was used to determine
the flattest region of the photocathode (see fig. 4). The effective detecting area of the integral-PMT
was limited by a 20 mm diameter diaphragm where the efficiency stays roughly constant when the
diaphragm size is increased. That way, a small error on the diaphragm diameter has a negligible
consequence on the PMT efficiency (it has however on the solid angle, but this is easy to control).
Figure 5. Fluorescence spectrum (vertical lines) with relative intensities and absolute response of the photon
detector (PMT + filter + diaphragm, dashed curve) (source of the fluorescence spectrum : [14]). The integral-
PMT efficiency for this spectrum is 17.8%.
The absolute spectral efficiency of the detector {PMT + filter + diaphragm} is the dashed line
superimposed on the fluorescence spectrum in fig. 5. Its value for photons at 377 nm is (18.9 ± 0.3)%,
and, when convoluted with the fluorescence bands [14] and the relative variations given by Photonis
[22], is (17.8± 0.4)%. The spectral-PMT is illuminated on a 10 x 18 mm area (smaller area than the 20
mm circle for the integral PMT, hence all photons at the exit slit of the spectrometer reaching different
parts of this area would here also have roughly the same probability to produce photoelectrons).
On their way from the fluorescence chamber to any of the photon-PMTs, integral or spectral-PMT,
the 20 cm long tubes are baffled to prevent reflected photons to reach the photocathode.
1.3. Gas
The gas used is either nitrogen for the adjustments or Messer dry air for the measurements. Fig. 6
shows the gas setup. It is possible to mix gases and to introduce controlled quantities of different
impurities and water vapour. Those measurements including pressure variations are planned for the
future.
The gas circulates at a rate of 1 L.h-1 through the chamber to avoid impurities build-up and
degassing of the walls. This circulation is controlled by a precise flow meter [27], followed by a
pressure controller [28] and an ultra-clean primary pump with an ultimate vacuum of 50 mbar [29].
Internal and external pressures and temperatures are regularly monitored during data taking.
Figure 6. The gas circulation scheme.
1.4. Data acquisition
Hardware data acquisition is basically the same that for other authors, but had to be adapted to
high counting rates. Fig. 7 shows its principle.
The lifetimes of excited levels of nitrogen molecules at atmospheric pressure are of the order
of a few nanoseconds [8, 14, 15, 18, 21]. A measurement based on the time coincidence between an
electron and a photon is made to discriminate fluorescence from background photons.
Figure 7. The data acquisition principle
Basically, an electron in the scintillator is the trigger for the photon measurements. An
electron produces about 0.16 photon in the 4 cm long fluorescence volume. Geometrical efficiencies
toward each photon-PMT are very small: 3.69·10-4 for the integral-PMT and 7.48·10-6 for the spectral-
PMT. As a consequence, only 5.9·10-5 and 1.2·10-6 photon respectively reach the PMTs. So, both
photon-PMTs work in single photoelectron mode. Their spectra are very “pure” in “1 photo-electron”,
with a negligible amount of “2”. So a discriminator is enough to select the “1 photoelectron” peak.
This can be seen in fig. 8.
ADCs (LRS 2249A, 0.25 pC per channel) record the electron and photon spectra. They are
used only to check the gain stability of all PMTs during data taking. The “detection inefficiency”, D,
defined as the proportion of lost fluorescence counts due to the mandatory discriminator threshold on
the single photoelectron spectrum (see fig. 8), has also been measured and been found to be 3.76 % of
the measured single photoelectron peak.
Figure 8. Single photoelectron spectrum, with and without the pedestal.
Time to Digital Converters (TDCs) (LRS 2228A, 0.3 ps per channel, monohit) are used to
record the time difference spectra between electrons and photons.
Both ADCs and TDCs could be triggered by the electron signal, but this would produce a very
high random rate. We cleaned up the trigger by replacing it by an electron-photon coincidence.
This coincidence is performed by a fast NIM coincidence unit. The photon pulse is 10 ns
wide. The electron pulse is 100 ns wide to take into account the delay in photon emission (lifetimes of
the levels). This was set to prepare acquisitions at low pressure, where the lifetimes are longer than at
atmospheric pressure. Even at very low pressure, 100 ns ensures the loss due to that effect will be
negligible. In other terms, a coincidence occurs when the photon arrives with less than 100 ns delay
after the electron.
The TDC is stopped by the same electron (see fig. 7), this time as a 10 ns wide pulse and
delayed by 150 ns. Therefore, the start has a rate much lower than the stop and this inverts the time
axis. Results can be seen in fig. 9. The useful signal is the peak in the spectrum. It can be easily
separated from the flat background. The effect on the peak left slope due to the lifetimes of the
nitrogen levels cannot be seen on this figure because it has been measured in air at atmospheric
pressure. The narrow peak on the far right-hand side is due to random photons arriving some
nanoseconds before electrons. These spectra suffer from an important dead time. But the ratio of the
peak to total spectrum is not affected. The real number of counts is extracted from scalers measuring
the total spectrum with a small and well corrected dead time. These fast scalers (CAMAC
CERNSPEC NE003, 25 MHz, and VME V560E, 100 MHz) count all electron, photon and
coincidence pulses.
Two thresholds, creating two triggers, are set on the 90Sr spectrum to study the possible
dependence of the fluorescence yield with the electron energy even if it is difficult to imagine an
influence other than energy deposit. One is set at ~ 600 keV and another at ~ 1.2 MeV. The
corresponding average energies are 1.1 and 1.5 MeV. There are around 1 000 photon pulses per
second counted by the integral-PMT (with a background of 300 Hz) and 40 by the spectral-PMT on at
the top of the 337 nm line (with a background of 30 Hz). Moreover, the dead times, TM, of both
electron scalers have been specifically measured and found to be 1.2 % (scaler rate 2·106 Hz) and
0.7 % (scaler rate 6·105 Hz).
2. Results
2.1. Integral measurement
Figure 9. Example of TDC spectrum for the integral measurement
The photon yield per unit length, Y, is derived from the signal portion of the TDC spectrum
through the following formula:
where:
• Π is the number of signal counts in the TDC spectrum (see fig. 9) ;
• H is the integral of this spectrum ;
• C is the number of coincidences sent to the TDC measured by a scaler. Only H coincidences have
effectively started the TDC due to its dead time ;
• 1 + D is the correction to the detection inefficiency ;
• Ne is the number of electrons according to the scaler ;
• 1 + TM is the correction to the dead time of the scaler itself ;
• Lmoy is the mean length of the electrons path in the fluorescence volume ;
• εPMT is the efficiency of the integral-PMT ;
• Tw is the transmission of the fused silica window closing the fluorescence chamber ;
• εgeo is the geometrical efficiency.
Systematic errors are presented in table 1. The main uncertainty of this experiment is due to
the high counting rate of the electrons, leading to a non-linear dead time dependence in the TDC
module. This effect, which varies from channel to channel, depends on the internal TDC time
constants and is not fully understood. It has been evaluated by using all the module’s channels, and
another TDC module (CAEN V1290N, multihit) to compare their results. All values are found equal
within 4% (at 1σ) and this uncertainty has been chosen in a conservative way. In the future, to further
reduce this uncertainty, fast flash ADCs will be used, allowing to discriminate pulses with and without
pile-up on an event per event basis.
On the contrary, the uncertainty on the efficiency of the integral-PMT is very low. This is due
to the photomultiplier efficiency measurement made especially for the fluorescence measurement by
the patented method.
Table 1. Systematic uncertainties of the experiment.
An example of typical data sample is presented in tab. 2. The low (1.1 MeV) and high (1.5 MeV)
energy measurements give respectively 3.95 and 4.34 photons per meter. Statistical uncertainties are
0.2 % and 0.8 %.
The energy normalization is made at 0.85 MeV using the dE/dX ratios with values
interpolated from NIST data [30] and yields respectively 4.05 and 4.41 photons per meter.
Table 2. Data sample for the low energy measurement.
The pressure and temperature normalizations are then applied. Each excited level has its own lifetime,
and the yield can be written with respect to pressure and temperature using the kinetic theory [6]. Thus
this normalization has been made for each band using previous parameterizations (which give the
same variations for the first kilometers above the ground [20]). Here, Nagano's model [21], who uses
the different bands yields is used. The normalized pressure and temperature are those of the US-
Standard 1976 model [19] at sea level: 760 mmHg and 15°C. In this model, for an electron energy of
0.85 MeV, one finds that the ratio of the yield at 753.8 mmHg and 295.95° K (the 1.1 MeV
conditions) to the yield at 760.0 mmHg and 288.15° K (the US Standard conditions), is 0.9863, and
the similar ratio for the 1.5 MeV conditions which are 751.8 mmHg and 296.05° K is 0.9860. These
ratios are then used to normalize the measured values.
Finally, the two normalized values are 4.05 and 4.42 ph/m and have, due to this normalization,
an added uncertainty of 0.6 %, setting the total relative uncertainty to 5.0 %. They are separated by
8.5%, inside an error bar (± 1σ). Hence, these are averaged to get the fluorescence yield, at
760 mmHg, 15°C and for 0.85 MeV electrons:
4.23 ± 0.21 photons / m.
This yield per meter is that of the primary particle, therefore it takes into account its
production of δ rays and their fluorescence. This number can also be written in units of photons per
deposited energy. For a given energy, these two quantities are strictly proportional, since:
where Φν is the fluorescence efficiency at the wavelength corresponding to the frequency ν, with the
yield per deposited energy being [31]:
The energy of the δ rays is naturally included in the dE/dX of the initial electron.
Nevertheless, in the context of this experiment, a special care must be taken for the δ rays. A Monte-
Carlo simulation has been performed and shows that here, with the geometry described earlier and at
atmospheric pressure, only 1 % of the δ rays have more than 5 keV and some of these could be lost in
the lead before having produced any fluorescence. The effective dE/dX should thus be reduced. This
simulation shows then that almost 67 % of these δ rays with 5keV or more will produce detectable
fluorescence. Therefore only 33 % of them are effectively lost and contribute to the reduction of the
dE/dX. Finally, this reduction, hence the reduction in yield, is 0.4 %. The central value is shifted from
0.4 % (we measured 4.21 photons/m before applying this correction). The uncertainty stays the same
at 5 %. This would be very different at lower pressures, where a more extensive Monte-Carlo
simulation would be required and the correction would be increased.
The energy deposited by a 0.85 MeV electron in a meter of the US Standard air is 0.2059 MeV
[30]. The fluorescence yield per deposited energy is thus 20.38 ± 0.98 photons per MeV.
The nitrogen yield to air yield ratio has also been measured, (at one electron energy only:
1.1 MeV), and found to be 4.90 ± 0.01, where the low uncertainty is only statistical: all systematic
errors compensate, the gas being the only element changed from one measurement to the other. This
ratio is compatible with what has been measured at different energies : 6 ± 2 at 28.5 GeV [13],
5.5 ± 0.3 at 1 MeV [21]
2.2. Spectral measurement
Spectral measurements are interesting for many purposes. The gas kinetic theory is valid for
individual bands. Is the sum of the different bands yield equal to the integrated yield used by the
majority of cosmic ray experiments using fluorescence? How do the different bands change yield
when temperature and pressure are modified? Up to now, the different bands have been observed
either with a grating spectrometer [6, 7, 14] in a first method, the electrons being produced by an
electron gun. Their energy (around 10 keV) is so small that the electron scattering in the gas prevents
any measurement of path length, hence cannot give a yield in photons per meter. The second method
[8,15, 18, 21] uses narrow interference filters to analyze the bands. Here, the electrons, as in this work,
come from a 90Sr source. Fig. 1 of [21] illustrates the method complexity. It is difficult to separate
overlapping bands in some filters. The only solution to have a good resolution on the bands is to use a
grating spectrometer, and the easiest way to have electrons with an energy large enough to know their
path is to use a 90Sr source. The acceptance of a spectrometer is roughly an order of magnitude smaller
than that of a filter. So the very first step is to prove the feasibility of such a method. This is what is
done here.
Hence, in this experiment, a 90Sr source 10 times stronger than in [8,15, 18, 21] was used, The
counterpart of the high number of incident electrons is that a very special attention has to be given to
pile-up effects and to the lead shield geometry to avoid the introduction of a large X-ray background.
Results of the spectral measurements are represented in fig. 10. The whole fluorescence spectrum has
been measured, from 300 to 436 nm, in dry air at room temperature and normal pressure, with
electrons of average energy of 1.1 MeV. The time allowed to measure this spectrum was short and the
spectrometer equipped with an output slit can measure only one wavelength at a time. So it was
decided to open the slits at their maximum, setting the 1σ resolution to 3 nm. Thus measurements
were made with steps of 3 nm. But even with 2·106 electrons detected every second, only
0.16 fluorescence photons per second are recorded in the signal part of the TDC spectrum for the most
intense line (337 nm). This is the reason why this spectrum has only been made for the “low” energy
threshold, as defined above.
Fluorescence lines are much narrower than the resolution set for the spectrometer as is seen in
[14]. The absolute yield is thus given by the height of the curve at a given wavelength, and not by the
area integrated over a bandwidth dλ. The sum of the yields of all lines indicated in fig. 10 is
3.9 ± 0.8 photons per meter. This value is in close agreement with the result of the integral
measurement. The 20% uncertainty is due most entirely to the bad knowledge of the spectrometer
absolute efficiency, especially in the UV.
Superimposed on this spectrum is Ulrich’s spectrum [14], convoluted with a 3 nm
resolution. As it is a relative yield spectrum, it is shown here normalized to the 337 nm line. Our
spectrum is well compatible with what would measure [14] with such a resolution.
Figure 10. Spectrum of the absolute fluorescence yield of nitrogen in air between 300 and 436 nm. The bold line
is the result of the present experiment, and the dashed line is Ulrich's spectrum [14] as if analyzed by our
spectrometer. Discrepancies could be explained by the ageing of our spectrometer.
The first observation is the confirmation of the feasibility of this measurement: all the main
lines are indeed observed. Evidence is given that an absolute spectrum can be measured with a basic
apparatus, and taking care of reducing the PMT background as explained earlier. The sum of the lines
yield is consistent with the previous integral measurement. Moreover, its uncertainty is 20 %, when
previous experiments do not give better than 15 %.
The second observation concerns the discrepancies, around 316 nm and beyond 375 nm. The
extraction of the absolute yield involves the efficiency of the spectrometer. Its spectral efficiency
curve is provided by the manufacturer with a low accuracy in the UV (which is true also for Ulrich's
results). Furthermore two effects induce a very important loss in our spectrometer:
• the ageing of each optical element (mirrors and grating). The spectrometer efficiency can be
reduced to a value around 15 % after ten years of use ;
• the extensive use with intense UV light before this experiment.
At 400 nm, the absolute efficiency of the spectrometer was measured with an accuracy of
about 2% to be only 15 %, instead of the 61 % given by the manufacturer (who confirms in a private
communication such a low value compatible with ageing). The method used is very similar to that
taken to determine the absolute efficiency of the photon-PMTs (comparison to a NIST photodiode).
This value of 15 % has been used to calculate the absolute yields in the entire spectral range. But there
is no reason that this loss is constant with respect to wavelength. The 20% uncertainty on the yield
arises from this unknown but limited variation. The question of getting the absolute efficiency of
spectrometers in the UV is quite challenging and is the object of specific attentions by the community
of atomic / spectral physics.
It is unfortunate that this spectrometer method does not yield yet an accuracy better than the
"integrated yield" method. It will if the spectrometer resolution can be made high enough to totally
separate the bands, which is possible if it is equipped with a CCD readout to minimize the experiment
duration. Then, the absolute efficiency of the spectrometer will have to be determined with a high
accuracy in the UV, not an easy task according to Ulrich, but possible through our patented method.
In the future, two measurements will be done:
• calibrate in an absolute way the old Jobin-Yvon;
• use the new spectrometer able to measure the full spectrum at once with a 0.1 nm resolution and
equipped with a light intensifier to measure the fluorescence yield.
3. Conclusion
The absolute fluorescence yield of nitrogen in dry air at atmospheric pressure has been
measured. The precision of the measurement is improved by a factor of three, which has an immediate
impact on the cosmic ray energies found by HiRes which uses Bunner's [6] yield. Their energies are
increased by 22%, hence their spectrum is much closer to AGASA’s (which incidentally have been
recently lowered the energy of their points by 10% [33]).
The first continuous fluorescence spectrum of nitrogen excited by electrons from a 90Sr source
was also measured.
Next steps are to introduce impurities in the gas, such as argon, water vapour and pollutants. A
pressure study of the total yield will be made. On another hand, the spectral measurement will be
improved thanks to a new spectrometer. Papers will follow to account for these measurements, which
will provide an overall and realistic view of the fluorescence phenomenon.
Acknowledgements
We would like to thank Bernard LEFIEVRE for his help with GEANT simulation of the setup,
and François LELONG and Jean-Paul RENY for their help in building the bench.
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|
0704.1533 | Concrete Classification and Centralizers of Certain $\mathbb{Z}^2
\rtimes {\rm SL}(2,\mathbb{Z})$-actions | CONCRETE CLASSIFICATION AND CENTRALIZERS OF
CERTAIN Z2 ⋊ SL(2,Z)-ACTIONS
HIROKI SAKO
Abstract. We introduce a new class of actions of the group Z2 ⋊ SL(2,Z) on
finite von Neumann algebras and call them twisted Bernoulli shift actions. We
classify these actions up to conjugacy and give an explicit description of their
centralizers. We also distinguish many of those actions on the AFD II1 factor in
view of outer conjugacy.
1. Introduction
We consider the classification of Z2 ⋊ SL(2,Z)-actions on finite von Neumann
algebras in this paper. Mainly, we concentrate on the case that the finite von
Neumann algebra is the AFD factor of type II1 or non-atomic abelian.
There are two difficulties for analyzing discrete group actions on operator algebras.
The first is that we do not have various ways to construct actions. The second is
that we can not analyze them by concrete calculation in most cases. To give many
examples of actions which admit concrete analysis, we introduce a class of trace
preserving Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras and call them
twisted Bernoulli shift actions. We classify those actions up to conjugacy and study
them up to outer conjugacy.
An action β(H, µ, χ) in the class is defined for a triplet (H, µ, χ), where H is
an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ
is a character of H . We obtain the action by restricting the so-called generalized
Bernoulli shift action to a subalgebra N(H, µ) and “twisting” it by the character χ.
The process of restriction has a vital role in concrete analysis of these actions.
A ∗-isomorphism which gives conjugacy between two twisted Bernoulli shift ac-
tions β(Ha, µa, χa) and β(Hb, µb, χb) must be induced from an isomorphism between
the two abelian groups Ha and Hb. We prove this by concrete calculation (Section
4). It turns out that there exist continuously many, non-conjugate Z2 ⋊ SL(2,Z)-
actions on the AFD factor of type II1 (Section 5). By using the same technique, we
describe the centralizers of all twisted Bernoulli shift actions. Here we should men-
tion that the present work was motivated by the previous ones [Ch], [NPS], where
similar studies were carried out in the case of SL(n,Z).
In Section 6, we distinguish many twisted Bernoulli shift actions in view of outer
conjugacy. The classification for actions of discrete amenable groups on the AFD
factor of type II1 was given by Ocneanu [Oc]. Outer actions of countable amenable
groups are outer conjugate. In the contrast to this, V. F. R. Jones [Jon] proved that
any discrete non-amenable group has at least two non outer conjugate actions on
2000 Mathematics Subject Classification. Primary 46L40; Secondary 46L10.
Key words and phrases. von Neumann algebras; automorphisms.
http://arxiv.org/abs/0704.1533v3
2 HIROKI SAKO
the AFD factor of type II1. S. Popa ([Po3], [Po4], [PoSa], etc.) used the malleabil-
ity/deformation arguments for the Bernoulli shift actions to study (weak) 1-cocycles
for the actions. For some of twisted Bernoulli shift actions, which we introduce in
this paper, it is shown that (weak) 1-cocycles are represented in simple forms under
some assumption on the (weak) 1-cocycles. We prove that there exist continuously
many twisted Bernoulli shift actions which are mutually non outer conjugate. This
strengthens the above mentioned result due to Jones in the Z2 ⋊ SL(2,Z) cases.
2. Preparations
2.1. Functions det and gcd. For the definition of twisted Bernoulli shift actions in
Section 3, we define two Z-valued functions det and gcd. The function det is given
by the following equation:
= qr0 − rq0,
∈ Z2.
The value of the function gcd at k ∈ Z2 is the greatest common divisor of the two
entries. For 0 ∈ Z2, let the value of gcd be 0.
Lemma 2.1.
(1) The action of SL(2,Z) on Z2 preserves the functions det and gcd, that is,
det(k, k0) = det(γ · k, γ · k0),
gcd(k) = gcd(γ · k), k, k0 ∈ Z2, γ ∈ SL(2,Z).
(2) The following equation holds true:
det(k, k0) = gcd(k) + gcd(k0)− gcd(k + k0) mod 2, k, k0 ∈ Z2.
Proof. The claim (1) is a well-known fact, so we prove the claim (2). For the function
gcd, we get
1 mod 2, (either q or r is odd),
0 mod 2, (both q and r are even).
Since the action of SL(2,Z) on Z2 preserves the functions det and gcd, it suffices to
show the desired equation against the following four pairs:
(k, k0) =
mod 2.
2.2. Scalar 2-cocycles for abelian groups. We fix some notations for countable
abelian groups and their scalar 2-cocycles. For the rest of this paper, let H be an
abelian countable discrete group and suppose that any scalar 2-cocycle µ : H×H →
T = {z ∈ C | |z| = 1} is normalized, that is, µ(g, 0) = 1 = µ(0, g) for g ∈ H . We
denote by µ∗ the 2-cocycle for H given by µ∗(g, h) = µ(h, g), g, h ∈ H . Let µ∗µ be
the function on H ×H defined by
µ∗µ(g, h) = µ(h, g)µ(g, h), g, h ∈ H.
CLASSIFICATION AND CENTRALIZERS 3
This is a bi-character, that is, µ∗µ(g, ·) and µ∗µ(·, h) are characters of H . By using
this function, we can describe the cohomology class of µ. See [OPT] for the proof
of the following Proposition:
Proposition 2.2. Two scalar 2-cocycles µ1 and µ2 of H are cohomologous if and
only if µ∗1µ1 = µ
Let Cµ(H) be the twisted group algebra of H with respect to the 2-cocycle µ. We
denote by {uh | h ∈ H} the standard basis for Cµ(H) as C-linear space. We recall
that the C-algebra Cµ(H) has a structure of ∗-algebra defined by
ug uh = µ(g, h) ug+h, u
g = µ(g,−g)u−g, g, h ∈ H.
Let µ̃ be the T-valued function on ⊕Z2H ×⊕Z2H defined by
µ̃(λ1, λ2) =
µ(λ1(k), λ2(k)), λ1, λ2 ∈ ⊕Z2H.(Eq1)
The function µ̃ is a normalized scalar 2-cocycle for ⊕Z2H . Let Λ(H) be the abelian
group defined by
Λ(H) =
λ : Z2 → H
∣∣∣∣∣ finitely supported and
λ(k) = 0
Its additive rule is defined by pointwise addition.
2.3. Definition of a Z2 ⋊ SL(2,Z)-action on Λ(H). The group SL(2,Z) acts on
Z2 as matrix-multiplication and the group Z2 also does on Z2 by addition. These
two actions define the action of Z2 ⋊ SL(2,Z) on Z2 which is explicitly described as
q + xq0 + yr0
r + zq0 + wr0
for all ((
∈ Z2 ⋊ SL(2,Z),
∈ Z2.
We define an action of Z2 ⋊ SL(2,Z) on ⊕Z2H as
(γ · λ)(k) = λ(γ−1 · k), k ∈ Z2,
for γ ∈ Z2 ⋊ SL(2,Z) and λ ∈ ⊕Z2H .
2.4. On the relative property (T) of Kazhdan. We give the definition of the
relative property (T) of Kazhdan for a pair of discrete groups.
Definition 2.3. Let G ⊂ Γ be an inclusion of discrete groups. We say that the pair
(Γ, G) has the relative property (T) if the following condition holds:
There exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a
unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying
‖π(g)ξ − ξ‖ < δ for g ∈ F , then there exists a non-zero vector η ∈ H such that
π(h)η = η for h ∈ G.
Instead of this original definition, we use the following condition.
4 HIROKI SAKO
Proposition 2.4. ([Jol]) Let G ⊂ Γ be an inclusion of discrete groups. The pair
(Γ, G) has the relative property (T) if and only if the following condition holds:
For any ǫ > 0, there exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H)
is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H
satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then ‖π(h)ξ − ξ‖ < ǫ for h ∈ G.
The pair (Z2⋊SL(2,Z),Z2) is a typical example of group with the relative property
(T). See [Bu] or [Sh] for the proof.
2.5. Weakly mixing actions. An action of a countable discrete group G on a von
Neumann algebra N is said to be ergodic if any G-invariant element of N is a scalar
multiple of 1. The weak mixing property is a stronger notion of ergodicity.
Definition 2.5. Let N be a von Neumann algebra with a faithful normal state φ.
A state preserving action (ρg)g∈G of a countable discrete group G on N is said to be
weakly mixing if for every finite subset {a1, a2, . . . , an} ⊂ N and ǫ > 0, there exists
g ∈ G such that |φ(aiρg(aj))− φ(ai)φ(aj)| < ǫ, i, j = 1, . . . , n.
The following is a basic characterization of the weak mixing property. Between
two von Neumann algebra N and M , N ⊗ M stands for the tensor product von
Neumann algebra.
Proposition 2.6 (Proposition D.2 in [Vaes]). Let a countable discrete group G act
on a finite von Neumann algebra (N, tr) by trace preserving automorphisms (ρg)g∈G.
The following statements are equivalent:
(1) The action (ρg) is weakly mixing.
(2) The only finite-dimensional invariant subspace of N is C1.
(3) For any action (αg) of G on a finite von Neumann algebra (M, τ), we have
(N ⊗ M)ρ⊗α = 1 ⊗ Mα, where (N ⊗ M)ρ⊗α and Mα are the fixed point
subalgebras.
2.6. A remark on group von Neumann algebras. Let Γ be a discrete group
and let µ be a scalar 2-cocycle of a countable group Γ. A group Γ acts on the Hilbert
space ℓ2Γ by the following two ways;
uγ(δg) = µ(γ, g)δγg, ργ(δg) = µ(g, γ
−1)δgγ−1 , γ, g ∈ Γ.
These two representations commute with each other. The von Neumann algebra
Lµ(Γ) generated by the image of u is called the group von Neumann algebra of
Γ twisted by µ. The normal state 〈·δe, δe〉 is a trace on Lµ(Γ). The vector δe is
separating for Lµ(Γ). For any element a ∈ Lµ(Γ), we define the square summable
function a(·) on Γ by aδe =
a(g)δg. The function a(·) is called the Fourier
coefficient of a. We write a =
g∈Γ a(g)ug and call this the Fourier expansion of
a. The Fourier expansion of a∗ is given by a∗ =
g∈Γ µ(g, g
−1)a(g−1)ug, since the
Fourier coefficient a∗(g) = 〈a∗δe, δg〉 is described as
〈δe, aρg−1δe〉 =
ρ∗g−1δe,
a(g)δg
= µ(g, g−1)a(g−1).
CLASSIFICATION AND CENTRALIZERS 5
Here we used the equation ρ∗
= µ(g, g−1)ρg, which is verified by direct computa-
tion. For two elements a, b, the Fourier coefficient of ab is given by
ab(γ) = 〈bδe, ργ−1a∗δe〉 =
a∗(g)µ(g, γ)b(gγ) =
µ(g−1, gγ)a(g−1)b(gγ).
This equation allows us to calculate the Fourier coefficient algebraically, that is,
µ(g−1, gγ)a(g−1)b(gγ)
a(g)b(h)
For a subgroup Λ ⊂ Γ, the subalgebra {uλ | λ ∈ Λ}′′ ⊂ Lµ(Γ) is isomorphic to
Lµ(Λ). We sometimes identify them. An element a ∈ Lµ(Γ) is in the subalgebra
Lµ(Λ) if and only if the Fourier expansion a(·) : Γ → C is supported on Λ, since
the trace 〈·δe, δe〉 preserving conditional expectation E from Lµ(Γ) onto Lµ(Λ) is
described as E(a) =
λ∈Λ a(λ)uλ.
3. Definition of twisted Bernoulli shift actions
In this section, we introduce twisted Bernoulli shift actions of Z2 ⋊ SL(2,Z) on
finite von Neumann algebras. The action is defined for a triplet i = (H, µ, χ), where
H 6= {0} is an abelian countable discrete group, µ is a normalized scalar 2-cocycle
of H and χ is a character of H . The finite von Neumann algebra, on which the
group Z2 ⋊ SL(2,Z) acts, is defined by the pair (H, µ).
We introduce a group structure on the set Γ0 = Ĥ × Z2 × SL(2,Z) as
(c1, k, γ1)(c2, l, γ2) =
c1c2χ
det(k,γ1·l), k + γ1 · l, γ1γ2
for any c1, c2 ∈ Ĥ, k, l ∈ Z2, γ1, γ2 ∈ SL(2,Z). The associativity is verified by Lemma
2.1. It turns out that the subsets Ĥ = Ĥ × {0} × {e} and G0 = Ĥ × Z2 × {e} are
subgroups in Γ0. It is easy to see that G0 is a normal subgroup of Γ0 and that Ĥ is a
normal subgroup of G0 and Γ0. We get a normal inclusion of groups G0/Ĥ ⊂ Γ0/Ĥ
and this is isomorphic to Z2 ⊂ Z2 ⋊ SL(2,Z).
Before stating the definition of the twisted Bernoulli shift action, we define a Γ0-
action ρ on the von Neumann algebra Lµ̃(⊕Z2H). We denote by u(λ) ∈ Lµ̃(⊕Z2H)
the unitary corresponding to λ ∈ ⊕Z2H . We define a faithful normal trace tr of
Lµ̃(⊕Z2H) in the usual way. For c ∈ Ĥ, k ∈ Z2, γ ∈ SL(2,Z), let ρ(c), ρ(k), ρ(γ) be
the linear transformations on Cµ̃(⊕Z2H) given by,
ρ(c)(u(λ)) =
c(λ(l))
u(λ),
ρ(k)(u(λ)) =
χ(λ(m))det(k,m)
u(k · λ),
ρ(γ)(u(λ)) = u(γ · λ), λ ∈ Λ(H),
These maps are compatible with the multiplication rule and the ∗-operation of
Cµ̃(⊕Z2H) . Since these maps preserve the trace, they extend to ∗-automorphisms
6 HIROKI SAKO
on Lµ̃(⊕Z2H). It is immediate to see that ρ(c) commutes with ρ(k) and ρ(γ). For
k, l ∈ Z2, we have the following relation:
ρ(k) ◦ ρ(l)(u(λ)) =
χ(λ(m))det(l,m)ρ(k)(u(l · λ))
χ(λ(m))det(l,m)
χ((l · λ)(m))det(k,m)u(k · (l · λ))
χ(λ(m))det(l,m)χ(λ(m))det(k,m+l)u((k + l) · λ).
By det(l, m) + det(k,m+ l) = det(k, l) + det(k + l, m), this equals to
χ(λ(m))
)det(k,l) ∏
χ(λ(m))det(k+l,m)u((k + l) · λ)
= ρ(χdet(k,l)) ◦ ρ(k + l)(u(λ)).
Since det is SL(2,Z)-invariant (Lemma 2.1), for k ∈ Z2, γ ∈ SL(2,Z), we get
ρ(γ · k) ◦ ρ(γ)(u(λ)) = ρ(γ · k)(u(γ · λ))
χ((γ · λ)(l))det(γ·k,l)u((γ · k) · (γ · λ))
χ(λ(l))det(γ·k,γ·l)u(γ · (k · λ))
χ(λ(l))det(k,l)ρ(γ)(u(k · λ))
= ρ(γ) ◦ ρ(k)(u(λ)), λ ∈ Λ(H).
By using the above two equations, ρ satisfies the following formula:
(ρ(c1) ◦ ρ(k) ◦ ρ(γ1)) ◦ (ρ(c2) ◦ ρ(l) ◦ ρ(γ2))
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1) ◦ ρ(l) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1 · l) ◦ ρ(γ1) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(χdet(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2)
= ρ(c1c2χ
det(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2).
With ρ(c, k, γ) = ρ(c) ◦ ρ(k) ◦ ρ(γ), ρ gives a Γ0-action on Lµ̃(⊕Z2H).
We define the finite von Neumann algebra N(H, µ) as the group von Neumann
algebra Lµ̃(Λ(H)). By using Fourier coefficients, we can prove that N(H, µ) is the
fixed point algebra under the Ĥ-action ρ(Ĥ, 0, e) on Lµ̃(⊕Z2H). We get a Z2 ⋊
SL(2,Z)-action on N(H, µ) by
β(k, γ)(x) = ρ(1, k, γ)(x), k ∈ Z2, γ ∈ SL(2,Z), x ∈ N(H, µ).
This is the definition of the twisted Bernoulli shift action β = β(H, µ, χ) on N(H, µ).
We obtained the actions β(H, µ, χ) not only by twisting generalized Bernoulli shift
actions but also restricting to subalgebras N(H, µ) ⊂ Lµ̂(⊕Z2H) =
Lµ(H).
This restriction allows us to classify the actions up to conjugacy in the next section.
CLASSIFICATION AND CENTRALIZERS 7
In order to give a variety of the actions, we twisted the shift actions by the character
χ of the abelian group H .
Remark 3.1. The action β|Z2 = β(H, µ, χ)|Z2 has the weak mixing property. In
definition 2.5, we may assume that the Fourier coefficients of ai (i = 1, 2, · · · , n) are
finitely supported, by approximating in the L2-norm. Then for appropriate k ∈ Z2,
we get tr(aiβ(k)(aj)) = tr(ai)tr(aj), i, j = 1, 2, · · · , n.
4. Classification up to conjugacy
In this section, we classify the twisted Bernoulli shift actions {β(H, µ, χ)} up to
conjugacy (Theorem 4.1). We prove that an isomorphism which gives conjugacy
between two twisted Bernoulli shift actions is of a very special form. In fact it
comes from an isomorphism in the level of base groups H . We also determine the
centralizer of the Z2 ⋊ SL(2,Z)-action β(H, µ, χ) on N(H, µ) (Theorem 4.4).
We fix some notations for the proofs. We define 0, e1, e2 ∈ Z2 as
, e1 =
, e2 =
Let ξ be the element of Z2 ⋊ SL(2,Z) satisfying
ξ · 0 = e1, ξ · e1 = e2, ξ · e2 = 0.
The elements ξ and ξ2 are explicitly described as
−1 −1
−1 −1
The order of ξ is 3. Let η, δ ∈ SL(2,Z) be given by η =
, δ =
Let D be the subset of all elements of Z2 fixed under the action of δ, that is,
n ∈ Z
Then we get
ξ ·D =
n ∈ Z
, ξ2 ·D =
n ∈ Z
We define the subgroup ΛD(H) of Λ(H) by
ΛD(H) = {λ ∈ Λ(H) | λ : Z2 → H is supported on D}.
Let (Ha, µa, χa) and (Hb, µb, χb) be triplets of countable abelian groups, their
normalized 2-cocycles and characters. For h ∈ Ha, we define λh ∈ ΛD(Ha) as
λh(k) =
h (k = e1),
−h (k = 0),
0 (k 6= e1, 0).
8 HIROKI SAKO
For g ∈ Hb, we define σg ∈ ΛD(Hb) as
σg(k) =
g (k = e1),
−g (k = 0),
0 (k 6= e1, 0).
We denote by v(σ) ∈ N(Hb, µb) the unitary corresponding to σ ∈ Λ(Hb).
Theorem 4.1. If π : N(Ha, µa) → N(Hb, µb) is a ∗-isomorphism giving conjugacy
between βa = β(Ha, µa, χa) and βb = β(Hb, µb, χb), then there exists a group isomor-
phism φ = φπ : Ha → Hb satisfying
(1) π(u(λ)) = v(φ ◦ λ) mod T for λ ∈ Λ(Ha),
(2) the 2-cocycles µa(·, ·) and µb(φ(·), φ(·)) of Ha are cohomologous,
(3) χ2a = (χb ◦ φ)2.
Conversely, given a group isomorphism φ : Ha → Hb satisfying (2) and (3), there
exists a ∗-isomorphism π = πφ : N(Ha, µa) → N(Hb, µb) which satisfies condition
(1) and gives conjugacy between βa, βb.
We note that by Proposition 2.2 condition (2) for φ is equivalent to
(2)′ µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), g, h ∈ Ha.
Proof for the first half of Theorem 4.1.
Suppose that there exists a (not necessarily trace preserving) ∗-isomorphism π from
N(Ha, µa) onto N(Hb, µb) such that π ◦ βa(γ) = βb(γ) ◦ π, γ ∈ Z2 ⋊ SL(2,Z).
We prove that for every h ∈ Ha there exists φ(h) ∈ Hb satisfying π(u(λh)) =
v(σφ(h)) mod T. Let Uh denote the unitary in N(Hb, µb)
Uh = π
µa(h,−h) u(λh)
, h ∈ Ha.
We identify N(H, µ) with the subalgebra of the infinite tensor product
Lµ(H),
which is canonically isomorphic to Lµ̃ (⊕Z2H). The preimage π−1(Uh) can be written
as u∗h ⊗ uh. Here uh is the unitary corresponding to h ∈ Ha and placed on 1 ∈ Z2
and the unitary u∗h is placed on 0 ∈ Z2. We describe Uh as the Fourier expansion
σ∈Λ(Hb)
c(σ)v(σ). Since e1 and 0 are fixed under the action of δ, one has
βb(δ)
n(Uh) = π ◦ βa(δ)n(π−1(Uh)) = Uh.
It follows that the Fourier expansion Uh =
σ∈Λ(Hb)
c(σ)v(σ) must satisfy that
c(σ) = c(δ−n · σ) for every σ ∈ Λ(Hb) and n ∈ Z. For σ ∈ Λ(Hb) \ ΛD(Hb), the
orbit of σ under the action of δ−1 is an infinite set, since the support supp(σ) ⊂ Z2
is not included in D. It turns out that c(σ) = 0 for all σ ∈ Λ(Hb) \ ΛD(Hb) due to
Σ|c(σ)|2 = 1 < +∞, so that Uh =
σ∈ΛD(Hb)
c(σ)v(σ).
The unitary χa(h)U
h is also fixed under the action of δ and can be written as
χa(h)U
h = π
χa(h)µa(h,−h)u(−λh)
χa(h)µa(h,−h)u(ξ · λh)
µa(h,−h)u(ξ2 · λh)
= βb(ξ)(Uh) βb(ξ
2)(Uh).
CLASSIFICATION AND CENTRALIZERS 9
Letting ne1 = (n, 0)
T ∈ Z2, we get
βb(ξ)(Uh) = βb(ξ)
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ · σ)
χb(σ(ne1))
2)(Uh) = βb(ξ
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ2 · σ).
Since Fourier expansion admits algebraical calculation as in subsection 2.6, the ex-
pansion of χa(h)U
χa(h)U
h = βb(ξ)(Uh) βb(ξ
2)(Uh)
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) v(ξ · σ1) v(ξ2 · σ2)
χb(σ1(ne1))
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) µ̃b(ξ · σ1, ξ2 · σ2)
χb(σ1(ne1))
n v(ξ · σ1 + ξ2 · σ2).
The map ΛD(Hb)×ΛD(Hb) ∋ (σ1, σ2) 7→ ξ ·σ1+ξ2·σ2 ∈ Λ(Hb) is injective. Indeed, σ1
is uniquely determined by ξ ·σ1+ξ2·σ2, since σ1(k) = (ξ ·σ1+ξ2·σ2)(ξ ·k), k ∈ D\{e1}
and σ1(e1) = −
k∈D\{e1}
σ1(k). Here we used the condition
σ1(k) = 0. The
element σ2 is also determined by ξ · σ1 + ξ2 · σ2. Thus the index (σ1, σ2) uniquely
determines ξ · σ1 + ξ2 · σ2.
We take arbitrary elements σ1, σ2 ∈ ΛD(Hb) and suppose that c(σ1) 6= 0, c(σ2) 6= 0.
Since the unitary χa(h)U
h is invariant under the action of δ and the coefficient of
ξ ·σ1+ξ2 ·σ2 is not zero, ξ ·σ1+ξ2 ·σ2 is supported on D. It follows that the elements
σ1 and σ2 can be written as σ1 = σφ(h) = σ2, by some φ(h) ∈ Hb. Indeed, since the
subsets D \ {0, e1}, ξD \ {e1, e2} and ξ2D \ {e2, 0} are mutually disjoint, the element
ξ · σ1 must be supported on {e1, e2} and the element ξ2 · σ2 must be supported on
{e2, 0}. By the assumption
k∈Z2 σi(k) = 0 (i = 1, 2), σi can be written as σφ(hi).
Then using the fact that (ξ · σ1 + ξ2 · σ2)(e2) = σ1(e1) + σ2(0) = 0, we get that
σ1 = σφ(h) = σ2 for some h ∈ Hb. This means that there exists only one σ ∈ Λ(Hb)
such that c(σ) 6= 0 and that it is of the form σ = σφ(h). Then the unitary Uh satisfies
Uh = π(u(λh)) = v(σφ(h)) mod T.
We claim that the map φ = φπ : Ha → Hb is a group isomorphism. For all
h1, h2 ∈ Ha, we get
π(u(λh1+h2)) = π(u(λh1)) π(u(λh2)) = v(σφ(h1)) v(σφ(h2))
= v(σφ(h1) + σφ(h2)) = v(σφ(h1)+φ(h2)) mod T.
On the other hand, we get π(u(λh1+h2)) = v(σφ(h1+h2)) mod T. Since {v(σ)} are
linearly independent, we get σφ(h1+h2) = σφ(h1)+φ(h2), and hence
φ(h1 + h2) = φ(h1) + φ(h2).
This means that the map φ is a group homomorphism. The bijectivity of the ∗-
isomorphism π leads to that of the group homomorphism φ = φπ. Since {γ ·λh | γ ∈
Z2⋊SL(2,Z), h ∈ Ha} ⊂ Λ(Ha) generates Λ(Ha), we get π(u(λ)) = v(φ◦λ) mod T
for λ ∈ Λ(Ha).
10 HIROKI SAKO
We prove that the group isomorphism φ = φπ satisfies conditions (2) and (3) in
the theorem. For all h ∈ Ha, there exists c(h) ∈ T satisfying
Uh = π
µa(h,−h) u(λh)
= c(h)µb(φ(h),−φ(h)) v(σφ(h)).
Since (e1, η) ∈ Z2 ⋊ SL(2,Z) acts on Z2 as (e1, η) · e1 = 0, (e1, η) · 0 = e1, we get
Uh βb(e1, η)(Uh) = π
µa(h,−h) u(λh)µa(h,−h) u(−λh)
= µa(h,−h)
µ̃a(λh,−λh) = 1.
The following equation also holds:
Uh βb(e1, η)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)µb(φ(h),−φ(h)) v(σ−φ(h))
= c(h)2 µb(φ(h),−φ(h))
µ̃b(σφ(h),−σφ(h)) = c(h)2.
Thus we have c(h) ∈ {1,−1} for h ∈ Ha.
Since ξ · e1 = e2, ξ · e2 = 0 and ξ · 0 = e1, we have
Uh βb(ξ)(Uh) βb(ξ
2)(Uh)
µa(h,−h) u(λh)
χa(h)µa(h,−h) u(ξ · λh)
µa(h,−h) u(ξ2 · λh)
= χa(h).
On the other hand, we have the following:
Uh βb(ξ)(Uh) βb(ξ
2)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h))
c(h)χb(φ(h))µb(φ(h),−φ(h)) v(ξ · σφ(h))
c(h)µb(φ(h),−φ(h)) v(ξ2 · σφ(h))
= c(h)3 χb(φ(h)) = c(h)χb(φ(h)).
It follows that
c(h) = χb(φ(h))χa(h)(Eq2)
and χb(φ(h))
2 = χa(h)
2, for all h ∈ Ha.
We recall that the algebra Lµ̃a(⊕Z2Ha) is canonically identified wit the infinite
tensor product
Lµa(Ha). The unitary π
−1(Uh) ∈ N(Ha, µa) ⊂
Lµa(Ha)
can be written as 1⊗ u∗h ⊗ uh, where 1 is placed on −e1 ∈ Z2, u∗h is placed on 0 and
uh is placed on e1. Since η ∈ Z2 ⋊ SL(2,Z) acts on Z2 as η · e1 = −e1, η · 0 = 0,
the unitary π−1(βb(η)(Ug)) can be written as ug ⊗ u∗g ⊗ 1. We have the following
equation:
Ug βb(η)(Uh)U
g βb(η)(Uh)
= π((1⊗ u∗g ⊗ ug)(uh ⊗ u∗h ⊗ 1)(1⊗ u∗g ⊗ ug)∗(uh ⊗ u∗h ⊗ 1)∗) = µ∗aµa(g, h).
The unitary Uh can be written as c(h)(1⊗ v∗φ(h) ⊗ vφ(h)) ∈ N(Hb, µb) ⊂
Lµb(Hb).
Here we write vφ(h) for the unitary in Lµb(Hb) corresponding to φ(h). The unitary
βb(η)(Ug) can be written as c(g)(vφ(g) ⊗ v∗φ(g) ⊗ 1). Then we get
Ug βb(η)(Uh)U
g βb(η)(U
= (1⊗ v∗φ(g) ⊗ vφ(g))(vφ(h) ⊗ v∗φ(h) ⊗ 1)(1⊗ v∗φ(g) ⊗ vφ(g))∗(vφ(h) ⊗ v∗φ(h) ⊗ 1)∗
= µ∗bµb(φ(g), φ(h)).
CLASSIFICATION AND CENTRALIZERS 11
Thus we get µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), for all g, h ∈ Ha. We proved that the
group isomorphism φ = φπ satisfies conditions (1), (2) and (3). �
From a group homomorphism which satisfies conditions (2) and (3), we construct
a ∗-homomorphism from N(Ha, µa) to N(Hb, µb) with condition (1). In the con-
struction, the function µ̂ on Λ(H) given below is useful. We fix an index for Z2 as
Z2 = {k0, k1, k2, · · · } throughout the rest of this section. For a scalar 2-cocycle µ of
H , we define the function µ̂ by
µ̂(λ) =
λ(ki), λ(kj)
, λ ∈ Λ(H),
where λ is supported on {k0, k1, k2, · · · , kn}. This definition depends on the choice
of an order on Z2. Since
i λ(ki) = 0, the function µ̂ is also given by the following
relation in Cµ(H):
µ̂(λ)1 = uλ(k0)uλ(k1)uλ(k2) · · ·uλ(kn), λ ∈ Λ(H).
If µ is a coboundary, then the definition of µ̂ does not depend on the order on Z2,
since Cµ(H) is commutative.
Lemma 4.2. Let µ0 be another normalized scalar 2-cocycle for H. Let µ̃0 be the
scalar 2-cocycle on Λ(H)×Λ(H) given in the same way as equation (Eq1) in subsec-
tion 2.2 and let µ̂0 be the function on Λ(H) constructed from µ0 in the above manner.
If the scalar 2-cocycles µ and µ0 are cohomologous, then for all λ1, λ2 ∈ Λ(H), we
have the equation
µ̃(λ1, λ2) µ̂(λ1) µ̂(λ2) µ̂(λ1 + λ2) = µ̃0(λ1, λ2) µ̂0(λ1) µ̂0(λ2) µ̂0(λ1 + λ2).
Proof. We denote by {ν(g, h)} the scalar 2-cocycle {µ0(g, h)µ(g, h)} of H . Since ν
is a 2-coboundary, there exists {c(g)}g∈H ⊂ T satisfying ν(g, h) = b(g)b(h)b(g + h).
Then the map ν̂ becomes ν̂(λ) =
b(λ(ki)). Since
ν̂(λ1) ν̂(λ2) =
b(λ1(ki)) b(λ2(ki)),
ν̃(λ1, λ2) =
b(λ1(ki)) b(λ2(ki)) b(λ1(ki) + λ2(ki)),
ν̂(λ1 + λ2) =
b(λ1(ki) + λ2(ki)),
we get ν̂(λ1) ν̂(λ2) = ν̃(λ1, λ2) ν̂(λ1 + λ2). By the definitions of µ̃, µ̃0, µ̂ and µ̂0, the
maps ν̂ and ν̃ are given by
ν̂(λ) = µ̂(λ) µ̂0(λ), ν̃(λ1, λ2) = µ̃(λ1, λ2) µ̃0(λ1, λ2),
Thus the desired equality immediately follows. �
Proof for the second half of Theorem 4.1.
Suppose that there exists a group isomorphism φ satisfying conditions (2) and (3)
in the theorem. We prove that there exists a ∗-isomorphism π = πφ from N(Ha, µa)
onto N(Hb, µb) preserving the Z
2 ⋊ SL(2,Z)-actions with condition (1).
We define a group homomorphism cφ from Ha to {1,−1} ⊂ T by
cφ(h) = χb(φ(h))χa(h), h ∈ Ha.
12 HIROKI SAKO
Let c̃φ be the group homomorphism from Λ(Ha) to {1,−1} ⊂ T given by
c̃φ(λ) =
cφ(λ(k))
gcd(k) =
χa(λ(k))
gcd(k)χb(φ(λ(k)))
gcd(k)
, λ ∈ Λ(Ha).
We define a linear map π from the group algebra Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) by
µ̂a(λ) u(λ)
= c̃φ(λ) µ̂b(φ ◦ λ) v(φ ◦ λ), λ ∈ Λ(Ha).
By direct computations, for all λ1, λ2 ∈ Λ(Ha), we get
µ̂a(λ1) u(λ1)
µ̂a(λ2)u(λ2)
= c̃φ(λ1) c̃φ(λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) v(φ ◦ λ1) v(φ ◦ λ2)
= c̃φ(λ1 + λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) v(φ ◦ (λ1 + λ2)).
On the other hand, we have the following equation:
µ̂a(λ1) µ̂a(λ2)u(λ1) u(λ2)
µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) u(λ1 + λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) π
µ̂a(λ1 + λ2) u(λ1 + λ2)
= c̃φ(λ1 + λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2)
µ̂b(φ ◦ (λ1 + λ2)) v(φ ◦ (λ1 + λ2)).
By Lemma 4.2 and condition (2) for the group isomorphism φ in the theorem, we
have that
µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1) + φ ◦ (λ2)).
Therefore we get π(u(λ1)) π(u(λ2)) = π(u(λ1) u(λ2)). The linear map π also pre-
serves the ∗-operation. As a consequence, π is a ∗-isomorphism from Cµ̃a(Λ(Ha))
onto Cµ̃b(Λ(Hb)) and this preserves the trace. The map π = πφ is extended to a
normal ∗-isomorphism from N(Ha, µa) onto N(Hb, µb).
We next prove that this π preserves the Z2 ⋊ SL(2,Z)-actions. The group ho-
momorphism c̃φ from Λ(Ha) to {1,−1} is invariant under the action of SL(2,Z),
by Lemma 2.1 (1). The scalar 2-cocycle ν(g, h) = µa(g, h)µb(φ(h), φ(g)) satisfies
ν(g, h) = ν(h, g) by condition (2), so the function ν̂(·) = µ̂a(·)µ̂b(φ ◦ ·) on Λ(Ha)
does not depend on the order on Z2 chosen before. Since
π ◦ βa(γ)(u(λ)) = π(u(γ · λ))
= c̃φ(γ · λ)µ̂a(γ · λ)µ̂b(φ ◦ (γ · λ))v(φ ◦ (γ · λ))
= c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(γ · (φ ◦ λ))
= βb(γ)
c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(φ ◦ λ)
= βb(γ) ◦ π(u(λ)), γ ∈ SL(2,Z), λ ∈ Λ(Ha),
it turns out that the ∗-isomorphism π preserves the SL(2,Z)-action.
CLASSIFICATION AND CENTRALIZERS 13
For all λ ∈ Λ(Ha) and k ∈ Z2, we have
π ◦ βa(k)(u(λ)) = π
χa(λ(l))
det(k,l)u(k · λ)
cφ((k · λ)(l))gcd(l)
χa(λ(l))
det(k,l)µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)).
Since cφ(h)
det(k,l)cφ(h)
gcd(k+l) = cφ(h)
gcd(k)cφ(h)
gcd(l), by Lemma 2.1 (2), the unitary
π ◦ βa(k)(u(λ)) equals to
cφ(λ(l))
gcd(k+l)
cφ(λ(l))
det(k,l)χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ))
cφ(λ(l))
gcd(k)
cφ(λ(l))
gcd(l)
χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(k · (φ ◦ λ))
= c̃φ(λ) µ̂a(k · λ) µ̂b(k · (φ ◦ λ))βb(k)(v(φ · λ))
= βb(k) ◦ π(u(λ)).
This means that the ∗-isomorphism π preserves the Z2-actions.
We get the ∗-isomorphism π = πφ fromN(Ha, µa) ontoN(Hb, µb) giving conjugacy
between βa and βb. �
Remark 4.3. The proof of the first half of Theorem 4.1 shows that any isomorphism
π giving conjugacy between βa and βb is of the form πφ. This means that an
isomorphism which gives conjugacy between two twisted Bernoulli shift actions must
be trace preserving.
This proof shows that an isomorphism giving conjugacy between the two actions
β(Ha, µa, χa), β(Hb, µb, χb) is of a very special form derived from a group isomor-
phism between Ha and Hb. Taking notice of this fact, we can describe the centralizer
of a twisted Bernoulli shift action. We define two topological groups before we state
Theorem 4.4.
Let β be a trace preserving action of some group Γ on a separable finite von
Neumann algebra (N, tr). We denote by Aut(N, β) the group of all automorphisms
which commute with the action β, that is,
{α ∈ Aut(N) | β(γ) ◦ α = α ◦ β(γ), γ ∈ Γ}.
We regard the group Aut(N, β) as a topological group equipped with the pointwise-
strong topology. When β is a twisted Bernoulli shift action on N , an automorphism
α commuting with β is necessarily trace preserving by Remark 4.3. We consider
that Aut(N, β) is equipped with the pointwise-2-norm topology.
Let Aut(H, µ, χ) be the group of all automorphisms of an abelian group H which
preserve its 2-cocycle µ and character χ, that is,
{φ ∈ Aut(H) | µ(g, h) = µ(φ(g), φ(h)), χ(g) = χ(φ(g)), g, h ∈ H}.
We define the topology of Aut(H, µ, χ) by pointwise convergence.
14 HIROKI SAKO
Theorem 4.4. For π ∈ Aut(N(H, µ), β(H, µ, χ)), there exists a unique element
φ = φπ ∈ Aut(H, µ∗µ, χ2) satisfying π(u(λ)) = u(φ ◦ λ) mod T for λ ∈ Λ(H). The
map π 7→ φπ gives an isomorphism between two topological groups
Aut(N(H, µ), β(H, µ, χ)) ∼= Aut(H, µ∗µ, χ2).
Proof. We use the notations in the proof of the previous theorem letting Ha = Hb =
H , µa = µb = µ and χa = χb = χ. Denote N = N(H, µ) and β = β(H, µ, χ). We
have already have shown the first claim. Let
Aut(H, µ∗µ, χ2) ∋ φ 7→ πφ ∈ Aut(N, β),
be the map given as in the proof of Theorem 4.1, that is,
µ̂(λ)u(λ)
= c̃φ(λ) µ̂(φ ◦ λ)u(φ ◦ λ), λ ∈ Λ(H),
where
c̃φ(λ) =
χ(λ(k))gcd(k) χ(φ ◦ λ(k))
gcd(k)
It is easy to prove that φπφ = φ by the definition. Thus the map π 7→ φπ is
surjective. This map is also injective. Let φ be an element of Aut(H, µ∗µ, χ2).
Suppose that π is an arbitrary element of Aut(N, β) satisfying φ = φπ. The set
{β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊ SL(2,Z)} generates N , so we have only to prove
the uniqueness of c(h) ∈ T satisfying
µ(h,−h) u(λh)
= c(h)µ(φ(h),−φ(h))u(λφ(h)),
for all h ∈ H . In the proof of the first half of the previous theorem (equation
(Eq2)), we have already shown that c(h) = χ(h)χ(φ(h)). Thus the ∗-isomorphism
π is uniquely determined and the map π 7→ φπ is injective.
We prove the two maps φ 7→ πφ and π 7→ φπ are continuous. Let (φi) be a net in
Aut(H, µ∗µ, χ2) converging to φ. For all h ∈ H , we have
χ(h)µ(h,−h) u(λh)
= χ(φi(h))µ(φi(h),−φi(h))u(λφi(h)).
The right side of the equation converges to
χ(φ(h))µ(φ(h),−φ(h))u(λφ(h)) = πφ
χ(h)µ(h,−h) u(λh)
This proves that πφi converges to πφ in pointwise 2-norm topology on the generating
set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊SL(2,Z)} of N . Thus πφi converges to πφ on N .
Conversely, let (πi) be a net in Aut(N, β) converging to π. For all h ∈ H ,
we get πi(u(λh)) = u(λφπi(h)) mod T. The left side of the equation converges to
π(u(λh)) = u(λφπ(h)). If φπi(h) 6= φπ(h), then the distance between Tu(λφπ(h)) and
Tu(λφπi (h)) is
2 in the 2-norm. Thus φπi(h) = φπ(h) for large enough i. This
means that (φπi) converges to φπ.
As a consequence, the two maps φ 7→ πφ and π 7→ φπ are continuous group
homomorphisms and inverse maps of each other. �
CLASSIFICATION AND CENTRALIZERS 15
5. Examples
5.1. Twisted Bernoulli shift actions on L∞(X). In this subsection, we consider
the case of µ = 1 and H 6= {0}. Then the algebra N(H, 1) is abelian and has a
faithful normal state, so it is isomorphic to L∞(X), whereX is a standard probability
space. The measure of X is determined by the trace on N(H, 1). Furthermore, X is
non-atomic, since N(H, 1) is infinite dimensional and the action β(H, 1, χ) is ergodic.
As corollaries of Theorems 4.1 and 4.4, we get trace preserving Z2⋊SL(2,Z)-actions
on L∞(X) whose centralizers are isomorphic to some prescribed groups.
Remark 5.1. The Z2 ⋊ SL(2,Z)-action on X defined by β = β(H, 1, χ) is free.
An automorphism α ∈ Aut(L∞(X), β) is free or the identity map for any twisted
Bernoulli shift action β on L∞(X). This is proved as follows. We identify β(γ)
(γ ∈ Z2 ⋊ SL(2,Z)) and α with measure preserving Borel isomorphisms on X here.
Suppose that there exists a non-null Borel subset Y ⊂ X whose elements are fixed
under α. All elements in Ỹ = ∪{β(γ)(Y ) | γ ∈ Z2⋊SL(2,Z)} are fixed under α. By
the ergodicity of β, the measure of Ỹ is 1. Then α is the identity map of L∞(X).
Corollary 5.2. For any abelian countable discrete group H 6= {0}, there exists a
trace preserving essentially free ergodic action β of Z2⋊SL(2,Z) on L∞(X) satisfying
Aut(L∞(X), β) ∼= Aut(H).
Proof. When we define β = β(H, 1, 1), we have the above relation by Theorem
4.4. �
In the next corollary we use the effect of twisting by a character χ.
Corollary 5.3. For every abelian countable discrete group H 6= {0}, there exist
continuously many trace preserving essentially free ergodic actions {βc} of Z2 ⋊
SL(2,Z) on L∞(X) which are mutually non-conjugate and satisfy
Aut(L∞(X), βc) ∼= H ⋊Aut(H).
Here the topology of H ⋊ Aut(H) is the product of the discrete topology on H and
the pointwise convergence topology on Aut(H).
Proof. Let c ∈ {eiπt | t ∈ (0, 1/2) \Q}. We put βc = β(H ⊕ Z, 1, 1× χc), where the
character χc of Z is defined as χc(n) = c
n. By Theorem 4.4, we get
Aut(L∞(X), βc) ∼= Aut(H ⊕ Z, 1, 1× χ2c).
Since the character χ2c is injective, a group automorphism α ∈ Aut(H⊕Z, 1, 1×χ2c)
preserves the second entry. For all α ∈ Aut(H⊕Z, 1, 1×χ2c), there exist φα ∈ Aut(H)
and hα ∈ H satisfying
α(h, n) = (φα(h) + nhα, n), (h, n) ∈ H ⊕ Z.
The map Aut(H ⊕ Z, 1, 1× χ2c) ∋ α 7→ (hα, φα) ∈ H ⋊ Aut(H) is a homeomorphic
group isomorphism.
If c1, c2 ∈ {eiπt | t ∈ (0, 1/2) \ Q} and c1 6= c2, then there exists no isomorphism
from H ⊕ Z to H ⊕ Z whose pull back of the character 1 × χ2c2 is equal to 1 × χ
The two actions βc1 and βc2 are not conjugate by Theorem 4.1. �
16 HIROKI SAKO
Corollary 5.4. There exist continuously many trace preserving essentially free er-
godic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate
and have the trivial centralizer Aut(L∞(X), βc) = {idL∞X}.
Proof. Let {χc | c = eiπt, t ∈ (0, 1/2)} be characters of Z such that χc(m) = cm.
Since χ2c(1) is in the upper half plane, the identity map is the only automorphism
of Z preserving χ2c . By Theorem 4.4, we get Aut(β(Z, 1, χc)) = {id}.
If β(Z, 1, χc1), β(Z, 1, χc2) are conjugate, then there exists a group isomorphism
on Z whose pull back of χ2c2 is χ
by Theorem 4.1. This means c1 = c2. Thus the
actions {β(Z, 1, χc)} are mutually non-conjugate. �
5.2. Twisted Bernoulli shift actions on the AFD factor of type II1. Firstly,
we find a condition that the finite von Neumann algebra N(H, µ) is the AFD factor
of type II1.
Lemma 5.5. For an abelian countable discrete group H 6= {0} and its normalized
scalar 2-cocycle µ, the following statements are equivalent:
(1) The algebra N(H, µ) is the AFD factor of type II1.
(2) The group von Neumann algebra Lµ(H) twisted by the scalar 2-cocycle µ is
a factor (of type II1 or In).
(3) For all g ∈ H \ {0}, there exists h ∈ H such that µ(g, h) 6= µ(h, g).
Proof. The amenability of the group Λ(H) leads the injectivity for N(H, µ). The
injectivity for N(H, µ) implies that N(H, µ) is approximately finite dimensional
([Co]). We have only to show the equivalence of conditions (2), (3) and
(1)′ The algebra N(H, µ) is a factor.
By using Fourier expansion it is easy to see that condition (2) holds true if and only
if for any g ∈ H \ {0} there exists h ∈ H satisfying uguh 6= uhug. This is equivalent
to condition (3). Similarly, condition (1)′ is equivalent to
(1)′′ For any λ1 ∈ Λ(H) \ {0}, there exists λ2 ∈ Λ(H) satisfying
µ̃(λ2, λ1)µ̃(λ1, λ2) 6= 1.
Suppose condition (3). For any λ1, choose element k, l ∈ Z2 so that k ∈ supp(λ1)
and l /∈ supp(λ1). By condition (3), there exists h ∈ H satisfying µ∗µ(λ1(k), h) 6= 1.
Let λ2 be the element in Λ(H) which takes h at k, −h at l and 0 for the other
places. The element λ2 satisfies µ̃(λ2, λ1)µ̃(λ1, λ2) = µ
∗µ(λ1(k), h) 6= 1. Here we get
condition (1)′′. The implication from (1)′′ to (3) is easily shown. �
Remark 5.6. The twisted Bernoulli shift action β = β(H, µ, χ) is an outer action
of Z2 ⋊ SL(2,Z). Any non-trivial automorphism in Aut(R, β(H, µ, χ)) is also outer.
This is proved by the weak mixing property of the action β(H, µ, χ) as follows. If
α ∈ Aut(R, β(H, µ, χ)) is an inner automorphism Ad(u), then we have
Ad(β(γ)(u))(x) = β(γ)(uβ(γ)−1(x)u∗) = β(γ) ◦ α ◦ β(γ)−1(x)
= α(x) = Ad(u)(x),
for all x ∈ R and γ ∈ Z2 ⋊ SL(2,Z). Since Ad(β(γ)(u)u∗) = id, Cu ⊂ R is an
invariant subspace of the action β. The only subspace invariant under the weakly
mixing action β is C1 (Proposition 2.6), thus we get α = id.
CLASSIFICATION AND CENTRALIZERS 17
Using Theorems 4.1 and 4.4, we give continuously many actions of Z2 ⋊ SL(2,Z)
on R such that there exists no commuting automorphism except for trivial one.
Corollary 5.7. There exist continuously many ergodic outer actions {βc} of Z2 ⋊
SL(2,Z) on the AFD factor R of type II1 which are mutually non-conjugate and
have the trivial centralizer Aut(R, βc) = {idR}.
Proof. We can choose and fix a character χ on Z2 such that χ2 is injective. Let
{µc | c = eiπt, t ∈ (0, 1/2) \Q} be scalar 2-cocycles for Z2 defined by
= cs1t2−t1s2, s1, t1, s2, t2 ∈ Z.
We put βc = β(Z
2, µc, χ). The 2-cocycle µc satisfies condition (3) in Lemma 5.5.
Thus βc defines a Z
2 ⋊ SL(2,Z)-action on R. By Theorem 4.4, we get the following
isomorphism between topological groups:
Aut(R, βc) ∼= Aut(Z2, µ∗cµc, χ2) = Aut(Z2, µc2, χ2).
Since the character χ2 of Z2 is injective, so the group of the right side is {id|Z2}.
This means that the action βc has trivial centralizers.
Finally, we prove that the actions {βc | c = eiπt, t ∈ (0, 1/2) \ Q} are mutually
non-conjugate. Suppose that actions βc1 and βc2 are conjugate. By Theorem 4.1,
there exists a group isomorphism φ of Z2 satisfying
(g, h) = µc2
(φ(g), φ(h)), g, h ∈ Z2.
A group isomorphism of Z2 is given by an element of GL(2,Z). If the automorphism
φ is given by an element of SL(2,Z), we get c22 = c
1. If φ is given by an element
of GL(2,Z) \ SL(2,Z), then we get c22 = −c21. Since both c1 and c2 have the form
eiπt, t ∈ (0, 1/2), we get c1 = c2. �
Any cyclic group of an odd order can be realized as the centralizer of a twisted
Bernoulli shift actions on R.
Corollary 5.8. Let q be an odd natural number ≥ 3 and denote by Hq the abelian
group (Z/qZ)2. We define the 2-cocycle µq and the character χq on Hq as
= exp (2πis1t2/q), χq
= exp (2πis1/q).
Then the algebra N(Hq, µq) is the AFD factor R of type II1 and the centralizer of
the twisted Bernoulli shift action βa = β(Hq, µq, χq) is isomorphic to Z/qZ.
Proof. By Lemma 5.5, it is shown that the algebra N(HQ, µQ) is the AFD factor of
type II1. Using Theorem 4.4, we have only to prove that Aut(Hq, µ
qµq, χ
∼= Z/qZ.
Let φ be in Aut(Hq, µ
qµq, χ
q). The automorphism φ of Hq is given by a 2 × 2
matrix A of Z/qZ. Since φ preserves µ∗qµq, the determinant of A must be 1. Since
q is odd, the value of χ2q determines the first entry of (Z/qZ)
2 and φ preserves χ2q .
The matrix A is of the form(
, tφ ∈ Z/qZ.
The map φ 7→ tφ is an isomorphism. In turn, if the matrix A is of this form, it
defines an element in Aut(Hq, µ
qµq, χ
q). �
18 HIROKI SAKO
Corollary 5.9. For a set Q consisting of odd prime numbers, let βQ be the tensor
product
q∈Q βq of the actions βq on the AFD factor of type II1. The centralizer
of βQ is isomorphic to
q∈Q Z/qZ.
Proof. The action βQ is the twisted Bernoulli shift action β(HQ, µQ, χQ), where HQ
is the abelian group ⊕q∈QHq and the scalar 2-cocycle µq and a character χQ on HQ
are given by
µQ((sq), (tq)) =
µq(sq, tq),
χQ((sq)) =
χq(sq), (sq), (tq) ∈ HQ, sq, tq ∈ Hq.
Using Theorem 4.4, we have only to prove
Aut(HQ, µ
QµQ, χ
Z/qZ.
A group automorphism φ of HQ = ⊕q∈QHq has a form φ((kq)) = (φq(kq)), for some
{φq ∈ Aut(Hq)}. Thus we get
Aut(HQ, µ
QµQ, χ
Aut(Hq, µ
qµq, χ
Together with the previous corollary, we get the conclusion. �
Remark 5.10. If Q1 6= Q2, then the two groups
Z/qZ and
Z/qZ are not
isomorphic. The continuously many outer actions {βQ} are distinguished in view of
conjugacy only by using the centralizers {Aut(R, βQ)}.
6. Malleability and rigidity arguments
In this section, we give malleability and rigidity type arguments invented by S.
Popa, in order to examine weak 1-cocycles for actions. See Popa [Po2], [Po3], [Po4]
and Popa–Sasyk [PoSa] for the references. S. Popa in [Po3] showed that every
1-cocycle for a Connes-Størmer Bernoulli shift by property (T) group (or w-rigid
group like Z2 ⋊ SL(2,Z)) vanishes modulo scalars. As a consequence, two such
actions are cocycle conjugate if and only if they are conjugate. In our case, 1-
cocycles do not vanish modulo scalars but they are still in the situation that cocycle
(outer) conjugacy implies conjugacy. We need the following notion to examine outer
conjugacy of two group actions.
Definition 6.1. Let α be an action of discrete group Γ on a von Neumann algebra
M. A weak 1-cocycle for α is a map w : Γ → U(M) satisfying
wgh = wgαg(wh) mod T, g, h ∈ Γ.
The weak 1-cocycle w is called a weak 1-coboundary if there exists a unitary v ∈
U(M) satisfying wg = vαg(v)∗ mod T. Two weak 1-cocycles w and w′ are said to be
equivalent when w′g = vwgαg(v)
∗ mod T for some v ∈ U(M).
Let N be a finite von Neumann algebra with a faithful normal trace. The following
is directly obtained by combining Lemmas 2.4 and 2.5 in [PoSa], although these
Lemmas were proved for Bernoulli shift actions on standard probability space. The
CLASSIFICATION AND CENTRALIZERS 19
following can be also regarded as a weak 1-cocycle version of Proposition 3.2 in
[Po4].
Proposition 6.2. Let G be a countable discrete group. Let β be a trace preserving
weakly mixing action of G on N . A weak 1-cocycle {wg}g∈G ⊂ N for β is a weak
1-coboundary if only if there exists a non-zero element x̃0 ∈ N ⊗N satisfying
(wg ⊗ 1)(βg ⊗ βg)(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
The following is a weak 1-cocycle version of Proposition 3.6.3◦ in [Po4].
Proposition 6.3. Let Γ be a countable discrete group and G be a normal subgroup
of Γ. The group Γ acts on a finite von Neumann algebra N in a trace-preserving
way by β. Suppose that the restriction of β to G is weakly mixing. Let {wγ}γ∈Γ be a
weak 1-cocycle for β. If w|G is a weak 1-coboundary, then w is a weak 1-coboundary
for the Γ-action.
Proof. Suppose that w|G is a weak 1-coboundary, that is, there exists a unitary
element v in N such that wg = vβg(v
∗) mod T for g ∈ G. It suffices to show
that {w′γ} = {v∗wγβγ(v)} is in T for all γ ∈ Γ. Take arbitrary γ ∈ Γ, g ∈ G. Write
h = γ−1gγ ∈ G. Let πγ be the unitary on L2(N) induced from βγ . Since w′h, w′g ∈ T,
we get
w′γπgw
= (w′γπγ)(w
hπh)(w
∗ = w′γhγ−1πγhγ−1 = πg mod T,
By applying these operators to 1̂ ∈ N̂ ⊂ L2(N), it follows that w′γβg(w′γ
) ∈ T.
Since the G-action is weakly mixing, we have w′γ ∈ T. �
By using the above propositions, we will “untwist” some weak 1-cocycles later.
We require some ergodicity assumption on the weak 1-cocycles.
Definition 6.4. Let Γ be a discrete group and G be a subgroup of Γ. Suppose that
its restriction to G is ergodic. Let β be a trace preserving action of Γ on N . A weak
1-cocycle w = {wg}g∈Γ for β is said to be ergodic on G, if the action βw of G is still
ergodic, where βw is defined by βwg = Adwg ◦ βg, g ∈ G.
Let β be a Γ-action onN . Suppose that the diagonal action β⊗β on (N⊗N, tr⊗tr)
has an extension β̃ on a finite von Neumann algebra (Ñ , τ). The algebra Ñ is not
necessarily identical with N⊗N . When the action β̃ is ergodic on a normal subgroup
G ⊂ Γ, we get the following:
Proposition 6.5. Let {wγ}γ∈Γ ⊂ N be a weak 1-cocycle for β. Let α be a trace
preserving continuous action of R on Ñ satisfying the following properties:
• α1(x⊗ 1) = 1⊗ x, for all x ∈ N .
• αt ◦ β̃(γ)(x̃) = β̃(γ) ◦ αt(x̃), for all t ∈ R, γ ∈ Γ and x̃ ∈ Ñ .
Suppose that the weak 1-cocycle {wγ⊗1} ⊂ Ñ is ergodic for the G-action β̃|G. If the
group inclusion G ⊂ Γ has the relative property (T) of Kazhdan, then there exists a
non-zero element x̃0 ∈ Ñ so that (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
This is proved in the same way for Bernoulli shift actions on the infinite tensor
product of abelian von Neumann algebras ([PoSa], Lemma 3.5). Since we are in-
terested in actions on the AFD II1 factor, we require the ergodicity assumption on
20 HIROKI SAKO
weak 1-cocycle {wγ ⊗ 1}. For the self-containedness and in order to make it clear
where the ergodicity assumption works, we write down a complete proof.
Proof. For t ∈ (0, 1], let Kt be the convex weak closure of
{(wg ⊗ 1)αt(w∗g ⊗ 1) | g ∈ G} ⊂ Ñ
and x̃t ∈ Kt be the unique element whose 2-norm is minimum in Kt. Since
(wg ⊗ 1)β̃g((wg1 ⊗ 1)αt(w∗g1 ⊗ 1))αt(w
g ⊗ 1)
= (wgβg(wg1)⊗ 1)αt(βg(w∗g1)w
g ⊗ 1)
= (wgg1 ⊗ 1)αt(w∗gg1 ⊗ 1), g, g1 ∈ G,
we have (wg ⊗ 1)β̃g(Kt)αt(w∗g ⊗ 1) = Kt, for g ∈ G. By the uniqueness of x̃t, we get
(wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1) = x̃t, g ∈ G.(Eq3)
By the assumption, the action (Ad(wg ⊗ 1) ◦ β̃g)g∈G is ergodic on Ñ . By the calcu-
lation
(wg ⊗ 1)β̃g(x̃tx̃t∗)(w∗g ⊗ 1)
= (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1)αt(wg ⊗ 1)β̃g(x̃t
)(w∗g ⊗ 1)
= x̃tx̃t
, g ∈ G,
we get x̃tx̃t
∗ ∈ C1. The element x̃t is a scalar multiple of a unitary in Ñ .
We shall next prove that x̃1/n is not zero for some positive integer n. The pair
(Γ, G) has the relative property (T) of Kazhdan. By proposition 2.4, we can find a
positive number δ and a finite subset F ⊂ Γ satisfying the following condition: If a
unitary representation (π,H) of Γ and a unit vector ξ of H satisfy ‖π(γ)ξ − ξ‖ ≤
δ (γ ∈ F ), then ‖π(g)ξ− ξ‖ ≤ 1/2 (g ∈ G). By the continuity of the action α, there
exists n such that
‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖tr,2 ≤ δ, γ ∈ F,
The actions β and (αl1/n)l∈Z on Ñ give a Γ× Z action on Ñ . Let P be the crossed
product von Neumann algebra P = Ñ ⋊ (Γ × Z). Let (Uγ)γ∈Γ and W be the
implementing unitaries in P for Γ and 1 ∈ Z respectively. We put Vγ = (wγ ⊗
1)Uγ , γ ∈ Γ. We regard AdV· as a unitary representation of Γ on L2(P ). Since
‖AdVγ(W )−W‖L2(P ) = ‖(wγ ⊗ 1)W (w∗γ ⊗ 1)W ∗ − 1‖L2(P )
= ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖L2(Ñ) ≤ δ, γ ∈ F,
we have the following inequality:
1/2 ≥ ‖AdVg(W )−W‖L2(P ) = ‖(wg ⊗ 1)α1/n(w∗g ⊗ 1)− 1‖L2(Ñ), g ∈ G.
We get 1/2 ≥ ‖x̃1/n − 1‖L2(Ñ) and x̃1/n 6= 0.
Let ũ1/n be the unitary of Ñ given by a scalar multiple of x̃1/n. By equation
(Eq3), the unitary satisfies
(wg ⊗ 1)β̃g(ũ1/n)α1/n(w∗g ⊗ 1) = ũ1/n, g ∈ G.
CLASSIFICATION AND CENTRALIZERS 21
Let x̃0 be the unitary defined by
x̃0 = ũ1/nα1/n(ũ1/n)α2/n(ũ1/n) . . . α(n−1)/n(ũ1/n).
By direct computations, we have the following desired equality:
(wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = (wg ⊗ 1)β̃g(x̃0)α1(w∗g ⊗ 1) = x̃0, g ∈ G.
Theorem 6.6. Let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ).
Suppose that N(H, µ) is the AFD factor of type II1 and that there exists a continuous
R-action (α
t )t∈R on Lµ(H)⊗ Lµ(H) satisfying the following properties:
• For any x ∈ Lµ(H), α(0)1 (x⊗ 1) = 1⊗ x,
• The automorphism α(0)t commutes with the diagonal action of Ĥ.
Let β(1) be another action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We deduce from outer conjugacy to conjugacy in the above situation. Let θ
be a ∗-isomorphism from N (1) onto N(H, µ) which gives the outer conjugacy of the
action β(1) and β = β(H, µ, χ). There exists a weak 1-cocycle {wγ}γ∈Z2⋊SL(2,Z) for β
satisfying
θ ◦ β(1)(γ) = Adwγ ◦ β(γ) ◦ θ, γ ∈ Z2 ⋊ SL(2,Z).
Since the action β(1) is ergodic on Z2, the weak 1-cocycle w is ergodic on Z2.
We use the notations Γ0, G0 given in Section 3. Let ρ̃ be the diagonal action ρ⊗ρ
of Γ0 on the tensor product algebra M̃ = Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H):
ρ̃(γ0)(a⊗ b) = ρ(γ0)(a)⊗ ρ(γ0)(b).
The fixed point algebra Ñ ⊂ M̃ of the diagonal Ĥ-action contains N(H, µ) ⊗
N(H, µ). Since Z2⋊SL(2,Z) = Γ0/Ĥ, the action ρ̃ gives a Z
2⋊SL(2,Z)-action β̃ on
Ñ . The action β̃ is the extension of the diagonal action β⊗β on N(H, µ)⊗N(H, µ).
We denote by αt the action on M̃ ∼=
(Lµ(H)⊗Lµ(H)) given by the infinite tensor
product of the R-action α
t . By the assumption on α
t , the R-action αt commutes
with the action ρ̃. It follows that the subalgebra Ñ is globally invariant under αt.
The set of unitary {Wγ = wγ ⊗ 1}γ∈Z2⋊SL(2,Z) ⊂ Ñ is a weak 1-cocycle for β̃. We
shall prove that this weak 1-cocycle is ergodic on Z2. Let a be an element in Ñ fixed
under β̃|Z2 . The element a can be written as a =
H aλ⊗u(λ) in L2M̃ , where
aλ ⊗ 1 = EM⊗C(a(1⊗ u(λ))∗). Since a is fixed under the action of Z2, we have
a = β̃W (k)(a) =
Adwk ◦ ρ(1, k)(aλ)⊗ ρ(1, k)(u(λ))
Adwk ◦ ρ(1, k)(aλ)⊗
χ(λ(l))det(k,l)u(k · λ).
Since Adwk ◦ ρ(1, k) preserves the 2-norm, we get ‖aλ‖2 = ‖ak−1·λ‖2. Since ‖a‖22 =∑
‖aλ‖22 < ∞ and the set {ak−1·λ | k ∈ Z2} is infinite for λ 6= 0, it turns out that
aλ = 0 for λ 6= 0 and thus a ∈ Ñ ∩ (M ⊗ C) = N(H, µ) ⊗ C. By the ergodicity of
22 HIROKI SAKO
the Z2-action {Adwk ◦ β(k)}, we get a ∈ C. We conclude that the weak 1-cocycle
{Wγ} ⊂ Ñ is ergodic on Z2.
By the relative property (T) for the inclusion Z2 ⊂ Z2⋊SL(2,Z) and Proposition
6.5, there exists a non-zero element x̃0 ∈ Ñ satisfying
(wk ⊗ 1)β̃(k)(x̃0)(1⊗ w∗k) = x̃0, k ∈ Z2.
The element x̃0 can be written as the following Fourier expansion:
x̃0 =
c(λ1, λ2)u(λ1)⊗ u(λ2) ∈ Ñ ⊂ Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H).
Here c(λ1, λ2) is a complex number and (λ1, λ2) ∈ (⊕Z2H)2 runs through all pairs
satisfying
k∈Z2(λ1(k) + λ2(k)) = 0. Choose and fix a pair (λ1, λ2) satisfying
λ1(k) = h =
λ2(k), c(λ1, λ2) 6= 0.
Let v′h ∈ M be the unitary written as v′h = uh ⊗ 1 ⊗ 1 ⊗ · · · , where uh ∈ Lµ(H) is
placed on 0 ∈ Z2. The following unitaries {w′γ} ⊂ N(H, µ) give a weak 1-cocycle
for β:
w′(k,γ0) = v
hw(k,γ0)ρ(1, k, γ0)(v
), (k, γ0) ∈ Z2 ⋊ SL(2,Z).
Letting ỹ = (v′h ⊗ 1)x̃0(1⊗ v′h)∗ ∈ M̃ , we get
ỹ = (w′k ⊗ 1)β̃(k)(ỹ)(1⊗ w′k
), k ∈ Z2.
Applying the trace preserving conditional expectation E = EN(H,µ)⊗N(H,µ), we get
E(ỹ) = (w′k ⊗ 1)E(β̃(k)(ỹ))(1⊗ w′k
= (w′k ⊗ 1)β̃(k)(E(ỹ))(1⊗ w′k
), k ∈ Z2.
Since the Fourier coefficient of x̃0 at (λ1, λ2) ∈ (⊕Z2H)2 is not zero, that of E(ỹ)
at (λ1 + δh,0, λ2 − δh,0) ∈ Λ(H)2 is also non-zero, where δh,0 ∈ ⊕Z2H is zero on
Z2 \ {0} and is h on 0 ∈ Z2. By Proposition 6.2, it follows that the weak 1-cocycle
{w′(k,e)}k∈Z2 ⊂ N(H, µ) is a weak 1-coboundary of β|Z2. Since the Z2-action β|Z2 is
weakly mixing, w′ is a weak 1-coboundary on Z2 ⋊ SL(2,Z), by Proposition 6.3. In
other words, there exists v ∈ N(H, µ) satisfying
w′γ = vβ(γ)(v
∗) mod T,
wγ = v
vρ(1, γ)(v∗v′h) mod T, γ ∈ Z2 ⋊ SL(2,Z).
Noting that u = v∗v′h ∈ M is a normalizer of N(H, µ), we get
(Ad(u) ◦ θ) ◦ β(1)(γ) = Ad(u) ◦ Ad(wγ) ◦ β(γ) ◦ θ
= Ad(ρ(1, γ)(u)) ◦ β(γ) ◦ θ
= ρ(1, γ) ◦ Ad(u) ◦ θ
= β(γ) ◦ (Ad(u) ◦ θ), γ ∈ Z2 ⋊ SL(2,Z).
Thus we get the conjugacy of two Z2 ⋊ SL(2,Z)-actions β(0) and β. �
We can always apply Theorem 6.6 if H is finite.
CLASSIFICATION AND CENTRALIZERS 23
Corollary 6.7. Let H be a finite abelian group and let β = β(H, µ, χ) be a twisted
Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type
II1. Let β
(1) be an action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We have only to construct an R-action on Lµ(H) ⊗ Lµ(H) satisfying the
properties in Theorem 6.6. Let U be an element of Lµ(H)⊗ Lµ(H) defined by
|H|1/2
uh ⊗ u∗h.
We note that µ∗µ(g, ·) is a character of H and that it is not identically 1 provided
g 6= 0 by Lemma 5.5. The element U is self-adjoint and unitary, since
|H|1/2
u∗h ⊗ uh =
|H|1/2
µ(h,−h)u−h ⊗ µ(h,−h)u∗−h = U,
g,h∈H
uguh ⊗ u∗gu∗h =
g,h∈H
µ∗µ(g, h)ug+h ⊗ u∗g+h
µ∗µ(g, h− g)
ug ⊗ u∗g = 1.
The operator U is a fixed point under the action of Ĥ , so the projections P1 =
(1 + U)/2 and P−1 = (1 − U)/2 are also fixed points. Thus the R-action α(0)t =
Ad(P1+exp (iπt)P−1) commutes with the Ĥ-action. The automorphism α
1 satisfies
1 (ug ⊗ 1) = U(ug ⊗ 1)U∗ = (1⊗ ug)UU∗ = 1⊗ ug, g ∈ H.
This verifies the first condition for α(0). �
Corollary 6.8. Let Q be a set consisting of odd prime numbers and βQ be the twisted
Bernoulli shift action defined in Corollary 5.9. Let β be a Z2 ⋊ SL(2,Z)-action on
the AFD factor of type II1 whose restriction to Z
2 is ergodic. The actions βQ and
β are outer conjugate if and only if they are conjugate. In particular, {βQ} is an
uncountable family of Z2⋊SL(2,Z)-actions which are mutually non outer conjugate.
Proof. We will use the notation given in Corollary 5.8 and 5.9. Let α
t be the R-
action on Lµq (Hq)⊗Lµq(Hq) constructed as in the previous corollary. We define the
R-action α(Q) on LµQ(HQ)⊗LµQ(HQ) by α
t (⊗q∈Qxq) = ⊗q∈Qα
t (xq), where xq ∈
Lµq (Hq) ⊗ Lµq (Hq) and xq 6= 1 only for finitely many q. The R-action satisfies the
conditions in Theorem 6.6. By Corollary 5.9, {βQ} are mutually non conjugate and
their restriction to Z2 is ergodic. Thus they are mutually non outer conjugate. �
Acknowledgment . The author would like to thank Professor Yasuyuki Kawahigashi
for helpful conversations. He thanks the referee for careful reading and numerous
detailed comments. He is supported by JSPS Research Fellowships for Young Sci-
entists.
24 HIROKI SAKO
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Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo,
153-8914, Japan
E-mail address : [email protected]
1. Introduction
2. Preparations
2.1. Functions det and gcd
2.2. Scalar 2-cocycles for abelian groups
2.3. Definition of a Z2 SL(2,Z)-action on (H)
2.4. On the relative property (T) of Kazhdan
2.5. Weakly mixing actions
2.6. A remark on group von Neumann algebras
3. Definition of twisted Bernoulli shift actions
4. Classification up to conjugacy
5. Examples
5.1. Twisted Bernoulli shift actions on L(X)
5.2. Twisted Bernoulli shift actions on the AFD factor of type II1
6. Malleability and rigidity arguments
References
|
0704.1534 | The Phase-resolved High Energy Spectrum of the Crab Pulsar | The Phase-resolved High Energy Spectrum of
the Crab Pulsar
J.J. Jia a, Anisia P.S. Tang a, J. Takata b, H.K. Chang c,
K.S.Cheng a
aDepartment of Physics, The University of Hong Kong, Hong Kong, China,
[email protected]
bASIAA/National Tsing Hua University - TIARA, PO Box 23-141, Taipei,
Taiwan
cDepartment of Physics and Institute of Astronomy, National Tsing Hua
University, Hsinchu 30013, Taiwan
Abstract
We present a modified outer gap model to study the phase-resolved spectra of the
Crab pulsar. A theoretical double peak profile of the light curve containing the
whole phase is shown to be consistent with the observed light curve of the Crab
pulsar by shifting the inner boundary of the outer gap inwardly to ∼ 10 stellar radii
above the neutron star surface. In this model, the radial distances of the photons
corresponding to different phases can be determined in the numerical calculation.
Also the local electrodynamics, such as the accelerating electric field, the curvature
radius of the magnetic field line and the soft photon energy, are sensitive to the
radial distances to the neutron star. Using a synchrotron self-Compton mechanism,
the phase-resolved spectra with the energy range from 100 eV to 3 GeV of the Crab
pulsar can also be explained.
Key words: neutron stars, pulsars, radiation mechanism, radiation processes
PACS: 97.60.Jd, 97.60.Gb, 95.30.Gv, 94.05.Dd
1 Introduction
There are eight pulsars have been detected in gamma-ray energy range (cf.
Thompson 2006 for a recent review) with period ranging from 0.033s to 0.237s
and age younger than million years old. Theoretically, it is suggested that
high-energy photons are produced by the radiation of charged particles that
are accelerated in the pulsar magnetosphere. There are two kinds of theoretical
Preprint submitted to Elsevier 18 November 2018
http://arxiv.org/abs/0704.1534v1
models: one is the polar gap model (e.g., Harding 1981, Daugherty & Hard-
ing 1996, for more detail review of polar cap model cf. Harding 2006), and
another is the outer gap model (e.g. Cheng, Ho, & Ruderman 1986a, 1986b;
Chiang & Romani 1994). Both models predict that electrons and positrons are
accelerated in a charge depletion region called a gap by the electric field along
the magnetic field lines and assume that charged particles lose their energies
via curvature radiation in both polar and outer gaps. The key differences are
polar gap are located near stellar surface and the outer gap are located near
the null charge surface, where are at least several tens stellar radii away the
star.
The continuous observations of powerful young pulsars, including the Crab,
the Vela and the Geminga, have collected large number of high energy photons,
which allow us to carry out much more detailed analysis. Fierro et al. (1998)
divided the whole phase into eight phase intervals, i.e. leading wing 1, peak 1,
trailing 1, bridge, leading wing 2, peak 2, trailing 2 and off-pulse. They showed
that the data in each of these phases can be roughly fitted with a simple power
law. However, the photon indices of these phases are very different, they range
from 1.6 to 2.6. Massaro et al. (2000) have shown that X-ray pulse profile is
energy dependent and the X-ray spectral index also depends on the phase of
the rotation.
The Crab pulsar has been extensively studied from the radio to the extremely
high energy ranges, and the phase of the double-pulse with separation of 144◦
is found to be consistent over all wavelengths. Recently Kuiper et al. (2001)
have combined the X-ray and gamma-ray data of the Crab pulsar, they showed
that the phase-dependent spectra exhibit a double-peak structure, i.e. one
very broad peak in soft gamma-rays and another broad peak in higher energy
gamma-rays. The position of these peaks depend on the phase. Although the
double-peak structure is a signature of synchrotron self-Compton mechanism,
it is impossible to fit the phase dependent spectrum by a simple particle energy
spectrum. Actually it is not surprised that the spectrum is phase dependent
because photons are emitted from different regions of the magnetosphere. The
local properties, e.g. electric field E(r), magnetic field B(r), particles density
and energy distribution are very much different for different regions. Therefore
these phase dependent data provide very important information for emission
region. Consequently, the phase-resolved properties provide very important
clues and constraints for the theoretical models. So far, the three-dimensional
outer gap model seems to be the most successful model in explaining both the
double-peak pulse profile and the phase-resolved spectra of the Crab pulsar
(e.g. Chiang & Romani 1992, 1994; Dyks & Rudak 2003; Cheng, Ruderman
& Zhang, 2000, hereafter CRZ). However, the leading-edge and trailing-edge
of the light curve cannot be given out, since the inner boundary of the outer
gap is located at the null charge surface in this model. Recently, the electro-
dynamics of the pulsar magnetosphere has been studied carefully by solving
the Poisson equation for electrostatic potential and the Boltzmann equations
for electrons/positrons (Hirotani & Shibata, 1999a,b,c; Takata et al. 2004,
2006; Hirotani 2005), and the inner boundary of the gap is shown to be located
near the stellar surface.
We will organize the paper as follows. We describe the modified outer gap
model in §2. In §3, we calculate the phase-resolved spectra and present the
fitting result of the spectra in different phase intervals. Finally, we discuss our
results and draw conclusions in §4.
2 A modified outer gap model
Originally proposed by Holloway (1973) that vacuum gaps may form in the
outer regions of pulsar’s magnetosphere, Cheng, Ho and Ruderman (1986a,
1986b; hereafter CHR) developed the idea of outer magnetosphere gaps and
explained the radiation mechanisms of the γ-rays from the Crab and Vela
pulsars. CHR argued that a global current flowing through the null surface
of a rapidly spinning neutron star would result in large regions of charge
depletion, which form the gaps in the magnetosphere. They assume the outer
gap should begin at the null charge surface and extend to the light cylinder.
In the gaps, a large electric field parallel to the magnetic field lines is induced
( ~E · ~B 6= 0), and it can accelerate the electrons or positrons to extremely
relativistic speed. Thus, those charges can emit high energy photons through
various mechanisms, and further produce copious e+e− pairs to sustain the
gaps and the currents.
Based on the CHR model, Chiang and Romani (1992, 1994) generated gamma
ray light curves for various magnetosphere geometries by assuming that gap-
type regions could be supported along all field lines which define the boundary
between the closed region and open field line region rather than just on the
bundle of field lines lying in the plane containing the rotation and magnetic
dipole axes. In their model, photons are generated and travel tangential to the
local magnetic field lines and there are beams in both the outward and inward
directions. They suggested that a single pole would produce a double-peak
emission profile when the line of sight crosses the enhanced regions of the γ-ray
beam, while the inner region of the beam results in the bridge emission between
these two pulses. The peak phase separation can be accommodated by choosing
a proper observer viewing angle. Because the location of emission of each point
in phase along a given line of sight can be mapped approximately in this model,
the outer gap is thus divided into small subzones. As the curvature radius,
photon densities and the local electrodynamics in different subzones are not
the same, the spectral variation of the high energy radiation in different phase
intervals varies. Later, Romani and Yadigaroglu (1995) developed the single
gap model by involving the effects of aberration, retarded potential and time of
flight across the magnetosphere. The light curve profiles in this modified model
is simply determined by only two parameters, which are magnetic inclination
angle α and the viewing angle ζ . They argued that the γ-ray emission can only
be observed from pulsars with large viewing angle (ζ ≥ 45◦), and we cannot
receive the γ photons but radio emissions from the aligned pulsars (α ≤ 35◦).
Furthermore, they showed the gap would grow larger as the pulsar slows down,
and more open field lines can occupy the outer gap, which means the older
pulsar are more efficient for producing GeV γ-ray photons (Yadigaroglu &
Romani, 1995).
However, the assumptions of the model proposed by Romani’s group are not
self-consistent. Why is there only a single pole and only outgoing current in
the magnetosphere? Cheng, Ruderman and Zhang (2000) proposed another
version of three dimensional outer gap model for high energy pulsars based
on the pioneering work of Romani, and made it more natural in physics. In
the CRZ model, two outer gaps and both outward and inward currents are
allowed (though it turns out that outgoing currents dominate the emitted ra-
diation intensities), and the azimuthal extension of the outer gap is restricted
on a bundle of fields instead of the whole lines. Like the previous work by
Yadigaroglu and Romani (1995), the CRZ model also contains the same two
parameters, but more self-consistent in gap geometry and radiation morphol-
ogy by using the pair production conditions. The electric field parallel to the
magnetic field lines is
ΩB(r)f 2(r)R2L
cs(r)
, (1)
where f(r) ∝ r3/2 and s(r) ∝ r1/2 are the fractional size of the outer gap and
the curvature radius at the distance r. The characteristic fractional size of the
outer gap evaluated at r ∼ rL, where rL is the light cylinder radius, can be
estimated by the condition of pair creation (Zhang & Cheng 1997; CRZ) and
is given by
f0 = 5.5P
26/21B
12 ∆Φ
1/7 , (2)
where ∆Φ is the azimuthal extension of the outer gap. CRZ estimates its value
by considering the local pair production condition and give ∆Φ ∼ 160◦ for the
Crab pular. It has been pointed out that if the inclination angle is small, f0
can be changed by a factor of several (Zhang et al. 2004). We want to remark
that equation (1) is the solution of vacuum solution, for regions near null
surface and the inward extension of the gap the electric field is shown to be
deviated from the vacuum solution (e.g Muslimov & Harding 2004; Hirotani
2006). Nevertheless for simplicity we shall assume the vacuum solution for the
entire gap.
In the numerical calculation, the outer gap should be divided into several layers
in space. The shape of each layer at the stellar surface is similar to that of the
polar cap, but smaller in size. Thus, for a thin gap, the calculation of only one
representative layer is enough; while for a thick one (e.g. Geminga), several
different layers should be added in the calculation (Zhang & Cheng, 2001). The
coordinate of the footprint of the last closed field lines on the stellar surface is
determined as (x0, y0, z0), then the coordinates values (x
0) of the inner
layers can be defined by x′0 = a1x0, y
0 = a1y0, and z
1− x′02 − y′02, where
a1 corresponds to the various layers in the open volume.
Inside the light cylinder, high energy photons will be emitted nearly tangent
to the magnetic field lines in the corotating frame because of the relativistic
1/γ beaming inherent in high energy processes unless |E×B| ∼ B2. Then the
propagation direction of each emitted photons by relativistic charged particles
can be expressed as (ζ ,Φ), where ζ is the polar angle from the rotation axis
and Φ is the phase of rotation of the star. Effects of the time of flight and
aberration are taken into account. A photon with velocity u = (ux, uy, uz)
along a magnetic field line with a relativistic addition of velocity along the
azimuthal angle gives an aberrated emission direction u′ = (u′x, u
z). The
time of flight gives a change of the phase of the rotation of the star. Combining
these two effects, and choosing Φ = 0 for radiation in the (x,z) plane from the
center of the star, ζ and Φ are given by cos ζ = u′z and Φ = −φu′ −~r · û′, where
φu′ is the azimuthal angle of û′ and ~r is the emitting location in units of RL.
In panel A of Fig. 1, the emission morphology in the (ζ , Φ) plane is shown.
For a given observer with a fixed viewing angle ζ , a double-pulsed structure is
observed because photons are clustered near two edges of the emission pattern
due to the relativistic effects (cf. panel B of Fig. 1).
In Fig. 1, we can see that this model can only produce radiation between two
peaks. However, the observed data of the Crab, Vela and Geminga indicate
that the leading wing 1 and the trailing wing 2 are quite strong, and even the
intensity in off-pulse cannot be ignored. Hirotani and his co-workers (Hirotani
& Shibata 2001; Hirotani, Harding & Shibata 2003) have pointed out that the
large current in the outer gap can change the boundary of the outer gap. They
solve the set of Maxwell and Boltzmann equations in pulsar magnetospheres
and demonstrate the existence of outer-gap accelerators, whose inner bound-
ary position depends the detail of the current flow and it is not necessarily
located at the null charge surface. For the gap current lower than 25% of the
Goldreich-Julian current, the inner boundary of the outer gap is very close to
the null surface (Hirotani 2005). On the other hand if the current is close to
the Goldreich-Julian current, the inner boundary can be as close as 10 stellar
radii. In Fig. 2, we show the light curve by assuming the inner boundary is
extended inward from the null charge surface to 10 stellar radii (cf. panel A
Fig. 1. Emission projection onto the (ζ,Φ) plane and pulse profile for the single pole
outer gap. The photons are emitted outwards from the outer gap. (a) The emission
projection (a1 = 0.9) and (b) the corresponding pulse profile (∆a1 = 0.03), for
Crab parameters α = 65◦ and ζ = 82◦.
of Fig. 2). In panel B of Fig. 2, the solid line represents emission trajectory of
outgoing radiation of one gap from the null surface to the light cylinder with
α = 50◦ and ζ = 75◦ and the dashed line represents the outgoing radiation
from another gap from the inner boundary to the null surface. In the presence
of the extended emission region from the near the stellar surface to the null
charge surface, leading wing 1, trailing wing 2 and the off-pulse components
can also be produced.
0 60 120 180 240 300 360
Phase
Fig. 2. Upper panel: the simulated pulse profile of the Crab pulsar; lower panel:
variation of radial distance with pulse phase for the Crab pulsar in units of RL,
where the bold line represents those in the outer magnetosphere, and the dashed
line represents those in the inner magnetosphere. The inclination angle is 50◦ and
the viewing angle is 75◦.
3 The phase-resolved spectra
3.1 radiation spectrum
The Crab pulsar has enough photons for its spectra to be analyzed, and the
phase-resolved spectra are useful for study of the local properties of the mag-
netosphere. Here, we summarize the calculation procedure of the radiation
spectrum given in CRZ, which is used to calculate the spectrum in different
phases.
The electric field of a thin outer gap (CHR) is given by E‖(r) =
ΩB(r)a2(r)
cs(r)
ΩB(r)f2(r)R2
cs(r)
, where a(r) is the thickness of the outer gap at position r, and
f(r) = a(r)/RL is the local fractional size of the outer gap. Assuming that
the magnetic flux subtended in the outer gap is constant in the steady state,
we get the local size factor f(r) ∼ f(RL)( rRL )
3/2, where f(RL) is estimated
by using the pair creation condition (cf. Zhang & Cheng 1997, CRZ). As the
equilibrium between the energy loss in radiation and gain from accelerating
electric field, the local Lorentz factor of the electrons/positrons in the outer
gap is γe(r) = (
eE‖(r)c)
For a volume element ∆V in the outer gap, the number of primary charged
particles can be roughly written as dN = nGJ∆A∆l, where nGJ =
is the
local Goldreich-Julian number density, B∆A is the magnetic flux through the
accelerator and ∆l is the path length along its magnetic field lines. (Here, We
would like to remark that this could overestimate the primary charge num-
ber density near the null surface, where the positronic charge density dom-
inates the Goldreich-Julian charge density. However, the observed radiation
comes from the wide range of magnetospheric region, an slight overestima-
tion of a small region should not cause a qualitative difference.) Thus, the
total number of the charged particles in the outer gap is N ∼ ΩΦ
RL, where
Φ ∼ f(RL)B(RL)R2L∆φ is the typical angular width of the magnetic flux
tube subtend in the outer gap. The primary e± pairs radiate curvature pho-
tons with a characteristic energy Ecur(r) =
~γ3e (r)
, and the power into
curvature radiation for dN e± pairs in a unit volume is dLcur
≈ lcurnGJ(r),
where lcur = eE‖c is the local power into the curvature radiation from a single
electron/positron. The spectrum of primary photons from a unit volume is
dV dEγ
≈ lcurnGJ
, Eγ ≤ Ecur. (3)
These primary curvature photons collide with the soft photons produced by
synchrotron radiation of the secondary e± pairs, and produce the secondary
e± pairs by photon-photon production. In a steady state, the distribution of
secondary electrons/positrons in a unit volume is
dV dEe
∫ d2Ṅ(E ′γ = 2E
dV dEγ
dE ′e ≈
lcurnGJ
), (4)
with Ėe the electron energy loss into synchrotron radiation, which is Ėe =
e4B2(r) sin2 β(r)
)2, where B(r) is the local magnetic field and β(r) the
local pitch angle, sin β(r) ∼ sin β(RL)( rRL )
1/2, sin β(RL) is the pitch angle at
the light cylinder. Therefore, the energy distribution of the secondary elec-
trons/positrons in volume ∆V (r) can be written as
dN(r)
dV dEe
∆V (r) ∼
lcurnGJ∆V (r)
). (5)
The corresponding photon spectrum of the synchrotron radiation is
Fsyn(Eγ , r) =
3e3B(r) sinβ
mec2h
dN(r)
F (x)dEe, (6)
where x = Eγ/Esyn, and Esyn(r) =
heB(r) sinβ(r)
is the typical photon
energy, and F (x) = x
x K5/3(y)dy, where K5/3(y) is the modified Bessel
function of order 5/3. Also, the spectrum of the inverse Compton scattered
photons in the volume ∆V (r) is
FICS(Eγ, r) =
dN(r)
d2NICS(r)
dEγdt
dEe, (7)
where
d2NICS(r)
dEγdt
nsyn(ǫ, r)F (%epsilon, Eγ , Ee)dǫ, and F (ǫ, Eγ , Ee) =
3σT c
4(Ee/mc2)2
[2q ln q+
(1 + 2q)(1− q) + (Γq)
2(1−q)
2(1+Γq)
], where Γ = 4ǫ(Ee/mec
2)/mec
2, q = E1/Γ(1− E1)
with E1 = Eγ/Ee and 1/4(Ee/mec
2) < q < 1. The number density of the
synchrotron photons with energy ǫ is nsyn(ǫ, r) =
Fsyn(ǫ)
cr2∆Ω
, where Fsyn is the
calculated synchrotron radiation flux, and ∆Ω is the usual beam solid angle.
Fig. 3 shows the observed data of the phase-resolved spectra from 100 eV
to 3 GeV of the Crab pulsar, and the theoretical fitting results calculated
by using the synchrotron self-Compton mechanism. The phase intervals are
defined by division given by Fierro (1998), and the amplitude of the spectrum
in each phase interval is proportional to the number of photons counted in it.
In this fitting, f(RL) = 0.21, and B = 3.0× 1012Gauss are used, which give a
consistent fitting of the phase-resolved spectra of the seven phase intervals. In
order to obtain a better fit, we treat the pitch angle (β) and the beam solid
angle (∆Ω) near the light cylinder as free parameters and vary from phase to
phase in the calculation. sin β(RL) = 0.06 and ∆Ω = 5.0 are chosen for trailing
wing 1, bridge and leading wing 2; sin β(RL) = 0.02,∆Ω = 1.0 for leading wing
1; sin β(RL) = 0.04,∆Ω = 3.5 for peak 1, sin β(RL) = 0.07,∆Ω = 3.0 for peak
2, and sin β(RL) = 0.03,∆Ω = 6.0 for trailing wing 2. Additionally, the phase-
averaged spectrum of the total pulse of the Crab pulsar is shown in Fig. 4,
where the parameters are chosen as sin β(RL) = 0.05 and ∆Ω = 5.0.
3.2 Analysis of the Phase-Resolved Spectra
The high energy spectra of the Crab pulsar is explained by using the syn-
chrotron self-Compton mechanism, which involves both the synchrotron radia-
tion and the Inverse Compton-Scattering (ICS) caused by the ultra-relativistic
electron/positron pairs created by the extremely high-energy curvature pho-
tons. The secondary e± pairs gyrate in the strong magnetic field and radiate
F(E) (MeV cm
Fig. 3. Phase resolved spectra of the Crab pulsar from 100 eV to 3 GeV in the 7
narrow pulse-phase intervals. Two spectra (for the TW1 and LW2) are displayed
twice. The curved line is calculated by the theoretical model, and the observed data
are taken from Kuiper et al. (2001).
synchrotron photons. While in the far regions of the magnetosphere where the
magnetic field decays rapidly, the relativistic pairs collide with the soft syn-
chrotron photons through the ICS process. Thus, the spectra of the radiation
contain two main components: one is the synchrotron radiation from the soft
X-ray to ∼10 MeV, and the other is the ICS component in the even higher
energy range. Usually, the synchrotron spectrum has stronger amplitude than
1x10
1x10
2
F
s
-
E (MeV)
Fig. 4. Phase-averaged spectrum of the Crab pulsar. The observed data are taken
from Kuiper et al. (2001).
that of ICS, and there is obvious turning frequency between these two com-
ponents, e.g. about 3MeV for peak 1. As we know, the power of synchrotron
radiation and ICS can be compared by the ratio of the local magnetic energy
density and the photon energy density, i.e.
∝ B(r)
ǫsyn(r)nsyn(ǫsyn, r)
, (8)
where ǫsyn(r) is the synchrotron photon energy in location r. In Fig. 3, the
spectra in trailing wing 1, bridge and leading wing 2 have broad synchrotron
spectra, which cover from 100 eV to ∼30 MeV. In Fig. 2, it is demonstrated
that the radiation of these three phase intervals are dominated by the photons
generated in the near surface region, where the magnetic field is so strong that
synchrotron radiation takes up the most emission. However, the radiation
of peak 1 and 2 are from the far regions near the light cylinder, where the
magnetic field decays rapidly (B ∝ r−3), thus, the ICS radiation becomes
more important above 3 MeV.
The peak of the synchrotron spectrum is determined by the characteristic
synchrotron photon energy. Since Esyn ∝ γ2eB sin β, where γe is the Lorentz
factor of the secondary pairs and β is the pitch angle of the electron/positron
to the magnetic field, the peak of the synchrotron spectrum can shift if the β
varies. Since the outward radiation direction covers a wider range than that of
the inward radiation, so the solid angle (∆Ω) is no longer the unity as assumed
in CRZ model. The solid angle can effect the amplitude of the ICS spectrum
because the number density of the synchrotron photons is proportional to 1
Therefore it is reasonable for us to choose β(RL) and ∆Ω as a set of parameters
in fitting the phase-resolved spectra of the Crab pulsar.
4 Conclusion and Discussion
We have tried to explain the high energy light curve and the phase-resolved
spectra in the energy range from 100 eV to 3 GeV of the Crab pulsar by mod-
ifying the three dimensional outer magnetosphere gap model. Compared to
the classical outer gap with the inner boundary at the null charge surface, the
modified model allows the outer gap to start at the region about several stel-
lar radii above the neutron star surface, and the ”inwardly-extended” part of
the outer gap contributes to the outer wings and off-pulse of the light curve.
Such modified outer gap geometry also plays a vital role in explaining the
optical polarization properties of the Crab pulsar (Takata et al. 2006). Two
adjustable parameters are used to simulate the light curve: one is the inclina-
tion angle of the magnetic axis to the rotational axis α, and the other is the
viewing angle also to the rotational axis ζ . As constrained by the phase sepa-
ration of the double peaks, we choose the values for these two parameters that
α = 50◦ and ζ = 75◦. So far, these two parameters have not been determined
from the observations. From radio observations, Rankin (1993) estimated that
α ≈ 84◦ and ζ is not known. Moffett and Hankins (1999) gave that α ≈ 56◦
and ζ = 117◦ by using the polarimetric observations at frequencies between
1.4 and 8.4 GHz. Of course, our values cannot be the true ones, and require
further observations to give strong restrictions of them.
In fitting the phase-resolved spectra of the Crab pulsar, our model performs
well from 100 eV to 1 GeV, but fails beyond 1 GeV. The inverse Compton
scattering spectrum of our results falls down quickly when the energy is over 1
GeV, but the observation data indicates that the spectrum still increases, es-
pecially in the first trailing wing, the bridge and the second leading wing phase
intervals. We have assumed that the curvature photons are all absorbed by
the magnetic field lines, however, some of these multi-GeV photons produced
near the light cylinder should be easily escaped from the photon-photon pair
creation process. In the spectrum fitting of peak 1, our result has a frequency
shift below 1 MeV, and we found that in order to well fit the spectrum we
should reduce the curvature photon energy by a quarter. The energy of the
curvature photon Ecur ∝ s−1(r), where s(r) is the local curvature radius. As
the high energy photons are produced in the far regions of the magnetosphere,
where s(r) maybe not follow the dipole form, we can change the photon energy
slightly.
Moreover, the stellar radius of a neutron star is usually treated as 106 cm when
calculating the strength of the surface magnetic field. However, the equation
of state inside the neutron star of the current theoretical models cannot give
a convincing value of the neutron star size. Thus, we can only determine the
magnetic moment, i.e. BpR
0, of the pulsar from the energy loss rate. Therefore,
we can rewrite the magnetic field of the Crab pulsar as B12R
6 = 3.8. In fitting
the phase-resolved spectra of the Crab pulsar, we choose Bp = 3×1012 G, not
the traditional value of 3.8× 1012 G, for it gives a better fitting on the lower
energy range below 10 keV.
Finally we want to emphasize that
This research is supported by a RGC grant of Hong Kong Government under
HKU 7015/05P.
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Introduction
A modified outer gap model
The phase-resolved spectra
radiation spectrum
Analysis of the Phase-Resolved Spectra
Conclusion and Discussion
|
0704.1535 | Massive N=1 supermultiplets with arbitrary superspins | Massive N = 1 supermultiplets
with arbitrary superspins
Yu. M. Zinoviev ∗
Institute for High Energy Physics
Protvino, Moscow Region, 142280, Russia
Abstract
In this paper we give explicit construction of massive N = 1 supermultiplets in
flat d = 4 Minkowski space-time. We work in a component on-shell formalism based
on gauge invariant description of massive integer and half-integer spin particles where
massive supermultiplets are constructed out of appropriate set of massless ones.
∗E-mail address: [email protected]
http://arxiv.org/abs/0704.1535v1
Introduction
In a flat space-time massive spin s particles in a massless limit decompose into massless spin
s, s− 1, . . . ones. This, in particular, leads to the possibility of gauge invariant description
of massive spin s particles, e.g. [1]-[13]. In this, two different approaches could be used.
From one hand, one can start with usual non gauge invariant description of massive particle
and achieve gauge invariance through the introduction of additional fields (thus promoting
second class constraints into the first class ones). From the other hand, one can start with
the appropriate set of massless particles having gauge invariance from the very beginning and
obtain massive particle description as a deformation of massless theory. This last approach
closely mimic situation in spontaneous gauge symmetry breaking where gauge field has to
eat some Goldstone field(s) to become massive.
In the supersymmetric theories all particles must belong to some supermultiplet, massive
or massless. Till now most of investigations in supersymmetric theories where bounded to
massless supermultiplets. Quite a few results on massive supermultiplets mainly devoted
to superspins 1 and 3/2 exist [14]-[21]. The aim of this paper is to extend these results to
include massive N = 1 supermultiplets with arbitrary superspins. Certainly, it would be
nice to have superfield off-shell description of such supermultiplets, but as previous results
clearly show it is a highly non-trivial task. So in this paper we restrict ourselves with
component on-shell formalism in terms of physical fields. The same reasoning on the massless
limit means that massive supermultiplets could (should) be constructed out of the massless
ones in the same way as massive particles out of the massless ones. So our approach will
be supersymmetric generalization of the second approach to massive particle description
mentioned above. Namely, we will start with appropriate set of massless supermultiplets
and obtain massive one as a smooth deformation.
The paper is organized as follows. Though our previous examples on massive superspin
1 [21] and superspin 3/2 [15] supermultiplets already give important hints on how general
case of arbitrary superspin could looks like, due to peculiarities of lower spin fields they are
not enough to achieve such generalization. Thus, in the first two sections we give two more
concrete examples, namely massive supermultiplets with superspin 2 and 5/2, correspond-
ingly. All these and subsequent results heavily depend on the gauge invariant description of
massive particles with integer [2, 3] and half-integer [10] spins as well as on the known form
of massless supermultiplets [22]. For reader convenience and to make paper self-contained,
in the next two sections we give all necessary formulas in compact condensed notations. One
of the lessons from previous investigations is that the structures of massive supermultiplets
with integer and half-integer superspins are different, so in the last two sections we consider
these two cases separately. We will see that, in spite of large number of fields, all calculations
are pretty straightforward and mainly combinatorical.
1 Superspin 2
Massive superspin 2 supermultiplet contains four massive particles with spins 5/2, 2, 2’ and
3/2, correspondingly. In the massless limit massive supermultilets must decompose into the
appropriate set of massless ones in the same way as massive spin s particles — into massless
spin s, s-1, ... ones. Simple counting of physical degrees of freedom immediately gives:
0, 0′
So we will start with five massless supermultiplets (Φµν , hµν), (fµν ,Ψµ), (Ωµ, Aµ), (Bµ, ψ)
and (χ, z). From our previous experience with massive superspin 1 and superspin 3/2 su-
permultiplets we know that it is crucial for the construction of massive supermultiplets to
make dual rotation of vector Aµ and axial-vector Bµ fields mixing massless supermultiplets
containing these fields. But now we have two tensor fields hµν and fµν as well, moreover they
necessarily must be tensor and pseudo-tensor ones. Thus we have to consider the possibility
to mix massless supermultiplets with these fields as well and the real structure of massless
supermultiplets we are going to work with looks like:
hµν fµν
Aµ Bµ
Then, introducing a sum of the massless Lagrangians for bosonic fields:
∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh−
∂µh∂µh+
∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf −
∂µf∂µf −
(∂µϕ)
(∂µπ)
2 (1)
as well as sum of the massless Lagrangians for fermionic fields:
Φ̄µν ∂̂Φµν − 2i(Φ̄γ)µ(∂Φ)µ + i(Φ̄γ)µ∂̂(γΦ)µ + i(Φ̄γ∂)Φ−
Φ̄∂̂Φ−
Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)−
(Ψ̄γ)∂̂(γΨ) +
ψ̄∂̂ψ −
Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω)−
(Ω̄γ)∂̂(γΩ) +
χ̄∂̂χ (2)
it is not hard to check that the most general supertransformations leaving sum of massless
Lagrangians invariant have the form (round brackets denote symmetrization):
δΦµν = −
σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η
δhµν =
2 cos(θ2)(Φ̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (3)
δfµν =
2 sin(θ2)(Φ̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η)
δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η
for the supermultiplets containing spin 2 fields and
δΩµ = −
σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη
δAµ =
2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (4)
δBµ =
2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η)
δψ = −
σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η
for those with (axial)vector ones. The last supermultiplets is simple:
δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η)
To construct massive supermultiplet we have to add mass terms for all fields as well as
appropriate corrections to fermionic supertransformations. In this, the most important ques-
tion is which lower spin fields play the roles of Goldstone ones and have to be eaten to make
main gauge fields massive. For the bosonic fields (taking into account parity conservation)
the choice is unambiguous: vector Aµ and scalar ϕ fields for tensor field hµν and axial-vector
Bµ and pseudo-scalar π — for pseudo-tensor fµν . Thus bosonic mass terms will be:
2[hµν∂µAν − h(∂A)] −
3Aµ∂µϕ+
2[fµν∂µBν − f(∂B)]−
3Bµ∂µπ
L2 = −
(hµνhµν − h2)−
hϕ+ ϕ2 −
(fµνhµν − f 2)−
fπ + π2 (5)
But for fermions we have two spin 3/2 and two spin 1/2 fields and there is no evident choice.
Thus we introduce the most general mass terms for the fermions:
Lm = −
Φ̄µνΦµν + (Φ̄γ)
µ(γΦ)µ +
−iα1[Φ̄µνγµΨν −
Φ̄(γΨ)]− iα2[Φ̄µνγµΩν −
Φ̄(γΩ)] +
+a1[Ψ̄
µΨµ − (Ψ̄γ)(γΨ)] + a2[Ω̄µΩµ − (Ω̄γ)(γΩ)] + a3[Ψ̄µΩµ − (Ψ̄γ)(γΩ)] +
+ia4(Ψ̄γ)ψ + ia5(Ψ̄γ)χ+ ia6(Ω̄γ)ψ + ia7(Ω̄γ)χ+
+a8ψ̄ψ + a9ψ̄χ+ a10χ̄χ (6)
and proceed with calculations. Cancellation of variations with one derivative gives:
sin(θ2) = cos(θ2) =
sin(θ1) = cos(θ1) =
, α2 =
a1 = −
, a3 = 1, a4 =
2, a5 = 0
a6 = a2
2, a7 =
3, a8 = 0, a9 = −
while variations without derivatives give:
, a10 = −
Resulting fermionic mass terms:
Lm = −
Φ̄µνΦµν + (Φ̄γ)
µ(γΦ)µ +
Φ̄µνγµΨν +
Φ̄(γΨ)− i
2Φ̄µνγµΩν +
Φ̄(γΩ) +
Ψ̄µΨµ +
(Ψ̄γ)(γΨ) +
Ω̄µΩµ −
(Ω̄γ)(γΩ) + Ψ̄µΩµ − (Ψ̄γ)(γΩ) +
2(Ψ̄γ)ψ +
(Ω̄γ)ψ + i
3(Ω̄γ)χ−
6ψ̄χ−
χ̄χ (7)
correspond to invariance of the Lagrangian (besides global supertransformations) under three
local spinor gauge transformations:
δΦµν = ∂(µξν) +
γ(µξν) +
gµνξ1 +
gµνξ2
δΨµ = ∂µξ1 +
γµξ1 +
δΩµ = ∂µξ2 +m
2ξµ +
γµξ1 +
γµξ2 (8)
δψ = m
2ξ1 +
ξ2 δχ = m
From these formula one can easily determine which combination of spin 3/2 fields Ψµ and
Ωµ plays the role of Goldstone field for spin 5/2 field Φµν . Indeed, if one introduces two
orthogonal combinations:
Ψ̃µ =
Ωµ, Ω̃µ = −
then after diagonalization of mass terms one finds that Ψ̃µ is a Goldstone field, while Ω̃µ —
physical field with the same mass as Φµν .
Similar to the case of massive supermultiplet with superspin 1, both mixing angles have
been fixed: θ1 = θ2 = π/4 and this, in turn, means that all bosonic fields enter through the
complex combinations only:
Hµν = hµν + γ5fµν , Cµ = Aµ + γ5Bµ, z = ϕ+ γ5π
Introducing gauge covariant derivatives:
∇µHαβ = ∂µHαβ −
Cµgαβ, ∇µz = ∂µz −m
we can write final form of fermionic supertransformations as:
δΦµν = [−
σαβ∇αH̄β(µγν) −mHµν +
γ(µ(γH)ν) +
gµνz]η
δΨµ = [−
σαβ∇αHβµ −
(γH)µ +
∇µz +
γµz]η
δΩµ = [−
σαβC̄αβγµ +
(γH)µ +
∇µz −
γµz]η (9)
δψ = −
σαβCαβη δχ = −iγµ∇µzη
Note also that due to complexification of bosonic fields the Lagrangian and supertrans-
formations are invariant under global axial U(1)A symmetry, axial charges for all fields being:
field η Φµν , Ψµ, Ωµ, ψ, χ Hµν , Cµ, z
qA +1 0 –1
2 Superspin 5/2
Our next example — massive supermultiplet with superpin 5/2. It also contains four massive
fields: with spin 3, 5/2, 5/2’ and 2 and in the massless limit it should reduce to six massless
supermultiplets:
5/2 5/2
0, 0′
By analogy with all previous cases we will take into account possible mixing for bosonic tensor
and vector fields, so we will start with the following structure of massless supermultiplets:
hµν fµν
Aµ Bµ
So we introduce sum of the massless Lagrangians for bosonic fields:
L0 = −
∂ρΦµνλ∂ρΦµνλ +
(∂Φ)µν(∂Φ)µν − 3(∂Φ)µν∂µΦν +
∂µΦν∂µΦν +
(∂Φ)2
∂αhµν∂αhµν − (∂h)µ(∂h)µ + (∂h)µ∂µh−
∂µh∂µh−
(∂µϕ)
∂αfµν∂αfµν − (∂f)µ(∂f)µ + (∂f)µ∂µf −
∂µf∂µf −
(∂µπ)
2 (10)
as well as sum of the massless Lagrangians for fermionic fields:
Ψ̄µν ∂̂Ψµν − 2i(Ψ̄γ)µ(∂Ψ)µ + i(Ψ̄γ)µ∂̂(γΨ)µ + i(Ψ̄γ∂)Ψ−
Ψ̄∂̂Ψ+
Ω̄µν ∂̂Ωµν − 2i(Ω̄γ)µ(∂Ω)µ + i(Ω̄γ)µ∂̂(γΩ)µ + i(Ω̄γ∂)Ω−
Ω̄∂̂Ω−
Ψ̄µ∂̂Ψµ + i(Ψ̄γ)(∂Ψ)−
(Ψ̄γ)∂̂(γΨ) +
ψ̄∂̂ψ −
Ω̄µ∂̂Ωµ + i(Ω̄γ)(∂Ω) −
(Ω̄γ)∂̂(γΩ) +
χ̄∂̂χ (11)
and start with the following global supertransformations:
δΦµνλ = i(Ψ̄(µνγλ)η) δΨµν = [−σαβ∂αΦβµν +
∂(µγν)(γΦ)]η (12)
for the supermultiplet (3, 5/2),
δΩµν = −
σαβ∂α(cos(θ2)hβ(µγν) − sin(θ2)γ5fβ(µγν))η
δhµν =
2 cos(θ2)(Ω̄µνη) + i sin(θ2)(Ψ̄(µγν)η) (13)
δfµν =
2 sin(θ2)(Ω̄µνγ5η) + i cos(θ2)(Ψ̄(µγν)γ5η)
δΨµ = −σαβ∂α(sin(θ2)hβµ + cos(θ2)fβµγ5)η
for the mixed (5/2, 2) and (2, 3/2) supermultiplets,
δΩµ = −
σαβ(cos(θ1)Aαβ − sin(θ1)γ5Bαβ)γµη
δAµ =
2 cos(θ1)(Ω̄µη) + i sin(θ1)(ψ̄γµη) (14)
δBµ =
2 sin(θ1)(Ω̄µγ5η) + i cos(θ1)(ψ̄γµγ5η)
δψ = −
σαβ(sin(θ1)Aαβ + cos(θ1)γ5Bαβ)η
for the mixed (3/2, 1) and (1, 1/2) supermultiplets and
δχ = −iγµ∂µ(ϕ+ γ5π)η, δϕ = (χ̄η), δπ = (χ̄γ5η)
for the last one.
By analogy with the superspin 3/2 case, we will assume that fermionic mass terms are
Dirac ones:
Lm = −Ψ̄µνΩµν + 2(Ψ̄γ)µ(γΩ)µ +
Ψ̄Ω +
[−Ψ̄µνγµΨν +
Ψ̄(γΨ)− Ω̄µνγµΩν +
Ω̄(γΩ)] +
Ψ̄µΩµ −
(Ψ̄γ)(γΩ) + 2i(Ψ̄γ)ψ + 2i(Ω̄γ)χ− 3ψ̄χ (15)
where all coefficients are completely fixed by the requirement that the Lagrangian has to be
invariant not only under the global supertransformations, but under four (by the number of
fermionic gauge fields) spinor gauge transformations:
δΨµν = ∂(µξν) +
γ(µην) +
gµνξ1, δΩµν = ∂(µην) +
γ(µξν) +
gµνξ2,
δΨµ = ∂µξ1 +m
ξµ + im
γµξ2, δΩµ = ∂µξ2 +m
ηµ + im
γµξ1,
δψ = 2mξ1, δχ = 2mξ2
As for the bosonic fields, here the roles of the fields are evident (again taking into account
parity conservation): we need tensor hµν , vector Aµ and scalar ϕ fields to make spin 3 field
Φµνλ massive, while pseudo-tensor fµν field needs to eat axial-vector Bµ and pseudo-scalar
π fields. So the bosonic mass terms are also completely fixed:
3[−Φµνλ∂µhνλ + 2Φµ(∂h)µ −
Φµ∂µh] +
5[hµν∂µAν − h(∂A)]−
6Aµ∂µϕ +
2[fµν∂µBν − f(∂B)]−
3Bµ∂µπ (16)
ΦµνλΦµνλ −
ΦµΦµ +
ΦµAµ −
(fµνfµν − f 2)−
fπ + π2 (17)
Now we require that the whole Lagrangian be invariant under global supertransformations
with appropriate corrections to fermionic transformations. This fixes both mixing angles:
sin(θ2) =
, cos(θ2) =
, sin(θ1) =
, cos(θ1) =
and gives the following form of additional terms for fermionic supertransformations:
δΨµν = [−
γ(µ(γh)ν) −
fµνγ5 +
γ(µ(γf)ν)γ5]η
δΩµν = i[(γΦ)µν +
gµν(γΦ)−
γ(µAν) +
gµνÂ−
γ(µBν)γ5 +
gµνB̂γ5]η
δΨµ = [
γµ(γΦ)−
γµÂ−
Bµγ5 −
γµB̂γ5]η (18)
δΩµ = i[
(γh)µ +
(γf)µγ5 − γµϕ− γµγ5π]η
δψ = [−ϕ− 2γ5π]η
δχ = i[
6Â +
3B̂γ5]η
The complete supertransformations for fermionic fields could be simplified by introduction
of gauge invariant derivatives;
∇µhαβ = ∂µhαβ −
Aµgαβ, ∇µϕ = ∂µϕ−m
∇µfαβ = ∂µfαβ −
Bµgαβ, ∇µπ = ∂µπ −m
This time bosonic fields do not combine into complex combinations, but due to the fact that
fermionic mass terms are Dirac ones the Lagrangian and supertransformations are invariant
under global axial U(1)A transformations, provided axial charges of all fields are assigned as
follows:
field η, Ψµν , Ψµ, ψ hµν , fµν , Aµ, Bµ, ϕ, π Ωµν , Ωµ, χ
qA +1 0 –1
3 Massive particles
All our previous and subsequent calculations heavily depend on the gauge invariant descrip-
tion of massive high spin particles. For reader convenience and to make paper self-contained
we will give here gauge invariant formulations for massive particles with arbitrary integer
[2, 3] and half-integer [10] spins. We restrict ourselves to flat d = 4 Minkowski space but all
results could be easily generalized to the case of (A)dS space with arbitrary dimension d.
3.1 Integer spin
The simplest way to describe massless bosonic field with arbitrary spin s is to use completely
symmetric rank s tensor Φ(α1α2...αs) which is double traceless. In what follows we will use
condensed notations where index denotes just number of free indices and not the indices
themselves. For example, the tensor field itself will be denoted as Φs, it’s contraction with
derivative as (∂Φ)s−1, it’s trace as Φ̃s−2 and so on. As we will see this does not lead to any
ambiguities then working with free Lagrangians quadratic in fields. In these notations the
Lagrangian for massless particles of arbitrary spin s could be written as:
L0 = (−1)s[
∂µΦs∂µΦ
(∂Φ)s−1(∂Φ)s−1 −
s(s− 1)
∂µΦ̃s−2∂µΦ̃
s−2 +
s(s− 1)
(∂Φ)s−1∂(1Φ̃s−2) −
s(s− 1)(s− 2)
(∂Φ̃)s−3(∂Φ̃)s−3] (19)
where
Φ = 0. This Lagrangian is invariant under the following gauge transformations:
δ0Φs = ∂(1ξs−1), ξ̃s−3 = 0
where parameter ξs−1 is completely symmetric traceless tensor of rank s− 1.
To construct gauge invariant Lagrangian for massive particle which has correct (i.e. with
right number of physical degrees of freedom) massless limit, we start with the sum of massless
Lagrangians with 0 ≤ k ≤ s:
(−1)k[
∂µΦk∂µΦ
(∂Φ)k−1(∂Φ)k−1 −
k(k − 1)
∂µΦ̃k−2∂µΦ̃
k(k − 1)
(∂Φ)k−1∂(1Φ̃k−2) −
k(k − 1)(k − 2)
(∂Φ̃)k−3(∂Φ̃)k−3] (20)
Then we add the following cross terms with one derivative as well as mass terms without
derivatives:
(−1)kak[(∂Φ)k−1Φk−1 + (k − 1)Φ̃k−2(∂Φ)k−2 +
(k − 1)(k − 2)
(∂Φ̃)k−3Φ̃k−3]
(−1)k[dkΦkΦk + ekΦ̃k−2Φ̃k−2 + fkΦ̃k−2Φk−2] (21)
and try to achieve gauge invariance with the help of appropriate corrections to gauge trans-
formations:
δΦk = αkξk + βkg(2ξk−2)
Straightforward but lengthy calculations give a number of algebraic equations on the un-
known coefficients which could be solved (and this is non-trivial because we obtain overde-
termined system of equations) and give us:
(s− k)(s+ k + 1)
2(k + 1)2
, βk+1 =
k + 1
αk, 0 ≤ k ≤ s− 1
ak = −
(s− k + 1)(s+ k)
, dk =
(s− k − 1)(s+ k + 2)
4(k + 1)
k(k − 1)
16(k + 1)
[(s− k + 2)(s+ k − 1) + 6]
fk = −
(s− k + 2)(s+ k − 1)(s− k + 1)(s+ k)
3.2 Half-integer spin
For the description of massless spin s+1/2 particles we will use completely symmetric rank
s tensor-spinor Ψs such that (γΨ̃)s−3 = 0 (in the same condensed notations as before). Then
Lagrangain for such field could be written as:
L0 = i(−1)s[
Ψ̄s∂̂Ψs − s(Ψ̄γ)s−1(∂Ψ)s−1 +
(Ψ̄γ)s−1∂̂(γΨ)s−1 +
s(s− 1)
(Ψ̄γ∂)s−2Ψ̃s−2 −
s(s− 1)
∂̂Ψ̃s−2] (22)
and is invariant under the following gauge transformations:
δ0Ψs = ∂(1ξs−1), (γξ) = 0,
where gauge parameter ξs−1 is a γ-traceless tensor-spinor of rank s− 1.
Once again we start with the sum of massless Lagrangians with 0 ≤ k ≤ s:
i(−1)k[
Ψ̄k∂̂Ψk − k(Ψ̄γ)k−1(∂Ψ)k−1 +
(Ψ̄γ)k−1∂̂(γΨ)k−1 +
k(k − 1)
(Ψ̄γ∂)k−2Ψ̃k−2 −
k(k − 1)
∂̂Ψ̃k−2] (23)
To combine all these massless fields into one massive particle we have to add the following
mass terms:
(−1)k
2(k + 1)
[Ψ̄kΨk − k(Ψ̄γ)k−1(γΨ)k−1 −
k(k − 1)
Ψ̃k−2]+
−ick[(Ψ̄γ)k−1Ψk−1 −
k − 1
(γΨ)k−2]
and corresponding corrections to gauge transformations:
δΨk = αkξk + iβkγ(1ξk−1) + ρkg(2ξk−2)
Then total Lagrangian will be gauge invariant provided:
(s+ 1)2 − k2
, αk =
k + 1
, βk =
2k(k + 1)
, ρk =
4 Massless supermultiplets
It is not easy to find in the recent literature the explicit component form of massless su-
permultiplets with arbitrary superspin [22], so for completeness we will give their short
description here. As we have already seen on the lower superspin cases, supermultiplets
with integer and half-integer superspins have different structure and have to be considered
separately.
(s, s+1/2). Supermultiplet with integer superspin s contains bosonic spin s field and
fermionic spin s+1/2 one. In this and in two subsequent sections we will use the same con-
densed notations as in the previous one. By analogy with superspins 1 and 2 supermultiplets,
we start with the following ansatz for the supertransformations:
δΨs = iα1σ
µν∂µΦν(s−1γ1)η δΦs = β(Ψ̄sη)
Indeed, calculating variations of the sum of two massless Lagrangians one can see that most
of variations cancel, provided one set α1 = −β2 . The residue:
δL = −(−1)sβ
(s− 1)(s− 2)
[2(Ψ̄∂∂)s−2Φ̃s−2 − ˜̄Ψ
∂2Φ̃s−2 − (s− 2)( ˜̄Ψ∂)s−3(∂Φ̃)s−3 −
−2(Ψ̄γ∂)s−2∂̂Φ̃s−2 + 2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3]
contains terms with Φ̃s−2 only, so we proceed by adding to fermionic supertransformations
one more term:
δ′Ψs = iα2∂(1γ1Φ̃s−2)
Then the choice α2 =
(s−1)(s−2)β
leaves us with:
δL = −(−1)sβ
(s− 1)(s− 2)
[2(Ψ̄γ∂∂)s−3(γΦ̃)s−3 − ( ˜̄Ψ∂)s−3∂̂(γΦ̃)s−3]
where the only terms are whose with (γΦ̃)s−3. So we make one more (last) correction to
supertransformations:
δ′′Ψs = iα3g(2∂1(γΦ̃)s−3)
and obtain full invariance with α3 = − (s−1)(s−2)β4s . To fix concrete normalization we will use
closure of the superalgebra. Calculating the commutator of two supertransformations we
obtain:
[δ1, δ2]Φs = −iβ2(η̄2γνη1)∂µΦs + . . .
where dots mean “up to gauge transformation”. So we set β =
2 and our final result looks
like:
δΨs = −
σµν∂µΦ̄ν(s−1γ1)η +
i(s− 1)(s− 2)
[∂(1γ1Φ̃s−2) − g(2∂1(γΦ̃)s−3)]η
δΦs =
2(Ψ̄sη) (25)
(s+1/2, s+1). Half-integer superspin multiplet contains fermionic spin s + 1/2 fields
and bosonic spin s + 1 one. Again by analogy with lower superspin case we will make the
following ansatz for supertransformations:
δΨs = α1σ
µν∂µΦν(s)η, δΦs+1 = iβ(Ψ̄(sγ1)η)
This time most of the variations cancel if one set α1 = −β leaving us with:
δL = i(−1)sβ
s(s− 1)
[2(Ψ̄∂∂)s−2(γΦ̃)s−2 − ˜̄Ψ
∂2(γΦ̃)s−2 −
−2(Ψ̄γ∂)s−2∂̂(γΦ̃)s−2 − (s− 2)( ˜̄Ψ∂)s−3(γ∂Φ̃)s−3]
Then the full invariance could be achieved with the following correction to supertransforma-
tions:
δ′Ψs = α2∂(1γ1(γΦ̃)s−2)
provided α2 =
. To check the closure of superalgebra and to choose normalization we
calculate commutator of two supertransformations:
[δ1, δ2]Φs+1 = −2iβ2(η̄2γµη1)∂µΦs+1 + . . .
Then our choice will be β = 1 and our final form:
δΨs = −σµν∂µΦν(s)η +
∂(1γ1(γΦ̃)s−2)η, δΦs+1 = i(Ψ̄(sγ1)η)
Note that starting with superspin 2 the structure of supertransformations are defined up to
possible field dependent gauge transformations and our choice differs from that of [22]. It
makes no difference for massless theories but for massive case the structure of corrections
for fermionic supertransformations depends on the choice made.
5 Integer superspin
Now, having in our disposal gauge invariant description of massive particles with arbitrary
(half-)integer spins, known form of supertransformations for massless arbitrary superspin
supermultiplets and concrete examples of massive supermultiplets with lower superspins, we
are ready to construct massive arbitrary superspin supermultiplets. As we have seen, integer
and half-integer cases have different structures and have to be considered separately.
In this section we consider massive supermultiplet with integer superspin. Such super-
multiplet also contains four massive fields: two bosonic spin s fields (with opposite parity)
and fermionic spin (s+1/2) and (s-1/2) ones. Calculating total number of physical degrees
of freedom and taking into account possible mixing of supermultiplets containing bosonic
fields with equal spins and opposite parity, we start with the following structure of massless
supermultiplets:
As Bs
Ak Bk
By analogy with superspin 1 and 2 cases, we will assume that all bosonic fields enter through
the complex combinations Ck = Ak + iBk only (so that all possible mixing angles are fixed
and equal π/4). Thus we choose the following form of supertransformations for massless
supermultiplets with 1 ≤ k ≤ s:
δΦk = −
σµν∂µC̄ν(k−1γ1)η +
i(k − 1)(k − 2)
[∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η
δC̄k = 2(Φ̄kη) + i
2(Ψ̄(k−1γ1)η) (26)
δΨk−1 = −
σµν∂µCν(k−1)η +
k − 2
∂(1γ1(γC̃)k−3)η
and also
δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η)
As a result of our assumption mass terms for bosonic fields are completely fixed:
(−1)kck[C̄k∂(1Ck−1) − (k − 1) ˜̄C
(∂C)k−2 +
(k − 1)(k − 2)
( ˜̄C
∂(1C̃k−3) + h.c.)]
(−1)k[dkC̄kCk + ek ˜̄C
C̃k−2 + fk(
Ck−2 + h.c.)] (27)
where
(s+ k)(s− k + 1)
, dk =
(s− k − 1)(s+ k + 2)
4(k + 1)
k(k − 1)
16(k + 1)
[(s− k + 2)(s+ k − 1) + 6]
fk = −
(s− k + 2)(s+ k − 1)(s− k + 1)(s+ k)
As for the fermionic mass terms, apriori we don’t have any restrictions on them so we have
to consider the most general possible form:
(−1)k
a1k[Φ̄
kΦk − k(Φ̄γ)k−1(γΦ)k−1 −
k(k − 1)
Φ̃k−2]+
+a2k[Φ̄
kΨk − k(Φ̄γ)k−1(γΨ)k−1 −
k(k − 1)
Ψ̃k−2] +
+a3k[Ψ̄
kΨk − k(Ψ̄γ)k−1(γΨ)k−1 −
k(k − 1)
Ψ̃k−2] +
+ib1k[(Φ̄γ)
k−1Φk−1 −
k − 1
(γΦ)k−2] +
+ib2k[(Φ̄γ)
k−1Ψk−1 −
k − 1
(γΨ)k−2] +
+ib3k[(Ψ̄γ)
k−1Φk−1 −
k − 1
(γΦ)k−2] +
+ ib4k[(Ψ̄γ)
k−1Ψk−1 −
k − 1
(γΨ)k−2]
Where:
a1s = −
, a2s = a3s = b3s = b4s = 0
The requirement that total Lagrangian be invariant under (appropriately corrected) super-
transformations gives:
a1k = −
, a2k = −
k + 1
ck+1, a3k =
2(k + 1)
b1k = −2ck, b2k = −
, b3k = 0, b4k = −2ck+1
In this, additional terms for fermionic supertransformations look like:
δ′Φk =
2ick+1
k + 1
[(γC)k +
k(k − 1)
4(k + 1)
γ(1C̃k−1) −
(k − 1)2(2k + 1)
8k(k + 1)
g(2(γC̃)k−2)]−
−Ck +
k − 1
γ(1(γC)k−1 −
(k − 1)(k − 2)
g(2C̃k−2) −
[γ(1Ck−1) − g(2(γC)k−2)] (29)
δ′Ψk =
kck+2
2(k + 1)
γ(1(γC̃)k−1) −
2(k + 1)
[(γC)k −
k(k − 1)
4(k + 1)
γ(1C̃k−1) +
(k − 1)(3k + 1)
8k(k + 1)
g(2(γC̃)k−2)]−
2ck+1
k + 1
[Ck +
γ(1(γC)k−1) +
(k − 1)(k − 2)
g(2C̃k−2)] (30)
Here the supertransformations for Φk field contain terms with Ck+1, Ck and Ck−1 fields in
the first, second and third lines correspondingly, while that of Ψk contain terms with Ck+2,
Ck+1 and Ck fields.
6 Half-integer superspin
Next we turn to the half-integer superspin case. This time we have two fermionic spin (s+1/2)
fields and bosonic ones with spins (s+1) and s. Usual reasoning on physical degrees of freedom
and possible mixings leads us to the following structure of massless supermultiplets we will
start with:
Φs Ψs
Ak Bk
We see that this structure is rather similar to that of integer superspin case. The main
difference (besides the presence of As+1, Ψs supermultiplet) comes from the mixing of bosonic
fields. We have no reasons to suggest that all mixing angles could be fixed from the very
beginning so we have to consider the most general possibility here. Let us denote:
Ck = cos(θk)Ak + γ5 sin(θk)Bk, Dk = sin(θk)Ak + γ5 cos(θk)Bk
In these notations supertransformations for massless supermultiplets could be written as
follows. Highest supermultiplet:
δAs+1 = i(Ψ̄(sγ1)η) δΨs = −σµν∂µAν(s)η +
∂(1γ1(γÃ)s−2)η
Main set (1 ≤ k ≤ s):
δΦk = −
σµν∂µC̄ν(k−1γ1)η +
i(k − 1)(k − 2)
[∂(1γ1C̃k−2) − g(2∂1(γC̃)k−3)]η
δAk =
2 cos(θk)(Φ̄kη) + i sin(θk)(Ψ̄(k−1γ1)η) (31)
δBk =
2 sin(θk)(Φ̄kγ5η) + i cos(θk)(Ψ̄(k−1γ1)γ5η)
δΨk−1 = −σµν∂µDν(k−1)η +
k − 2
∂(1γ1(γD̃)k−3η
and the last supermultiplet:
δΦ0 = −i∂̂zη, δz̄ = 2(Φ̄0η)
By analogy with superspins 3/2 and 5/2 cases we will assume that fermionic mass terms are
Dirac ones. This immediately gives:
(−1)k
s + 1
k + 1
[Ψ̄kΦk − k(Ψ̄γ)k−1(γΦ)k−1 −
k(k − 1)
Φ̃k−2]−
− ick[(Ψ̄γ)k−1Ψk−1 −
k − 1
(γΨ)k−2 + (Ψ → Φ)]
where:
(s+ k + 1)(s− k + 1)
The choice for the bosonic mass terms (taking into account parity) is also unambiguous:
(−1)kak[Ak∂(1Ak−1) − (k − 1)Ãk−2(∂A)k−2 +
(k − 1)(k − 2)
Ãk−2∂(1Ãk−3)] +
(−1)kbk[Bk∂(1Bk−1) − (k − 1)B̃k−2(∂B)k−2 +
(k − 1)(k − 2)
B̃k−2∂(1B̃k−3)] (33)
for the terms with one derivative, where:
(s+ k + 1)(s− k + 2)
, bk =
(s+ k)(s− k + 1)
and the following terms without derivatives:
(−1)k[d̂kAkAk + êkÃk−2Ãk−2 + f̂kÃk−2Ak−2] +
(−1)k[dkBkBk + ekB̃k−2B̃k−2 + fkB̃k−2Bk−2] (34)
Here:
d̂k =
(s− k)(s+ k + 3)
4(k + 1)
, êk =
k(k − 1)
16(k + 1)
[(s− k + 3)(s+ k) + 6]
f̂k = −
(s− k + 3)(s+ k)(s− k + 2)(s+ k + 1)
(s− k − 1)(s+ k + 2)
4(k + 1)
, ek =
k(k − 1)
16(k + 1)
[(s− k + 2)(s+ k − 1) + 6]
fk = −
(s− k + 2)(s+ k − 1)(s− k + 1)(s+ k)
Note that hatted coefficients differ from the unhatted ones by replacement s→ s+ 1.
Now we require that total Lagrangian be invariant under the supertransformations. First
of all this fixes all mixing angles:
sin(θk) =
s+ k + 1
2(s+ 1)
, cos(θk) =
s− k + 1
2(s+ 1)
and gives us additional terms for fermionic supertransformations:
δ′Ψk = α1Ak + α2γ(1(γA)k−1) + α3g(2Ãk−2) +
+β1Bk + β2γ(1(γB)k−1) + β3g(2B̃k−2) +
kck+1
4(k + 1)
[sin(θk+2)γ(1(γÃ)k−1) + cos(θk+1)γ(1(γB̃)k−1)]
δ′Φk = α4(γA)k + α5γ(1Ãk−1) + α6g(2(γÃ)k−2) +
+β4(γB)k + β5γ(1B̃k−1) + β6g(2(γB̃)k−2) −
cos(θk)[γ(1Ak−1) − g(2(γA)k−2]−
sin(θk)[γ(1Bk−1) − g(2(γB)k−2)]
Where:
α1 = −
s− k√
2(k + 1)
cos(θk), α2 = −
k2 + s+ k + 1√
2k(k + 1)
cos(θk)
α3 = −
(s + 1)(k − 1)(k − 2)
2k2(k + 1)
cos(θk)
β1 = −
s+ k + 2√
2(k + 1)
sin(θk), β2 =
k2 − s+ k − 1√
2k(k + 1)
sin(θk)
β3 = −
(s + 1)(k − 1)(k − 2)
2k2(k + 1)
sin(θk)
k + 1
sin(θk+1), α5 =
k(k − 1)(s− k)
4(k + 1)2
sin(θk+1)
(k − 1)[(k + 1)(s+ 1)− 2k2(s− k)]
8k(k + 1)2
sin(θk+1)
k + 1
cos(θk+1), β5 =
k(k − 1)(s+ k + 2)
4(k + 1)2
cos(θk+1)
(k − 1)[(k + 1)(s+ 1)− 2k2(s+ k + 2)]
8k(k + 1)2
cos(θk+1)
We have explicitely checked that (rather complicated) formulas from this and previous
sections correctly reproduce all lower superspins results.
Conclusion
Thus, using supersymmetric generalization of gauge invariant description for massive parti-
cles, we managed to show that all massive N = 1 supermultiplets could be constructed out
of appropriate set of massless ones. In this, in spite of large number of fields involved, all
calculations are pretty straightforward and mainly combinatorical. Certainly, using gauge
invariance one can fix the gauge where all but four physical massive fields are equal to
zero. But in this case all supertransformations must be supplemented with field dependent
gauge transformations restoring the gauge. So the structure of resulting supertransformation
become very complicated and will contain higher derivative terms.
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Superspin 2
Superspin 5/2
Massive particles
Integer spin
Half-integer spin
Massless supermultiplets
Integer superspin
Half-integer superspin
|
0704.1536 | Critical Current of Type-II Superconductors in a Broken Bose Glass State | Critical Current of Type-II Superconductors in a Broken Bose
Glass State
J. P. Rodriguez1
1Department of Physics and Astronomy,
California State University, Los Angeles, California 90032
(Dated: November 1, 2018)
Abstract
The tilt modulus of a defective Abrikosov vortex lattice pinned by material line defects is com-
puted using the boson analogy. It tends to infinity at long wavelength, which yields a Bose glass
state that is robust to the addition of weak point-pinning centers, and which implies a restoring
force per vortex line for rigid translations about mechanical equilibrium that is independent of
magnetic field. It also indicates that the Bose glass state breaks into pieces along the direction of
the correlated pinning centers if the latter have finite length. The critical current is predicted to
crossover from two dimensional to three dimensional behavior as a function of sample thickness
along the correlated pinning centers in such case. That crossover notably can occur at a film
thickness that is much larger than that expected from point pins of comparable strength. The
above is compared to the dependence on thickness shown by the critical current in certain films of
high-temperature superconductors currently being developed for wire technology.
http://arxiv.org/abs/0704.1536v6
INTRODUCTION
It well known that thin films of high-temperature superconductors exhibit larger critical
currents than their single-crystal counterparts. Thin films of the high-temperature supercon-
ductor YBa2Cu3O7−δ (YBCO) grown by pulsed laser deposition (PLD), which are actively
being developed for wire technology, achieve critical currents that are a significant fraction
of the maximum depairing current, for example[1]. Evidence exists that lines of dislocations
that run parallel to the crystalline c axis in PLD-YBCO act as correlated pinning centers for
vortex lines inside of the superconducting state[2], and thereby give rise to such high critical
currents. This is confirmed by the peak observed in the critical current of PLD-YBCO at
orientations of the c axis aligned parallel to an applied magnetic field, as well as by the dra-
matic enhancement of the former peak after more material defects in the form of nano-rods
aligned parallel to the c axis are added[3][4].
The microstructure described above for PLD-YBCO films immediately suggests that
the vortex lattice that emerges from the superconducting state in applied magnetic field
aligned parallel to the c axis is some form of Bose glass characterized by a divergent tilt
modulus[5]. In the limit of a rigid vortex lattice, to be expected at large magnetic fields,
two-dimensional (2D) collective pinning of vortex lines by the material line defects then
determines the critical current [6][7][8][9]. Recent theoretical calculations that follow this
line of reasoning find moderate quantitative agreement with the critical current measured
in films of PLD-YBCO in c-axis magnetic fields of a few to many kG, at liquid nitrogen
temperature[10]. A potential problem with the Bose glass hypothesis, however, is that the
material line defects in PLD-YBCO films can meander, or they can be of relatively short
length[3][4]. That question is addressed in this paper, where we find that the Bose glass
breaks up into pieces along the direction of the correlated pinning centers when the effective
length of the latter is less than the film thickness. In particular, the profile of the critical
current versus film thickness is predicted to reflect a crossover from two-dimensional to three-
dimensional (3D) collective pinning of the vortex lines[11]. This cross-over can occur at a
length scale that is notably much larger than that expected from point pins of comparable
strength[12]. We also find that the unbroken Bose glass state is robust to the addition of
weak point pins. Good agreement is achieved between the dependence on thickness shown
by the critical current in certain films of PLD-YBCO at applied magnetic field[12] and that
predicted for the true Bose glass state[10]. Last, the effective restoring force per vortex line
due to a rigid translation of the Bose glass about mechanical equilibrium, which is gauged
by the Labusch parameter[13], is found to depend only weakly on applied magnetic field.
This prediction agrees with recent measurements of the microwave surface resistance on
PLD-YBCO films with nano-rod inclusions[14].
TILT MODULUS OF BOSE GLASS
Material line defects in thin enough films of PLD-YBCO can be considered to be per-
fectly parallel to the c-axis. They notably arrange themselves in a manner that resembles a
snapshot of a 2D liquid[2], as opposed to a gas[15][16]. In particular, such correlated pin-
ning centers do not show clusters or voids. A defective vortex lattice that assumes a hexatic
Bose glass state will then occur for external magnetic fields aligned in parallel to such a
microstructure, in the limit of weak correlated pinning. It is characterized by parallel lines
of edge dislocation defects that are injected into the pristine vortex lattice in order to relieve
shear stress due to the correlated pins. The former do not show any intrinsic tendency to
arrange themselves into grain boundaries, however, due to the absence of clusters and voids
in the “liquid” arrangement of linear pinning centers. (Cf. refs. [15] and [16].) This results
in a vortex lattice whose translational order is destroyed at long range by the isolated lines
of edge dislocations, but which retains long-range orientational order. Collective pinning
of the dislocation defects by the correlated pins then results both in an elastic response
to shear and in a net superfluid density. Theoretical calculations[17] and numerical Monte
Carlo simulations (see fig. 1 and caption) of the corresponding 2D Coulomb gas ensemble[18]
confirm this picture.
The correlation length Lc(|) for order along the magnetic field direction is infinite for
a Bose glass state. In the limit of weak correlated pins, the density of dislocation defects
that thread the corresponding vortex lattice can then be obtained by applying the theory
of 2D collective pinning[6][7][8]. Each line of unbound edge dislocations is in one-to-one
correspondence with a well-ordered bundle of vortex lattice, or Larkin domain, that has
dimensions Rc(|) × Rc(|) in the directions transverse to the magnetic field. The injection
of the dislocation lines into the pristine vortex lattice then results in plastic creep of each
Larkin domain by a Burgers vector[19]. The transverse Larkin scale is hence obtained by
minimizing the sum of the elastic energy cost due to the edge dislocations with the gain in
pinning energy due to the translation of a Larkin domain by an elementary Burgers vector
of the triangular vortex lattice, b = a△. This yields the estimate[6][7][8]
Rc(|)−2 = C20np(fp/c66b)2, (1)
for the density of Larkin domains, which coincides with the density of lines of unbound
dislocations. Here np denotes the density of pinned vortex lines, fp denotes the maximum
pinning force along a material line defect per unit length, and c66 denotes the elastic shear
modulus of the pristine vortex lattice. The prefactor above is of order[8] C0 ∼= π/ln(Rc/a′df)2,
where a′df is the core diameter of a dislocation in the vortex lattice. Consider now the
limit of weak pinning centers that do not crowd together: fp → 0 and πr2p · nφ ≪ 1,
respectively, where nφ denotes the density of material line defects, and where rp denotes
their range. Simple considerations of probability then yield the identity np/nφ = nB · πr2p
between the fraction of occupied pinning centers and the product of the density of vortex
lines, nB, with the effective area of each pinning center. Substituting it plus the estimate
c66 = (Φ0/8πλL)
2nB for the shear modulus of the pristine vortex lattice[21] into Eq. (1)
then yields the result Rc(|)−2 ∼= (
3π/2)C20(4fprp/ε0)
2nφ for the density of Larkin domains,
which depends only weakly on magnetic field. Here, λL denotes the London penetration
depth and ε0 = (Φ0/4πλL)
2 is the maximum tension of a flux line in the superconductor. All
of the above is valid in the 2D collective pinning regime that exists at perpendicular magnetic
fields beyond the threshold Bcp = C
3/2)(4fp/ε0)
2Φ0, in which case many vortex lines are
pinned by material line defects within a Larkin domain of transverse dimensions Rc(|)×Rc(|)
[10].
We will now exploit the boson analogy for vortex matter in order to compute the uniform
tilt modulus of the hexatic Bose glass in the absence of point pinning centers[5]. It amounts
to a London model set by the free-energy density
gB({r}, z) =
V0(ri, rj) +
Vp(ri) (2)
for vortex lines located at transverse positions {ri(z)}, at a coordinate z along the field
direction. Here ε̃l denotes the tension of an isolated vortex line, while the pair potential
V0(r, r
′) describes the interaction between vortex lines at the same longitudinal coordinate
z. The energy landscape for the correlated pinning centers is described by the potential
energy Vp(r), which is independent of the coordinate z along the field direction. The ther-
modynamics of this system in the presence of an external tilt stress nBa is then set by the
partition function
ZB[a] = (Πi
D[ri(z)])exp[−(kBT )−1
dz[gB({r}, z)−
d2r jB · a]], (3)
which under periodic boundary conditions, ri(z+Lz) = ri(z), is equivalent to a system of 2D
bosons. Above, jB(r, z) =
(2)[r − ri(z)](dri/dz) is the current density within the boson
analogy. Observe now that the kernel Πµ,ν(ω) of the uniform electromagnetic response for
alternating current (AC) is connected to this partition function through the proportionality
relationship
ZB[a] ∝ exp
(kBT )
a ·Π · a
in the limit that the corresponding uniform tilt stress vanishes, a → 0. Above, the Matsubara
frequencies iωn are given by the allowed wavenumbers along the magnetic field, qz = 2πn/Lz,
and V = LxLyLz is the volume of the system. The fact that the tilt stress is given by nBa
then yields the identity
C44(qz) = n
B/Π⊥(ωn) (5)
between the uniform tilt modulus and the uniform AC electromagnetic response of the
2D Bose glass. The subscript “⊥” above is a tag for the pure shear component of the
electromagnetic kernel Πµ,ν (Cf. ref. [5]).
The hexatic Bose glass state is clearly a 2D dielectric insulator within the boson analogy.
Its electromagnetic response can therefore be modeled by the kernel
Π⊥(ω) = (nB/ε̃l)ω
2/(ω2 − ω20), (6)
which is dielectric in the low-frequency limit, and which conserves charge by satisfying the
oscillator f-sum rule. Above, ω0 is the natural frequency of the electric dipole degrees of free-
dom in the Bose glass. The latter correspond to the 2D Larkin domains in reality, since they
represent the smallest units of well-ordered vortex lattice that can respond independently
to an applied force. We then have that the above natural frequency is of order the resonant
frequency for transverse sound inside a Larkin domain of the 2D Bose glass: ω0 ∼ γ/Rc(|),
where γ = (c66/c44)
1/2 is the effective mass anisotropy parameter equal to the transverse
sound speed within the boson analogy. Here c44 = nBε̃l is the tilt modulus due to isolated
flux lines. After substitution into Eqs. (6) for the AC response, the identity (5) then yields
a divergent tilt modulus for the Bose glass at long wavelength
C44(qz) = nBε̃l[1 + (qzL∗)
−2], (7)
with a longitudinal scale L∗ = ω
0 that is related to the transverse Larkin length by the
anisotropic scale transformation L∗ ∼ Rc(|)/γ.
Expression (7) for the uniform tilt modulus of a Bose glass is the central result of the
paper. We shall first extract the Labusch parameter[13] from the singular behavior that it
shows at long wavelength. In particular, observe that the elastic energy density for a periodic
tilt of the Bose glass by a displacement u0 at long wavelength,
C44(qz)q
0, acquires a
contribution of the form 1
0 from this divergence, with k0 = c66/Rc(|)2. The latter is
simply the spring constant per unit volume of the restoring force for a rigid translation of
the Bose glass state about mechanical equilibrium. Using the estimate c66 =
ε0nB for the
shear modulus of the vortex lattice[21] yields an effective spring constant per vortex line due
to 2D collective pinning limited by plastic creep, k0/nB, that is given by kp =
ε0/Rc(|)2.
Notice that kp depends only weakly on magnetic field. The Labusch parameter extracted
from the microwave surface resistance on PLD-YBCO films with nano-rod inclusions also
exhibits only a weak dependence on external magnetic field aligned parallel to the nano-rods
(or c-axis)[14]! The above should be compared to the corresponding Labusch parameter due
to point pins[13], which depends strongly on magnetic field. Indeed, given the conjecture
kp = (c66/R
c + c44/L
c)/nB for the Labusch parameter due to 3D collective-pinning implies
that it decays with increasing magnetic field instead as 1/B2 in such case. Here we have
assumed anisotropic scaling, and we have used[9] Rc ∝ B.
We shall next use Eq. (7) for the uniform tilt modulus to test how robust the hexatic Bose
glass is to the addition of point pinning centers. The hexatic Bose glass shows long-range
orientational order in all directions (see fig. 1). The addition of point pins will therefore
break it up into Larkin domains, of dimensions R′c × R′c × Lc transverse and parallel to the
magnetic induction, that tilt in the transverse direction by a distance of order the size of
the vortex core, ξ. Such a break-up then has an elastic energy cost per unit volume and a
pinning energy gain per unit volume that sum to[6][9]
R′2c Lc
f ′0ξ. (8)
Here C66 denotes the shear modulus of the hexatic Bose glass, which can be approximated
by c66 in the limit of weak correlated pinning, while the corresponding tilt modulus C44 is
given by expression (7) evaluated at wavenumber qz = 1/Lc. Also, n
0 denotes the density of
point pins, while f ′0 denotes the magnitude of their characteristic force. Minimizing δu with
respect to the dimensions of the Larkin domains then yields standard results for these[9]:
Lc = Lc(·)[1 + (qzL∗)−2] and R′c = Rc(·)[1 + (qzL∗)−2]1/2, where Lc(·) = 2c44c66ξ2/n′0f ′20 and
Rc(·) = 21/2c1/244 c
2/n′0f
0 are respectively the longitudinal Larkin scale and the transverse
Larkin scale in the absence of correlated pinning centers. The first equation is quadratic in
terms of the variable L−1c , and it has a formal solution L
c = (2Lc(·))−1+[(2Lc(·))−2−L−2∗ ]1/2.
The hexatic Bose glass is therefore robust to the addition of weak point pins. In particular,
the longitudinal Larkin scale Lc remains divergent for Lc(·) > L∗/2, which is equivalent to
the inequality 23/2Rc(·) > Rc(|).
2D-3D CROSSOVER IN CRITICAL CURRENT
The critical current density of the above hexatic Bose glass, which is robust to the addition
of point pins, can be obtained by applying 2D collective pinning theory[10]. All vortex lines
can be considered to be rigid rods. Balancing the Lorentz force against the collective pinning
force over a Larkin domain[6][9] yields the identity JcB/c = [(npf
p + n
p )/R
1/2 for the
product of the critical current density along the film, Jc, with the perpendicular magnetic
field, B, aligned parallel to the material line defects[10]. Here np and n
p denote the density of
vortex lines pinned by material line defects and the density of interstitial vortex lines pinned
by material point defects, while fp and f
p are the maximum force exerted on the respective
vortex lines per unit length. Again, it is important to observe that the critical current
is limited by plastic creep of the vortex lattice due to slip of the quenched-in lines of edge
dislocations along their respective glide planes[19]. Minimization of the sum of the elastic and
pinning energy densities then yields a higher density of Larkin domains in the hexatic Bose
glass with material point defects added by comparison (1): R−2c = C
0 (npf
p )/(c66b)
This reflects the injection of extra lines of edge dislocations that relieve shear stress caused
by point pins. Substitution in turn yields a critical current density, Jc = jc + j
c, that has a
component due to correlated pins set by the identity
jcB/c = C0npf
p/c66b, (9)
and that has a component due to point pins set by the ratio j′c/jc = n
p /npf
p . The
critical current density notably varies as Jc ∝ B−1/2 with magnetic field in the limit of weak
pinning[10].
Consider now a film geometry of thickness τ along the axis of the material line defects.
The forces due to point pins add up statistically along a rigid interstitial vortex line. The
effective pinning force per unit length experienced by an interstitial vortex line is then given
by[7] f ′p = f
1/2 at film thicknesses τ that are much greater than the average separation
τ ′p between such pins along the field direction. Again, f
0 denotes the maximum force exerted
by a point pin. The relative contribution by point pins to the critical current density is then
predicted to show an inverse dependence on film thickness, j′c/jc = τ0/τ , that is set by the
scale τ0 = (n
p/np)(f
p). We thereby obtain the linear dependence on film thickness
I(2D)c = (τ0 + τ)jc for the net critical current per unit width, τJc. Last, comparison of Eq.
(1) with Eq. (9) yields the useful expression
L∗ = γ
−1[(35/4/27/2C0)(ξ · avx)(j0/jc)]1/2 (10)
for the longitudinal scale characteristic of the Bose glass as a function of the bulk critical
current density, jc. Here avx = n
B is the average distance between vortex lines and
j0 is the depairing current density (see ref. [9]). Figure 2 shows a fit to data for the
critical current versus thickness obtained from a thin film of PLD-YBCO at liquid-nitrogen
temperature in 1T magnetic field aligned parallel to the c axis[12]. A bulk critical current
density jc = 0.22MA/cm
is extracted from it. Using a value of j0 = 36MA/cm
for the
depairing current (ref. [12]), of ξ = 11 nm for the coherence length, of γ = 7 for the mass
anisotropy parameter, and setting C0 = 1 yields a longitudinal scale L∗ = 24 nm.
Finally, consider again an arrangement of material line defects that are aligned along the
c axis, that show no voids or clusters in the transverse directions, but that are broken up
into relatively long rods of length L0 ≫ L∗. It can be realized by meandering material line
defects[1][2], where L0 is the correlation length for alignment along the c-axis, or by artificial
nano-rod defects[3][4]. Expression (7) for the divergent tilt modulus indicates proximity to
the regime of 2D collective pinning, where both Lc and L0 are divergent. It hence indicates
that Lc is also large compared to L∗ here. Second, observe that L∗ ∼ Rc(|)/γ coincides
with the longitudinal Larkin scale if the rods are considered to be point defects. The
previous ultimately implies the chain of inequalities L0 ≥ Lc ≫ L∗. They are consistent
with a broken Bose glass state for the vortex lattice, which is threaded by isolated lines
of edge dislocations of length Lc along the c-axis that are connected together by lines of
screw-dislocations[19] along the transverse directions[11]. As in the limiting case of the true
Bose glass state (fig. 1), the lines of edge dislocations are injected into the pristine vortex
lattice in order to relieve shear stress due to the correlated pins. Larkin domains hence
are finite volumes[6][9]. In contrast to their transverse dimensions, however, Larkin domains
exhibit well defined boundaries along the direction parallel to the correlated pinning centers,
across which the vortex lattice slips by a Burger’s vector due to the presence of the screw
dislocations[19]. The critical current density expected from this peculiar example of 3D
collective pinning therefore coincides with that of a Bose glass of thickness Lc: Jc = jc + j
with a bulk component due to interstitial vortex lines j′c/jc = τ0/Lc. The critical current
per unit width then varies with film thickness as I(3D)c = τJc, showing no offset. Figure 2
depicts the predicted dependence on film thickness for the critical current of such a broken
Bose glass. It notably exhibits 2D-3D cross-over at film thicknesses in the vicinity of Lc [11].
CONCLUDING REMARKS
The dependence of the critical current on thickness τ shown by certain films of PLD-
YBCO is in fact consistent with dimensional cross-over at τ ∼ 1µm. A previous attempt
by Gurevich to account for such behavior by collective pinning of individual vortex lines at
point defects yields a longitudinal Larkin scale Lc ∼ 10 nm that is too small, however[12].
We find here, on the other hand, that pinning due to correlated material defects of length L0
yields a Larkin scale Lc that is much longer than that [L∗ = 24 nm, see Eq. (10)] expected
from point pins of comparable strength if L0 is much longer than that scale as well. Indeed,
we predict here that the film thickness at which the critical current crosses over from 2D
to 3D behavior is of order the effective length of the correlated pinning centers when that
length satisfies L0 ≫ L∗.
The author thanks Leonardo Civale, Chandan Das Gupta and Sang-il Kim for discussions.
This work was supported in part by the US Air Force Office of Scientific Research under
grant no. FA9550-06-1-0479.
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FIG. 1: Shown is a Delaunay triangulation of a low-temperature groundstate made up of 2016
vortices that interact logarithmically over a 336 × 336 grid with periodic boundary conditions,
and that experience an equal number of identically weak δ-function pinning centers arranged in
a “liquid” fashion. (See ref. [18].) The state shows macroscopic phase coherence and long-range
hexatic order, with a supefluid density and a hexatic order parameter, respectively, that are 29%
and 58% of the maximum possible values attained by the perfect triangular vortex lattice. It was
obtained after simulated annealing from the liquid state down to low temperature, which resulted
in 371 pinned vortices. A red and green pair of disclinations forms a dislocation (see ref. [19]).
The above Monte Carlo simulation results are consistent with theoretical predictions of a hexatic
vortex glass state in two dimensions, in the zero-temperature limit, for pinning arrangements that
do not show any clusters or voids (ref. [17]). Josephson coupling between layers then produces a
Bose glass transition above zero temperature as long as the 2D glass transtion is second order (ref.
[20]).
-0.1 0 0.1 0.2 0.3 0.4 0.5
film thickness (microns)
PLD-YBCO in liq. N @ H=1T (ref. 12)
2D COLLECTIVE PINNING
3D COLLECTIVE PINNING
2D-3D CROSSOVER
FIG. 2: Shown is the dependence on film thickness predicted for the critical current of a defective
vortex lattice found in a broken Bose glass state (solid line). The dashed and dotted lines are
extrapolated from the 2D and 3D behaviors, respectively. Measurements made by Sang Kim
(circles, ref. [12]) of the critical current on a thin film of PLD-YBCO at liquid nitrogen temperature
subject to 1T magnetic field aligned parallel to the c-axis are fit to the straight dashed line
predicted by 2D collective pinning (ref. [10]). This yields an intercept −τ0 = −69 nm and a
slope jc = 0.22MA/cm
. Although the value of Lc shown here is indeed larger than the lower
bound L∗ = 24nm [see Eq. (10)] and is consistent with the fit to 2D collective pinning, it is only
hypothetical.
Introduction
Tilt modulus of Bose glass
2D-3D crossover in critical current
Concluding remarks
Acknowledgments
References
|
0704.1537 | Estimates for singular integrals and extrapolation | arXiv:0704.1537v1 [math.CA] 12 Apr 2007
ESTIMATES FOR SINGULAR
INTEGRALS AND EXTRAPOLATION
Shuichi Sato
Abstract. In this note, we study singular integrals with rough kernels, which belong
to a class of singular Radon transforms. We prove certain estimates for the singular
integrals that are useful in an extrapolation argument. As an application, we prove
Lp boundedness of the singular integrals under a certain sharp size condition on their
kernels.
1. Introduction
Let Ω be a function in L1(Sn−1) satisfying
(1.1)
Ω(θ) dσ(θ) = 0,
where dσ denotes the Lebesgue surface measure on the unit sphere Sn−1 in Rn. In
this note we assume n ≥ 2. For s ≥ 1, let ∆s denote the collection of measurable
functions h on R+ = {t ∈ R : t > 0} satisfying
‖h‖∆s = sup
∫ 2j+1
|h(t)|s dt/t
where Z denotes the set of integers. We note that ∆s ⊂ ∆t if s > t. In this note
we always assume h ∈ ∆1. Let P (y) = (P1(y), P2(y), . . . , Pd(y)) be a polynomial
mapping, where each Pj is a real-valued polynomial on R
n. We consider a singular
integral operator of the form:
(1.2) T (f)(x) = p. v.
f(x− P (y))K(y) dy = lim
|y|>ǫ
f(x− P (y))K(y) dy,
for an appropriate function f on Rd, where K(y) = h(|y|)Ω(y′)|y|−n, y′ = |y|−1y.
Then, T (f) belongs to a class of singular Radon transforms. See Stein [17], Fan-Pan
[8] and Al-Salman-Pan [1] for this singular integral.
When h = 1 (a constant function), n = d and P (y) = y, we also write T (f) =
S(f). Let f̂(ξ) =
f(x)e−2πi〈x,ξ〉 dx be the Fourier transform of f , where 〈·, ·〉
Key words and phrases. Singular integrals, singular Radon transforms, maximal functions,
extrapolation.
2000 Mathematics Subject Classification. Primary 42B20, 42B25
Typeset by AMS-TEX
http://arxiv.org/abs/0704.1537v1
2 SHUICHI SATO
denotes the inner product in Rd. Then it is known that (Sf )̂ (ξ) = m(ξ′)f̂(ξ),
where
m(ξ′) = −
sgn(〈ξ′, θ〉) + log |〈ξ′, θ〉|
dσ(θ).
Using this, we can show that S extends to a bounded operator on L2 if Ω ∈
L logL(Sn−1), where L logL(Sn−1) denotes the Zygmund class of all those func-
tions Ω on Sn−1 which satisfy
|Ω(θ)| log(2 + |Ω(θ)|) dσ(θ) <∞.
Furthermore, if Ω ∈ L logL(Sn−1), by the method of rotations of Calderón-Zygmund
(see [2]) it can be shown that S extends to a bounded operator on Lp for all
p ∈ (1,∞).
When n = d and P (y) = y, R. Fefferman [10] proved that if h is bounded and Ω
satisfies a Lipschitz condition of positive order on Sn−1, then the singular integral
operator T in (1.2) is bounded on Lp for 1 < p < ∞. Namazi [13] improved this
result by replacing the Lipschitz condition by the condition that Ω ∈ Lq(Sn−1)
for some q > 1. In [7], Duoandikoetxea and Rubio de Francia developed methods
which can be used to study mapping properties of several kinds of operators in
harmonic analysis including the singular integrals considered in [13]. Also, see [6,
22] for weighted Lp boundedness of singular integrals, and [18, 19] for background
materials.
For the rest of this note we assume that the polynomial mapping P in (1.2)
satisfies P (−y) = −P (y) and P 6= 0. We shall prove the following:
Theorem 1. Let Ω ∈ Lq(Sn−1), q ∈ (1, 2] and h ∈ ∆s, s ∈ (1, 2]. Suppose Ω
satisfies (1.1). Let T be as in (1.2). Then we have
‖T (f)‖Lp(Rd) ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd)
for all p ∈ (1,∞), where the constant Cp is independent of q, s,Ω and h. Also, the
constant Cp is independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d).
In Al-Salman-Pan [1], the Lp boundedness of T was proved under the condition
that Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ ∆s for some s > 1 ([1,
Theorem 1.3]). Also it is noted there that estimates like those in Theorem 1 (with
s being fixed) can be used to prove the same result by applying an extrapolation
method, but such estimates are yet to be proved (see [1, p. 156]). In [1], the authors
also considered singular integrals defined by certain polynomial mappings P which
do not satisfy the condition P (−y) = −P (y).
As a consequence of Theorem 1 we can give a different proof of [1, Theorem 1.3]
via an extrapolation method; in fact, we can get an improved result. For a positive
number a and a function h on R+, let
La(h) = sup
∫ 2j+1
|h(r)| (log(2 + |h(r)|))
dr/r.
We define a class La to be the space of all those measurable functions h on R+
which satisfy La(h) <∞. Also, let
Na(h) =
ma2mdm(h),
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 3
where dm(h) = supk∈Z 2
−k|E(k,m)| with E(k,m) = {r ∈ (2k, 2k+1] : 2m−1 <
|h(r)| ≤ 2m} for m ≥ 2, E(k, 1) = {r ∈ (2k, 2k+1] : |h(r)| ≤ 2}. We denote by Na
the class of all those measurable functions h on R+ such that Na(h) < ∞. Then
we readily see that Na(h) < ∞ implies La(h) < ∞. Conversely, if La+b(h) < ∞
for some b > 1, then Na(h) <∞. To see this, note that
2mma+b2−k|E(k,m)| ≤ C
E(k,m)
|h(r)| (log(2 + |h(r)|))
dr/r ≤ CLa+b(h)
for m ≥ 2; thus Na(h) ≤ 2d1(h) + CLa+b(h)
−b < ∞. By Theorem 1 and
an extrapolation method we have the following:
Theorem 2. Suppose Ω is a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1.
Let T be as in (1.2). Then
‖T (f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd)
for all p ∈ (1,∞), where Cp is independent of polynomials Pj if the polynomials are
of fixed degree.
By Theorem 2 and the remark preceding it we see that T is bounded on Lp for
all p ∈ (1,∞) if Ω is as in Theorem 2 and h ∈ La for some a > 2.
When n = d, P (y) = y, Ω is as in Theorem 2 and h is a constant function, it is
known that T is of weak type (1, 1); see [5, 15]. Also, see [4, 9, 11, 12, 16, 20, 21]
for related results.
In Section 2, we shall prove Theorem 1. Applying the methods of [7] involving the
Littlewood-Paley theory and using results of [8, 14], we shall prove Lp estimates
for certain maximal and singular integral operators related to the operator T in
Theorem 1 (Lemmas 1 and 2). Lemma 1 is used to prove Lemma 2. By Lemma 2
we can easily prove Theorem 1. A key idea of the proof of Theorem 1 is to apply a
Littlewood-Paley decomposition adapted to a suitable lacunary sequence depending
on q and s for which Ω ∈ Lq(Sn−1) and h ∈ ∆s. The method of appropriately
choosing the lacunary sequence was inspired by [1], where, in a somewhat different
way from ours, a similar method was used to study several classes of singular
integrals.
We shall prove Theorem 2 in Section 3. Finally, in Section 4, we consider the
maximal operator
(1.3) T ∗(f)(x) = sup
N,ǫ>0
ǫ<|y|<N
f(x− P (y))K(y) dy
where P and K are as in (1.2). We shall prove analogs of Theorems 1 and 2 for the
operator T ∗.
Throughout this note, the letter C will be used to denote non-negative constants
which may be different in different occurrences.
2. Proof of Theorem 1
Let Ω, h be as in Theorem 1. We consider the singular integral T (f) defined
in (1.2). Let ρ ≥ 2 and Ek = {x ∈ R
n : ρk < |x| ≤ ρk+1}. Then T (f)(x) =
−∞ σk ∗ f(x), where {σk} is a sequence of Borel measures on R
d such that
(2.1) σk ∗ f(x) =
f(x− P (y))K(y) dy.
4 SHUICHI SATO
We note that
(σk ∗ f )̂ (ξ) = f̂(ξ)
e−2πi〈P (y),ξ〉K(y) dy.
We write
P (y) =
Qj(y), Qj(y) =
|γ|=N(j)
γ (aγ ∈ R
where Qj 6= 0, 1 ≤ N(1) < N(2) < · · · < N(ℓ), γ = (γ1, . . . , γn) is a multi-
index, yγ = y
1 . . . y
n and |γ| = γ1 + · · · + γn. Let βm = ρ
N(m) and αm =
(q − 1)(s− 1)/(2qsN(m)) for 1 ≤ m ≤ ℓ. Put P (m)(y) =
j=1Qj(y) and define a
sequence µ(m) = {µ
k } of positive measures on R
k ∗ f(x) =
x− P (m)(y)
|K(y)| dy
for m = 1, 2, . . . , ℓ. Also, define µ(0) = {µ
k } by µ
k = (
|K(y)| dy)δ, where δ
is Dirac’s delta function on Rd. For a sequence ν = {νk} of finite Borel measures
on Rd, we define the maximal operator ν∗ by ν∗(f)(x) = supk ||νk| ∗ f(x)|, where
|νk| denotes the total variation. We consider the maximal operators
m ≤ ℓ). We also write
= µ∗ρ.
Lj(ξ) = (〈aγ(j,1), ξ〉, 〈aγ(j,2), ξ〉, . . . , 〈aγ(j,rj), ξ〉),
where {γ(j, k)}
k=1 is an enumeration of {γ}|γ|=N(j) for 1 ≤ j ≤ ℓ. Then Lj is a
linear mapping from Rd to Rrj . Let sj = rankLj. There exist non-singular linear
transformations Rj : R
d → Rd and Hj : R
sj → Rsj such that
Rj(ξ)| ≤ |Lj(ξ)| ≤ C|Hjπ
Rj(ξ)|,
where πdsj (ξ) = (ξ1, . . . , ξsj ) is the projection and C depends only on rj (a proof
can be found in [8]). Let {σ
k } (0 ≤ m ≤ ℓ) be a sequence of Borel measures on
d such that
k ∗ f(x) =
x− P (m)(y)
K(y) dy
for m = 1, 2, . . . , ℓ, while σ
k = 0. Let ϕ ∈ C
0 (R) be supported in {|r| ≤ 1} and
ϕ(r) = 1 for |r| < 1/2. Define a sequence τ (m) = {τ
k } of Borel measures by
(2.2) τ̂
k (ξ) = σ̂
k (ξ)Φk,m(ξ)− σ̂
(m−1)
k (ξ)Φk,m−1(ξ)
for m = 1, 2, . . . , ℓ, where
Φk,m(ξ) =
j=m+1
βkj |Hjπ
Rj(ξ)|
if 0 ≤ m ≤ ℓ− 1 and Φk,ℓ = 1. Then σk = σ
m=1 τ
k . We note that
Φk,m(ξ)ϕ
βkm|Hmπ
Rm(ξ)|
= Φk,m−1(ξ) (1 ≤ m ≤ ℓ).
For 1 ≤ m ≤ ℓ, let T
ρ (f) =
k ∗ f . Then T =
m=1 T
For p ∈ (1,∞) we put p′ = p/(p − 1) and δ(p) = |1/p − 1/p′|. Let θ ∈ (0, 1).
Then we have the following Lp estimates for (µ(m))∗ and T
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 5
Lemma 1. For p > 1 + θ and 0 ≤ j ≤ ℓ, we have
(2.3)
(µ(j))∗(f)
Lp(Rd)
≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s
1− ρ−θ/(2q
)−2/p
‖f‖Lp(Rd).
Lemma 2. For p ∈ (1 + θ, (1 + θ)/θ) and 1 ≤ m ≤ ℓ, we have
‖T (m)ρ (f)‖Lp(Rd) ≤ C(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s
1− ρ−θ/(2q
)−1−δ(p)
‖f‖Lp(Rd).
The constants C in Lemmas 1 and 2 are independent of q, s ∈ (1, 2], Ω ∈
Lq(Sn−1), h ∈ ∆s, ρ and the coefficients of the polynomials Pk (1 ≤ k ≤ d).
We prove Lemma 2 first, taking Lemma 1 for granted for the moment. Let A =
(log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s and B =
1− β−θαmm
1− ρ−θ/(2q
. Then, we
have the following estimates:
(2.4) ‖τ
k ‖ ≤ c1A (‖τ
k ‖ = |τ
k |(R
(2.5) |τ̂
k (ξ)| ≤ c2A
βkm|Lm(ξ)|
(2.6) |τ̂
k (ξ)| ≤ c3A
βk+1m |Lm(ξ)|
(2.7)
(τ (m))∗(f)
≤ CpAB
2/p‖f‖p for p > 1 + θ,
for some constants ci (1 ≤ i ≤ 3) and Cp, where we simply write ‖f‖Lp(Rd) = ‖f‖p.
Now we prove the estimates (2.4)–(2.7). First we see that
k ‖ ≤ C
k ‖+ ‖σ
(m−1)
(2.8)
≤ C‖Ω‖1
∫ ρk+1
|h(r)| dr/r ≤ C(log ρ)‖Ω‖1‖h‖∆1.
From this (2.4) follows. To prove (2.5), define
F (r, ξ) =
Ω(θ) exp(−2πi〈ξ, P (m)(rθ)〉) dσ(θ).
Then, via Hölder’s inequality, for s ∈ (1, 2] we see that
k (ξ)| =
∫ ρk+1
h(r)F (r, ξ) dr/r
(2.9)
∫ ρk+1
|h(r)|s dr/r
)1/s(
∫ ρk+1
|F (r, ξ)|
)1/s′
≤ C(log ρ)1/s‖h‖∆s‖Ω‖
(s′−2)/s′
∫ ρk+1
|F (r, ξ)|
)1/s′
We need the following estimates for the last integral:
6 SHUICHI SATO
Lemma 3. Let 1 < q ≤ 2 and Ω ∈ Lq(Sn−1). Then there exists a constant C > 0
independent of q, ρ,Ω and the coefficients of the polynomial components of P (m)
such that
∫ ρk+1
|F (r, ξ)|
dr/r ≤ C(log ρ)
βkm|Lm(ξ)|
)−1/(2q′N(m))
‖Ω‖2q.
Proof. Take an integer ν such that 2ν < ρ ≤ 2ν+1. By the proof of Proposition 5.1
of [8] we have
∫ ρk+1
|F (r, ξ)|
dr/r =
∣F (ρkr, ξ)
dr/r ≤
∫ 2j+1
∣F (ρkr, ξ)
∣F (2jρkr, ξ)
)2/q′
2jN(m)ρkN(m)|Lm(ξ)|
)−1/(2N(m)q′)
‖Ω‖2q
≤ C(log ρ)
ρkN(m)|Lm(ξ)|
)−1/(2N(m)q′)
‖Ω‖2q.
This completes the proof of Lemma 3.
By (2.9) and Lemma 3 we have |σ̂
k (ξ)| ≤ CA
βkm|Lm(ξ)|
. Also, we have
(m−1)
k ‖ ≤ CA by (2.8). We can prove the estimate (2.5) by using these estimates
in the definition of τ
k in (2.2) and by noting that ϕ is compactly supported.
Next, to prove (2.6), using (1.1) when m = 1, we see that
k (ξ)| ≤
k (ξ)− σ̂
(m−1)
k (ξ)
Φk,m(ξ)
(Φk,m(ξ) − Φk,m−1(ξ)) σ̂
(m−1)
k (ξ)
≤ C‖Ω‖1β
m |Lm(ξ)|
∫ ρk+1
|h(r)| dr/r + C‖σ
(m−1)
m|Lm(ξ)|
≤ C(log ρ)‖Ω‖1‖h‖∆1β
m |Lm(ξ)|,
where to get the last inequality we have used (2.8). By this and (2.8), we have
k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1
βk+1m |Lm(ξ)|
for all c ∈ (0, 1], which implies (2.6). Finally, the estimate (2.7) follows from Lemma
1 since
(τ (m))∗(f)
(µ(m))∗(|f |)
(µ(m−1))∗(|f |)
≤ CAB2/p‖f‖p
for p > 1 + θ, where the first inequality can be seen by change of variables and a
well-known result on maximal functions (see [8]).
Let {ψk}
−∞ be a sequence of functions in C
∞((0,∞)) such that
supp(ψk) ⊂ [β
m , β
ψk(t)
2 = 1, |(d/dt)jψk(t)| ≤ cj/t
j (j = 1, 2, . . . ),
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 7
where the constants cj are independent of βm. Define an operator Sk by (Sk(f)) (̂ξ) =
Rm(ξ)|
f̂(ξ) and let
j (f) =
k ∗ Sj+k(f)
Then by Plancherel’s theorem and the estimates (2.4)–(2.6) we have
j (f)
D(j+k)
k (ξ)|
2|f̂(ξ)|2 dξ
(2.10)
≤ CA2 min
1, β−2αm(|j|−2)m
D(j+k)
|f̂(ξ)|2 dξ
≤ CA2 min
1, β−2αm(|j|−2)m
‖f‖22,
where D(k) = {β−k−1m ≤ |Hmπ
Rm(ξ)| ≤ β
Applying the proof of Lemma in [7, p. 544] and using the estimates (2.4) and
(2.7), we can prove the following.
Lemma 4. Let u ∈ (1 + θ, 2]. Define a number v by 1/v− 1/2 = 1/(2u). Then we
have the vector valued inequality
(2.11)
k ∗ gk|
≤ (c1Cu)
1/2AB1/u
where the constants c1 and Cu are as in (2.4) and (2.7), respectively.
By the Littlewood-Paley theory we have
j (f)‖p ≤ cp
k ∗ Sj+k(f)|
,(2.12)
|Sk(f)|
≤ cp‖f‖p,(2.13)
where 1 < p < ∞ and cp is independent of βm and the linear transformations
Rm, Hm. Suppose that 1 + θ < p ≤ 4/(3− θ). Then we can find u ∈ (1 + θ, 2] such
that 1/p = 1/2 + (1 − θ)/(2u). Let v be defined by u as in Lemma 4. Then by
(2.11)–(2.13) we have
(2.14) ‖V
j (f)‖v ≤ CAB
1/u‖f‖v.
Since 1/p = θ/2 + (1− θ)/v, interpolating between (2.10) and (2.14), we have
j (f)‖p ≤ CAB
(1−θ)/u min
1, β−θαm(|j|−2)m
‖f‖p.
8 SHUICHI SATO
It follows that
‖T (m)ρ (f)‖p ≤
j (f)‖p ≤ CAB
(1−θ)/u(1− β−θαmm )
−1‖f‖p
(2.15)
≤ CAB2/p‖f‖p,
where we have used the inequality
−θαm(|j|−2)
1− β−θαmm
We also have ‖T
ρ (f)‖2 ≤
j (f)‖2 ≤ CAB‖f‖2 by (2.10), since B ≥
(1− β−αmm )
. By duality and interpolation, we can now get the conclusion of
Lemma 2.
Next, we give a proof of Lemma 1. We prove Lemma 1 by induction on j. Now
we assume (2.3) for j = m − 1, 1 ≤ m ≤ ℓ, and prove (2.3) for j = m. Let
ϕ ∈ C∞0 (R) be as above. Define a sequence η
(m) = {η
k } of Borel measures on R
k (ξ) = ϕ
βkm|Hmπ
Rm(ξ)|
(m−1)
k (ξ).
Then by (2.3) with j = m− 1, we have
(2.16)
(η(m))∗(f)
(µ(m−1))∗(f)
≤ CAB2/p‖f‖p
for p > 1 + θ. Furthermore, we have the following:
k ‖+ ‖µ
k ‖ ≤ C‖µ
(m−1)
k ‖+ ‖µ
k ‖ ≤ C‖Ω‖1
∫ ρk+1
|h(r)| dr/r
(2.17)
≤ C(log ρ)‖Ω‖1‖h‖∆1 ≤ CA,
k (ξ)− η̂
k (ξ)| ≤ C(log ρ)‖Ω‖1‖h‖∆1
βk+1m |Lm(ξ)|
(2.18)
βk+1m |Lm(ξ)|
(2.19) |µ̂
k (ξ)| ≤ CA
βkm|Lm(ξ)|
(2.20) |η̂
k (ξ)| ≤ C(log ρ)‖h‖∆1‖Ω‖1
βkm|Lm(ξ)|
βkm|Lm(ξ)|
To see (2.18) we note that
k (ξ)−η̂
k (ξ)| ≤ |µ̂
k (ξ)−µ̂
(m−1)
k (ξ)|+
βkm|Hmπ
Rm(ξ)|
(m−1)
k (ξ)
Thus arguing as in the proof of (2.6), we have the first inequality of (2.18). The
estimate (2.19) follows from the arguments used to prove (2.5). Also, we can see
the first inequality of (2.20) by the definition of η
k and (2.17).
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 9
Since ‖(µ(m))∗(f)‖∞ ≤ CA‖f‖∞, by taking into account an interpolation, it
suffices to prove (2.3) with j = m for p ∈ (1+θ, 2]. Define a sequence ν(m) = {ν
of Borel measures by ν
k = µ
k − η
k . Let
gm(f)(x) =
k ∗ f(x)
(2.21) (µ(m))∗(f) ≤ gm(f) + (η
(m))∗(|f |).
Thus, by (2.16), to get (2.3) with j = m it suffices to prove ‖gm(f)‖p ≤ CAB
2/p‖f‖p
for p ∈ (1 + θ, 2] with an appropriate constant C. By a well-known property of
Rademacher’s functions, this follows from
(2.22)
U (m)ǫ (f)
≤ CAB2/p‖f‖p
for p ∈ (1 + θ, 2], where U
ǫ (f) =
k ǫkν
k ∗ f with ǫ = {ǫk}, ǫk = 1 or −1, and
the constant C is independent of ǫ.
The estimate (2.22) is a consequence of the following:
Lemma 5. We define a sequence {pj}
1 by p1 = 2 and 1/pj+1 = 1/2+(1−θ)/(2pj)
for j ≥ 1. (We note that 1/pj = (1 − a
j)/(1 + θ), where a = (1 − θ)/2, so {pj} is
decreasing and converges to 1 + θ.) Then, for j ≥ 1 we have
U (m)ǫ (f)
≤ CjAB
2/pj ‖f‖pj .
Proof. Let
j (f) =
ǫkSj+k
k ∗ Sj+k(f)
Then by Plancherel’s theorem and the estimates (2.17)–(2.20), as in (2.10) we have
(2.23)
j (f)
≤ CAmin
1, β−αm(|j|−2)m
‖f‖2.
It follows that
ǫ (f)
j (f)‖2 ≤ CAB‖f‖2. If we denote by A(s)
the assertion of Lemma 5 for j = s, this proves A(1).
Now we derive A(s+1) from A(s) assuming that A(s) holds, which will complete
the proof of Lemma 5 by induction. Using (2.21), we see that
(ν(m))∗(f) ≤ (µ(m))∗(|f |) + (η(m))∗(|f |) ≤ gm(|f |) + 2(η
(m))∗(|f |).
Note that A(s) implies ‖gm(f)‖ps ≤ CAB
2/ps‖f‖ps . By this and (2.16) we have
(2.24)
(ν(m))∗(f)
≤ ‖gm(|f |)‖ps + 2
(η(m))∗(|f |)
≤ CAB2/ps‖f‖ps .
10 SHUICHI SATO
By (2.17), (2.23) and (2.24), we can now apply the arguments used in the proof of
(2.15) to get A(s+ 1). This completes the proof of Lemma 5.
Now we prove (2.22) for p ∈ (1 + θ, 2]. Let {pj}
1 be as in Lemma 5. Then we
have pN+1 < p ≤ pN for some N . Thus, interpolating between the estimates of
Lemma 5 for j = N and j = N + 1, we have (2.22). This proves (2.3) for j = m.
Finally, we can easily see that (µ(0))∗(f) ≤ C(log ρ)‖Ω‖1‖h‖∆1|f | (see (2.17)),
which implies the estimate (2.3) for j = 0. Therefore, by induction we have (2.3)
for all 0 ≤ j ≤ ℓ. This completes the proof of Lemma 1.
Now we can prove Theorem 1. Since θ ∈ (0, 1) is arbitrary, by taking ρ = 2q
in Lemma 2 we have
(f)‖p ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p
for all p ∈ (1,∞). This completes the proof of Theorem 1, since T =
m=1 T
3. Proof of Theorem 2
Theorem 2 can be proved by Theorem 1 and an extrapolation argument. Let
T (f) be the singular integral in (1.2). We also write T (f) = Th,Ω(f). We fix
q ∈ (1, 2], Ω ∈ Lq(Sn−1), p ∈ (1,∞) and a function f with ‖f‖p ≤ 1 and put
S(h) = ‖Th,Ω(f)‖p. Then we have the following subadditivity:
(3.1) S(h+ k) ≤ S(h) + S(k).
Set E1 = {r ∈ R+ : |h(r)| ≤ 2} and Em = {r ∈ R+ : 2
m−1 < |h(r)| ≤ 2m} for
m ≥ 2. Then, applying Theorem 1, we see that
(3.2) S (hχEm) ≤ C(q − 1)
−1(s− 1)−1‖Ω‖q‖hχEm‖∆s
for s ∈ (1, 2], where χE denotes the characteristic function of a set E. Now we follow
the extrapolation argument of Zygmund [23, Chap. XII, pp. 119–120]. First, note
‖hχEm‖∆1+1/m ≤ 2
mdm/(m+1)m (h)
for m ≥ 1, where dm(h) is as in Section 1. Using this and (3.2) we see that
S (hχEm) ≤ C(q − 1)
−1‖Ω‖q
m‖hχEm‖∆1+1/m
≤ C(q − 1)−1‖Ω‖q
m2mdm/(m+1)m (h).
Recalling the definition of Na(h), we have
m2mdm/(m+1)m (h) =
dm(h)<3−m
m2mdm/(m+1)m (h) +
dm(h)≥3−m
m2mdm/(m+1)m (h)
m2m3−m
2/(m+1) +
m2mdm(h)3
m/(m+1) ≤ C(1 +N1(h)).
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 11
Therefore, by (3.1) we see that
(3.3) S(h) ≤
S (hχEm) ≤ C(q − 1)
−1‖Ω‖q (1 +N1(h)) .
Next, fix h ∈ N1, p ∈ (1,∞) and f with ‖f‖p ≤ 1 and let R(Ω) = ‖Th,Ω(f)‖p.
Put em = σ(Fm) for m ≥ 1, where Fm = {θ ∈ S
n−1 : 2m−1 < |Ω(θ)| ≤ 2m} for
m ≥ 2 and F1 = {θ ∈ S
n−1 : |Ω(θ)| ≤ 2}. We decompose Ω as Ω =
m=1 Ωm,
where Ωm = ΩχFm − σ(S
n−1)−1
Ω dσ. We note that
Ωm dσ = 0, ‖Ωm‖r ≤
m for 1 < r <∞. Now, by (3.3) and the subadditivity of R(Ω) we see that
R(Ω) ≤
R(Ωm) ≤ C (1 +N1(h))
m‖Ωm‖1+1/m
≤ C (1 +N1(h))
m2mem/(m+1)m = C (1 +N1(h))
em<3−m
em≥3−m
≤ C (1 +N1(h))
m2m3−m
2/(m+1) +
m2mem3
m/(m+1)
≤ C (1 +N1(h)))
|Ω(θ)| log(2 + |Ω(θ)|) dσ(θ)
This completes the proof of Theorem 2.
4. Estimates for maximal functions
For the maximal operator T ∗ in (1.3) we have a result similar to Theorem 1.
Theorem 3. Let q ∈ (1, 2], s ∈ (1, 2] and Ω ∈ Lq(Sn−1), h ∈ ∆s. Suppose Ω
satisfies (1.1). Then we have
‖T ∗(f)‖Lp(Rd) ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖Lq(Sn−1)‖h‖∆s‖f‖Lp(Rd)
for all p ∈ (1,∞), where Cp is independent of q, s, Ω and h.
As Theorem 1 implies Theorem 2, we have the following as a consequence of
Theorem 3.
Theorem 4. Let Ω be a function in L logL(Sn−1) satisfying (1.1) and h ∈ N1.
‖T ∗(f)‖Lp(Rd) ≤ Cp‖f‖Lp(Rd)
for all p ∈ (1,∞).
As in the cases of Theorems 1 and 2, the constants Cp of Theorems 3 and 4 are
also independent of polynomials Pj if we fix deg(Pj) (j = 1, 2, . . . , d). When Ω is
as in Theorem 4 and h ∈ ∆s for some s > 1, the L
p boundedness of T ∗ was proved
in [1]. When n = d, P (y) = y, Ω ∈ Lq for some q > 1 and h is bounded, the Lp
boundedness of T ∗ is due to [3].
We use the following to prove Theorem 3.
12 SHUICHI SATO
Lemma 6. Let τ (m) = {τ
k } (1 ≤ m ≤ ℓ), where the measures τ
k are as
in (2.2). Let θ ∈ (0, 1) and let positive numbers A = (log ρ)‖Ω‖Lq(Sn−1)‖h‖∆s,
1− β−θαmm
be as above. We define
(4.1) T ∗ρ,m(f)(x) = sup
j ∗ f(x)
Then, for p ∈ (2(1 + θ)/(θ2 − θ + 2), (1 + θ)/θ) =: Iθ we have
‖T ∗ρ,m(f)‖p ≤ CA
B1+δ(p) +B2/p+1−θ/2
‖f‖p,
where C is independent of q, s ∈ (1, 2], Ω ∈ Lq(Sn−1), h ∈ ∆s, ρ and the coefficients
of the polynomials Pj (1 ≤ j ≤ d).
Proof. Let T
ρ (f) =
k ∗ f be as in Lemma 2. Let a function ϕ be as in the
definition of τ
k in (2.2). Define ϕk by ϕ̂k(ξ) = ϕ
βkm|Hmπ
Rm(ξ)|
. Let δ be
the delta function as above. Following [8], we decompose
j ∗ f = ϕk ∗ T
ρ (f)− ϕk ∗
j ∗ f
+ (δ − ϕk) ∗
j ∗ f
It follows that
(4.2) T ∗ρ,m(f) ≤ sup
ϕk ∗ T
ρ (f)
j (f),
whereN
j (f) = supk
k−j−1 ∗ f
+supk
(δ − ϕk) ∗
j+k ∗ f
. By Lemma
2 we have
(4.3)
ϕk ∗ T
ρ (f)
≤ CAB1+δ(p)‖f‖p for p ∈ (1 + θ, (1 + θ)/θ).
Also, by (2.7) we see that
(4.4) ‖N
j (f)‖r ≤ CAB
2/r‖f‖r for r > 1 + θ.
On the other hand, we have
j (f) ≤
(δ − ϕk) ∗
j+k ∗ f
k−j−1 ∗ f
Therefore, by the estimates (2.5), (2.6) and Plancherel’s theorem, as in [8, p. 820]
we see that
(4.5) ‖N
j (f)‖2 ≤ CAβ
1− β−2αmm
)−1/2
‖f‖2.
ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION 13
For p ∈ Iθ we can find r ∈ (1 + θ, 2(1 + θ)/θ) such that 1/p = (1 − θ)/r + θ/2, so
an interpolation between (4.4) and (4.5) implies that
(4.6) ‖N
j (f)‖p ≤ CAB
2(1−θ)/r
1− β−2αmm
)−θ/2
β−αmθjm ‖f‖p.
Therefore, by (4.2), (4.3) and (4.6), for p ∈ Iθ we have
‖T ∗ρ,m(f)‖p ≤ CA
B1+δ(p) +B2(1−θ)/r+1
1− β−2αmm
)−θ/2
‖f‖p.
This implies the conclusion of Lemma 6, since
1− β−2αmm
≤ B and 2(1−θ)/r+
θ/2 + 1 = 2/p+ 1− θ/2.
Proof of Theorem 3. Note that T ∗(f) ≤ 2T ∗0 (f) + 2µ
ρ(|f |), where T
0 (f) is defined
by the formula in (4.1) with {τ
j } replaced by the sequence {σj} of measures in
(2.1) and µ∗ρ = (µ
(ℓ))∗ is as in Lemma 1. We note that T ∗0 (f) ≤
m=1 T
ρ,m(f).
Now, Lemma 6 implies that
‖T ∗ρ,m(f)‖p ≤ C(log ρ)
1− ρ−θ/(2q
‖Ω‖q‖h‖∆s‖f‖p
for p ∈ Iθ. By using this with ρ = 2
q′s′ , since θ ∈ (0, 1) is arbitrary, we can conclude
′s′ ,m
(f)‖p ≤ Cp(q − 1)
−1(s− 1)−1‖Ω‖q‖h‖∆s‖f‖p
for p ∈ (1,∞). Also, by Lemma 1 µ∗ρ satisfies a similar estimate when ρ = 2
q′s′ .
Collecting results, we have Theorem 3.
Remark. Let
M(f)(x) = sup
|y|<t
|f(x− P (y))||Ω(y′)||h(|y|)| dy.
It is easy to see that M(f) ≤ Cµ∗ρ(f), where C is independent of ρ ≥ 2. Therefore,
by Lemma 1 we can prove results similar to Theorems 1 and 2 for the maximal
operator M . In [1], Lp boundedness of M was proved under the condition that
Ω ∈ L logL(Sn−1) and h ∈ ∆s for some s > 1. When n = d, P (y) = y, it is known
that M is of weak type (1, 1) if Ω ∈ L logL(Sn−1) and h is bounded (see [5, 4]).
References
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(2001), 381–415.
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Univ. Press, Princeton, NJ, 1971.
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Math. J. 48 (1999), 1547–1584.
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New York and Melbourne, 1977.
Department of Mathematics
Faculty of Education
Kanazawa University
Kanazawa, 920-1192
Japan
E-mail address: [email protected]
|
0704.1538 | Rounding of first-order phase transitions and optimal cooperation in
scale-free networks | Rounding of first-order phase transitions and optimal cooperation in scale-free
networks
M. Karsai,1,2 J-Ch. Anglès d’Auriac,2 and F. Iglói3, 1
Institute of Theoretical Physics, Szeged University, H-6720 Szeged, Hungary
Centre de Recherches sur les Trés Basses Températures, B. P. 166, F-38042 Grenoble, France
Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.Box 49, Hungary
(Dated: November 20, 2018)
We consider the ferromagnetic large-q state Potts model in complex evolving networks, which is
equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum
of pair cooperation benefits and the supports of independent projects. The agents are found to be
typically of two kinds: a fraction of m (being the magnetization of the Potts model) belongs to a
large cooperating cluster, whereas the others are isolated one man’s projects. It is shown rigorously
that the homogeneous model has a strongly first-order phase transition, which turns to second-order
for random interactions (benefits), the properties of which are studied numerically on the Barabási-
Albert network. The distribution of finite-size transition points is characterized by a shift exponent,
1/ν̃′ = .26(1), and by a different width exponent, 1/ν′ = .18(1), whereas the magnetization at the
transition point scales with the size of the network, N , as: m ∼ N−x, with x = .66(1).
I. INTRODUCTION
Complex networks have been used to describe the
structure and topology of a large class of systems in dif-
ferent fields of science, technics, transport, social and
political life, etc, see Refs.1,2,3,4 for recent reviews. A
complex network is represented by a graph5, in which the
nodes stand for the agents and the edges denote the possi-
ble interactions. Realistic networks generally have three
basic properties. The average distance between the nodes
is small, which is called the small-world effect6. There is a
tendency of clustering and the degree-distribution of the
edges, P (k), has a power-law tail, PD(k) ≃ Ak
−γ , k ≫ 1.
Thus the edge distribution is scale free7, which is usually
attributed to growth and preferential attachment during
the evaluation of the network.
In reality there is some sort of interaction between the
agents of a network which leads to some kind of coop-
erative behavior in macroscopic scales. One throughly
studied question in this field is the spread of infections
and epidemics in networks8,9,10, which problem is closely
related to another non-equilibrium processes, such as
percolation11, diffusion12, the contact process13 or the
zero-range process14, etc. In another investigations one
considers simple magnetic models15,16,17,18,19, in which
the agents are represented by classical (Ising or Potts)
spin variables, the interactions are described by ferro-
magnetic couplings, whereas the temperature plays the
rôle of a disordering field.
In theoretical investigations of the cooperative behav-
ior one usually resort on some kind of approximations.
For example the sites of the networks are often consid-
ered uncorrelated, which is generally not true for evolv-
ing networks, such as the Barabási-Albert (BA) network.
However this effect is expected to be irrelevant, as far as
the singularities in the system are considered. Also the
simple mean-field approach could lead to exact results
due to long-range interactions in the networks, which has
been checked by numerical simulations15 and by another,
more accurate theoretical methods19 (Bethe-lattice ap-
proach, replica method, etc.). In these calculations the
critical behavior of the network is found to depend on the
value of the degree exponent, γ. For sufficiently weakly
connected networks with γ > γu (γu = 5 for the Ising
model) there are conventional mean-field singularities. In
the intermediate or unconventional mean-field regime, for
γu > γ > γc, the critical exponents are γ dependent. Fi-
nally, for γ ≤ γc, when the average of k
2, defined by
〈k2〉 =
PD(k)k
2dk, as well as the strength of the aver-
age interaction becomes divergent the scale-free network
remains in the ordered state at any finite temperature.
Since γc = 3, in realistic networks with homogeneous
interactions always this type of phenomena is expected
to occur. In weighted networks, however, in which the
strengths of the interaction is appropriately rescaled with
the degrees of the connected vertices, γc is shifted to
larger values and therefore the complete phase-transition
scenario can be tested13,19. We note that the properties
of the phase transitions are generally different for undi-
rected (as we consider here) and directed networks20.
In several models the phase transition in regular lat-
tices is of first order, such as for the q-state Potts model
for sufficiently large value of q. Putting these models on a
complex network the inhomogeneities of the lattice play
the rôle of some kind of disorder and it is expected that
the value of the latent heat is reduced or even the tran-
sition is smoothened to a continuous one. This type of
scenario is indeed found in a mean-field treatment17, in
which the transition is of first-order for γ > γ(q) and be-
comes continuous for γc < γ < γ(q), where 3 < γ(q) < 4.
On the other hand in an effective medium Bethe lattice
approach one has obtained γ(q) = 3, thus the unconven-
tional mean-field regime is absent in this treatment18.
The interactions considered so far were homogeneous,
however, in realistic situations the disorder is inevitable,
which has a strong influence on the properties of the
http://arxiv.org/abs/0704.1538v1
phase transition. In regular lattices and for a second-
order transition Harris-type relevance-irrelevance crite-
rion can be used to decide about the stability of the pure
system’s fixed point in the presence of weak disorder. On
the contrary for a first-order transition such type of cri-
terion does not exist. In this case rigorous results asserts
that in two dimensions (2d) for any type of continuous
disorder the originally first order transition softens into a
second order one21. In three dimensions there are numer-
ical investigations which have shown22,23,24,25,26 that this
kind of softening takes place only for sufficiently strong
disorder.
In this paper we consider interacting models with ran-
dom interactions on complex networks and in this way we
study the combined effect of network topology and bond
disorder. The particular model we consider is the ran-
dom bond ferromagnetic Potts model (RBPM) for large
value of q. This model besides its relevance in ordering-
disordering phenomena and phase transitions has an
exact relation with an optimal cooperation problem27.
This mapping is based on the observation that in the
large-q limit the thermodynamic properties of the sys-
tem are dominated by one single diagram28 of the high-
temperature expansion29 and its calculation is equiva-
lent to the solution of an optimization problem. This
optimization problem can be interpreted in terms of co-
operating agents which try to maximize the total sum of
benefits received for pair cooperations plus a unit sup-
port which is paid for each independent projects. For a
given realization of the interactions the optimal state is
calculated exactly by a combinatorial optimization algo-
rithm which works in strongly polynomial time27. The
optimal graph of this problem consists of connected com-
ponents (representing sets of cooperating agents) and
isolated sites and its temperature (support) dependent
topology contains information about the collective be-
havior of the agents. In the thermodynamic limit one
expects to have a sharp phase transition in the system,
which separates the ordered (cooperating) phase with a
giant clusters from a disordered (non-cooperating) phase,
having only clusters of finite extent.
The structure of the paper is the following. The model
and the optimization method used in the study for large
q is presented in Sec. II. The solution for homogeneous
non-random evolving networks can be found in Sec. III,
whereas numerical study of the random model on the
Barabási-Albert network is presented in Sec. IV. Our
results are discussed in Sec. V.
II. THE MODEL AND ITS RELATION WITH
OPTIMAL COOPERATION
The q-state Potts model30 is defined by the Hamilto-
nian:
H = −
〈i,j〉
Jijδ(σi, σj) (1)
in terms of the Potts-spin variables, σi = 0, 1, · · · , q − 1.
Here i and j are sites of a lattice, which is represented by
a complex network in our case and the summation runs
over nearest neighbors, i.e. pairs of connected sites.
The couplings, Jij > 0, are ferromagnetic and they
are either identical, Jij = J , which is the case of homo-
geneous networks, or they are identically and indepen-
dently distributed random variables. In this paper we
use a quasi-continuous distribution:
P (Jij) =
1 + ∆
2i− l − 1
− Jij
which consists of large number of l equally spaced discrete
values within the range J(1 ± ∆/2) and 0 ≤ ∆ ≤ 2
measures the strength of disorder.
For a given set of couplings the partition function of
the system is convenient to write in the random cluster
representation29 as:
qc(G)
qβJij − 1
where the sum runs over all subset of bonds, G and c(G)
stands for the number of connected components of G. In
Eq. (3) we use the reduced temperature, T → T ln q and
its inverse β → β/ ln q, which are of O(1) even in the
large-q limit31. In this limit we have qβJij ≫ 1 and the
partition function can be written as
qφ(G), φ(G) = c(G) + β
Jij (4)
which is dominated by the largest term, φ∗ = maxG φ(G).
Note that this graph, which is called the optimal set,
generally depends on the temperature. The free-energy
per site is proportional to φ∗ and given by −βf = φ∗/N
where N stands for the number of sites of the lattice.
As already mentioned in the introduction the optimiza-
tion in Eq. (4) can be interpreted as an optimal coopera-
tion problem27 in which the agents, which cooperate with
each other in some projects, form connected components.
Each cooperating pair receives a benefit represented by
the weight of the connecting edge (which is proportional
to the inverse temperature) and also there is a unit sup-
port to each component, i.e. for each projects. Thus by
uniting two projects the support will be reduced but at
the same time the edge benefits will be enhanced. Finally
one is interested in the optimal form of cooperation when
the total value of the project grants is maximal.
In a mathematical point of view the cost-function in
Eq. (4), −φ(G), is sub-modular32 and there is an effi-
cient combinatorial optimization algorithm which calcu-
lates the optimal set (i.e. set of bonds which minimizes
the cost-function) exactly at any temperature in strongly
polynomial time27. In the algorithm the optimal set is
calculated iteratively and at each step one new vertex of
the lattice is taken into account. Having the optimal set
at a given step, say with n vertices, its connected compo-
nents have the property to contain all the edges between
their sites. Due to the submodularity of −φ(G) each con-
nected component is contracted into a new vertex with
effective weights being the sum of individual weights in
the original representation. Now adding a new vertex one
should solve the optimization problem in terms of the ef-
fective vertices, which needs the application of a standard
maximum flow algorithm, since any contractions should
include the new vertex. After making the possible new
contractions one repeats the previous steps until all the
vertices are taken into account and the optimal set of the
problem is found.
This method has already been applied for 2d31,33 and
3d25,26 regular lattices with short range random inter-
actions. As a general result the optimal graph at low
temperatures is compact and the largest connected sub-
graph contains a finite fraction of the sites, m(T ), which
is identified by the orderparameter of the system. In the
other limit, for high temperature, most of the sites in the
optimal set are isolated and the connected clusters have
a finite extent, the typical size of which is used to define
the correlation length, ξ. Between the two phases there is
a sharp phase transition in the thermodynamic limit, the
order of which depends on the dimension of the lattice
and the strength of disorder, ∆.
In the following the optimization problem is solved ex-
actly for homogeneous evolving networks in Sec.III and
studied numerically in random Barabási-Albert networks
in Sec.IV.
III. EXACT SOLUTION FOR HOMOGENEOUS
EVOLVING NETWORKS
In regular d-dimensional lattices the solution of the
optimization problem in Eq. (4) is simple, since there are
only two distinct optimal sets, which correspond to the
T = 0 and T → ∞ solutions, respectively. For T <
Tc(0) it is the fully connected diagram, E, with a free-
energy: −βNf = 1 + NβJd and for T > Tc(0) it is the
empty diagram, Ø, with −βNf = N . In the proof we
make use of the fact, that any edge of a regular lattice,
e1, can be transformed to any another edge, e2, through
operations of the automorphy group of the lattice. Thus
if e1 belongs to some optimal set, then e2 belongs to
an optimal set, too. Furthermore, due to submodularity
the union of optimal sets is also an optimal set, from
which follows that at any temperature the optimal set
is either Ø or E. By equating the free energies in the
two phases we obtain for the position of the transition
point: Tc(0) = Jd/(1 − 1/N) whereas the latent heat is
maximal: ∆e/Tc(0) = 1− 1/N .
In the following we consider the optimization problem
in homogeneous evolving networks which are generated
by the following rules:
• we start with a complete graph with 2µ vertices
• at each timestep we add a new vertex
• which is connected to µ existing vertices.
In definition of these networks there is no restric-
tion in which way the µ existing vertices are selected.
These could be chosen randomly, as in the Erdős-Rényi
model34, or one can follow some defined rule, like the
preferential attachment in the BA network7. In the
following we show that for such networks the phase-
transition point is located at Tc(0) = Jµ and for T <
Tc(0) (T > Tc(0)) the optimal set is the fully connected
diagram (empty diagram), as for the regular lattices.
Furthermore, the latent heat is maximal: ∆e/Tc(0) = 1.
In the proof we follow the optimal cooperation
algorithm27 outlined in Sec.II, and in application of the
algorithm we add the vertices one by one in the same or-
der as in the construction of the network. First we note,
that the statement is true for the initial graph, which is a
complete graph, thus the optimal set can be either fully
connected, having a free-energy: −β2µf = 1 + µ(2µ −
1)βJd, or empty, having a free-energy: −β2µf = 2µ,
thus the transition point is indeed at T = Tc(0). We
suppose then that the property is satisfied after n steps
and add a new vertex, v0. Here we investigate the two
cases, T ≤ Tc(0) and T ≥ Tc(0) separately.
• If T ≤ Tc(0), then according to our statement all
vertices of the original graph are contracted into
a single vertex, s, which has an effective weight,
µ × J/T > µJ/Tc(0) = 1, to the new vertex v0.
Consequently in the optimal set s and v0 are con-
nected, in accordance with our statement.
• If T ≥ Tc(0), then all vertices of the original graph
are disconnected, which means that for any subset,
S, having ns ≤ n vertices and es edges one has:
ns ≥ esJT + 1. Let us denote by µs ≤ µ the
number of edges between v0 and the vertices of S.
One has µsJ/T ≤ µJ/T ≤ µJ/Tc(0) = 1, so that
for the composite S+ v0 we have: ns+1 ≥ esJT +
1+µsJ/T , which proves that the vertex v0 will not
be connected to any subset S and thus will not be
contracted to any vertex.
This result, i.e. a maximally first-order transition of
the large-q state Potts model holds for a wide class of
evolving networks, which satisfy the construction rules
presented above. This is true, among others, for ran-
domly selected sites, for the BA evolving network which
has a degree exponent γ = 3 and for several generaliza-
tions of the BA network1 including nonlinear preferential
attachment, initial attractiveness, etc. In these latter
network models the degree exponent can vary in a range
of 2 < γ < ∞. It is interesting to note that for uncorre-
lated random networks with a given degree distribution
the q-state Potts model is in the ordered phase17,18 for
any γ ≤ 3. This is in contrast to evolving networks in
which correlations in the network sites results in the ex-
istence of a disordered phase for T > Tc(0), at least for
large q.
3.12.92.72.5
Tc(∞)
N=256
N=512
N=1024
N=2048
N=4096
0 0.5 1 1.5 2 2.5 3
∆=0.2
∆=0.8
∆=1.4
∆=2.0
FIG. 1: (Color online) Temperature dependence of the av-
erage magnetization in a BA network of N = 1024 sites for
different strength of the disorder, ∆. At T = Tc(0) = 2 the
magnetization is independent of ∆ > 0 and its value is in-
dicated by an arrow. Inset: The average magnetization for
uniform disorder, ∆ = 2, close to the transition point for dif-
ferent finite sizes. The arrow indicates the critical point of
the infinite system.
IV. NUMERICAL STUDY OF RANDOM
BARABÁSI-ALBERT NETWORKS
In this section we study the large-q state Potts model
in the BA network with a given value of the connectiv-
ity, µ = 2, and the size of the network varies between
N = 26 to N = 212. The interactions are independent
random variables taken from the quasi-continuous distri-
bution in Eq.(2) having l = 1024 discrete peaks and we fix
J = 1. The advantage of using quasi-continuous distri-
butions is that in this way we avoid extra, non-physical
singularities, which could appear for discrete (e.q. bi-
modal) distributions31. For a given size we have gener-
ated 100 independent networks and for each we have 100
independent realizations of the disordered couplings.
A. Magnetization and structure of the optimal set
In Fig.1 the temperature dependence of the average
magnetization is shown for various strength of disorder,
∆, for a BA network ofN = 1024 sites. It is seen that the
sharp first-order phase transition of the pure system with
∆ = 0 is rounded and the magnetization has considerable
variation within a temperature range of ∼ ∆. The phase
transition seems to be continuous even for weak disorder.
Close to the transition point the magnetization curves
for uniform disorder (∆ = 2) are presented in the inset
of Fig.1, which are calculated for different finite systems.
Some features of the magnetization curves and the
properties of the phase transition can be understood
by analysing the structure of the optimal set. For low
enough temperature this optimal set is fully connected,
i.e. the magnetization is m = 1, which happens for
T < Tc(0)−∆. Indeed, the first sites with k = µ = 2 (i.e.
those which have only outgoing edges) are removed from
the fully connected diagram, if the sum of the connected
bonds is
i=1 Ji < T , which happens within the tem-
perature range indicated above. From a similar analysis
follows that the optimal set is empty for any finite system
for T > Tc(0) + ∆. The magnetization can be estimated
for t = T − (Tc(0)−∆) ≪ 1, and the correction is given
by: 1−m ∼ tµ. For the numerically studied model with
µ = 2 and ∆ = 2, we have m(T ) ≈ 1 − T 2/8, which is
indeed a good approximation for T < 1. In the tempera-
ture range Tc(0)−∆ < T < Tc(0)+∆ typically the sites
are either isolated or belong to the largest cluster. There
are also some clusters with an intermediate size, which
are dominantly two-site clusters for T < Tc(0) and their
fraction is less then 1%, as shown in Fig.2. The fraction
of two-site clusters for ∆ = 2 and T < Tc(0) = 2 can
be estimated as follows. First, we note that since they
are not part of the biggest cluster they can be taken out
of a fraction of p1 = 1 − m(T ) sites. Before being dis-
connected a two-site cluster has typically three bonds to
the biggest cluster, denoted by J1, J2 and J3. When it
becomes disconnected we have J1 + J2 + J3 < T , which
happens with a probability p2 = T
3/48. At the same
time the coupling within the two-site cluster should be
J4 > T , which happens with probability p3 = (2− T )/2.
Thus the fraction of two-site clusters is approximately:
n2 ≈ p1 × p2 × p3 ≈ T
5(2− T )/768, which describes well
the general behavior of the distribution in Fig.2.
In the temperature range T > Tc(0) the intermedi-
ate clusters have at least three sites and their fraction is
negligible, which is seen in Fig.2. Consequently the in-
termediate size clusters do not influence the properties of
the phase transition in the system. In the ordered phase,
T < Tc, the largest connected cluster contains a finite
fraction of m(T ) < 1 of the sites. We have analyzed the
degree distribution of this connected giant cluster in Fig.
3, which has scale-free behavior and for any tempera-
ture T < Tc there is the same degree exponent, γ = 3,
as for the original BA network. We note an interesting
feature of the magnetization curves in Fig.1 that cross
each other at the transition point of the pure system, at
Tc(0) = 2, having a value of m(Tc(0)) = 0.58, for any
strength of disorder. This property follows from the fact
that for a given realization of the disorder the optimal
set at T = Tc(0) only depends on the sign of the sum
of fluctuations of given couplings (c.f. some set of sites
is connected (disconnected) to the giant cluster only for
positive (negative) accumulated fluctuations) and does
not depend on the actual value of ∆ > 0.
We can thus conclude the following picture about the
evaluation of the optimal set. This is basically one large
connected cluster with N sites, immersed in the see of
isolated vertices. With increasing temperature more and
more loosely connected sites are dissolved from the clus-
ter, but for T < Tc we have N/N = m(T ) > 0 and the
N=128
N=256
N=512
N=1024
FIG. 2: (Color online) Fraction of intermediate size clusters
as a function of the temperature.
0 1 2 3 4 5 6
ln(k)
T=1.957
T=2.464
T=2.738
T=2.816
T=2.894
T=2.972
T=3.030
T=3.089
T=3.128
FIG. 3: (Color online) Degree distribution of the largest
cluster at different temperatures in a finite network with
N = 2048. The dashed straight line indicates the range of
the critical temperature.
cluster has the same type of scale-free character as the
underlaying network. On the contrary above the phase-
transition point, Tc(0) + ∆ > T > Tc, the large clus-
ter has only a finite extent, N < ∞. The order of the
transition depends on the way how N behaves close to
Tc. A first-order transition, i.e. phase-cooexistence at
Tc does not fit to the above scenario. Indeed, as long
as N ∼ N the same type of continuous erosion of the
large cluster should take place, i.e. the transition is of
second order for any strength, ∆ > 0 of the disorder.
Approaching the critical point one expects the following
singularities: m(T ) ∼ (Tc − T )
β and N ∼ (T − Tc)
−ν′ .
Finally, at T = Tc the large cluster has N ∼ N
1−x sites,
with x = β/ν′.
B. Distribution of the finite-size transition
temperatures
The first step in the study of the critical singularities
is to locate the position of the phase-transition point.
In this respect it is not convenient to use the magne-
tization, which approaches zero very smoothly, see the
inset of Fig.1, so that there is a relatively large error by
calculating Tc in this way. One might have, however, a
better estimate by defining for each given sample, say
α, a finite-size transition temperature Tc(α,N), as has
been made for regular lattices25,26,31. For a network we
use a condition for the size of the connected component:
N (T ) ≃ AN1−x , in which x is the magnetization critical
exponent and A = O(1) is a free parameter, from which
the scaling form of the distribution is expected to be in-
dependent. The calculation is made self-consistently: for
a fixed A and a starting value of xs = x1 we have deter-
mined the distribution of the finite-size transition tem-
peratures and at their average value we have obtained
an estimate for the exponent, x = x2. Then the whole
procedure is repeated with xs = x2, etc. until a good
convergence is obtained. Fortunatelly the distribution
function, p(Tc, N), has only a weak x-dependence thus it
was enough to make only two iterations. We have started
with a logarithmic initial condition, N (T ) ≃ A lnN ,
which means formally x1 = 1 and we have obtained
x2 = .69. Then in the next step the critical exponents
are converged within the error of the calculation and they
are found to be independent of the value of A, which has
been set to be A = 1, 2 and 3.
The distribution of the finite-size critical temperatures
calculated with x2 = 0.69 and A = 3. are shown in Fig.4
for different sizes of the network. One can observe a
shift of the position of the maxima as well as a shrinking
of the width of the distribution with increasing size of
the network. The shift of the average value, T avc (N), is
asymptotically given by:
T avc (N)− Tc(∞) ∼ N
−1/ν̃′ , (5)
whereas the width, characterized by the mean standard
deviation, ∆Tc(N), scales with another exponent, ν
′, as:
∆Tc(N) ∼ N
−1/ν′ . (6)
Using Eq.(5) from a three-point fit we have obtained
ν̃′ = 3.8(2) and Tc(∞) = 3.03(2). We have determined
the position of the transition point in the infinite sys-
tem, Tc(∞), in another way by plotting the difference
T avc (N) − Tc(∞) vs. N in a log-log scale for different
values of Tc(∞), see Fig.5. At the true transition point
according to Eq.(5) there is an asymptotic linear depen-
dence, which is indeed the case around Tc(∞) = 3.03(2)
and the slope of the line is compatible with 1/ν̃′ = .27(1).
For the width exponent, ν′, we obtained from Eq.(6)
with two-point fit the estimate: ν′ = 5.6(2). With
these parameters the data in Fig.4 can be collapsed to
a master curve as shown in the inset of Fig.4. This
1.8 2 2.2 2.4 2.6 2.8 3 3.2
N=128
N=256
N=512
N=1024
N=2048
N=4096
-3 -2 -1 0 1 2 3 4 5
FIG. 4: (Color online) Distribution of the finite-size transi-
tion temperatures for different sizes of the BA network. In-
set: scaling collapse of the data in terms of t = (Tc(N) −
(N))/∆Tc(N), using the scaling form in Eqs.(5) and (6)
with ν′ = 3.8(2), and ν̃′ = 5.6(2).
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5
ln(N)
Tc=3.17
Tc=3.14
Tc=3.11
Tc=3.08
Tc=3.05
Tc=3.03
Tc=3.01
Tc=2.98
Tc=2.96
FIG. 5: (Color online) Shift of the average finite-size transi-
tion temperatures, T av
(N) − Tc(∞), vs. N in a log-log scale
plotted for different values of Tc(∞). The lines connecting
the points at the same Tc(∞) are guide for the eye. At the
true transition point the asymptotic behavior is linear which
is indicated by a dotted straight line.
master curve looks not symmetric, at least for the fi-
nite sizes used in the present calculation, and can be
well fitted by a modified Gumbel distribution, Gω(−y) =
ωω/Γ(ω)(exp(−y−e−y))ω , with a parameter ω = 4.2. We
note that the same type of fitting curve has already been
used in Ref.35. For another values of the initial parame-
ter, A = 1 and 2 the estimates of the critical exponents
as well as the position of the transition point are found
to be stable and stand in the range indicated by the error
bars.
The equations in Eqs.(5) and (6) are generalizations
of the relations obtained in regular d-dimensional lat-
tices36,37,38,39,40 in which N is replaced by Ld, L being
the linear size of the system and therefore instead of ν′
and ν̃′ we have ν = ν′/d and ν̃ = ν̃′/d, respectively. Gen-
erally at a random fixed point the two characteristic ex-
ponents are equal and satisfy the relation41 ν′ = ν̃′ ≥ 2.
This has indeed been observed for the 2d31 and 3d25,26
random bond Potts models for large q at disorder induced
critical points. On the other hand if the transition stays
first-order there are two distinct exponents35,42 ν̃′ = 1
and ν′ = 2.
Interestingly our results on the distribution of the
finite-size transition temperatures in networks are differ-
ent of those found in regular lattices. Here the transition
is of second order but still there are two distinct critical
exponents, which are completely different of that at a
disordered first-order transition. For our system ν′ > ν̃′,
which means that disorder fluctuations in the critical
point are dominant over deterministic shift of the tran-
sition point. Similar trend is observed about the finite-
size transition parameters in the random transverse-field
Ising model43, the critical behavior of which is controlled
by an infinite disorder fixed point. In this respect the
RBPM in scale-free networks can be considered as a new
realization of an infinite disorder fixed point.
C. Size of the critical cluster
Having the distribution of the finite-size transition
temperatures we have calculated the size of the largest
cluster at T avc (N), which is expected to scale as
N [N, T avc (N)] ∼ N
1−x. Then from two-point fit we
have obtained an estimate for the magnetization expo-
nent: x = .66(1). We have also plotted N [N, T avc (N)] vs.
N1−x in Fig.6 for different initial parameters, A. Here
we have obtained an asymptotic linear dependence with
an exponent, x = .65(1), which agrees with the previous
value within the error of the calculation.
V. DISCUSSION IN TERMS OF OPTIMAL
COOPERATION
In this paper we have studied the properties of the
Potts model for large value of q on scale-free evolving
complex networks, such as the BA network, both for ho-
mogeneous and random ferromagnetic couplings. This
problem is equivalent to an optimal cooperation prob-
lem in which the agents try to optimize the total sum of
the benefits coming from pair cooperations (represented
by the Potts couplings) and the total sum of the support
which is the same for each cooperating projects (given by
the temperature of the Potts model). The homogeneous
problem is shown exactly to have two distinct states: ei-
ther all the agents cooperate with each other or there is
no cooperation between any agents. There is a strongly
first-order phase transition: by increasing the support
the agents stop cooperating at a critical value.
0 10 20 30 40 50
[0 , 0]
0 10 20 30 40 50
[0 , 0]
FIG. 6: (Color online) Size dependence of the critical cluster
at the average finite-size critical temperature as a function
of N1−x with x = .65. The date points for different initial
parameters, A, are well described by straight lines, which are
guides to the eye.
In the random problem, in which the benefits are ran-
dom and depend on the pairs of the cooprerating agents,
the structure of the optimal set depends on the value of
the support. Typically the agents are of two kinds: a frac-
tion of m belongs to a large cooperating cluster whereas
the others are isolated, representing one man’s projects.
With increasing support more and more agents are split
off the cluster, thus its size, as well as m is decreasing,
but the cluster keeps its scale-free topology. For a critical
value of the support m goes to zero continuously and the
corresponding singularity is characterized by non-trivial
critical exponents. This transition, as shown by the nu-
merically calculated critical exponents for the BA net-
work, belongs to a new universality class. One interesting
feature of it is that the distribution of the finite-size tran-
sition points is characterized by two distinct exponents
and the width of the distribution is dominated over the
shift of the average transition point, which is character-
istic at an infinite disorder fixed point43.
We thank for useful discussions with C. Monthus and
for previous cooperation on the subject with M.-T. Mer-
caldo. This work has been supported by the National
Office of Research and Technology under Grant No.
ASEP1111, by the Hungarian National Research Fund
under grant No OTKA TO48721, K62588, MO45596 and
M36803. M.K. thanks the Ministère Français des Affaires
Étrangères for a research grant.
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|
0704.1539 | A New Monte Carlo Method and Its Implications for Generalized Cluster
Algorithms | A New Monte Carlo Method and Its Implications for Generalized Cluster Algorithms
C. H. Mak and Arun K. Sharma
Department of Chemistry, University of Southern California,
Los Angeles, California 90089-0482, USA
(Dated: November 13, 2018)
We describe a novel switching algorithm based on a “reverse” Monte Carlo method, in which the
potential is stochastically modified before the system configuration is moved. This new algorithm
facilitates a generalized formulation of cluster-type Monte Carlo methods, and the generalization
makes it possible to derive cluster algorithms for systems with both discrete and continuous de-
grees of freedom. The roughening transition in the sine-Gordon model has been studied with this
method, and high-accuracy simulations for system sizes up to 10242 were carried out to examine the
logarithmic divergence of the surface roughness above the transition temperature, revealing clear
evidence for universal scaling of the Kosterlitz-Thouless type.
PACS numbers: 05.10.Ln, 05.50.+q, 64.60.Ht, 75.40.Mg
Large-scale Monte Carlo (MC) simulations are often
plagued by slow sampling problems. These problems are
especially severe in systems near the critical point or in
those with strong correlations. Slow sampling problems
manifest themselves as poor scaling of the dynamic relax-
ation time with the system size, making large-size sim-
ulations extremely slow to converge. The cause of these
problems is that most MC simulations are based on lo-
cal moves, and when the correlation length of the system
grows or as relaxation modes of the system become heav-
ily entangled, local moves become increasingly inefficient.
But if nonlocal MC moves are used [1], their acceptance
ratios are often found to be exceedingly low when system
correlations are strong.
One way to circumvent these problems was suggested
by Swendsen and Wang [2], who devised a clever scheme
where large-scale nonlocal MC moves may be constructed
to achieve high sampling efficiencies by exploiting certain
geometric symmetries in the system. This algorithm led
to a marked reduction in the dynamical scaling exponent
in the 2-dimensional Ising model near criticality. Since
the nonlocal moves in this algorithm update a large num-
ber of degrees of freedom at the same time, the Swendsen-
Wang method and others inspired by it are also often
referred to as “cluster Monte Carlo” methods.
Since Swendsen and Wang’s paper in 1987, many
cluster-type MC algorithms have appeared [3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. But
the success of cluster MC has not been universal because
the proper cluster moves needed seem to be highly de-
pendent on the system, and efficient cluster MC meth-
ods have been found for only a small number of models
[3, 5, 6, 7, 11, 14, 15, 18, 19] so far. The difficulty of
formulating a generalized MC method that could work
for any system seems to be associated with the appar-
ent geometric nature of the cluster-type MC methods –
all existing cluster MC methods have been derived in
one way or another by using certain geometric features
of the system. For example, in the original Swendsen-
Wang formulation a mapping between the Ising model
and the percolation model originally described by For-
tuin and Kasteleyn [20] was exploited to effect cluster
spin flips. In other models, the requisite mapping is not
always obvious, making cluster MC methods difficult to
implement for general systems.
In this letter, we will show that the derivations of clus-
ter MC methods do not have to be based on geometric
features of the systems. Instead, they may be more con-
veniently formulated based on algebraic features of the
system potential V (C). We will exploit this algebraic for-
mulation and suggest a way to generalize cluster Monte
Carlo methods to systems with any potential.
We focus on classical systems with partition function
Z = Tr e−V (C), where V (C) is the potential in units of
the temperature T . Acceptable Monte Carlo methods
to sample the system configurations can be constructed
using any transition probabilities W (C → C′) as long as
the detailed balance condition
W (C → C′)e−V (C) = W (C′ → C)e−V (C
′) (1)
is satisfied. Conventional MC methods such as Metropo-
lis [21] accomplishes this in two steps: a trial move is
made from C to C′ with some transition probability, and
the move is then accepted or rejected according to an
acceptance probability based on V (C′), V (C) or both, so
that the composite process satisfies Eqn.(1). This way
of constructing the Markov chain – trial moves followed
by acceptance/rejection – has been the accepted “stan-
dard” method for doing MC since the inception of the MC
method [22]. Other MC methods do exist, such as the
heat bath algorithm [23], which follow alternative strate-
gies, but by far the standard method is conceptually the
simplest and most convenient in practice.
In the Monte Carlo method we are proposing, we will
reverse the order of the two steps in the standard method.
That is, we will first determine an acceptable way to mod-
ify the potential V and then find a transition C → C′ that
is consistent with the new potential. To our knowledge,
http://arxiv.org/abs/0704.1539v1
the basic elements of this “reverse MC” idea were first
suggested by Kandel et al. [4], who used it to stochasti-
cally remove interaction terms from the system’s poten-
tial in an Ising model to arrive at an alternative deriva-
tion of the Swendsen-Wang method. The formulation of
Kandel et al. was limited to discrete systems like the
Ising model. In the following, we will show how the re-
verse MC idea may be formulated more generally for any
system, discrete or continuous, and how it may then be
used as a framework to construct generalized cluster al-
gorithms.
Consider a system with potential V =
i vi+Vr, con-
sisting of a number of “interaction terms” vi plus a “resid-
ual” Vr . These interactions may be bonds between parti-
cles, interactions of the particles with a field, or any other
additive terms in V . We consider replacing each interac-
tion term vi by some pre-selected ṽi with a “switching”
probability
Si(C) = cie
∆vi(C), (2)
where ∆vi = vi− ṽi, ci = e
i and ∆v∗i = maxC vi(C).
The outcome of the switches defines two complemen-
tary sets of interactions – the switched ones σ̃ and the
unswitched ones σ̄. Using the outcome of the switches,
we define a stochastically modified potential Ṽ as follows:
ṽi +
v̄j + Vr, (3)
with v̄i = vi − ln(1 − Si). An MC pass starts with an
attempt to switch every interaction vi to the new ṽi us-
ing the Si defined in Eqn.(2). If the switch is successful,
the interaction is replaced by ṽi. If not, the interaction
is replaced by another interaction v̄i. This is followed
by an update in the configuration of the entire system
from C to C′ using a transition probability W̃ (C → C′)
that satisfies detailed balance on the modified potential
Ṽ . This constitutes one pass. The move from C to C′ can
of course be carried out using any conventional MC move
that satisfies detailed balance on the modified potential.
But the reverse formulation of the MC method now offers
possibilities that were previously unavailable to conven-
tional MC methods — if a simple scheme can be devised
to update the configuration of the entire system on the
stochastically modified potential, one can envision de-
signing global moves for the system to accelerate its sam-
pling, and our freedom in choosing the ṽi can be actively
exploited to facilitate this. Within this context, the origi-
nal formulation of Kandel et al. corresponds to switching
vi to ṽi = 0, i.e. simplifying the potential by deleting in-
teractions from it. Kandel et al. showed that for the Ising
model they could easily construct global moves on this
stochastically simplified potential and their formulation
regenerates the Swendsen-Wang method. But compared
to the deletion formulation of Kandel et al., the switch-
ing implementation of the reverse MC method now offers
a much wider set of possibilities because the form of the
“switch to” interactions is completely arbitrary. Whereas
previously there may not be an obvious way to globally
update the configuration of the system on the original po-
tential, with the proper choices for ṽi large-scale moves
may now become possible on the stochastically modified
potential. Indeed, we have shown that the switching idea
may be used to formulate a cluster MC algorithm for a
Lennard-Jones fluid [24].
Equations (2), (3) and the transition probability W̃
define the switching algorithm. To prove detailed bal-
ance Eqn.(1) for the switching algorithm, it is suffi-
cient to treat a case where there are only two interac-
tion terms. Extension to any number of interactions
is straightforward. Starting with C, with two interac-
tion terms v1 and v2, there are four possible outcomes
from the switch: I. both 1 and 2 are switched, which oc-
curs with probability PI = S1(C)S2(C), II. 1 is switched
and 2 is unswitched, with PII = S1(C)[1 − S2(C)], III.
1 is unswitched and 2 is switched, with PIII = [1 −
S1(C)]S2(C), and IV. both 1 and 2 are unswitched, with
PIV = [1 − S1(C)][1 − S2(C)]. After the switch, an up-
date C → C′ is made with a transition probability W̃
that satisfies detailed balance on the modified potential
Ṽ defined in Eqn.(3). Each of the four channels will
have a different W̃ : W̃I, W̃II, etc., and W (C → C
′) in
Eqn.(1) is the sum PIW̃I + PIIW̃II + PIIIW̃III + PIVW̃IV
over all four channels. For the reverse transition, we
start with C′ and consider switching v1(C
′) → ṽ1(C
′) and
′) → ṽ2(C
′). Again there are four possible outcomes
and we call these scenarios I′, II′, III′ and IV′ as for
the forward transition. W (C′ → C) in Eqn.(1) is again
the sum PI′W̃I′ +PII′W̃II′ +PIII′W̃III′ +PIV′W̃IV′ . Using
the choice of S and Ṽ in Eqs.(2) and (3), it is easy to
show that detailed balance is obeyed along each chan-
nel, i.e. PIW̃I = PI′W̃I′ , PIIW̃II = PII′W̃II′ , etc. Of
course, detailed balance only requires the total W to sat-
isfy Eqn.(1), and it is possible to choose alternate forms
of S and Ṽ to do that, which may provide further flexi-
bilities.
In the rest of this letter, we will illustrate the effec-
tiveness of the switching implementation of the reverse
MC method, and show how it can be used to easily de-
rive a cluster MC method in a system with continuous
degrees of freedom. Previously, it has been extremely dif-
ficult to design cluster MC algorithms for systems with
continuous degrees of freedom. The few that have been
reported to date [3, 5, 6, 7, 11, 14, 19] were mainly based
on embedding discrete degrees of freedom into continu-
ous ones. The only exception is the recent discovery of a
geometric MC algorithm by Liu and Luitjen [19] where
they formulated a rejection-free MC method to sample
the Lennard-Jones fluid at its critical point.
The switching algorithm we have proposed makes the
process of deriving cluster-type MC methods much more
straightforward compared to those based on geometric
features of the system. We will illustrate this using
the sine-Gordon model, which can be used to study the
roughening transition on 2-dimensional surfaces. The
sine-Gordon (SG) model has the potential
VSG = T
〈i,j〉
|φi − φj |
cos(φi)
, (4)
where φi are continuous variables on a 2-dimensional
square lattice, the second sum is over all sites and the
first sum is over all nearest-neighbor pairs. The SG
model is often considered to be a coarse-grained ver-
sion of the discrete Gaussian (DG) model with poten-
tial VDG = T
〈i,j〉 |hi − hj |
2, where hi are integers.
The DG model can in turn be mapped directly onto
the Coulomb gas model [25], and as a result, the SG
model should belong in the same universality class as the
Kosterlitz-Thouless (KT) transition [26, 27].
Roughening is expected to be a weak transition. The
only easily discernible divergence is exhibited in a loga-
rithmic dependence of the surface roughness σ2 = 〈|φi −
〈φ〉|2〉 on the system size L at the roughening tempera-
ture TR. Below TR, σ
2 is expected to approach a finite
value as L → ∞. In addition to this, since the divergence
is slow, large lattice sizes are needed to reach the scaling
limit. All of these features of the SG model make it hard
to accurately study the roughening transition using MC
simulations. Previous simulations have been limited to
small systems [14, 28, 29, 30, 31, 32].
In order to locate TR and study the scaling behavior
at the roughening transition, we make use of the switch-
ing algorithm of the reverse MC method proposed above.
The essential difficulty in treating the SG model is due
to the nonlinear cosine terms in the potential in Eqn.(4).
If these nonlinearities could be removed, the residual po-
tential becomes a simple Gaussian and we could move
the system configuration effectively using uncoupled sur-
face modes. With this in mind, we separate the potential
into two parts and treat the cosine terms as interactions
vi = −T
−1 cosφi and the harmonic part as the resid-
ual Vr. Each of the interactions is switched to a uni-
form potential ṽi = −T
−1 with Si = e
[1−cosφi]/T . After
the switches, a number of φi would have effectively lost
their couplings to the cosine potential, while the rest have
their interactions with the cosine potential replaced by
v̄i = − ln[e
cosφi/T − e−1/T ]. In the ensuring MC move,
we can update the unswitched φi which are now cou-
pled to the replacement interactions v̄i using conventional
methods, but try to formulate an update scheme where
the rest of the φi, now forming a constrained Gaussian
field, may be updated globally. A Gaussian field sub-
ject to linear constraints is still Gaussian, and in prin-
ciple we can diagonalize the potential to obtain all the
normal modes and then move each one independently.
This problem is the subject of fracton dynamics and has
been studied previously [33]. However, the cost of ob-
64 128 256 512 1024
64 128 256 512 1024
(b)(a)
FIG. 1: (a) Surface thickness σ2 as a function of the log
of lattice size L for different temperatures T . (b) Expanded
view of (a) for several temperatures near TR shifted vertically
to coincide at L = 64. Dashed line is the expected KT slope
at TR, showing that TR is slightly above T = 25 but below
taining all the normal modes of the constrained surface
and their frequencies will grow rapidly with the size of
the lattice and will only be feasible for small-size simula-
tions. Since the scaling limit in the SG model can only
be reached with large system sizes, we will need an al-
ternative method. The method we have used to update
the constrained Gaussian fields is based on the method
of Hoffman and Ribak [34]. Since the statistics of the
fluctuations of a Gaussian field from its mean is indepen-
dent of the value of the mean field, the fluctuations from
a free Gaussian field can be transferred to a constrained
field with a different mean. Near the roughening tem-
perature, the switching procedure produces roughly 5%
unswitched field points, and the corresponding mean field
with these constraints can easily be determined using a
steepest descent molecular dynamics method. To ensure
ergodicity, a conventional Metropolis move is also carried
out with every reverse MC move.
Figure 1(a) shows simulation results for the scaling of
the surface roughness σ2 with the length L of the lattice
in simulations with different lattice sizes L2 up to 10242
and at several temperatures T from 16 to 30. KT theory
[26, 27] predicts a logarithmic divergence for σ2 with a
universal slope at TR
2(L) = σ20(TR) +
lnL, (5)
where a is the lattice constant of the surface, and in the
units of Eqn.(4), a = 2π. Therefore, at TR the slope of
Fig. 1(a) should be equal to 4. Above TR, the logarith-
mic behavior of σ2 continues to hold except the constant
σ20 as well as the slope both increase with T . The data
in Fig. 1(a) show that for T = 21 and below, σ2 appears
to approach a finite value as L → ∞. Therefore, it is
64 128 256 512 1024
10000
Metropolis
cluster MC
ξ = 1.4
ξ = 2.5
FIG. 2: Dynamic scaling for the relaxation time τ (in units of
MC passes) of σ2 as a function of lattice size L in Metropolis
versus cluster MC, with their corresponding exponent ξ.
clear that TR > 21. The most recent simulation of the
SG model by Sanchez et al. [32] (referred to as the “or-
dered SG model”, OSGM, in this paper) suggested that
TR ≈ 16. Our data show that this is incorrect, and their
error is likely due to slow sampling problems. Locating
the precise value of TR is more involved, since the data
for T > 21 show no obvious tendency toward a finite σ2.
There are two possibilities: either these temperatures are
above TR or the system size may not be large enough to
have reached the scaling limit for these temperatures.
To determine which one is the case, we must resort to
a comparison between the simulation data with KT the-
ory. Figure 1(b) shows an expanded view of Fig. 1(a)
for a few temperatures 23 < T < 30, but for each T
the curve has been shifted vertically to remove the off-
set σ20 so that they all coincide at L = 64. The heavy
dashed line indicates the KT slope at TR according to
Eqn.(5). The data therefore suggest that TR is slightly
larger than 25 but less than 26, which is consistent with
the RG prediction for TR = 8π in the continuum model
[35, 36]. The apparent lack of an asymptotic σ2 in the
data for 21 < T < TR implies that even for L = 1024,
these lattice sizes are not yet large enough to be in the
scaling limit for those temperatures. Finally, to compare
the dynamic scaling behavior of the switching algorithm
with Metropolis, Fig. 2 shows the relaxation time in the
measurement of σ2 with the lattice size L slightly above
TR. Compared with the dynamic exponent ξ ≈ 2.5 in
Metropolis, the switching algorithm shows a markedly
improved ξ ≈ 1.4.
This work was supported by the National Science
Foundation under grant CHE-9970766.
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|
0704.1540 | The Orbifolds of Permutation-Type as Physical String Systems at
Multiples of $c=26$ III. The Spectra of $\hat{c}=52$ Strings | The Orbifolds of Permutation-Type as
Physical String Systems
at Multiples of c = 26
III. The Spectra of ĉ = 52 Strings
M.B.Halpern∗
Department of Physics, University of California
and Theoretical Physics Group,
Lawrence Berkeley National Laboratory
University of California, Berkeley CA 94720 USA
November 15, 2018
Abstract
In the second paper of this series, I obtained the twisted BRST systems
and extended physical-state conditions of all twisted open and closed
ĉ = 52 strings. In this paper, I supplement the extended physical-state
conditions with the explicit form of the extended (twisted) Virasoro
generators of all ĉ = 52 strings, which allows us to discuss the physical
spectra of these systems. Surprisingly, all the ĉ = 52 spectra admit an
equivalent description in terms of generically-unconventional Virasoro
generators at c = 26. This description strongly supports our prior
conjecture that the ĉ = 52 strings are free of negative-norm states,
and moreover shows that the spectra of some of the simpler cases
are equivalent to those of ordinary untwisted open and closed c = 26
strings.
∗[email protected]
http://arxiv.org/abs/0704.1540v1
Table of Contents
1. Introduction
2. The Extended Virasoro Generators of ĉ = 52 Strings
3. First Discussion of the ĉ = 52 String Spectra
4. Equivalent c = 26 Description
of the ĉ = 52 Spectra
5. Conclusions
1 Introduction
Opening another chapter in the orbifold program [1-11,12-15], this is the third
in a series of papers which considers the critical orbifolds of permutation-
type as candidates for new physical string systems at higher central charge.
In the first paper [16] of this series, we found that the twisted sectors of
these orbifolds are governed by new, extended (permutation-twisted) world-
sheet gravities – which indicate that the free-bosonic orbifold-string systems
of permutation-type can be free of negative-norm states at critical central
charge ĉ = 26K. Correspondingly-extended world-sheet permutation super-
gravities are expected in the twisted sectors of the superstring orbifolds of
permutation-type, where superconformal matter lives at higher multiples of
critical superstring central charges.
In the second paper [17] of the series, we found the corresponding twisted
BRST systems for all sectors of the free-bosonic orbifolds which couple to the
simple case of Z2-twisted permutation gravity, i.e. for all the twisted strings
with ĉ = 52 matter. The new BRST systems also implied the following
extended physical-state conditions for the physical states {|χ〉} of each of the
ĉ = 52 strings:
|χ〉 = 0, m ∈ Z, u = 0, 1 (1.1a)
, L̂v
m− n+ u−v
L̂u+v
m+ n + u+v
(1.1b)
δm+n+u+v
The algebra in Eq. (1.1b) is called an order-two orbifold Virasoro algebra (or
extended, twisted Virasoro algebra) and general orbifold Virasoro algebras
[1,18,9,12,16,17] are known to govern all the twisted sectors of the orbifolds
of permutation-type at higher multiples of c = 26.
The set of all ĉ = 52 orbifold-strings is a very large class of fractional-
moded free-bosonic string systems, including e.g. the twisted open-string
sectors of the orientation orbifolds, the twisted closed-string sectors of the
generalized Z2-permutation orbifolds and many others (see Refs. [16,17] and
Sec. 2). Starting from the extended physical-state conditions (1.1) (and
a right-mover copy of (1.1) on the same {|χ〉}for the twisted closed-string
sectors) this paper begins the concrete study of the physical spectrum of
each ĉ = 52 string.
As the prerequisite for this analysis, I first provide in Sec. 2 the explicit
form – in terms of twisted matter fields – of the extended Virasoro generators
, u = 0, 1
of all ĉ = 52 strings. This construction allows us
to begin the study of the general ĉ = 52 string spectra in Sec. 3. The
same subject is further considered in Sec. 4, where I point out that all
the ĉ = 52 spectra admit an equivalent description in terms of generically-
unconventional Virasoro generators at c = 26. This description allows us to
see clearly a number of spectral regularities which are only glimpsed in Sec. 3,
including strong further evidence that the critical orbifolds of permutation-
type can be free of negative-norm states. Moreover, although the generic
ĉ = 52 spectrum is apparently new, we are able to show that some of the
simpler spectra are equivalent to those of ordinary untwisted open and closed
critical strings at c = 26.
Based on these results, the discussion in Sec. 5 raises some interesting
questions about these theories at the interacting level, and speculates on
the form of the extended physical-state conditions for more general orbifold-
strings of permutation-type. I will return to both of these subjects in suc-
ceeding papers of the series.
2 The Twisted Virasoro Generators
of ĉ = 52 Strings
As emphasized in Ref. [17], the universal form of the twisted BRST systems
and the extended physical-state conditions (1.1) are consequences of their
origin in Z2-twisted permutation gravity, which governs all twisted ĉ = 52
matter.
There are however many distinct ĉ = 52 strings, including the twisted
open-string sectors of the orientation orbifolds [12,13,15-17]
U(1)26
, H− = Z2(w.s.)×H (2.1)
and the twisted closed-string sectors of the generalized Z2-permutation orb-
ifolds [15-17]
U(1)26 × U(1)26
, H+ = Z2(perm)×H
′ (2.2)
as well as the generalized open-string Z2-permutation orbifolds and their T -
duals [15-17]. For the orientation orbifolds in Eq. (2.1), I remind that H− is
any automorphism group of the untwisted closed string U(1)26 which includes
world-sheet orientation-reversing automorphisms. Indeed the twisted open-
string orientation-orbifold sectors correspond to the orientation-reversing au-
tomorphisms, which have the form τ− × ω, ω ∈ H where the basic automor-
phism τ− exchanges the left- and right-movers of the closed string and ω is
an extra automorphism which acts uniformly on the left- and right-movers
of the closed string. Similarly, the automorphism group H+ of the general-
ized Z2-permutation orbifolds in (2.2) is generated by elements of the form
τ+×ω, ω ∈ H
′, where the basic automorphism τ+ exchanges the two copies
of the closed string and the extra automorphism ω again acts uniformly on
the left- and right-movers of each closed string. In both cases, the extra
automorphisms ω in τ × ω may or may not form a group (see the examples
at the end of this section).
The spectra of different ĉ = 52 strings are characterized by their extended
(twisted) Virasoro generators, all of which can in fact be written in the
following unified form:
Gn(r)µ;−n(r)ν(σ)
× (2.3a)
×:Ĵn(r)µv
Ĵ−n(r),ν,u−v
m− p−
+ u−v
+δm+u
,0 ∆̂0(σ)
Ĵn(r)µu
, Ĵn(s)νv
= (2.3b)
δn(r)+n(s),0modρ(σ)δm+n+n(r)+n(s)
Gn(r),µ;−n(r),ν(σ)
, Ĵn(r)µv
(2.3c)
Ĵn(r)µ,u+v
m+ n+
+ u+v
∆̂0(σ) =
dim[n(r)]
(2.3d)
dim[n(r)] = 26. (2.3e)
Each set of extended Virasoro generators in Eq. (2.3a) satisfies the order-two
orbifold Virasoro algebra (1.1b) at ĉ = 52, and the current algebras in Eq.
(2.3b) are of the type called doubly-twisted in the orbifold program.
For those unfamiliar with the program, I first give a short summary of the
standard notation in the result (2.3) – followed by the derivation of the result.
As in the extended Virasoro generators themselves, the indices u, v with
fundamental range ū, v̄ ∈ {0, 1} describe the twist of the basic permutations
τ∓ in each H∓. For each extra automorphism ω(σ) in each τ∓ × ω(σ), the
spectral indices {n(r)} and the degeneracy indices {µ ≡ µ(n(r))} of each
twisted sector σ are determined by the so-called H-eigenvalue problem [3,5,6]
of ω(σ)
ω(σ)a
U †(σ)b
n(r)µ
= U †(σ)a
n(r)µ
−2πin(r)
ρ(σ) , ω(σ) ∈ H or H ′ (2.4a)
ω(σ)a
ω(σ)b
Gcd = Gab, Gab = G
(2.4b)
a, b = 0, 1, . . . , 25, n(r) ∈ (0, 1, . . . , ρ(σ)− 1) (2.4c)
where G is the untwisted target-space metric of U(1)26. The quantity ρ(σ)
is the order of ω(σ) and all indices {n(r)µ} are periodic modulo ρ(σ), with
{n(r)} the pullback to the fundamental region and dim[n(r)] the size of the
subspace n(r). The index r is summed once over the fundamental region in
Eqs. (2.3a), (2.3d) and (2.3e). The twisted metric G.(σ) and its inverse G
are defined in terms of the unitary eigenvectors U(σ) of the H-eigenvalue
problem
Gn(r)µ;n(s)ν(σ) = χn(r)µχn(s)νU(σ)n(r)µ
U(σ)n(s)ν
Gab (2.5a)
= δn(r)+n(s),0modρ(σ)Gn(r)µ;−n(r),ν(σ) (2.5b)
Gn(r)µ;n(s)ν(σ) = χ−1
n(r)µ
n(s)ν
GabU †(σ)a
n(r)µ
U †(σ)b
n(s)ν
(2.5c)
= δn(r)+n(s),0modρ(σ)G
n(r),µ;−n(r),ν (2.5d)
where G is again the untwisted metric and the χ’s are essentially-arbitrary
normalization constants. Finally, the standard mode normal-ordering in Eq.
(2.3a) is:
:Ĵn(r)µu
Ĵn(s)νv
:M (2.6)
Ĵn(s)νv
Ĵn(r)µu
Ĵn(r)µu
Ĵn(s)νv
It follows that the quantity ∆0(σ) in Eqs. (2.3a) and (2.3d)
Ĵn(r)µu
|0〉σ = 0 (2.7a)
→ L̂u
|0〉σ = ∆̂0(σ) δm+u
,0|0〉σ (2.7b)
is the conformal weight of the scalar twist-field state |0〉σ of sector σ.
I comment briefly on the derivation of the unified form (2.3) of the
ĉ = 52 extended Virasoro generators. Essentially this result was given for
the twisted open-string sectors of the non-abelian orientation orbifolds in
Subsecs. 3.4, 3.5 of Ref. [12], and that result is easily reduced for our
abelian case U(1)26/H− in Eq. (2.1). With a right-mover copy of the ex-
tended Virasoro generators (and u → ̂ = 0, 1), the result also hold for
the twisted closed-string sectors of the generalized Z2-permutation orbifolds
(U(1)26 × U(1)26)/H+ in Eq. (2.2). This follows by the substitution
G → G, u
(2.8)
into the known results for the ordinary Z2-permutation orbifolds with trivial
H ′ (see Ref. [perm] and Subsec. 4.2 of Ref. [16]). Finally, a single copy of
the unified form (2.3) holds as well for each twisted sector of the generalized
open-string Z2-permutation orbifolds (U(1)
26 × U(1)26)open/H+ and all pos-
sible T -dualizations of each of these sectors. This conclusion follows because
the left-mover extended Virasoro generators of the closed-string orbifolds for
each H+ are the input data for the construction of the correponding open-
string orbifolds [14], and the twisted-current form of each set of extended
Virasoro generators is independent of T-dualization [15]. The branes, quasi-
canonical algebra and non-commutative geometry of the twisted open strings
[13-15,16,17] depend of course on the particular T-dualization, but these will
not be needed here.
In what follows I will consider each twisted ĉ = 52 string separately, but
the reader may find it helpful to bear in mind the complete sector structure of
these orbifold-string systems as labelled by the elements of the automorphism
groups H∓. Given a particular extra automorphism ωn ∈ H or H
′ of order
n, one may list the following low-order examples:
(1; τ∓) (2.9a)
(1; τ∓ × ω2) (2.9b)
(1, ω3, ω
3; τ∓, τ∓ × ω3, τ∓ × ω
3) (2.9c)
(1, ω24; τ∓ × ω4, τ∓ × ω
4) (2.9d)
(1, ω26, ω
6; τ∓ × ω6, τ∓ × ω
6, τ∓ × ω
6). (2.9e)
For the generalized Z2-permutation orbifolds (τ+) all of these sectors are
twisted closed strings at ĉ = 52, while all the sectors of the generalized open-
string Z2-permutation orbifolds (τ+) and their T-dualizations are twisted
open strings at ĉ = 52. For the orientation orbifolds (τ−) the sectors before
the semicolon are twisted closed strings at c = 26 (which form an ordinary
space-time orbifold) while the sectors after the semicolon are twisted open
strings at ĉ = 52. More generally, orientation orbifolds always contain an
equal number of twisted open and closed strings. In all cases, the twisting is
of course trivial for sectors corresponding to the unit element.
3 First Discussion of
the ĉ = 52 String Spectra
To frame this discussion, I remind [1] the reader that the Virasoro primary
states of our orbifold CFT’s are defined by the integral Virasoro subalgebra
(generated by {L̂0(m)}) of the extended Virasoro algebra. Then the extended
physical-state conditions (1.1a) tell us that all the physical states {|χ〉} of
each ĉ = 52 orbifold-string are Virasoro primary
L̂0(m > 0)|χ〉 = 0 (3.1)
but only a small subset of these primary states are selected by the rest of the
physical-state conditions:
L̂0(0)−
|χ〉 = L̂1
|χ〉 = 0. (3.2)
In what follows, I will refer to the L̂0(0) condition in Eq. (3.2) as the spectral
condition, since it will determine the allowed values of momentum-squared
for each ĉ = 52 string.
The space of physical states of each orbifold-string is then much smaller
than the space of states of the underlying orbifold conformal field theory. For
the experts, I remark in particular that the extended physical-state condi-
tions generically disallow the characteristic sequence [19] of Virasoro primary
states known as the principle-primary states [1,9]. This follows first by the
spectral condition (which fixes the conformal weight), and second because
the physical-state condition {L̂u
≃ 0} is stronger than the
principle-primary state condition [1,9]
|p.p.s.〉 = 0, u = 0, 1, m > 0 (3.3)
which does not extend to m = 0.
I turn now to concretize the spectral condition of each twisted ĉ = 52
string, using the explicit form (2.3) of its extended Virasoro generators. For
this, recall [12,15] first that these generators contain in general two kinds
of commuting zero modes (dimensionless momenta), namely {Ĵ0µ0(0)} and
{Ĵρ(σ)/2,µ,1(0)}, where the latter is relevant only when the order ρ(σ) of ω(σ)
is even. In what follows, I often refer to these zero modes collectively as
{Ĵ(0)}. It is then natural to define the “momentum-squared” operator P̂ 2
as follows:
L̂0(0) =
−P̂ 2 + R̂(σ)
+ ∆̂0(σ) (3.4a)
P̂ 2 ≡ −
G0µ;0ν(σ)Ĵ0µ0(0)Ĵ0ν0(0) + (3.4b)
,µ;− ρ(σ)
,ν(σ)Ĵρ(σ)/2,µ,1(0)Ĵ−ρ(σ)/2,ν,−1(0)
R̂(σ) ≡
r,µ,ν
Gn(r)µ;−n(r),ν(σ)× (3.4c)
×:Ĵn(r)µu
Ĵ−n(r),ν,−u
Here the primed sum in the “level-number” operator R̂(σ) indicates omission
of the zero modes.
With this decomposition, the spectral condition in Eq. (3.2) takes the
simple form:
P̂ 2|χ〉 =
P̂ 2(0) + R̂(σ)
|χ〉 (3.5a)
P̂ 2(0) ≡ 2
δ̂0(σ)− 1
(3.5b)
δ̂0(σ) =
dim[n(r)]
≥ 0. (3.5c)
Although I will continue the discussion primarily in this form, in fact Eqs.
(3.4a) and (3.5a) hold only for the twisted open-string sectors of the orbifolds.
For the twisted closed-string sectors, we also have right-mover copies of the
extended Virasoro generators (2.3), and a corresponding right-mover copy of
the extended physical-state conditions (1.1) on the same {|χ〉}. For simplicity
I will limit the discussion of these sectors here to the case of decompactified
zero modes, for which it is appropriate to equate the left and right movers
ĴR(0) = ĴL(0) = 1√
Ĵ(0) → R̂R(σ) = R̂L(σ) (3.6)
where the last equality is level-matching in each twisted sector. Keeping the
same definition of the operator P̂ 2 in Eq. (3.4b), the correct closed-string
ĉ = 52 spectral condition is then obtained by the substitution
P̂ 2 → 1
P̂ 2 (3.7)
in both Eqs. (3.4a) and (3.5a). These identifications, and hence P̂ 2
→ 2P̂ 2
can be used at any point in the discussion below to obtain the corresponding
closed-string results.
Returning to the open-string case, one simple solution of the extended
physical-state conditions is the ground state |0, Ĵ(0)〉σ of twisted sector σ:
R̂(σ)|0, Ĵ(0)〉σ = L̂u
|0, Ĵ(σ)〉σ = 0 (3.8a)
P̂ 2|0, Ĵ(0)〉σ = P̂
(0)|0, Ĵ(0)〉σ, P̂
(0) = −2 + 2δ̂0(σ) (3.8b)
This is the “momentum-boosted” twist-field state (see Eq. (2.7)) of that
sector, with ground-state mass-squared P̂ 2
. Moreover Eq. (3.4c) and the
commutator (2.3c) give the increments
∆(P̂ 2) = ∆(R̂(σ)) = 4
(3.9)
obtained by adding the negatively-moded current
Ĵn(r)µu
to any previous state. The precise content of these excited levels must of
course be determined from the remainder of the extended physical-state con-
ditions.
I continue this discussion with some specific examples of ĉ = 52 strings,
beginning with the simplest twisted open-string orientation-orbifold sectors
[12,13,15,16,17]:
ω = 1l : ρ = 1, n = 0, U = 1l, G = G, Ĵ0au
(3.10a)
∆(P̂ 2) = 4
∣m+ u
∣ (u = 0 is DD, u = 0 is ND) (3.10b)
ω = −1l : ρ = 2, n = 1, U = 1l, G = G, Ĵ1au
m+ u+1
(3.11a)
∆(P̂ 2) = 4
∣m+ u+1
∣ (u = 0 is DN, u = 0 is NN). (3.11b)
In these cases, the extra automorphisms ω act uniformly on the labels a =
0, . . . 25 and G is the untwisted target space metric in Eq. (2.4b). Although
both twisted strings have (26+26) = 52 matter degrees of freedom, note that
each example has only one of the two types of zero modes {Ĵ(0)}: 26DD zero
modes {Ĵ0a0(0)} for ω = 1l and 26NN zero modes {Ĵρ/2,a,1(0)} for ω = −1l.
In both cases, the momentum-squared (3.4b) has the schematic form
P̂ 2 = ηabĴa(0)Ĵb(0), η =
0 −1l
(3.12)
where η = −G is the standard (west-coast) 26-dimensional target-space met-
ric. Then we compute from Eqs. (3.5b) and (3.5c) that both strings share
the same tachyonic ground-state mass-squared
δ̂0(σ) = 0, ∆̂0(σ) =
, P̂ 2(0) = −2 (3.13)
and the first excited state of each is massless:
Ĵ0a1
Ĵ1a0
|0, Ĵ(0)〉σ = 0 for ω =
(3.14)
For this level, I have checked that the L̂1
≃ 0 gauge eliminates the
longitudinal parts of the 26-dimensional “photons”, and moreover the L̂1
and L̂0 (1) gauges together eliminate the negative-norm states at the next
level:
+ βĴ(−1)
|0, Ĵ(0)〉σ, P̂
2 = 2. (3.15)
Since the increments ∆(P 2) in Eqs. (3.10b) and (3.11b) are even integers,
we are led to suspect that the spectra of these two twisted ĉ = 52 strings are
nothing but the spectrum of an ordinary open c = 26 string in disguise 1. I
will return to this question in the following section.
A larger subset of twisted ĉ = 52 strings is the following. For a particular
twisted sector σ, suppose that ω = ±1l acts uniformly on a set of d labels
a = 0, 1, . . . d − 1, d ≥ 4 while a non-trivial element ω(perm) of some per-
mutation group acts non-trivially on the other 26 − d spatial labels. Then
Eqs. (2.4),(2.5) and standard results [3,5-7,9] in the orbifold program give
the following explicit form of the extended Virasoro generators (2.3) in this
sector:
∆0(σ) + (3.16a)
Gab(d)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p + u−v−ǫ
fj(σ)
fj(σ)−1
×:Ĵ̂jv
p+ ̂
fj(σ)
Ĵ−̂,j,u−v
m− p− ̂
fj(σ)
+ u−v
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
fj(σ)
≥ 0 (3.16b)
fj(σ) = 26− d, 4 ≤ d ≤ 26. (3.16c)
Here ǫ = 0 or 1 for ω = 1l or −1l, G(d) is the restriction of the flat target-space
metric (2.4b) to the first d labels, fj(σ) is the size of the jth cycle in ω(perm),
1The spectra of these two ĉ = 52 strings look even more familiar in terms of the
dimensionful momenta k ≡ Ĵ(0)/
, where α′
is the conventional open-string Regge
slope.
and the previous cases with δ̂0(σ) = 0 are included when d = 26. The half-
integer moded currents in the second term of (3.16) satisfy the twisted current
algebra (2.3b) with G → G(d). For the permutation-twisted currents in the
last term of (3.16), I have used the standard relation (n(r)/ρ(σ)) = (̂/fj(σ))
and (the inverse of) the twisted metric [3,5-7,9]
̂j;l̂l
(σ) = δjlfj(σ)δ̂+l̂,0modfj(σ) (3.17)
which also determines the twisted current algebra (2.3b) for these currents.
Using Eq. (3.16b), we see that the non-trivial element of Z2 on two labels
also gives δ̂0(σ) = 0 and a P̂
= −2 ground state, but a non-trivial element
of Z3 on three labels gives a slightly-raised ground state
δ̂0(σ) =
, ∆̂0(σ) =
, P̂ 2(0) = −
(3.18)
and no photons.
Given the cycle-structure {fj(σ)} of any extra automorphism w(perm)
(see e.g. Eq. (3.4) of Ref. [16]), it is straightforward to evaluate the sum in
Eq. (3.16b). As an illustration, one finds the simple tachyonic ground-state
mass-squares
5 ≤ (d = prime) ≤ 23 : P̂ 2(0) = −
(d− 2 +
26− d
) (3.19)
in twisted sectors which correspond to the action of any non-trivial element
of the cyclic group Zλ of prime order on 3 ≤ (λ = 26−d) ≤ 21 spatial labels.
The result (3.19) includes Eq. (3.18) when d = 23, but does not extend to
the cases d = 26, 24 with P̂ 2(0) = −2 discussed above. I remind that this
result applies only to the open orbifold-strings, while twice these values of
P̂ 2(0) are obtained for the closed-string versions.
Further analysis of the ĉ = 52 strings, including the “larger subset” of
examples (3.16), is found in the following section.
4 Equivalent c = 26 Description
of the ĉ = 52 Spectra
In fact, there exists an entirely equivalent description of all the ĉ = 52 string
spectra in terms of generically-unconventional Virasoro generators at c = 26.
To obtain the c = 26 description, I first define the relabelled (unhatted)
operators
Jn(r)µ
2m+ u+
2n(r)
≡ Ĵn(r)µu
, u = 0, 1 (4.1a)
L(2m+ u) ≡ 2L̂u
(4.1b)
in terms of the hatted operators above. This 1−1 map is recognized as
a modest generalization of (the inverse of) the order-two orbifold-induction
procedure of Borisov, Halpern and Schweigert [1]. SinceM ≡ 2m+u, u = 0, 1
covers the integers once, we then find from (2.3) the explicit form of the c = 26
generators:
L(M) = δ̂0(σ)δM,0 + (4.2a)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)
:Jn(r)µ
2n(r)
J−n(r),ν
M −Q−
2n(r)
δ̂0(σ) =
dim[n(r)]
(4.2b)
[L(M), L(N)] = (M −N)L(M +N) + 26
M(M2 − 1)δM+N,0 (4.2c)
L(M), Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
M +N +
2n(r)
(4.2d)
Jn(r)µ
2n(r)
, Jn(s)ν
2n(s)
(4.2e)
= δn(r)+n(s),0modρ(σ)δM+N+2(n(r)+n(s)ρ(σ) ),0
Gn(r)µ;−n(r),ν(σ).
The expression (4.2b) for δ̂0(σ) is the same as above, and the mode-normal
ordering in Eq. (4.2a)
:Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
:M (4.3)
2n(r)
Jn(s)ν
2n(s)
Jn(r)µ
2n(r)
2n(r)
Jn(r)µ
2n(r)
Jn(s)ν
2n(s)
follows from the ĉ = 52 ordering (2.6) because the map (4.1) preserves the
sign of all arguments.
I emphasize that the c = 26 Virasoro generators in Eq. (4.2) are generically-
unconventional because the twisted matter is now summed over the fractions
{2n/ρ} instead of the conventional orbifold-fractions {n/ρ}. This distortion
of the “extra twist” is the price we must pay in order to unwind the “basic
twist” associated to the basic permutations τ∓ of H∓.
The map (4.1) also tells us that the ĉ = 52 momenta {Ĵ(0)} and the
c = 26 momenta {J(0)} are identical, and we may record
J(0) = Ĵ(0) : J0µ(0) = Ĵ0µ0(0), Jρ(σ)/2,µ(0) = Ĵρ(σ)/2,µ,1(0) (4.4a)
P 2 = P̂ 2 (4.4b)
G0µ:0ν(σ)J0µ(0)J0ν(0) +
,µ;− ρ(σ)
,ν(σ)Jρ(σ)/2,µ(0)J−ρ(σ)/2,ν(0)
where the ĉ = 52 form of P̂ 2 was given in Eq. (3.4b). Similarly, the “level-
number” operator R(σ) in the decomposition of L(0) is the same
L(0) = −1
P 2 +R(σ)
+ δ̂0(σ) (4.5a)
R(σ) = R̂(σ) (4.5b)
r,µ,ν
Gn(r)µ;−n(r),ν(σ)×
×:Jn(r)µ
2n(r)
J−n(r),ν
2n(r)
where the ĉ = 52 form of R̂(σ) was given in Eq. (3.4c).
By itself, the inverse orbifold-induction procedure (4.1) is only a rela-
belling of the operators of the permutation-orbifold CFT’s. The central point
of this discussion however is that for the orbifold-string theories – restricted
by the extended physical state conditions (1.1) – the map also gives us a
completely equivalent c = 26 description of the physical spectrum of each
ĉ = 52 orbifold-string. Indeed, it is easily checked that both components
ū = 0, 1 of the ĉ = 52 extended physical-state condition (1.1a) map directly
onto the simpler and in fact conventional physical-state condition
L(M ≥ 0)|χ〉 = δM,0|χ〉 (4.6)
in the 26-dimensional description! A right- mover copy of Eq. (4.6) on the
same physical states {|χ〉} is similarly obtained in the equivalent c = 26
description of the closed orbifold-strings.
I emphasize that the physical states {|χ〉} of the 26-dimensional descrip-
tion (4.6) are exactly the original physical states (1.1a) of the ĉ = 52 string.
Indeed, each physical state |χ〉 can be regarded as invariant under the map,
or each can now be rewritten in 26-dimensional form. In further detail, Eqs.
(4.5) and (4.6) give the same spectral condition P 2 ≃ P 20 + R(σ), the same
physical ground state 2
|0, J(0)〉σ ≡ |0, Ĵ(0)〉σ, P
0 = P̂
0 = −2 + 2δ̂0(σ) (4.7)
and each negatively-moded hatted current in any physical state can be re-
placed according to Eq. (4.1a) by the corresponding unhatted current mode.
Note finally that the commutator (4.2d) and the decomposition (4.5a) give
the 26-dimensional increment
∆(P̂ 2) = ∆(R(σ)) = 2
2n(r)
(4.8)
which results from the addition of Jn(r)µ
2n(r)
to any previous
state. With M = 2m+ n, these are recognized as the same increments (3.9)
obtained in the ĉ = 52 description.
As simple examples, consider the “larger subset” (3.16) of ĉ = 52 strings
– whose equivalent c = 26 physical state condition (4.6) now involves the
following subset of the c = 26 Virasoro generators (4.2):
L(M) = δM,0δ̂0(σ) +
Gab(d)
:Jǫa(Q + ǫ)J−ǫ,b(M −Q− ǫ):M +
fj(σ)
fj(σ)−1
:Ĵj
Q + 2̂
fj(σ)
J−̂,j
M −Q− 2̂
fj(σ)
:M (4.9a)
δ̂0(σ) =
fj(σ)−1
fj(σ)
fj(σ)
− 2̂
fj(σ)
(4.9b)
2Although it is not directly relevant in either description of the ĉ = 52 strings, one
notes that the conformal weight of the scalar twist-field state |0〉
of sector σ has now
shifted from ∆̂0(σ) to δ̂0(σ) in the c = 26 description.
a, b = 0, . . . , d− 1,
fj(σ) = 26− d, 4 ≤ d ≤ 26. (4.9c)
Recall for the larger subset that ǫ = 0, 1 corresponds in the symmetric theory
to the action of the extra automorphism ω = ±1l on the first d ≥ 4 labels
{a}, while fj(σ) is the length of the j-th cycle of the extra permutation
ω(perm) which acts on the remaining 26 − d spatial labels. Shifting the
dummy integer Q by the integer ǫ, we note that the second term in Eq.
(4.9a) is a set of ordinary Virasoro generators for d untwisted bosons with
the ordinary current algebra
[Ja(Q), Jb(P )] = G
ab QδQ+P,0 (4.10)
for both values of ǫ. The currents in the third term satisfy the twisted current
algebra (4.2e) with the permutation-twisted metric (3.17), and the value of
δ̂0(σ) in Eq. (4.9b) is only a slightly-rewritten form of that given in Eq.
(3.16b).
We are now in a position to confirm our suspicions in the previous sec-
tion about the simplest orbifold-strings, described earlier at ĉ = 52 by the
extended Virasoro generators:
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
,0, u = 0, 1, ǫ = 0, 1. (4.11)
These are now equivalently described by the choice d = 26 in Eq. (4.9), in
which case only the second (ordinary) term of Eq. (4.9a) is non-zero – and
then the equivalent physical-state condition (4.6) verifies that the physical
spectrum of each of these particular twisted ĉ = 52 strings is indeed equiva-
lent to that of an ordinary untwisted c = 26 string! These cases include the
open-string orientation-orbifold sectors corresponding to τ− × (ω = ±1l) in
Eq. (3.10) and their T-duals, as well as the twisted closed-string sectors of
the generalized Z2-permutation orbifolds corresponding to τ+ × (ω = ±1l).
Additionally, consider the following special cases of the extended Virasoro
generators (3.16) at ĉ = 52
= δm+u
+ (4.12)
Gab(24)
:Ĵǫav
p+ v+ǫ
Ĵ−ǫ,b,u−v
m− p+ u−v−ǫ
:Ĵ̂v
p+ ̂+v
Ĵ−̂,u−v
m− p+ u−v−̂
which result when the extra automorphism in the symmetric theory acts as
ω = ±1l on the first d = 24 labels and the non-trivial element of a Z2 on the
remaining 2 spatial labels. I have noted in Sec. 3 that δ̂0(σ) = 0 for these
cases as well, and indeed the equivalent c = 26 description (4.6) and (4.9)
at d = 24 now shows that the open and closed orbifold- strings of this type
also have the spectrum of ordinary untwisted c = 26 strings. The common
thread for the orbifold-strings in Eqs. (4.10) and (4.12) is that they are at
most half-integer moded, so that the shift {n/ρ} → {2n/ρ} gives integer
moding in the c = 26 description.
Beyond these simple cases, the ĉ = 52 strings are apparently new – with
δ̂0(σ) 6= 0, unfamiliar ground-state mass-squares, and fractional modeing
(and increments) in either description.
5 Conclusions
We have discussed the physical spectrum of the general ĉ = 52 orbifold-string,
as well as an equivalent but unconventionally-twisted c = 26 description
of the twisted ĉ = 52 matter. The equivalent c = 26 description holds
only for the orbifold-string theories – restricted by the extended physical-
state conditions (1.1) – and not in the larger Hilbert space of the underlying
orbifold conformal field theories.
In general we have found that the spectra of these orbifold-string systems
are unfamiliar. One simple and unexpected conclusion however is that, as
string theories restricted by the extended physical-state conditions, the single
twisted ĉ = 52 sector of each of the simplest orbifolds of permutation-type
(see Eq. (2.9))
(1; τ∓) (5.1a)
(1; τ∓ × ω2), ω
2 = 1 (5.1b)
have the same physical spectra as ordinary untwisted c = 26 strings. No such
equivalence is found of course in the half-integer moded Hilbert space of the
full orbifold CFT’s. The list in Eq. (5.1) includes the simplest orientation
orbifolds (with τ−) and their T-duals, as well as the simplest generalized
Z2-permutation orbifolds (with τ+).
For the simplest orientation orbifolds in particular, the string theories in
Eq. (5.1) consist of an ordinary unoriented closed string (the unit element)
at c = 26 and a ĉ = 52 twisted open string whose physical spectrum is
equivalent to that of an ordinary untwisted c = 26 critical open string. Since
both the closed- and open-string spectra of these simple orientation orbifolds
are equivalent to those of the archtypal orientifold (without Chan-Paton fac-
tors), we are led to suspect that orientation orbifolds include orientifolds. I
will return in the next paper of this series to consider this question at the
interacting level, where we will also be able to ask about the decoupling of
null physical states. Following that, I will consider in a succeeding paper the
corresponding situation and modular invariance for the simplest permutation
orbifold-string systems.
More generally, we have seen that there are many other orientation orb-
ifolds, open-string Z2-permutation orbifolds and generalized Z2-permutation
orbifolds whose ĉ = 52 spectra show fractional modeing in both the ĉ = 52
and the c = 26 descriptions. These include in particular the orbifolds in Eq.
(2.9) when the order n of the extra automorphism is greater than two.
There is more to say about no-ghost theorems for the general twisted
ĉ = 52 string. The original intuition [16] was that the doubled gauges ū = 0, 1
of the extended physical state condition (1.1) could remove the doubled set
of negative-norm states (time-like modes) of the ĉ = 52 strings – which
are also associated with ū = 0, 1. For the simplest ĉ = 52 strings in Eq.
(5.1), this intuition is certainly born out [20]. More generally, the equivalent
c = 26 description of each spectrum shows that both aspects of the doubling
are indeed eliminated at the same time, leaving us with the conventional
physical state condition (4.6) and only a single set of time-like modes. This
is clearly visible in the set of examples (4.9), where the only time-like modes
(a = 0) are included in the second term. For the general ĉ = 52 string,
the reader should bear in mind that the twisted metric G in Eq. (4.2) is
only a unitary transformation (2.5) of the untwisted metric G with a single
time-like direction. Although not yet a proof, and illustrated here only for
ĉ = 52, I consider this a stronger form of the original arguments [16] that
all the critical orbifolds of permutation-type should be free of negative-norm
states.
The next question I wish to address is the following: I have empha-
sized that the equivalent c = 26 Virasoro generators (4.2) are generically-
unconventional, being summed over the matter-field fractions {2n/ρ} instead
of the conventional orbifold fractions {n/ρ}, but are they actually new Vira-
soro generators? I do not know the answer to this question in general, but
at least some of them can in fact be re-expressed by further mode-relabeling
in terms of more familiar Virasoro generators. As examples, consider the
special case of the “larger subset” (4.9) when ω(perm) is one of the elements
of order λ of each cyclic group Zλ. (These are the particular, single-cycle
elements of Zλ with f0(σ) = λ.) When λ is odd, one finds that the first
and third terms of (4.9) can in fact be re-expressed in terms of the conven-
tional Virasoro generators associated to a twisted sector of an ordinary cyclic
permutation orbifold U(1)λ/Zλ [christ]
Lλ(M) =
Q+ ̂
M −Q− ̂
:M (5.2)
+δM,0
, c = λ = 2l + 1
where I have relabeled the currents Ĵ ≡ Ĵ0. To obtain this result from (4.9),
one needs the fact that {2̂/λ} ≃ {̂/λ} modulo the integers when λ is odd.
This observation is consistent with the ground-state mass-squares for prime
λ in Eq. (3.19). When λ is even, I have also checked that the first and third
terms of (4.9) can be re-expressed as the sum of two identical commuting
Virasoro generators of this type
Lλ(M) = Lλ
(M) + L̃λ
(M), c = λ = 2l (5.3)
each of which is associated to a twisted sector of U(1)λ/2/Zλ/2. This result
is also obtained by relabeling the modes modulo the integers, and provides
us with another way to understand that the ground-state mass-squared is
unshifted when ω(perm) is the non-trivial element of a Z2.
My final remark is a conjecture, that the extended physical-state condi-
tions for the twisted strings at ĉ = 26λ, λ prime will in fact read
m+ ̂
|χ〉 = 0, ̂ = 0, 1, . . . , λ− 1 (5.4a)
âλ ≡
13λ2 − 1
(5.4b)
m+ ̂
n+ l̂
m− n+ ̂−l̂
̂+l̂
m+ n + ̂+l̂
+ (5.4c)
+ 26λ
m+ ̂
m+ ̂
̂+l̂
where Eq. (5.4c) is an orbifold Virasoro algebra [1,18,9] of order λ. This form
includes the correct generators {L̂̂} corresponding to the classical extended
Polyakov constraints of Ref. [16], and includes the correct value â2 = 17/8
studied here for the ĉ = 52 strings. I obtained the system (5.4) by requiring
(as we now know for λ = 2) that it map by the inverse of the order-λ
orbifold-induction procedure [1] to the conventional physical-state condition
(4.6) with â1 = 1 at c = 26. One way to test this conjecture would be the
construction of the corresponding twisted BRST systems [17] for these higher
values of ĉ.
Extensions to include winding number and twisted B fields at ĉ = 52 are
also deferred to another time and place.
Acknowledgements
For helpful information, discussions and encouragement, I thank L. Alvarez-
Gaumé, K. Bardakci, I. Brunner, J. de Boer, D. Fairlie, O. Ganor, E. Gi-
mon, C. Helfgott, E. Kiritsis, R. Littlejohn, S. Mandelstam, J. McGreevy, N.
Obers, A. Petkou, E. Rabinovici, V. Schomerus, K. Schoutens, C. Schweigert
and E. Witten. This work was supported in part by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division of
High Energy Physics of the U.S. Department of Energy under Contract DE-
AC02-O5CH11231 and in part by the National Science Foundation under
grant PHY00-98840.
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http://arxiv.org/abs/hep-th/0309101
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Introduction
The Twisted Virasoro Generators of =52 Strings
First Discussion of the = 52 String Spectra
Equivalent c = 26 Description of the = 52 Spectra
Conclusions
|
0704.1541 | Riemannian and Lorentzian structures on the non symmetric space
SO(2m)/Sp(m) | L’espace Riemannien et pseudo-Riemannien
non symétrique SO(2m)/Sp(m).
Elisabeth REMM ∗- Michel GOZE †
Université de Haute Alsace, F.S.T.
4, rue des Frères Lumière - 68093 MULHOUSE - France
2000 Mathematics Subject Classification. Primary 53C30, Secondary 53C20, 53C50,
17Bxx
Mots clés. Homogeneous manifolds, Riemannian structures, non symmetric spaces.
Abstract
In this work, we are interested in a non symmetric homogeneous space, namely
SO(2m)/Sp(m). We show that this space admits a structure of (Z2)
2-symmetric
space. We describe all the non degenerated metrics and classify the Riemannian and
Lorentzian ones.
Dans ce travail, nous nous intéressons à un espace homogène non symétrique, à
savoir SO(2m)/Sp(m). Nous montrons que cet espace admet une structure d’espace
2-symétrique. Nous décrivons toutes les métriques non dégénérées et classons les
métriques riemanniennes et lorentziennes.
1 Rappel. Espaces Γ-symétriques
Soit Γ un groupe abélien fini. Un espace Γ-symétrique est un espace homogène M = G/H
où G est un groupe de Lie connexe et H un sous-groupe fermé de G tel qu’il existe un
homomorphisme injectif de groupes
ρ : γ ∈ Γ −→ ρ(γ) ∈ Aut(G),
tel que le groupe H vérifie (GΓ)1 ⊂ H ⊂ G
Γ où GΓ = {g ∈ G/∀γ ∈ Γ, ρ(γ)(g) = g} et (GΓ)1
sa composante connexe passant par l’élément neutre 1 de G. On a en particulier:
ρ(γ1) ◦ ρ(γ2) = ρ(γ1 · γ2),
ρ(eΓ) = Id où eΓ est l’élément neutre de Γ,
∀γ ∈ Γ, ρ(γ)(g) = g ⇐⇒ g ∈ H dès que H est connexe.
∗corresponding author: e-mail: [email protected]
†[email protected].
http://arxiv.org/abs/0704.1541v1
Si M = G/H est un espace Γ-symétrique, alors l’algèbre de Lie g de G est Γ-graduée:
g = ⊕γ∈Γgγ
[gγ1 , gγ2 ] ⊂ gγ1·γ2
geΓ = h
où h est l’algèbre de Lie de H . On dit que la paire (g, h) est l’espace Γ-symétrique local de
G/H . Si G est simplement connexe alors toute algèbre de Lie Γ-graduée g = ⊕γ∈Γgγ définit
une paire (g, ge) qui est l’espace Γ-symétrique local d’un espace Γ-symétrique G/H .
Soit M = G/H un espace Γ-symétrique. En tout point x de M on peut définir un
sous-groupe Γx de Diff(M) isomorphe à Γ tel que x soit le seul point fixe commun à tous
les éléments sγ,x de Γx.
Les premiers exemples d’espaces Γ-symétriques lorsque Γ n’est pas cyclique correspon-
dent au cas où Γ = (Z2)
2. Rappelons que tout espace Z2-symétrique n’est rien d’autre qu’un
espace symétrique.
Exemple. La sphère S3.
Nous savons que la sphère S3 (ou plus généralement Sn) est munie d’une structure
d’espace symétrique. Il suffit de considérer un difféomorphisme de Sn sur l’espace homogène
SO(n+ 1)/SO(n). L’algèbre de Lie so(n+ 1) admet une Z2-graduation
so(n+ 1) = so(n)⊕m
où so(n) est l’ensemble des points fixes de l’automorphisme involutif
τa : so(n+ 1) → so(n+ 1)
donné par
τa(A) = SAS
avec S =
−1 0n
0n In
et In est la matrice identité d’ordre n.
Nous pouvons munir la sphère S3 d’une structure d’espace (Z2)
2- symétrique en con-
sidérant un difféomorphisme de S3 sur l’espace homogène SO(4)/Sp(2). Considérons sur
l’algèbre de Lie so(4) les automorphismes τa, τb, τc = τa ◦ τb donnés par
τa : M ∈ SO(4) 7→ J
a MJa
τb : M ∈ SO(4) 7→ J
0 −1 0 0
1 0 0 0
0 0 0 1
0 0 −1 0
et Jb =
0 0 0 1
0 0 −1 0
0 1 0 0
−1 0 0 0
La famille {Id, τa, τb, τc} détermine un sous-groupe de Aut(so(4)) isomorphe à (Z2)
2. Si
ga = {M ∈ so(4) / τa(M) = M, τb(M) = −M} ,
gb = {M ∈ so(4) / τa(M) = −M, τb(M) = M}
gc = {M ∈ so(4) / τa(M) = −M, τb(M) = −M} ,
on a la décomposition (Z2)
2-symétrique de so(4):
so(4) = sp(2)⊕ ga ⊕ gb ⊕ gc.
En effet l’ensemble de points fixes pour τa, τb et τc est la sous-algèbre de so(4) dont les
éléments sont les matrices
0 −a2 −a3 −a4
a2 0 −a4 a3
a3 a4 0 −a2
a4 −a3 a2 0
Cette sous-algèbre est isomorphe à sp(2). Ainsi (so(4), sp(2)) est un espace local (Z2)
symétrique déterminant une structure d’espace (Z2)
2-symétrique sur la shpère
S3 = SO(4)/Sp(2).
La structure d’espace symétrique sur S3 est liée à l’existence en tout point x ∈ S3 d’une
symétrie sx ∈ Diff(S
3) vérifiant s2x = Id et x est le seul point tel que sx(x
′) = x′ (seul
point fixe).
La structure d’espace (Z2)
2-symétrique de S3 donne l’existence d’un sous-groupe Γx de
Diff(S3) de ”symétries” de S3:
Γx = {id, sa,x, sb,x, sc,x}
s2x,a = s
= s2x,c = Id
sx,a ◦ sx,b = sx,b ◦ sx,a = sx,c
sx,b ◦ sx,c = sx,c ◦ sx,b = sx,a
sx,a ◦ sx,c = sx,c ◦ sx,a = sx,b
et x est le seul point fixe commun à toutes les symétries:
(sx,a(x
′) = sx,b(x
′) = sx,c(x
′) = x′) ⇔ x = x′.
Notons que chacune des symétries peut avoir un ensemble de points fixes non réduit à {x}
et donc chacune des symétries ne déterminent pas une structure d’espace symétrique sur
S3. Déterminons ces symétries.
Pour tout γ ∈ Γ = (Z2)
2 = {e, a, b, c}, considérons l’automorphisme de SO(4) donné par
ρa(A) = J
a AJa
ρb(A) = J
ρc(A) = ρa ◦ ρb(A)
ρe(A) = A
Si x = [A] désigne la classe dans l’espace homogène SO(4)/Sp(2) de la marice A ∈ SO(4),
alors sa,x[A] = [J
a AJa], sb,x[A] = [J
AJb], sc,x[A] = [J
c AJc], où Jc = JaJb. Ainsi
chacune des symétries a un grand cercle comme variété de points invariants.
2 Espaces riemanniens Γ-symétriques
Soit (M = G/H,Γ) un espace homogène Γ-symétrique.
Définition 1 Une métrique riemannienne g sur M est dite adaptée à la structure Γ-
symétrique si chacune des symétries sγ,x est une isométrie.
Si ▽g est la connexion de Levi-Civita de g, cette connexion ne cöıncide pas en général
avec la connexion canonique ▽ de l’espace homogène (Γ-symétrique). Ces deux connexions
cöıncident si et seulement si g est naturellement réductive.
Par exemple dans le cas de la sphère S3 considérée comme espace (Z2)
2-symétrique,
les métriques adaptées à cette structure sont les métriques sur SO(4)/Sp(2) invariantes
par SO(4) chacune étant définie par une forme bilinéaire symétrique B sur so(4) qui est
ad(sp(2))-invariante. Si so(4) = sp(2) ⊕ ga ⊕ gb ⊕ gc est la décomposition (Z2)
2-graduée
correspondante, le fait de dire que sur S3 les symétries sγ,x sont des isométries est équivalent
à dire que les espaces ge, ga, gb, gc sont deux à deux orthogonaux pour B. Décrivons en détail
cette graduation:
sp(2) =
0 −a2 −a3 −a4
a2 0 −a4 a3
a3 a4 0 −a2
a4 −a3 a2 0
, ga =
0 0 0 x
0 0 −x 0
0 x 0 0
−x 0 0 0
0 y 0 0
−y 0 0 0
0 0 0 −y
0 0 y 0
et gc =
0 0 z 0
0 0 0 z
−z 0 0 0
0 −z 0 0
Si {A1, A2, A3, X, Y, Z} est une base adaptée à cette graduation et si {α1, α2, α3, ω1, ω2, ω3}
en est la base duale alors
B |ga⊕gb⊕gc= λ
1 + λ
2 + λ
La métrique correspondante sera naturellement réductive si et seulement si λ1 = λ2 = λ3
et dans ce cas-là elle correspond à la restriction de la forme de Killing Cartan.
3 Métriques adaptées à la structure (Z2)
2-symétrique de
SO(2m)/Sp(m)
3.1 La graduation (Z2)
2-symétrique
Considérons les matrices
−In 0
, Xa =
, Xb =
, Xc =
Soit M ∈ so(2m). Les applications
τa(M) = J
a MJa
τb(M) = J
τa(M) = J
c MJc
où Ja = Sm ⊗ Xa, Jb = Sm ⊗ Xb, Jc = Sm ⊗ Xc sont des automorphismes involutifs de
so(2m) qui commutent deux à deux. Ainsi {Id, τa, τb, τc} est un sous groupe de Aut(so(2m))
isomorphe à (Z2)
2. Il définit donc une (Z2)
2-graduation
so(2m) = ge ⊕ ga ⊕ gb ⊕ gc
ge = {M ∈ so(2m) / τa(M) = τb(M) = τc(M) = M}
ga = {M ∈ so(2m) / τa(M) = τc(M) = −M, τb(M) = M}
gb = {M ∈ so(2m) / τb(M) = τc(M) = −M, τa(M) = M}
gc = {M ∈ so(2m) / τa(M) = τb(M) = −M, τc(M) = M}
Ainsi
A1 B1 A2 B2
−B1 A1 B2 −A2
−tA2 −
tB2 A1 B1
tA2 −B1 A1
tA1 = −A1,
tB1 = B1
tA2 = A2,
tB2 = B2
X1 Y1 Z1 T1
Y1 −X1 −T1 Z1
tT1 −X1 −Y1
−tT1 −
tZ1 −Y1 X1
tX1 = −X1,
tY1 = −Y1
tZ1 = −Z1,
tT1 = T1
X2 Y2 Z2 T2
−Y2 X2 T2 Z2
−tZ2 −
tT2 −X2 −Y2
−tT2 −
tZ2 Y2 −X2
tX2 = −X2,
tY2 = Y2
tZ2 = −Z2,
tT2 = −T2
X3 Y3 Z3 T3
Y3 −X3 −T3 Z3
tT3 X3 Y3
−tT3 −
tZ3 Y3 −X3
tX3 = −X3,
tY3 = −Y3
tZ3 = Z3,
tT3 = −T3
Notons que dimge = m(2m+ 1), dimga = dimgb = dimgc = m(2m− 1).
Proposition 2 Dans cette graduation ge est isomorphe à sp(m) et toute (Z2)
2-graduation
de so(2m) telle que ge soit isomorphe à sp(m) est équivalente à la graduation ci-dessus.
En effet ge est simple de rang m et de dimension m(2m+1). La deuxième partie résulte
de la classification donnée dans [1] et [2].
Corollaire 3 Il n’existe, à équivalence près, qu’une seule structure d’espace homogène (Z2)
symétrique sur l’espace homogène compact SO(2m)/Sp(m).
Cette structure est associée à l’existence en tout point x de SO(2m)/Sp(m) d’un sous-
groupe de Diff(M) isomorphe à (Z2)
2. Notons Γx ce sous-groupe. Il est entièrement défini
dès que l’on connait Γ1̄ où 1̄ est la classe dans SO(2m)/Sp(m) de l’élément neutre 1 de
SO(2m). Notons
Γ1̄ =
se,1̄, sa,1̄, sb,1̄, sc,1̄
les symétries sγ,1̄(x) = π(ργ(A)) où π : SO(2m) → SO(2m)/Sp(m) est la submersion
canonique, x = π(A) et ργ est un automorphisme de SO(2m) dont l’application tangente
en 1 coincide avec τγ . Ainsi
ρa(A) = J
a AJa
ρb(A) = J
ρc(A) = J
c AJc
Si B ∈ π(A) alors il existe P ∈ Sp(m) tel que B = AP . On a J−1a BJa = J
a AJaJ
a PJa =
J−1a AJa car P est invariante pour tous les automorphismes ρa, ρb, ρc.
3.2 Structure métrique (Z2)
2-symétrique
Une métrique non dégénérée g invariante par SO(2m) sur SO(2m)/Sp(m) est adaptée à la
2-structure si les symétries sx,γ sont des isométries c’est-à-dire si les automorphismes
ργ induisent des isométries linéaires.
Ceci implique que g soit définie par une forme bilinéaire symétrique non dégénérée B sur
ga ⊕ gb ⊕ gc telle que les espaces ga, gb, gc soient deux à deux orthogonaux. Déterminons
toutes les formes bilinéaires B vérifiant les hypothèses ci-dessus. Une telle forme s’écrit donc
B = Ba +Bb +Bc
où Ba(resp. Bb, resp. Bc ) est une forme bilinéaire symétrique non dégénérée invariante par
ge dont le noyau contient gb ⊕ gc (resp. ga ⊕ gc, resp. ga ⊕ gb).
3.3 Exemples
1) Dans le cas de la sphère SO(4)/Sp(2) la métrique adaptée à la structure (Z2)
2-symétrique
est définie par la forme bilinéaire B sur ga⊕gb⊕gc qui est ad(sp(2))-invariante. Nous avons
vu qu’une telle forme s’écrivait
B = λ21ω
1 + λ
2 + λ
Elle est définie positive si et seulement si les coefficients λi sont positifs ou nuls.
2) Considérons l’espace (Z2)
2-symétrique compact SO(8)/Sp(4). Afin de fixer les nota-
tions écrivons la (Z2)
2-graduation de so(8) ainsi:
X1 Y1 Z1 T1
Y1 −X1 −T1 Z1
tT1 −X1 −Y1
−tT1 −
tZ1 −Y1 X1
tX1 = −X1,
tY1 = −Y1
tZ1 = −Z1,
tT1 = T1
X2 Y2 Z2 T2
−Y2 X2 T2 Z2
−tZ2 −
tT2 −X2 −Y2
tZ2 Y2 −X2
tX2 = −X2,
tY2 = Y2
tZ2 = −Z2,
tT2 = −T2
X3 Y3 Z3 T3
Y3 −X3 −T3 Z3
tT3 X3 Y3
−tT3 −
tZ3 Y3 −X3
tX3 = −X3,
tY3 = −Y3
tZ3 = Z3,
tT3 = −T3
et pour la matrice Xi (resp. Yi, Zi, Ti) on notera Xi =
−xi 0
si elle est anti-
symétrique ou Xi =
x1i x
x2i x
si elle est symétrique, c’est à dire Xi =
Enfin on notera par les lettres αi, βi, γi, δi les formes linéaires duales des vecteurs définis
respectivement par les matrices Xi, Yi, Zi, Ti. Ainsi si Xi est antisymétrique, la forme duale
correspondante sera notée αi, et si Xi est symétrique, les formes duales α
i , α
i , α
i cor-
respondent aux vecteurs
. Ceci étant la forme B s’écrit
Ba+Bb+Bc où la forme Bγ a pour noyau gγ1 ⊕gγ2 avec gγ 6= gγ1 et gγ 6= gγ2 . Déterminons
Ba. Comme elle est invariante par ad(sp(2)) on obtient:
Ba(X1, Y1) = Ba(X1, Z1) = Ba(X1, T
1) = 0
Ba(Y1, Z1) = Ba(Y1, T
1) = 0
Ba(Z1, T
1) = Ba(T
1 ) = 0pour i = 1, 3
Ba(X1, X1) = Ba(Y1, Y1) = Ba(Z1, Z1) = Ba(T
1 , T
1 ) = 0
1 , T
1 ) = Ba(T
1 , T
Ba(X1, X1) = 2Ba(T
1 , T
1 )− 2Ba(T
1 , T
Ainsi la forme quadratique associée s’écrit
qga = λ1(α
1 + β
1 + γ
1 + (δ
2) + λ2((δ
2) + (δ31)
2) + (λ2 −
)((δ11)(δ
qga = λ1(α
1 + β
1 + γ
1 + (δ
2) + (
)(δ11 + δ
2 + (
)(δ11 − δ
De même nous aurons
qgb = λ3(α
2 + (β
2 + γ22 + δ
2) + (
)(β12 + β
2 + (
)(β12 − β
qgc = λ5(α
3 + β
3 + (γ
2 + δ23) + (
)(γ13 + γ
2 + (
)(γ13 − γ
Remarques. 1. La forme B définit une métrique riemannienne si et seulement si
λ2p >
λ2p−1
pour p = 1, 2, 3. Si cette contrainte est relachée, la forme B, supposée non dégénérée, peut
définir une métrique pseudo riemannienne sur l’espace (Z2)
2-symétrique. Nous verrons cela
dans le dernier paragraphe.
2. Considérons le sous-espace ge ⊕ ga. Comme [ga, ga] ⊂ ge c’est un sous-algèbre de
so(8) (ou plus généralement de g) admettant une stucture symétrique. La forme Ba induit
donc une structure riemannienne ou pseudo-riemannienne sur l’espace symétrique associé à
l’espace symétrique local (ge, ga). Dans l’exemple précédent ge ⊕ ga est la sous-algèbre de
so(8) donnée par les matrices:
X1 X3 X4 X5
−tX3 X2 X6 −
−tX4 −
tX6 X2
−tX5 X4 −
tX3 X2
tX1 = −X1,
tX2 = −X2
tX5 = X5,
tX6 = X6
Dans [3], on détermine les espaces réels en étudiant ces structures symétriques ge ⊕ ga
données par deux automorphismes commutant de g. En effet si g est simple réelle et si
σ est un automorphisme involutif de g, il existe une sous-algèbre compacte maximale g1
de g qui est invariante par σ et l’étude des espaces locaux symétriques (g, ge) se ramène
à l’étude des espaces locaux symétriques (g1, g11) où g1 est compacte. Dans ce cas g est
définie à partir de g1 par un automorphisme involutif τ commutant avec l’automorphisme
σ. Ici notre approche est en partie similaire mais le but est de regarder la structure des
espaces non symétrique associés aux paires (g, ge).
Dans le cas particulier de l’espace (Z2)
2-symétrique compact SO(8)/Sp(4) l’algèbre de
Lie ge⊕ga est isomorphe à so(4)⊕R où R désigne l’algèbre abélienne de dimension 1. Notons
également que chacun des espaces symétriques ge⊕ga, ge⊕gb, ge⊕gc est isomorphe à so(4)⊕
R. Mais ceci n’est pas général, les algèbres symétriques peuvent ne pas être isomorphes
ni même de même dimension. L’espace symétrique compact connexe associé est l’espace
homogène Su(4)/Sp(2) × T où T est le tore à une dimension. C’est un espace riemannien
symétrique compact non irréductible. La métrique qga définie précédemment correspond
à une métrique riemannienne ou pseudo riemannienne sur cet espace. La restriction au
premier facteur correspond à la métrique associée à la forme de Killing Cartan sur su(4).
Elle correspond à λ2 =
3.4 Cas général: métriques adaptées sur SO(2m)/Sp(m)
Notations. Nous avons écrit une matrice générale de ga sous la forme (3.1). Si on note
(X1, Y1, Z1, T1) un élément de ga, on considère la base de ga, {X1,ij , Y1,ij , Z1,ij , T1,ij} cor-
respondant aux matrices élémentaires . La base duale sera notée (αa,ij , βa,ij , γa,ij , δa,ij).
Rappelons que X1, Y1, Z1 sont antisymétriques alors que T1 est symétrique. Les crochets
correspondent aux représentations de so(m
) sur lui-même ou de so(m
) sur l’espace des
matrices symétriques. On aura donc
qga = λ
(α2a,ij + β
a,ij + γ
a,ij) +
δ2a,ij) + λ
a,ii) + (λ
(δa,iiδa,jj).
Les formes qgb et qgc admettent une décomposition analogue, en tenant compte du fait que
dans gb ce sont les matrices Y2 qui sont symétriques et pour ga les matrices Z1 (3.1). On note
(αb,ij , βb,ij , γb,ij , δb,ij) la base duale de {X2,ij, Y2,ij , Z2,ij, T2,ij} et par (αc,ij , βc,ij, γc,ij , δc,ij)
la base duale de {X3,ij , Y3,ij , Z3,ij , T3,ij}.
Proposition 4 Toute métrique non dégénéré adaptée à la structure (Z2)
2-symétrique de
l’espace homogène SO(2m)/Sp(m) est définie à partir de la forme bilinéaire ad(ge)-invariante
sur ga ⊕ gb ⊕ gc B = qga + qgb + qgb avec
qga = λ
(α2a,ij + β
a,ij + γ
i6=j δ
a,ij) + λ
a,ii) + (λ
(δa,iiδa,jj)
qgb = λ
+ γ2ij) + δ
i6=j β
) + λb2(β
) + (λb2 −
(βb,iiβb,jj)
qgc = λ
(β2c,ij + γ
c,ij) + δ
c,ij +
i6=j α
c,ij) + λ
c,ii) + (λ
(αc,iiαc,jj)
4 Métriques pseudo-riemanniennes (Z2)
2-symétriques sur
SO(2m)/Sp(m)
4.1 Signature des formes qgγ
Soit γ ∈ {a, b, c}. Les valeurs propres de la forme qgγ sont
µ1,γ = λ
1 , µ2,γ = λ
2/2 + λ
1/4, µ3,γ = λ
r + 1
r − 1
où r est l’ordre commun des matrices symétriques X4, Y2, Z1. Ces valeurs propres sont
respectivement de multiplicité dimgγ − r, r − 1, 1. Le signe des valeurs propres µ2,γ et µ3,γ
est donc
µ2,γ > 0 ⇐⇒ λ
2 > −λ
µ3,γ > 0 ⇐⇒ λ
2 > −λ
2(r+1)
On en déduit, si s(q) désigne la signature de la forme quadratique q :
s(qgγ ) = (dimgγ , 0) ⇔ (λ
1 > 0, λ
2 > λ
2(r+1)
= (dimgγ − 1, 1) ⇔ (λ
1 > 0, −λ
1/2 < λ
2 < λ
2(r+1)
= (dimgγ − r, r) ⇔ (λ
1 > 0, λ
2 < −λ
= (r, dimgγ − r) ⇔ (λ
1 < 0, λ
2 > −λ
= (1, dimgγ − 1) ⇔ (λ
1 < 0, λ
2(r+1)
2 < −λ
= (0, dimgγ) ⇔ (λ
1 < 0, λ
2 < λ
2(r+1)
Notons que µ2,γ = µ3,γ si et seulement si λ
1 = 2λ
4.2 Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m)
Comme r = m
2+m−2
m2+m+2
on a le résultat suivant
Théorème 5 Toute métrique riemannienne sur SO(2m)/Sp(m) adaptée à la structure
2-symétrique est définie à partir de la forme bilinéaire sur gaø
plusgb ⊕ gc
B = qga(λ
1 , λ
2) + qgb(λ
2) + qgb(λ
1 > 0
2 > λ
2+m−2
2(m2+m+2
pour tout γ ∈ {a, b, c}.
Pour une telle métrique, la connexion de Levi-Civita ne cöıncide pas en général avec la
connexion canonique associée à la structure (Z2)
2-symétrique ( [2]). Ces deux connexions
sont les mêmes si et seulement si la mt́rique riemannienne est naturellement réductive. Elle
correspond donc à la restriction de la forme de Killing (au signe près) de SO(2m). Cette
métrique correspond à la forme bilinéaire B définie par les paramètres
λa1 = λ
1 = λ
1 = 2λ
2 = 2λ
2 = 2λ
4.3 Classification des métriques lorentziennes adaptées sur l’espace
SO(2m)/Sp(m)
Les métriques lorentziennes adaptées à la structure (Z2)
2-symétrique sont définies par les
formes bilinéaires B, définies dans la section précédente, dont la signature est (dim(ga) +
dim(gb) + dim(gc)− 1, 1). On a donc
Théorème 6 Toute métrique lorentzienne sur SO(2m)/Sp(m) adaptée à la structure (Z2)
symétrique est définie par l’une des formes bilinéaires
B = qga(λ
1 , λ
2) + qgb(λ
2) + qgb(λ
∀γ ∈ {a, b, c}, λ
1 > 0
∃γ0 ∈ {a, b, c} tel que − λ
1 /2 < λ
2 < λ
2(r+1)
∀γ 6= γ0, λ
2 > λ
2(r+1)
References
[1] Bahturin Y., Goze M., Γ-symmetric homogeneous spaces. Preprint Mulhouse (2006).
[2] Bouyakoub A., Goze M., Remm E. On Riemannian nonsymmetric spaces and flag
manifolds. Preprint Mulhouse (2006).
[3] Berger M., Les espaces symétriques non compacts, Ann.E.N.S. 74, 2, (1957), 85-177.
Rappel. Espaces -symétriques
Espaces riemanniens -symétriques
Métriques adaptées à la structure (Z2)2-symétrique de SO(2m)/Sp(m)
La graduation (Z2)2-symétrique
Structure métrique (Z2)2-symétrique
Exemples
Cas général: métriques adaptées sur SO(2m)/Sp(m)
Métriques pseudo-riemanniennes (Z2)2-symétriques sur SO(2m)/Sp(m)
Signature des formes qg
Classification des métriques riemanniennes adaptées sur SO(2m)/Sp(m)
Classification des métriques lorentziennes adaptées sur l'espace SO(2m)/Sp(m)
|
0704.1542 | Quasiparticles in Neon using the Faddeev Random Phase Approximation | Quasiparticles in Neon using the Faddeev Random Phase Approximation
C. Barbieri
Gesellschaft für Schwerionenforschung, Planckstr. 1, D-64291, Darmstadt, Germany
D. Van Neck
Laboratory of Theoretical Physics, Ghent University, Proeftuinstraat 86, B-9000 Gent, Belgium
W.H. Dickhoff
Department of Physics, Washington University, St. Louis, MO 63130, USA
(Dated: November 2, 2018)
The spectral function of the closed-shell Neon atom is computed by expanding the electron self-energy
through a set of Faddeev equations. This method describes the coupling of single-particle degrees of free-
dom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using
the Random Phase Approximation, rather than the Tamm-Dancoff framework employed in the third-order al-
gebraic diagrammatic contruction [ADC(3)] method. The difference between these two approaches is studied,
as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained
for the ionization energies as well as the energy of the ground state with the Faddeev-RPA scheme that is also
appropriate for the high-density electron gas.
PACS numbers: 31.10.+z,31.15.Ar
I. INTRODUCTION
Ab initio treatments of electronic systems become unwork-
able for sufficiently complex systems. On the other hand,
the Kohn-Sham formulation [1] of density functional the-
ory (DFT) [2] incorporates many-body correlations (beyond
Hartree-Fock), while only single-particle (sp) equations must
be solved. Due to this simplicity DFT is the only feasible
approach in some modern applications of electronic structure
theory. There is therefore a continuing interest both in devel-
oping new and more accurate functionals and in studying con-
ceptual improvements and extensions to the DFT framework.
In particular it is found that DFT can handle short-range inter-
electronic correlations quite well, while there is room for im-
provements in the description of long-range (van der Waals)
forces and dissociation processes.
Microscopic theories offer some guidance in the devel-
opment of extensions to DFT. Orbital dependent function-
als can be constructed using many-body perturbation theory
(MBPT) [3, 4]. More recently, the development of general
ab initio DFT [5, 6] addressed the lack of a systematic im-
provement in DFT methods. In this approach one considers
an expansion of the exact ground-state wave function (e.g.,
MBPT or coupled cluster) from a chosen reference determi-
nant. Requiring that the correction to the density vanishes at a
certain level of perturbation theory allows one to construct the
corresponding approximation to the Kohn-Sham potential.
A different route has been proposed in Ref. [7] by devel-
oping a quasi-particle (QP)-DFT formalism. In the QP-DFT
approach the full spectral function is decomposed in the con-
tribution of the QP excitations, and a remainder or background
part. The latter part is complicated, but does not need to be
known accurately: it is sufficient to have a functional model
for the energy-averaged background part to set up a single-
electron selfconsistency problem that generates the QP exci-
tations. Such an approach is appealing since it contains the
well-developed standard Kohn-Sham formulation of DFT as
a special case, while at the same time emphasis is put on the
correct description of QPs, in the Landau-Migdal sense [8].
Hence, it can provide an improved description of the dynam-
ics at the Fermi surface. Given the close relation between QP-
DFT and the Green’s function (GF) formulation of many-body
theory [9, 10], it is natural to employ ab initio calculations
in the latter formalism to investigate the structure of possible
QP-DFT functionals. In this respect it is imperative to iden-
tify which classes of diagrams are responsible for the correct
description of the QP physics.
Some previous calculations, based on GF theory, have fo-
cused on a self-consistent treatment of the self-energy at the
second order [11, 12, 13] for simple atoms and molecules. For
the atomic binding energies it was found that the bulk of corre-
lations, beyond Hartree-Fock, are accounted for while signifi-
cant disagreement with experiment persists for QP properties
like ionization energies and electron affinities. The formal-
ism beyond the second-order approximation was taken up in
Ref. [14, 15, 16, 17, 18] by employing a self-energy of the
GW type [19]. In this approach, the random phase approx-
imation (RPA) in the particle-hole (ph) channel is adopted
to allow for possible collective effects on the atomic excited
states. The latter are coupled to the sp states by means of dia-
grams like the last two in Fig. 1(c). Two variants of the G0W0
formalism were employed in Ref. [14] (where the subscript
“0” indicates that non-dressed propagators are used). In the
first only the direct terms of the interelectron Coulomb po-
tential are taken into account. In the second version, also the
exchange terms are included when diagonalizing the ph space
[generalized RPA (GRPA)] and in constructing the self-energy
[generalized GW (GGW)]. Although the exchange terms are
known to be crucial in order to reproduce the experimentally
observed Rydberg sequence in the excitation spectrum of neu-
tral atoms, they were found to worsen the agreement between
the theoretical and experimental ionization energies [14].
http://arxiv.org/abs/0704.1542v2
In the GW approach the sp states are directly coupled
with the two-particle–one-hole (2p1h) and the two-hole–one-
particle (2h1p) spaces. However, only partial diagonalizations
(namely, in the ph subspaces) are performed. This procedure
unavoidably neglects Pauli correlations with the third particle
(or hole) outside the subspace. In the case of the GGW ap-
proach, this leads to a double counting of the second order
self-energy which must be corrected for explicitly [20, 21].
We note that simply subtracting the double counted diagram is
not completely satisfactory here, since it introduces poles with
negative residues in the self-energy. More important, the inter-
action between electrons in the two-particle (pp) and two-hole
(hh) subspaces are neglected altogether in (G)GW. Clearly, it
is necessary to identify which contributions, beyond GGW,
are needed to correctly reproduce the QP spectrum.
In this respect, it is known that highly accurate descrip-
tions of the QP properties in finite systems can be obtained
with the algebraic diagrammatic construction (ADC) method
of Schirmer and co-workers [22]. The most widely used third-
order version [ADC(3)] is equivalent to the so-called extended
2p1h Tamm-Dancoff (TDA) method [23] and allows to pre-
dict ionization energies with an accuracy of 10-20 mH in
atoms and small molecules. Upon inspection of its diagram-
matic content, the ADC(3) self-energy is seen to contain all
diagrams where TDA excitations are exchanged between the
three propagator lines of the intermediate 2p1h or 2h1p prop-
agation. The TDA excitations are constructed through a diag-
onalization in either 2p1h or 2h1p space, and neglect ground-
state correlations. However, it is clear that use of TDA leads
to difficulties for extended systems. In the high-density elec-
tron gas e.g., the correct plasmon spectrum requires the RPA
in the ph channel, rather than TDA.
In order to bridge the gap between the QP description in
finite and extended systems, it seems therefore necessary to
develop a formalism where the intermediate excitations in the
2p1h/2h1p propagator are described at the RPA level. This
can be achieved by a formalism based on employing a set of
Faddeev equations, as proposed in Ref. [24] and subsequently
applied to nuclear structure problems [25, 26, 27]. In this ap-
proach the GRPA equations are solved separately in the ph and
pp/hh subspaces. The resulting polarization and two-particle
propagators are then coupled through an all-order summation
that accounts completely for Pauli exchanges in the 2p1h/2h1p
spaces. This Faddeev-RPA (F-RPA) formalism is required if
one wants to couple propagators at the RPA level or beyond.
Apart from correctly incorporating Pauli exchange, F-RPA
takes the explicit inclusion of ground-state correlations into
account, and can therefore be expected to apply to both finite
and extended systems. The ADC(3) formalism is recovered
as an approximation by neglecting ground-state correlations
in the intermediate excitations (i.e. replacing RPA with TDA
phonons).
In this work we consider the Neon atom and apply the F-
RPA method to a nonrelativistic electronic problem for the
first time. The relevant features of the F-RPA formalism (also
extensively treated in Ref. [24]), are introduced in Sect. II.
The application to the Neon atom is discussed in Sec. III,
where we also investigate the separate effects of the ladder
and ring series on the self-energy, as well as the differences
between including TDA and RPA phonons. Our findings are
summarized in Sec. IV. Some more technical aspects are rele-
gated to the appendix, where the interested reader can find the
derivation of the Faddeev expansion for the 2p1h/2h1p propa-
gator, adapted from Ref. [24]. In particular, the approach used
to avoid the multiple-frequency dependence of the Green’s
functions is discussed in App. A 1, along with its basic as-
sumptions. The explicit expressions of the Faddeev kernels
are given in App. A 3. Together with Ref. [24], the appendix
provides sufficient information for an interested reader to ap-
ply the formalism.
II. FORMALISM
The theoretical framework of the present study is that of
propagator theory, where the object of interest is the sp prop-
agator, instead of the many-body wave function. In this paper
we will employ the convention of summing over repeated in-
dices, unless specified otherwise. Given a complete orthonor-
mal basis set of sp states, labeled by α,β,..., the sp propagator
can be written in its Lehmann representation as [9, 10]
gαβ(ω) =
ω − ε+n + iη
ω − ε−k − iη
, (1)
where Xnα = 〈Ψ
0 〉 (Y
α = 〈Ψ
|cα|Ψ
0 〉) are the
spectroscopic amplitudes, cα (c
) are the second quantization
destruction (creation) operators and ε+n = E
n − E
EN0 − E
). In these definitions, |ΨN+1n 〉, |Ψ
〉 are the
eigenstates, and EN+1n , E
k the eigenenergies of the (N ± 1)-
electron system. Therefore, the poles of the propagator reflect
the electron affinities and ionization energies.
The sp propagator solves the Dyson equation
gαβ(ω) = g
αβ(ω) + g
αγ(ω)Σ
γδ(ω)gδβ(ω) , (2)
which depends on the irreducible self-energy Σ⋆(ω). The lat-
ter can be written as the sum of two terms
αβ(ω) = Σ
Vαλ,µν Rµνλ,γδε(ω) Vγδ,βε , (3)
where ΣHF represents the Hartree-Fock diagram for the self-
energy. In Eqs. (2) and (3), g0(ω) is the sp propagator for
the system of noninteracting electrons, whose Hamiltonian
contains only the kinetic energy and the electron-nucleus at-
traction. The Vαβ,γδ represent the antisymmetrized matrix el-
ements of the interelectron (Coulomb) repulsion. Note that
in this work we only consider antisymmetrized elements of
the interaction, hence, our result for the ring summation al-
ways compare to the generalized GW approach. Equation (3)
introduces the 2p1h/2h1p-irreducible propagator R(ω), which
carries the information concerning the coupling of sp states to
more complex configurations. Both Σ⋆(ω) and R(ω) have a
perturbative expansion as a power series in the interelectron
FIG. 1: (Color online) a) Diagrammatic expansion of R(ω) in terms
of the (antisymmetrized) Coulomb interaction and undressed prop-
agators. b) R(ω) is related to the self-energy according to Eq. (3).
c) By substituting the diagrams a) in the latter equation, one finds the
perturbative expansion of the self-energy.
interaction V̂ . Some of the diagrams appearing in the expan-
sion of R(ω) are depicted in Fig. 1, together with the corre-
sponding contributions to the self-energy. Note that already at
zero order in R(ω) (three free lines with no mutual interaction)
the second order self-energy is generated.
Different approximations to the self-energy can be con-
structed by summing particular classes of diagrams. In this
work we are interested in the summation of rings and ladders,
through the (G)RPA equations. In order to include such effects
in R(ω), we first consider the polarization propagator describ-
ing excited states in the N-electron system
Παβ,γδ(ω) =
〈ΨN0 |c
n 〉 〈Ψ
γcδ|Ψ
ENn − E
〈ΨN0 |c
γcδ|Ψ
n 〉 〈Ψ
ENn − E
, (4)
and the two-particle propagator, that describes the addi-
tion/removal of two electrons
gIIαβ,γδ(ω) =
〈ΨN0 |cβcα|Ψ
n 〉 〈Ψ
|ΨN0 〉
EN+2n − E
〈ΨN0 |c
|ΨN−2
〉 〈ΨN−2
|cβcα|Ψ
EN0 − E
. (5)
We note that the expansion of R(ω) arises from applying the
equations of motion to the sp propagator (1), which is associ-
ated to the ground state |ΨN0 〉. Hence, all the Green’s functions
appearing in this expansion will also be ground state based, in-
cluding Eqs. (4) and (5). However the latter contain, in their
Lehmann representations, all the relevant information regard-
ing the excitation of ph and pp/hh collective modes. The ap-
proach of Ref. [24] consists in computing these quantities by
=g II
Π(ph)
(pp/hh)II
(pp/hh)
FIG. 2: (Color online) Diagrammatic equations for the polarization
(above) and the two-particle (below) propagators in the (G)RPA ap-
proach. Dashed lines are always antisymmetrized Coulomb matrix
elements and the full lines represent free (undressed) propagators.
solving the ring-GRPA and the ladder-RPA equations [10],
which are depicted for propagators in Fig. 2. In the more
general case of a self-consistent calculation, a fragmented in-
put propagator can be used and the corresponding dressed
(G)RPA [D(G)RPA] equations [10, 28] solved [see Eqs. (A2a)
and (A2b)]. Since the propagators (4) and (5) reflect two-body
correlations, they still have to be coupled to an additional sp
propagator in order to obtain the corresponding approximation
for the 2p1h and 2h1p components of R(ω). This is achieved
by solving two separate sets of Faddeev equations.
Taking the 2p1h case as an example, one can split R(2p1h)(ω)
in three different components R̄(i)(ω) (i = 1, 2, 3) that differ
from each other by the last pair of lines that interact in their
diagrammatic expansion,
R̄(2p1h)
αβγ,µνλ
(ω) =
αβγ,µνλ(ω) −G
βαγ,µνλ(ω)
i=1,2,3
R̄(i)
αβγ,µνλ
(ω) ,
where G0
(ω) is the 2p1h propagator for three freely propa-
gating lines. These components are solutions of the following
set Faddeev equations [29]
R̄(i)
αβγ,µνλ
(ω) = G0
αβγ,µ′ν′λ′(ω) Γ
µ′ν′λ′,µ′′ν′′λ′′
µ′′ν′′λ′′ ,µνλ
(ω) + R̄(k)
µ′′ν′′λ′′ ,µνλ
(ω) (7)
µ′′ν′′λ′′ ,µνλ(ω) −G
ν′′µ′′λ′′ ,µνλ(ω)
, i = 1, 2, 3
where (i, j, k) are cyclic permutations of (1, 2, 3). The inter-
action vertices Γ(i)(ω) contain the couplings of a ph or pp/hh
collective excitation and a freely propagating line. These are
given in the Appendix in terms of the polarization (4) and
two-particle (5) propagators. Equations. (7) include RPA-like
phonons and fully describe the resulting energy dependence
of R(ω). However, they still neglect energy-independent
contributions–even at low order in the interaction–that also
correspond to relevant ground-state correlations. The latter
can be systematically inserted according to
R(2p1h)
αβγ,µνλ
(ω) = Uµνλ,µ′ν′λ′ R̄
(2p1h)
µ′ν′λ′,µ′′ν′′λ′′
(ω) U†
µ′′ν′′λ′′ ,µνλ
, (8)
where R(ω) is the propagator we employ in Eq. (3), R̄(ω) is
the one obtained by solving Eqs. (7), U ≡ I + ∆U, and I is the
(pp/hh)
Π(ph)
g II (pp/hh)
Π(ph)
FIG. 3: (Color online) Example of one of the diagrams that are
summed to all orders by means of the Faddeev Eqs. (7) (left). The
corresponding contribution to the self-energy, obtained upon inser-
tion into Eq. (3), is also shown (right).
identity matrix. Following the algebraic diagrammatic con-
struction method [22, 23], the energy independent term ∆U
was determined by expanding Eq. (8) in terms of the inter-
action and imposing that it fulfills perturbation theory up to
first order (corresponding to third order in the self-energy).
The resulting ∆U, employed in this work, is the same as in
Ref. [23] and is reported in App. A 3 for completeness. It
has been shown that the additional diagrams introduced by
this correction are required to obtain accurate QP properties.
Equations. (7) and (8) are valid only in the case in which a
mean-field propagator is used to expand R(ω). This is the case
of the present work, which employs Hartree-Fock sp propaga-
tors as input. The derivation of these equations for the gen-
eral case of a fragmented propagator is given in the appendix.
More details about the actual implementation of the Faddeev
formalism to 2p1h/2h1p propagation have been presented in
Ref. [24]. The calculation of the 2h1p component of R(ω)
follows completely analogous steps.
It is important to note that the present formalism includes
the effects of ph and pp/hh motion to be included simulta-
neously, while allowing interferences between these modes.
These excitations are evaluated here at the RPA level and are
then coupled to each other by solving Eqs. (7). This generates
diagrams as the one displayed in Fig. 3, with the caveat that
two phonons are not allowed to propagate at the same time.
Equations. (7) also assure that Pauli correlations are properly
taken into account at the 2p1h/2h1p level. In addition, one can
in principle employ dressed sp propagators in these equations
to generate a self-consistent solution. If we neglect the lad-
der propagator gII (ω) (5) in this expansion, we are left with
the ring series alone and the analogous physics ingredients as
for the generalized GW approach. However, this differs from
GGW due to the fact that no double counting of the second-
order self-energy occurs, since the Pauli exchanges between
the polarization propagator and the third line are properly ac-
counted for (see Fig. 3). Alternatively, one can suppress the
polarization propagator to investigate the effects of pp/hh lad-
ders alone.
It is instructive to replace in the above equations all RPA
phonons with TDA ones; this amounts to allowing only
forward-propagating diagrams in Fig. 2, and is equivalent to
separate diagonalisations in the spaces of ph, pp and hh con-
l 0 1 2 3 4 5 6
rw 2.0 4.0 0.0 0.0 0.0 0.0 0.0
no 12 21 10 10 5 5 5
TABLE I: Parameters that define the sp basis: radius of the confining
wall rw (in atomic units) and number of orbits no used for different
partial waves l. The value of cw is always set to 5 a.u..
figurations, relative to the HF ground state. It can be shown
that using these TDA phonons to sum all diagrams of the type
in Fig. 3 reduces to one single diagonalization in the 2p1h or
2h1p spaces. Therefore, Eqs. (7) and (8) with TDA phonons
lead directly to the “extended” 2p1h TDA of Ref. [23], which
was later shown to be equivalent to ADC(3) in the general
ADC framework [22]. The Faddeev expansion formalism of
Ref. [24] creates the possibility to go beyond ADC(3) by in-
cluding RPA phonons. This is more satisfactory in the limit
of large systems. At the same time, the computational cost
remains modest since only diagonalizations in the 2p1h/2h1p
spaces are required.
Note that complete self-consistency requires the use of
fragmented (or dressed) propagators in the evaluation of all in-
gredients leading to the self-energy. This is outside the scope
of the present paper, but we included partial selfconsistency
by taking into account the modifications to the HF diagram
by employing the correlated one-body density matrix and it-
erating to convergence. This is relatively simple to achieve,
since the 2p1h/2h1p propagator is only evaluated once with
the input HF propagators. Below we will give results with and
without this partial selfconsistency at the HF level.
III. RESULTS
Calculations have been performed using two different
model spaces: (1) a standard quantumchemical Gaussian ba-
sis set, aug-cc-pVTZ for Neon [30], with Cartesian repre-
sentation of the d and f functions; (2) a numerical basis set
based on HF and subsequent discretization of the continuum,
to be detailed below. The aug-cc-pVTZ basis set was used
primarily to check our formalism with the ADC(3) result in
literature (i.e. [31], where this basis was employed). The
HF+continuum basis allows to approach, at least for the ion-
ization energies, the results for the full sp space (basis set
limit).
The HF+continuum is the same discrete model space em-
ployed previously in Refs. [11, 14]. It consists of: (1) Solv-
ing on a radial grid the HF problem for the neutral atom; (2)
Adding to this fixed nonlocal HF potential a parabolic poten-
tial wall of the type U(r) = θ(r − rw)cw(r − rw)
2, placed at a
distance rw of the nucleus. The latter eigenvalue problem has
a basis of discrete eigenstates. This basis is truncated by spec-
ifying some largest angular momentum lmax and the number
of virtual states for each value of l ≤ lmax. (3) Solve the HF
problem again, without the potential wall, in this truncated
discrete space. The resulting basis set is used for the subse-
F-TDA F-RPA F-TDAc F-RPAc Expt.
2p -0.799 -0.791 -0.803 -0.797 (0.94) -0.793 (0.92)
2s -1.796 -1.787 -1.802 -1.793 (0.90) -1.782 (0.85)
1s -32.126 -32.087 -32.140 -32.102 (0.86) -31.70
Etot -128.778 -128.772 -128.836 -128.840 -128.928
TABLE II: Results with the aug-cc-pVTZ basis. The first three rows
list the energies of the main sp fragments below the Fermi level, as
predicted by different self-energies. F-TDA/F-RPA refers to the Fad-
deev summation with TDA/RPA phonons, respectively. In all cases
the self-energy was corrected at third order through Eq. (8). The
suffix “c” refers to partial selfconsistency, when the static (HF-type)
self-energy is consistent with the correlated density matrix. Without
“c” the pure HF self-energy was taken. In the F-RPAc column the
strength of the fragment is indicated between brackets. The last row
is the total electronic binding energy. The experimental values are
taken from Refs. [32, 33]. All energies are in atomic units.
quent Green’s function calculations.
When a sufficiently large number of states is retained after
truncation, the final results should approach the basis set limit.
In particular the results should not depend on the choice of the
auxiliary confining potential. This was verified in Ref. [11] for
the second-order, and in Ref. [14] for the G0W0 self-energy; in
these cases the self-energy is sufficiently simple that extensive
convergence checks can be made for various choices of the
auxiliary potential. The parameters of the confining wall and
the number of sp states kept in the basis set was optimized in
Ref. [11], by requiring that the ionization energy is converged
to about 1 mH for the second-order self-energy. In Ref. [14]
the same choice of basis set was also seen to bring the ioniza-
tion energy for the G0W0 self-energy near convergence. For
completeness, the details of this basis are reported in Table I.
While the self-energy in the present paper is too complicated
to allow similar convergence checks, it seems safe to assume
that basis set effects will affect the calculated ionization ener-
gies by at most 5 mH.
In Table II we compare, for the aug-cc-pVTZ basis, the ion-
ization energies of the main single-hole configurations when
TDA or RPA phonons are employed in the Faddeev construc-
tion (this is labeled F-TDA and F-RPA, respectively, in the
table). Note that use of TDA phonons corresponds to the
usual ADC(3) self-energy. We find that the replacement of
TDA with RPA phonons provides more screening, leading to
slightly less bound poles which are shifted towards the exper-
imental values. This shift increases with binding energy. As
discussed at the end of Sec. I, one can include consistency of
the static part of the self-energy. About eight iterations are
needed for convergence. This is a nonnegligible correction,
providing about 5 mH more binding (i.e. larger ionization en-
ergies) for the valence/subvalence 2p and 2s, 15 mH for the
deeply bound 1s, and 60 mH to the total binding energy. Our
converged result for the Faddeev-TDA self-energy (labeled F-
TDAc in Table II) is in good agreement with the ADC(3) value
for the 2p ionization energy (-0.804 H) quoted in [31], as it
should be.
The analogous results obtained with the larger
F-TDA F-RPA F-TDAc F-RPAc Expt.
2p -0.807 -0.799 -0.808 -0.801 (0.94) -0.793 (0.92)
2s -1.802 -1.792 -1.804 -1.795 (0.91) -1.782 (0.85)
1s -32.136 -32.097 -32.142 -32.104 (0.81) -31.70
Etot -128.863 -128.857 -128.883 -128.888 -128.928
TABLE III: Results with the HF+continuum basis set from Table I.
See also the caption of Table II.
-40 -35 -30
[a.u.]
F-RPA(ring)
F-RPA(ladder)
F-RPA
FIG. 4: (Color online) Spectral function for the s states in Ne
obtained with various self-energy approximations. From the top
down: the second-order (Σ(2)) self-energy, the F-RPA(ring), the F-
RPA(ladder), and the full F-RPA self-energy. The strength is given
relative to the Hartree-Fock occupation of each shell. Only fragments
with strength larger than Z > 0.005 are shown.
HF+continuum basis are given in Table III, which al-
lows to assess overall stability and basis set effects. We find
exactly the same trends as for aug-cc-pVTZ. In particular
the reduction of ionization energies from the replacement of
TDA with RPA phonons is almost independent of the basis
set used, while the effect of including partial consistency
is roughly halved. Overall, the ionization states are always
more bound with the larger basis set; while the basis set
limit could be still more bound than the present results with
the HF+continuum basis set, it is likely (based on the G0W0
extrapolation in Ref. [14]) that the difference does not exceed
5 mH.
As discussed in Sec. I, the the F-RPA self-energy contains
RPA excitations of both ph type (ring diagrams) and pp/hh
type (ladder diagrams). It is instructive to analyze their sepa-
rate contributions to the final ionization energies, in order to
1s 2s 2p
HF -32.77 (1.00) -1.931 (1.00) -0.850 (1.00)
Σ(2) -31.84 (0.74) -1.736 (0.88) -0.747 (0.91)
G0W0 -31.14 (0.85) -1.774 (0.91) -0.801 (0.94)
F-RPA (ring) -31.82 (0.73) -1.636 (0.56) -0.730 (0.80)
F-RPA (ladder) -32.04 (0.87) -1.802 (0.95) -0.781 (0.96)
F-RPA -32.10 (0.81) -1.792 (0.91) -0.799 (0.94)
Exp. -31.70 -1.782 (0.85) -0.793 (0.92)
TABLE IV: Energy (in a.u.) and strength (bracketed numbers) of the
main fragments in the spectral function of Neon, generated by differ-
ent self-energies. Results for the HF+continuum basis. Consecutive
rows refer to: (1) HF; (2) second-order self-energy; (3) G0W0 results
from Ref. [14]; (4) F-RPA self-energy with only ph rings retained;
(5) F-RPA self-energy with only pp/hh ladders retained; (6) Com-
plete F-RPA self-energy. In all F-RPA results the self-energy was
corrected at third order through Eq. (8). The static self-energy was
pure HF (no partial self-consistency). The experimental values are
taken from Refs. [32, 33].
understand how the F-RPA self-energy is related to the stan-
dard (G)GW self-energy. Table IV compares the results for
the ionization energies, obtained with the second-order self-
energy, to different approximations for including the ring sum-
mations. As one can see, the second-order self-energy gener-
ates an l=1 sp energy of -0.747 mH, which is 46 mH above
the empirical 2p ionization energy. The G0W0 self-energy,
which includes the ring summation with only direct Coulomb
matrix elements, improves this result and brings it close to ex-
periment. The 2s behaves in a similar way. Unfortunately,
including the exchange terms of the interelectron repulsion in
the GG0W0 method turns out to have the opposite effect (the
2p ionization energy becomes -0.712 H [14] [41]) and the
agreement with experiment is lost. Obviously, GG0W0 is too
simplistic to account for exchange in the ph channel.
With the F-RPA(ring) self-energy one can go one step fur-
ther and employ the Faddeev expansion to also force proper
Pauli exchange correlations in the 2p1h/2h1p spaces. As
shown in Table IV, this enhances the screening due to the
exchange interaction terms, leading to even less binding for
the 2s and 2p. The corrections relative to the second-order
self-energy can be large (100 mH for the 2s state) and in the
direction away from the experimental value. We also note that
the larger shift, in the 2s orbit, is accompanied by an increase
of the fragmentation (see Fig. 4 and Tab. IV). Similar observa-
tions were also made in Ref. [14] for other atoms: in general
ring summations in the direct channel alone bring the quasi-
hole peaks close to the experiment. This agreement is then
spoiled as soon as one includes proper exchange terms in the
self-energy. On the other hand, exchange in the ph channel
is required to reproduce the correct Rydberg sequence in the
excitation spectrum of neutral atoms. So further corrections
must arise from other diagrams, and obviously the summation
of ladder diagrams can play a relevant role, since these con-
tribute to the expansion of the self-energy at the same level as
that of the ring diagrams.
The result when only including ladder-type RPA phonons
in the F-RPA self-energy is also shown in Table IV. One can
see that pp/hh ladders do actually work in the opposite way
as the ph channel ring diagrams, and have the same order of
magnitude with, e.g., a shift of 66 mH for the 2s relative to
the second-order result. When combined with the ring dia-
grams in the full F-RPA self-energy, the agreement with ex-
periment is restored again. Note that the final result cannot be
obtained by adding the contributions of rings and ladders, but
depends nontrivially on the interplay between these classes of
diagrams thereby pointing to significant interference effects.
With the F-RPA(ring) self-energy, where only the contribu-
tions of the ph channel are included, the main peaks listed in
Table IV are not only considerably shifted but also strongly
depleted, e.g. a strength of only 0.56 for the main 2s peak.
The complete spectral function for the l = 0 strength in Fig. 4
shows that the depletion of the main fragment is accompanied
by strong fragmentation over several states. While correlation
effects are overestimated in F-RPA(ring), they are suppressed
in F-RPA(ladder), where only the pp/hh ladders are included
in the self-energy. In this case one finds a spectral distribu-
tion closer to the HF one, with a main 2s fragment of strength
0.95 and less fragmentation than the the second-order self-
energy. The spectral distribution generated by the complete
F-RPA self-energy is again a combination of the above ef-
fects. The strength of the deeply bound 1s orbital behaves in
an analogous way. The strength of the main peak is reduced
but several satellite levels appear due to the mixing with 2h1p
configurations. In all the calculations reported in Fig. 4 we
found a summed l = 0 strength exceeding 0.98 in the interval
[-40 H, -30 H] which can be associated with the 1s orbital,
and this remains true even in the presence of strong correla-
tions using the F-RPA(ring) self-energy. Of course, the mix-
ing with 3h2p configurations, not included in this work, may
further contribute to the fragmentation pattern in this energy
region.
IV. CONLUSIONS AND DISCUSSION
In conclusion, the electronic self-energy for the Ne atom
was computed by the F-RPA method which includes –
simultaneously– the effects of both ring and ladder diagrams.
This was accomplished by employing an expansion of the
self-energy based on a set of Faddeev equations. This tech-
nique was originally proposed for nuclear structure applica-
tions [24] and is described in the appendix. At the level of
the self-energy one sums all diagrams where the three propa-
gator lines of the intermediate 2p1h or 2h1p propagation are
connected by repeated exchange of RPA excitations in both
the ph and the pp/hh channel. This differs from the ADC(3)
formalism in the fact that the exchanged excitations are of
the RPA type, rather than the TDA type, and therefore take
ground-state correlations effects into account. The coupling
to the external points of the self-energy uses the same modi-
fied vertex as in ADC(3), which must be introduced to include
consistently all third-order perturbative contributions.
The resulting main ionization energies in the Neon atom
are at least of the same quality, and even somewhat improved,
compared to the ADC(3) result. Note that, numerically, F-
RPA can be implemented as a diagonalization in 2p1h/2h1p
space implying about the same cost as ADC(3). The present
study also shows that in localized electronic systems subtle
cancellations occur between the ring and ladder series. In par-
ticular, only a combination of the ring and ladder series leads
to sensible results, as the separate ring series tends to correct
the second-order result in the wrong direction.
Since the limit to extended systems requires an RPA treat-
ment of excitations, the F-RPA method holds promise to
bridge the gap between an accurate description of quasiparti-
cles in both finite and extended systems. In particular, the GW
treatment of the electron gas has been shown to yield excellent
binding energies, but poor quasiparticle properties [34, 35].
Further progress beyond GW theory requires a consistent in-
corporation of exchange in the ph channel. The F-RPA tech-
nique may be highly relevant in this respect. A common
framework for calculating accurate QP properties in both fi-
nite and extended systems, is also important for constrain-
ing functionals in quasiparticle density functional theory (QP-
DFT) [7].
Finally, complete self-consistency requires sizable compu-
tational efforts for bases as large as the HF+continuum basis
used here. It would nevertheless represent an important exten-
sion of the present work, since it is related to the fulfillment
of conservation laws [36, 37]. These issues will be addressed
in future work.
Acknowledgments
This work was supported by the U.S. National Science
Foundation under grant PHY-0652900.
APPENDIX A: FADDEEV EXPANSION OF THE 2P1H/2H1P
PROPAGATOR
Although only the one-energy (or two-time) part of the
2p1h/2h1p propagator enters the definition of the self energy,
Eq. (3), a full resummation of all its diagrammatic contribu-
tions would require to treat explicitly the dependence on three
separate frequencies, corresponding to the three final lines in
the expansion of R(ω). For example, inserting the RPA ring
(ladder) series in R(ω) implies the propagation of a ph (pp/hh)
pair of lines both forward and backward in time, while the
third line remains unaffected. A way out of this situation is
to solve the Bethe-Salpeter-like equations for the polarization
and ladder propagators separately and then to couple them to
the additional line. If it is assumed that different phonons do
not overlap in time, the three lines in between phonon struc-
tures will propagate only in one time direction [see figures (3)
and (5)]. In this situation the integration over several frequen-
cies can be circumvented following the prescription detailed
in the next subsection. This approach will be discussed in
the following for the general case of a fully fragmented prop-
gator, in order to derive a set of Faddeev equations capable
FIG. 5: (Color online) Diagrammatic representation of Eq. (A3).
Double lines represent fully dressed sp Green’s funcions which, how-
ever, are restricted to propagate only in one time direction [i.e., only
one of the two terms on the r.h.s. of Eq. (1) is retained]. The Faddeev
Eqs. (A9) and (7) allow for both forward and backward propagation
of the phonons Γ(π)(ω) and Γ(II)(ω) as long as these do not overlap
in time. For the propagators, time ordereing is asumed with forwad
propagation in the upward direction.
of dressing the sp propagator self-consistently. Since the for-
ward (2p1h) and the backward (2h1p) parts of R(ω) decouple
in two analogous sets of equations, it is sufficient to focus on
the first case alone.
1. Multiple frequencies integrals
We start by considering the effective interactions in the ph
and pp/hh channels that correspond to Eqs. (4) and (5) stripped
of the external legs. In the present work, these are the follow-
ing two-time objects:
αβ,γδ
(ω) = Vαδ,βγ + Vαν,βµ Π
µν,ρσ(ω) Vρδ,σγ (A1a)
= Vαδ,βγ +
ω − επn + iη
ω + επn′ − iη
αβ,γδ
(ω) = Vαβ,γδ + Vαβ,µν g
µν,ρσ(ω) Vρσ,γδ (A1b)
= Vαδ,βγ +
ω − εΓ+n + iη
ω − εΓ−
where the residues and poles for the ring series are Ωn
〈ΨNn |c
µcν|Ψ
0 〉Vµβ,να and ε
n = E
0 . For the ladders, ∆
〈ΨN+2n |c
0 〉Vµν,αβ and ∆
= Vαβ,µν〈Ψ
k |cµcν|Ψ
0 〉, with
poles εΓ+n = E
0 and ε
k = E
. Equations. (A1)
solve the ring and ladder RPA equations, respectively
αβ,γδ
(ω) = Vαδ,βγ (A2a)
αβ,µν
gµρ(ω + ω1)gσν(ω1) Vρδ,σγ ,
αβ,γδ
(ω) = Vαβ,γδ (A2b)
αβ,µν
gµρ(ω − ω1)gνσ(ω1) Vρσ,γδ .
To display how the phonons (A1a) and (A1b) enter the ex-
pansion of R(ω), we perform explicitly the frequency integrals
for the diagram of Fig. 5. Since it is assumed that the separate
propagators lines evolve only in one time direction, only the
forwardgoing (g>(ω)) or the backwardgoing (g<(ω)) part of
Eq. (1) must be included for particles and holes, respectively.
After some algebra, one obtains
∆Rαβγ,µνλ(ω) =
g>αα1 (ω −Ω) g
(ω1) g
(ω1 −Ω) Γ
β1γ1,σ1λ1
(Ω) g>σ1σ2 (s + Ω − ω)
α1σ2,µ1ν1
(s) g>µ1µ(s − ω2) g
(ω2) g
(s − ω)
ω − (ε+n1 + ε
) + iη
Vβ1λ1,γ1σ1 +
ω − (ε+n1 + ε
nπ) + iη
[ω − επ
− ε+n1 − ε
+ ε−k3
− ε+n4 + ε
− ε+n2 + ε
][−επ
− ε+n4 + ε
ω − (ε+n1 + ε
) + iη
Vα1σ2,µ1ν1 +
+,nII
+,nII
ω − (εΓ+nII − ε
) + iη
[ω + εΓ−kII − ε
− ε+n4 − ε
− ε+n6 + ε
] ∆−,kIIα1σ2
−,kII
− ε+n1 − ε
][εΓ−
− ε+n5 − ε
ω − (ε+n5 + ε
− ε−k7
) + iη
ω − (εΓ−
− ε+n4 − ε
) − iη
−,kII
−,kII
− ε+n2 + ε
] [−επ
− ε+n4 + ε
] [εΓ−
− ε+n1 − ε
] [εΓ−
− ε+n5 − ε
The last term in this expression contains an energy denom-
inator that involves the simultaneous propagation of two
phonons. Thus, it will be discarded in accordance with our
assumptions. It must be stressed that similar terms, with over-
lapping phonons, imply the explicit contribution of at least
3p2h/3h2p. A proper treatment of these would require a non
trivial externsion of the present formalism, which is beyond
the scope of this paper.
The remaining part in Eq. (A3) is the relevant contribution
for our purposes. This has the correct energy dependence of
a product of denominators that correspond to the intermediate
steps of propagation. All of these involve configurations that
have at most 2p1h character. Although, ground state corre-
lations are implicitely included by having already resummed
the RPA series. Still, this term does not factorize in a prod-
uct of separate Green’s functions due to the summations over
the fragmentation indices ni and ki [labeling the eigenstates of
the (N±1)-electron systems]. This is overcome if one defines
the matrices G0>(ω), Γ(1,2)(ω) and Γ(3)(ω), with elements (no
implicit summation used)
αnαβnβγkγ; µnµνnνλkλ
(ω) = δnα,nµ δnβ,nν δkγ,kλ
ω − (ε+nα + ε
− ε−kγ
) + iη
, (A4a)
αnαβnβγkγ; µnµνnνλkλ
(ω) = Γ(2)>
βnβαnαγkγ; νnνµnµλkλ
(ω) =
δα,µ δnα ,nµ
Vβλ,γν +
ω − (ε+nα + ε
nπ) + iη
[ω − επ
− ε+nα − ε
+ ε−kγ
− ε+nν + ε
− ε+nβ + ε
][−επ
− ε+nν + ε
, (A4b)
αnαβnβγkγ; µnµνnνλkλ
(ω) =
δγ,λ δkγ ,kλ
Vαβ,µν +
+,nII
+,nII
ω − (εΓ+nII − ε
) + iη
[ω + εΓ−kII − ε
− ε+nβ − ε
− ε+nν + ε
] ∆−,kII
−,kII
− ε+nα − ε
][εΓ−
− ε+nµ − ε
. (A4c)
In these definitions, the row and column indices are ordered to
represent at first two quasiparticle lines and then a quasihole.
The index ‘i’ in Γ(i)> refer to the line that propagates indepen-
dently along with the phonon. Using Eqs.(A4), the first term
on the r.h.s. of Eq. (A3) can be written as
∆R(2p1h)
αβγ,µνλ
(ω) = (A5)
nα nβ kγ
nµ nν kλ
G0>(ω)Γ(1)>(ω)G0>(ω)Γ(3)>(ω)G0>(ω)
αnαβnβγkγ; µnµνnνλkλ
Eq. (A5) generalizes to diagrams involving any number of
phonon insertions, as long as the terms involving two or more
simultaneous phonons are dropped. Based on this relation, we
use the following prescription to avoid performing integrals
over frequencies. One extends all the Green’s functions to ob-
jects depending not only on the sp basis’ indices (α, β, γ) but
also on the indices labeling quasi-particles and holes (ni and
ki). Whether a given argument represents a particle or an hole
depends on the type of line being propagated. At this point one
can perform calculations working with only two-time quanti-
ties. The standard propagator is recovered at the end by sum-
ming the “extended” one over the quasi-particle/hole indices.
2. Faddeev expansion
The 2p1h/2h1p propagator that includes the full resumma-
tion of both the ladder and ring diagrams at the (G)RPA level
is the solution of the following Bethe-Salpeter-like equation,
Rαβγ,µνλ(ω1, ω2, ω3) = (A6)
gαµ(ω1)gβν(ω2) − gβµ(ω2)gαν(ω1)
gλγ(−ω3) +
gββ1(ω2)gγ1γ(−ω3)Vβ1σ,γ1ρ
Rαρσ,µνλ(ω1, s, ω2 + ω3 − s)
+ gαα1 (ω1)gγ1γ(−ω3)Vα1σ,γ1ρ
Rρβσ,µνλ(s, ω2, ω1 + ω3 − s)
gαα1 (ω1)gββ1(ω2)Vα1β1,ρσ
Rρσγ,µνλ(s, ω1 + ω2 − s, ω3)
If this equation is solved, a double integration of R(ω1, ω2, ω3)
would yield the two-time propagator R(ω) contributing to
Eq. (3). However, the numerical solution of Eq. (A6) appears
beyond reach of the present day computers and one needs
to avoid dealing directly with multiple frequencies integrals.
The strategy used is to first solve the RPA equations (A2a)
and (A2b) separately. Once this is done it is necessary to re-
arrange the series (A6) in such a way that only the resummed
phonons appear. Following the formalism introduced by Fad-
deev [29, 38], we identify the components R(i)(ω) with the
three terms between curly brakets in Eq. (A6). By employ-
ing Eqs. (A2a) and (A2b) one is lead to the following set of
equations [42],
αβγ,µνλ
(ω1, ω2, ω3) = gαα1 (ω1)gββ1(ω2)gγ1γ(−ω3) (A7)
ds1 ds2 ds3
α1β1γ1,µ1ν1λ1
(ω1, ω2, ω3; s1, s2, s3)
gµ1µ(s1)gν1ν(s2) − gν1µ(s2)gµ1ν(s1)
gλλ1(−s3)
µ1ν1λ1,µνλ
(s1, s2, s3) + R
µ1ν1λ1,µνλ
(s1, s2, s3)
, i = 1, 2, 3 ,
where (i,j,k) are cyclic permutations of (1,2,3) and the inter-
action vertices Γ(i)(ω1, ω2, ω3) are given by
αβγ,µνλ
(ω1, ω2, ω3;ω4, ω5, ω6) = (A8a)
βαγ,νµλ
(ω2, ω1, ω3;ω5, ω4, ω6) =
= δ(ω1 − ω4)δ(ω2 + ω3 − ω5 − ω6)g
αµ(ω1)Γ
βγ,νλ
(ω2 + ω3) ,
αβγ,µνλ
(ω1, ω2, ω3;ω4, ω5, ω6) = (A8b)
δ(ω3 − ω6)δ(ω1 + ω2 − ω4 − ω5)g
λγ (−ω3)Γ
αβ,µν
(ω1 + ω2) .
Finally, we apply the prescription of Sec. A 1 and substitute
R(ω1, ω2, ω3) with its extended but two-time version R(ω).
This leads to the following set of Faddeev equations which
propagate 2p1h forward in time,
R̄(i)
αnαβnβγkγ; µnµνnνλkλ
(ω) =
αnαβnβγkγ; α′n′αβ′n
γ′k′γ
(ω) Γ(i)
α′n′αβ′n
γ′k′γ; µ′n
′n′νλ′k
αnαβnβγkγ; µnµνnνλkλ
(ω) −G0
αnαβnβγkγ; µnµνnνλkλ
× R̄( j)
µ′n′µν′n
; µnµνnνλkλ
(ω) + R̄(k)
µ′n′µν′n
; µnµνnνλkλ
i = 1, 2, 3 . (A9)
Since the full energy dependence is retained in Eq. (A7),
the self-energy corresponding to its solution, R(ω1, ω2, ω3), is
complete up to third order [see Eq. (3)]. This is no longer the
case after the reduction to a two-time propagator. In particu-
lar, the approximation that only forward 2p1h propagation is
allowed between different phonons implies that all diagrams
with different time propagation of their external lines are ne-
glected in Eqs. (A9). However, these terms are not energy
dependent and can be can be reinserted in a systematic way a
posteriori as in Eq. (8). In the general case,
Rαβγ,µνλ(ω) = (A10)
U (2p1h)
αβγ; α′n′αβ′n
γ′k′γ
R̄(2p1h)
α′n′αβ′n
γ′k′γ; µ′n
′n′νλ′k
(ω) U (2p1h) †
µ′n′µν′n
; µνλ
U (2p1h)
αβγ; µnµνnνλkλ
= δαµδβνδγλ + ∆U
(2p1h)
αβγ; µnµνnνλkλ
, (A11)
where the correction ∆U can be determined by comparison
with perturbation theory.
The vertices (A4), that appear in Eqs. (A9), and U (2p1h) are
expressed in terms of the fully fragmented propagator. There-
fore, this approach allows to obtain self-consistent solutions
of the sp Green’s function [25]. Whenever, like in this work,
only a mean-field propagator is employed as input there ex-
ist a one-to-one correspondence between the fragmentation
indices and the sp basis. This is expressed by the relations
Xnα = δn,α(1 − δα∈F) and Y
α = δk,αδα∈F , where F represents
the set of occupied orbits. In this case, it is possible to drop
one set of indices so that Eqs. (A9) and (A10) simplify into
the form (7) and (8).
3. Faddeev vertices
In practical applications, it is worth to note that the poles
of the free propagator G0(ω), Eq. (A4a), do not contribute to
the kernel of Eqs. (A9). This can be proven by employing the
closure relations for the RPA problem, in the form obtained
by extracting the free poles in Eqs. (A2). As an example, for
the forward poles of the ladder propagator these are
ω→ε+n1
+ε+n2
[(ω − ε+n1 − ε
) × (Eq. A2b)] =⇒ (A12)
µν,γδ
(ω = ε+n1 + ε
) = 0 , ∀n1, n2 ,
and similarly for other cases. Making use of these relations
one can derive the following working expression of the ker-
nels of the 2p1h Faddeev equations (no implicit summations
used)
G0>(ω)Γ(1)>(ω)
αnαβnβγkγ; µnµνnνλkλ
G0>(ω)Γ(2)>(ω)
βnβαnαγkγ; νnνµnµλkλ
= (A13a)
= δnα,nµ
[επnπ − ε
][ω − (ε+nα + ε
nπ) + iη]
− ε+nβ + ε
][−επ
− ε+nν + ε
G0>(ω)Γ(3)>(ω)
αnαβnβγkγ; µnµνnνλkλ
= (A13b)
= δkγ,kλ
+,nII
+,nII
[εΓ+nII − ε
− ε+nβ][ω − (ε
) + iη]
−,kII
−,kII
− ε+nα − ε
][εΓ−
− ε+nµ − ε
After substituting Eq. (A10) into (3), one needs the working expression for the matrix product V U(2p1h) (where V is the inter-
electron interaction). The minimum correction that guaranties to reproduce all third order self-energy diagrams is
V U(2p1h)
α; µnµνnνλkλ
= Vαλ,µν +
Vαλ,γ1δ1 Y
Vγ2δ2,µν
2 [ε−
− ε+nµ − ε
(A14)
Vαδ1,µγ1 Y
Vγ2λ,δ2ν
− ε+nδ − ε
Vαδ1,νγ1 Y
Vγ2λ,δ2µ
− ε+nδ − ε
The case of 2h1p is handled in a completely analogous way along the steps of Secs. (A 1) and (A 2). After extending
R(ω1, ω2, ω3) to depend on the fragmentation indices (k1,k2,n), the 2h1p equivalent of Eq. (A9) is obtained with the follow-
ing definitions of the kernels,
G0>(ω)Γ(1)>(ω)
αkαβkβγnγ; µkµνkνλnλ
G0>(ω)Γ(2)>(ω)
βkβαkαγnγ; νkνµkµλnλ
= (A15a)
= δkα,kµ
+ ε+nγ ][ω − (ε
) − iη]
[επnπ − ε
+ ε+nγ ][ε
nπ − ε
+ ε+nλ]
G0>(ω)Γ(3)>(ω)
αkαβkβγnγ; µkµνkνλnλ
= (A15b)
= δnγ,nλ
−,nII
−,nII
][ω − (εΓ−
− ε+nγ ) − iη]
+,kII
+,kII
[εΓ+nII − ε
− ε−kβ
][εΓ+nII − ε
− ε−kν
and correction to the external legs,
V U(2h1p)
α; µkµνkνλnλ
= Vαλ,µν +
Vαλ,γ1δ1 X
Vγ2δ2,µν
2 [ε−kµ + ε
− ε+nγ − ε
(A16)
Vαδ1,µγ1 X
Vγ2λ,δ2ν
[ε−kδ + ε
− ε+nγ − ε
Vαδ1,νγ1 X
Vγ2λ,δ2µ
[ε−kδ + ε
− ε+nγ − ε
It should be pointed out that while the prescription of
Sec. A 1 allows sp lines to propagate only in one time di-
rection, it allows for backward propagation of the phonons.
These contributions translate directly into the energy indepen-
dent terms of Eqs. (A13) and (A15) and are a direct conse-
quence of the inversion pattern typical of RPA theory. These
terms have normally a weaker impact than the direct ones on
the solutions of Eqs. (A9). However, it is show in Ref. [24]
that they are crucial to guarantee the exact separation of the
spurious solutions—always introduced by the Faddeev for-
malism [39, 40]—if RPA phonons are used. For the same rea-
sons, the last terms in curly brackets of Eqs. (A13) and (A15)
should be dropped whenever Tamm-Dancoff (TDA) phonons
are propagated.
The approach followed in this work for solving Eqs. (A9)
is to transform them into a matrix representations [24].
Once this is done, one is left with an eigenvalue prob-
lem that depends only on the 2p1h (2h1p) configurations
(n, n′, k) [(k, k′, n)]. The spurious states are known ex-
actly [24] and can be projected out analytically to reduce the
computational load. In any case, they would give vanishing
contributions to Eq. (3).
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http://physics.nist.gov/PhysRefData/ASD/in-dex.html
|
0704.1543 | Discrete Nonholonomic Lagrangian Systems on Lie Groupoids | DISCRETE NONHOLONOMIC LAGRANGIAN SYSTEMS ON LIE
GROUPOIDS
DAVID IGLESIAS, JUAN C. MARRERO, DAVID MARTÍN DE DIEGO,
AND EDUARDO MARTÍNEZ
Abstract. This paper studies the construction of geometric integrators for
nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange
equations in a setting which permits to deduce geometric integrators for con-
tinuous nonholonomic systems (reduced or not). The formalism is given in
terms of Lie groupoids, specifying a discrete Lagrangian and a constraint sub-
manifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie
algebroid associated to the Lie groupoid. We also discuss the existence of non-
holonomic evolution operators in terms of the discrete nonholonomic Legendre
transformations and in terms of adequate decompositions of the prolongation
of the Lie groupoid. The characterization of the reversibility of the evolution
operator and the discrete nonholonomic momentum equation are also consid-
ered. Finally, we illustrate with several classical examples the wide range of
application of the theory (the discrete nonholonomic constrained particle, the
Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on
a rotating table and the two wheeled planar mobile robot).
Contents
1. Introduction 2
2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids 5
2.1. Lie algebroids 5
2.2. Lie groupoids 6
2.3. Discrete Unconstrained Lagrangian Systems 9
3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie
groupoids 11
3.1. Discrete Generalized Hölder’s principle 11
3.2. Discrete Nonholonomic Legendre transformations 14
3.3. Nonholonomic evolution operators and regular discrete nonholonomic
Lagrangian systems 20
3.4. Reversible discrete nonholonomic Lagrangian systems 22
3.5. Lie groupoid morphisms and reduction 23
3.6. Discrete nonholonomic Hamiltonian evolution operator 24
3.7. The discrete nonholonomic momentum map 24
4. Examples 26
4.1. Discrete holonomic Lagrangian systems on a Lie groupoid 26
4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid 27
This work has been partially supported by MICYT (Spain) Grants MTM 2006-03322, MTM
2004-7832, MTM 2006-10531 and S-0505/ESP/0158 of the CAM. D. Iglesias thanks MEC for a
“Juan de la Cierva” research contract.
2 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
4.3. Discrete nonholonomic Lagrangian systems on a Lie group 29
4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid 32
4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid 35
4.6. Discrete Chaplygin systems 39
5. Conclusions and Future Work 43
References 43
1. Introduction
In the paper of Moser and Veselov [40] dedicated to the complete integrability
of certain dynamical systems, the authors proposed a discretization of the tangent
bundle TQ of a configuration space Q replacing it by the product Q×Q, approx-
imating a tangent vector on Q by a pair of ‘close’ points (q0, q1). In this sense,
the continuous Lagrangian function L : TQ −→ R is replaced by a discretization
Ld : Q×Q −→ R. Then, applying a suitable variational principle, it is possible to
derive the discrete equations of motion. In the regular case, one obtains an evolu-
tion operator, a map which assigns to each pair (qk−1, qk) a pair (qk, qk+1), sharing
many properties with the continuous system, in particular, symplecticity, momen-
tum conservation and a good energy behavior. We refer to [32] for an excellent
review in discrete Mechanics (on Q×Q) and its numerical implementation.
On the other hand, in [40, 44], the authors also considered discrete Lagrangians
defined on a Lie group G where the evolution operator is given by a diffeomorphism
of G.
All the above examples led to A. Weinstein [45] to study discrete mechanics on
Lie groupoids. A Lie groupoid is a geometric structure that includes as particular
examples the case of cartesian products Q × Q as well as Lie groups and other
examples as Atiyah or action Lie groupoids [26]. In a recent paper [27], we studied
discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids, deriving from
a variational principle the discrete Euler-Lagrange equations. We also introduced
a symplectic 2-section (which is preserved by the Lagrange evolution operator)
and defined the Hamiltonian evolution operator, in terms of the discrete Legendre
transformations, which is a symplectic map with respect to the canonical symplectic
2-section on the prolongation of the dual of the Lie algebroid of the given groupoid.
These techniques include as particular cases the classical discrete Euler-Lagrange
equations, the discrete Euler-Poincaré equations (see [5, 6, 29, 30]) and the discrete
Lagrange-Poincaré equations. In fact, the results in [27] may be applied in the
construction of geometric integrators for continuous Lagrangian systems which are
invariant under the action of a symmetry Lie group (see also [18] for the particular
case when the symmetry Lie group is abelian).
From the perspective of geometric integration, there are a great interest in intro-
ducing new geometric techniques for developing numerical integrators since stan-
dard methods often introduce some spurious effects like dissipation in conservative
systems [16, 42]. The case of dynamical systems subjected to constraints is also
of considerable interest. In particular, the case of holonomic constraints is well
established in the literature of geometric integration, for instance, in simulation of
molecular dynamics where the constraints may be molecular bond lengths or angles
and also in multibody dynamics (see [16, 20] and references therein).
DISCRETE NONHOLONOMIC MECHANICS 3
By contrast, the construction of geometric integrators for the case of nonholo-
nomic constraints is less well understood. This type of constraints appears, for
instance, in mechanical models of convex rigid bodies rolling without sliding on a
surface [41]. The study of systems with nonholonomic constraints goes back to the
XIX century. The equations of motion were obtained applying either D’Alembert’s
principle of virtual work or Gauss principle of least constraint. Recently, many
authors have shown a new interest in that theory and also in its relation to the
new developments in control theory and robotics using geometric techniques (see,
for instance, [2, 3, 4, 8, 19, 22, 24]).
Geometrically, nonholonomic constraints are globally described by a submanifold
M of the velocity phase space TQ. If M is a vector subbundle of TQ, we are dealing
with the case of linear constraints and, in the case M is an affine subbundle, we are in
the case of affine constraints. Lagrange-D’Alembert’s or Chetaev’s principles allow
us to determine the set of possible values of the constraint forces only from the set
of admissible kinematic states, that is, from the constraint manifold M determined
by the vanishing of the nonholonomic constraints φa. Therefore, assuming that the
dynamical properties of the system are mathematically described by a Lagrangian
function L : TQ −→ R and by a constraint submanifold M, the equations of motion,
following Chetaev’s principle, are[
δqi = 0 ,
where δqi denotes the virtual displacements verifying
δqi = 0. By using the
Lagrange multiplier rule, we obtain that
= λ̄a
, (1.1)
with the condition q̇(t) ∈ M, λ̄a being the Lagrange multipliers to be determined.
Recently, J. Cortés et al [9] (see also [11, 38, 39]) proposed a unified framework for
nonholonomic systems in the Lie algebroid setting that we will use along this paper
generalizing some previous work for free Lagrangian mechanics on Lie algebroids
(see, for instance, [23, 33, 34, 35]).
The construction of geometric integrators for Equations (1.1) is very recent. In
fact, in [37] appears as an open problem:
...The problem for the more general class of non-holonomic con-
straints is still open, as is the question of the correct analogue
of symplectic integration for non-holonomically constrained La-
grangian systems...
Numerical integrators derived from discrete variational principles have proved their
adaptability to many situations: collisions, classical field theory, external forces...[28,
32] and it also seems very adequate for nonholonomic systems, since nonholonomic
equations of motion come from Hölder’s variational principle which is not a stan-
dard variational principle [1], but admits an adequate discretization. This is the
procedure introduced by J. Cortés and S. Mart́ınez [8, 10] and followed by other
authors [12, 14, 15, 36] extending, moreover, the results to nonholonomic systems
defined on Lie groups (see also [25] for a different approach using generating func-
tions).
In this paper, we tackle the problem from the unifying point of view of Lie
groupoids (see [9] for the continuous case). This technique permits to recover all
the previous methods in the literature [10, 14, 36] and consider new cases of great
4 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
importance in nonholonomic dynamics. For instance, using action Lie groupoids,
we may discretize LR-nonholonomic systems such as the Veselova system or us-
ing Atiyah Lie groupoids we find discrete versions for the reduced equations of
nonholonomic systems with symmetry.
The paper is structured as follows. In section 2 we review some basic results on
Lie algebroids and Lie groupoids. In particular, we describe the prolongation of a
Lie groupoid [43], which has a double structure of Lie groupoid and Lie algebroid.
Then, we briefly expose the geometric structure of discrete unconstrained mechanics
on Lie groupoids: Poincaré-Cartan sections, Legendre transformations... The main
results of the paper appear in section 3, where the geometric structure of discrete
nonholonomic systems on Lie groupoids is considered. In particular, given a discrete
Lagrangian Ld : Γ → R on a Lie groupoid Γ, a constraint distribution Dc in the
Lie algebroid EΓ of Γ and a discrete constraint submanifold Mc in Γ, we obtain
the nonholonomic discrete Euler-Lagrange equations from a discrete Generalized
Hölder’s principle (see section 3.1). In addition, we characterize the regularity of the
nonholonomic system in terms of the nonholonomic Legendre transformations and
decompositions of the prolongation of the Lie groupoid. In the case when the system
is regular, we can define the nonholonomic evolution operator. An interesting
situation, studied in in Section 3.4, is that of reversible discrete nonholonomic
Lagrangian systems, where the Lagrangian and the discrete constraint submanifold
are invariants with respect to the inversion of the Lie groupoid. The particular
example of reversible systems in the pair groupoid Q×Q was first studied in [36].
We also define the discrete nonholonomic momentum map. In order to give an
idea of the breadth and flexibility of the proposed formalism, several examples are
discussed, including their regularity and their reversibility:
- Discrete holonomic Lagrangian systems on a Lie groupoid, which are a
generalization of the Shake algorithm for holonomic systems [16, 20, 32];
- Discrete nonholonomic systems on the pair groupoid, recovering the equa-
tions first considered in [10]. An explicit example of this situation is the
discrete nonholonomic constrained particle.
- Discrete nonholonomic systems on Lie groups, where the equations that
are obtained are the so-called discrete Euler-Poincaré-Suslov equations (see
[14]). We remark that, although our equations coincide with those in [14],
the technique developed in this paper is different to the one in that paper.
Two explicit examples which we describe here are the Suslov system and
the Chaplygin sleigh.
- Discrete nonholonomic Lagrangian systems on an action Lie groupoid.
This example is quite interesting since it allows us to discretize a well-
known nonholonomic LR-system: the Veselova system (see [44]; see also
[13]). For this example, we obtain a discrete system that is not reversible
and we show that the system is regular in a neighborhood around the
manifold of units.
- Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid.
With this example, we are able to discretize reduced systems, in particular,
we concentrate on the example of the discretization of the equations of
motion of a rolling ball without sliding on a rotating table with constant
angular velocity.
- Discrete Chaplygin systems, which are regular systems (Ld,Mc,Dc) on
the Lie groupoid Γ ⇒ M , for which (α, β) ◦ iMc : Mc → M × M is
a diffeomorphism and ρ ◦ iDc : Dc → TM is an isomorphism of vector
bundles, (α, β) being the source and target of the Lie groupoid Γ and ρ
DISCRETE NONHOLONOMIC MECHANICS 5
being the anchor map of the Lie algebroid EΓ. This example includes a
discretization of the two wheeled planar mobile robot.
We conclude our paper with future lines of work.
2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids
2.1. Lie algebroids. A Lie algebroid E over a manifold M is a real vector bundle
τ : E →M together with a Lie bracket [[·, ·]] on the space Sec(τ) of the global cross
sections of τ : E → M and a bundle map ρ : E → TM , called the anchor
map, such that if we also denote by ρ : Sec(τ) → X(M) the homomorphism of
C∞(M)-modules induced by the anchor map then
[[X, fY ]] = f [[X,Y ]] + ρ(X)(f)Y, (2.1)
for X,Y ∈ Sec(τ) and f ∈ C∞(M) (see [26]).
If (E, [[·, ·]], ρ) is a Lie algebroid over M then the anchor map ρ : Sec(τ) →
X(M) is a homomorphism between the Lie algebras (Sec(τ), [[·, ·]]) and (X(M), [·, ·]).
Moreover, one may define the differential d of E as follows:
dµ(X0, . . . , Xk) =
(−1)iρ(Xi)(µ(X0, . . . , X̂i, . . . , Xk))
(−1)i+jµ([[Xi, Xj ]], X0, . . . , X̂i, . . . , X̂j , . . . , Xk),
(2.2)
for µ ∈ Sec(∧kτ∗) and X0, . . . , Xk ∈ Sec(τ). d is a cohomology operator, that is,
d2 = 0. In particular, if f : M −→ R is a real smooth function then df(X) = ρ(X)f,
for X ∈ Sec(τ).
Trivial examples of Lie algebroids are a real Lie algebra of finite dimension (in
this case, the base space is a single point) and the tangent bundle of a manifold M.
On the other hand, let (E, [[·, ·]], ρ) be a Lie algebroid of rank n over a manifold
M of dimension m and π : P →M be a fibration. We consider the subset of E×TP
TEP = { (a, v) ∈ E × TP | (Tπ)(v) = ρ(a) } ,
where Tπ : TP → TM is the tangent map to π. Denote by τπ : TEP → P the
map given by τπ(a, v) = τP (v), τP : TP → P being the canonical projection. If
dimP = p, one may prove that TEP is a vector bundle over P of rank n + p −m
with vector bundle projection τπ : TEP → P .
A section X̃ of τπ : TEP → P is said to be projectable if there exists a section X
of τ : E →M and a vector field U ∈ X(P ) which is π-projectable to the vector field
ρ(X) and such that X̃(p) = (X(π(p)), U(p)), for all p ∈ P . For such a projectable
section X̃, we will use the following notation X̃ ≡ (X,U). It is easy to prove that
one may choose a local basis of projectable sections of the space Sec(τπ).
The vector bundle τπ : TEP → P admits a Lie algebroid structure ([[·, ·]]π, ρπ).
Indeed, if (X,U) and (Y, V ) are projectable sections then
[[(X,U), (Y, V )]]π = ([[X,Y ]], [U, V ]), ρπ(X,U) = U.
(TEP, [[·, ·]]π, ρπ) is the E-tangent bundle to P or the prolongation of E
over the fibration π : P →M (for more details, see [23]).
Now, let (E, [[·, ·]], ρ) (resp., (E′, [[·, ·]]′, ρ′)) be a Lie algebroid over a manifold M
(resp., M ′) and suppose that Ψ : E → E′ is a vector bundle morphism over the map
Ψ0 : M →M ′. Then, the pair (Ψ,Ψ0) is said to be a Lie algebroid morphism if
d((Ψ,Ψ0)
∗φ′) = (Ψ,Ψ0)
∗(d′φ′), for all φ′ ∈ Sec(∧k(τ ′)∗) and for all k, (2.3)
6 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where d (resp., d′) is the differential of the Lie algebroid E (resp., E′) (see [23]). In
the particular case when M = M ′ and Ψ0 = Id then (2.3) holds if and only if
[[Ψ ◦X,Ψ ◦ Y ]]′ = Ψ[[X,Y ]], ρ′(ΨX) = ρ(X), for X,Y ∈ Sec(τ).
2.2. Lie groupoids. A Lie groupoid over a differentiable manifold M is a differ-
entiable manifold Γ together with the following structural maps:
• A pair of submersions α : Γ → M , the source, and β : Γ → M, the
target. The maps α and β define the set of composable pairs
Γ2 = { (g, h) ∈ G×G | β(g) = α(h) } .
• A multiplication m : Γ2 → Γ, to be denoted simply by m(g, h) = gh,
such that
– α(gh) = α(g) and β(gh) = β(h).
– g(hk) = (gh)k.
• An identity section � : M → Γ such that
– �(α(g))g = g and g�(β(g)) = g.
• An inversion map i : Γ → Γ, to be simply denoted by i(g) = g−1, such
– g−1g = �(β(g)) and gg−1 = �(α(g)).
A Lie groupoid Γ over a set M will be simply denoted by the symbol Γ ⇒ M .
On the other hand, if g ∈ Γ then the left-translation by g and the right-
translation by g are the diffeomorphisms
lg : α−1(β(g)) −→ α−1(α(g)) ; h −→ lg(h) = gh,
rg : β−1(α(g)) −→ β−1(β(g)) ; h −→ rg(h) = hg.
Note that l−1g = lg−1 and r
g = rg−1 .
A vector field X̃ on Γ is said to be left-invariant (resp., right-invariant) if
it is tangent to the fibers of α (resp., β) and X̃(gh) = (Thlg)(X̃h) (resp., X̃(gh) =
(Tgrh)(X̃(g))), for (g, h) ∈ Γ2.
Now, we will recall the definition of the Lie algebroid associated with Γ.
We consider the vector bundle τ : EΓ → M , whose fiber at a point x ∈ M is
(EΓ)x = V�(x)α = Ker(T�(x)α). It is easy to prove that there exists a bijection
between the space Sec(τ) and the set of left-invariant (resp., right-invariant) vector
fields on Γ. If X is a section of τ : EΓ →M , the corresponding left-invariant (resp.,
right-invariant) vector field on Γ will be denoted
X (resp.,
X ), where
X (g) = (T�(β(g))lg)(X(β(g))), (2.4)
X (g) = −(T�(α(g))rg)((T�(α(g))i)(X(α(g)))), (2.5)
for g ∈ Γ. Using the above facts, we may introduce a Lie algebroid structure
([[·, ·]], ρ) on EΓ, which is defined by
←−−−−
[[X,Y ]] = [
Y ], ρ(X)(x) = (T�(x)β)(X(x)), (2.6)
for X,Y ∈ Sec(τ) and x ∈M . Note that
−−−−→
[[X,Y ]] = −[
Y ], [
Y ] = 0, (2.7)
(for more details, see [7, 26]).
Given two Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′, a morphism of Lie groupoids
is a smooth map Φ : Γ→ Γ′ such that
(g, h) ∈ Γ2 =⇒ (Φ(g),Φ(h)) ∈ (Γ′)2
DISCRETE NONHOLONOMIC MECHANICS 7
Φ(gh) = Φ(g)Φ(h).
A morphism of Lie groupoids Φ : Γ → Γ′ induces a smooth map Φ0 : M → M ′ in
such a way that
α′ ◦ Φ = Φ0 ◦ α, β′ ◦ Φ = Φ0 ◦ β, Φ ◦ � = �′ ◦ Φ0,
α, β and � (resp., α′, β′ and �′) being the source, the target and the identity section
of Γ (resp., Γ′).
Suppose that (Φ,Φ0) is a morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒
M ′ and that τ : EΓ → M (resp., τ ′ : EΓ′ → M ′) is the Lie algebroid of Γ (resp.,
Γ′). Then, if x ∈ M we may consider the linear map Ex(Φ) : (EΓ)x → (EΓ′)Φ0(x)
defined by
Ex(Φ)(v�(x)) = (T�(x)Φ)(v�(x)), for v�(x) ∈ (EΓ)x. (2.8)
In fact, we have that the pair (E(Φ),Φ0) is a morphism between the Lie algebroids
τ : EΓ →M and τ ′ : EΓ′ →M ′ (see [26]).
Trivial examples of Lie groupoids are Lie groups and the pair or banal groupoid
M ×M , M being an arbitrary smooth manifold. The Lie algebroid of a Lie group
Γ is just the Lie algebra g of Γ. On the other hand, the Lie algebroid of the pair
(or banal) groupoid M ×M is the tangent bundle TM to M .
Apart from the Lie algebroid EΓ associated with a Lie groupoid Γ ⇒ M , other
interesting Lie algebroids associated with Γ are the following ones:
• The EΓ- tangent bundle to E∗Γ:
Let TEΓE∗Γ be the EΓ-tangent bundle to E
Γ, that is,
E∗Γ =
(vx, XΥx) ∈ (EΓ)x × TΥxE
∣∣ (TΥxτ∗)(XΥx) = (T�(x)β)(vx)}
for Υx ∈ (E∗Γ)x, with x ∈M. As we know, T
EΓE∗Γ is a Lie algebroid over E
We may introduce the canonical section Θ of the vector bundle (TEΓE∗Γ)
∗ → E∗Γ
as follows:
Θ(Υx)(ax, XΥx) = Υx(ax),
for Υx ∈ (E∗Γ)x and (ax, XΥx) ∈ T
E∗Γ. Θ is called the Liouville section as-
sociated with EΓ. Moreover, we define the canonical symplectic section Ω
associated with EΓ by Ω = −dΘ, where d is the differential on the Lie algebroid
TEΓE∗Γ → E
Γ. It is easy to prove that Ω is nondegenerate and closed, that is, it is
a symplectic section of TEΓE∗Γ (see [23]).
Now, if Z is a section of τ : EΓ → M then there is a unique vector field Z∗c on
E∗Γ, the complete lift of X to E
Γ, satisfying the two following conditions:
(i) Z∗c is τ∗-projectable on ρ(Z) and
(ii) Z∗c(X̂) = ̂[[Z,X]]
for X ∈ Sec(τ) (see [23]). Here, if X is a section of τ : EΓ → M then X̂ is the
linear function X̂ ∈ C∞(E∗) defined by
X̂(a∗) = a∗(X(τ∗(a∗))), for all a∗ ∈ E∗.
Using the vector field Z∗c, one may introduce the complete lift Z∗c of Z as the
section of τ τ
: TEΓE∗Γ → E
Γ defined by
Z∗c(a∗) = (Z(τ∗(a∗)), Z∗c(a∗)), for a∗ ∈ E∗. (2.9)
8 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Z∗c is just the Hamiltonian section of Ẑ with respect to the canonical symplectic
section Ω associated with EΓ. In other words,
iZ∗cΩ = dẐ, (2.10)
where d is the differential of the Lie algebroid τ τ
: TEΓE∗Γ → E
Γ (for more details,
see [23]).
• The Lie algebroid τ̃Γ : TΓΓ→ Γ :
Let TΓΓ be the Whitney sum V β ⊕Γ V α of the vector bundles V β → Γ and
V α → Γ, where V β (respectively, V α) is the vertical bundle of β (respectively,
α). Then, the vector bundle τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ admits a Lie algebroid
structure ([[·, ·]]T
ΓΓ, ρT
ΓΓ). The anchor map ρT
ΓΓ is given by
ΓΓ)(Xg, Yg) = Xg + Yg
and the Lie bracket bracket [[·, ·]]T
ΓΓ on the space Sec(τ̃Γ) is characterized for the
following relation
Y ), (
Y ′)]]T
ΓΓ = (−
−−−−−→
[[X,X ′]],
←−−−−
[[Y, Y ′]]),
for X,Y,X ′, Y ′ ∈ Sec(τ) (for more details, see [27]).
On other hand, if X is a section of τ : EΓ → M , one may define the sections
X(1,0), X(0,1) (the β and α-lifts) and X(1,1) (the complete lift) of X to τ̃Γ : TΓΓ→ Γ
as follows:
X(1,0)(g) = (
X (g), 0g), X
(0,1)(g) = (0g,
X (g)), and X(1,1)(g) = (−
X (g),
X (g)).
We have that
[[X(1,0), Y (1,0)]]T
ΓΓ = −[[X,Y ]](1,0) [[X(0,1), Y (1,0)]]T
ΓΓ = 0,
[[X(0,1), Y (0,1)]]T
ΓΓ = [[X,Y ]](0,1),
and, as a consequence,
[[X(1,1), Y (1,0)]]T
ΓΓ = [[X,Y ]](1,0), [[X(1,1), Y (0,1)]]T
ΓΓ = [[X,Y ]](0,1),
[[X(1,1), Y (1,1)]]T
ΓΓ = [[X,Y ]](1,1).
Now, if g, h ∈ Γ one may introduce the linear monomorphisms (1,0)h : (EΓ)
(TΓhΓ)
∗ ≡ V ∗h β ⊕ V
h α and
(0,1)
g : (EΓ)∗β(g) → (T
∗ ≡ V ∗g β ⊕ V ∗g α given by
(1,0)
h (Xh, Yh) = γ(Th(i ◦ rh−1)(Xh)), (2.11)
γ(0,1)g (Xg, Yg) = γ((Tglg−1)(Yg)), (2.12)
for (Xg, Yg) ∈ TΓg Γ and (Xh, Yh) ∈ TΓhΓ.
Thus, if µ is a section of τ∗ : E∗Γ → M , one may define the corresponding lifts
µ(1,0) and µ(0,1) as the sections of τ̃Γ
∗ : (TΓΓ)∗ → Γ given by
µ(1,0)(h) = µ(1,0)h , for h ∈ Γ,
µ(0,1)(g) = µ(0,1)g , for g ∈ Γ.
Note that if g ∈ Γ and {XA} (respectively, {YB}) is a local basis of Sec(τ) on an
open subset U (respectively, V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V )
then {X(1,0)A , Y
(0,1)
B } is a local basis of Sec(τ̃Γ) on the open subset α
−1(U)∩β−1(V ).
In addition, if {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively,
{YB}) then {(XA)(1,0), (Y B)(0,1)} is the dual basis of {X
(1,0)
A , Y
(0,1)
DISCRETE NONHOLONOMIC MECHANICS 9
2.3. Discrete Unconstrained Lagrangian Systems. (See [27] for details) A
discrete unconstrained Lagrangian system on a Lie groupoid consists of a
Lie groupoid Γ ⇒ M (the discrete space) and a discrete Lagrangian Ld : Γ→
2.3.1. Discrete unconstrained Euler-Lagrange equations. An admissible sequence
of order N on the Lie groupoid Γ is an element (g1, . . . , gN ) of ΓN ≡ Γ× · · · ×Γ
such that (gk, gk+1) ∈ Γ2, for k = 1, . . . , N − 1.
An admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete
unconstrained Euler-Lagrange equations for Ld if
do[Ld ◦ lgk + Ld ◦ rgk+1 ◦ i](�(xk))|(EΓ)xk = 0
where β(gk) = α(gk+1) = xk and do is the standard differential on Γ, that is, the
differential of the Lie algebroid τΓ : TΓ→ Γ (see [27]).
The discrete unconstrained Euler-Lagrange operator DDELLd : Γ2 → E∗Γ
is given by
(DDELLd)(g, h) = d
o[Ld ◦ lg + Ld ◦ rh ◦ i](�(x))|(EΓ)x = 0,
for (g, h) ∈ Γ2, with β(g) = α(h) = x ∈M (see [27]).
Thus, an admissible sequence (g1, . . . , gN ) of order N is a solution of the discrete
unconstrained Euler-Lagrange equations if and only if
(DDELLd)(gk, gk+1) = 0, for k = 1, . . . , N − 1.
2.3.2. Discrete Poincaré-Cartan sections. Consider the Lie algebroid τ̃Γ : TΓΓ ≡
V β ⊕Γ V α → Γ, and define the Poincaré-Cartan 1-sections Θ−Ld ,Θ
Sec((τ̃Γ)∗) as follows
Θ−Ld(g)(Xg, Yg) = −Xg(Ld), Θ
(g)(Xg, Yg) = Yg(Ld), (2.13)
for each g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα.
Since dLd = Θ
−Θ−Ld and so, using d
2 = 0, it follows that dΘ+Ld = dΘ
. This
means that there exists a unique 2-section ΩLd = −dΘ
= −dΘ−Ld , which will be
called the Poincaré-Cartan 2-section. This 2-section will be important to study
the symplectic character of the discrete unconstrained Euler-Lagrange equations.
If g is an element of Γ such that α(g) = x and β(g) = y and {XA} (respectively,
{YB}) is a local basis of Sec(τ) on the open subset U (respectively, V ) of M , with
x ∈ U (respectively, y ∈ V ), then on α−1(U) ∩ β−1(V ) we have that
Θ−Ld = −
XA(L)(XA)(1,0), Θ
YB(L)(Y B)(0,1),
ΩLd = −
YB(Ld))(XA)(1,0) ∧ (Y B)(0,1)
(2.14)
where {XA} (respectively, {Y B}) is the dual basis of {XA} (respectively, {YB})
(for more details, see [27]).
2.3.3. Discrete unconstrained Lagrangian evolution operator. Let Υ : Γ → Γ be a
smooth map such that:
- graph(Υ) ⊆ Γ2, that is, (g,Υ(g)) ∈ Γ2, for all g ∈ Γ (Υ is a second order
operator) and
- (g,Υ(g)) is a solution of the discrete unconstrained Euler-Lagrange equa-
tions, for all g ∈ Γ, that is, (DDELLd)(g,Υ(g)) = 0, for all g ∈ Γ.
10 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
In such a case ←−
X (g)(Ld)−
X (Υ(g))(Ld) = 0, (2.15)
for every section X of τ : EΓ → M and every g ∈ Γ. The map Υ : Γ→ Γ is called
a discrete flow or a discrete unconstrained Lagrangian evolution operator
for Ld.
Now, let Υ : Γ → Γ be a second order operator. Then, the prolongation TΥ :
TΓΓ ≡ V β ⊕Γ V α → TΓΓ ≡ V β ⊕Γ V α of Υ is the Lie algebroid morphism over
Υ : Γ→ Γ defined as follows (see [27]):
TgΥ(Xg, Yg) = ((Tg(rgΥ(g) ◦ i))(Yg), (TgΥ)(Xg)
+(TgΥ)(Yg)− Tg(rgΥ(g) ◦ i)(Yg)),
(2.16)
for all (Xg, Yg) ∈ TΓg Γ ≡ Vgβ ⊕ Vgα. Moreover, from (2.4), (2.5) and (2.16), we
obtain that
X (g),
Y (g)) = (−
Y (Υ(g)), (TgΥ)(
X (g) +
Y (g)) +
Y (Υ(g))), (2.17)
for all X,Y ∈ Sec(τ).
Using (2.16), one may prove that (see [27]):
(i) The map Υ is a discrete unconstrained Lagrangian evolution operator for
Ld if and only if (TΥ,Υ)∗Θ
= Θ+Ld .
(ii) The map Υ is a discrete unconstrained Lagrangian evolution operator for
Ld if and only if (TΥ,Υ)∗Θ
−Θ−Ld = dLd.
(iii) If Υ is discrete unconstrained Lagrangian evolution operator then
(TΥ,Υ)∗ΩLd = ΩLd .
2.3.4. Discrete unconstrained Legendre transformations. Given a Lagrangian Ld :
Γ → R we define the discrete unconstrained Legendre transformations
F−Ld : Γ→ E∗Γ and F
+Ld : Γ→ E∗Γ by (see [27])
(F−Ld)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ (EΓ)α(h),
(F+Ld)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ (EΓ)β(g).
Now, we introduce the prolongations TΓF−Ld : TΓΓ ≡ V β ⊕Γ V α → TEΓE∗Γ and
TΓF+Ld : TΓΓ ≡ V β ⊕Γ V α→ TEΓE∗Γ by
−Ld(Xh, Yh) = (Th(i ◦ rh−1)(Xh), (ThF−Ld)(Xh) + (ThF−Ld)(Yh)),(2.18)
TΓg F
+Ld(Xg, Yg) = ((Tglg−1)(Yg), (TgF+Ld)(Xg) + (TgF+Ld)(Yg)), (2.19)
for all h, g ∈ Γ and (Xh, Yh) ∈ TΓhΓ ≡ Vhβ ⊕ Vhα and (Xg, Yg) ∈ T
g Γ ≡ Vgβ ⊕
Vgα (see [27]). We observe that the discrete Poincaré-Cartan 1-sections and 2-
section are related to the canonical Liouville section of (TEΓE∗Γ)
∗ → E∗Γ and the
canonical symplectic section of ∧2(TEΓE∗Γ)
∗ → E∗Γ by pull-back under the discrete
unconstrained Legendre transformations, that is (see [27]),
(TΓF−Ld,F−Ld)∗Θ = Θ−Ld , (T
ΓF+Ld,F+Ld)∗Θ = Θ+Ld , (2.20)
(TΓF−Ld,F−Ld)∗Ω = ΩLd , (T
ΓF+Ld,F+Ld)∗Ω = ΩLd . (2.21)
2.3.5. Discrete regular Lagrangians. A discrete Lagrangian Ld : Γ→ R is said to be
regular if the Poincaré-Cartan 2-section ΩLd is nondegenerate on the Lie algebroid
τ̃Γ : TΓΓ ≡ V β ⊕Γ V α → Γ (see [27]). In [27], we obtained some necessary and
sufficient conditions for a discrete Lagrangian on a Lie groupoid Γ to be regular
that we summarize as follows:
Ld is regular ⇐⇒ The Legendre transformation F+Ld is a local diffeomorphism
⇐⇒ The Legendre transformation F−Ld is a local diffeomorphism
DISCRETE NONHOLONOMIC MECHANICS 11
Locally, we deduce that Ld is regular if and only if for every g ∈ Γ and every
local basis {XA} (respectively, {YB}) of Sec(τ) on an open subset U (respectively,
V ) of M such that α(g) ∈ U (respectively, β(g) ∈ V ) we have that the matrix
Y B(Ld))) is regular on α−1(U) ∩ β−1(V ).
Now, let Ld : Γ→ R be a discrete Lagrangian and g be a point of Γ. We define
the R-bilinear map GLdg : (EΓ)α(g) ⊕ (EΓ)β(g) → R given by
GLdg (a, b) = ΩLd(g)((−T�(α(g))(rg ◦ i)(a), 0), (0, (T�(β(g))lg)(b))). (2.22)
Then, using (2.14), we have that
Proposition 2.1. The discrete Lagrangian Ld : Γ → R is regular if and only if
GLdg is nondegenerate, for all g ∈ Γ, that is,
GLdg (a, b) = 0, for all b ∈ (EΓ)β(g) ⇒ a = 0
(respectively, GLdg (a, b) = 0, for all a ∈ (EΓ)α(g) ⇒ b = 0).
On the other hand, if Ld : Γ → R is a discrete Lagrangian on a Lie groupoid Γ
then we have that
τ∗ ◦ F−Ld = α, τ∗ ◦ F+Ld = β,
where τ∗ : E∗Γ → M is the vector bundle projection. Using these facts, (2.18) and
(2.19), we deduce the following result.
Proposition 2.2. Let Ld : Γ → R be a discrete Lagrangian function. Then, the
following conditions are equivalent:
(i) Ld is regular.
(ii) The linear map TΓhF
−Ld : Vhβ ⊕ Vhα → TEΓF−Ld(h)E
Γ is a linear isomor-
phism, for all h ∈ Γ.
(iii) The linear map TΓg F
+Ld : Vgβ ⊕ Vgα → TEΓF+Ld(g)EΓ
∗ is a linear isomor-
phism, for all g ∈ Γ.
Finally, let Ld : Γ→ R be a regular discrete Lagrangian function and (g0, h0) ∈
Γ×Γ be a solution of the discrete Euler-Lagrange equations for Ld. Then, one may
prove (see [27]) that there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0
and h0 ∈ V0, and there exists a (local) discrete unconstrained Lagrangian evolution
operator ΥLd : U0 → V0 such that:
(i) ΥLd(g0) = h0,
(ii) ΥLd is a diffeomorphism and
(iii) ΥLd is unique, that is, if U
0 is an open subset of Γ, with g0 ∈ U ′0, and
Υ′Ld : U
0 → Γ is a (local) discrete Lagrangian evolution operator then
ΥLd|U0∩U ′0 = Υ
Ld|U0∩U ′0
3. Discrete Nonholonomic (or constrained) Lagrangian systems on
Lie groupoids
3.1. Discrete Generalized Hölder’s principle. Let Γ be a Lie groupoid with
structural maps
α, β : Γ→M, � : M → Γ, i : Γ→ Γ, m : Γ2 → Γ.
Denote by τ : EΓ → M the Lie algebroid associated to Γ. Suppose that the rank
of EΓ is n and that the dimension of M is m.
A generalized discrete nonholonomic (or constrained) Lagrangian system on Γ
is determined by:
12 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
- a regular discrete Lagrangian Ld : Γ −→ R,
- a constraint distribution, Dc, which is a vector subbundle of the bundle
EΓ → M of admissible directions. We will denote by τDc : Dc → M the
vector bundle projection and by iDc : Dc → EΓ the canonical inclusion.
- a discrete constraint embedded submanifold Mc of Γ, such that
dimMc = dimDc = m + r, with r ≤ n. We will denote by iMc : Mc → Γ
the canonical inclusion.
Remark 3.1. Let Ld : Γ→ R be a regular discrete Lagrangian on a Lie groupoid
Γ and Mc be a submanifold of Γ such that �(M) ⊆ Mc. Then, dimMc = m + r,
with 0 ≤ r ≤ m. Moreover, for every x ∈M , we may introduce the subspace Dc(x)
of EΓ(x) given by
Dc(x) = T�(x)Mc ∩ EΓ(x).
Since the linear map T�(x)α : T�(x)Mc → TxM is an epimorphism, we deduce that
dimDc(x) = r. In fact, Dc =
x∈M Dc(x) is a vector subbundle of EΓ (over M) of
rank r. Thus, we may consider the discrete nonholonomic system (Ld,Mc,Dc) on
the Lie groupoid Γ. �
For g ∈ Γ fixed, we consider the following set of admissible sequences of order
CNg =
(g1, . . . , gN ) ∈ ΓN
∣∣ (gk, gk+1) ∈ Γ2, for k = 1, .., N − 1 and g1 . . . gN = g } .
Given a tangent vector at (g1, . . . , gN ) to the manifold CNg , we may write it as the
tangent vector at t = 0 of a curve in CNg , t ∈ (−ε, ε) ⊆ R −→ c(t) which passes
through (g1, . . . , gN ) at t = 0. This type of curves is of the form
c(t) = (g1h1(t), h
1 (t)g2h2(t), . . . , h
N−2(t)gN−1hN−1(t), h
N−1(t)gN )
where hk(t) ∈ α−1(β(gk)), for all t, and hk(0) = �(β(gk)) for k = 1, . . . , N − 1.
Therefore, we may identify the tangent space to CNg at (g1, . . . , gN ) with
T(g1,g2,..,gN )C
g ≡ { (v1, v2, . . . , vN−1) | vk ∈ (EΓ)xk and xk = β(gk), 1 ≤ k ≤ N − 1 } .
Observe that each vk is the tangent vector to the curve hk at t = 0.
The curve c is called a variation of (g1, . . . , gN ) and (v1, v2, . . . , vN−1) is called
an infinitesimal variation of (g1, . . . , gN ).
Now, we define the discrete action sum associated to the discrete Lagrangian
Ld : Γ −→ R as
SLd : CNg −→ R
(g1, . . . , gN ) 7−→
Ld(gk).
We define the variation δSLd : T(g1,...,gN )C
g → R as
δSLd(v1, . . . , vN−1) =
SLd(c(t))
Ld(g1h1(t)) + Ld(h
1 (t)g2h2(t))
+ . . .+ Ld(h
N−2(t)gN−1hN−1(t)) + Ld(h
N−1(t)gN )
do(Ld ◦ lgk)(�(xk))(vk) + d
o(Ld ◦ rgk+1 ◦ i)(�(xk))(vk)
where do is the standard differential on Γ, i.e., do is the differential of the Lie
algebroid τΓ : TΓ → Γ. It is obvious from the last expression that the definition
DISCRETE NONHOLONOMIC MECHANICS 13
of variation δSLd does not depend on the choice of variations c of the sequence g
whose infinitesimal variation is (v1, . . . , vN−1).
Next, we will introduce the subset (Vc)g of T(g1,...,gN )C
g defined by
(Vc)g =
(v1, . . . , vN−1) ∈ T(g1,...,gN )C
∣∣ ∀k ∈ {1, . . . , N − 1}, vk ∈ Dc } .
Then, we will say that a sequence in CNg satisfying the constraints determined
by Mc is a Hölder-critical point of the discrete action sum SLd if the restriction
of δSLd to (Vc)g vanishes, i.e.
(Vc)g
Definition 3.2 (Discrete Hölder’s principle). Given g ∈ Γ, a sequence (g1, . . . , gN )
∈ CNg such that gk ∈ Mc, 1 ≤ k ≤ N , is a solution of the discrete nonholo-
nomic Lagrangian system determined by (Ld,Mc,Dc) if and only if (g1, . . . , gN ) is
a Hölder-critical point of SLd.
If (g1, . . . , gN ) ∈ CNg ∩ (Mc × · · · ×Mc) then (g1, . . . , gN ) is a solution of the
nonholonomic discrete Lagrangian system if and only if
(do(Ld ◦ lgk) + d
o(Ld ◦ rgk+1 ◦ i))(�(xk))|(Dc)xk = 0,
where β(gk) = α(gk+1) = xk. For N = 2, we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc)
(with β(g) = α(h) = x) is a solution if
do(Ld ◦ lg + Ld ◦ rh ◦ i)(�(x))|(Dc)x = 0.
These equations will be called the discrete nonholonomic Euler-Lagrange
equations for the system (Ld,Mc,Dc).
Let (g1, . . . , gN ) be an element of CNg . Suppose that β(gk) = α(gk+1) = xk,
1 ≤ k ≤ N − 1, and that {XAk} = {Xak, Xαk} is a local adapted basis of Sec(τ)
on an open subset Uk of M , with xk ∈ Uk. Here, {Xak}1≤a≤r is a local basis of
Sec(τDc) and, thus, {Xαk}r+1≤α≤n is a local basis of the space of sections of the
vector subbundle τD0c : D
c →M , where D0c is the annihilator of Dc and {Xak, Xαk}
is the dual basis of {Xak, Xαk}. Then, the sequence (g1, . . . , gN ) is a solution of
the discrete nonholonomic equations if (g1, . . . , gN ) ∈Mc×· · ·×Mc and it satisfies
the following closed system of difference equations
gk)(Ld)−
gk+1)(Ld)
〈dLd, (Xak)(0,1)〉(gk)− 〈dLd, (Xak)(1,0)〉(gk+1)
for 1 ≤ a ≤ r, d being the differential of the Lie algebroid πτ : TΓΓ ≡ V β⊕ΓV α −→
Γ. For N = 2 we obtain that (g, h) ∈ Γ2 ∩ (Mc ×Mc) (with β(g) = α(h) = x) is a
solution if
Xa(g)(Ld)−
Xa(h)(Ld) = 0
where {Xa} is a local basis of Sec(τDc) on an open subset U of M such that x ∈ U .
Next, we describe an alternative version of these difference equations. First
observe that using the Lagrange multipliers the discrete nonholonomic equations
are rewritten as
do [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))(v) = λαXα(x)(v),
14 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
for v ∈ (EΓ)x, with (g, h) ∈ Γ2 ∩ (Mc ×Mc) and β(g) = α(h) = x. Here, {Xα} is
a local basis of sections of the annihilator D0c .
Thus, the discrete nonholonomic equations are:
Y (g)(Ld)−
Y (h)(Ld) = λα(X
α)(Y )|β(g), (g, h) ∈ Γ2 ∩ (Mc ×Mc),
for all Y ∈ Sec(τ) or, alternatively,
〈dLd − λα(Xα)(0,1), Y (0,1)〉(g)− 〈dLd, Y (1,0)〉(h) = 0, (g, h) ∈ Γ2 ∩ (Mc ×Mc),
for all Y ∈ Sec(τ).
On the other hand, we may define the discrete nonholonomic Euler-Lagrange
operator DDEL(Ld,Mc,Dc) : Γ2 ∩ (Mc ×Mc)→ D∗c as follows
DDEL(Ld,Mc,Dc)(g, h) = d
o [Ld ◦ lg + Ld ◦ rh ◦ i] (�(x))|(Dc)x ,
for (g, h) ∈ Γ2 ∩ (Mc ×Mc), with β(g) = α(h) = x ∈M .
Then, we may characterize the solutions of the discrete nonholonomic equations
as the sequences (g1, . . . , gN ), with (gk, gk+1) ∈ Γ2 ∩ (Mc × Mc), for each k ∈
{1, . . . , N − 1}, and
DDEL(Ld,Mc,Dc)(gk, gk+1) = 0.
Remark 3.3. (i) The set Γ2 ∩ (Mc ×Mc) is not, in general, a submanifold
of Mc ×Mc.
(ii) Suppose that αMc : Mc → M and βMc : Mc → M are the restrictions
to Mc of α : Γ → M and β : Γ → M , respectively. If αMc and βMc are
submersions then Γ2∩(Mc×Mc) is a submanifold of Mc×Mc of dimension
m+ 2r.
3.2. Discrete Nonholonomic Legendre transformations. Let (Ld,Mc,Dc) be
a discrete nonholonomic Lagrangian system. We define the discrete nonholo-
nomic Legendre transformations
F−(Ld,Mc,Dc) : Mc → D∗c and F
+(Ld,Mc,Dc) : Mc → D∗c
as follows:
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ rh ◦ i), for v�(α(h)) ∈ Dc(α(h)),(3.1)
F+(Ld,Mc,Dc)(g)(v�(β(g))) = v�(β(g))(Ld ◦ lg), for v�(β(g)) ∈ Dc(β(g)).(3.2)
If F−Ld : Γ→ E∗Γ and F
+Ld : Γ→ E∗Γ are the standard Legendre transformations
associated with the Lagrangian function Ld and i∗Dc : E
Γ → D
c is the dual map of
the canonical inclusion iDc : Dc → EΓ then
F−(Ld,Mc,Dc) = i∗Dc ◦ F
−Ld ◦ iMc , F
+(Ld,Mc,Dc) = i
◦ F+Ld ◦ iMc . (3.3)
Remark 3.4. (i) Note that
τ∗Dc ◦ F
−(Ld,Mc,Dc) = αMc , τ
◦ F+(Ld,Mc,Dc) = βMc . (3.4)
(ii) If DDEL(Ld,Mc,Dc) is the discrete nonholonomic Euler-Lagrange opera-
tor then
DDEL(Ld,Mc,Dc)(g, h) = F+(Ld,Mc,Dc)(g)− F−(Ld,Mc,Dc)(h),
for (g, h) ∈ Γ2 ∩ (Mc ×Mc).
DISCRETE NONHOLONOMIC MECHANICS 15
On the other hand, since by assumption Ld : Γ → R is a regular discrete La-
grangian function, we have that the discrete Poincaré-Cartan 2-section ΩLd is sym-
plectic on the Lie algebroid τ̃Γ : TΓΓ → Γ. Moreover, the regularity of L is equiv-
alent to the fact that the Legendre transformations F−Ld and F+Ld to be local
diffeomorphisms (see Subsection 2.3.5).
Next, we will obtain necessary and sufficient conditions for the discrete non-
holonomic Legendre transformations associated with the system (Ld,Mc,Dc) to be
local diffeomorphisms.
Let F be the vector subbundle (over Γ) of τ̃Γ : TΓΓ→ Γ whose fiber at the point
h ∈ Γ is
(1,0)
∣∣∣ γ ∈ Dc(α(h))0 }0 ⊆ TΓhΓ.
In other words,
F 0h =
(1,0)
∣∣∣ γ ∈ Dc(α(h))0 } .
Note that the rank of F is n+ r.
We also consider the vector subbundle F̄ (over Γ) of τ̃Γ : TΓΓ→ Γ of rank n+ r
whose fiber at the point g ∈ Γ is
F̄g =
γ(0,1)g
∣∣∣ γ ∈ Dc(β(g))0 }0 ⊆ TΓg Γ.
Lemma 3.5. F (respectively, F̄ ) is a coisotropic vector subbundle of the symplectic
vector bundle (TΓΓ,ΩLd), that is,
F⊥h ⊆ Fh, for every h ∈ Γ
(respectively, F̄⊥g ⊆ F̄g, for every g ∈ Γ), where F⊥h (respectively, F̄
g ) is the
symplectic orthogonal of Fh (respectively, F̄g) in the symplectic vector space (TΓhΓ,
ΩLd(h)) (respectively, (T
g Γ,ΩLd(g))).
Proof. If h ∈ Γ we have that
F⊥h = [
ΩLd (h)
(F 0h ),
[ΩLd (h)
: TΓhΓ→ (T
∗ being the canonical isomorphism induced by the symplectic
form ΩLd(h). Thus, using (2.14), we deduce that
F⊥h =
ΩLd (h)
(γ(1,0)h )
∣∣∣ γ ∈ Dc(α(h))0 } ⊆ {0} ⊕ Vhα ⊆ Fh.
The coisotropic character of F̄g is proved in a similar way. �
We also have the following result
Lemma 3.6. Let TΓF−Ld : TΓΓ→ TEΓE∗Γ (respectively, T
ΓF+Ld : TΓΓ→ TEΓE∗Γ)
be the prolongation of the Legendre transformation F−Ld : Γ → E∗Γ (respectively,
F+Ld : Γ→ E∗Γ). Then,
(TΓhF
−Ld)(Fh) = T
F−Ld(h)
E∗Γ =
(vα(h), XF−Ld(h)) ∈ T
F−Ld(h)
∣∣∣ vα(h) ∈ Dc(α(h))} ,
for h ∈Mc (respectively,
(TΓg F
+Ld)(F̄g) = T
F+Ld(g)
E∗Γ =
(vβ(g), XF+Ld(g)) ∈ T
F+Ld(g)
∣∣∣ vβ(g) ∈ Dc(β(g))} ,
for g ∈Mc).
Proof. It follows using (2.11), (2.18) (respectively, (2.12), (2.19)) and Proposition
2.2. �
Now, we may prove the following theorem.
16 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Theorem 3.7. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re-
spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism.
(ii) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0} (3.5)
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
Proof. (i) ⇒ (ii) If h ∈ Mc and (Xh, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h then, using the
fact that F⊥h ⊆ {0} ⊕ Vhα (see the proof of Lemma 3.5), we have that Xh = 0.
Therefore,
Yh ∈ Vhα ∩ ThMc. (3.6)
Next, we will see that
(ThF−(Ld,Mc,Dc))(Yh) = 0. (3.7)
From (3.4) and (3.6), it follows that (ThF−(Ld,Mc,Dc))(Yh) is vertical with
respect to the projection τ∗Dc : D
c →M .
Thus, it is sufficient to prove that
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = 0, for all Z ∈ Sec(τDc).
Here, Ẑ : D∗c → R is the linear function on D∗c induced by the section Z.
Now, using (3.3), we deduce that
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = d(Ẑ ◦ i∗Dc)((F
−Ld)(h))(0, (ThF−Ld)(Yh)),
where d is the differential of the Lie algebroid τ τ
: TEΓE∗Γ → E
Consequently, if Z∗c : E∗Γ → T
EΓE∗Γ is the complete lift of Z ∈ Sec(τ), we have
that (see (2.10)),
((ThF−(Ld,Mc,Dc))(Yh))(Ẑ) = Ω(F−Ld(h))(Z∗c(F−Ld(h)),
(0, (ThF−Ld)(Yh)),
(3.8)
Ω being the canonical symplectic section associated with the Lie algebroid EΓ.
On the other hand, since Z ∈ Sec(τDc), it follows that Z∗c(F−Ld(h)) is in
F−Ld(h)
E∗Γ and, from Lemma 3.6, we conclude that there exists (X
h) ∈ Fh
such that
(TΓhF
−Ld)(X
h) = Z
∗c((F−Ld)(h)). (3.9)
Moreover, using (2.18), we obtain that
(TΓhF
−Ld)(0, Yh) = (0, (ThF−Ld)(Yh)). (3.10)
Thus, from (2.21), (3.8), (3.9) and (3.10), we deduce that
((ThF−(Ld,M,Dc))(Yh))(Ẑ) = −ΩLd(h)((0, Yh), (X
Therefore, since (0, Yh) ∈ F⊥h , it follows that (3.7) holds, which implies that Yh = 0.
This proves that (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
If F+(Ld,Mc,Dc) is a local diffeomorphism then, proceeding as above, we have
that (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
(ii) ⇒ (i) Suppose that h ∈Mc and that Yh is a tangent vector to Mc at h such
(ThF−(Ld,Mc,Dc))(Yh) = 0. (3.11)
DISCRETE NONHOLONOMIC MECHANICS 17
We have that (Thα)(Yh) = 0 and, thus,
(0, Yh) ∈ (ρT
ΓΓ)−1(ThMc).
We will see that (0, Yh) ∈ F⊥h , that is,
ΩLd(h)((0, Yh), (X
h)) = 0, for (X
h) ∈ Fh. (3.12)
Now, using (2.18) and (2.21), we deduce that
ΩLd(h)((0, Yh), (X
h)) = Ω(F
−Ld(h))((0, (ThF−Ld)(Yh)), (TΓhF
−Ld)(X
Therefore, from Lemma 3.6, we obtain that
ΩLd(h)((0, Yh), (X
h)) = Ω(F
−Ld(h))(0, (ThF−Ld)(Yh)), (vα(h), YF−Ld(h)))
with (vα(h), YF−Ld(h)) ∈ T
F−Ld(h)
Next, we take a section Z ∈ Sec(τDc) such that Z(α(h)) = vα(h). Then (see
(2.9)),
(vα(h), YF−Ld(h)) = Z
∗c(F−Ld(h)) + (0, Y ′F−Ld(h)),
where Y ′F−Ld(h) ∈ TF−Ld(h)E
Γ and Y
F−Ld(h)
is vertical with respect to the projection
τ∗ : E∗Γ →M .
Thus, since (see Eq. (3.7) in [23])
Ω(F−Ld(h))((0, (ThF−Ld)(Yh)), (0, Y ′F−Ld(h))) = 0,
we have that
ΩLd(h)((0, Yh), (X
h)) = −Ω(F
−Ld(h))(Z∗c(F−Ld(h)), (0, (ThF−Ld)(Yh)))
= −d(Ẑ ◦ i∗Dc)(F
−Ld(h))(0, (ThF−Ld)(Yh))
and, from (3.11), we deduce that (3.12) holds.
This proves that Yh ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h which implies that Yh = 0.
Therefore, F−(Ld,Mc,Dc) is a local diffeomorphism.
If (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0} for all g ∈ Mc then, proceeding as above, we
obtain that F+(Ld,Mc,Dc) is a local diffeomorphism. �
Now, let ρT
ΓΓ : TΓΓ→ TΓ be the anchor map of the Lie algebroid πτ : TΓΓ→ Γ.
Then, we will denote by Hh the subspace of TΓhΓ given by
Hh = (ρ
TΓΓ)−1(ThMc) ∩ Fh, for h ∈Mc.
In a similar way, for every g ∈Mc we will introduce the subspace H̄g of TΓg Γ defined
H̄g = (ρ
TΓΓ)−1(TgMc) ∩ F̄g.
On the other hand, let h be a point of Mc and G
h : (EΓ)α(h) ⊕ (EΓ)β(h) → R
be the R-bilinear map given by (2.22). We will denote by (
h the subspace of
(EΓ)β(h) defined by
b ∈ (EΓ)β(h)
∣∣ (T�(β(h))lh)(b) ∈ ThMc }
and by GLdch : (Dc)α(h)× (
h → R the restriction to (Dc)α(h)× (
h of the
R-bilinear map GLdh .
In a similar way, if g is a point of Γ we will consider the subspace (
E Γ)Mcg of
(EΓ)α(g) defined by
a ∈ (EΓ)α(g)
∣∣ (T�(α(g))(rg ◦ i))(a) ∈ TgMc }
18 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
and the restriction ḠLdcg : (
E Γ)Mcg × (Dc)β(g) → R of GLdg to the space (
E Γ)Mcg ×
(Dc)β(g).
Proposition 3.8. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
(ii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub-
space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle
H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener-
(iii) For every h ∈Mc (respectively, g ∈Mc){
b ∈ (
∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0}
(respectively,
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}).
Proof. (i) ⇒ (ii) Assume that h ∈Mc and that
ΓΓ)−1(ThMc) ∩ F⊥h = {0}. (3.13)
Let U be an open subset of Γ, with h ∈ U , and {φγ}γ=1,...,n−r a set of independent
real C∞-functions on U such that
Mc ∩ U = {h′ ∈ U | φγ(h′) = 0, for all γ } .
If d is the differential of the Lie algebroid τ̃Γ : TΓΓ → Γ then it is easy to prove
ΓΓ)−1(ThMc) =< {dφγ(h)} >0 .
Thus,
dim((ρT
ΓΓ)−1(ThMc)) ≥ n+ r. (3.14)
On the other hand, dimF⊥h = n−r. Therefore, from (3.13) and (3.14), we obtain
dim((ρT
ΓΓ)−1(ThMc)) = n+ r
TΓhΓ = (ρ
TΓΓ)−1(ThMc)⊕ F⊥h .
Consequently, using Lemma 3.5, we deduce that
Fh = Hh ⊕ F⊥h . (3.15)
This implies that dimHh = 2r. Moreover, from (3.15), we also get that
Hh ∩H⊥h ⊆ Hh ∩ F
and, since Hh ∩ F⊥h = (ρ
TΓΓ)−1(ThMc) ∩ F⊥h (see Lemma 3.5), it follows that
Hh ∩H⊥h = {0}.
Thus, we have proved that Hh is a symplectic subspace of the symplectic vector
space (TΓhΓ,ΩLd(h)).
If (ρT
ΓΓ)−1(TgMc)∩ F̄⊥g = {0}, for all g ∈Mc then, proceeding as above, we ob-
tain that H̄g is a symplectic subspace of the symplectic vector space (TΓg Γ,ΩLd(g)),
for all g ∈Mc.
(ii) ⇒ (i) Suppose that h ∈ Mc and that Hh is a symplectic subspace of the
symplectic vector space (TΓhΓ,ΩLd(h)).
DISCRETE NONHOLONOMIC MECHANICS 19
If (Xh, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h then, using Lemma 3.5, we deduce that
(Xh, Yh) ∈ Hh.
Now, if (X ′h, Y
h) ∈ Hh then, since (Xh, Yh) ∈ F
h , we conclude that
ΩLd(h)((Xh, Yh), (X
h)) = 0.
This implies that
(Xh, Yh) ∈ Hh ∩H⊥h = {0}.
Therefore, we have proved that (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
If H̄g ∩ H̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we obtain that
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
(i) ⇒ (iii) Assume that
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
and that b ∈ (
h satisfies the following condition
h (a, b) = 0, ∀a ∈ (Dc)α(h).
Then, Yh = (T�(β(h))lh)(b) ∈ ThMc ∩ Vhα and (0, Yh) ∈ (ρT
ΓΓ)−1(ThMc).
Moreover, if (X ′h, Y
h) ∈ Fh, we have that
X ′h = −(T�(α(h))(rh ◦ i))(a), with a ∈ (Dc)α(h).
Thus, using (2.14) and (2.22), we deduce that
ΩLd(h)((X
h), (0, Yh)) = ΩLd(h)((X
h, 0), (0, Yh)) = G
h (a, b) = 0.
Therefore,
(0, Yh) ∈ (ρT
ΓΓ)−1(ThMc) ∩ F⊥h = {0},
which implies that b = 0.
If (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈ Mc then, proceeding as above, we
obtain that{
a ∈ (
∣∣∣ GLdcg (a, b) = 0, for all b ∈ (Dc)β(g) } = {0}.
(iii) ⇒ (i) Suppose that h ∈Mc, that{
b ∈ (
∣∣∣ GLdh (a, b) = 0, ∀a ∈ (Dc)α(h) } = {0}
and let (Xh, Yh) be an element of the set (ρT
ΓΓ)−1(ThMc) ∩ F⊥h .
Then (see the proof of Lemma 3.5), Xh = 0 and Yh ∈ ThMc∩Vhα. Consequently,
Yh = (T�(β(h)lh)(b), with b ∈ (
Now, if a ∈ (Dc)α(h), we have that
X ′h = (T�(α(h))(rh ◦ i))(a) ∈ Vhβ and (X
h, 0) ∈ Fh.
Thus, from (2.22) and since (0, Yh) ∈ F⊥h , it follows that
h (a, b) = ΩLd(h)((X
h, 0)(0, Yh)) = 0.
Therefore, b = 0 which implies that Yh = 0.
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}, for all g ∈ Mc, then
proceeding as above we obtain that (ρT
ΓΓ)−1(TgMc)∩F̄⊥g = {0}, for all g ∈Mc. �
Using Theorem 3.7 and Proposition 3.8, we conclude
20 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Theorem 3.9. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) (re-
spectively, F+(Ld,Mc,Dc)) is a local diffeomorphism.
(ii) For every h ∈Mc (respectively, g ∈Mc)
ΓΓ)−1(ThMc) ∩ F⊥h = {0}
(respectively, (ρT
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}).
(iii) For every h ∈ Mc (respectively, g ∈ Mc) the dimension of the vector sub-
space Hh (respectively, H̄g) is 2r and the restriction to the vector subbundle
H (respectively, H̄) of the Poincaré-Cartan 2-section ΩLd is nondegener-
(iv) For every h ∈Mc (respectively, g ∈Mc){
b ∈ (
∣∣∣ GLdch (a, b) = 0,∀a ∈ (Dc)α(h) } = {0}
(respectively,
a ∈ (
E Γ)Mcg
∣∣∣ GLdcg (a, b) = 0,∀b ∈ (Dc)β(g) } = {0}).
3.3. Nonholonomic evolution operators and regular discrete nonholo-
nomic Lagrangian systems. First of all, we will introduce the definition of a
nonholonomic evolution operator.
Definition 3.10. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system
and Υnh : Mc →Mc be a differentiable map. Υnh is said to be a discrete nonholo-
nomic evolution operator for (Ld,Mc,Dc) if:
(i) graph(Υnh) ⊆ Γ2, that is, (g,Υnh(g)) ∈ Γ2, for all g ∈Mc and
(ii) (g,Υnh(g)) is a solution of the discrete nonholonomic equations, for all
g ∈Mc, that is,
do(Ld ◦ lg + Ld ◦ rΥnh(g) ◦ i)(�(β(g)))|Dc(β(g)) = 0, for all g ∈Mc.
Remark 3.11. If Υnh : Mc → Mc is a differentiable map then, from (3.1), (3.2)
and (3.4), we deduce that Υnh is a discrete nonholonomic evolution operator for
(Ld,Mc,Dc) if and only if
F−(Ld,Mc,Dc) ◦Υnh = F+(Ld,Mc,Dc).
Now, we will introduce the notion of a regular discrete nonholonomic Lagrangian
system.
Definition 3.12. A discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is said
to be regular if the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc)
and F+(Ld,Mc,Dc) are local diffeomorphims.
From Theorem 3.9, we deduce
Corollary 3.13. Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system.
Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) The following relations hold
ΓΓ)−1(ThMc) ∩ F⊥h = {0}, for all h ∈Mc,
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}, for all g ∈Mc.
DISCRETE NONHOLONOMIC MECHANICS 21
(iii) H and H̄ are symplectic subbundles of rank 2r of the symplectic vector
bundle (TΓMcΓ,ΩLd).
(iv) If g and h are points of Mc then the R-bilinear maps GLdch and Ḡ
g are
right and left nondegenerate, respectively.
The map GLdch (respectively, Ḡ
g ) is right nondegenerate (respectively, left non-
degenerate) if
h (a, b) = 0,∀a ∈ (Dc)α(h) ⇒ b = 0
(respectively, ḠLdcg (a, b) = 0,∀b ∈ (Dc)β(g) ⇒ a = 0).
Every solution of the discrete nonholonomic equations for a regular discrete
nonholonomic Lagrangian system determines a unique local discrete nonholonomic
evolution operator. More precisely, we may prove the following result:
Theorem 3.14. Let (Ld,Mc,Dc) be a regular discrete nonholonomic Lagrangian
system and (g0, h0) ∈Mc×Mc be a solution of the discrete nonholonomic equations
for (Ld,Mc,Dc). Then, there exist two open subsets U0 and V0 of Γ, with g0 ∈ U0
and h0 ∈ V0, and there exists a local discrete nonholonomic evolution operator
Υ(Ld,Mc,Dc)nh : U0 ∩Mc → V0 ∩Mc such that:
(i) Υ(Ld,Mc,Dc)nh (g0) = h0;
(ii) Υ(Ld,Mc,Dc)nh is a diffeomorphism and
(iii) Υ(Ld,Mc,Dc)nh is unique, that is, if U
0 is an open subset of Γ, with g0 ∈ U ′0,
and Υnh : U ′0 ∩ Mc → Mc is a (local) discrete nonholonomic evolution
operator then
(Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc = (Υnh)|U0∩U ′0∩Mc .
Proof. From remark 3.4, we deduce that
F+(Ld,Mc,Dc)(g0) = F−(Ld,Mc,Dc)(h0) = µ0 ∈ D∗c .
Thus, we can choose two open subsets U0 and V0 of Γ, with g0 ∈ U0 and h0 ∈ V0,
and an open subset W0 of E∗Γ such that µ0 ∈W0 and
F+(Ld,Mc,Dc) : U0 ∩Mc →W0 ∩D∗c , F
−(Ld,Mc,Dc) : V0 ∩Mc →W0 ∩D∗c
are diffeomorphisms. Therefore, from Remark 3.11, we deduce that
Υ(Ld,Mc,Dc)nh = (F
−(Ld,Mc,Dc)
−1 ◦ F+(Ld,Mc,Dc))|U0∩Mc : U0 ∩Mc → V0 ∩Mc
is a (local) discrete nonholonomic evolution operator. Moreover, it is clear that
Υ(Ld,Mc,Dc)nh (g0) = h0 and it follows that Υ
(Ld,Mc,Dc)
nh is a diffeomorphism.
Finally, if U ′0 is an open subset of Γ, with g0 ∈ U ′0, and Υnh : U ′0 ∩Mc → Mc
is another (local) discrete nonholonomic evolution operator then (Υnh)|U0∩U ′0∩Mc
is also a (local) discrete nonholonomic evolution operator. Consequently, from
Remark 3.11, we conclude that
(Υnh)|U0∩U ′0∩Mc = [F
−(Ld,Mc,Dc)−1 ◦ F+(Ld,Mc,Dc)]|U0∩U ′0∩Mc
= (Υ(Ld,Mc,Dc)nh )|U0∩U ′0∩Mc .
22 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
3.4. Reversible discrete nonholonomic Lagrangian systems. Let (Ld,Mc,
Dc) be a discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M .
Following the terminology used in [36] for the particular case when Γ is the pair
groupoid M ×M , we will introduce the following definition
Definition 3.15. The discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is
said to be reversible if
Ld ◦ i = Ld, i(Mc) = Mc,
i : Γ→ Γ being the inversion of the Lie groupoid Γ.
For a reversible discrete nonholonomic Lagrangian system we have the following
result:
Proposition 3.16. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys-
tem on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The discrete nonholonomic Legendre transformation F−(Ld,Mc,Dc) is a
local diffeomorphism.
(ii) The discrete nonholonomic Legendre transformation F+(Ld,Mc,Dc) is a
local diffeomorphism.
Proof. If h ∈Mc then, using (3.1) and the fact that Ld ◦ i = Ld, it follows that
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −v�(α(h))(Ld ◦ l−1h )
for v�(α(h)) ∈ (Dc)α(h). Thus, from (3.2), we obtain that
F−(Ld,Mc,Dc)(h)(v�(α(h))) = −F+(Ld,Mc,Dc)(h−1)(v�(β(h−1))).
This implies that
F+(Ld,Mc,Dc) = −F−(Ld,Mc,Dc) ◦ i.
Therefore, since the inversion is a diffeomorphism (in fact, we have that i2 = id),
we deduce the result �
Using Theorem 3.9, Definition 3.12 and Proposition 3.16, we prove the following
corollaries.
Corollary 3.17. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system
on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) For all h ∈Mc,
ΓΓ)−1(ThMc) ∩ F⊥h = {0}.
(iii) H = (ρT
ΓΓ)−1(TMc)∩F is a symplectic subbundle of the symplectic vector
bundle (TΓMcΓ,ΩLd).
(iv) The R-bilinear map GLdch : (
h ×(Dc)α(h) → R is right nondegenerate,
for all h ∈Mc.
Corollary 3.18. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian system
on a Lie groupoid Γ. Then, the following conditions are equivalent:
(i) The system (Ld,Mc,Dc) is regular.
(ii) For all g ∈Mc,
ΓΓ)−1(TgMc) ∩ F̄⊥g = {0}.
(iii) H̄ = (ρT
ΓΓ)−1(TMc)∩ F̄ is a symplectic subbundle of the symplectic vector
bundle (TΓMcΓ,ΩLd).
DISCRETE NONHOLONOMIC MECHANICS 23
(iv) The R-bilinear map ḠLdcg : (Dc)β(g) × (
E Γ)Mcg → R is left nondegenerate,
for all g ∈Mc.
Next, we will prove that a reversible nonholonomic Lagrangian system is dynam-
ically reversible.
Proposition 3.19. Let (Ld,Mc,Dc) be a reversible nonholonomic Lagrangian sys-
tem on a Lie groupoid Γ and (g, h) be a solution of the discrete nonholonomic Euler-
Lagrange equations for (Ld,Mc,Dc). Then, (h−1, g−1) is also a solution of these
equations. In particular, if the system (Ld,Mc,Dc) is regular and Υ
(Ld,Mc,Dc)
nh is the
(local) discrete nonholonomic evolution operator for (Ld,Mc,Dc) then Υ
(Ld,Mc,Dc)
is reversible, that is,
Υ(Ld,Mc,Dc)nh ◦ i ◦Υ
(Ld,Mc,Dc)
nh = i.
Proof. Using that i(Mc) = Mc, we deduce that
(h−1, g−1) ∈ Γ2 ∩ (Mc ×Mc).
Now, suppose that β(g) = α(h) = x and that v ∈ (Dc)x. Then, since Ld ◦ i = Ld,
it follows that
do[Ld ◦ lh−1 + Ld ◦ rg−1 ◦ i](ε(x))(v) = v(Ld ◦ i ◦ rh ◦ i) + v(Ld ◦ i ◦ lg)
= v(Ld ◦ lg) + v(Ld ◦ rh ◦ i) = 0.
Thus, we conclude that (h−1, g−1) is a solution of the discrete nonholonomic
Euler-Lagrange equations for (Ld,Mc,Dc).
If the system (Ld,Mc,Dc) is regular and g ∈Mc, we have that (g,Υ
(Ld,M,Dc)
nh (g))
is a solution of the discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc).
Therefore, (i(Υ(Ld,M,Dc)nh (g)), i(g)) is also a solution of the dynamical equations
which implies that
Υ(Ld,M,Dc)nh (i(Υ
(Ld,M,Dc)
nh (g))) = i(g).
Remark 3.20. Proposition 3.19 was proved in [36] for the particular case when Γ
is the pair groupoid. �
3.5. Lie groupoid morphisms and reduction. Let (Φ,Φ0) be a Lie groupoid
morphism between the Lie groupoids Γ ⇒ M and Γ′ ⇒ M ′.
Denote by (E(Φ),Φ0) the corresponding morphism between the Lie algebroids
EΓ and EΓ′ of Γ and Γ′, respectively (see Section 2.2).
If Ld : Γ→ R and L′d : Γ
′ → R are discrete Lagrangians on Γ and Γ′ such that
Ld = L
d ◦ Φ
then, using Theorem 4.6 in [27], we have that
(DDELLd)(g, h)(v) = (DDELL
d)(Φ(g),Φ(h))(Ex(Φ)(v))
for (g, h) ∈ Γ2 and v ∈ (EΓ)x, where x = β(g) = α(h) ∈M.
Using this fact, we deduce the following result:
Corollary 3.21. Let (Φ,Φ0) be a Lie groupoid morphism between the Lie groupoids
Γ ⇒ M and Γ′ ⇒ M ′. Suppose that L′d : Γ
′ → R is a discrete Lagrangian on Γ′,
that (Ld = L′d ◦Φ,Mc,Dc) is a discrete nonholonomic Lagrangian system on Γ and
that (g, h) ∈ Γ2 ∩ (Mc ×Mc). Then:
24 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
(i) The pair (g, h) is a solution of the discrete nonholonomic problem (Ld,Mc,
Dc) if and only if (DDELL′d)(Φ(g),Φ(h)) vanishes over the set
(Eβ(g)Φ)((Dc)β(g)).
(ii) If (L′d,M
c) is a discrete nonholonomic Lagrangian system on Γ
′ such
that (Φ(g),Φ(h)) ∈M′c×M′c and (Eβ(g)(Φ))((Dc)β(g)) = (D′c)Φ0(β(g)) then
(g, h) is a solution for the discrete nonholonomic problem (Ld,Mc,Dc) if
and only if (Φ(g),Φ(h)) is a solution for the discrete nonholonomic problem
(L′d,M
3.6. Discrete nonholonomic Hamiltonian evolution operator. Let (Ld,Mc,
Dc) a regular discrete nonholonomic system. Assume, without the loss of gener-
ality, that the discrete nonholonomic Legendre transformations F−(Ld,Mc,Dc) :
Mc −→ D∗c and F+(Ld,Mc,Dc) : Mc −→ D∗c are global diffeomorphisms. Then,
(Ld,Mc,Dc)
nh = F
−(Ld,Mc,Dc)−1◦F+(Ld,Mc,Dc) is the discrete nonholonomic evo-
lution operator and one may define the discrete nonholonomic Hamiltonian
evolution operator, γ̃nh : D∗c → D∗c , by
γ̃nh = F+(Ld,Mc,Dc) ◦ γ
(Ld,Mc,Dc)
nh ◦ F
+(Ld,Mc,Dc)
−1 . (3.16)
From Remark 3.11, we have the following alternative definitions
γ̃nh = F−(Ld,Mc,Dc) ◦ γ
(Ld,Mc,Dc)
nh ◦ F
−(Ld,Mc,Dc)
γ̃nh = F+(Ld,Mc,Dc) ◦ F−(Ld,Mc,Dc)−1
of the discrete Hamiltonian evolution operator. The following commutative diagram
illustrates the situation
Mc Mc
(Ld,Mc,Dc)
D∗c D
F−(Ld, Mc, Dc)
F+(Ld, Mc, Dc)
F−(Ld, Mc, Dc)
F+(Ld, Mc, Dc)
γ̃nh γ̃nh
Remark 3.22. The discrete nonholonomic evolution operator is an application
from D∗c to itself. It is remarkable that D
c is also the appropriate nonholonomic
momentum space for a continuous nonholonomic system defined by a Lagrangian
L : EΓ → R and the constraint distribution Dc. Therefore, in the regular case, the
solution of the continuous nonholonomic Lagrangian system also determines a flow
from D∗c to itself. We consider that this would be a good starting point to compare
the discrete and continuous dynamics and eventually to establish a backward error
analysis for nonholonomic systems. �
3.7. The discrete nonholonomic momentum map. Let (Ld,Mc,Dc) be a reg-
ular discrete nonholonomic Lagrangian system on a Lie groupoid Γ ⇒ M and
τ : EΓ →M be the Lie algebroid of Γ.
Suppose that g is a Lie algebra and that Ψ : g→ Sec(τ) is a R-linear map. Then,
for each x ∈M, we consider the vector subspace gx of g given by
gx = { ξ ∈ g | Ψ(ξ)(x) ∈ (Dc)x }
DISCRETE NONHOLONOMIC MECHANICS 25
and the disjoint union of these vector spaces
gDc =
We will denote by (gDc)∗ the disjoint union of the dual spaces, that is,
(gDc)∗ =
(gx)∗.
Next, we define the discrete nonholonomic momentum map Jnh : Γ →
(gDc)∗ as follows: Jnh(g) ∈ (gβ(g))∗ and
Jnh(g)(ξ) = Θ+Ld(Ψ(ξ)
(1,1))(g) =
Ψ(ξ)(g)(Ld), for g ∈ Γ and ξ ∈ gβ(g).
If ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for all x ∈ M, then we may
consider the smooth function Jnheξ : Γ→ R defined by
Jnheξ (g) = Jnh(g)(ξ̃(β(g))), ∀g ∈ Γ.
Definition 3.23. The Lagrangian Ld is said to be g-invariant with respect Ψ if
Ψ(ξ)(1,1)(Ld) =
Ψ(ξ)(Ld)−
Ψ(ξ)(Ld) = 0, ∀ξ ∈ g.
Now, we will prove the following result
Theorem 3.24. Let Υ(Ld,Mc,Dc)nh : Mc → Mc be the local discrete nonholonomic
evolution operator for the regular system (Ld,Mc,Dc). If Ld is g-invariant with
respect to Ψ : g→ Sec(τ) and ξ̃ : M → g is a smooth map such that ξ̃(x) ∈ gx, for
all x ∈M, then
Jnheξ (Υ(Ld,Mc,Dc)nh (g))− Jnheξ (g) =
←−−−−−−−−−−−−−−−−−−−−−−−−−−
Ψ(ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g)))(Υ
(Ld,Mc,Dc)
nh (g))(Ld)
for g ∈Mc.
Proof. Using that the Lagrangian Ld is g-invariant with respect to Ψ, we have that
−−−−−−−−−−−−−−−−−−→
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld) =
←−−−−−−−−−−−−−−−−−−
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld).
(3.17)
Also, since (g,Υ(Ld,Mc,Dc)nh (g)) is a solution of the discrete nonholonomic equations:
←−−−−−−−
Ψ(ξ̃(β(g)))(g)(Ld) =
−−−−−−−−−−−−−−−−−−→
Ψ(ξ̃(α(Υ(Ld,Mc,Dc)nh (g))))(Υ
(Ld,Mc,Dc)
nh (g))(Ld).(3.18)
Thus, from (3.17) and (3.18), we find that
←−−−−−−−
Ψ(ξ̃(β(g))(g)(Ld) =
←−−−−−−−
Ψ(ξ̃(β(g)))(Υ(Ld,Mc,Dc)nh (g))(Ld).
26 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Therefore,
(Υ(Ld,Mc,Dc)nh (g))− J
(g) =
←−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))
(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−−
ξ̃(β(g))
(g)(Ld)
←−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))
(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−
Ψ(ξ̃(β(g))(Υ(Ld,Mc,Dc)nh (g))(Ld)
←−−−−−−−−−−−−−−−−−−−−−−−−−−−
ξ̃(β(Υ(Ld,Mc,Dc)nh (g)))− ξ̃(β(g))
(Υ(Ld,Mc,Dc)nh (g))(Ld).
Theorem 3.24 suggests us to introduce the following definition
Definition 3.25. An element ξ ∈ g is said to be a horizontal symmetry for
the discrete nonholonomic system (Ld,Mc,Dc) and the map Ψ : g→ Sec(τ) if
Ψ(ξ)(x) ∈ (Dc)x, for all x ∈M.
Now, from Theorem 3.24, we conclude that
Corollary 3.26. If Ld is g-invariant with respect to Ψ and ξ ∈ g is a horizontal
symmetry for (Ld,Mc,Dc) and Ψ : g → Sec(τ) then Jnhξ̃ : Γ → R is a constant of
the motion for Υ(Ld,Mc,Dc)nh , that is,
◦Υ(Ld,Mc,Dc)nh = J
4. Examples
4.1. Discrete holonomic Lagrangian systems on a Lie groupoid. Let us
examine the case when the system is subjected to holonomic constraints.
Let Ld : Γ → R be a discrete Lagrangian on a Lie groupoid Γ ⇒ M . Suppose
that Mc ⊆ Γ is a Lie subgroupoid of Γ over M ′ ⊆M , that is, Mc is a Lie groupoid
over M ′ with structural maps
α|Mc : Mc →M
′, β|Mc : Mc →M
′, �|M ′ : M
′ →Mc, i|Mc : Mc →Mc,
the canonical inclusions iMc : Mc −→ Γ and iM ′ : M ′ −→ M are injective immer-
sions and the pair (iMc , iM ′) is a Lie groupoid morphism. We may assume, without
the loss of generality, that M ′ = M (in other case, we will replace the Lie groupoid
Γ by the Lie subgroupoid Γ′ over M ′ defined by Γ′ = α−1(M ′) ∩ β−1(M ′)).
Then, if LMc = Ld ◦ iMc and τMc : EMc → M is the Lie algebroid of Mc, we
have that the discrete (unconstrained) Euler-Lagrange equations for the Lagrangian
function LMc are:
X (g)(LMc)−
X (h)(LMc) = 0, (g, h) ∈ (Mc)2, (4.1)
for X ∈ Sec(τMc).
We are interested in writing these equations in terms of the Lagrangian Ld
defined on the Lie groupoid Γ. From Corollary 4.7 (iii) in [27], we deduce that
(g, h) ∈ (Mc)2 is a solution of Equations 4.1 if and only if DDELLd(g, h) vanishes
over Im(Eβ(g)(iMc)). Here, E(iMc) : EMc → EΓ is the Lie algebroid morphism
induced between EMc and EΓ by the Lie groupoid morphism (iMc , Id). Therefore,
we may consider the discrete holonomic system as the discrete nonholonomic system
(Ld,Mc,Dc), where Dc = (E(iMc))(EMc) ∼= EMc .
DISCRETE NONHOLONOMIC MECHANICS 27
In the particular case, when the subgroupoid Mc is determined by the vanishing
set of n− r independent real C∞-functions φγ : Γ→ R:
Mc = { g ∈ Γ | φγ(g) = 0, for all γ } ,
then the discrete holonomic equations are equivalent to:
Y (g)(Ld)−
Y (h)(Ld) = λγd
oφγ(�(β(g)))(Y (β(g)),
φγ(g) = φγ(h) = 0,
for all Y ∈ Sec(τ), where do is the standard differential on Γ. This algorithm is
a generalization of the Shake algorithm for holonomic systems (see [10, 20, 32, 36]
for similar results on the pair groupoid Q×Q).
4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid.
Let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the pair group-
oid Q×Q ⇒ Q and suppose that (q0, q1) is a point of Mc. Then, using the results
of Section 3.1, we deduce that ((q0, q1), (q1, q2)) ∈ (Q × Q)2 is a solution of the
discrete nonholonomic Euler-Lagrange equations for (Ld,Mc,Dc) if and only if
(D2Ld(q0, q1) +D1Ld(q1, q2))|Dc(q1) = 0,
(q1, q2) ∈Mc,
or, equivalently,
D2Ld(q0, q1) +D1Ld(q1, q2) =
j(q1),
(q1, q2) ∈Mc,
where λj are the Lagrange multipliers and {Aj} is a local basis of the annihilator
D0c . These equations were considered in [10] and [36].
Note that if (q1, q2) ∈ Γ = Q×Q then, in this particular case, GLd(q1,q2) : Tq1Q×
Tq2Q→ R is just the R-bilinear map (D2D1Ld)(q1, q2).
On the other hand, if (q1, q2) ∈Mc we have that
(TQ)Mc
(q1,q2)
vq2 ∈ Tq2Q
∣∣ (0, vq2) ∈ T(q1,q2)Mc } ,
(TQ)Mc
(q1,q2)
vq1 ∈ Tq1Q
∣∣ (vq1 , 0) ∈ T(q1,q2)Mc } .
Thus, the system (Ld,Mc,Dc) is regular if and only if for every (q1, q2) ∈ Mc
the following conditions hold:
If vq1 ∈
(TQ)Mc
(q1,q2)
〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq2 ∈ Dc(q2)
=⇒ vq1 = 0,
If vq2 ∈
(TQ)Mc
(q1,q2)
〈D2D1Ld(q1, q2)vq1 , vq2〉 = 0, ∀vq1 ∈ Dc(q1)
=⇒ vq2 = 0.
The first condition was obtained in [36] in order to guarantee the existence
of a unique local nonholonomic evolution operator Υ(Ld,Mc,Dc)nh for the system
(Ld,Mc,Dc). However, in order to assure that Υ
(Ld,Mc,Dc)
nh is a (local) diffeomor-
phism one must assume that the second condition also holds.
28 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Example 4.1 (Discrete Nonholonomically Constrained particle). Consider
the discrete nonholonomic system determined by:
a) A discrete Lagrangian Ld : R3 × R3 → R:
Ld(x0, y0, z0, x1, y1, z1) =
x1 − x0
y1 − y0
z1 − z0
b) A constraint distribution of Q = R3,
Dc = span
,X2 =
c) A discrete constraint submanifold Mc of R3 × R3 determined by the con-
straint
φ(x0, y0, z0, x1, y1, z1) =
z1 − z0
y1 + y0
x1 − x0
The system (Ld,Mc,Dc) is a discretization of a classical continuous nonholonomic
system: the nonholonomic free particle (for a discussion on this continuous system
see, for instance, [4, 8]). Note that if E(R3×R3) ∼= TR3 is the Lie algebroid of the
pair groupoid R3 × R3 ⇒ R3 then
T(x1,y1,z1,x1,y1,z1)Mc ∩ E(R3×R3)(x1, y1, z1) = Dc(x1, y1, z1).
Since
X1 = −
X2 = −
then, the discrete nonholonomic equations are:(
x2 − 2x1 + x0
z2 − 2z1 + z0
= 0, (4.2)
y2 − 2y1 + y0
= 0, (4.3)
which together with the constraint equation determine a well posed system of dif-
ference equations.
We have that
D2D1Ld = − 1h{dx0 ∧ dx1 + dy0 ∧ dy1 + dz0 ∧ dz1}
TR3)Mc
(x0,y0,z0,x1,y1,z1)
= {a0 ∂∂x0 + b0
+ c0 ∂∂z0 ∈ T(x0,y0,z0)R
c0 = 12 (a0(y1 + y0)− b0(x1 − x0))}.
TR3)Mc
(x0,y0,z0,x1,y1,z1)
= {a1 ∂∂x1 + b1
+ c1 ∂∂z1 ∈ T(x1,y1,z1)R
c1 = 12 (a1(y1 + y0) + b1(x1 − x0))}.
Thus, if we consider the open subset of Mc defined by{
(x0, y0, z0, x1, y1, z1) ∈Mc
∣∣ 2 + y21 + y1y0 6= 0, 2 + y20 + y0y1 6= 0}
then in this subset the discrete nonholonomic system is regular.
Let Ψ : g = R2 −→ X(R3) given by Ψ(a, b) = a ∂
+ b ∂
. Then gDc =
span{Ψ(ξ̃) = X1}, where ξ̃ : R3 → R2 is defined by ξ̃(x, y, z) = (1, y). More-
over, the Lagrangian Ld is g-invariant with respect to Ψ. Therefore,
(x1, y1, z1, x2, y2, z2)− Jnhξ̃ (x0, y0, z0, x1, y1, z1)
←−−−−−−−−−
Ψ(0, y2 − y1)(x1, y1, z1, x2, y2, z2)(Ld),
DISCRETE NONHOLONOMIC MECHANICS 29
that is,(
x2 − x1
z2 − z1
x1 − x0
z1 − z0
= (y2 − y1)
z2 − z1
This equation is precisely Equation (4.2).
4.3. Discrete nonholonomic Lagrangian systems on a Lie group. Let G be
a Lie group. G is a Lie groupoid over a single point and the Lie algebra g of G is
just the Lie algebroid associated with G.
If g, h ∈ G, vh ∈ ThG and αh ∈ T ∗hG we will use the following notation:
gvh = (Thlg)(vh) ∈ TghG, vhg = (Thrg)(vh) ∈ ThgG,
gαh = (T ∗ghlg−1)(αh) ∈ T
ghG, αhg = (T
hgrg−1)(αh) ∈ T
Now, let (Ld,Mc,Dc) be a discrete nonholonomic Lagrangian system on the Lie
group G, that is, Ld : G → R is a discrete Lagrangian, Mc is a submanifold of G
and Dc is a vector subspace of g.
If g1 ∈ Mc then (g1, g2) ∈ G × G is a solution of the discrete nonholonomic
Euler-Lagrange equations for (Ld,Mc,Dc) if and only if
g−11 dLd(g1)− dLd(g2)g
gk ∈Mc, k = 1, 2
(4.4)
where λj are the Lagrange multipliers and {µj} is a basis of the annihilator D0c of
Dc. These equations were obtained in [36] (see Theorem 3 in [36]).
Taking pk = dLd(gk)g
k , k = 1, 2 then
p2 −Ad∗g1p1 = −
λjµj ,
gk ∈Mc, k = 1, 2
(4.5)
where Ad : G × g −→ g is the adjoint action of G on g. These equations were
obtained in [14] and called discrete Euler-Poincaré-Suslov equations.
On the other hand, from (2.14), we have that
ΩLd((
−→η ,←−µ ), (−→η ′,←−µ ′)) = −→η ′(←−µ (Ld))−−→η (←−µ ′(Ld)).
Thus, if g ∈ G then, using (2.22), it follows that the R-bilinear map GLdg : g×g→ R
is given by
GLdg (ξ, η) = −
←−η (g)(
ξ (Ld)).
Therefore, the system (Ld,Mc,Dc) is regular if and only if for every g ∈ Mc the
following conditions hold:
η ∈ g/←−η (g) ∈ TgMc and ←−η (g)(
ξ (Ld)) = 0,∀ξ ∈ Dc =⇒ η = 0,
ξ ∈ g/
ξ (g) ∈ TgMc and ←−η (g)(
ξ (Ld)) = 0,∀η ∈ Dc =⇒ ξ = 0.
We illustrate this situation with two simple examples previously considered in
[14].
30 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
4.3.1. The discrete Suslov system. (See [14]) The Suslov system studies the motion
of a rigid body suspended at its centre of mass under the action of the following
nonholonomic constraint: the body angular velocity is orthogonal to some fixed
direction.
The configuration space is G = SO(3) and the elements of the Lie algebra so(3)
may be identified with R3 and represented by coordinates (ωx, ωy, ωz). Without
loss of generality, let us choose as fixed direction the third vector of the body frame
ē1, ē2, ē3. Then, the nonholonomic constraint is ωz = 0.
The discretization of this system is modelled by considering the discrete La-
grangian Ld : SO(3) −→ R defined by Ld(Ω) = 12Tr (ΩJ), where J represents the
mass matrix (a symmetric positive-definite matrix with components (Jij)1≤i,j≤3).
The constraint submanifold Mc is determined by the constraint Tr (ΩE3) = 0
(see [14]) where
0 0 00 0 −1
0 1 0
, E2 =
0 0 10 0 0
−1 0 0
, E3 =
0 −1 01 0 0
0 0 0
is the standard basis of so(3), the Lie algebra of SO(3).
The vector subspace Dc = span{E1, E2}. Therefore, D0c = span{E3}. Moreover,
the exponential map of SO(3) is a diffeomorphism from an open subset of Dc (which
contains the zero vector) to an open subset of Mc (which contains the identity
element I). In particular, TIMc = Dc.
On the other hand, the discrete Euler-Poincaré-Suslov equations are given
Ei(Ω1)(Ld)−
Ei(Ω2)(Ld) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}.
After some straightforward operations, we deduce that the above equations are
equivalent to:
Tr ((EiΩ2 − Ω1Ei)J) = 0, Tr (ΩiE3) = 0, i ∈ {1, 2}
or, considering the components Ωk = (Ω
ij ) of the elements of SO(3), we have that:(
33 − J33Ω
32 + J22Ω
−J23Ω
22 + J12Ω
13 − J13Ω
−J23Ω
33 − J22Ω
32 − J12Ω
+J33Ω
23 + J23Ω
22 + J13Ω
−J13Ω
33 + J33Ω
31 − J12Ω
+J23Ω
21 − J11Ω
13 + J13Ω
33 + J12Ω
32 + J11Ω
−J33Ω
13 − J23Ω
12 − J13Ω
Ω(1)12 = Ω
21 , Ω
12 = Ω
Moreover, since the discrete Lagrangian verifies that
Ld(Ω) =
Tr (ΩJ) =
Tr (ΩtJ) = Ld(Ω
and also the constraint satisfies Tr (ΩE3) = −Tr (Ω−1E3), then this discretization
of the Suslov system is reversible. The regularity condition in Ω ∈ SO(3) is in this
particular case:
η ∈ so(3) /Tr (E1ΩηJ) = 0, Tr (E2ΩηJ) = 0 and Tr (ΩηE3) = 0 =⇒ η = 0
It is easy to show that the system is regular in a neighborhood of the identity I.
DISCRETE NONHOLONOMIC MECHANICS 31
4.3.2. The discrete Chaplygin sleigh. (See [12, 14]) The Chaplygin sleigh system
describes the motion of a rigid body sliding on a horizontal plane. The body is
supported at three points, two of which slide freely without friction while the third
is a knife edge, a constraint that allows no motion orthogonal to this edge (see [41]).
The configuration space of this system is the group SE(2) of Euclidean motions
of R2. An element Ω ∈ SE(2) is represented by a matrix
cos θ − sin θ xsin θ cos θ y
0 0 1
with θ, x, y ∈ R.
Thus, (θ, x, y) are local coordinates on SE(2).
A basis of the Lie algebra se(2) ∼= R3 of SE(2) is given by
0 −1 01 0 0
0 0 0
, e1 =
0 0 10 0 0
0 0 0
, e2 =
0 0 00 0 1
0 0 0
and we have that
[e, e1] = e2, [e, e2] = −e1, [e1, e2] = 0.
An element ξ ∈ se(2) is of the form
ξ = ω e+ v1 e1 + v2 e2
and the exponential map exp : se(2) ∼= R3 → SE(2) of SE(2) is given by
exp(ω, v1, v2) = (ω, v1
+ v2(
cosω − 1
),−v1(
cosω − 1
) + v2
), if ω 6= 0,
exp(0, v1, v2) = (0, v1, v2).
Note that the restriction of this map to the open subset U =] − π, π[×R2 ⊆ R3 ∼=
se(2) is a diffeomorphism onto the open subset exp(U) of SE(2).
A discretization of the Chaplygin sleigh may be constructed as follows:
- The discrete Lagrangian Ld : SE(2) −→ R is given by
Ld(Ω) =
Tr (ΩJΩT )− Tr (ΩJ),
where J is the matrix:
(J/2) +ma2 mab mamab (J/2) +mb2 mb
ma mb m
(see [14]).
- The vector subspace Dc of se(2) is
Dc = span {e, e1} = { (ω, v1, v2) ∈ se(2) | v2 = 0 } .
- The constraint submanifold Mc of SE(2) is
Mc = exp(U ∩Dc). (4.6)
Thus, we have that
Mc = { (θ, x, y) ∈ SE(2) | − π < θ < π, θ 6= 0, (1− cos θ)x− y sin θ = 0 }
∪ { (0, x, 0) ∈ SE(2) | x ∈ R } .
32 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
Figure 1. Submanifold Mc
From (4.6) it follows that I ∈Mc and TIMc = Dc. In fact, one may prove
T(0,x,0)Mc = span {
∂θ |(0,x,0)
∂y |(0,x,0)
∂x |(0,x,0)
for x ∈ R.
Now, the discrete Euler-Poincaré-Suslov equations are:
←−e (θ1, x1, y1)(Ld)−−→e (θ2, x2, y2)(Ld) = 0,
←−e1(θ1, x1, y1)(Ld)−−→e1(θ2, x2, y2)(Ld) = 0,
and the condition (θk, xk, yk) ∈Mc, with k ∈ {1, 2}. We rewrite these equations as
the following system of difference equations:(
−am cos θ1 − bm sin θ1 + am
+mx1 cos θ1 +my1 sin θ1
mx2 + am cos θ2
−bm sin θ2 − am
amy1 cos θ1 − amx1 sin θ1 − bmx1 cos θ1
−bmy1 sin θ1 + (a2m+ b2m+ J) sin θ1
amy2 − bmx2
+(a2m+ b2m+ J) sin θ2
together with the condition
(θk, xk, yk) ∈Mc, k ∈ {1, 2}.
On the other hand, one may prove that the discrete nonholonomic Lagrangian
system (Ld,Mc,D) is reversible.
Finally, consider a point (0, x, 0) ∈ Mc and an element η ≡ (ω, v1, v2) ∈ se(2)
such that
←−η (0, x, 0) ∈ T(0,x,0)Mc, ←−η (0, x, 0)(−→e (Ld)) = 0, ←−η (0, x, 0)(−→e1(Ld)) = 0.
Then, if we assume that a2m+ J + amx
6= 0 it follows that η = 0.
Thus, the discrete nonholonomic Lagrangian system (Ld,Mc,Dc) is regular in a
neighborhood of the identity I.
4.4. Discrete nonholonomic Lagrangian systems on an action Lie group-
oid. Let H be a Lie group with identity element e and · : M ×H → M , (x, h) ∈
M × H 7→ xh, a right action of H on M . Thus, we may consider the action Lie
groupoid Γ = M ×H over M with structural maps given by
α̃(x, h) = x, β̃(x, h) = xh, �̃(x) = (x, e),
m̃((x, h), (xh, h′)) = (x, hh′), ĩ(x, h) = (xh, h−1).
(4.7)
DISCRETE NONHOLONOMIC MECHANICS 33
Now, let h = TeH be the Lie algebra of H and Φ : h→ X(M) the map given by
Φ(η) = ηM , for η ∈ h,
where ηM is the infinitesimal generator of the action · : M×H →M corresponding
to η. Then, Φ is a Lie algebra morphism and the corresponding action Lie algebroid
pr1 : M × h→M is just the Lie algebroid of Γ = M ×H.
We have that Sec(pr1) ∼= { η̃ : M → h | η̃ is smooth } and that the Lie algebroid
structure ([[·, ·]]Φ, ρΦ) on pr1 : M ×H →M is defined by
[[η̃, µ̃]]Φ(x) = [η̃(x), µ̃(x)]+(η̃(x))M (x)(µ̃)−(µ̃(x))M (x)(η̃), ρΦ(η̃)(x) = (η̃(x))M (x),
for η̃, µ̃ ∈ Sec(pr1) and x ∈M. Here, [·, ·] denotes the Lie bracket of h.
If (x, h) ∈ Γ = M ×H then the left-translation l(x,h) : α̃−1(xh) → α̃−1(x) and
the right-translation r(x,h) : β̃−1(x)→ β̃−1(xh) are given
l(x,h)(xh, h
′) = (x, hh′), r(x,h)(x(h
′)−1, h′) = (x(h′)−1, h′h). (4.8)
Now, if η ∈ h then η defines a constant section Cη : M → h of pr1 : M × h→M
and, using (2.4), (2.5), (4.7) and (4.8), we have that the left-invariant and the
right-invariant vector fields
C η and
C η, respectively, on M ×H are defined by
C η(x, h) = (−ηM (x),−→η (h)),
C η(x, h) = (0x,
←−η (h)), (4.9)
for (x, h) ∈ Γ = M ×H.
Note that if {ηi} is a basis of h then {Cηi} is a global basis of Sec(pr1).
On the other hand, we will denote by expΓ : EΓ = M × h → Γ = M × H the
map given by
expΓ(x, η) = (x, expH(η)), for (x, η) ∈ EΓ = M × h,
where expH : h → H is the exponential map of the Lie group H. Note that if
Φ(x,e) : R → Γ = M ×H is the integral curve of the left-invariant vector field
on Γ = M ×H such that Φ(x,e)(0) = (x, e) then (see (4.9))
expΓ(x, η) = Φ(x,e)(1).
Next, suppose that Ld : Γ = M × H → R is a Lagrangian function, Dc is a
constraint distribution such that {Xα} is a local basis of sections of the annihilator
D0c , and Mc ⊆ Γ is the discrete constraint submanifold.
For every h ∈ H (resp., x ∈ M) we will denote by Lh (resp., Lx) the real
function on M (resp., on H) given by Lh(y) = Ld(y, h) (resp., Lx(h′) = Ld(x, h′)).
A composable pair ((x, hk), (xhk, hk+1)) ∈ Γ2 ∩ (Mc × Mc) is a solution of the
discrete nonholonomic Euler-Lagrange equations for the system (Ld,Mc,Dc) if
C η(x, hk)(Ld)−
C η(xhk, hk+1)(Ld) = λαX
α(xhk)(η), for all η ∈ h,
or, in other terms (see (4.9))
{(Telhk)(η)}(Lx)− {(Terhk+1)(η)}(Lxhk) + ηM (xhk)(Lhk+1) = λαX
α(xhk)(η),
for all η ∈ h.
4.4.1. The discrete Veselova system. As a concrete example of a nonholonomic
system on a transformation Lie groupoid we consider a discretization of the Veselova
system (see [44]). In the continuous theory [9], the configuration manifold is the
transformation Lie algebroid pr1 : S2 × so(3)→ S2 with Lagrangian
Lc(γ, ω) =
ω · Iω −mglγ · e,
34 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where S2 is the unit sphere in R3, ω ∈ R3 ' so(3) is the angular velocity, γ is the
direction opposite to the gravity and e is a unit vector in the direction from the
fixed point to the center of mass, all them expressed in a frame fixed to the body.
The constants m, g and l are respectively the mass of the body, the strength of
the gravitational acceleration and the distance from the fixed point to the center
of mass. The matrix I is the inertia tensor of the body. Moreover, the constraint
subbundle Dc → S2 is given by
γ ∈ S2 7→ Dc(γ) =
ω ∈ R3 ' so(3)
∣∣ γ · ω = 0} .
Note that the section φ : S2 → S2 × so(3)∗, (x, y, z) 7→ ((x, y, z), xe1 + ye2 + ze3),
where {e1, e2, e3} is the canonical basis of R3 and {e1, e2, e3} is the dual basis, is a
global basis for D0c .
If ω ∈ so(3) and ω̂ is the skew-symmetric matrix of order 3 such that ω̂v = ω×v
then the Lagrangian function Lc may be expressed as follows
Lc(γ, ω) =
Tr(ω̂IIω̂T )−mg l γ · e,
where II = 1
Tr(I)I3×3− I. Here, I3×3 is the identity matrix. Thus, we may define
a discrete Lagrangian Ld : Γ = S2 × SO(3)→ R for the system by (see [27])
Ld(γ,Ω) = −
Tr(IIΩ)− hmg l γ · e.
On the other hand, we consider the open subset of SO(3)
V = {Ω ∈ SO(3) | Tr Ω 6= ±1 }
and the real function ψ : S2 × V → R given by
ψ(γ,Ω) = γ · (Ω̂− ΩT ).
One may check that the critical points of ψ are
(γ,Ω) ∈ S2 × V
∣∣ Ωγ − γ = 0} .
Thus, the subset Mc of Γ = S2 × SO(3) defined by
(γ,Ω) ∈ (S2 × V )− Cψ
∣∣∣ γ · (Ω̂− ΩT ) = 0} ,
is a submanifold of Γ of codimension one. Mc is the discrete constraint submanifold.
We have that the map expΓ : S
2× so(3)→ S2×SO(3) is a diffeomorphism from
an open subset of Dc, which contains the zero section, to an open subset of Mc,
which contains the subset of Γ given by
�̃(S2) = {(γ, e) ∈ S2 × SO(3)}.
So, it follows that
(Dc)(γ) = T(γ,e)Mc ∩ EΓ(γ), for γ ∈ S2.
Following the computations of [27] we get the nonholonomic discrete Euler-Lagrange
equations, for ((γk,Ωk), (γkΩk,Ωk+1)) ∈ Γ2
Mk+1 − ΩTkMkΩk +mglh
2( ̂γk+1 × e) = λγ̂k+1,
γk( ̂Ωk − ΩTk ) = 0, γk+1(
̂Ωk+1 − ΩTk+1) = 0,
where M = ΩII − IIΩT . Therefore, in terms of the axial vector Π in R3 defined by
Π̂ = M , we can write the equations in the form
Πk+1 = ΩTk Πk −mglh
2γk+1 × e + λγk+1,
γk( ̂Ωk − ΩTk ) = 0, γk+1(
̂Ωk+1 − ΩTk+1) = 0.
DISCRETE NONHOLONOMIC MECHANICS 35
Note that, using the expression of an arbitrary element of SO(3) in terms of the
Euler angles (see Chapter 15 of [31]), we deduce that the discrete constraint sub-
manifold Mc is reversible, that is, i(Mc) = Mc. However, the discrete nonholonomic
Lagrangian system (Ld,Mc,Dc) is not reversible. In fact, it is easy to prove that
Ld ◦ i 6= Ld.
On the other hand, if γ ∈ S2 and ξ, η ∈ R3 ∼= so(3) then it follows that
C ξ(γ, I3)(
C η(Ld)) = −ξ · Iη.
Consequently, the nonholonomic system (Ld,Mc,Dc) is regular in a neighborhood
(in Mc) of the submanifold �̃(S2).
4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie group-
oid. Let p : Q → M = Q/G be a principal G-bundle and choose a local trivial-
ization G× U , where U is an open subset of M . Then, one may identify the open
subset (p−1(U) × p−1(U))/G ' ((G × U) × (G × U))/G of the Atiyah groupoid
(Q×Q)/G with the product manifold (U ×U)×G. Indeed, it is easy to prove that
the map
((G× U)× (G× U))/G→ (U × U)×G,
[((g, x), (g′, y))]→ ((x, y), g−1g′)),
is bijective. Thus, the restriction to ((G × U) × (G × U))/G of the Lie groupoid
structure on (Q × Q)/G induces a Lie groupoid structure in (U × U) × G with
source, target and identity section given by
α : (U × U)×G→ U ; ((x, y), g)→ x,
β : (U × U)×G→ U ; ((x, y), g)→ y,
� : U → (U × U)×G; x→ ((x, x), e),
and with multiplication m : ((U × U) × G)2 → (U × U) × G and inversion i :
(U × U)×G→ (U × U)×G defined by
m(((x, y), g), ((y, z), h)) = ((x, z), gh),
i((x, y), g) = ((y, x), g−1).
(4.10)
The Lie algebroid A((U×U)×G) may be identified with the vector bundle TU×g→
U . Thus, the fibre over the point x ∈ U is the vector space TxU × g. Therefore, a
section of A((U ×U)×G) is a pair (X, ξ̃), where X is a vector field on U and ξ̃ is a
map from U on g. The space Sec(A((U × U)×G)) is generated by sections of the
form (X, 0) and (0, Cξ), with X ∈ X(U), ξ ∈ g and Cξ : U → g being the constant
map Cξ(x) = ξ, for all x ∈ U (see [27] for more details).
Now, suppose that Ld : (U ×U)×G→ R is a Lagrangian function, Dc a vector
subbundle of TU×g and Mc a constraint submanifold on (U×U)×G. Take a basis
of sections {Y α} of the annihilator Doc . Then, the discrete nonholonomic equations
are ←−−−−−
(Xα, η̃α)((x, y), gk)(Ld)−
−−−−−→
(Xα, η̃α)((y, z), gk+1)(Ld) = 0,
with (Xα, η̃α) : U → TU × g a basis of the space Sec(τDc) and (((x, y), gk), ((y, z),
gk+1)) ∈ (Mc×Mc)∩ ((U ×U)×G)2. The above equations may be also written as
(X, 0)((x, y), gk)(Ld)−
(X, 0)((y, z), gk+1)(Ld) = λαY α(y)(X(y)),←−−−−
(0, Cξ)((x, y), gk)(Ld)−
−−−−→
(0, Cξ)((y, z), gk+1)(Ld) = λαY α(y)(Cξ(y)),
with X ∈ X(U), ξ ∈ g and (((x, y), gk), ((y, z), gk+1)) ∈ (Mc×Mc)∩((U×U)×G)2.
An equivalent expression of these equations is
D2Ld((x, y), gk) +D1Ld((y, z), gk+1) = λαµα(y),
pk+1(y, z) = Ad∗gkpk(x, y)− λαη̃
α(y),
(4.11)
36 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
where pk(x̄, ȳ) = d(r∗gkL(x̄,ȳ, ))(e) for (x̄, ȳ) ∈ U × U and we write Y
α ≡ (µα, η̃α),
µα being a 1-form on U and η̃α : U → g∗ a smooth map.
4.5.1. A discretization of the equations of motion of a rolling ball without sliding on
a rotating table with constant angular velocity. A (homogeneous) sphere of radius
r > 0, mass m and inertia about any axis I rolls without sliding on a horizontal
table which rotates with constant angular velocity Ω about a vertical axis through
one of its points. Apart from the constant gravitational force, no other external
forces are assumed to act on the sphere (see [41]).
The configuration space for the continuous system isQ = R2×SO(3) and we shall
use the notation (x, y;R) to represent a typical point in Q. Then, the nonholonomic
constraints are
Tr(ṘRTE2) = −Ωy,
Tr(ṘRTE1) = Ωx,
where {E1, E2, E3} is the standard basis of so(3).
The matrix ṘRT is skew symmetric, therefore we may write
ṘRT =
0 −w3 w2w3 0 −w1
−w2 w1 0
where (w1, w2, w3) represents the angular velocity vector of the sphere measured
with respect to the inertial frame. Then, we may rewrite the constraints in the
usual form:
ẋ− rw2 = −Ωy,
ẏ + rw1 = Ωx.
The Lagrangian for the rolling ball is:
Lc(x, y;R, ẋ, ẏ; Ṙ) =
m(ẋ2 + ẏ2) +
I Tr(ṘRT (ṘRT )T )
m(ẋ2 + ẏ2) +
I(ω21 + ω
2 + ω
Moreover, it is clear that Q = R2 × SO(3) is the total space of a trivial princi-
pal SO(3)-bundle over R2 and the bundle projection φ : Q → M = R2 is just the
canonical projection on the first factor. Therefore, we may consider the correspond-
ing Atiyah algebroid E′ = TQ/SO(3) over M = R2. We will identify the tangent
bundle to SO(3) with so(3)× SO(3) by using right translation.
Under this identification between T (SO(3)) and so(3)×SO(3) the tangent action
of SO(3) on T (SO(3)) ∼= so(3)× SO(3) is the trivial action
(so(3)× SO(3))× SO(3)→ so(3)× SO(3), ((ω,R), S) 7→ (ω,RS). (4.12)
Thus, the Atiyah algebroid TQ/SO(3) is isomorphic to the product manifold
TR2×so(3) and the vector bundle projection is τR2 ◦pr1, where pr1 : TR2×so(3)→
TR2 and τR2 : TR2 → R2 are the canonical projections.
A section of E′ = TQ/SO(3) ∼= TR2 × so(3) → R2 is a pair (X,u), where X is
a vector field on R2 and u : R2 → so(3) is a smooth map. Therefore, a global basis
of sections of TR2 × so(3)→ R2 is
s′1 = (
, 0), s′2 = (
, 0),
s′3 = (0, E1), s
4 = (0, E2), s
5 = (0, E3).
DISCRETE NONHOLONOMIC MECHANICS 37
The anchor map ρ′ : E′ = TQ/SO(3) ∼= TR2 × so(3) → TR2 is the projection
over the first factor and if [[·, ·]]′ is the Lie bracket on the space Sec(E′ = TQ/SO(3))
then the only non-zero fundamental Lie brackets are
[[s′3, s
′ = s′5, [[s
′ = s′3, [[s
′ = s′4.
Moreover, the Lagrangian function Lc = T and the constraint functions are
SO(3)-invariant. Consequently, Lc induces a Lagrangian function L′c on E
TQ/SO(3)
L′c(x, y, ẋ, ẏ;ω) =
m(ẋ2 + ẏ2) +
I Tr(ωωT ),
m(ẋ2 + ẏ2)−
I Tr(ω2),
where (x, y, ẋ, ẏ) are the standard coordinates on TR2 and ω ∈ so(3). The con-
straint functions defined on E′ = TQ/SO(3) are:
ẋ+ r
Tr(ωE2) = −Ωy,
ẏ − r
Tr(ωE1) = Ωx.
(4.13)
We have a nonholonomic system on the Atiyah algebroid E′ = TQ/SO(3) ∼= TR2×
so(3). This kind of systems was recently analyzed by J. Cortés et al [9] (in particular,
this example was carefully studied).
Eqs. (4.13) define an affine subbundle of the vector bundle E′ ∼= TR2× so(3)→
R2 which is modelled over the vector subbundle D′c generated by the sections
D′c = {s
5, rs
1 + s
4, rs
2 − s
Our objective is to discretize this example directly on the Atiyah algebroid. The
Atiyah groupoid is now identified to R2 × R2 × SO(3) ⇒ R2. We may construct
the discrete Lagrangian by
L′d(x0, y0, x1, y1;W1) = L
c(x0, y0,
x1 − x0
y1 − y0
; (logW1)/h)
where log : SO(3) −→ so(3) is the (local)-inverse of the exponential map exp :
so(3) −→ SO(3). For simplicity instead of this procedure we use the following
approximation:
logW1/h ≈
W1 − I3×3
where I3×3 is the identity matrix.
L′d(x0, y0, x1, y1;W1) = L
c(x0, y0,
x1 − x0
y1 − y0
W1 − I3×3
x1 − x0
y1 − y0
(2h)2
Tr(I3×3 −W1)
Eliminating constants, we may consider as discrete Lagrangian
L′d =
x1 − x0
y1 − y0
Tr(W1)
The discrete constraint submanifold M′c of R
2 × R2 × SO(3) is determined
by the constraints:
x1 − x0
Tr(W1E2) = −Ω
y1 + y0
y1 − y0
Tr(W1E1) = Ω
x1 + x0
38 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
We have that the system (L′d,M
c) is not reversible. Note that the Lagrangian
function L′d is reversible. However, the constraint submanifold M
c is not reversible.
The discrete nonholonomic Euler-Lagrange equations for the system (L′d, ,M
D′c) are:
s′5(x0, y0, x1, y1;W1)(L
s′5(x1, y1, x2, y2;W2)(L
d) = 0
←−−−−−−
(rs′1 + s
4)(x0, y0, x1, y1;W1)(L
−−−−−−→
(rs′1 + s
4)(x1, y1, x2, y2;W2)(L
d) = 0
←−−−−−−
(rs′2 − s
3)(x0, y0, x1, y1;W1)(L
−−−−−−→
(rs′2 − s
3)(x1, y1, x2, y2;W2)(L
d) = 0
with the constraints defining Mc.
On the other hand, the vector fields ←−s ′5,
−→s ′5,
←−−−−−
rs′1 + s
−−−−−→
rs′1 + s
←−−−−−
rs′2 − s′3 and−−−−−→
rs′2 − s′3 on (R2 × R2)× SO(3) are given by
←−s ′5 = ((0, 0),
E 3),
−→s ′5 = ((0, 0),
E 3),←−−−−−
rs′1 + s
4 = ((0, r
E 2),
−−−−−→
rs′1 + s
4 = ((−r
, 0),
E 2),←−−−−−
rs′2 − s′3 = ((0, r
E 1),
−−−−−→
rs′2 − s′3 = ((0,−r
E 1),
where
E i (respectively,
E i) is the left-invariant (respectively, right-invariant) vec-
tor field on SO(3) induced by Ei ∈ so(3), for i ∈ {1, 2, 3}. Thus, we deduce the
following system of equations:
Tr ((W1 −W2)E3) = 0,
x2 − 2x1 + x0
Tr ((W1 −W2)E2) = 0,
y2 − 2y1 + y0
Tr ((W1 −W2)E1) = 0,
x2 − x1
Tr(W2E2) + Ω
y2 + y1
y2 − y1
Tr(W2E1)− Ω
x2 + x1
where (x0, x1, y0, y1;W1) are known. Simplifying we obtain the following system of
equations:
x2 − 2x1 + x0
I +mr2
y2 − y0
= 0 (4.14)
y2 − 2y1 + y0
I +mr2
x2 − x0
= 0 (4.15)
Tr ((W1 −W2)E3) = 0 (4.16)
x2 − x1
Tr(W2E2) + Ω
y2 + y1
= 0, (4.17)
y2 − y1
Tr(W2E1)− Ω
x2 + x1
= 0. (4.18)
Now, consider the open subset U of R2 × R2 × SO(3)
U = (R2 × R2)× {W ∈ SO(3) | W − Tr(W )I3×3 is regular } .
Then, using Corollary 3.13 (iv), we deduce that the discrete nonholonomic La-
grangian system (L′d,M
c) is regular in the open subset U
′ of M′c given by
U ′ = U ∩M′c.
DISCRETE NONHOLONOMIC MECHANICS 39
If we denote by uk = (xk+1 − xk)/h and vk = (yk+1 − yk)/h, k ∈ N then from
Equations (4.14) and (4.15) we deduce that(
4 + α2h2
4− α2h2 −4αh
4αh 4− α2h2
or in other terms
x(k + 2) =
8x(k + 1) + (α2h2 − 4)x(k)− 4αh(y(k + 1)− y(k))
α2h2 + 4
y(k + 2) =
8y(k + 1) + (α2h2 − 4)y(k) + 4α(x(k + 1)− x(k))
α2h2 + 4
where α = IΩ
I+mr2
. Since A ∈ SO(2), the discrete nonholonomic model predicts that
the point of contact of the ball will sweep out a circle on the table in agreement
with the continuous model. Figure 2 shows the excellent behaviour of the proposed
numerical method
Figure 2. Orbits for the discrete nonholonomic equations of mo-
tion (left) and a standard numerical method (right) (initial condi-
tions x(0) = 0.99, y(0) = 1, x(1) = 1, y(1) = 0.99 and h = 0.01
after 20000 steps).
4.6. Discrete Chaplygin systems. Now, we present the theory for a particu-
lar (but typical) example of discrete nonholonomic systems: discrete Chaplygin
systems. This kind of systems was considered in the case of the pair groupoid in
[10].
For any groupoid Γ ⇒ M , the map χ : Γ → M × M , g 7→ (α(g), β(g)) is a
morphism over M from Γ to the pair groupoid M ×M (usually called the anchor
of Γ). The induced morphism of Lie algebroids is precisely the anchor ρ : EΓ → TM
of EΓ (the Lie algebroid of Γ).
Definition 4.2. A discrete Chaplygin system on the groupoid Γ is a discrete
nonholonomic problem (Ld,Mc,Dc) such that
- (Ld,Mc,Dc) is a regular discrete nonholonomic Lagrangian system;
- χMc = χ ◦ iMc : Mc −→M ×M is a diffeomorphism;
- ρ ◦ iDc : Dc −→ TM is an isomorphism of vector bundles.
Denote by L̃d : M ×M −→ R the discrete Lagrangian defined by L̃d = Ld ◦
iMc ◦ (χMc)−1.
In the following, we want to express the dynamics on M×M , by finding relations
between de dynamics defined by the nonholonomic system on Γ and M ×M .
40 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
From our hypothesis, for any vector field Y ∈ X(M) there exists a unique section
X ∈ Sec(τDc) such that ρ ◦ iDc ◦X = Y .
Now, using (2.4), (2.5) and (2.6), it follows that
X (g)) = −Y (α(g)) and Tgβ(
X (g)) = Y (β(g))
with some abuse of notation. In other words,
Tgχ(X
(1,0)(g)) = Y (1,0)(α(g), β(g)) and Tgχ(X
(0,1)(g)) = Y (0,1)(α(g), β(g))
for g ∈ Mc, where Tχ : TΓΓ ∼= V β ⊕Γ V α → TM×M (M ×M) ∼= T (M ×M) is the
prolongation of the morphism χ given by
(Tgχ)(Xg, Yg) = ((Tgα)(Xg), (Tgβ)(Yg)),
for g ∈ Γ and (Xg, Yg) ∈ TΓg Γ ∼= Vgβ ⊕ Vgα.
Since χMc is a diffeomorphism, there exists a unique X
g ∈ TgMc (respectively,
X̄ ′g ∈ TgMc) such that
(TgχMc)(X
g) = Y
(1,0)(α(g), β(g)) = (−Y (α(g)), 0β(g))
(respectively, (TgχMc)(X̄
g) = Y
(0,1)(α(g), β(g)) = (0α(g), Y (β(g)))) for all g ∈Mc.
Thus,
X ′g ∈ TgMc ∩ Vgβ,
X (g)−X ′g = Z ′g ∈ Vgα ∩ Vgβ,
X̄ ′g ∈ TgMc ∩ Vgα,
X (g)− X̄ ′g = Z̄ ′g ∈ Vgα ∩ Vgβ,
for all g ∈Mc.
Now, if (g, h) ∈ Γ2 ∩ (Mc ×Mc) then
X (g)(Ld)−
X (h)(Ld) = X̄
g(Ld) + Z̄
g(Ld)−X
h(Ld)− Z
h(Ld)
Y (α(g), β(g))(L̃d)−
Y (α(h), β(h))(L̃d)
+Z̄ ′g(Ld)− Z
h(Ld).
Therefore, if we use the following notation
(α(g), β(g)) = (x, y), (α(h), β(h)) = (y, z)
F+Y (x, y) = −Z̄
(x,y)
(Ld), F
Y (y, z) = Z
(y,z)
(Ld),
X (g)(Ld)−
X (h)(Ld) =
Y (x, y)(L̃d)−
Y (y, z)(L̃d)
−F+Y (x, y) + F
Y (y, z).
In conclusion, we have proved that (g, h) is a solution of the discrete nonholonomic
Euler-Lagrange equations for the system (Ld,Mc,Dc) if and only if ((x, y), (y, z))
is a solution of the reduced equations
Y (x, y)(L̃d)−
Y (y, z)(L̃d) = F
Y (x, y)− F
Y (y, z), Y ∈ X(M).
Note that the above equations are the standard forced discrete Euler-Lagrange
equations (see [32]).
4.6.1. The discrete two wheeled planar mobile robot. We now consider a discrete
version of the two-wheeled planar mobile robot [8, 9]. The position and orientation
of the robot is determined, with respect a fixed cartesian reference, by an element
Ω = (θ, x, y) ∈ SE(2), that is, a matrix
cos θ − sin θ xsin θ cos θ y
0 0 1
DISCRETE NONHOLONOMIC MECHANICS 41
Moreover, the different positions of the two wheels are described by elements
(φ, ψ) ∈ T2. Therefore, the configuration space is SE(2) × T2. The system is
subjected to three nonholonomic constraints: one constraint induced by the condi-
tion of no lateral sliding of the robot and the other two by the rolling conditions of
both wheels.
It is well known that this system is SE(2)-invariant and then the system may
be described as a nonholonomic system on the Lie algebroid se(2)×TT2 → T2 (see
[9]). In this case, the Lagrangian is
Jω2 +m(v1)2 +m(v2)2 + 2m0lωv
2 + J2φ̇
2 + J2ψ̇
Tr(ξJξT ) +
φ̇2 +
where
ξ = ω e+ v1 e1 + v
2 e2 =
0 −ω v1ω 0 v2
0 0 0
and J =
J/2 0 m0l0 J/2 0
m0l 0 m
Here, m = m0 + 2m1, where m0 is the mass of the robot without the two wheels,
m1 the mass of each wheel, J its the moment of inertia with respect to the vertical
axis, J2 the axial moments of inertia of the wheels and l the distance between the
center of mass of the robot and the intersection point of the horizontal symmetry
axis of the robot and the horizontal line connecting the centers of the two wheels.
The nonholonomic constraints are
v1 + R
φ̇+ R
ψ̇ = 0,
v2 = 0,
ω + R
φ̇− R
ψ̇ = 0,
(4.19)
determining a submanifold M of se(2) × TT2, where R is the radius of the two
wheels and 2c the lateral length of the robot.
In order to discretize the above nonholonomic system, we consider the Atiyah
groupoid Γ = SE(2)×(T2×T2) ⇒ T2. The Lie algebroid of SE(2)×(T2×T2) ⇒ T2
is TT2 × se(2)→ T2. Then:
- The discrete Lagrangian Ld : SE(2)× (T2 × T2)→ R is given by:
Ld(Ωk, φk, ψk, φk+1, ψk+1) = 12h2 Tr ((Ωk − I3×3)J(Ωk − I3×3)
(∆φk)
(∆ψk)
where I3×3 is the identity matrix, ∆φk = φk+1 − φk, ∆ψk = ψk+1 − ψk
cos θk − sin θk xksin θk cos θk yk
0 0 1
We obtain that
mx2k +my
k − 2lm0xk(1− cos θk)
+2J(1− cos θk) + 2lm0yk sin θk) +
(∆φk)2
(∆ψk)2
- The constraint vector subbundle of se(2)×TT2 is generated by the sections:{
, s2 =
42 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND E. MARTÍNEZ
- The continuous constraints of the two-wheeled planar robot are written in
matrix form (see 4.19):
0 −ω v1ω 0 v2
0 0 0
0 R2c φ̇− R2c ψ̇ −R2 φ̇− R2 ψ̇− R
φ̇+ R
ψ̇ 0 0
0 0 0
We discretize the previous constraints using the exponential on SE(2)
(see Section 4.3.2) and discretizing the velocities on the right hand side
cos( R2c∆φk− R2c∆ψk) sin( R2c∆φk− R2c∆ψk) −c
∆φk+∆ψk
∆φk−∆ψk
sin( R2c∆φk− R2c∆ψk)
− sin( R2c∆φk− R2c∆ψk) cos( R2c∆φk− R2c∆ψk) c
∆φk+∆ψk
∆φk−∆ψk
(1−cos( R2c∆φk− R2c∆ψk))
0 0 1
if ∆φk 6= ∆ψk and
1 0 −R∆φk0 1 0
0 0 1
if ∆φk = ∆ψk.
Therefore, the constraint submanifold Mc is defined as
θk = −
∆φk +
∆ψk (4.20)
xk = −c
∆φk + ∆ψk
∆φk −∆ψk
∆φk −
(4.21)
yk = c
∆φk + ∆ψk
∆φk −∆ψk
1− cos
∆φk −
(4.22)
if ∆φk 6= ∆ψk and θk = 0, xk = −R∆φk and yk = 0 if ∆φk = ∆ψk.
We have that the discrete nonholonomic system (Ld,Mc,Dc) is reversible. More-
over, if �Γ : T2 → SE(2) × (T2 × T2) is the identity section of the Lie groupoid
Γ = SE(2)× (T2 × T2) then it is clear that
�Γ(T2) = {I3×3} ×∆T2×T2 ⊆Mc.
Here, ∆T2×T2 is the diagonal in T2 × T2. In addition, the system (Ld,Mc,Dc) is
regular in a neighborhood U of the submanifold �Γ(T2) = {I3×3} ×∆T2×T2 in Mc.
Note that
T(I3×3,φ1,ψ1,φ1,ψ1)Mc ∩ EΓ(φ1, ψ1) = Dc(φ1, ψ1),
for (φ1, ψ1) ∈ T2, where EΓ = se(2)× TT2 is the Lie algebroid of the Lie groupoid
Γ = SE(2)× (T2 × T2).
On the other hand, it is easy to show that the system (Ld, U,Dc) is a discrete
Chaplygin system.
The reduced Lagrangian on T2 × T2 is
L̃d =
(mc2(
∆φk + ∆ψk
∆φk −∆ψk
)2(1− cos(
∆φk −
∆ψk))
+J(1− cos(
∆φk −
∆ψk))) +
(∆φk)2
(∆ψk)2
if ∆φk 6= ∆ψk
(J1 +
(∆φk)2
, if ∆φk = ∆ψk
The discrete nonholonomic equations are:
(Ω1,φ1,ψ1,φ2,ψ2)
(Ld)−−→s1
(Ω2,φ2,ψ2,φ3,ψ3)
(Ld) = 0
(Ω1,φ1,ψ1,φ2,ψ2)
(Ld)−−→s2
(Ω2φ2,ψ2,φ3,ψ3)
(Ld) = 0
DISCRETE NONHOLONOMIC MECHANICS 43
These equations in coordinates are:
2J1(φ3 − 2φ2 + φ1) = lRm0(cos θ2 + cos θ1) +
(sin θ2 − sin θ1)
R cos θ1
(lm0y1 + cmx1) +
R sin θ1
(lm0x1 − cmy1)
(cmx2 + lm0(y2 − 2c)) (4.23)
2J1(ψ3 − 2ψ2 + ψ1) = lRm0(cos θ2 + cos θ1)−
(sin θ2 − sin θ1)
R cos θ1
(lm0y1 − cmx1)−
R sin θ1
(lm0x1 + cmy1)
(cmx2 − lm0(y2 + 2c)) (4.24)
Substituting constraints (4.20), (4.21) and (4.22) in Equations (4.23) and (4.24)
we obtain a set of equations of the type 0 = f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) and 0 =
f1(φ1, φ2, φ3, ψ1, ψ2, ψ3) which are the reduced equations of the Chaplygin system.
5. Conclusions and Future Work
In this paper we have elucidated the geometrical framework for nonholonomic
discrete Mechanics on Lie groupoids. We have proposed discrete nonholonomic
equations that are general enough to produce practical integrators for continuous
nonholonomic systems (reduced or not). The geometric properties related with
these equations have been completely studied and the applicability of these devel-
opments has been stated in several interesting examples.
Of course, much work remains to be done to clarify the nature of discrete non-
holonomic mechanics. Many of this future work was stated in [36] and, in particular,
we emphasize:
- a complete backward error analysis which explain the very good energy
behavior showed in examples or the preservation of a discrete energy (see
[14]);
- related with the previous question, the construction of a discrete exact
model for a continuous nonholonomic system (see [17, 32, 36]);
- to study discrete nonholonomic systems which preserve a volume form on
the constraint surface mimicking the continuous case (see, for instance,
[13, 46] for this last case);
- to analyze the discrete hamiltonian framework and the construction of
integrators depending on different discretizations;
- and the construction of a discrete nonholonomic connection in the case of
Atiyah groupoids (see [21, 27]).
Related with some of the previous questions, in the conclusions of the paper of R.
McLachlan and M. Perlmutter [36], the authors raise the question of the possibility
of the definition of generalized constraint forces dependent on all the points qk−1,
qk and qk+1 (instead of just qk) for the case of the pair groupoid. We think that
the discrete nonholonomic Euler-Lagrange equations can be generalized to consider
this case of general constraint forces that, moreover, are closest to the continuous
model (see [25, 36]).
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http://arxiv.org/abs/math-ph/0512003
http://www.math.lsa.umich.edu/~mleok
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D. Iglesias: Instituto de Matemáticas y F́ısica Fundamental, Consejo Superior de
Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain
E-mail address: [email protected]
Juan C. Marrero: Departamento de Matemática Fundamental, Facultad de Matemá-
ticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain
E-mail address: [email protected]
D. Mart́ın de Diego: Instituto de Matemáticas y F́ısica Fundamental, Consejo Supe-
rior de Investigaciones Cient́ıficas, Serrano 123, 28006 Madrid, Spain
E-mail address: [email protected]
Eduardo Mart́ınez: Departamento de Matemática Aplicada, Facultad de Ciencias,
Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail address: [email protected]
1. Introduction
2. Discrete Unconstrained Lagrangian Systems on Lie Groupoids
2.1. Lie algebroids
2.2. Lie groupoids
2.3. Discrete Unconstrained Lagrangian Systems
3. Discrete Nonholonomic (or constrained) Lagrangian systems on Lie groupoids
3.1. Discrete Generalized Hölder's principle
3.2. Discrete Nonholonomic Legendre transformations
3.3. Nonholonomic evolution operators and regular discrete nonholonomic Lagrangian systems
3.4. Reversible discrete nonholonomic Lagrangian systems
3.5. Lie groupoid morphisms and reduction
3.6. Discrete nonholonomic Hamiltonian evolution operator
3.7. The discrete nonholonomic momentum map
4. Examples
4.1. Discrete holonomic Lagrangian systems on a Lie groupoid
4.2. Discrete nonholonomic Lagrangian systems on the pair groupoid
4.3. Discrete nonholonomic Lagrangian systems on a Lie group
4.4. Discrete nonholonomic Lagrangian systems on an action Lie groupoid
4.5. Discrete nonholonomic Lagrangian systems on an Atiyah Lie groupoid
4.6. Discrete Chaplygin systems
5. Conclusions and Future Work
References
|
0704.1544 | Pseudogap and charge density waves in two dimensions | Pseudogap and charge density waves in two dimensions
S. V. Borisenko1, A. A. Kordyuk1,2, A. N. Yaresko3, V. B. Zabolotnyy1, D. S. Inosov1, R. Schuster1,
B. Büchner1, R. Weber4, R. Follath4, L. Patthey5, H. Berger6
1Leibniz-Institute for Solid State Research, IFW-Dresden, D-01171, Dresden, Germany
2Institute of Metal Physics, 03142 Kyiv, Ukraine
3Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany
4BESSY, Berlin, Germany
5Swiss Light Source, Paul Scherrer Institut, CH-5234 Villigen, Switzerland
6Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland
An interaction between electrons and lattice vibrations (phonons) results in two fundamental
quantum phenomena in solids: in three dimensions it can turn a metal into a superconductor
whereas in one dimension it can turn a metal into an insulator1, 2, 3. In two dimensions (2D)
both superconductivity and charge-density waves (CDW)4, 5 are believed to be anomalous. In
superconducting cuprates, critical transition temperatures are unusually high and the energy
gap may stay unclosed even above these temperatures (pseudogap). In CDW-bearing
dichalcogenides the resistivity below the transition can decrease with temperature even faster
than in the normal phase6, 7 and a basic prerequisite for the CDW, the favourable nesting
conditions (when some sections of the Fermi surface appear shifted by the same vector), seems
to be absent8, 9, 10. Notwithstanding the existence of alternatives11, 12, 13, 14, 15 to conventional
theories1, 2, 3, both phenomena in 2D still remain the most fascinating puzzles in condensed
matter physics. Using the latest developments in high-resolution angle-resolved
photoemission spectroscopy (ARPES) here we show that the normal-state pseudogap also
exists in one of the most studied 2D examples, dichalcogenide 2H-TaSe2, and the formation of
CDW is driven by a conventional nesting instability, which is masked by the pseudogap. Our
findings reconcile and explain a number of unusual, as previously believed, experimental
responses as well as disprove many alternative theoretical approaches12, 13, 14, 15. The
magnitude, character and anisotropy of the 2D-CDW pseudogap are intriguingly similar to
those seen in superconducting cuprates.
Variations of the electron density in a metal are highly unfavourable because of the Coulomb
repulsion. Though, some low-dimensional systems such as transition-metal dichalcogenides
spontaneously develop a static periodic modulation (known as charge-density wave) of the electron
gas below a certain temperature. A typical representative of the transition metal dichalcogenides is a
2H (trigonal prismatic) polytype of TaSe2 which exhibits two CDW phase transitions at accessible
temperatures: a second-order one at TNIC=122 K from normal to an incommensurate CDW state and
a first-order lock-in transition at TICC=90 K from the incommensurate to a 3x3 commensurate
CDW phase16. In the one-dimensional case, where all points of the Fermi surface (FS) can be
connected by the same vector (perfect nesting), the CDW transition occurs when the energy gain
due to opening of a gap at the Fermi level exceeds the energy costs to distort the lattice2, 4, 17. In real
2D materials the energy balance is more delicate since all FS points cannot be connected by the
same vector (non-perfect nesting) and thus the FS can be gapped only partially. To understand the
CDW mechanism in 2H-TaSe2, a detailed knowledge of the low-energy electronic structure is
required. We therefore start with an overview of its temperature evolution in Figs. 1 and 2.
The upper panel of Fig. 1 shows the topology of the Fermi surface in the normal state together with
the CDW vectors Mqn Γ= 3
defined by other experiments5, 16. Contrary to the earlier band
structure calculations8, but in accordance with a recent study10, the FS consists of single hole-like
“barrels” centred at Γ and K points, and electron-like “dogbones” around the M-point. The upper
row of panels in Fig. 2a shows the corresponding dispersions of the electronic states crossing the
Fermi level. FS sheets originate from two bands: one is responsible for the Γ and K barrels with a
saddle point in between, the other one supports the dogbone with M being another saddle point. The
second row of panels in Fig. 2a showing the data below the first phase transition, suggests that the
normal ( 290 K ) and incommensurate CDW state ( TICC < 107 K < TNIC ) dispersions are
qualitatively similar, except for the naturally different temperature broadening and weaker crossing
#3 (Fig. 2a). This seems to be in agreement with the surprising earlier ARPES results, when
virtually no change of the electronic structure has been detected upon entering the incommensurate
CDW phase9, 10, 18, thus clearly contradicting the concept of the energy gain that should accompany
a CDW transition. In contrast, the lock-in transition to the commensurate CDW state at TICC=90 K
is much more pronounced (see lower panel of Fig.1). The new folded FS is schematically shown in
Fig. 1 as a set of nearly circles around new Γ’-points and rounded triangles around new K’-points.
This topology of the folded FS, though natural, as suggested by the normal state FS (see
Supplementary material), has never been detected before. A possible reason could be a rather weak
umklapp potential due to small lattice distortion of the order of 0.05 Å (Ref 16, 17), which
consequently results in the intensity distribution along the FS that still reminds the one seen in the
normal state. Note, that without such an overview of the large portion of the k-space, it is
problematic to understand what exactly happens to the electronic structure below 90 K. The energy-
momentum intensity distributions in the lower row of panels in Fig. 2a are also considerably
modified by the 3x3 folding and show clear signatures of a strong hybridization. Exactly this
hybridization, when occurs in the vicinity of the Fermi level, can lower the energy of the system
(see right inset to Fig. 3c). This observation of strong changes at TICC is again paradoxical.
According to other experimental techniques, the CDW phase transition with the critical energy
lowering occurs at TNIC=122 K as is clearly seen in e.g. temperature dependences of the specific
heat or resistivity6, 7. The lock-in transition at TICC=90 K, in contrast, is hardly detectable in the
mentioned curves6, 7 and appears as a small break in the superlattice strength as seen by neutron
scattering16.
More detailed data analysis clarifies the situation. ARPES offers a unique opportunity to find out
whether the energy gap opens up at the Fermi level anywhere on the FS by tracking the binding
energy of the leading edge of an energy distribution curve (EDC) taken at kF. In Fig. 2b we show
EDCs corresponding exactly to kF for selected high symmetry cuts in the k-space (Fig. 2a). Both
panels unambiguously signal the leading edge shift (~15 meV) of the EDC #6. In both cases a clear
suppression (not absence) of the spectral weight at the Fermi level is evident also from the
corresponding energy-momentum distributions (rightmost panels in Fig. 2a), which is reinforced by
an obviously more diffused appearance of the K-barrels on the normal-state FS map (Fig. 1). In a
striking analogy with the superconducting cuprates, we thus conclude the presence of a pseudogap
in both, normal and incommensurate CDW states of 2H-TaSe2.
In order to understand whether necessary energy gain can come from the pseudogap, we further
characterize the pseudogap as a function of temperature and momentum in Fig. 3. The so called
maps of gaps19, the plots of binding energies of the EDC leading edges as a function of k (i.e. not
only kF), are shown for the normal and incommensurate CDW states in Fig. 3a,b. These maps are a
visual demonstration of the energy lowering of the system and are ideal for the determination of the
anisotropy of the gap as they make the analysis of the behaviour of the leading edge in the vicinity
of the Fermi surface (shown as dotted lines) possible. While the normal state map (Fig. 3a) reveals
an isotropic pseudogap detectable only on the K-barrel, in the CDW state a pseudogap opens, in
addition, on the dogbone FS and is anisotropic. As a quantitative measure of the pseudogap we take
the difference of the leading edge binding energies of the kF-EDCs of M-dogbone and K-barrel
which belong to the M-K cut, as is sketched in the left inset to the Fig. 3c. The sharp reproducible
increase of the pseudogap magnitude below ~122 K, which escaped the detection before, is
distinctly seen in Fig. 3c. It is now clear that it is the NIC transition (TNIC=122K) at which the
critical lowering of the electronic energy occurs, and not only because of the larger pseudogap on
the K-barrel, but also owing to the opening of the anisotropic pseudogap on the dogbone M-centred
FS. In the commensurate CDW phase one can no longer characterize the energy gap by the leading
edge position because of the interference with the folded bands --- the rounded corners of the new
small triangular FS fall right where the normal state K-barrel was located (see Fig. 1). Instead, we
plot in Fig. 3c also the values determined as shown in the right inset. The “band-gap” is a direct
consequence of the hybridization, distinctly observed below TICC (Figs. 1, 2). In order to emphasize
the existence of a crossover regime where the pseudogap evolves into a band-gap we also plot in a
limited T- interval the leading edge positions below TICC and the band-gap above TICC. Both gaps
do not exhibit any anomalies at TICC monotonically increasing upon cooling deeper in the CDW
state.
Presented data already put certain limitations on several alternative theoretical approaches that were
stimulated by the earlier experiments. The locations of the saddle points in the momentum-energy
space obviously do not support the saddle point nesting scenario13 , proposed as an alternative to the
conventional nesting, as both points are far from the Fermi level (280 meV and 330 meV) and are
not connected by any of the CDW vectors (the first one is located between Γ- and K-barrels, and the
second is the M-point itself) . The pseudogap supported by the K-centred FS stays isotropic (within
~ 2 meV) thus ruling out a six-fold symmetric CDW gap with nodes as suggested in Ref. 14. The
value of the band-gap saturates at low temperatures at ~33 meV which is nearly a factor of 5
smaller than the one (~ 150 meV) obtained in the strong-coupling approach12. A recent proposal15 to
consider two components of the electronic structure, one of which is gapped and the other one is
not, seems to be not supported by the data as well.
What is then responsible for the CDW in two dimensions? In the following we demonstrate that the
conventional FS nesting scenario, though modified by the presence of the normal-state pseudogap,
is perfectly applicable. From our high-resolution measurements, we have analysed the nesting
properties of the FS quantitatively, which made it possible to explain the temperature evolution of
the electronic structure of TaSe2 step by step. The quantitative measure of the nesting is the charge
susceptibility. Here we approximate the charge susceptibility by the autocorrelation of the FS
map20. In order to avoid the influence of the matrix elements, for further processing we take the
model FS map shown in Fig. 4a, which is an exact copy of the experimental one as far as the locus
of kF-points is concerned. In addition, this gives possibility to investigate the FS nesting properties
of a hypothetical compound, with the electronic structure that yet have not reacted to the nesting
instability (i.e. without gaps). The resulting autocorrelation maps as a function of temperature are
shown in Fig. 4b, while the corresponding cuts along the ΓM direction -- in Fig. 4d. The sharp peak,
seen in the 290 K curve exactly at 2/3 ΓM in Fig. 4d, is the first clear evidence for the nearly perfect
nesting in 2D chalcogenides. This is at variance with the previous calculations17 which have found
susceptibility for 2H polytypes to take a broadly humped form. Thus, the Fermi surface shape alone,
without taking into account the pseudogap, results in the peak in the charge susceptibility at a wave
vector ~2/3ΓM. According to the neutron scattering16, exactly at this wave vector there is a strong
Kohn-like anomaly4 of the Σ1 phonon branch already at 300 K, and the matching phonon softens
even more as the transition is approached. We consider therefore the formation of both, normal state
pseudogap and anomaly of the Σ1 phonon branch as respective reactions of the electronic and lattice
subsystems to the instability caused by the strong scattering channel that appears due to the nesting
and a presence of the suitable mediating phonon. Such a mutual response can signify a strong
electron-phonon interaction in 2H-TaSe2. It is interesting, that despite the favourable conditions, the
system does not escape the instability by developing a static commensurate CDW order,
presumably because of still too high temperature, which would effectively close not large enough
band-gaps. Instead, it opens up a pseudogap. Upon cooling we observe correlated changes of the
susceptibility, pseudogap magnitude, and energy of the softened phonon16: the peak in the
susceptibility splits into two (middle panel in Fig. 4d), the pseudogap slowly closes (as suggested
by the EDC shift at 290 K in Fig. 2b and a weak but detectable trend of the gap to decrease in Fig.
3c) , the phonon energy becomes lower. This worsening of the nesting conditions is fundamentally
different from the 1D case, where the susceptibility exhibits a peak, which becomes sharper with
lowering the temperature4. We have found out that such anomalous behaviour of the susceptibility,
i.e. the splitting and the size of the peak at ~2/3ΓM, is very sensitive to the shape of the Fermi
surface itself, namely to the distance between the dogbone and K-centred FSs (distance, marked ‘D’
in Fig. 4a). The data plotted in Fig. 4c clearly suggest that the FS itself is temperature dependent
explaining the variation of the charge susceptibility. Moreover, a close correlation with the
temperature dependence of the pseudogap magnitude (Fig. 3c) implies that the pseudogap itself is
directly related to the charge susceptibility. Nevertheless, the transition finally occurs at TNIC=122
K , but now only into the incommensurate CDW phase as dictated by the split peak in the
susceptibility. Further reduction of the temperature results in a reversed modification of the
obviously correlated quantities: pseudogap opens up more, peaks in susceptibility move towards
each other (right panel in Fig. 4d), which agrees well with the dynamics of the superlattice peaks
seen by neutrons, and phonon energy starts to increase again16. In such a manner the system arrives
at the transition into the commensurate CDW state at 90 K.
Our findings provide a natural explanation not only for the neutron scattering experiments16. Nearly
linear in-plane resistivity in the normal state6, 7 is similar to the resistivity of an optimally doped
cuprate superconductor21 and shows a typical behaviour of a pseudo-gapped metal. Below 122 K
the slope is increasing in close correspondence to the larger pseudogap, and below 90 K the
resistivity resembles the one of a normal metal. Optical measurements on the same single crystals7
have indirectly suggested the presence of a pseudogap already at 300 K as well, though with
somewhat different energy scale. We also notice an excellent agreement with the Hall coefficient
measurements22. According to Ref.22, the Hall coefficient starts to decrease sharply from its
positive value below ~120 K, and changes sign at 90 K. The positive value in the normal state is
explained by the larger volume of the Γ- and K-centred hole-like barrels in comparison with the
electron-like dogbones around the M-points. The sharp increase of the pseudogap at 122 K results
in the reduction of the number of charge carriers and finally in the commensurate phase (below 90
K), the area enclosed by the electron-like circular FS around new Γ’-points is clearly larger than the
two (per new BZ) hole-like triangular FS, which leads to the negative sign of the Hall coefficient.
It was suggested before23 that the HTSCs and 2D chalcogenides have similar phase diagrams and
that the pseudogap regime in cuprates is very similar to the CDW regime of chalcogenides. Now,
after observation of the pseudogap in TaSe2 we can directly compare both pseudogaps. The energy-
momentum distribution of the photoemission intensity near kF of the K-barrel (crossing #6, Fig. 2a)
is very similar to the one measured for the bonding barrel in the underdoped Bi2212-cuprate24, 25. In
both cases one can still track the dispersion up to the Fermi level, but the spectral weight is
significantly suppressed resulting in the shift of the kF-EDC’s leading edge. Furthermore, the
detected anisotropy of the pseudogap on the dogbone FS is reminiscent of the famous anisotropic
behaviour of the pseudogap in cuprates. Finally, the highly inhomogeneous intensity distribution
along some of the small triangular FSs and especially along incomplete circles around new Γ’
points cannot escape a comparison with the famous Fermi surface ‘arcs’. We believe that this series
of remarkable similarities to the high-temperature superconducting cuprates calls for more careful
comparative studies of the pseudogap phenomenon in these materials.
It is interesting, that unlike in the 1D case where a pseudogap has been reported26, 27, 28 to be a
consequence of fluctuations (potentially able to suppress the transition temperature up to a quarter
of the mean-field value29), the pseudogap in 2D shows unusual non-monotonic behaviour clearly
tracking the temperature evolution of both, the bare susceptibility and phonon spectrum, and thus
seems to represent a natural response of the system to a nesting instability.
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Acknowledgements: The project was supported, in part, by the DFG under Grant No. KN393/4. We thank R. Hübel for
technical support. This work was partially performed at the Swiss Light Source, Paul Scherrer Institut, Villigen,
Switzerland.
Correspondence and requests for materials should be addressed to S.V.B. (S. [email protected]).
Figure 1. Fermi surfaces. Momentum distribution of the photoemission intensity at the
Fermi level at 180 K and 30 K. Short dashed lines are the BZ boundaries.
Figure 2. Electronic structure and leading edge gap. a) Photoemission intensity as a
function of energy and momentum in the normal (upper row of panels), incommensurate
CDW (middle row) and commensurate CDW (lower row) states. Sketches of the FS on the
middle panels show cuts in momentum space along which the data were taken and are
valid for all panels in the same column. Hybridization effects in the lower row are seen as
“repulsions” of the bands, which occur when a folded band is supposed to cross the
original one, as schematically shown in the right inset to Fig. 3b. Since the spectral weight
of the folded band is lower, these effects appear as breaks in the intensity of the original
bands. Numbers correspond to the different kF. b) kF EDCs from the datasets shown in a).
Leading edge gap is clearly seen as a shift of the EDC#6 to the higher binding energies at
290 K and 107 K.
Figure 3. Momentum and temperature dependence of the pseudogap. a,b) Binding
energies of the leading edges of all EDCs from the momentum range close to the
irreducible parts of the M-dogbone and K-barrel FS. Colour scales reproduce the vertical
coordinate. Anisotropy of the pseudogap on the M-dogbone FS is seen as a changing
colour when going along the dashed line which correspond to FS, i.e. kF points. c)
Difference between the binding energies of the leading edges of the EDC#5 and #6 from
Fig. 2b as a function of temperature as shown schematically in the left inset (pseudogap)
when cycling the temperature (filled symbols). Shift of the EDC maximum (bandgap) which
corresponds to the top of the hybridized band as shown in the right inset with respect to its
position at TNIC=122 K (open symbols). Note that the given leading edge gap values below
90 K cannot be considered as a measure of the pseudogap because of folding.
Figure 4. Nesting properties. a) A model copy of the FS map from Fig. 1 with the
homogeneous intensity distribution along the FS. b) Autocorrelation maps of this model at
different temperatures. c) Temperature dependence of the momentum distance between
the M-dogbone and K-barrel (T > 90 K). The distance below 90 K is just a difference
between corresponding maxima at the FS map. d) Intensity of the maps from b) along the
cuts shown by dotted lines.
Supplementary information.
Here we explain in more details the nesting properties of the normal state FS and how to understand
the 3x3 folding upon entering the commensurate CDW regime.
In panel a we show only Γ- and K-centred barrels and their first order ( 1q
± , 2q
± , 3q
± ) and
second order ( 21 qq
+ , 32 qq
+ ) replica. One can see that the nesting conditions are nearly perfect
for all points of the K-barrel: shifting the Γ-barrel by one of the CDW vectors results in an overlap
with the corresponding part of the K-barrel with opposite Fermi velocities. Panel b shows an
idealized situation, which is actually very close to the real one, since a more careful examination of
Fig. 1a reveals the straight sections of both Γ- and K-barrels.
This is, however, not enough to explain the fully gapped K-barrel in the CDW state.
In panel a it is seen that 3x3 folding produces a double-walled barrels in the centre of the new BZ
which are supposed to interact with the single copies of Γ-barrel. This is not surprising as the
original BZ contains only one complete Γ-barrel and two complete K-barrels. It means, that the K-
barrels of a system with the FS schematically shown in panel b, will not disappear after a 3x3
folding.
Panel c shows the dogbone FS centred around M-points together with its first- and second order
replica. Folding of this FS sheet results in the same triangular FSs as in panel a, a rather
complicated set of features around the centres of the new BZ and nearly exact copies of the Γ-
barrels and its replica. It is these copies of the Γ-barrels which were missing in panel a to interact
with double-walled K-barrels.
Panel d, where all FSs and their folded replica are shown, summarizes all mentioned above. When
hybridization effects are switched on, the former K-barrel completely disappears, doubly degenerate
triangular FSs emerge around the corners of the new BZ and the complicated set of features seems
to evolve into a four-times degenerate nearly circle FSs around the Γ’- points (at least
experimentally we can currently resolve only these semi-circular FSs and not two doubly
degenerate hexagons with rounded corners as is suggested by the panel c).
|
0704.1545 | Supersymmetric Field Theory Based on Generalized Uncertainty Principle | RIKEN-TH-96
Supersymmetric Field Theory Based on Generalized Uncertainty
Principle
SHIBUSA Yuuichirou
Theoretical Physics Laboratory,
RIKEN (The Institute of Physical and Chemical Research),
Wako, 351-0198, Japan
Abstract
We construct a quantum theory of free fermion field based on the deformed Heisenberg algebra
[x̂, p̂] = i~(1+βp̂2) where β is a deformation parameter using supersymmetry as a guiding principle.
A supersymmetric field theory with a real scalar field and a Majorana fermion field is given explicitly
and we also find that the supersymmetry algebra is deformed from an usual one.
PACS numbers: 03.65.Ca, 03.70.+k, 11.10.Ef, 11.30.Pb
http://arxiv.org/abs/0704.1545v1
I. INTRODUCTION
Physics in extremely high energy regions is particularly of interest to particle physics. In
particular, when we discuss gravity, it is expected that there is a minimal length in principle.
String theory which has a characteristic scale
α′, is one of the most successful theoretical
frameworks which overcome the difficulty of ultra-violet divergence in quantum theory of
gravity. However, string theory has many difficulties in performing practical computations.
Therefore if we construct a field theory which captures some stringy nature and/or includes
stringy corrections, it would play a pivotal role in investigating physics in high energy regions
even near the Planck scale.
Some of the stringy corrections appear as α′ corrections. In other words, it often takes the
form as higher derivative corrections i.e. higher order polynomial of momentum. One way to
discuss these corrections is deforming the Heisenberg uncertainty principle to a generalized
uncertainty principle (GUP):
∆x̂ ≥ ~
∆p̂, (1.1)
where β is a deforming parameter and corresponds to the square of the minimal length scale.
If GUP is realized in a certain string theory context, β would take a value of order the string
scale (β ∼ α′). This relation comes from various types of studies such as on high energy or
short distance behavior of strings [1], [2], gedanken experiment of black hole [3], de Sitter
space [4], the symmetry of massless particle [5] and wave packets [6].
There are several canonical commutation algebra which lead to the GUP. Among these
algebra we will focus on the algebra;
[x̂, p̂] = i~(1 + βp̂2). (1.2)
This algebra is investigated in [7]-[10] and an attempt to construct a field theory with
minimal length scale is made in [11] by using the Bargmann-Fock representation in 1+1
dimensional spacetime. It has also been used in cosmology, especially in physics at an early
universe (see for example, [12]-[15] and references therein).
In our previous paper [16], we investigated the quantization of fields based on the de-
formed algebra (1.2) in the canonical formalism in 1+1 dimensions and in the path integral
formalism as well. Using the path integral formalism we constructed a quantum theory of
scalar field in arbitrary spacetime dimensions. This theory has a non-locality which stems
from the existence of a minimal length.
In this paper, we construct a quantum theory of free fermion field based on the deformed
Heisenberg algebra. Where, we respect supersymmetry as a guiding principle. This is be-
cause a string theory has this symmetry and we intend to construct a field theory which con-
tains the stringy corrections. Moreover, supersymmetry is also an useful tool to understand
physics in ultra-violet momentum regions. It manages a behavior of system in extremely
high energy regions and eases ultra-violet divergence in quantum theory. Therefore we pro-
pose a quantum field theory of fermion to have a supersymmetry for a scalar system which
was given in [16]. In two and three-dimensional spacetime, we give a system with one real
scalar and one Majorana fermion explicitly. This system has a special symmetry between
a boson and a fermion which corresponds to supersymmetry. Although, this symmetry is
deformed from ordinary supersymmetry. From the fermionic part of this system, we propose
an action of fermionic fields based on GUP in general dimensional spacetime.
II. SCALAR FIELD THEORY
In the paper [16], we proposed a field theory of scalar based on GUP in the path-integral
formalism. We begin with a review of this theory.
Our theory is based on the following algebra [7]:
x̂i, p̂j
= i~(1 + βp̂2)δij . (2.1)
This is an extension to higher dimensional spacetime of deformed Heisenberg algebra (1.2).
Here i, j run from 1 to d which is the number of spatial coordinates and p̂2 ≡
(p̂i)
Hereinafter, we use index i, j for spatial coordinates and a, b for all spacetime coordinates.
Jacobi identity determines the full algebra:
x̂i, x̂j
= −2i~β(1 + βp̂2)L̂ij . (2.2)
p̂i, p̂j
= 0. (2.3)
Here L̂ij are angular momentum like operators L̂ij ≡ 1
2(1+βp̂
(x̂ip̂j − x̂j p̂i + p̂j x̂i − p̂ix̂j).
Because operators p̂i commute with each other, we construct a theory in momentum space
representation. In momentum space representation, momentum operators are diagonalized
simultaneously and we do not distinguish eigenvalues of momentum pi from operators p̂i. In
the following, we set Planck constant ~ to be 1 for simplicity.
Lagrangian in d+ 1 dimensional spacetime [16] is
L = −
ddp(1 + βp2)−1φ(−p, t)
∂2t + p
2 +m2
φ(p, t), (2.4)
where, p2 ≡
The difference from ordinary quantum field theory is a prefactor (1+βp2)−1 in Lagrangian.
Using the Bjorken-Johnson-Low prescription[17], from behavior of T∗-product between
φ(p, t) and φ(p′, t′), we obtain canonical commutation relation:
[φ(p, t), ∂tφ(p
′, t)] = i(1 + βp2)δd(p+ p′). (2.5)
As we can see from this equation, a deforming prefactor (1 + βp̂2) of Heisenberg algebra
in the first quantization (2.1) also appears in canonical commutation relation of the second
quantized field theory.
In a fermion field case, we encounter a difficulty at constructing the second quantized
Hilbert space which does not appear in a scalar system. Note that a system of spin 0
particles contains only spin 0 particle. By contrast, a system of spin 1
particles is not closed
with only fermions in the sense that it contains bosons as bound states. Therefore algebra
of fermion fields must be introduced to be consistent with that of bosons fields. Because the
scalar fields in our theory have a different commutation relation (2.5) from ordinary one,
we must construct fermion fields so that the composite fields which correspond to scalar
particles have the same commutation relations. Or, in two-dimensional ordinary quantum
field theory we could use the concepts of bosonization and fermionization which associate
fermion fields with boson fields. However, it is obscure which of these principles which
relate bosons and fermions remains unchanged in GUP or in extremely high energy regions.
Instead of handling this problem directly, we use supersymmetry to construct quantized field
theory of fermion. This is because string theory accommodates this symmetry and therefore
it is expected that this symmetry is reflected in GUP or in extremely high energy regions.
In the next section we construct a quantum field theory of fermions which is consistent
with the above scalar theory by using supersymmetry.
III. SUPERSYMMETRY IN GUP
In two and three-dimensional spacetime, a system with a real scalar and a Majorana
fermion has a special symmetry between a boson and a fermion, namely supersymmetry.
Thus we construct a quantum field theory of fermion in GUP to have a similar symme-
try between bosons and fermions with an above-reviewed scalar system in two and three-
dimensional spacetime.
Our notation for two and three-dimensional spacetime is as follows: In those dimensional
spacetime (with signature −+ or − + +) the Lorentz group has a real (Majorana) two-
component spinor representation ψα. In the following, we explain the notation of three-
dimensional spacetime. Reduction to two-dimensional spacetime is trivial. We define a
representation of Gamma matrices by Pauli matrices1 as follows:
{Γa,Γb} = 2ηab = 2diag(−++), (3.1)
Γ0 = −iσ2,Γ1 = σ1,Γ2 = −σ3. (3.2)
Spinor indices are lowered and raised by charge conjugation matrix Cαβ ≡ Γ0 and its inverse
matrix C−1:
ψα = ψ
βCβα(= ψ̄α), ψ
α = ψβ(C
−1)βα. (3.3)
Because the algebra of scalar field (2.5) is deformed from usual one, it is natural to
expect that supersymmetry algebra may also be deformed from ordinary one. We generalize
supersymmetry algebra and its actions on a scalar field φ, a Majorana fermion ψ and an
auxiliary field F with parameter ǫαas follows:
ǭ1Q̂, ǭ2Q̂
= 2∆ǭ1Γ
aǫ2P̂a, (3.4)
δφ(p, t) = iǭψ(p, t), (3.5)
δψα(p, t) = A1F (p, t)ǫ
α − A2{(ǭΓ0C−1)α∂t + (ǭΓjC−1)α(ipj)}φ(p, t), (3.6)
δF (p, t) = A3iǭ(Γ
0∂t + Γ
j(ipj))ψ(p, t). (3.7)
Here, we introduce factors ∆, Ai as functions of a deforming parameter β and momentum.
These factors should reduce to 1 in the limit of β → 0 and will be determined later by
consistency conditions.
1 Pauli matrices are σ1 =
, σ2 =
, σ3 =
From the closeness of algebra on each fields, we obtain conditions
A1A3 = A2 = ∆. (3.8)
We also generalize a Lagrangian by introducing factorsBi, which are functions of a deforming
parameter β and momentum and are to be determined as well:
φ(−p, t)(∂2t + p2)φ(p, t)−
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
B1B3mφ(−p, t)F (p, t) +
F (−p, t)F (p, t)
. (3.9)
Here d is the number of spatial coordinates (1 or 2). By integrating out the field F , we
obtain Lagrangian with the scalar field and the Majorana field:
φ(−p, t)(∂2t + p2 +m2)φ(p, t)
− iB2
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
. (3.10)
Invariance of Lagrangian (3.9) under supersymmetry variations (3.5)-(3.7) leads following
conditions;
A1B2 =
B1B3,
A1B2 = A3B3,
B1 = A1A3B2. (3.11)
From conditions (3.8) and (3.11), only B1 and B2 remain to be determined. (Factor A1 can
be absorbed into normalization of a field F and we set it to be 1 for a field F to be an
auxiliary field.) Noether’s current for supersymmetry can be calculated from Lagrangian
(3.10) and supersymmetry charge is found to be
dtdpdB1{ − ψα(−p, t)∂tφ(p, t) + (ΓiΓ0ψ(−p, t))α(ipi)φ(p, t)
+ m(Γ0ψ(−p, t))αφ(p, t)}. (3.12)
Then, we obtain Hamiltonian of this system from supersymmetry charge and algebra (3.4),
H = P 0 = −1
(CΓ0)αβ{Qα, Qβ}. (3.13)
Using the Bjorken-Johnson-Low prescription, from behaviors of T∗-product between fields,
we obtain canonical commutation relations as follows,
[φ(p, t), ∂tφ(q, t)] =
δ(p+ q), (3.14)
α(p, t), ψβ(q, t)
(Γ0C−1)αβ
δ(p+ q). (3.15)
Thus we can write the Hamiltonian in the following form;
{π(−p, t)π(p, t) + φ(−p, t)(p2 +m2)φ(p, t)}
ψ̄(−p, t)((ipi)Γi +m)ψ(p, t). (3.16)
Here, we use conjugate momentum π(p, t) = ∂tφ(−p, t) and indices i runs from 1 to d.
There is another condition which can be used to determine the factors B1 and B2. It
comes from the free energy of supersymmetric vacuum. From algebra (3.4), supersymmetric
state has zero energy:
TrB ln(B1(E
2 + p2 +m2))− 1
TrF ln(B
2 + p2 +m2)). (3.17)
This fact leads to the condition;
B1 = B
2 . (3.18)
Here TrB and TrF represent trace in bosonic and fermionic Hilbert space respectively.
Lastly, we set B1 = (1+βp
2)−1 as we can see from the scalar action (2.4). This determines
all of the introduced factors as follows;
∆ = A2 = A3 = B2 = (1 + βp
2 , (3.19)
A1 = B3 = 1, (3.20)
B1 = (1 + βp
2)−1. (3.21)
Thus we construct quantized fields of fermion which is consistent with scalar fields (2.5) as
ψα(p, t), ψβ(q, t)
= −(1 + βp2)
2 (Γ0C−1)αβδ(p+ q). (3.22)
Note that a factor ∆ is not equal to 1 no matter how we set A1. Therefore this super-
symmetry algebra is deformed from an usual one as
[ǭ1Q, ǭ2Q] = 2(1 + βp
2 ǭ1Γ
aǫ2Pa. (3.23)
There is no difficulty in generalizing the above quantum fields of fermion to higher d+ 1
dimensions than three dimensions. The action is as follows:
(1 + βp2)
ψ̄(−p, t)(Γ0∂t + (ipi)Γi +m)ψ(p, t)
. (3.24)
There appears a universal prefactor (1+βp2)−
2 comparing with usual fermion action regard-
less as to whether there were supersymmetry or not. This prefactor ensures that fermion
fields are compatible with the scalar fields which had been constructed in our previous paper
[16].
From the actions (2.4) and (3.24), we also have supersymmetric field theory in four
dimensions with an complex scalar and a Majorana (or Weyl) fermion just as a corresponding
ordinary field theory has supersymmetry in four dimensions.
IV. CONCLUSION AND DISCUSSIONS
In summary, we have constructed a quantum theory of free fermion field based on the
deformed Heisenberg algebra. It is consistent with already proposed scalar theory through
supersymmetry. We start with a system with an real scalar and a Majorana fermion in two-
and three-dimensional spacetime and determine supersymmetric action. We found that
supersymmetry algebra is deformed from an usual one. An extension to higher dimensions
are trivial and there is also supersymmetric theory in four-dimensional spacetime.
We conclude with a brief discussion on Lorentz invariant extension of our theory. Lorentz
invariant extension of deformed Heisenberg algebra (2.1) is known as a sort of ‘doubly special
relativity’ or ‘κ-deformation’ (for example, see [18], [19] and references therein),
[x̂a, p̂b] = i~(1 + βp̂
2)δab . (4.1)
Here a, b run from 0 to d and p̂2 ≡ −(p̂0)2 +
i=0(p̂i)
2. Thus we claim that an action where
the factor (1+βp2) is replaced with a new factor (1+βp2) describes quantum field theory of
doubly special relativity. In such case, time slice is not well-defined because of the existence
of minimal time interval. Therefore there is no canonical formalism.
Acknowledgments
The author is grateful to T. Matsuo, K. Oda, T. Tada, and N. Yokoi for valuable dis-
cussions. The author is supported by the Special Postdoctoral Researchers Program at
RIKEN.
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http://arxiv.org/abs/hep-th/9301067
http://arxiv.org/abs/hep-th/0505183
http://arxiv.org/abs/gr-qc/0610056
http://arxiv.org/abs/hep-th/9412167
http://arxiv.org/abs/hep-th/0305262
http://arxiv.org/abs/hep-ph/0405127
http://arxiv.org/abs/hep-th/0510245
http://arxiv.org/abs/hep-th/9602085
http://arxiv.org/abs/astro-ph/0410139
http://arxiv.org/abs/gr-qc/0410053
http://arxiv.org/abs/gr-qc/0411056
http://arxiv.org/abs/gr-qc/0504135
http://arxiv.org/abs/hep-th/0511031
http://arxiv.org/abs/hep-th/0203065
http://arxiv.org/abs/gr-qc/0210063
Introduction
Scalar Field Theory
Supersymmetry in GUP
Conclusion and Discussions
Acknowledgments
References
|
0704.1546 | Optimal flexibility for conformational transitions in macromolecules | Optimal flexibility for conformational transitions in macromolecules
Richard A. Neher1, Wolfram Möbius1, Erwin Frey1, and Ulrich Gerland1,2
1Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for Nanoscience (CeNS),
LMU München, Theresienstraße 37, 80333 München, Germany
2Institute for Theoretical Physics, University of Cologne, Germany
(Dated: November 20, 2018)
Conformational transitions in macromolecular complexes often involve the reorientation of lever-
like structures. Using a simple theoretical model, we show that the rate of such transitions is
drastically enhanced if the lever is bendable, e.g. at a localized “hinge”. Surprisingly, the transition
is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition
rate is also least sensitive to the amount of “cargo” attached to the lever arm, which could be
exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for
multi-dimensional barrier crossing to configuration dependent mobility matrices.
Many biological functions depend on transitions in the
global conformation of macromolecules, and the associ-
ated kinetic rates can be under strong evolutionary pres-
sure. For instance, the directed motion of molecular mo-
tors is based on power strokes [1], protein binding to
DNA can require DNA bending [2] or spontaneous partial
unwrapping of DNA from histones [3, 4], and the func-
tioning of some ribozymes depends on global transitions
in the tertiary structure [5]. These and other examples
display two generic features: (i) A long segment within
the molecule or complex is turned during the transition,
e.g. an RNA stem in a ribozyme, the DNA as it unwraps
from histones or bends upon protein binding, or the lever
arm of a molecular motor relative to the attached head.
(ii) The segment has a certain bending flexibility. Here,
we use a minimal physical model to study the coupled
dynamics of the transition and the bending fluctuations.
Our model, illustrated in Fig. 1, demonstrates explic-
itly how even a small bending flexibility can drastically
accelerate the transition. Furthermore, if the flexibil-
ity arises through a localized “hinge”, e.g. in the protein
structure of some molecular motors [6, 7] or an interior
loop in an RNA stem, we find that the transition rate
is maximal at an intermediate hinge stiffness. Thus, in
situations where rapid transition rates are crucial, molec-
ular evolution could tune a hinge stiffness to the optimal
value. We find that an intermediate stiffness is optimal
also from the perspective of robustness, since it renders
the transition rate least sensitive to changes in the drag
on the lever arm, incurred e.g. by different cargos trans-
ported by a molecular motor.
Our finding of an optimal rate is reminiscent of a phe-
nomenon known as resonant activation [8, 9], where a
transition rate displays a peak as a function of the charac-
teristic timescale of fluctuations in the potential barrier.
However, we will see that the peak in our system has
a different origin: a trade-off between the accelerating
effect of the bending fluctuations and a decreasing av-
erage mobility of the reaction coordinate. The standard
Kramers-Langer theory [10] for multi-dimensional transi-
tion processes is not sufficient to capture this trade-off. A
generalization of the theory to the case of configuration-
dependent mobility matrices turns out to be essential to
understand the peak at intermediate stiffness.
Model.— We model the conformational transition as
a thermally activated change in the attachment angle ϕ
of a macromolecular lever, see Fig. 1. The lever has two
segments connected by a hinge with stiffness ǫ, which ren-
ders the lever preferentially straight, but allows thermal
fluctuations in the bending angle θ. The energy function
V (ϕ, θ) of this ‘Two-Segment Lever’ (TSL) is
V (ϕ, θ)
= ǫ(1− cos θ)−
(aϕ)3
− b(aϕ)
, (1)
where kBT is the thermal energy unit. The hinge, de-
scribed by the first term, serves not only as a sim-
ple model for a protein or RNA hinge, but also as a
zeroth-order approximation to a more continuously dis-
tributed flexibility; see below. The second term is the
potential on the attachment angle ϕ, which produces a
metastable minimum at (ϕ, θ) = (0, 0). The thermally-
assisted escape from this minimum passes through the
transition state at (ϕ, θ) = (b/a, 0) with a barrier height
∆V = b3kBT/6 [20].
a) b)
FIG. 1: Schematic illustration of the ‘Two-Segment Lever’
(TSL) model for conformational transitions. (a) The two
segments of lengths 1 and ρ are connected by a hinge and
attached to the origin. The viscous drag acts on the ends
of the segments as indicated by the beads. (b) Schematic
illustration of the barrier crossing processes. The external
meta-stable potential V (ϕ) is indicated by shading (top; dark
corresponds to high energy) and is also sketched below.
http://arxiv.org/abs/0704.1546v1
In the present context, inertial forces are negligible,
i.e. it is sufficient to consider the stochastic dynamics of
the TSL in the overdamped limit. We localize the fric-
tion forces to the ends of the two segments, as indicated
by the beads in Fig. 1(a). The length of the first segment
defines our length unit and ρ denotes the relative length
of the second segment. Similarly, we choose our time
unit such that the friction coefficient of the first bead is
unity, and denote the coefficient of the second bead by
ξ. To describe the Brownian dynamics of the TSL, we
derive the Fokker-Planck equation for the time-evolution
of the configurational probability density p(ϕ, θ, t). In
general, the derivation of the correct dynamic equations
can be a nontrivial task for stochastic systems with con-
straints [11, 12]. For instance, implementing fixed seg-
ment lengths through the limit of stiff springs, leads to
Fokker-Planck equations with equilibrium distributions
that depend on the way in which the limit is taken [12].
However, for our overdamped system, we can avoid this
problem by imposing the desired equilibrium distribu-
tion, i.e. the Boltzmann distribution p = exp(−V/kBT ),
which together with the well-defined deterministic equa-
tions of motion uniquely determines the Fokker-Planck
equation for the TSL.
The deterministic equations of motion take the form
q̇k = −Mkl ∂V/∂ql with the coordinates (q1, q2) = (ϕ, θ)
and a mobility matrix M. We obtain M with a standard
Lagrange procedure: Given linear friction, M is the in-
verse of the friction matrix, which in turn is the Hessian
matrix of the dissipation function [13]. This yields
1 + ξ sin2 θ
1 ρ+cos θ
ρ+cos θ
ρ+2 cos θ
+ 1+ξ
. (2)
The Fokker-Planck equation then follows from the conti-
nuity equation ∂tp({qi}, t) = −∂kjk({qi}, t) together with
jk({qi}, t) = −Mkl
+ kBT
p({qi}, t) (3)
as the probability flux density. Our analytical analysis
below is based directly on Eqs. (2) and (3), while we
perform all Brownian dynamics simulations with a set of
equivalent stochastic differential equations [14].
Transition rate.— To explore the phenomenology of
the TSL, we performed simulations to determine its av-
erage dwell time τ in the metastable state, for a range of
hinge stiffnesses ǫ. The rate for the conformational tran-
sition is related to the dwell time by k(ǫ) = 1/τ(ǫ). Fig. 2
shows k(ǫ) (circles) for a barrier ∆V =12 kBT , a distance
∆ϕ=0.4 to the transition state, and ξ= ρ=1 (data for
different parameter values behaves qualitatively similar,
as long as the process is reaction-limited, i.e. ∆V is suf-
ficiently large that τ is much longer than the time for the
TSL to freely diffuse over an angle ∆ϕ). We observe a
significant flexibility-induced enhancement of the transi-
tion rate over a broad range of stiffnesses, compared to
0 10 20 30 40 50 60 70
stiffness ε [kT]
Langer rate
gen. Langer rate
0 50 100 150
stiff limit
FIG. 2: Simulation data of the barrier crossing rate normal-
ized by k0 display a prominent peak at finite stiffness (cir-
cles, each obtained from 20000 simulation runs initialized at
the metastable minimum). The conventional Langer theory
fails to describe the non-monotonicity of the rate and over-
estimates the rate at small ǫ. The generalized Langer theory
captures the non-monotonicity of the rate and describes the
simulations data accurately; parameters see main text.
the dynamics in the stiff limit (ǫ→∞), see inset. Note
that the enhancement persists even at relatively large ǫ,
where typical thermal bending fluctuations δϕ ∼
ǫ are
significantly smaller than ∆ϕ. Surprisingly, the acceler-
ation is strongest at an intermediate stiffness (ǫ ≈ 10).
This observation suggests that the stiffness of molecular
hinges could be used, by evolution or in synthetic con-
structs, to tune and optimize reaction rates.
When the friction coefficient ξ of the outer bead is
increased, the rate of the conformational transition de-
creases; see Fig. 3a. This decrease is most dramatic in
the stiff limit (dash-dotted line). In the flexible limit (dia-
monds) the decrease is less pronounced. Notably, the rate
appears least sensitive to the viscous drag on the outer
bead at intermediate ǫ (circles). Indeed, Fig. 3b shows
that the ǫ-dependence of this sensitivity (measured as the
slope of the curves in Fig. 3a at ξ=1) has a pronounced
minimum at ǫ ≈ 20. Hence, intermediate hinge stiffnesses
in the TSL lead to maximal robustness, which is an im-
portant design constraint for many biomolecular mecha-
nisms in the cellular context. For instance, as molecular
motors transport various cargos along one-dimensional
filaments, it may be advantageous to render their speed
insensitive to the cargo size, e.g. to avoid “traffic jams”.
In the remainder of this letter, we seek a theoretical
understanding of the above phenomenology. First, it is
instructive to consider simple bounds on the transition
rate. An upper bound is obtained by completely elimi-
nating the outer bead. The Kramers rate [15] for the re-
maining 1D escape process, k0 = (a
2b/2π) e−∆V/kBT , is
used in Figs. 2 and 3 to normalize the transition rates. At
the optimal stiffness, the transition rate in Fig. 2 comes
within 20% of this upper bound. An obvious lower bound
is the stiff limit: For ǫ→∞, the second segment increases
1 10 100
friction ξ
0 20 40 60 80
stiffness ε [kT]
a) b)
ε=25
stiff
FIG. 3: The sensitivity of the rate to the friction coefficient
ξ is minimal at intermediate stiffness. (a) Simulation results
at ǫ = 0 and ǫ = 25 as well as the theoretical estimates of the
rate at ǫ = 0 and in the stiff limit. (b) The derivative of ln k
with respect to ln ξ evaluated at ξ = 1, i.e. the slope of the
curves in a), is minimal in an intermediate stiffness range.
the rotational friction by a factor ζ = 1 + (1 + ρ)2ξ, so
that the 1D Kramers rate becomes k∞ = k0/ζ, as shown
by the dash-dotted line in Fig. 2 and Fig. 3a. However,
to understand how the dynamics of the bending fluctua-
tions affects the transition rate, we must consider the full
2D dynamics of the TSL. The multi-dimensional gener-
alization of Kramers theory is Langer’s formula for the
escape rate over a saddle in a potential landscape [10],
kLanger =
det e(w)
| det e(s)|
. (4)
Here, e(w) and e(s) denote the Hessian matrix of the po-
tential energy, ∂2V/∂qk∂ql, evaluated at the well bot-
tom and the saddle point, respectively, whereas λ is the
unique negative eigenvalue of the product of the mobility
matrix M and e(s). Eq. (4) can be made plausible in sim-
ple terms: Given a quasi-equilibrium in the metastable
state, the second factor represents the probability of be-
ing in the transition region, i.e. the region within ∼ kBT
of the saddle. The escape rate is then given by this prob-
ability multiplied by the rate λ at which the system re-
laxes out of the transition state, analogous to Michaelis-
Menten reaction kinetics.
For our potential (1), the determinants in (4) can-
cel. The eigenvalue can be determined analytically (the
dashed line in Fig. 2 shows the resulting kLanger), but for
the present purpose it is more instructive to consider the
expansions for large and small stiffness. In the stiff limit,
the natural small parameter is the stiffness ratio γ/ǫ,
where γ = a2b is the absolute curvature or “stiffness” of
the external potential at the transition state. The ex-
pansion yields kLanger/k∞ = 1+ (ρ
2ξ/ζ) γ/ǫ+O(γ2/ǫ2).
As expected, the rate approaches k∞, but the stiff limit
is attained only when the bending fluctuations ∼
ǫ are
small compared to the width of the barrier ∼ √γ. In the
opposite limit, ǫ≪ γ, the rate is given by kLanger/k0 =
1 + ρ−1
ǫ/γ + O(ǫ2/γ). Since the linear term is
negative, Langer theory predicts that the transition rate
peaks at zero stiffness, with a peak value equal to the
Kramers rate k0 for the lever without the second segment.
a) b)
FIG. 4: The friction opposing rotation of the attachment an-
gle ϕ depends on the bending angle θ, since the outer bead is
moved by different amounts in different configurations. For an
infinitesimal displacement dϕ, the displacement of the outer
bead is sin θ dϕ. The projection of the resulting friction force
onto the direction of motion adds another factor sin θ, yielding
a friction coefficient for ϕ of 1 + ξ sin2 θ.
This prediction is clearly at variance with the simulation
results. It is interesting to note, however, that the slope
of the linear decay is independent of ξ. This is consistent
with our observation that the transition rate is insensi-
tive to ξ in the intermediate stiffness regime. Indeed,
Fig. 2 shows that Langer theory (dashed line) describes
the simulation data (circles) reasonably well for interme-
diate and large hinge stiffness.
To understand the origin of the peak at intermediate
stiffness, it is useful to consider the flexible limit (ǫ = 0).
In this limit, the transition state is degenerate in θ, and
it seems plausible to estimate the transition rate by using
a θ-averaged mobility for the reaction coordinate ϕ,
k(ǫ = 0) ≈ k0
M11(θ) =
1 + ξ
. (5)
This estimate agrees well with the simulation data,
see the dashed line in Fig. 3a, indicating that the
configuration-dependent mobility (2) plays an important
role for the transition rate. In contrast, the conventional
Langer theory assumes the mobility matrix to be con-
stant in the relevant region near the transition state.
Fig. 4 illustrates why the mobility M11 of the coordinate
ϕ is affected by the bending angle θ and gives a graphical
construction for M11.
Generalized Langer theory.— To account for the mo-
bility effect identified above, we must generalize the
Langer theory to configuration-dependent mobility ma-
trices. The special case where the mobility varies only
along the reaction coordinate has already been studied in
[16], however the main effect in our case is due to the vari-
ation in the transverse direction. In the following, we out-
line the derivation of the central result, while all details
will be presented elsewhere. Near the saddle, the mobil-
ity matrix takes the form Mij({qi}) = M (s)ij + 12A
ij q̂lq̂k,
where q̂i are deviations from the saddle and A
ij denotes
the tensor of second derivatives of the mobility matrix
(we assume that the first derivatives of M vanish at the
saddle, which is the case for the TSL). The escape rate
is given by the probability flux out of the metastable
well, divided by the population inside the well. To cal-
culate the flux, we construct a steady state solution to
the Fokker-Planck equation in the vicinity of the saddle,
as described in [15] for the conventional Langer theory.
We use the Ansatz p({qi}) = 12peq({qi}) erfc(u), where
peq({qi}) = Z−1e−V ({qi})/kBT and erfc(u) is the comple-
mentary error function with argument u = Uk q̂k. Insert-
ing the Ansatz into the Fokker-Planck equation yields an
equation for the vector U,
Ui(−Mije(s)jk +Bik)− UiMijUj Uk = 0 , (6)
where Bik = kBT
ni . Bik q̂k is the noise induced
drift, which is absent in the conventional Langer theory.
Ignoring higher order terms, this equation determines U
to be the left eigenvector of −M(s)e(s)+B to the unique
positive eigenvalue λ, and requires U to be normalized
such that UiM
ij Uj = λ. The directions of the left and
right eigenvectors of −M(s)e(s) + B have a physical in-
terpretation: U is perpendicular to the stochastic sepa-
ratrix, while the corresponding right eigenvector points
in the direction of the diffusive flux at the saddle [17].
From p({qi}), the flux density is determined by (3) and
the total flux is obtained by integrating the flux density
over a plane containing the saddle; a convenient choice is
the plane u = 0. Evaluation of the integral is particularly
simple in a coordinate system, where the first coordinate
is parallel toU, and the remaining coordinates are chosen
such that e(s) is diagonal in this subspace, e
ij = µiδij
for i, j > 1. In this coordinate system, the generalized
Langer rate takes the simple form
1 + 1
det e(w)
| det e(s)|
T , (7)
where c = Uie
ij Uj + 1 = B1ie
i1 /M
11 and e
−1 denotes
the inverse matrix of e(s). Eq. (7) contains three correc-
tions to (4), all of which vanish whenM({qi}) is constant:
The most important one is given by
l>1 A
11/µl, which
changes the mobility M11 in the direction of U to an ef-
fective mobility that is averaged over the separatrix with
respect to the Boltzmann distribution. In addition, there
are two corrections incurred by the noise-induced drift:
the factor
1− c and a change due to the fact that λ is
now the eigenvalue to M(s)e(s) −B instead of M(s)e(s).
The solid line in Fig. 2 shows the application of the
generalized Langer formula to the TSL. We observe that
it captures the peak in the transition rate and thus the es-
sential phenomenology of the TSL. Obviously, the evalu-
ation of the Gaussian integral that leads to Eq. (7) is only
meaningful, if the harmonic approximation of the mobil-
ity matrix is reasonable within the relevant saddle point
region. This integral diverges as the saddle point degen-
erates, which explains the behavior for ǫ → 0. At high
ξ, the very anisotropic friction can also render Langer
theory invalid [18, 19].
Conclusion.— We have introduced the “Two-Segment
Lever” as a simple model for a class of conformational
transitions in biomolecules. The model clearly demon-
strates how flexibility can enhance the rate of a confor-
mational transition. This remains true, if the hinge in the
TSL is replaced by a more continuous bendability. In-
terestingly, a discrete hinge has a stiffness regime, where
the rate is large and robust against cargo variation, which
raises the question, whether these effects are exploited by
evolution, for example in the design of molecular motors.
To understand these effects theoretically, we derived a
generalized Langer theory that takes into account con-
figuration dependent mobility matrices. We hope that
this theory will find applications also in other fields.
We thank the German Excellence Initiative for finan-
cial support via the program NIM. RN and UG are grate-
ful for the hospitality of the CTBP at UCSD, where part
of this work was done, and for financial support by the
CeNS in Munich and the DFG.
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|
0704.1547 | AFM Imaging of SWI/SNF action: mapping the nucleosome remodeling and
sliding | Montel_2007_condmat
AFM imaging of SWI/SNF action : mapping the
nucleosome remodeling and sliding
Fabien MONTEL, Emeline FONTAINE, Philippe ST-JEAN, Martin CASTELNOVO
and Cendrine FAIVRE-MOSKALENKO
Laboratoire Joliot-Curie (CNRS USR 3010) et Laboratoire de Physique (CNRS UMR 5672),
Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69007 Lyon, France
ABSTRACT
We propose a combined experimental (Atomic Force Microscopy) and theoretical study of the
structural and dynamical properties of nucleosomes. In contrast to biochemical approaches,
this method allows to determine simultaneously the DNA complexed length distribution and
nucleosome position in various contexts. First, we show that differences in the nucleo-proteic
structure observed between conventional H2A and H2A.Bbd variant nucleosomes induce
quantitative changes in the in the length distribution of DNA complexed with histones. Then,
the sliding action of remodeling complex SWI/SNF is characterized through the evolution of
the nucleosome position and wrapped DNA length mapping. Using a linear energetic model
for the distribution of DNA complexed length, we extract the net wrapping energy of DNA
onto the histone octamer, and compare it to previous studies.
Keywords : Atomic Force Microscopy, mono-nucleosome, H2A.Bbd, length distribution of
wrapped DNA, nucleosome position distribution, chromatin remodeling factor, histone
variant
INTRODUCTION
DNA is packaged into chromatin in the cell nucleus. The chromatin repeating unit,
called the nucleosome, consists of an octamer of the core histones (two each of H2A, H2B,
H3 and H4) around which about two superhelical turns of DNA are wrapped (1). The
Nucleosome Core Particle (NCP) represents a barrier for the transcription factors binding to
their target DNA sequences and interferes with several basic cellular processes (2). Histone
modifications, ATP-remodeling machines and the incorporation of histone variants within
chromatin are used by the cell to overcome the nucleosomal barrier and modulate DNA
accessibility by the control of nucleosome dynamics (3-6). In this work, we use a single
molecule technique (Atomic Force Microscopy) to visualize isolated mono-nucleosomes, to
quantify the influence of histone octamer composition (H2A-Bbd variant) on the equilibrium
nucleosome conformation and to map nucleosome mobility induced by a remodeling complex
(SWI/SNF).
Chromatin remodeling complexes are used by the cell to overcome the general
repression of transcription associated with the DNA organization into chromatin (7-9). In
order to destabilize histone-DNA interaction, remodeling factors (like SWI/SNF) consume the
energy from ATP hydrolysis to relocate the histone octamer along the DNA sequence (10, 11)
and in some cases, the ejection of the octamer from the DNA template is observed (12). The
molecular motor SWI/SNF is known to mobilize the histone octamer from a central to an end-
position on short DNA templates (13). Nevertheless, the molecular mechanisms involved in
the nucleosome remodeling process have not yet been elucidated .
Histone variants are nonallelic isoforms of the conventional histones. The function of the
different histone variants is far from clear, but the emerging general picture suggests that the
incorporation of histone variants (14-19) in the nucleosome has serious impacts on several
processes, including transcription and repair, and it may have important epigenetic
consequences (20-23). H2A.Bbd (Barr body deficient) is an unusual histone variant whose
primary sequence shows only 48% identity compared to its conventional H2A counterpart
(24). The current view is that H2A.Bbd is enriched in nucleosomes associated with
transcriptionally active regions of the genome (24). In recent studies, the unusual properties of
this variant nucleosome were described (16, 25) using a combination of physical methods and
molecular biology approaches. Those results were mainly focused on the biological role of the
various histone fold domains of H2A.Bbd on the overall structure, stability and dynamics of
the nucleosome, whereas we concentrate here on the quantification of the subtle modifications
in the nucleosome conformation induced by the presence of this histone variant.
Different experimental approaches have been used so far to study the structure and
dynamics of the nucleosome, including crystallographic studies by Luger et al. (26) ,
restriction enzyme accessibility assays (27, 28), and FRET measurements (29, 30).
Additionally, physical models (31) and recent computational efforts were developed to
describe the nucleosome dynamics and energetics (32-35). Following these numerous
contributions, the present study combines experimental (Atomic Force Microscopy) and
theoretical tools to bring complementary information regarding the interplay between
nucleosome position dynamics and DNA wrapping energetics.
Atomic Force Microscopy (36) allows direct visualization of chromatin fibers and
isolated nucleosomes (37). Several experimental procedures allow to depose and observe
reproducibly, all this without any fixing agent, DNA or chromatin samples (38-44). By
scanning the sample with an apex of very high aspect ratio mounted on a flexible lever, the
topography of a surface at the nanometric scale can be acquired. Moreover, computer analysis
of AFM images enables the extraction of systematic and statistically relevant distributions of
structural parameters describing these biological objects (45-47). As nucleosome is a complex
and very dynamic structure, it has been observed that, for a given DNA template, the position
of the octamer relative to the sequence (13, 48-50) and the length of DNA wrapped around the
histone octamer (27-29, 51, 52) both could change drastically in time.
This paper is organized as follows. First, we show that mapping the nucleosome
position along with the length of DNA complexed with histones within individual nucleosome
is a powerful tool to discriminate between conventional and variant nucleosomes. A model is
then proposed to explain quantitatively these differences and to calculate the wrapping energy
of nucleosomes in each case. Next, we have studied nucleosomes in a more dynamic context
by observing the action of chromatin remodeling factor SWI/SNF. To do so, similar mapping
of the nucleosome position and DNA complexed length was used to quantify the impact of
ATP-activated remodeling and sliding of nucleosomes. The results suggest experimental
insights into the processivity of SWI/SNF on mono-nucleosomes.
MATERIALS AND METHODS
Preparation of DNA fragments
The 255 bp and 356 bp DNA fragments, containing the 601 nucleosome positioning
sequence(53), were obtained by PCR amplification from plasmid pGem-3Z-601. For the
255 bp template, 147 bp long 601 positioning sequence is flanked by 52 bp on one side and
56 bp on the other side. For the 356 bp template, 147 bp long-601 positioning sequence is
flanked by 127 bp on one side and 82 bp on the other side. As both 601 DNA templates are
built from the same plasmid, the DNA flanking sequences of the short template are included
in the long DNA template.
Protein purification, nucleosome reconstitution and remodeling
Recombinant Xenopus laevis full-length histone proteins were produced in bacteria
and purified as described (54). For the H2A.Bbd protein, the coding sequences for the H2A
and for H2A.Bbd were amplified by PCR and introduced in the pET3a vector. Recombinant
proteins were purified as previously described (55).
Yeast SWI/SNF complex was purified as described previously (56) and its activity
was normalized by measuring its effect on the sliding of conventional nucleosomes : 1 unit
being defined as the amount of ySWI/SNF required to mobilize 50% of input nucleosomes
(~50 ng) at 29°C during 45 minutes. Nucleosome reconstitution was performed by the salt
dialysis procedure (57). Nucleosomes reconstituted on a 601 nucleosome positioning
sequence (20 ng) were incubated with SWI/SNF as indicated at 29°C and in remodeling
buffer (RB) containing 10 mM Tris-HCl, pH = 7.4, 2.5 mM MgCl2, and 1 mM ATP. The
reaction was stopped after the time as indicated by diluting about 10 times in TE buffer (Tris-
HCl 10 mM, pH = 7.4, EDTA 1 mM) and NaCl 2 mM and deposing the sample onto the
functionalized APTES-mica surface.
Atomic Force Microscopy and surface preparation
For the AFM imaging the conventional and variant nucleosomes were immobilized
onto APTES-mica surfaces. The functionalization of freshly cleaved mica disks (muscovite
mica, grade V-1, SPI) was obtained by self-assembly of a monolayer of APTES under Argon
atmosphere for 2 hours (39). Nucleosomes (DNA concentration ~ 75 ng/µl) were filtered and
concentrated using Microcon® centrifugal filters to remove free histones from the solution,
and diluted 10 times in TE buffer, just prior to deposition onto APTES-Mica surfaces. A 5 µl
droplet of the nucleosome solution is applied on the surface for 1 min, rinsed with 1 mL of
milliQ-Ultrapure © water and gently dried by nitrogen flow. The samples were visualized by
using a Nanoscope III AFM (Digital Instruments™, Veeco, Santa Barbara, CA). The images
were obtained in Tapping Mode in air, with silicon tips (resonant frequency 250-350 kHz) or
Diamond Like Carbon Spikes tips (resonant frequency ~150 kHz) at scanning rates of 2 Hz
over scan areas of 1 µm wide.
This surface functionalization was chosen because it is known to trap 3D conformation
of naked DNA molecule on a 2D surface (58, 59). Moreover, under such experimental
conditions, rinsing and drying are thought to have little effect on the observed conformation
of biomolecule (60).
Image analysis
We have extracted parameters of interest from the AFM images using a MATLAB©
(The Mathworks, Natick, MA) script essentially based on morphological tools such as binary
dilatation and erosion (61-64), and height/areas selections . The aim of the first three steps of
this algorithm is to select relevant objects :
1. In order to remove the piezoelectric scanner thermal drift, flatten of the image is
performed. The use of a height criteria (h>0.5nm where h is the height of the object)
allows to avoid the shadow artifact induced by high objects on the image.
2. Building of a binary image using a simple thresholding (h > 0.25 nm where h is the
height of the object)) and then selection of the binary objects in the good area range
(500 < A < 2000 nm² where A is the area of the object)).
3. Selection of the objects in the good height range using a hysteresis thresholding (65)
(hmin1 = 0.25 nm and h
2 = 1.4 nm, where h
1 and h
2 are the height of the two
thresholds).
These three steps leads to the selection of binary objects whose area is between 500
and 2000 nm² and corresponds in the AFM image to a group of connected pixels whose
minimun height is more than 0.25 nm and maximum height is above 1.6 nm. For example a
height criterion is used to reject tetrasomes while events with SWI/SNF still complexed with
nucleosomes are removed from analysis by a size criterion. The next steps correspond to
measurements in itself :
4. Detection of the NCP centroid by shrinking the objects in the binary image.
5. Building of a distance map inside the nucleosome with respect to their NCP centroid
using a pseudo-euclidian dilatation based algorithm.
6. Selection of the non-octamer parts of the nucleosomes (d > dc , where d is the
constraint distance to the NCP centroid and dc ~ 7.5 nm is the apparent nucleosome
radius) and then thinning of the free arm regions using a commercial MATLAB©
script optimised to avoid most of the branching in the skeleton.
7. Selection of the free arm ends and measurement of the free arm lengths.
8. Measurement of other parameters of interest like areas, volumes and mean height of
the nucleosomes and the octamers (see supplemental materials).
These last 5 steps lead to quick and robust measurements. Indeed the use of morphological
tools allows parallel calculation simultaneously on all the objects. Moreover, erosion is a
good approximation for the inverse operation of the AFM dilatation due to the finite tip radius
and leads to a partial removal of the tip effect (66, 67).
The longest arm is named L+ and the shortest L-. DNA complexed length is deduced
by Lc = Ltot - L- - L+ where Ltot is either 255 bp for short conventional and variant
nucleosomes or 356 bp for long conventional nucleosomes. The position of the nucleosome
relatively to the DNA template center is calculated as ∆L = (L+ - L-)/2. Notice that the
position defined this way corresponds to the location of the most deeply buried base pair,
which might differ from dyad axis position (strictly defined for symmetric nucleosomes).
Complexed DNA length and nucleosome position distribution construction
For the distribution of DNA complexed length, well centered nucleosomes were
selected (∆L* - σ∆L/2 ~ 0 bp < ∆L < 12 bp ~ ∆L
* + σ∆L/2 for the 255 bp mono-nucleosomes
where ∆L* is the most probable nucleosome position and σ∆L is the standard deviation of the
∆L distribution). To construct the histogram a 20 bp-sliding box was used. For each L0 in [0,
300 bp], nucleosomes with a DNA complexed length included in the range [L0 – 10 bp,
L0 + 10 bp] were counted. After normalization, a smooth distribution is obtained that
represents mathematically the convolution of the real experimental distribution with a
rectangular pulse of 20 bp long.
To obtain the nucleosome position distribution we have selected nucleosomes with a
DNA complexed length Lc in a range of width σLc around L
* = 146bp
(123 bp ~ L* - σLc < Lc < L
* - σLc 169 bp for canonical nucleosomes). Then, the same 20-bp
sliding box protocol was used to construct the nucleosome position distribution. The error on
the distribution function mean value (standard error) is given by σexp/√N, where σexp is the
standard deviation of the experimental distribution, and N the number of analyzed
nucleosomes (central limit theorem).
2D distribution Lc/∆∆∆∆L construction
To construct the 2D-histogram a 10 bp-sliding box was used. For each coordinates
(∆L0, L0) in [0, 75 bp]×[0, 300 bp], nucleosomes with a DNA complexed length included in
the range [L0 – 5 bp, L0 + 5 bp] and a position included in the range [∆L0 – 5bp, ∆L0 + 5 bp]
were counted. After normalization a smooth distribution is obtained that represents
mathematically the convolution of the real experimental 2D-distribution with a 10 bp square
rectangular pulse.
Reproducibility and experimental errors
We have checked that different batches of APTES, nucleosome reconstitutions,
ySWI/SNF and mica surfaces lead to similar results for the sliding assays and for the 2D
mapping within the experimental uncertainty. Moreover we have checked by image analysis
of the same naked DNA on the same surface and within the same experimental conditions
(data not shown) that the whole measurement and analysis process have an experimental error
of about 10 bp in DNA length measurement. Notice that uncertainty on the mean value of
length measurements can be much smaller than this resolution as it is explained in the
supplemental material S3.
RESULTS AND DISCUSSION
Simultaneous measurements of DNA complexed length and
nucleosome position.
Several biochemical approaches allow accessing either the nucleosome position along
a DNA template, or the length of DNA wrapped around the histone octamer, but using AFM,
we were able to measure them simultaneously. The results are conveniently plotted as 2D
histograms of nucleosome position versus DNA complexed length.
For short and long arm mononucleosomes
We first investigated the influence of the DNA template length on the nucleosome
complexed length distribution for conventional nucleosomes. Indeed, one could expect that
the nucleosome positioning efficiency for the 601 DNA template and/or the range of wrapped
DNA length could depend on the length of free DNA arms. Using purified conventional
recombinant histones, nucleosomes were reconstituted by salt dialysis on 255 bp (short
nucleosomes) or 356 bp (long nucleosomes) DNA fragments containing the 601 positioning
sequence. Tapping Mode AFM in air was used to visualize the reconstituted particles
adsorbed on APTES-mica surfaces and images of 1 µm2 were recorded. A representative
image of long mono-nucleosomes (Ltot = 356 bp) is displayed on Figure 1a. Such an image
enables to clearly distinguish the nucleosome core particle (red part of the complex : hNCP ~ 2
nm) from the free DNA arms (yellow part of the complex, hDNA ~ 0.7 nm) entering and
exiting the complex.
Precise measurement of the length of each DNA fragment (respectively L+ and L- for
the longer and shorter arm) exiting the nucleosome have been performed. To measure each
“arm” of the mono-nucleosome, the octamer part is excluded and the free DNA trajectory is
obtained (Fig.1b) using morphological tools avoiding false skeletonization by heuristic
algorithm ( cf Material and Methods). From the total DNA length that is un-wrapped around
the histone octamer, we get the length of DNA organized by the histone octamer
(Lc = Ltot - L+ - L-) as well as the nucleosome position with respect to the center of the
sequence (∆L = (L+ - L-)/2).
The 2D histogram Lc/ ∆L is plotted on Fig. 1c for 702 conventional short nucleosomes
using a 2D sliding box as described in the Material and Methods section. The maximum of
the 2D distribution is positioned at L* = 145 bp and ∆L = 15 bp, in qualitative agreement with
the DNA template construction. The 2D mapping is an important tool to study nucleosome
mobilization (see the SWI/SNF sliding section), since both variables are highly correlated
during nucleosome sliding/remodeling. Quantitative information can be however also
obtained by projecting such a 2D histogram on each axis. First, we have selected well
positioned nucleosomes according to the expected position given by the DNA 601 template
construction (0 bp < ∆L < 12 bp for short DNA fragments) and shown their DNA complexed
length probability density function (red line, Fig. 1d). This distribution of the DNA length,
organized by conventional octamer peaks at L* = 146 ± 2 bp, in quantitative agreement with
the crystal structure of the nucleosome (26) and cryoEM measurements (25). The broadness
of this distribution (σ = 23bp) might be explained by different nucleosomes wrapping
conformations. We will explain later on, how this dispersion relates to DNA-histone
interaction energies using a simple model.
We have used the same approach to study long nucleosomes (2D histogram not
shown). Well positioned long nucleosomes according to the DNA sequence
(12 bp < ∆L < 32 bp) have very similar probability distribution (blue line on Fig. 1d) than that
obtained for short nucleosomes showing that the free linker DNA does not affect significantly
the organization of complexed DNA for such nucleosomes.
We now select nucleosomes that have a complexed length in the range L* ± σLc, where
σLc is the standard deviation of the Lc distribution, and their position distribution is displayed
on Figure 1e. The peak values for each DNA fragment (9 ± 2 bp and 24 ± 2 bp for short and
long nucleosomes respectively) is close to the expected value from the DNA template
construct (2 bp and 22 bp for short and long DNA fragments respectively). Both distributions
have a full width at half maximum that exceeds 20 bp. This width might arise from several
features : asymmetric unwrapping of one of the two DNA arms, AFM uncertainty and
dispersion in octamer position. However, it is not possible with these measurements to
determine what is the contribution of each phenomenon. Next, we can see that the distribution
width for longer fragments seems greater. After corrections of artifacts inherent to L+/L-
labeling (cf Supplemental Figure 2) these two position distributions are very similar showing
that the free linker DNA does not affect either the DNA complexed length nor the positioning
of such nucleosomes significantly.
We have shown in this section that AFM measurements give comparable estimations
with other methods for both the positioning and the DNA wrapping of short 601
mononucleosomes. Furthermore, our experimental approach showed no difference in
complexed length probability or nucleosome positioning dynamics for long and short DNA
templates.
For conventional and H2A.Bbd variant mononucleosomes
In order to investigate the influence of the octamer composition on the wrapping of
DNA around the histone octamer, a H2A.Bbd histone variant was used instead of
conventional H2A, in order to reconstitute mono-nucleosomes on a 255 bp DNA fragment.
The H2A.Bbd variant nucleosomes were imaged by AFM (25) and using the same analysis as
described above, only the well positioned nucleosomes (∆L < 12 bp) were selected. Their
DNA complexed length distribution is plotted on Fig. 2a where it is compared to conventional
mononucleosomes reconstituted on the same 601 positioning sequence, 255 bp long, with the
same position range selection (∆L < 12 bp).
The average length of wrapped DNA is clearly different for the variant H2A.Bbd
nucleosomes as the distribution peak value is L*H2A.Bbd = 130 ± 3bp instead of
L*H2A = 146 ± 2bp for the conventional nucleosomes. Moreover the standard deviation of the
distribution is clearly larger for the H2A.Bbd variant (σ = 41 bp to be compared to σ = 23 bp
for the conventional nucleosomes). These differences show that the H2A.Bbd variant
nucleosome is a more labile complex with less DNA wrapped around the octamer, in
agreement with previous observations by AFM and cryo-EM (25). The difference in DNA
complexed length suggests that ∼10 bp at each end of nucleosomal DNA are released from the
octamer. Therefore, AFM allows visualizing subtle differences in the nucleosome structure.
Finally, the DNA complexed length distribution is asymmetric for canonical
nucleosomes. This asymmetry can be quantified by measuring their skewness 3µɶ , defined as:
3 c c
2 22 c c
(L L )
((L L ) )
.We find 3µɶ = -0.57 ± 0.09, the negative sign meaning that
nucleosome conformations with sub-complexed DNA, as compared to the mean value 146 bp,
are energetically more favorable than with over-complexed DNA. This can be interpreted
within the simple model proposed below, based on relevant structural data information (26).
Notice that for variant nucleosomes, the complexed length distribution is nearly symmetric
( 3µɶ ≈ 0.01 ± 0.16), and this feature will also be discussed in the modeling section.
Simple model of DNA complexed length distribution
It has been shown that 14 discrete contacts between DNA and histone octamer are
responsible for the stability of the nucleosome (26).The energetic gain at these sites is made
through electrostatic interactions and hydrogen bonding. At the length scale of the present
analysis, the discreteness of binding sites is not relevant, and it will be replaced by a uniform
effective adsorption energy εa< per unit length, in units of kT/bp. The finite number of binding
sites, or equivalently the finite DNA length L* complexed through these sites (146 bp for
canonical nucleosomes, as determined both by the present experiments and crystal structure),
is due to the specific locations of favorable interactions located at the surface of the histone
octamer, forming a superhelical trajectory on which DNA is complexed. DNA wrapping
around the histone core involves additional bending penalty characterized by the energy per
unit length :
ε = where Lp is the persistence length of DNA within classical linear
elasticity and R the radius of the histone octamer. The stability of the nucleosome requires
that the net energy per unit length is negative (energetic gain), and therefore : εb < εa< .
The experimental distributions of DNA complexed length show that more DNA can be
wrapped around the octamer. For these additional base pairs, the net energy per unit length
has to be positive, due mainly to bending cost. However, to allow for the possibility of some
residual non specific (mainly electrostatic) attractive interactions beyond the 14 binding sites,
the energetic gain of DNA contacting the octamer surface outside of the 14 sites superhelical
path has a different value denoted εa> . The difference εa< - εa> is then representative of the
specificity of the 14 sites region.
Assuming that the energy reference is given by un-complexed straight DNA and
octamer, the total energy for nucleosome is given by
* * *
(sub-complexed nucleosome)
(over-complexed nucleosome)
( ) L if L <LE(L )
( ) L + ( ) (L -L ) if L >L
− ⋅ − ⋅
a ab b
ε ε ε ε
(1)
The distribution of DNA complexed length is given by cc
-E(L ) / (kT)
(L ) ∝P e . It is
maximum for the characteristic length L*, which characterizes the region of specific contacts.
This length may vary for canonical and variant nucleosomes. The assumptions of energy
linearity in wrapped DNA length and of the existence of L*, lead to a double exponential
distribution. By construction, one has the following constraints between effective energies
εa> < εb < εa< .
It should be kept in mind that the effective values εa>, εa< and εb are representative of
nucleosomes adsorbed on a charged flat surface. These values might differ for nucleosomes in
bulk solution, as discussed below.
Extraction of the DNA complexed length parameters
It is possible to extract some parameters from each distribution by using the physical
model presented below, in order to interpret the experimental distribution of DNA complexed
length. We found it more reliable to use global procedure for parameter determination, instead
of fitting the multivariate distribution. Since we expect the DNA complexed length
distribution to be described by a simple double-exponential model, the probability density
function can be written as a skew-Laplace distribution which moments are calculated as :
*2 (1 )
2 2 2
2 c c
*2 (1 ) 3 2 3
3 3/ 2 3/ 2 2 3/ 2
L 2 2
, for L>L1
( ) and then (L L ) 4 (1 )
, for L<L (L L ) 4 50 2 12 48 2
4 (1 )
= = −
= = = − = +
− − + − = =
P L L
µ σ ε
ε ε ε
where L* is the most probable complexed length, ε is the relative asymmetry of the skew-
Laplace distribution and σ is the mean decay length. The distribution normalization is taken
on full real axis as a first approximation, thus neglecting finite size effects. Given the
experimentally determined µ
and µ
parameters, we extract straightforwardly the
parameter L*, ε and σ by numerically solving the equation system (2).
Hence, we are able to measure without any fitting the parameters L*, ε and σ by
calculating the first three moments µ1, µ2 and µ3 of the DNA complexed length statistical
series. In our case we thus have :
2(1 ) 2
and then
1 1 (1 )
2(1 )
− = − = − + = − =
− − = =
specific
ads a a
To see the adequacy of this model with the experimental distribution, the function P(Lc=L) is
drawn for the parameters extracted from the experimental data using the same 20 bp-sliding
box protocol as for the experimental complexed length distribution (Fig.2)
The results are summarized in table 1. The values of energies are expressed in units of
kT per binding site, assuming 14 such sites along the 147 base pairs of DNA for canonical
nucleosomes. Several comments are to be made on these values. First, the measured
characteristic decay lengths corresponding to sub- (L<) and over-complexed (L>) DNA
lengths (Table 1, (b) and (d)) are clearly higher than the intrinsic resolution of our AFM
measurements (related to the tip size that correspond to ~ 10 bp, as checked by image analysis
of the same naked DNA on the same surface and within the same experimental conditions -
data not shown) for both conventional and variant nucleosomes, showing therefore the
significance of the parameters extracted here. Hence, we are able to quantify the energetic of
both sub- and over-complexed DNA length in a mono-nucleosome. For over-complexed DNA
length, the energy has been converted artificially into units of kT per binding site for the sake
of comparison, although the model assumes that there are no such binding sites beyond the 14
sites found in the crystal structure (26). If one assumes that over-complexed DNA length
results solely from bending around the histone core (εa> = 0), the value found for εb leads to a
persistence length Lp ~ 3.5 bp, a value definitely too small for double stranded DNA. Even
more so, this energy is similar in amplitude to the energy of sub-complexed DNA length but
with an opposite sign (Table 1, (c) and (e)). We conclude that it cannot simply be associated
to a bending penalty, therefore justifying a posteriori the assumption of residual attractive
interaction between DNA extra length and histone octamer.
The combination of experimental asymmetry of DNA complexed length distribution
and the simple model allows quantifying the specificity of the 14 binding sites in the
nucleosomes (Table 1, (f)). In particular this can be interpreted as a rough estimation of non-
electrostatic contribution to adhesion energy between DNA and histone octamer.
Comparison of model parameters extracted from data.
These values have to be compared to other estimates reported in the literature. The net
energetic gain per site can be compared to values extracted from experiments done in the
group of J. Widom (68-70). The spirit of these experiments was to probe the transient
exposure of DNA complexed length in a nucleosome by using different restriction enzymes
acting at various well-defined sites along the DNA. The experimental results clearly
demonstrate that DNA accessibility is strongly reduced when restriction sites are located far
away from entry or exit of nucleosomal DNA, towards the dyad axis. From the experimental
data, the authors extract a Boltzmann weight for different site exposures. This distribution
should a priori be similar to the DNA complexed length distribution obtained in our work,
except that only sub-complexed nucleosomes are probed. However, due to the use of different
restriction enzymes with different sizes and mechanisms of action, there is an inherent
uncertainty in the assignment of precise DNA complexed length with a free energy of the
Boltzmann weight. In other words, only a range of energy per binding site can be extracted
from these data. This has to be contrasted with most of previous works using Polach and
Widom's data, which quote a single value of 2 kT per binding site (31). The range of net
energetic gain we are able to estimate out of these data is between 0,5 to 3 kT per binding site.
The value we extracted from our own measurements coincides therefore with the lower bound
of this range. This might be due to the difference in the type of experiments used.
First, our observations are made on nucleosomes adsorbed on a charged substrate. This
might change the energetics of nucleosome opening as compared to its value in solution. A
theoretical estimation of this change is currently under progress (Castelnovo et al, work in
preparation). Another significant difference between Polach and Widom's experiments and
our work is the composition of the buffer, which is known to affect the nucleosome stability.
In particular, the buffer used for restriction enzyme assays contains more magnesium ions
(about 10 mM MgCl2).
The specificity of DNA binding sites on histone octamer, as determined in Table 1 (e)
can also be compared to values extracted from X-ray experiments performed in the group of
T.J. Richmond (71). Indeed, by counting the hydrogen bonds per binding site found in this
structure, one can estimate the specific contribution to the binding energy. These
contributions range between 0.8 and 2 kT per binding site (72). Our estimate for conventional
nucleosome falls in this range (1.1 kT per binding site).
Finally, the comparison between canonical and variant nucleosome shows that both
the average complexed length and the energy per binding site are different. The most probable
length L* = 127 bp (Table 1 (0)) for the variant claims for either the absence or the strong
weakening of at least 2 binding sites. Furthermore, the energy and therefore the stability of the
nucleosome for the remaining binding sites is reduced (εH2A.Bbd ~ 2/3 εH2A), in accordance
with other experimental observations (16, 25, 73). We have shown in this section that a
simple model using a linear energy for the DNA-histone interaction can be used to extract
from the AFM data two important energetic parameters : the net energetic gain per site and
the specific interaction between the DNA and the histone octamer per site. These values are in
good agreement with previous biochemical and X-Ray studies done on conventional
nucleosomes and for the first time are measured on a variant nucleosome.
Visualization of nucleosome sliding and remodeling by
SWI/SNF for conventional and variant nucleosomes.
After studying the nucleosomes in their equilibrium state, the same mononucleosomes
were visualized in the presence of the SWI/SNF remodeling factor to validate the possibility
for this direct imaging approach to acquire new information on the mechanism and dynamics
of nucleosome sliding.
Centrally positioned conventional and variant mononucleosomes (Ltot = 255 bp) were
incubated with SWI/SNF at 29°C in the presence or absence of ATP and then adsorbed on
APTES-mica surfaces for AFM visualization. On figure 3, we report AFM images of
mononucleosomes incubated without (Fig. 3a) and with (Fig. 3b) ATP for one hour. The
sample containing no ATP is the control experiment to account for possible nucleosome
thermally driven diffusion when incubated 1 h at 29 °C. The representative chosen set of
AFM images of Fig.3 clearly shows that most of the nucleosomes are centered on the DNA
template in the negative control (-ATP) whereas they rather exhibit end-position when
SWI/SNF and ATP are present.
On AFM images, SWI/SNF motor is sometimes visible as a very large proteic
complex, and if still attached to nucleosome prevents any image analysis of such objects. Our
protocol does not include removing of SWI/SNF before deposition, even if by diluting the
nucleosome/motor mix, one could expect detaching of some motors. Therefore, the motor per
nucleosome ratio used in the sliding experiments is kept low with respect to biochemical
assays (55) (roughly five time less SWI/SNF per nucleosome).
Using the same type of image analysis we were able to reconstruct 2D histograms
Lc/∆L (using a 2D sliding box as described in the Material and Methods section) at various
time steps during nucleosome sliding : 0 (-ATP), 20 min and 1 hour (Fig. 4). We first notice
that in the absence of ATP, SWI/SNF has apparently no effect on the Lc/∆L map. The 2D
distribution exhibits a single peak corresponding to the canonical nucleosome positioned as
expected from the DNA template (α state). As a function of time in the presence of
remodeling complex and ATP, new states appear : (β) corresponds to an over-complexed
nucleosome having the same mean position ∆L value as (α); this state could result from the
capture of extra DNA (a loop of ~ 40 bp) inside the NCP induced by SWI/SNF. (β)-state is
spread in the ∆L direction showing that this extra complexed DNA length (~ 40 bp) seems to
exist for various positions of the nucleosome (0 < ∆L < 30 bp). (γ) is the slided end-positioned
nucleosome (∆L ~ 50 bp) having slightly less DNA wrapped around the histone octamer
(Lc ~ 125 bp). The ∆L distance separating (α) and (γ) states is close to the Lc distance
between (β) and (α) states, meaning that the slided (γ)-state most likely results from the
release of the (β)-state DNA loop (~ 40 bp). The fact that slided nucleosomes are sub-
complexed i.e. their dyad has been moved beyond the expected end-position, has already been
observed in other biochemical studies (74). Similarly, the anisotropic spreading of the
(γ)-peak towards higher ∆L and lower Lc is also consistent with this feature. We cannot
exclude that a finite size effect of the DNA template could account for this feature. Finally,
(δ) is a wide state with a sub-complexed Lc ~ 75 bp, that could correspond to a tetrasome or
hexasome. This state could be due to the loss of one wrapped DNA turn either from the α
state or the (γ)-state. Nevertheless, one could notice that the (δ)-state is missing on the ‘+ATP
20 min’ map (Fig. 4b) where only few nucleosomes have been slided (weak γ peak) whereas
(δ)-state nucleosomes are clearly visible on Fig. 4c (‘+ATP 1 h’). This tends to show that (δ)-
state nucleosomes more likely arise from the loss of one DNA turn of the end-positioned
nucleosomes ((γ)-state).
We have seen that the 2D-mapping of nucleosome position and DNA complexed
length allows characterizing the new states resulting from the ATP-dependent action of
SWI/SNF on our 601 nucleosomes : an over-complexed state close to the 601 template center
(β), a slided state (γ) and a sub-complexed state (δ).
Again, more information can be gained by appropriate projections of these 2D-
histograms. Nucleosomes having their DNA complexed length in the range L* ± σLc were
selected (L* and σLc are respectively the maximum value and the standard deviation of the
corresponding complexed length distribution) and their position distribution is plotted on
Fig. 5a. For conventional nucleosomes with SWI/SNF but no ATP, the distributions obtained
for nucleosome position (Fig. 5a) and DNA complexed length (Fig. 5b) are very similar to the
case without any remodeling complex (Fig. 1b), showing no effect of thermally driven
diffusion of mononucleosomes reconstituted on 601 positioning sequence in our conditions.
When incubation is increased in the presence of ATP (20 minutes and 1 hour), the
position distribution of conventional nucleosomes is clearly changed (Fig. 5a). Indeed, as a
function of incubation time, a second peak appears corresponding to the end-positioned
nucleosomes (∆L ~ 50 bp, cf (γ)−state in Fig. 4c). After one hour of SWI/SNF action in
presence of ATP the second peak height has increased at the expense of the primary peak.
This corresponds to the situation were one third of the mono-nucleosomes are positioned at
the end of the DNA template. It is interesting to note that during the remodeling factor action
we do not see any significant increase in the amount of nucleosomes in an intermediate
position (20 bp < ∆L < 40 bp). This provides experimental evidence that this remodeling
factor moves centrally positioned nucleosomes directly to the end of our short DNA template.
Mainly, two situations can explain the bimodal position distribution of nucleosomes
after the action of SWI/SNF. The first hypothesis is that the SWI/SNF complex is a
processive molecular motor. As it will not detach from the nucleosome before it reaches the
end of the DNA template, the elementary step of the SWI/SNF induced sliding might not be
accessible. Indeed, in our experimental conditions, only nucleosomes without SWI/SNF
complex attached can be analyzed. The other possibility is that SWI/SNF is weakly
processive (SWI/SNF turnover rate is unknown) but with an elementary step of the order of
50 bp, which corresponds to the value measured by us and other approaches (75-77), and
happens to be the length of free DNA arms in our case. Therefore, a single step would be
enough for the motor to slide a nucleosome to an end-position and release the complex.
Nevertheless, another mechanism cannot be excluded by our data, where SWI/SNF
action would consist of octamer destabilization followed by thermally driven diffusion
towards the end-positioned entropically favored. In this situation, ATP-hydrolysis would only
be involved in the nucleosome ‘destabilization’ step.
In Fig. 5b, we show projections of the previous 2D-histograms along the DNA
complexed length axis without any selection on their position. For conventional nucleosomes
in the presence of SWI/SNF but no ATP, the complexed length distribution is similar to case
with neither SWI/SNF nor ATP. However, the former distribution is larger due to the
contribution of different nucleosome positioning. Then after 20 min, the distribution is
broader (roughly twice) and shifted towards higher Lc. This might be attributed to the
contributions of the different states (β, γ, δ) identified in the Fig. 4b/c. The increase in Lc
mean value is likely due to the statistical weight of the over-complexed (β)-state.
The same sliding experiment was performed on H2A.Bbd variant nucleosomes in
absence and in presence of ATP and analyzed through the projection of the 2D-histogram
Lc/∆L. No significant effect of SWI/SNF complexes in presence of ATP on the position
distribution of H2A.Bbd variant nucleosomes is observed (Fig. 5c). This corroborates
previous findings using biochemical sliding assay done on 5S and 601 positioning sequence
(55). However in AFM measurements, the full position distribution is accessed directly with a
resolution better than 10 bp (the size of AFM tip). This variant nucleosome sliding assay
shows the reproducibility of our experimental approach as not only the position distribution
mean value is constant during one hour in the presence of SWI/SNF and ATP, but also the
complete position distribution remains constant (Fig. 5c). Similarly, SWI/SNF in presence of
ATP does not seem to influence the DNA complexed length distribution of H2A.Bbd
nucleosomes (Fig. 5d).
CONCLUSION
In summary, we have shown that AFM combined with a systematic computer analysis
is a powerful tool to determine the structure of conventional and variant mononucleosomes at
equilibrium and after the action of ATP-dependent cellular machineries. With this technique
we have quantified simultaneously two important and closely coupled variables : the DNA
complexed length and the position of mono-nucleosomes along the 601 DNA template. For
each of these two distributions, the most probable value is in perfect agreement with
measurements done by other methods that give access to one of these two parameters only. In
addition, to explain the experimental complexed length distribution, we have developed a
simple model that uses the experimental shape of DNA complexed length distributions to
quantify the interaction of DNA with histones. With this model, we extract both the net
energetic gain for sub-complexed nucleosomes and the estimation of the non-electrostatic
contribution to the adhesion energy between DNA and histone octamer .
We further show that H2A.Bbd variant and conventional nucleosomes exhibit clear
differences in DNA complexed length and in their ability to be slided by SWI/SNF. Indeed,
these variant nucleosomes organize less DNA on average than conventional nucleosomes, and
present larger opening and closing fluctuations. Moreover, the whole position distribution as
well as complexed length distribution remain unchanged showing H2A.Bbd variant is neither
displaced nor remodeled by SWI/SNF complex.
Finally, we have plotted Lc/∆L as a 2D map of the nucleosome states. This
representation is well suited to highlight the various nucleosome states that appear during the
SWI/SNF action. For example, as a function of time, we have evidenced the formation of an
over-complexed state followed by the appearance of a slided state. More quantitative
information can be obtained by appropriate projections of the 2D-histograms, as for instance
the bimodal position distribution induced by SWI/SNF sliding on conventional nucleosomes,
suggesting two possible scenarii : a processive action of the molecular motor (no intermediate
position visualized) or an elementary stepping length (~ 40 bp) of the size of the free DNA
arms (~ 50 bp). The short length of DNA templates and lack of directionality in our position
analysis prevent us from discriminating between these two hypotheses, and further
experiments on long oriented mononucleosomes are needed to get more insights into the
molecular mechanism of SWI/SNF action. The present results as well as preliminary data on
longer oriented templates prove nevertheless that this extension will provide useful
information on remodeling mechanisms of SWI/SNF.
A further perspective of this AFM study will be to test the effect of the flanking DNA
sequences on the conformation and dynamics of 601 nucleosomes. Nevertheless, in order to
test sequence effect, nucleosomes should be reconstituted on less positioning sequences (5S
rDNA for example) or non-positioning sequences, but this will complicate significantly the
nucleosome sliding analysis as the initial position distribution of the nucleosome is expected
to be broader in this case.
ACKNOWLEDGMENTS
We thank Dimitar Angelov and Hervé Ménoni for various forms of help with
nucleosome reconstitution and sliding assays and Cécile-Marie Doyen for producing
H2A.Bbd variant histones. We are grateful to Stefan Dimitrov, Dimitar Angelov, Françoise
Argoul, Alain Arneodo, Cédric Vaillant and Phillipe Bouvet for fruitful discussions. We thank
Ali Hamiche for providing us with the ySWI/SNF complex. P.St-J. acknowledges CRSNG for
financial support. This work was supported by the CPER ‘Nouvelles Approches Physiques
des Sciences du Vivant’.
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TABLE 1
<Lc>
(bp)
decay length L<
(bp)
εεεεb - εεεεa<
(kT per site)
decay length L>
(bp)
εεεεb - εεεεa>
(kT per site)
εεεεa< - εεεεa>
(kT per site)
conventional
nucleosome 146 ± 2 22 ± 1.6 -0.479 ± 0.045 17 ± 1.4 0.61 ± 0.064 1.1 ± 0.072
variant
nucleosome
127 ± 3 31 ± 1.5 -0.33 ± 0.022 27 ± 1.5 0.39 ± 0.026 0.72 ± 0.021
site exposure
model (g)
-3 <...< -0.5
crystal
structure (h)
147 0.8<…<2
Caption Table1 : Summary of model parameters extracted from experimental data as
explained in Materials and methods. All energies are expressed in units of kT per binding site.
DNA lengths are expressed in bp. (a) Average complexed length (b) Characteristic length L<
of exponential decay towards sub-complexed DNA length. (c) Energy per binding site (1/ L<)
for sub-complexed DNA length. (d) Characteristic length L> of exponential decay, towards
over-complexed DNA length. (e) Energy per binding site (1/L>) for over-complexed DNA
length. (f)Asymmetry of adhesion energy per binding site between sub- and over-complexed
DNA length. (g) Range of values extracted from Polach and Widom (27, 69) data using the
site exposure model. (h) Range of values extracted from Davey and Richmond (71) data using
X-ray crystal structure of the nuclear core particle. Uncertainty values are determined using
the central limit theorem and a propagation of uncertainty calculus detailed in supplemental
data. N(H2A conventional) = 301 nucleosomes. N(H2A.Bbd variant) = 252 nucleosomes.
FIGURE CAPTIONS
Figure 1 : AFM visualization of centered mononucleosomes with short and long arms.
(a) AFM topography image of mono-nucleosomes reconstituted on 356 bp 601 positioning
sequence. Color scale : from 0 to 1.5 nm. X/Y scale bar : 100 nm. (b) Zoom in the AFM
topography image of a centered mono-nucleosome and the result of the image analysis. Black
line : contour of the mono-nucleosome. Blue point : centroid of the histone octamer. Blue dot
circle : excluded area of the histone octamer. Blue line : skeletons of the free DNA arms.
Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm. The longest arm is named L+ and the
shortest L-. DNA complexed length is deduced by Lc = Ltot - L- - L+ where Ltot is in this case
356 bp. The position of the nucleosome relatively to the center of the sequence is calculated
by ∆L = (L+ - L-)/2. (c) 2D histogram Lc/∆L representing the DNA complexed length Lc along
with the nucleosome position ∆L for a short DNA fragment of 255 bp
(N = 702 nucleosomes). (d) Probability density function of the DNA complexed length Lc for
a short DNA fragment (255 bp, purple line) and for a long DNA fragment (356 bp, blue line)
obtained by selecting the well positioned nucleosomes (0 < ∆L < 12 bp and 12 < ∆L < 32 bp
for the short and long fragments respectively) and projecting the 2D map along the y-axis. (e)
Probability density function of the ∆L nucleosome position for a short DNA fragment
(255 bp, purple line) and for a long DNA fragment (356 bp, blue line) obtained by selecting
nucleosomes having their DNA complexed length Lc in the range 123 bp < Lc < 169 bp for
both fragments, and projecting the 2D map along the x-axis.
Figure 2 : AFM Visualization of centered H2A.Bbd variant and H2A conventional
mononucleosome.
(a) Probability density function of the DNA complexed length Lc for a short DNA fragment
(255 bp) with conventional H2A (solid thick line) and with variant H2A.Bbd (dotted thick
line) nucleosome. Simple model for conventional and variant nucleosomes (respectively solid
and dashed thin lines). (b) Description of the model used to measure the DNA-histone
adsorption energies per bp (εa< and εa>) and the DNA bending energy per bp (εb) (dotted line).
Representation of the model using the 20bp-sliding-box procedure (dotted dashed line). L*
corresponds to the most probable DNA complexed length of the distribution.
Figure 3 : AFM Visualization of the sliding of centered mononucleosomes by the
remodeling complex SWI/SNF.
AFM topography image of mononucleosomes reconstituted on 255 bp 601 positioning
sequence, incubated at 29°C with SWI/SNF for one hour (a) in the absence and (b) in the
presence of ATP. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 150 nm. Zoom in the AFM
topography image of a (c) centered mononucleosome and (d) end-positioned
mononucleosome the result of the image analysis. Black line : contour of the mono-
nucleosome. Blue point : centroid of the histone octamer. Blue line : skeletons of the free
DNA arms. Color scale : from 0 to 1.5 nm. X/Y Scale bar : 20 nm.
Figure 4 : Evolution of nucleosome Lc/∆∆∆∆L map during nucleosome sliding by SWI/SNF
complex for conventional nucleosome.
2D histogram Lc/∆L representing the DNA complexed length Lc along with the nucleosome
position ∆L for a conventional nucleosome reconstituted on a short DNA fragment (255 bp)
in the presence of remodeling complex SWI/SNF (a) without ATP (1h at 29°C) (b) with ATP
(20 min at 29°C) and (c) with ATP (1h at 29°C). (d) Representation of the nucleosome Lc/∆L
states for (α), (β), (χ) and (δ) positions as pointed on the 2D maps. N(-ATP, 1h at
29°C) = 692 nucleosomes, N(+ATP, 20 min at 29°C) = 245 nucleosomes, N(+ATP, 1h at
29°C) = 655 nucleosomes.
Figure 5 : Evolution of nucleosome position and DNA complexed length distributions
during nucleosome sliding by SWI/SNF complex, for conventional and variant
nucleosome.
Nucleosome position ∆L (a) and DNA complexed length Lc (b) distributions as a function of
time (0, 20 min, 1 hour) in the presence of SWI/SNF, for conventional mono-nucleosomes
reconstituted on 255 bp long 601 positioning sequence. Nucleosome position ∆L (c) and DNA
complexed length Lc (d) distributions as a function of time (0, 1 hour) in the presence of
SWI/SNF, for H2A.Bbd variant mono-nucleosomes reconstituted on the same DNA template.
The zero time is given by the control in the absence of ATP (solid purple line). For each ∆L
position distribution, only nucleosomes having their complexed length in the range Lc
* ± σLc
are selected. For the sake of figure clarity, error bars are only depicted on 2 distributions of
graph (a).
Lc = Ltot - L-- L+
∆∆∆∆L = (L+-L-) / 2
Figure 1
(a)(a) (b)(b)
L-=86bp
L+=130bp
Ltot = 255 bp,
0 < ∆L< 12 bp
Ltot = 356 bp,
12 < ∆L< 32 bp
0 0.01 0.02
Probability
Ltot = 255 bp,
123 < Lc <169 bp
123 < Lc <169 bp
Ltot = 356 bp,
Lc (bp)
∆∆ ∆∆
50 150 250200100
12e-4
probability scale
Figure 2
Lcomplexed
c( ) L( ) b aP ε εα <+ ⋅ *c cL e if L < L c( ) L( ) b aP ε εα >+ ⋅ *c cL e if L > L
-50 0 50 100 150 200 250 300
0.005
0.015
complexed
H2A, L
=255bp, ∆L<12bp
H2A-Bbd, L
=255bp, ∆L<12bp
Model H2A
Model H2A.Bbd
+ ATP + SWI/SNF
- ATP + SWI/SNF
Figure 3
Figure 4
((((a))))
((((b))))
(α)(α)
(α)(α)
(α)(α) (β)(β)
(γ)(γ)
(δ)(δ)
LLLLcccc (bp)(bp)(bp)(bp)
(γ)(γ)
147bp (601)
12e-4
probability
scale
7.5e-4
2.5e-4
7.5e-4
2.5e-4
0 50 100 150 200 250 300
0.002
0.004
0.006
0.008
0.012
-ATP t=1h
+ATP t=20min
+ATP t=1h
H2A nucleosomes
0 25 50 75 100
0.005
0.015
∆∆∆∆ L (bp)
-ATP 1h
+ATP 1h
H2A.Bbd nucleosomes (d)
0 50 100 150 200 250 300
0.002
0.004
0.006
0.008
0.012
-ATP t=1h
+ATP t=1h
H2A.Bbd nucleosomes
(a) H2A nucleosomes
0 25 50 75 100
0.005
0.015
∆∆∆∆L (bp)
-ATP, t=1h
+ATP, t=20min
+ATP, t=1h
Figure 5
|
0704.1548 | When the orbit algebra of group is an integral domain? Proof of a
conjecture of P.J. Cameron | WHEN THE ORBIT ALGEBRA OF GROUP IS AN INTEGRAL
DOMAIN? PROOF OF A CONJECTURE OF P.J. CAMERON
MAURICE POUZET
Abstract. P.J.Cameron introduced the orbit algebra of a permutation group and
conjectured that this algebra is an integral domain if and only if the group has no
finite orbit. We prove that this conjecture holds and in fact that the age algebra of a
relational structure R is an integral domain if and only if R is age-inexhaustible. We
deduce these results from a combinatorial lemma asserting that if a product of two
non-zero elements of a set algebra is zero then there is a finite common tranversal of
their supports. The proof is built on Ramsey theorem and the integrity of a shuffle
algebra.
Introduction
In 1981, P.J.Cameron [4] (see also [9] p.86) associated a graded algebra A[G] to a
permutation group G acting on an infinite set E. He formulated two conjectures on
the integrity of this algebra. The purpose of this paper is to present a solution to the
first of these conjectures. Consequences on the enumeration of finite substructures of
a given structure are mentionned. Some problems are stated.
0.1. The conjectures. Here is the content of these conjectures, freely adapted from
Cameron’s web page (see Problem 2 [10]). The graded algebra A[G] is the direct sum
A[G]n
where A[G]n is the set of all G-invariant functions f from the set [E]
n of n-element
subsets of E into the field C of complex numbers. Multiplication is defined by the
rule that if f ∈ A[G]m, g ∈ A[G]n and Q is an (m+ n)-element subset of E then
(1) (fg)(Q) :=
P∈[Q]m
f(P )g(Q \ P )
As shown by Cameron, the constant function e in A[G]1 (with value 1 on every one
element set) is not a zero-divisor (see Theorem 0.8 below). The group G is entire if
A[G] is an integral domain, and strongly entire if A[G]/eA[G] is an integral domain.
Date: November 6, 2018.
1991 Mathematics Subject Classification. 03 C13, 03 C52, 05 A16, 05 C30, 20 B27.
Key words and phrases. Relational structures, ages, counting functions, oligomorphic groups, age
algebra, Ramsey theorem, integral domain.
Research done under the auspices of Intas programme 03-51-4110 ”Universal algebra and lattice
theory” .
http://arxiv.org/abs/0704.1548v1
2 MAURICE POUZET
Conjectures 0.1. G is (strongly) entire if and only if it has no finite orbit on E.
The condition that G has no finite orbit on E is necessary. We prove that it suffices
for G to be entire. As it turns out, our proof extends to the algebra of an age, also
invented by Cameron [10].
0.2. The algebra of an age. A relational structure is a realization of a language
whose non-logical symbols are predicates. This is a pair R := (E, (ρi)i∈I) made of a
set E and a family of mi-ary relations ρi on E. The set E is the domain or base of
R; the family µ := (mi)i∈I is the signature of R. The substructure induced by R on
a subset A of E, simply called the restriction of R to A, is the relational structure
R↾A := (A, (A
mi ∩ ρi)i∈I). Notions of isomorphism, as well as isomorphic type, are
defined in natural way (see Subsection 1.1).
A map f : [E]m → C, where m is a non negative integer, is R-invariant if f(P ) =
f(P ′) whenever the restrictions R|P and R|P ′ are isomorphic. The R-invariant maps
can be multiplied. Indeed, it is not difficult to show that if f : [E]m → C and
g : [E]n → C are R-invariant, the product defined by Equation (1) is R-invariant.
Equipped with this multiplication, the C-vector space spanned by the R-invariant
maps becomes a graded algebra, the age algebra of R, that we denote by C.A(R).
The name, coined by Cameron, comes from the notion of age defined by Fräıssé [13].
Indeed, the age of R is the collection A(R) of substructures of R induced on the
finite subsets of R, isomorphic substructures being identified. And it can be shown
that two relational structures with the same age yields the same algebra (up to an
isomorphism of graded algebras).
The algebra associated to a group is a special case of age algebra. Indeed, to a
permutation group G acting on E we may associate a relational structure R with
base E such that the G-invariant maps coincide with the R-invariant maps.
Our criterium for the integrity of the age algebra is based on the notion of kernel:
The kernel of a relational structure R is the subset K(R) of x ∈ E such that
A(R|E\{x}) 6= A(R).
The emptyness of the kernel R is a necessary condition for the integrity of the
age algebra. Indeed, if K(R) 6= ∅, pick x ∈ K(R) and F ∈ [E]<ω such that R↾F ∈
A(R) \ A(R|E\{x}). Let P ∈ [E]
<ω. Set f(P ) := 1 if R↾P is isomorphic to R↾F ,
otherwise set f(P ) := 0. Then f 2 := ff = 0.
Theorem 0.2. Let R be a relational structure with possibly infinitely many non iso-
morphic types of n-element substructures. The age algebra C.A(R) is an integral
domain if and only if the kernel of R is empty.
The application to the conjecture of Cameron is immediate. LetG be a permutation
group acting on E and let R be a relational structure encoding G. Then, the kernel
of R is the union of the finite G-orbits of the one-element sets. Thus, if G has no
finite orbit, the kernel of R is empty. Hence from Theorem 0.2, A[G] is an integral
domain, as conjectured by Cameron.
We deduce Theorem 0.2 from a combinatorial property of a set algebra over a field
(Theorem 0.3 below). This property does not depends upon the field, provided that
its characteristic is zero. The proof we give in Section 1.1 is an extension of our
A CONJECTURE OF P.J. CAMERON 3
1970 proof that the profile of an infinite relational structure does not decrease (see
Theorem 0.5 below). The key tool we used then was Ramsey’s theorem presented in
terms of a property of almost-chainable relations. Here, these relations are replaced by
F −L-invariant relational structures, structures which appeared, under other names,
in several of our papers (see [24], [25], [27]). The final step is reminiscent of the proof
of the integrity of a shuffle algebra.
We introduced the notion of kernel in[24] and studied it in several papers [25] [26],
[27] and [29]. As it is easy to see (cf [25][29]), the kernel of a relational structure
R is empty if and only if for every finite subset F of E there is a disjoint subset F ′
such that the restrictions R|F and R|F ′ are isomorphic. Hence, relational structures
with empty kernel are those for which their age has the disjoint embedding property,
meaning that two arbitrary members of the age can be embedded into a third in
such a way that their domain are disjoint. In Fräıssé’s terminology, ages with the
disjoint embedding property are said inexhaustible and relational structures whose
age is inexhaustible are said age-inexhaustible; we say that relational structures with
finite kernel are almost age-inexhaustible. 1
0.3. A transversality property of the set algebra. Let K be a field with charac-
teristic zero. Let E be a set and let [E]<ω be the set of finite subsets of E (including
the empty set ∅). Let K[E]
be the set of maps f : [E]<ω → K. Endowed with the
usual addition and scalar multiplication of maps, this set is a vector space over K.
Let f, g ∈ K[E]
and Q ∈ [E]<ω. Set:
(2) fg(Q) =
P∈[Q]<ω
f(P )g(Q \ P )
With this operation added, the above set becomes a K-algebra. This algebra is
commutative and it has a unit, denoted by 1. This is the map taking the value 1
on the empty set and the value 0 everywhere else. The set algebra is the subalgebra
made of maps f such that f(P ) = 0 for every P ∈ [E]<ω with |P | large enough.
This algebra is graded, the homogeneous component of degree n being made of maps
which take the value 0 on every subset of size different from n (see Cameron [6]). If
f and g belong to two homogeneous components, their product is given by Equation
(1), thus an age algebra, or a group algebra, A, as previously defined, is a subalgebra
of this set algebra. The set algebra is far from to be an integral domain. But, with
the notion of degree, the integrity of A will reduce to the fact that if m and n are
two non negative integers and f : [E]m → K, f : [E]n → K are two non-zero maps
belonging to A, their product fg is non zero.
Let H be a family of subsets of E, a subset T of E is a transversal of H if F ∩T 6= ∅
for every F ∈ H; the transversality of H, denoted τ(H), is the minimum of the
cardinalities (possibly infinite) of transversals of H. We make the convention that
τ(H) = 0 if H is empty.
Let f : [E]m → K, denote supp(f) := {P ∈ [V ]m : f(P ) 6= 0}.
1In order to agree with Fräıssé’s terminology, we disagree with the terminology of our papers, in
which inexhaustibility, resp. almost inexhaustibility, is used for relational structures with empty,
resp. finite, kernel, rather than for their ages.
4 MAURICE POUZET
Here is our combinatorial result:
Theorem 0.3. Let m,n be two non negative integers. There is an integer t such that
for every set E with at least m + n elements, every field K with characteristic zero,
every pair of maps f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are
not, then τ(supp(f) ∪ supp(g)) ≤ t.
With this result, the proof of Theorem 0.2 is immediate. Indeed, let R be a rela-
tional structure with empty kernel. If K.A(R), the age algebra of R over K, is not an
integral domain there are two non-zero maps f : [E]m → K, f : [E]n → K belonging
to K.A(R), whose product fg is zero. Since K is an integral domain, none of the in-
tegers m and n can be zero. Since f is R-invariant, m is positive and the kernel K(R)
of R is empty, it turns out that τ(supp(f)) is infinite. Hence τ(supp(f)∪ supp(g)) is
infinite, contradicting the conclusion of Theorem 0.3.
An other immediate consequence of Theorem 0.3 is the fact, due to Cameron, that
on an infinite set E, e is not a zero-divisor (see Theorem 0.8 below).
0.3.1. Existence and values of τ . The fact the size of a transversal can be bounded
independently of f and g, and the value of the least upper bound, seem to be of
independent interest.
So, let τ(m,n) be the least t for which the conclusion of Theorem 0.3 holds.
Trivially, we have τ(m,n) = τ(n,m). We have τ(0, n) = τ(m, 0) = 0. Indeed,
if m = 0, f is defined on the empty set only, an thus fg(Q) = f(∅)g(Q). Since
K has no non zero divisors, fg is non zero provided that f and g are non zero.
The fact that there is no pair f, g such that fg is zero, but f and g are not, yields
τ(supp(f) ∪ supp(g)) = 0.
We have τ(1, n) = 2n (Theorem 2.4). This is a non-trivial fact which essentially
amounts to a weighted version of the Gottlieb-Kantor Theorem on incidence matrices
([15], [19], see subsection 0.4 and Theorem 2.3). These are the only exact values
we know. We prove that τ(m,n) exists, by supposing that τ(m − 1, n) exists. Our
existence proof relies in an essential way on Ramsey theorem. It yields astronomical
upper bounds. For example, it yields τ(2, 2) ≤ 2(R2k(4) + 2) , where k = 5
30 and
R2k(4) is the Ramsey number equal to the least integer p such that for every colouring
of the pairs of {1, . . . , p} into k colors there are four integers whose all pairs have
the same colour. The only lower bound we have is τ(2, 2) ≥ 7 and more generally
τ(m,n) ≥ (m+ 1)(n + 1)− 2. We cannot preclude a extremely simple upper bound
for τ(m,n), eg quadratic in n+m.
0.4. Age algebra and profile of a relational structure. The group agebra was
invented by Cameron in order to study the behavior of the function θG which counts
for each integer n the number θG(n) of orbits of n-subsets of a set E on which acts a
permutation groupG, a function that we call the orbital profile ofG. Groups for which
the orbital profile takes only finite values are quite important. Called oligomorphic
groups by Cameron, they are an objet of study by itself (see Cameron’s book[5]).
We present first some properties of the profile, a counting function somewhat more
general. Next, we present the link with the age algebra, then we gives an illustration
of Theorem 0.2. We conclude with some problems.
A CONJECTURE OF P.J. CAMERON 5
0.4.1. Profile of a relational structure. The profile of a relational structure R with
base E is the function ϕR which counts for every integer n the number (possibly
infinite) ϕR(n) of substructures of R induced on the n-element subsets, isomorphic
substructures being identified. Clearly, if R encodes a permutation groups G, ϕR(n)
is the number θG(n) of orbits of n-element subsets of E.
If the signature µ is finite (in the sense that I is finite), there are only finitely
many relational structures with signature µ on an n-element domain, hence ϕR(n) is
necessarily an integer for each integer n. In order to capture examples coming from
algebra and group theory, one cannot preclude I to be infinite. But then, ϕR(n) could
be an infinite cardinal. As far as one is concerned by the behavior of ϕR, this case
can be excluded:
Fact 0.4. [28] Let n < |E|. Then
(3) ϕR(n) ≤ (n + 1)ϕR(n+ 1)
In particular:
(4) If ϕR(n) is infinite then ϕR(n + 1) is infinite too and ϕR(n) ≤ ϕR(n+ 1).
Inequality (3) can be substantially improved:
Theorem 0.5. If R is a relational structure on an infinite set then ϕR is non-
decreasing.
This result was conjectured with R.Fräıssé [14]. We proved it in 1971; the proof -
for a single relation- appeared in 1971 in R.Fräıssé’s book [12], Exercise 8 p. 113; the
general case was detailed in [26]. The proof relies on Ramsey theorem [32].
More is true:
Theorem 0.6. If R is a relational structure on a set E having at least 2n + m
elements then ϕR(n) ≤ ϕR(n+m).
Meaning that if |E| := ℓ then ϕR increases up to
; and, for n ≥ ℓ
the value in n
is at least the value of the symmetric of n w.r.t. ℓ
The result is a straightforward consequence of the following property of incidence
matrices.
Let m,n, ℓ be three non-negative integers and E be an ℓ-element set. Let Mn,n+m
be the matrix whose rows are indexed by the n-element subsets P of E and columns
by the n +m-element subsets Q of E, the coefficient aP,Q being equal to 1 if P ⊆ Q
and equal to 0 otherwise.
Theorem 0.7. If 2n+m ≤ l then Mn,n+m has full row rank (over the field of rational
numbers).
Theorem 0.7 is in W.Kantor 1972 [19], with similar results for affine and vector
subspaces of a vector space. Over the last 30 years, it as been applied and redis-
covered many times; recently, it was pointed out that it appeared in a 1966 paper
of D.H.Gottlieb [15]. Nowadays, this is one of the fundamental tools in algebraic
combinatorics. A proof, with a clever argument leading to further developments, was
given by Fräıssé in the 1986’s edition of his book, Theory of relations, see [13].
6 MAURICE POUZET
We proved Theorem 0.6 in 1976 [23]. The same conclusion was obtained first
for orbits of finite permutation groups by Livingstone and Wagner, 1965 [20], and
extended to arbitrary permutation groups by Cameron, 1976 [3]. His proof uses the
dual version of Theorem 0.7. Later on, he discovered a nice translation in terms of
his age algebra, that we present now.
For that, observe that ϕR only depends upon the age of R and, moreover, if ϕR take
only integer values, then K.A(R) identifies with the set of (finite) linear combinations
of members of A(R). In this case, as pointed out by Cameron, ϕR(n) is the dimension
of the homogeneous component of degree n of K.A(R).
Let e ∈ K[E]
be the map which is 1 on the one-element subsets of E and 0
elsewhere. Let U be the subalgebra generated by e. We can think of e as the sum of
isomorphic types of the one-element restrictions of R. Members of U are then of the
form λme
m + · · ·+ λ1e + λ01 where 1 is the isomorphic type of the empty relational
structure and λm, . . . , λ0 are in K. Hence U is graded, with Un, the homogeneous
component of degree n, equals to K.en.
Here is the Cameron’s result:
Theorem 0.8. If R is infinite then, for every u ∈ K.A(R), eu = 0 if and only if
u = 0
This innocent looking result implies that ϕR is non decreasing. Indeed, the image of
a basis of K.A(R)n by multiplication by e
m is an independent subset of K.A(R)n+m.
0.4.2. Growth rate of the profile. Infinite relational structures with a constant profile,
equal to 1, were called monomorphic and characterized by R. Fräıssé who proved that
they were chainable. Later on, those with bounded profile, called finimorphic, were
characterized as almost chainable [14]. Groups with orbital profile equal to 1 were
described by P.Cameron in 1976 [3]. From his characterization, Cameron obtained
that an orbital profile is ultimately constant, or grows as fast as a linear function with
slope 1
The age algebra can be also used to study the growth of the profile.
If A is a graded algebra, the Hilbert function hA of A is the function which associates
to each integer n the dimension of the homogeneous component of degree n. So,
provided that it takes only finite values, the profile ϕR is the Hilbert function of the
age algebra C.A(R). In [10], Cameron made the following important observation
about the behavior of the Hilbert fonction.
Theorem 0.9. Let A be a graded algebra over an algebraically closed field of charac-
teristic zero. If A is an integral domain the values of the Hilbert function hA satisfy
the inequality
(5) hA(n) + hA(m)− 1 ≤ hA(n +m)
for all non-negative integers n and m.
This result has an immediate consequence on the growth of the profile:
Theorem 0.10. [26] The growth of the profile of a relational structure with empty
kernel is at least linear provided that it is unbounded.
A CONJECTURE OF P.J. CAMERON 7
In fact, provided that the relational structures satisfy some mild conditions, the
existence of jumps in the behavior of the profile extends.
Let ϕ : N → N and ψ : N → N. Recall that ϕ = O(ψ) and ψ grows as fast as ϕ
if ϕ(n) ≤ aψ(n) for some positive real number a and n large enough. We say that
ϕ and ψ have the same growth if ϕ grows as fast as ψ and ψ grows as fast as ϕ.
The growth of ϕ is polynomial of degree k if ϕ has the same growth as n →֒ nk; in
other words there are positive real numbers a and b such that ank ≤ ϕ ≤ bnk for n
large enough. Note that the growth of ϕ is as fast as every polynomial if and only if
limn→+∞
= +∞ for every non negative integer k.
Theorem 0.11. Let R := (E, (ρi)i∈I) be a relational structure. The growth of ϕR
is either polynomial or as fast as every polynomial provided that either the signature
µ := (ni)i∈I is bounded or the kernel K(R) of R is finite.
Theorem 0.11 is in [24]. An outline of the proof is given in [28]. A part appeared in
[26], with a detailed proof showing that the growth of unbounded profiles of relational
structures with bounded signature is at least linear.
The kernel of any relational structure which encodes an oligomorphic permutation
group is finite (indeed, as already mentionned, if R encodes a permutation group G
acting on a set E then K(R) is the set union of the finite orbits of the one-element
subsets of E. Since the number of these orbits is at most θG(1), K(R) is finite if G is
oligomorphic). Hence:
Corollary 0.12. The orbital profile of an oligomorphic group is either polynomial or
faster than every polynomial.
For groups, and graphs, there is a much more precise result than Theorem 0.11. It
is due to Macpherson, 1985 [22].
Theorem 0.13. The profile of a graph or a permutation groups grows either as a
polynomial or as fast as fε, where fε(n) = e
, this for every ε > 0.
0.4.3. Growth rate and finite generation. A central question in the study of the profile,
raised first by Cameron in the case of oligomorphic groups, is this:
Problem 1. If the profile of a relational structures R with finite kernel has polynomial
growth, is ϕR(n) ≃ cn
k′ for some positive real c and some non-negative integer k′?
Let us associate to a relational structure R whose profile takes only finite values
its generating series
HϕR :=
ϕR(n)x
Problem 2. If R has a finite kernel and ϕR is bounded above by some polynomial,
is the series HϕR a rational fraction of the form
P (x)
(1− x)(1− x2) · · · (1− xk)
with P ∈ Z[x]?
8 MAURICE POUZET
Under the hypothesis above we do not know if HϕR is a rational fraction.
It is well known that if a generating function is of the form
P (x)
(1−x)(1−x2)···(1−xk)
for n large enough, an is a quasi-polynomial of degree k
′, with k′ ≤ k − 1, that is
a polynomial ak′(n)n
k′ + · · ·+ a0(n) whose coefficients ak′(n), . . . , a0(n) are periodic
functions. Hence, a subproblem is:
Problem 3. If R has a finite kernel and ϕR is bounded above by some polynomial,
is ϕR(n) a quasi-polynomial for n large enough?
Remark 0.14. Since the profile is non-decreasing, if ϕR(n) is a quasi-polynomial for
n large enough then ak′(n) is eventually constant. Hence the profile has polynomial
growth in the sense that ϕR(n) ∼ cn
k′ for some positive real c and k′ ∈ N. Thus, in
this case, Problem 1 has a positive solution.
A special case was solved positively with N.Thiéry [30].
These problems are linked with the structure of the age algebra. Indeed, if a
graded algebra A is finitely generated, then, since A is a quotient of a polynomial
ring K[x1, . . . , xd], its Hilbert function is bounded above by a polynomial. And, in
fact, as it is well known, its Hilbert series is a fraction of form
P (x)
(1−x)d
, thus of the form
given in (6). Moreover, one can choose a numerator with non-negative coefficients
whenever the algebra is Cohen-Macaulay. Due to Problem 2, one could be tempted
to conjecture that these sufficient conditions are necessary in the case of age agebras.
Indeed, from Theorem 0.8 one deduces easily:
Theorem 0.15. The profile of R is bounded if and only if K.A(R) is finitely generated
as a module over U , the graded algebra generated by e. In particular, if one of these
equivalent conditions holds, K.A(R) is finitely generated
But this case is exceptional. The conjecture can be disproved with tournaments.
Indeed, on one hand, there are tournaments whose profile has arbitrarily large poly-
nomial growth rate and, on an other hand, the age algebra of a tournament is finitely
generated if and only if the profile of the tournament is bounded (this result was
obtained with N.Thiery, a proof is presented in [28]).
0.4.4. Initial segments of an age and ideals of a ring. No concrete description of
relational structures with bounded signature, or finite kernel, which have polynomial
growth is known. In [24] (see also [28]) we proved that if a relational structure R
has this property then its age, A(R), is well-quasi-ordered under embeddability, that
is every final segment of A(R) is finitely generated, which amounts to the fact that
the collection F (A(R)) of final segments of A(R) is noetherian, w.r.t. the inclusion
order. Since the fundamental paper of Higman[17], applications of the notion of
well-quasi-ordering have proliferated (eg see the Robertson-Seymour’s theorem for an
application to graph theory [11] ). Final segments play for posets the same role than
ideals for rings. Noticing that an age algebra is finitely generated if and only if it is
noetherian, we are lead to have a closer look at the relationship between the basic
objects of the theory of relations and of ring theory, particularly ages and ideals.
We mention the following result which will be incorporated into a joint paper with
N.Thiéry.
A CONJECTURE OF P.J. CAMERON 9
Proposition 0.16. Let A be the age of a relational structure R such that the profile
of R takes only finite values and K.A be its age algebra. If A′ is an initial segment
of A then:
(i) The vector subspace J := K.(A \ A′) spanned by A \ A′ is an ideal of K.A.
Moreover, the quotient of K.A by J is a ring isomorphic to the ring K.A′.
(ii) If this ideal is irreducible then A′ is a subage of A.
(iii) This is a prime ideal if and only if A′ is an inexhaustible age.
The proof of Item (i) and Item (ii) are immediate. The proof of Item (iii) is
essentially based on Theorem 0.2.
According to Item (i), F (A) embeds into the collection of ideals of K.A). Conse-
quently:
Corollary 0.17. If an age algebra is finitely generated then the age is well-quasi-
ordered by embeddability.
Problem 4. How the finite generation of an age algebra translates in terms of em-
beddability between members of the ages?
0.4.5. Links with language theory. In the theory of languages, one of the basic results
is that the generating series of a regular language is a rational fraction (see [1]). This
result is not far away from our considerations. Indeed, if A is a finite alphabet, with
say k elements, and A∗ is the set of words over A, then each word can be viewed as a
finite chain coloured by k colors. Hence A∗ can be viewed as the age of the relational
structure R made of the chain Q of rational numbers divided into k colors in such
a way that, between two distinct rational numbers, all colors appear. Moreover, as
pointed out by Cameron [6], the age algebra Q.A(R) is isomorphic to the shuffle
algebra over A, an important object in algebraic combinatorics (see [21]).
Problem 5. Does the members of the age of a relational structure with polynomial
growth can be coded by words forming a regular language?
Problem 6. Extend the properties of regular languages to subsets of the collection
Ωµ made of isomorphic types of finite relational structures with signature µ.
1. Proof of Theorem 0.3
The proof idea of Theorem 0.3 is very simple and we give it first.
We prove the result by induction. We suppose that it holds for pairs (m − 1, n).
Now, let f : [E]m → K and g : [E]n → K such that fg is zero, but f and g are not.
As already mentionned, m and n are non zero, hence members of supp(f) ∪ supp(g)
are non empty. Let sup(f, g) := {(A,B) ∈ supp(f) × supp(g) : A ∩ B = ∅}. We
may suppose sup(f, g) 6= ∅, otherwise the conclusion of Theorem 0.3 holds with t :=
m+ n− 1. For the sake of simplicity, we suppose that K := Q. In this case, we color
elements A of [E]m into three colors :-,0, +, according to the value of f(A). We do the
same with elements B of [E]n and we color each member (A,B) of supp(f, g) with the
colors of its components. With the help of Ramsey’ theorem and a lexicographical
ordering, we prove that if the transversality is large enough there is an (m+n)-element
subset Q such that all pair (A,B) ∈ supp(f, g)(Q) := supp(f, g)∩ ([Q]m × [Q]n) have
10 MAURICE POUZET
the same color. This readily implies that fg(Q) 6= 0, a contradiction. If K 6= Q, we
may replace the three colors by five, as the following lemma indicates.
Lemma 1.1. Let K be a field with characteristic zero. There is a partition of K∗ :=
K \ {0} into at most four blocks such that for every integer k and every k-element
sequences (α1, . . . , αk) ∈ D
k , (β1, . . . , βk) ∈ D
′k, where D, D′ are two blocks of the
partition of K∗, then
i=1 αiβi ∈ K
Proof. This holds trivially if K := C. For an example, divide C∗ into the sets
Di := {z ∈ C
∗ : πi
≤ Argz <
π(i+1)
} (i < 4). If K is arbitrary, use the Com-
pactness theorem of first-order logic, under the form of the ”diagram method” of
A.Robinson [18]. Namely, to the language of fields, add names for the elements of
K, a binary predicate symbol, and axioms, this in such a way that a model, if any,
of the resulting theory T will be an extension of K with a partition satisfying the
conclusion of the lemma. According to the Compactness theorem of first-order logic,
the existence of a model of T , alias the consistency of T , reduces to the consistency of
every finite subset A of T . A finite subset A of T leads to a finitely generated subfield
of K. Such subfield is isomorphic to a subfield of C (see [18] Example 2, p.99, or [2]
Proposition 1, p. 108). This latter subfield equipped with the partition induced by
the partition existing on C∗ satisfies the conclusion of the lemma, hence is a model
of T , proving that A is consistent.
Let T∗ be the set of these four blocks, let T := T∪{0} and let χ be the map from
K onto T.
1.1. Invariant relational structures and their age algebra.
1.1.1. Isomorphism, local isomorphism. Let R := (E, (ρi)i∈I) and R
′ := (E ′, (ρ′i)i∈I)
be two relational structures having the same signature µ := (mi)i∈I . A map h : E →
E ′ is an isomorphism from R onto R′ if
(1) h is bijective,
(2) (x1, . . . , xmi) ∈ ρi if and only if (h(x1), . . . , h(xmi)) ∈ ρ
i for every (x1, . . . , xmi) ∈
Emi , i ∈ I.
A partial map of E is a map h from a subset A of E onto a subset A′ of E, these
subsets are the domain and codomain of h. A local isomorphism of R if a partial map
h which is an isomorphism from R↾A onto R↾A′ (where A and A
′ are the domain an
codomain of h).
1.1.2. Invariant relational structures. A chain is a pair L := (C,≤) where ≤ is a
linear order on C. Let L be a chain. Let V be a non-empty set, F be a set disjoint
from V ×C and let E := F ∪ (V ×C). Let R be a relational structure with base set
E. Let r be a non-negative integer, r ≤ |C|. Let X,X ′ ∈ [C]r. Let ℓ be the unique
order isomorphism from L↾X onto L↾X′ and let ℓ := 1F ∪ (1V , ℓ) be the partial map
such that ℓ(x) = x for x ∈ F and ℓ(x, y) = (x, ℓ(y)) for (x, y) ∈ V ×X .
We say that X and X ′ are equivalent if ℓ is an isomorphism of H↾F∪V×X onto
H↾F∪V×X′. This defines an equivalence relation on [C]
A CONJECTURE OF P.J. CAMERON 11
We say that R is r−F−L-invariant if two arbitrary members of [C]r are equivalent.
We say that R is F − L-invariant if it is r − F − L-invariant for every non-negative
integer r, r ≤ |C|.
It is easy to see that if the signature µ of R is bounded and r :=Max({mi : i ∈ I}),
R is F −L-invariant if and only if it is r′ − F − L-invariant for every r′ ≤ r. In fact:
Lemma 1.2. If |C| > r :=Max({mi : i ∈ I}), R is F −L-invariant if and only if it
is r − F − L-invariant.
This is an immediate consequence of the following lemma:
Lemma 1.3. If R is r−F −L-invariant and r < |C| then R is r′ −F −L-invariant
for all r′ ≤ r.
Proof. We only prove that R is (r− 1)− F − L-invariant. This suffices. Let X,X ′ ∈
[C]r−1. Since r < |C|, we may select Z ∈ [C]r such that the last element of Z (w.r.t.
the order L) is strictly below some element c ∈ C.
Claim 1.4. There are Y, Y ′ ∈ [Z]r−1 which are equivalent to X and X ′ respectively.
Proof of Claim 1.4. Extend X and X ′ to two r-element subsets X1 and X
1 of C.
Since R is r− F −L-invariant, X1 is equivalent to Z, hence the unique isomorphism
from L↾X1 onto L↾Z carries X onto an equivalent subset Y of Z. By the same token,
X ′1 is equivalent to a subset Y
′ of Z.
Claim 1.5. Y and Y ′ are equivalent.
Proof of Claim 1.5. The unique isomorphism from L↾Y ∪{c} onto L↾Y ′∪{c} carries
Y onto Y hence T and Y ′ are equivalent.
From the two claims above X and X ′ are equivalent. Hence, R is (r− 1)− F −L-
invariant.
1.1.3. Coding by words. Let A := P(V ) \ {∅}. Let A∗ :=
Ap be the set of
finite sequences of members of A. A finite sequence u being viewed as a word on the
alphabet A, we write it as a juxtaposition of letters and we denote by λ the empty
sequence; the length of u, denoted by |u| is the number of its terms. Let p be a non
negative integer. If X is a subset of p := {0, . . . , p − 1} and u a word of length p,
the restriction of u to X induces a word that we denote by t(u↾X). We suppose that
V is finite and we equip A with a linear order. We compare words with the same
length with the lexicographical order, denoted by ≤lex. We record without proof the
following result.
Lemma 1.6. Let p, q be two non negative integers and X be an p-element subset of
p + q := {0, . . . , p + q − 1}. The map from Ap × Aq into Ap+q which associates to
every pair (u, v) ∈ Ap × Aq the unique word w ∈ Ap+q such that t(w↾X) = u and
t(w↾p+q\X) = v is strictly increasing (w.r.t. the lexicographical order).
This word w is a shuffle of u and v that we denote uX v. We denote by u ̂ v the
largest word of the form uX v.
We order A∗ with the radix order defined as follows: if u and v are two distincts
words, we set u < v if and only if either |u| < |v| or |u| = |v| et u <lex v. We suppose
12 MAURICE POUZET
that F is finite and we order P(F ) in such a way that X < Y implies |X| ≥ |Y |.
Finally, we order P(F )×A∗ lexicographically.
Let L := (C,≤). Let Q be a finite subset of E := F ∪ (V × C). Let proj(Q) :=
{i ∈ C : Q ∩ V × {i} 6= ∅}. Let i0, . . . , ip−1 be an enumeration of proj(Q) in an
increasing order (w.r.t L) and let w(Q \ F ) be the word u0 . . . up−1 ∈ A
∗ such that
Q \ F = u0 ×{i0} ∪ · · · ∪ up−1 × {ip−1}. We set w(Q) := (Q∩ F,w(Q \ F )). If Q is a
subset of [E]<ω, we set w(Q) := {w(Q) : Q ∈ Q}. If f : [E]m → K, let lead(f) := −∞
if f = 0 and otherwise let lead(f) be the largest element of w(supp(f)). We show
below that this latter parameter behaves as the degree of a polynomial.
We start with an easy fact.
Lemma 1.7. Let m and n be two non negative integers, A ∈ [E\F ]m and B ∈ [E]n. If
|C| ≥ m+n there is A′ ∈ [E\F ]m such that proj(A′)∩proj(B) = ∅ and w(A′) = w(A).
Lemma 1.8. Let R be an F − L-invariant structure on E. Let m and n be two
non negative integers; let f : [E]m → K, g : [E]n → K be two non zero members
of K.A(R). Let A0 ∈ supp(f), and B0 ∈ supp(g) such that w(A0) = lead(f) and
w(B0) = lead(g). Suppose that F and V are finite, that |C| ≤ n +m and supp(f) ∩
[E \ F ]m 6= ∅. Then:
(7) supp(f, g) 6= ∅.
(8) (w(A), w(B)) = (lead(f), lead(g)).
for all (A,B) ∈ supp(f, g)(Q0), where w(Q0) = lead(f, g) and lead(f, g) is the largest
element of w({A ∪ B : (A,B) ∈ supp(f, g)}).
(9) (f(A), g(B)) = (f(A0), g(B0))
for every (A,B) ∈ supp(f, g)(Q0).
(10) fg(Q0) = |supp(f, g)(Q0)|f(A0)g(B0).
(11) lead(fg) = lead(f, g) = (Q0 ∩ F,w(A0)̂w(B0 \ F )).
Proof. (1) Proof of (7). Since supp(f) ∩ [E \ F ]m 6= ∅ and the order on P(F )
decreases with the size, A0 is disjoint from F . Let B ∈ supp(g). According to
Lemma 1.7 there is A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) = w(A0).
We have A′ ∩ B = ∅ and, since f ∈ K.A(R) and R is F − L-invariant,
f(A′) = f(A0). Thus (A
′, B) ∈ supp(f, g).
(2) Proof of (8). Let (A,B) ∈ supp(f, g)(Q0). Since A ∈ supp(f) and B ∈
supp(g), we have trivially:
(12) w(A) ≤ lead(f) and w(B) ≤ lead(g).
Claim 1.9. B ∩ F = Q0 ∩ F and A ∩ F = ∅.
A CONJECTURE OF P.J. CAMERON 13
The pair (A′, B) obtained in the proof of (7) belongs to supp(f, g). Let
Q′ := A′ ∪ B. By maximality of w(Q0), we have w(Q
′) ≤ w(Q0). If B ∩ F 6=
Q0 ∩ F , then |Q
′ ∩ F | < |Q0 ∩ F |, hence w(Q0) < w(Q
′). A contradiction.
The fact that A ∩ F = ∅ follows.
Claim 1.10. proj(A) ∩ proj(B) = ∅ and |proj(A)| = |proj(B).|
Apply Lemma 1.7. Let A′ such that proj(A′) ∩ proj(B) = ∅ and w(A′) =
w(A). Since f ∈ K.A(R) and R is F − L-invariant, f(A′) = f(A) thus
(A′, B) ∈ supp(f, g). Set Q′ := A′ ∪ B. We have w(Q′ \ F ) ≤ w(Q0 \ F )
hence |w(Q′)| ≤ |w(Q0)|. Since |w(Q
′ \ F )| = |proj(A′)| + |proj(B)| and
|w(Q0\F )| ≤ |proj(A)|+|proj(B)|, we get |w(Q0\F )| = |proj(A)|+|proj(B)|.
This proves our claim.
Let i0, . . . , ir−1 be an enumeration of proj(Q0 \ F ) in an increasing order.
Let X := {j ∈ r : ij ∈ proj(A)}. Since proj(A)∩proj(B) = ∅, we have w(Q0\
F ) = w(A)X w(B \ F ). Since w(A) ≤ w(A0) and |w(A)| = |w(A0)|, Lemma
1.6 yields w(Q0 \ F ) ≤ w(A0)X w(B). As it is easy to see, there is A
0 such
that w(A′0) = w(A0) and Q
′ := A′0 ∪ B satisfies w(Q
′ \ F ) = w(A0)X w(B).
Since (A′, B) ∈ supp(f, g), we have w(Q′) = w(Q0) by maximality of w(Q0).
With Lemma 1.6 again, this yields w(A) = w(A0). Hence w(A) = w(A0). A
similar argument yields w(B \F ) = w(B0 \F ) and also w(B \F ) = w(B0 \F ).
(3) Proof of (9). Since R is F −L-invariant, from w(A) = lead(f) := w(A0) we
get f(A) = f(A0). By the same token, we get g(B) = g(B0).
(4) Proof of (10). Since fg(Q0) =
(A,B)∈sup(f,g) f(A)g(B) the result follows
from (9).
(5) Proof of (11). From (10), fg(Q0) 6= 0, the equality lead(fg) = lead(f, g)
follows. The remaining equality follows from (8).
With this, the proof of the lemma is complete.
As far as invariant structures are concerned, we can retain this:
Corollary 1.11. Under the hypotheses of Lemma 1.12, fg 6= 0.
1.1.4. An application. Letm,n be two positive integers, E be a set and f : [E]m → K,
g : [E]n → K. Let R := (E, (ρ(i,j))(i,j)∈T∗×2)) be the relational structure made of the
four m-ary relations ρ(i,0) := {(x1, . . . , xm) : χ ◦ f({x1, . . . , xm}) = i} and the four n-
ary relations ρ(i,1) := {(x1, . . . , xn) : χ ◦ g({x1, . . . , xn}) = i}. A map h from a subset
A of E onto a subset A′ of E is a local isomorphism of R if χ ◦ f(P ) = χ ◦ f(h[P ])
and χ ◦ f(R) = χ ◦ f(h[R]) for every P ∈ [A]m, every R ∈ [A]n. This fact allows us
to consider the pair H := (E, (χ ◦ f, χ ◦ g)) as a relational structure. In the sequel we
suppose that E = F ∪ (V × C) with F and V finite; we fix a chain L := (C,≤).
Lemma 1.12. Suppose that there are P ∈ supp(f) ∩ [V × C]m and R ∈ supp(g) ∩
[E \ P ]n. If H is F − L-invariant and |C| ≥ m+ n. Then fg 6= 0.
Proof. Let lead(f, g) be the largest element of w({A ∪ B : (A,B) ∈ supp(f, g)}) and
let Q0 such that w(Q0) = lead(f, g).
14 MAURICE POUZET
Claim 1.13. (χ ◦ f(A), χ ◦ g(B)) is constant for (A,B) ∈ sup(f, g)(Q0).
Proof of Claim 1.13. Let s : T → K be a section of χ. Let f ′ := s ◦ χ ◦ f and
let g′ := s ◦ χ ◦ g. Then f ′, g′ ∈ K.A(H) and supp(f ′, g′) = supp(f, g). According to
Equation (9) of Lemma 1.8, (f ′(A), g′(B)) is constant for (A,B) ∈ supp(f ′, g′)(Q0).
The result follows.
From Lemma 1.1, fg(Q0) :=
(A,B)∈sup(f,g)(Q0)
f(A)g(B) 6= 0.
We recall the finite version of the theorem of Ramsey [32], [16].
Theorem 1.14. For every integers r, k, l there is an integer R such that for every
partition of the r-element subsets of a R-element set C into k colors there is a l-
element subset C ′ of C whose all r-element subsets have the same color.
The least integer R for which the conclusion of Theorem 1.14 holds is a Ramsey
number that we denote Rrk(l).
Let m and n be two non negative integers. Set r := Max({m,n}), s :=
, k := 5s and, for an integer l, l > r, set ν(l) := Rrk(l).
Lemma 1.15. If |F | = n, |V | = m and |C| ≥ ν(l) there is an l-element subset C ′ of
C such that H↾F∪V×C′ is F − L↾C′-invariant.
The proof is a basic application of Ramsey theorem. We give it for reader conve-
nience. See [13] 10.9.4 page 296, or [27] Lemme IV.3.1.1 for a similar result).
Proof. The number of equivalence classes on [C]r is at most k := 5s (indeed, this
number is bounded by the number of distinct pairs (χ◦f ′, χ◦g′) such that f ′ ∈ K[E
g′ ∈ K[E
′]n and |E ′| = n+mr). Thus, according to Theorem 1.14, there is a l-element
subset C ′ of C whose all r-element subsets are equivalent. This means that H↾F∪V×C′
is r − F − L↾C′ -invariant. Now, since r < l, Lemma 1.3 asserts that H↾F∪V×C′ is
r′ − F − L↾C′-invariant for all r
′ ≤ r. Since the signature of H is bounded by r,
H↾F∪V×C′ is F − L↾C′ -invariant from Lemma 1.2.
1.2. The existence of τ(m,n). Let m and n be two non negative integers. Suppose
1 ≤ m ≤ n and that τ(m− 1, n) exists. Let E be a set with at least m+ n elements
and f : [E]m → K, g : [E]n → K such that fg is zero, but f and g are not.
Lemma 1.16. Let A be a transversal for supp(f). Then there is a transversal B for
supp(g) such that:
(13) |B \ A| ≤ τ(m− 1, n)
Proof. Among the sets P ∈ supp(f) select one, say P0, such that F0 := P0 ∩ A has
minimum size, say r0.
Case 1. A∪P0 is also a transversal for supp(g). In this case, set B := A∪P0. We
have |B \ A| ≤ m− 1, hence inequality (13) follows from inequality (14) below:
(14) m− 1 ≤ τ(m− 1, n)
This inequality is trivial for m = 1. Let us prove it for m > 1 (a much better
inequality is given in Lemma 3.1). Let E ′ be an m + n − 1-element set, let f ′ :
[E ′]m−1 → K and g′ : [E ′]n → K, with f ′ non zero on a single m − 1-element set A′,
A CONJECTURE OF P.J. CAMERON 15
g′(B′) = 1 if A′ ∩ B′ 6= ∅ and g′(B′) = 0 otherwise. Since m − 1 and n are not 0, f ′
and g′ are not 0. Trivially, f ′g′ = 0 and, as it easy to check, A′ is a transversal for
supp(f ′) ∪ supp(g′) having minimum size, hence τ(supp(f ′) ∪ supp(g′)) = m− 1.
Case 2. Case 1 does not hold. In this case, pick x0 ∈ F0 and set E
′ := (E \ A) ∪
(F0 \ {x0}). Let f
′ : [E ′]m−1 → K be defined by setting f(P ′) := f(P ′ ∪ {x0}) for all
P ′ ∈ [E ′]m−1 and let g′ := g↾[E′]n .
Claim 1.17. |E ′| ≥ m+ n− 1 and f ′g′ = 0.
Proof of Claim 1.17. The inequality follows from the fact that A ∪ P0 is not a
transversal for supp(g). Now, let Q′ ∈ [E ′]m+n−1 and Q := Q′ ∪ {x0}.
(15) f ′g′(Q′) =
P ′∈[Q′]m−1
f ′(P ′)g′(Q′ \ P ′) =
x0∈P∈[Q]m
f(P )g(Q \ P )
Since fg = 0 we have:
0 = fg(Q) =
P∈[Q]m
f(P )g(Q \P ) =
x0∈P∈[Q]m
f(P )g(Q \P ) +
x0 6∈P∈[Q]m
f(P )g(Q \P )
If x0 6∈ P , |P ∩ A| < r0, hence f(P ) = 0. This implies that the second term in the
last member of (16) is zero, hence the second member of (15) is zero. This proves our
claim.
From our hypothesis A ∪ P0 is not a tranversal for g. Hence, we have
Claim 1.18. f ′ and g′ are not zero.
The existence of τ(m − 1, n) insures that there is a transversal H for supp(f ′) ∪
supp(g′) of size at most τ(m − 1, n). The set B := A ∪ H is a transversal for
supp(f) ∪ supp(g).
Lemma 1.19. Let l be a positive integer. If τ(supp(f) ∪ supp(g)) > n +m(l − 1) +
τ(m − 1, n) then for every F ∈ supp(g) there is a subset P ⊆ supp(f) ∩ [E \ F ]m
made of at least l pairwise disjoint sets.
Proof. Fix F ∈ supp(g). Let P ⊆ supp(f) ∩ [E \ F ]m be a finite subset made of
pairwise disjoint sets and let p := |P|. If the conclusion of the lemma does not hold,
we have p < l. Select then P with maximum size and set A := F ∪
P. Clearly A is
a transversal for supp(f). According to Lemma 1.16 above, τ(supp(f) ∪ supp(g)) ≤
|A|+ τ(m − 1, n) = n +mp + τ(m− 1, n). Thus, according to our hypothese, l ≤ p.
A contradiction.
Let ϕ(m,n) := n+m(ν(n +m)− 1) + τ(m− 1, n).
Lemma 1.20.
(17) τ(m,n) ≤ ϕ(m,m)
Proof. Suppose τ(supp(f) ∪ supp(g)) > ϕ(m,n). Let F ∈ supp(g). According to
Lemma 1.19 there is a subset P ⊆ supp(f) ∩ [E \ F ]m made of at least ν(n + m)
pairwise disjoint sets. With no loss of generality, we may suppose that ∪P is a set
16 MAURICE POUZET
of the form V × C where |V | = m and |C| = ν(m,n). Let ≤ be a linear order on C
and L := (C,≤). According to Lemma 1.15 there is an n +m-element subset C ′ of
C such that H↾F∪V×C′ is F − L↾C′-invariant. According to Lemma 1.12 fg 6= 0. A
contradiction.
With Lemma 1.20, the proof of Theorem 0.3 is complete.
Note that ϕ(1, 2) = 1 + R2
(3) + τ(0, n) = R2
(3), whereas τ(1, 2) = 4. Also
ϕ(2, 2) = 2R2
(4) + τ(1, 2) = 2(R2
(4) + 2).
Our original proof of Theorem 0.3 was a bit simpler. Instead of an m-element set
F and several pairwise disjoint n-element sets, we considered several pairwise n+m-
element sets. In the particular case of m = n = 2, we got τ(2, 2) ≤ 4R2
(4) + 1. In
term of concrete upper-bounds, we are not convinced that the improvement worth
the effort.
2. The Gottlieb-Kantor theorem and the case m = 1
Let E be a set. To each x ∈ E associate an indeterminate Xx. Let K[E] be the
algebra over the field K of polynomials in these indeterminates. Let f : [E]1 → K. Let
Df be the derivation on this algebra which is induced by f , that is Df(Xx) := f({x})
for every x ∈ E. Let ϕ : K[E] → K[E] be the ring homomorphism such that ϕf (1) :=
1 and ϕf(Xx) := f({x})Xx. Let e : [E]
1 → K be the constant map equal to 1 and
let De be the corresponding derivation. For example Df(XxXyXz) = f({x})XyXz +
f({y})XxXz + f({z})XxXy whereas De(XxXyXz) = XyXz + XxXz + XxXy. It is
easy to check that:
(18) De ◦ ϕf = ϕf ◦Df .
Let n be a non negative integer; let K[n][E] be the vector space generated by the
monomials made of n distinct variables. From equation (18), we deduce:
Corollary 2.1. If f does not take the value zero on [E]1, the surjectivity of the maps
from K[n+1][E] into K[n][E] induced by Df and De are equivalent.
Suppose that E is finite. In this case, the matrix of the restriction ofDe to K[n+1][E]
identifies to Mn,n+1. Thus, according to Theorem 0.7, De is surjective provided that
|E| ≥ 2n+ 1. Corollary asserts that in this case, Df is surjective too. This yields:
Lemma 2.2. If f does not take the value zero on a subset E ′ of E of size at least
2n + 1 but fg = 0 for some g : [E]n → K then g is zero on the n-element subsets of
Proof. Suppose that fg = 0. Let g : K[n][E] → K be the linear form defined by
setting g(Πx∈BXx) := g(B) for each B ∈ [E]
n. Then g ◦Df is 0 on K[n+1][E]. From
Corollary 2, the map from K[n+1][E
′] into K[n][E
′] induced by Df is surjective. Hence
g is 0 on K[n][E
′]. Thus g is zero on [E ′]n as claimed.
Going a step further, we get a weighted version of Gottlieb-Kantor theorem:
Theorem 2.3. Let f : [E]1 → K and g : [E]n → K. If f does not take the value zero
on a subset of size at least 2n+ 1 and if fg = 0 then g est identically zero on [E]n.
A CONJECTURE OF P.J. CAMERON 17
Proof. Set E ′ := supp(f). According to our hypothesis, E ′ 6= ∅. We prove the lemma
by induction on n. If n = 0, pick x ∈ E ′. We have fg({x}) = f({x})g(∅). Since
f({x}) 6= 0 and K is an integral domain, g(∅) = 0. Thus g = 0 and the conclusion of
the lemma holds for n = 0. Let n ≥ 1. Let B ∈ [E]n. We claim that g(B) = 0. Let
F := B \E ′ and r := |F |. If r = 0, that is B ⊆ E ′, we get g(B) = 0 from Lemma 2.2.
If r 6= 0, we define gr on [E
′]n−r, setting gr(B
′) := g(B′ ∪ F ) for each B′ ∈ [E ′]n−r.
Let Q′ ∈ [E ′]n−r+1 and Q := Q′ ∪ F . We have fgr(Q
x∈Q′ f({x})g(Q \ {x}) =∑
x∈Q f({x})g(Q \ {x}) −
x∈F f({x})g(Q \ {x}). From our hypothesis on g and
the fact that f({x}) = 0 for all x /∈ E ′, both terms on the right hand side of the
latter equality are 0, thus fgr(Q
′) = 0. Since |E ′| ≥ 2(n − r) + 1, induction on n
applies. Hence gr is 0 on [E
′]n−r. This yields g(B) = 0, proving our claim. Hence the
conclusion of the lemma holds for n.
Theorem 2.4. τ(1, n) = 2n
Proof. Trivially, the formula holds if n = 0. Hence, in the sequel, we suppose n ≥ 1.
Claim 2.5. τ(1, n) ≤ 2n
Proof of Claim 2.5.Let f and g be non identically zero such that fg = 0. From
Theorem 2.3, the support S of f has at most 2n elements. From Lemma 1.16
|supp(f) ∪ (supp(g)| ≤ |S|+ τ(0, n) = |S| ≤ 2n.
For the converse inequality, we prove that:
Claim 2.6. There is a 2n element set E and a map g : [E]n → K such that eg = 0
and g 6= 0.
Proof of Claim 2.6 Let E := {0, 1} × {0, . . . , n− 1}. Set Ei := {0, 1} × {i} for
i < n. Let B ∈ [E]n. Set g(B) := 0 if B is not a transversal of the Ei’s, g(B) := −1 if
B is a transversal containing an odd number of elements of the form (0, i), g(B) = 1
otherwise. Let Q ∈ [E]n+1. If Q is not a transversal of the Ei’s then g(B) = 0 for
every B ∈ [Q]n hence eg(Q) = 0. If Q is a transversal, then there is a unique index i
such that Ei ⊆ Q. In this case, the only members of [Q]
n on which g is non-zero are
Q\{(0, i)} and Q\{(1, i)}; by our choice, they have opposite signs, hence eg(Q) = 0.
Since in the example above τ(supp(e)) = 2n, we have τ(1, n) ≥ 2n. With this
inequality, the proof of Theorem 2.4 is complete.
3. A lower bound for τ(m,n)
Lemma 3.1. τ(m,n) ≥ (m+ 1)(n+ 1)− 2 for all m,n ≥ 1.
Proof. For m = 1 this inequality was obtained in Claim 2.6. For the case m > 1, we
need the following improvement of Claim 2.6.
Claim 3.2. Let n ≥ 1. There is a 2n element set E and a map g : [E]n → K such
that eg = 0 and supp(g) = [E]n.
Proof of Claim 3.2. Fix a 2n-element set E. From Claim 2.6 and the fact that the
symmetric group SE acts transitively on [E]
n, we get for each B ∈ [E]n some gB such
that egB = 0 and B ∈ supp(gB). Next, we observe that a map g : [E]
n → K satisfies
18 MAURICE POUZET
eg = 0 if and only if g belongs to the kernel of the linear map T : K[E]
→ K[E]
defined by setting T (g)(Q) :=
B∈[Q]n g(B) for all g, Q ∈ [E]
n+1. To conclude, we
apply the claim below with k :=
Claim 3.3. Let e1 := (1, 0 . . . , 0), . . . , ei := (0, . . . , 1, 0, . . . , 0), . . . , ek := (0, . . . , 1) be
the canonical basis of Kk and let H be a subspace of Kk. Then H contains a vector
with all its coordinates which are non-zero if and only if for every coordinate i it
contains some vector with this coordinate non-zero.
Proof of Claim 3.3. This assertion amounts to the fact that a vector space on
an infinite field is not the union of finitely many proper subspaces.
Let E be the disjoint union of m sets E0, . . . , Ei, . . . Em−1, each of size 2n. For
each i, let gi : [Ei]
n → K such that
B∈[Q]n gi(B) = 0 for each Q ∈ [Ei]
n+1 and
supp(gi) = [Ei]
n (according to Claim 3.2 such a gi exists). Let g : [E]
n → K be
the ”direct sum” of the gi’s: g(B) := gi(B) if B ∈ [Ei]
n, g(B) := 0 otherwise. Let
f : [E]m → K defined by setting f(A) := 1 if A ∩ Ei 6= ∅ for all i < m and 0
otherwise. Then, by a similar argument as in Claim 2.6, fg = 0. Next, a transversal
of supp(f) must contains some Ei (thus τ(supp(f)) = 2n). And, also, a transversal
of supp(g) must be a transversal of each of the supp(gi)’s. Since supp(gi) = [Ei]
τ(supp(gi)) = n + 1, hence τ(supp(g)) = (n + 1)m. We get easily that τ(supp(f) ∪
supp(g)) = 2n + (n + 1)(m − 1) = mn + m + n − 1. This completes the proof of
Lemma 3.1.
Example 3.4. The lemma above gives τ(2, 2) ≥ 7. An example illustrating this
inequality is quite simple: let E be made of two squares, let f be the map giving value
−1/2 on each side of the squares, value 1 on the diagonals; let g be giving value 1
on each pair meeting the two squares. Then for every x ∈ E, E \ {x} is a minimal
transversal of supp(f) ∪ supp(g).
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A CONJECTURE OF P.J. CAMERON 19
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Hammamet, March 20-23, 2006). ArXiv math.CO/0703211.
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ICJ, Mathématiques, Université Claude-Bernard Lyon1, Domaine de Gerland, Bât.
Recherche [B], 50 avenue Tony-Garnier, F69365 Lyon cedex 07, France, e-mail: pouzet@univ-
lyon1.fr
http://arxiv.org/abs/math/0703211
http://arxiv.org/math.CO/0601256
Introduction
0.1. The conjectures
0.2. The algebra of an age
0.3. A transversality property of the set algebra
0.4. Age algebra and profile of a relational structure
1. Proof of Theorem ??
1.1. Invariant relational structures and their age algebra
1.2. The existence of (m,n)
2. The Gottlieb-Kantor theorem and the case m=1
3. A lower bound for (m,n)
References
|
0704.1549 | Saturated actions by finite dimensional Hopf *-algebras on C*-algebras | SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF
∗-ALGEBRAS ON C∗-ALGEBRAS
JA A JEONG† AND GI HYUN PARK‡
Abstract. If a finite group action α on a unital C∗-algebra M is saturated,
the canonical conditional expectation E : M → Mα onto the fixed point
algebra is known to be of index finite type with Index(E) = |G| in the sense
of Watatani. More generally if a finite dimensional Hopf ∗-algebra A acts on
M and the action is saturated, the same is true with Index(E) = dim(A).
In this paper we prove that the converse is true. Especially in case M is
a commutative C∗-algebra C(X) and α is a finite group action, we give an
equivalent condition in order that the expectation E : C(X) → C(X)α is of
index finite type, from which we obtain that α is saturated if and only if G acts
freely on X. Actions by compact groups are also considered to show that the
gauge action γ on a graph C∗-algebra C∗(E) associated with a locally finite
directed graph E is saturated.
1. Introduction
It is known [17] that if α is an action by a compact group G on a C∗-algebra M ,
the fixed point algebra Mα is isomorphic to a hereditary subalgebra e(M ×αG)e of
the crossed product M ×αG for a projection e in the multiplier algebra of M ×αG.
If e(M ×α G)e is full in M ×α G (that is, e(M ×α G)e generates M ×α G as a
closed two-sided ideal), the action α is said to be saturated (the notion of saturated
action was introduced by Rieffel [14, Chap.7]). Every action α with a simple crossed
product M ×α G is obviously saturated.
On the other hand, an action of a finite dimensional Hopf ∗-algebra A on a unital
C∗-algebra M is considered in [18] and it is shown that if the action is saturated,
the canonical conditional expectation E : M → MA onto the fixed point algebra
MA is of index finite type in the sense of Watatani [19] and Index(E) = (dimA)1.
The main purpose of the present paper is to prove that the converse is also true.
We see from our result that for an action α by a finite group G, α is saturated
if and only if the canonical expectation E : M → Mα is of index-finite type with
index Index(E) = |G|.
Besides, we consider actions by compact groups to study the saturation property
of a gauge action γ on a C∗-algebra C∗(E) associated with a locally finite directed
graph E with no sinks or sources. This paper is organized as follows.
In section 2, we review the C∗-basic construction from [19] and finite dimensional
Hopf ∗-algebras from [18] setting up our notations. Then we prove in section 3 that
if A is a finite dimensional Hopf ∗-algebra acting on a unital C∗-algebra M such
that E : M → MA is of index finite type with Index(E) = (dimA)1, then the
action is saturated (Theorem 3.3).
In section 4, we deal with the crossed product M ×α G by a finite group in
detail and give other equivalent conditions in order that α be saturated. From the
Research supported by KRF-ABRL-R14(2003-2008)† and Hanshin University Research Grant‡.
http://arxiv.org/abs/0704.1549v1
2 JA A JEONG AND GI HYUN PARK
conditions one easily see that an action with the Rokhlin property [7] is always
saturated. Also we shall show that if M has the cancellation, an action with the
tracial Rokhlin property [12] on M is saturated.
Note that even for an action α by the finite group Z2, the expectation E :
M → Mα may not be of index finite type in general [19, Example 2.8.4]. For a
commutative C∗-algebra C(X) and a finite group action α, we give a necessary and
sufficient condition that E : C(X) → C(X)α is of index finite type (Theorem 4.10)
and provide a formula for Index(E). Then as a corollary we obtain that α is
saturated if and only if G acts freely on X .
In section 5, we consider a compact group action α and investigate the ideal
Jα of M ×α G generated by the hereditary subalgebra e(M ×α G)e. Then we
apply the result on Jα to the gauge action on a graph C∗-algebra in section 6. As
a generalization of the Cuntz-Krieger algebras [5], the class of graph C∗-algebras
C∗(E) associated with directed graphs E has been studied in various directions by
considerably many authors (for example see the bibliography in the book [15] by
Raeburn). In [9], Kumjian and Pask show among others that if γ is the gauge action
on C∗(E), then C∗(E)γ is stably isomorphic to the crossed product C∗(E) ×γ T,
which was done by hiring the notions of skew product of graphs and groupoid C∗-
algebras. In Theorem 6.3 we shall directly show that the gauge action is actually
saturated (this implies that C∗(E)γ and C∗(E)×γ T are stably isomorphic).
2. Preliminaries
Watatani’s index theory for C∗-algebras. In [19], Watatani developed the
index theory for C∗-algebras, and here we briefly review the basic construction
C∗(B, eA). Let B be a C
∗-algebra and A its C∗-subalgebra containing the unit of
B. Let E : B → A be a faithful conditional expectation. If there exist finitely
many elements {vi}ni=1 in B satisfying the following
E(bvi)v
viE(v
i b), for every b ∈ B,
E is said to be of index-finite type and {(vi, v∗i )}ni=1 is called a quasi-basis for E.
The positive element
i is then the index of E, Index(E), which is known to
be an element in the center of B and does not depend on the choice of quasi-bases
for E ([19, Proposition 1.2.8]). Let B be the completion of the pre-Hilbert module
B0 = {η(b) | b ∈ B} over A with an A-valued inner product
〈η(x), η(y)〉 = E(x∗y), η(x), η(y) ∈ B0.
Let LA(B) be the C∗-algebra of all (right) A-module homomorphisms on B with
adjoints. For T ∈ LA(B), the norm ‖T ‖ = sup{‖Tx‖ : ‖x‖ = 1} is always bounded.
Each b ∈ B is regarded as an operator Lb in LA(B) defined by Lb(η(x)) = η(bx)
for η(x) ∈ B0. By eA : B → B we denote the projection in LA(B) such that
eA(η(x)) = η(E(x)), η(x) ∈ B0. Then the C∗-basic construction C∗(B, eA) is the
C∗-subalgebra of LA(B) in which the linear span of elements LbeALb′ (b, b′ ∈ B) is
dense.
Finite dimensional Hopf ∗-algebras. As in [18], a finite dimensional Hopf ∗-
algebra is a finite matrix pseudogroup of [20]. We review from [18] the definition
and some basic properties of a finite dimensional Hopf ∗-algebra which we need in
the following section.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 3
Definition 2.1. ([18, Proposition 2.1]) A finite dimensional C∗-algebra is called a
finite dimensional Hopf ∗-algebra if there exist three linear maps,
∆ : A → A⊗A, ǫ : A → C, and S : A → A
which satisfy the following properties
(i) ∆(comultiplication) and ǫ(counit) are ∗-homomorphisms, and S(antipode)
is a ∗-preserving antimultiplicative involution,
(ii) ∆(1) = 1⊗ 1, ǫ(1) = 1, S(1) = 1,
(iii) (∆⊗ id)∆ = (id⊗∆)∆,
(iv) (ǫ ⊗ id)∆ = ∆(ǫ⊗ id),
(v) m(S⊗id)(∆(a)) = ǫ(a)1 = m(id⊗S)(∆(a)) for a ∈ A, wherem : A⊗A → A
is the multiplication.
Proposition 2.2. ([18], [20]) Let A be a finite dimensional Hopf ∗-algebra. Then
the following properties hold.
(i) For a ∈ A, with the notation ∆(a) =
aLi ⊗ aRi , we have
ǫ(aLi )a
i = a =
ǫ(aRi )a
aLi S(a
i ) = ǫ(a)1 =
S(aLi )a
aRi S(a
i ) = ǫ(a)1 =
S(aRi )a
(ii) There is a unique normalized trace (called the Haar trace) τ on A such that
τ(aLi )a
i = τ(a)1 =
τ(aRi )a
i , a ∈ A.
(iii) There exists a minimal central projection e ∈ A (called the distinguished
projection) such that ae = ǫ(a)e, a ∈ A. We have
ǫ(a) = 1, S(e) = e, and τ(e) = (dimA)−1.
3. Actions by finite dimensional Hopf ∗-algebras
Throughout this section A will be a finite dimensional Hopf ∗-algebra. An action
of A on a unital C∗-algebra M is a bilinear map · : A × M → M such that for
a, b ∈ A, x, y ∈ M ,
1 · x = x,
a · 1 = ǫ(a)1,
ab · x = a · (b · x),
a · xy =
(aLi · x)(aRi · y),
(a · x)∗ = S(a∗) · x∗.
4 JA A JEONG AND GI HYUN PARK
Then the crossed product M ⋊A is the algebraic tensor product M ⊗A as a vector
space with the following multiplication and ∗-operation:
(x⊗ a)(y ⊗ b) :=
x(aLi · y)⊗ aRi b,
(x⊗ a)∗ :=
(aLi )
∗ · x∗ ⊗ (aRi )∗.
Identifying a ∈ A with 1⊗ a and x ∈ M with x⊗ 1, we see [18] that
M ⋊A = span{xa | x ∈ M, a ∈ A}.
For the definition of saturated action of A on M , refer to section 4 of [18].
Proposition 3.1. ([18]) Let MA = {x ∈ M | a · x = ǫ(a)x, for all a ∈ A} be the
fixed point algebra for the action of A on a unital C∗-algebra M .
(i) The action is saturated if and only if
M ⋊A = span{xey | x, y ∈ M},
where e ∈ A is the distinguished projection.
(ii) The map E : M → MA, E(x) = e · x, is a faithful conditional expectation
onto the fixed point algebra such that
E((a · x)y) = E(x(S(a) · y)), a ∈ A, x, y ∈ M.
(iii) The linear map F : M ⋊A → M , F (xa) = τ(a)x, is a faithful conditional
expectation onto M .
Recall that M0 := M is an MA-valued inner product module by
〈η(x), η(y)〉MA = E(x∗y)
(here we use the convention in [19] for the inner product as in section 2). Since
every norm bounded MA-module map on M0 extends uniquely to the Hilbert MA-
moduleM, we may identify the ∗-algebra End(M0) (in [18]) of norm bounded right
MA-module endomorphisms ofM0 having an adjoint with the C∗-algebra LMA(M)
explained in section 2.
Remark 3.2. ([19, Proposition 1.3.3]) If E : M → MA is of index-finite type, then
C∗(M, eMA) = span{LxeMALy | x, y ∈ M} = LMA(M).
In fact, we see from the proof of [19, Proposition 2.1.5] that C∗(M, eMA) contains
the unit of LMA(M). Thus the ideal span{LxeMALy | x, y ∈ M} which is dense in
C∗(M, eMA) must contain the unit of LMA(M).
Theorem 3.3. Let A be a finite dimensional Hopf ∗-algebra acting on a unital
C∗-algebra M . Then the following are equivalent:
(i) The action is saturated.
(ii) E : M → MA is of index finite type with Index(E) = (dimA)1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 5
Proof. (i)=⇒ (ii) is shown in [18, Proposition 4.5].
(ii)=⇒ (i). By Remark 3.2, C∗(M, eMA) = span{LxeMALy | x, y ∈ M}. Consider
a map ϕ : C∗(M, eMA) → M ⋊A given by
LxieMALyi) =
xieyi.
To see that ϕ is well defined, let
LxieMALyi = 0. Then for each z ∈ M ,
LxieMALyi)(η(z)) =
η(xiE(yiz)) = η(
xi(e · (yiz))) = 0,
hence by the injectivity of η ([19, 2.1]),
xi(e·(yiz)) = 0 inM . Since (a·x)e = axe
for a ∈ A, a ∈ M (see (7) of [18]), we thus have
xi(e · (yiz))e =
(xieyi)ze = 0
in M ⋊A for every z ∈ M , which then implies that
xieyi)(zez
′) = 0, z, z′ ∈ M.
Particulary, (
xieyi)(
xieyi)
∗ = 0, so that
xieyi = 0 (in M ⋊ A). Thus ϕ
is well defined. It is tedious to show that ϕ is a ∗-homomorphism such that the
range ϕ(C∗(M, eMA)) = MeM is an ideal of M ⋊A; if x, y, and z ∈ M and a ∈ A,
(za)(xey) = (z(a · x))ey ∈ MeM.
Hence it suffices to show that ϕ(1) = 1. If {(ui, u∗i )}ni=1 is a quasi-basis for E, then
LuieMALu∗i (η(z)) =
η(uiE(u
i z)) = η(z), z ∈ M,
which means that
i LuieMALu∗i = 1 ∈ LMA(M). Therefore by Proposition 2.2(iii)
and Proposition 3.1(iii)
F (ϕ(1)) = F (
i ) =
τ(e)uiu
i = 1.
Since ϕ is a ∗-homomorphism, ϕ(1) is a projection in M⋊A such that F (1−ϕ(1)) =
0. But F is faithful, and ϕ(1) = 1 follows. �
4. Actions by finite groups
Throughout this section G will denote a finite group. As is well known the group
C∗-algebra C∗(G) generated by the unitaries {λg | g ∈ G} is a finite dimensional
Hopf ∗-algebra with
∆(λg) = λg ⊗ λg, ǫ(λg) = 1, S(λg) = λg−1 for λg ∈ C∗(G).
The Haar trace τ is given by τ(λg) = διg, where ι is the identity of G, and the
distinguished projection is e = 1
g λg.
Let α be an action of G on a unital C∗-algebra M . Then it is easy to see that
λg · x := αg(x) for g ∈ G, x ∈ M,
6 JA A JEONG AND GI HYUN PARK
defines an action of C∗(G) on M . Furthermore M ⋊ C∗(G) is nothing but the
usual crossed product M ×α G = span{xλg | x ∈ M, g ∈ G}, and the expectations
E : M → Mα(= MC∗(G)), F : M ×α G → M of Proposition 3.1 are given by
E(x) =
αg(x) and F (
xgλg) = xι (x, xg ∈ M, g ∈ G). (1)
Note that for each
xhλh ∈ M ×α G and g ∈ G,
‖xg‖ = ‖F ((
xhλh)λg−1)‖ ≤ ‖(
xhλh)λg−1‖ = ‖
xhλh‖. (2)
If Jα denotes the closed ideal ofM×αG generated by the distinguished projection
e, then Proposition 3.1(i) says that α is saturated if and only if Jα = M ×α G. We
will see in Proposition 5.4 that
Jα = span{
xαg(y)λg | x, y ∈ M} = span{
xαg(x
∗)λg | x ∈ M}. (3)
The ∗-homomorphism ϕ : C∗(M, eMα) → M ×α G we discussed in the proof of
Theorem 3.3 can be rewritten as follows.
ϕ(LxeMαLy) =
xαg(y)λg, x, y ∈ M (4)
because ϕ(LxeMαLy) = xey and e =
λg. If {(ui, u∗i )} is a quasi-basis for E,
we see from
LuieMαLu∗i = 1 and (4) that
ϕ(1) =
uiαg(u
i ))λg (5)
is a projection in M ×α G. Recall that ϕ(1) = 1 holds if α is saturated.
Theorem 4.1. Let M be a unital C∗-algebra and α be an action of a finite group
G on M . Then the following are equivalent:
(i) α is saturated, that is, Jα = M ×α G.
(ii) E : M → Mα is of index finite type with Index(E) = |G|.
(iii) E : M → Mα is of index finite type with Index(E) = |G| and
uiαg(u
i ) = 0, g 6= ι (6)
for a quasi-basis {(ui, u∗i )} for E.
(iv) There exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1 such that
(a) αg(b
) = b
, for j = 1, . . . ,m and g, h ∈ G.
bjg(b
)∗ = δgh.
(v) For every ε > 0, there exist {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} for some m ≥ 1
such that
‖αg(bjh)− b
‖ < ε,
(b) ‖
bjg(b
)∗ − δgh‖ < ε.
Proof. (i) ⇐⇒ (ii) follows from Theorem 3.3.
(i) =⇒ (iii). If {(ui, u∗i )} is a quasi-basis for E, we have from (5) that
uiαg(u
i ) =
0 for g 6= ι since ϕ(1) = 1.
(iii) =⇒ (ii). Obvious.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 7
(i) =⇒ (iv). Suppose Jα = M ×α G. By (3) there exist m ∈ N and bj ∈ M ,
1 ≤ j ≤ m, such that
bjαg(b
λg = 1.
j = 1 and
bjαg(b
j ) = 0 for g 6= ι. (7)
Set bjg := αg(bj). Then
) = αg(αh(bj)) = αgh(bj) = b
bjg(b
αg(bj)αh(b
bjαg−1h(b
= δgh by (7).
(iv) =⇒ (v). Obvious.
(v) =⇒ (i). Let ε > 0 and let {bjg ∈ M | g ∈ G, 1 ≤ j ≤ m} satisfy (a) and
(b) of (v). Note that (b) implies ‖bjg‖ < 1 + ε for g ∈ G, 1 ≤ j ≤ m. Indeed from
bjg(b
∥ ≤ ‖
bjg(b
∗−1‖ < ε, we have ‖bjg‖2 ≤ ‖
bjg(b
∗‖ < 1+ε.
αg((b
)∗)λg)− |G| ‖
αg((b
)∗)λg − |G| ‖
)∗ − |G| ‖+ ‖
g 6=ι
αg((b
)∗)λg)‖
)∗ − 1 ‖+
g 6=ι
αg((b
< ε|G|+
g 6=ι
αg((b
)∗)− (bj
g 6=ι
|G|+ |G|2 max
‖bjg‖+ |G|2
|G|+ |G|2(1 + ε) + |G|2
Since
αg((b
)∗)λg) ∈ Jα and ε can be chosen to be arbitrarily small,
we conclude that Jα = M ×α G. �
Example 4.2. Let w =
be a unitary with wn = 1 and define an
automorphism α on M2(C)) by α(a) = waw
∗, a ∈ M2(C). We will show that α is
saturated if and only if z2 = −z1. For this, recall from (3) that α is saturated if
and only if there exist xj ∈ M2(C), 1 ≤ i ≤ m, satisfying
k(x∗j )λk = 1M2(C). (8)
8 JA A JEONG AND GI HYUN PARK
Hence, particularly for k = 0, 1, we have
xjα(x
j ) =
With xj =
aj bj
cj dj
and zi = e
iθi , i = 1, 2, this means
|aj |2 + |bj |2 aj c̄j + bj d̄j
cj āj + dj b̄j |cj |2 + |dj |2
|aj |2 + ei(θ2−θ1)|bj |2 ei(θ1−θ2)aj c̄j + bj d̄j
|aj |2 + ei(θ2−θ1)dj b̄j ei(θ1−θ2)aj c̄j + |dj |2
. (9)
Therefore, by comparing (1,1) entries of each matrices, it follows that if α is satu-
rated, then there exist positive real numbers a (=
|aj |2) > 0, b (=
|bj |2) > 0
such that
a+ b = 1 and a+ ei(θ2−θ1)b = 0. (10)
Note that b 6= 0 since b = 0 implies a = 0 from
|aj |2 + ei(θ2−θ1)dj b̄j
= 0 in
(9). There are three possible cases for θ1, θ2 as follows.
(i) If θ2 − θ1 ≡ 0(mod 2π), that is, α is trivial, then (10) is not possible.
(ii) If θ2 − θ1 ≡ π(mod 2π), then
and x2 =
satisfy (8) with m = 2. Thus α is saturated.
(iii) If θ2− θ1 6= 0, π(mod 2π), then (10) is not possible for any a, b > 0. Hence
α is not saturated.
Remark 4.3. Let α, β ∈ Aut(M) satisfy αn = βn = idM for some n ≥ 1. If
there is a unitary u ∈ M such that β = Ad(u) ◦ α, then α and β are said to
be exterior equivalent, and if this is the case the crossed products are isomorphic,
M×αG ∼= M×βG, [14, p.45]. Example 4.2 says that the property of being saturated
may not be preserved under exterior equivalence. Also the case (iii) of Example 4.2
above with w = diag(λ, λ̄), λ = e
3 (hence θ1− θ2 = 2π3 −
6= π(mod 2π)), shows
that Index(E) < |G| is possible even when E is of index-finite type. In fact, if
and u2 =
, then {(ui, u∗i )}2i=1 forms a quasi-basis for E,
but Index(E) = 2 < |Z3|.
Remark 4.4. Recall that the Rokhlin property and the tracial Rokhlin property
(weaker than the Rokhlin property) are defined as follows and considered intensively
in [7] and [12], respectively:
(a) ([7]) α is said to have the Rokhlin property if for every finite set F ⊂ M ,
every ε > 0, there are mutually orthogonal projections {eg | g ∈ G} in M
such that
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii)
g∈G eg = 1.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 9
(b) ([12]) α is said to have the tracial Rokhlin property if for every finite set
F ⊂ M , every ε > 0, every n ∈ N, and every nonzero positive element
x ∈ M , there are mutually orthogonal projections {eg | g ∈ G} in M such
that:
(i) ‖αg(eh)− egh‖ < ε for g, h ∈ G.
(ii) ‖egx− xeg‖ < ε for g ∈ G and all x ∈ F .
(iii) With e =
g∈G eg, the projection 1 − e is Murray-von Neumann
equivalent to a projection in the hereditary subalgebra of M generated
by x.
The following proposition is actually observed in [12, Lemma 1.13], and we put a
proof for reader’s convenience.
Proposition 4.5. Let M be a unital C∗-algebra and α be an action of a discrete
group G on M . Suppose that for every ε > 0 and every finite subset F ⊂ M , there
exist a family of projections {eg}g∈G such that
(1) ‖αg(eh)− egh‖ < ε.
(2) ‖egx− xeg‖ < ε for each x ∈ F .
Then α is an outer action.
Proof. Suppose there is a unitary u ∈ M such that αg(x) = uxu∗ for every x ∈ M
(g 6= ι). Put F = {u} and 0 < ε < 1/4. Then there exist mutually orthogonal
projections {eg}g∈G such that ‖αg(eh)− egh‖ < ε < 1/4. Thus ‖egu− ueg‖ < ε <
1/4. Then ‖αg(eι)− ueιu∗‖ = 0. But
‖αg(eι)− ueιu∗‖ = ‖αg(eι)− eg + eg − eι + eι − ueιu∗‖
≥ ‖eg − eι‖ − ‖αg(eι)− eg‖ − ‖eι − ueι1u∗‖
≥ 1− 1
which is a contradiction. �
Remark 4.6. If M ×αG is simple, α is obviously saturated, and this is the case if G
is a finite group, M is α-simple, and T̃(αg) 6= {1} for all g 6= ι ([8, Theorem 3.1]).
In particular, α is saturated if M is simple and α is outer.
But for a nonsimple M , this may not hold. In fact, if α is an outer action of Zn
on M and u is a unitary in M with un = 1 such that the action Ad(u) on M is not
saturated (as in Example 4.2), then the action α ⊕ Ad(u) on M ⊕M is outer but
not saturated.
Now we show that if α satisfies the Rokhlin property (or satisfies the tracial Rokhlin
property and M has cancellation) then α is saturated. For this we first review the
cancellation property of C∗-algebras. For projections p, q in a C∗-algebra, we write
p ⊥ q if pq = 0, and p ∼ q if they are Murrey-von Neumann equivalent.
Definition 4.7. A unital C∗-algebra M has the cancellation if, whenever p, q, r
are projections in Mn(M) for some n, with p ⊥ r, q ⊥ r, and (p + r) ∼ (q + r),
then p ∼ q.
10 JA A JEONG AND GI HYUN PARK
Remark 4.8. (1) If M has the cancellation and p, q are projections in M such
that (1− p) ∼ (1− q), then p ∼ q ([4, V.2.4.14]).
(2) It is well known that every C∗-algebra with stable rank one has the cancel-
lation ([4, V.3.1.24]).
Proposition 4.9. Let α be an action of a finite group G on a unital C∗-algebra
M . Then α is saturated if one of the following holds.
(i) α has the Rokhlin property.
(ii) α has the tracial Rokhlin property and M has the cancellation.
Proof. (i) For an ε > 0, there exist mutually orthogonal projections {eg}g such that
g eg = 1 and ‖αg(eh)− egh‖ < ε. Then, with m = 1, the elements b1g := eg satisfy
(v) of Theorem 4.1.
(ii) Now suppose α has the tracial Rokhlin property and M has the cancellation.
We shall show that Jα contains the unit of M ×α G. Let 0 < ε < 1. For each
g ∈ G, choose mutually orthogonal projections {eg
}h∈G such that
‖αk(egh)− e
‖ < ε
2|G|2 ,
and put eg :=
h∈G e
. If eg = 1, for some g, then b1h := e
(h ∈ G) will satisfy
(v) of Theorem 4.1 as in (i). If eg 6= 1 for every g ∈ G, then by the tracial Rokhlin
property of α there exist mutually orthogonal projections {fg
}h∈G in M such that
‖αk(fgh)− f
‖ < ε
2|G|2
) ∼ (eg)′ < eg
for a subprojection (eg)′ of eg. Put fg :=
. Then since M has cancellation, it
follows that fg ∼
1−(eg)′
> (1−eg). Let vg ∈ M be a partial isometry satisfying
v∗gvg = f
g, vgv
g = 1− (eg)′,
and set
xg :=
(ekhαg(e
h) + (1 − ek)vkfkhαg(fkhαg−1 (v∗k))
, g ∈ G.
Now we show that the element x :=
xgλg ∈ Jα satisfies ‖x − 1‖ < ε. In fact,
for g 6= ι,
‖xg‖ ≤
‖(ekhαg(ekh) + (1− ek)vkfkhαg(fkhαg−1(v∗k))‖
≤ 1|G|
‖(ekhαg(ekh)‖ + ‖fkhαg(fkh )‖
≤ 1|G|
‖ekh(αg(ekh)− ekgh)‖ + ‖ekhekgh‖+ ‖fkh (αg(fkh )− fkgh)‖+ ‖fkhfkgh‖
≤ 1|G|
2|G|2 +
2|G|2 )
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 11
|G| (
(1− ek)vkfkv∗k)
(1 − ek)(1 − (ek)′)
For the rest of this section we consider a finite group action on a commutative
C∗-algebra C(X). If G acts on a compact Hausdorff space X , it induces an action,
say α, on C(X) by
αg(f)(x) = f(g
−1x), f ∈ C(X).
For each x ∈ X , let Gx = {g ∈ G : gx = x} be the isotropy group of x and for a
subgroup H of G (H < G), put
XH = {x ∈ X : Gx = H}.
It is readily seen that XH and XH′ are disjoint if H 6= H ′, and X is partitioned as
Theorem 4.10. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action of G on C(X), the following are equivalent:
(i) E : C(X) → C(X)α is of index finite type.
(ii) XH is closed for each H < G.
Moreover, if this is case the index of E is Index(E) =
χXH , where χXH
is the characteristic function on XH .
Proof. (i) =⇒ (ii). If E is of index-finite type and {(ui, u∗i )}ki=1 is a quasi-basis for
E, then
uiE(u
i f) = f,
that is,
ui(x)
u∗i (g
−1x)f(g−1x)
= f(x), (11)
for f ∈ C(X) and x ∈ X . For each x ∈ X , choose a continuous function fx ∈ C(X)
satisfying fx|Gx\{x} ≡ 0 and fx(x) = 1. Then (11) with fx in place of f gives
ui(x)
u∗i (g
−1x)fx(g
= fx(x), (12)
and so we have
ui(x)u
i (x) = 1. (13)
12 JA A JEONG AND GI HYUN PARK
To show that each XH is closed, let {xn ∈ XH : n = 1, 2, . . .} be a sequence of
elements in XH with limit x ∈ XH′ . Then (11) gives
fx(xn) =
ui(xn)
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
u∗i (g
−1xn)fx(g
−1xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
ui(xn)
|H |u∗i (xn)fx(xn) +
g 6∈H
u∗i (g
−1xn)fx(g
−1xn)
Taking the limit as n → ∞, we have
fx(x) =
ui(x)
|H |u∗i (x)fx(x) +
g 6∈H
u∗i (g
−1x)fx(g
ui(x)
|H |u∗i (x)fx(x) + |H ′ \H |u∗i (x)fx(x)
|H |+ |H ′ \H |
ui(x)u
i (x)fx(x).
Therefore, comparing with (13), we obtain
|H ′| = |H |+ |H ′ \H |
since Gx = H
′ and fx(x) = 1. Hence
H ⊂ H ′.
On the other hand, since Gxn = H , again by (13),
ui(xn)u
i (xn) = 1 with
the limit
ui(x)u
i (x) = 1 as n → ∞. But also
ui(x)u
i (x) = 1 by
(13), and thus |H | = |H ′|. Consequently we have
H = H ′
because H ⊂ H ′. This shows that XH is closed.
(ii) =⇒ (i). Assume that XH is closed for every subgroup H of G. Then XH
is open since there are only finitely many such subsets. Let UH = {UH,iH : iH =
1, 2, . . . , nH} be an open covering of XH such that
x ∈ UH,iH =⇒ g−1x 6∈ UH,iH or g−1x 6∈ XH whenever g−1x 6= x.
Let {vH,iH} be a partition of unity subordinate to UH . We understand that the
domain of vH,iH is X by assigning 0 to x 6∈ XH . Let uH,iH =
vH,iH .
We claim that
|H | uH,iH ,
|H | u
: H < G, iH = 1, 2, . . . , nH
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 13
is a quasi-basis for E. For f ∈ C(X) and x ∈ X , let F < G and 1 ≤ j ≤ nF be
such that x ∈ XF and x ∈ UF,j. Then
|H | uH,iHE
|H | u
|H |uH,iH (x)
|H |u
(g−1x)f(g−1x)
|F |uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)
u∗F,iF (g
−1x)f(g−1x)
uF,iF (x)|F |u∗F,iF (x)f(x)
vF,iF (x)f(x)
= f(x),
as claimed.
Recall that an action G on X is free if gx 6= x for g ∈ G, g 6= 1, and x ∈ X
Corollary 4.11. Let X be a compact Hausdorff space and G a finite group acting
on X. If α is the induced action on C(X), the following are equivalent.
(i) G acts freely on X.
(ii) E : C(X) → C(X)α is of index-finite type with Index(E) = |G|.
(iii) α is saturated.
Proof. (i) =⇒ (ii) is proved in [19, Proposition 2.8.1].
To show (ii) =⇒ (i), let E be of index-finite type with Index(E) = |G|. Then
from Theorem 4.10, we have
|G| = Index(E) =
|H |χXH ,
which implies that H = {ι} is the only subgroup of G such that XH 6= ∅. Hence
X = X{ι}, that is, G acts freely on X . (ii) ⇐⇒ (iii) comes from Theorem 4.1. �
5. Saturated actions by compact groups
Notation 5.1. Let M be a C∗-algebra and α be an action of a compact group G on
M . For x, y ∈ M , define continuous functions fx,y, fx,1, f1,y ∈ C(G,M) from G to
14 JA A JEONG AND GI HYUN PARK
M as follows:
fx,y(t) = xαt(y),
fx,1(t) = x, f1,y(t) = αt(y) for t ∈ G.
Then it is easily checked that fx,y = fx,1 ∗ f1,y and f∗x,y = fy∗,x∗ .
Recall that C(G,M) is a dense ∗-subalgebra of M ×α G with the multiplication
and involution defined by
f ∗ g(t) =
f(s)αs(g(s
−1t))ds,
f∗(t) = αt(f(t
−1)∗),
where dg is the normalized Haar measure ([13, 7.7], [6, 8.3.1]). Hence if G is a finite
group, fx,y can be written as fx,y =
xαg(y)λg .
If M̃ denotes the smallest unitization of M (so M̃ = M if M is unital), the
function
e : G → M̃, e(s) = 1, for every s ∈ G
is a projection of the multiplier algebra of M ×α G ([17]).
Proposition 5.2. ([17]) Let α be an action of a compact group G on a C∗-algebra
M . Then identifying x ∈ Mα and the constant function in C(G,M) with the value
x everywhere we see that
x 7→ fx,1 : Mα → e(M ×α G)e
is an isomorphism of Mα onto the hereditary subalgebra e(M×αG)e of the crossed
product M ×α G.
The notion of saturated action is introduced by Rieffel for a compact group action
on a C∗-algebra, and we adopt the following equivalent condition as the definition.
Definition 5.3. (Rieffel, see [14, 7.1.9 Lemma]) Let M be a C∗-algebra and α be
an action of compact group G on M . α is said to be saturated if the linear span of
{fa,b | a, b ∈ M} is dense in M ×α G (see Notation 5.1). We denote
Jα = span{fa,b | a, b ∈ M}.
Proposition 5.4. Let α be an action of a compact group G on a C∗-algebra M .
Then Jα is the ideal of M×αG generated by the hereditary subalgebra e(M ×αG)e.
Moreover Jα = span{fa,a∗ ∈ C(G,M) | a ∈ M}.
Proof. We first show that Jα is an ideal ofM×αG. Let x ∈ C(G,M) and a, b ∈ M .
Then x ∗ fa,b ∈ Jα. Indeed,
(x ∗ fa,b)(t) =
x(s)αs(fa,b(s
−1t))ds
x(s)αs(a)αt(b)ds
x(s)αs(a)ds
αt(b),
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 15
hence x∗fa,b = fc,b ∈ Jα, where c =
x(s)αs(a)ds ∈ M . Also f∗a,b = fb∗,a∗ implies
that Jα = Jα∗ is an ideal of M ×α G.
Let J := (M ×α G)e(M ×α G) be the closed ideal generated by e(M ×α G)e.
Now we show that Jα ⊂ J . From
(fa,b ∗ e)(t) =
fa,b(s)αs(e(s
−1t)) ds =
aαs(b) ds = a
αs(b)ds,
we have fa,b ∗ e = faE(b),1, where E(b) =
αs(b)ds ∈ Mα. Hence for a, b, c, and d
in M , we have
fa,b ∗ e ∗ fc,d = (fa,b ∗ e) ∗ (fd∗,c∗ ∗ e)∗
= faE(b),1 ∗ (fd∗E(c∗),1)∗
= faE(b),1 ∗ f1,E(c∗)d
= faE(b),E(c∗)d,
which means that fax,yd ∈ J for any a, d ∈ M and x, y ∈ Mα. Since Mα contains
an approximate identity for M , it follows that fa,b ∈ J for a, b ∈ A.
For the converse inclusion J ⊂ Jα, note that if x ∈ C(G,M), then (x ∗ e)(t) =
x(s)ds for t ∈ G. With notations x′ =
x(s)ds and x′′ :=
αs(x(s
−1))ds (∈ M),
we see that
(x ∗ e ∗ y)(t) =
(x ∗ e)(s)αs(y(s−1t))ds
αs(y(s
−1t))ds
= x′αt(
αs(y(s
−1))ds)
= x′αt(y
= fx′,y′′(t)
belongs to Jα for x, y ∈ C(G,M).
Finally the following polarization identity proves the last assertion.
aαt(b) =
ik(b + ika∗)∗αt(b+ i
ka∗).
6. The gauge action γ on a graph C∗-algebra
By a (directed) graph E we mean a quadruple E = (E0, E1, r, s) consisting of
the vertex set E0, the edge set E1, and the range, source maps r, s : E1 → E0. If
each vertex of E emits only finitely many edges E is called row finite and a row
finite graph E is locally finite if each vertex receives only finitely many edges. By
En we denote the set of all finite paths α = e1 · · · en (r(ei) = s(ei+1), 1 ≤ i ≤ n−1)
of length n (|α| = n). Each vertex is regarded as a finite path of length 0. Then
E∗ = ∪n≥0En is the set of all finite paths and the maps r and s naturally extend
to E∗. A vertex v is called a sink if s−1(v) = ∅ and a source if r−1(v) = ∅.
If E is a row finite graph, we call a family {se, pv | e ∈ E1, v ∈ E0} of operators a
Cuntz-Krieger(CK) E-family if {se}e are partial isometries and {pv}v are mutually
16 JA A JEONG AND GI HYUN PARK
orthogonal projections such that
s∗ese = pr(e) and pv =
s(e)=v
e if s
−1(v) 6= ∅.
It is now well known that there exists a C∗-algebra C∗(E) generated by a universal
CK E-family {se, pv | e ∈ E1, v ∈ E0}, in this case we simply write C∗(E) =
C∗(se, pv). For the definition and basic properties of graph C
∗-algebras, see, for
example, [1, 2, 10, 11, 15] among others. If α = α1α2 · · ·α|α| (αi ∈ E1) is a finite
path, by sα we denote the partial isometry sα1sα2 · · · sα|α| (sv = s∗v = pv, for v ∈
We will consider only locally finite graphs and it is helpful to note the following
properties of graph C∗-algebras.
Remark 6.1. (i) Let C∗(E) = C∗(se, pv) be the graph C
∗-algebra associated
with a row finite graph E, and let α, β ∈ E∗ be finite paths in E. Then
s∗αsβ =
s∗µ, if α = βµ
sν , if β = αν
0, otherwise.
Therefore C∗(E) = span{sαs∗β | α, β ∈ E∗}.
(ii) Note that sαs
β = 0 for α, β ∈ E∗ with r(α) 6= r(β).
(iii) If α, β, µ, and ν in En are the paths of same length,
β)(sµs
ν) = δβ,µsαs
Thus for each n ∈ N and a vertex v in a locally finite graph E, we see that
span{sαs∗β | α, β ∈ En and r(α) = r(β) = v}
is a ∗-algebra which is isomorphic to the full matrix algebraMm = (Mm(C)),
where m =
∣{α ∈ En | r(α) = v}
Recall that the gauge action γ of T on C∗(E) = C∗(se, pv) is given by
γz(se) = zse, γz(pv) = pv, z ∈ T.
γ is well defined by the universal property of the CK E-family {se, pv}. Since
γz(sαs
β)dz =
z|α|−|β|(sαs
β)dz = 0, |α| 6= |β|,
one sees that
C∗(E)γ = span{sαs∗β | α, β ∈ E∗, |α| = |β|}.
If Z denotes the following graph:
• • • • •______ // ______ // ______ //______ // · · · ,· · ·Z :
−2 −1 0 1 2
then C∗(Z) is isomorphic to the C∗-algebra K of compact operators on an infinite
dimensional separable Hilbert space, hence C∗(Z) is itself a simple AF algebra. But
C∗(Z)γ coincides with the commutative subalgebra span{sαs∗α | α ∈ Z∗} which is
far from being simple, and thus we know that the simplicity of C∗(E) does not
imply that of C∗(E)γ in general.
SATURATED ACTIONS BY FINITE DIMENSIONAL HOPF ∗-ALGEBRAS 17
In [9], the Cartesian product of two graphs E and F is defined to be the graph E×
F = (E0 ×F 0, E1 ×F 1, r, s), where r(e, f) = (r(e), r(f)) and s(e, f) = (s(e), s(f)).
Since the graph Z × E has no loops for any row-finite graph E, we know that
C∗(E)γ is an AF algebra ([10]) by the following proposition.
Proposition 6.2. ([9]) Let E be a row finite graph with no sources. Then the
following hold:
(a) C∗(E)γ is stably isomorphic to C∗(E)×γ T.
(b) C∗(E)×γ T ∼= C∗(Z× E).
Now we show that a gauge action is saturated. For this, note that the linear
span of the continuous functions of the form
t 7→ f(t)x, f ∈ C(G), x ∈ A
is dense in C(G,A) [13, 7.6.1]. Hence by Remark 6.1(i), one sees that
C∗(E)×γ T = span{znsαs∗β | α, β ∈ E∗ n ∈ Z}. (15)
Theorem 6.3. Let E be a locally finite graph with no sinks and no sources. Then
the gauge action γ on C∗(E) is saturated.
Proof. We show that Jγ = C∗(E)×γ T. By (15) it suffices to see that
znsαs
β ∈ Jγ for all α, β ∈ E∗, n ≥ 0
(because z−nsαs
β = (z
∗ for n ≥ 0).
Now fix α, β ∈ E∗ and n ≥ 0. Put l = n− (|α| − |β|). There are two cases.
(i) l ≥ 0: One can choose a path µ such that |µ| = l and r(µ) = s(α). Then
znsαs
β = z
l+|α|−|β|s∗µsµsαs
β = s
µγz(sµαs
β) = fs∗µ,sµαs∗β (z),
where the function fs∗µ,sµαs∗β belongs to Jγ .
(ii) l < 0: Choose a path ν with |ν| = |β|+n and r(ν) = r(α). With a = sαs∗ν ,
b = sνs
β , we have fa,b ∈ Jγ and
znsαs
β = sαs
νγz(sνs
β) = fa,b(z).
Acknowledgements. The first author would like to thank Hiroyuki Osaka and Ta-
motsu Teruya for valuable discussions.
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Keywords: Finite dimensional Hopf ∗-algebra; saturated action; conditional ex-
pectation of index-finite type.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul
National University, Seoul, 151–747, Korea
E-mail address: [email protected]
Department of Mathematics, Hanshin University, Osan, 447–791, Korea
E-mail address: [email protected]
1. Introduction
2. Preliminaries
3. Actions by finite dimensional Hopf *-algebras
4. Actions by finite groups
5. Saturated actions by compact groups
6. The gauge action on a graph C*-algebra
References
|
0704.1550 | The electronic structures, the equilibrium geometries and finite
temperature properties of Na_n (n=39-55) | The electronic structures, the equilibrium geometries and finite temperature
properties of Na
(n=39-55)
Shahab Zorriasatein1,2, Mal-Soon Lee1, and D. G. Kanhere1
Department of Physics, and Center for Modeling and Simulation,
University of Pune, Ganeshkhind, Pune–411 007, India
Department of Physics, Islamic Azad University, Tehran south branch, Tehran, Iran
Density-functional theory has been applied to investigate systematics of sodium clusters Nan in the
size range of n= 39-55. A clear evolutionary trend in the growth of their ground-state geometries
emerges. The clusters at the beginning of the series (n=39-43) are symmetric and have partial
icosahedral (two-shell) structure. The growth then goes through a series of disordered clusters
(n=44-52) where the icosahedral core is lost. However, for n ≥53 a three shell icosahedral structure
emerges. This change in the nature of the geometry is abrupt. In addition, density-functional
molecular dynamics has been used to calculate the specific heat curves for the representative sizes
n= 43, 45, 48 and 52. These results along with already available thermodynamic calculations for n=
40, 50, and 55 enable us to carry out a detailed comparison of the heat capacity curves with their
respective geometries for the entire series. Our results clearly bring out strong correlation between
the evolution of the geometries and the nature of the shape of the heat capacities. The results also
firmly establish the size-sensitive nature of the heat capacities in sodium clusters.
PACS numbers: 31.15.Ew, 31.15.Qg, 36.40.Ei, 36.40.Qv
I. INTRODUCTION
Physics and chemistry of clusters are very active ar-
eas of research especially because of the emergence of
nano science and nano technology.1 Although major ef-
forts have been spent into ground-state investigations,
finite temperature properties are turning out to be very
interesting. Such investigations are challenging, both ex-
perimentally as well as theoretically. One of the first de-
tailed measurements providing much impetus for theoret-
ical work was on free sodium clusters by Haberland and
co-workers.2 These measurements reported the melting
temperatures (Tm) of sodium clusters in the size range
between 55 and 350 and remained unexplained for almost
about a decade.
The main puzzle was related to the irregular behav-
ior of the melting temperature and the absence of any
correlation between the peaks and the magic numbers ei-
ther geometric or electronic. A good deal of simulation
works has been carried out to explain the sodium data,
most of the early work being with classical inter atomic
potentials.3,4 It turned out that none of these could ob-
tain qualitative and quantitative agreement with the ex-
perimental data. Thus, it needed an ab initio density-
functional method to achieve this. Indeed, much insight
and excellent quantitative agreement has been obtained
by density-functional molecular dynamics (DFMD) sim-
ulations.5,6,7,8,9,10
Recently a very different aspect of finite temperature
behavior has been brought out by the experimental and
later by theoretical work on gallium clusters.11,12 The
experimental reports of Breaux and co-workers11 showed
that in the size range of N=30 to 55, free clusters of gal-
lium melt much above their bulk melting temperature.
Interestingly, their experiment also showed that the na-
ture of heat capacity is size sensitive. In fact, addition of
even one atom changes the shape of the heat capacity dra-
matically, e.g for Ga30 and Ga31. A similar experimental
observation has been reported for aluminum clusters in
the size range n=49-62.13 Such a size sensitive behavior
has also been observed in DFMD simulation of Au19 and
Au20.
14 A detailed analysis of the ground-state geome-
tries of these clusters brought out the role of order and
disorder in their geometries on the shape of the melting
curve. A disordered system is shown to display a con-
tinuous melting transition leading to a very broad heat
capacity curve. However, the effect is subtle and descrip-
tion of order and disorder needs careful qualifications.
In spite of substantial experimental works on sodium
clusters over a period of 10 years and or so, there is no
firm and systematic evidence of size sensitivity. This is
mainly due to the fact that the reported experimental
data2 is for the size range of n=55-350 at discrete sizes.
It is necessary to investigate the effect of addition of few
atoms in a continuous manner in appropriate size range.
However, it is not clear whether larger clusters having
sizes of n >100 will also show this effect.
An extensive ab initio study on the structural proper-
ties of small Na clusters up to N=20 has been reported
by Röthlisberger and Andreoni.15 The study reveals that
pentagonal motifs dominate the structures above N=7.
As expected most of the atoms in these clusters lie on
the surface and a discernible core develops after about
N=15-16. The shapes after this sizes show signature of
icosahedral structures.
The finite temperature behavior of sodium clusters in
the size range of n=8 to 55 has been reported.7 The
study reveals that it is not easy to discern any melt-
ing peaks below n=25. However, the simulation data
available at rather coarse sizes above n=40 already shows
size-sensitive feature. In addition to this feature, our re-
cent investigation8 on Na57 and Na58 does bring out the
http://arxiv.org/abs/0704.1550v1
role of geometry and electronic structure on the melt-
ing. Nevertheless, in order to draw definitive conclusions
it is necessary to investigate the effect of addition of a
few atoms in a continuous manner in appropriate size
range. Therefore, we have chosen the size range of n=39-
55 and have carried out detailed density-functional inves-
tigations. The purpose of the present work is two fold.
First, to obtain reliable equilibrium geometries for all the
clusters in the size range of n=39-55 and to discern evo-
lutionary trends. We note that Na40 has a symmetric
partially icosahedron core and Na55 is a complete icosa-
hedron. Thus it is of considerable interest to examine
the growth pattern from n=39 to n=55. The second
purpose is to seek correlation between the nature of the
ground-state and the evolutionary trends observed in the
nature of their specific heats. Towards this end we have
carried out extensive finite temperature simulations on
representative clusters of size n= 43, 45, 48, and 52.
Together with the already published results, this gives
us access to specific heats for Na40, Na43, Na45, Na48,
Na50, Na52 and Na55, a reasonable representation across
the series under investigation. Finally, we note that all
the DFMD simulations reported so far have yielded ex-
cellent agreement with the experimental data6,8,9 These
reports demonstrate the reliability of density-functional
molecular dynamics in describing the finite temperature
properties.
The plan of the paper is as follows. In the next sec-
tion (Sec. II) we note the computational details. Sec. III
presents equilibrium geometries and their shape system-
atic. Sec. IV presents the finite temperatures behavior of
Na43, Na45, Na48, Na52 and finally we discuss the corre-
lation between the ground states and nature of the spe-
cific heats for all the available thermodynamics data. We
conclude our discussion in Sec. V.
II. COMPUTATIONAL DETAILS
We have carried out Born-Oppenheimer molecular dy-
namics (BOMD) simulations16 using Vanderbilt’s ultra-
soft pseudopotentials17 within the local-density approxi-
mation (LDA), as implemented in the VASP package.18
We have optimized about 300 geometries for each of
the sodium clusters in the size range between n=39 and
n=55. The initial configuration for the optimization of
each cluster were obtained by carrying out a constant
temperature dynamics simulation of 60 ps each at vari-
ous temperatures between 300 to 400 K. For many of the
geometries we have also employed basin hopping19 and
genetic20,21 algorithms using Gupta potential4 for gen-
erating initial guesses. Then we optimized these struc-
tures by using the ab initio density-functional method.22
For computing the heat capacities, the iso-kinetic BOMD
calculations were carried out at 14 different temperatures
for each cluster of Na43, Na45, Na48, Na52 in the range
between 100 K and 460 K, each with the time dura-
tion of 180 ps or more. Thus, it results in the total
simulation time of 2.5 ns per system. In order to get
converged heat capacity curve especially in the region
of co-existence, more temperatures were required with
longer simulation times. We have discarded at least first
30 ps for each temperature for thermalization. To ana-
lyze the thermodynamic properties, we first calculate the
ionic specific heat by using the Multiple Histogram (MH)
technique.23,24 We extract the classical ionic density of
states (Ω(E)) of the system, or equivalently the classi-
cal ionic entropy, S(E) = kB lnΩ(E), following the MH
technique. With S(E) in hand, one can evaluate ther-
modynamic averages in a variety of ensembles. We focus
in this work on the ionic specific heat. In the canon-
ical ensemble, the specific heat is defined as usual by
C(T ) = ∂U(T )/∂T , where U(T ) =
E p(E, T ) dE is
the average total energy. The probability of observing
an energy E at a temperature T is given by the Gibbs
distribution p(E, T ) = Ω(E) exp(−E/kBT )/Z(T ), with
Z(T ) the normalizing canonical partition function. We
normalize the calculated canonical specific heat by the
zero-temperature classical limit of the rotational plus vi-
brational specific heat, i.e., C0 = (3N − 9/2)kB.
We have calculated a number of thermodynamic indi-
cators such as root-mean-square bond length fluctuations
(δrms), mean square displacements (MSD) and radial dis-
tribution function (g(r)). The δrms is defined as
δrms =
N(N − 1)
(〈r2ij〉t − 〈rij〉
〈rij〉t
, (1)
where N is the number of atoms in the system, rij is
the distance between atoms i and j, and 〈. . .〉t denotes
a time average over the entire trajectory. MSDs for in-
dividual atoms is another traditional parameter used for
determining phase transition and is defined as,
〈r2I(t)〉 =
[RI(t0m + t)−RI(t0m)]
where RI is the position of the I
th atom and we aver-
age over M different time origins t0m spanning over the
entire trajectory. The interval between the consecutive
t0m for the average was taken to be about 1.5 ps. The
MSDs of a cluster indicate the displacement of atoms in
the cluster as a function of time. The g(r) is defined
as the average number of atoms within the region r and
r + dr.
We have also calculated the shape deformation param-
eter (εdef ), to analyze the shape of the ground state for
all the clusters. The εdef is defined as,
εdef =
Qy +Qz
, (3)
where Qx ≥ Qy ≥ Qz are the eigenvalues, in descend-
ing order, of the quadrupole tensor
39 40 41
42 43
FIG. 1: The ground-state geometries of Nan (n=39-43).
44 45 46
47 48 49
50 51 52
FIG. 2: The ground-state geometries of Nan (n=44-52).
Qij =
RIiRIj . (4)
Here i and j run from 1 to 3, I runs over the number
of ions, and RIi is the i
th coordinate of ion I relative
to the center of mass (COM) of the cluster. A spherical
system (Qx = Qy = Qz) has εdef=1 and larger values of
εdef indicates deviation of the shape of the cluster from
sphericity.
53 54 55
FIG. 3: The ground-state geometries of Nan (n=53-55).
5551474339
Size (No. of Atoms)
FIG. 4: The shape deformation parameter for Nan (n=39-55)
as a function of cluster size.
III. GEOMETRIES
The lowest energy geometries of sodium clusters (Nan
n=39-55) are shown in Fig. 1 (n=39-43), Fig. 2 (n=44-
52), and Fig. 3 (n=53-55). We have also plotted the
shape deformation parameter εdef and the eigenvalues
of quadrupole tensor for the ground state geometries of
these clusters in Fig. 4 and Fig. 5, respectively.
It is convenient to divide these clusters into three
groups. The clusters in the first group, shown in Fig. 1
are nearly spherical. The ground-state geometry for Na39
555351494745434139
Cluster Size
FIG. 5: The eigenvalues of the quadrupole tensor for Nan
(n=39-55) as a function of cluster size.
40200
40200
6040200
° A
Atom number
FIG. 6: The distance from the center of mass for each of the atoms ordered in the increasing fashion for Nan (n=39-55). The
formation of the shells is evident from the sharp steps.
is highly symmetric. This structure has three identical
units, each based on the icosahedral motive and these
three units are arranged as shown in the Fig. 1. It is
interesting to note that an addition of an extra atom
changes the structure dramatically. It can be seen that
the geometry of Na40 is based on icosahedral structure
with missing 12 corner atoms and (111) facet as reported
by Rytkönen et al .5 An extra atom added to this struc-
ture is accommodated near the surface and deforms the
structure slightly. In addition to the deformation the dis-
tance between the two shells is reduced by 0.3 Å as com-
pared to that in Na40. A single atom added to Na41 is not
accommodated in the structure, instead it caps the sur-
face. However, the low-lying geometries for Na42 have a
spherical shape without any cap (figure not shown). The
lowest-energy structure of Na43 shows two caps symmet-
rically placed on the opposite side of Na41, accompanied
by distortion of the icosahedral core. The second group in
Fig. 2 consisting of the clusters with n=44-52 shows sub-
stantial distortion of the icosahedral core and even loss
of this core structure. These clusters essentially repre-
sent the transition region from the two shell icosahedron,
Na40 to three shell complete icosahedron, Na55. It can
be seen that by adding one atom to Na43 the two shell
core is destroyed. The growth from n=44 to n=52 shows
successive stages of capping followed by rearrangement in
the inner core. There is a dramatic change in the struc-
ture as soon as we add one more atom to Na52. All the
atoms rearrange to form an icosahedral structure as seen
in Na53. Thus, the clusters in the last group namely,
Na53 and Na54 differ from a perfect icosahedron of Na55
by an absence of two and one atom(s), respectively (Fig.
The nature of the changes in the shape of the clus-
ters during the growth can be seen in Fig. 4 and Fig.
5. The shape deformation parameter (εdef ) increases to
a value about 2 up to Na52 with slightly higher values
for Na45 and Na49 (Fig. 4). However, this value drops
suddenly for Na53. It is interesting to examine the three
eigenvalues of quadrupole tensor (Qx, Qy, Qz) shown in
Fig. 5. It can be seen that two of the eigenvalues are
nearly same up to Na52 while the third one continuously
grows and indicates that the growth dominantly takes
place along one of the directions. A prolate configura-
tion has Qz ≫ Qx ≈ Qy. Thus, the majority of the
clusters in the second group are prolate. The formation,
the destruction and reformation of the shell structure is
clearly seen in Fig. 6. In this figure we have plotted the
420340260180100
Temperature (K)
420340260180100
Temperature (K)
FIG. 7: (a) The normalized heat capacity and (b) the δrms
for Na43. The peak in heat capacity curve is at 270 K.
∆E=0.033 eV ∆E=0.067 eV
(a) (b)
FIG. 8: Two low lying isomers of Na43. ∆E represents the
energy difference with respect to the ground-state.
distance of each atom from the center of mass arranged
in the increasing fashion. Clearly small rearrangement of
atoms yields a change in the structure from Na39 to Na40.
The two shell structures are observed till Na43. The for-
mation of the shell structure is reflected in the formation
of the sharp steps in the graph. As the size increases
from Na43 to Na44 the shell structure is destroyed and
seen again at Na53. Thus, three shells begin to emerge
at Na53.
IV. THERMODYNAMICS
We have calculated the ionic heat capacity and indi-
cators like mean square displacements (MSD) and root-
mean-square bond length fluctuations (δrms) for four
of the representative clusters in the investigated series
which are Na43, Na45, Na48, and Na52.
FIG. 9: The radial distribution function calculated for Na43
at five different temperatures.
We note that the thermodynamics of Na40
5,7, Na50
and Na55
6,9 has already been reported. Thus it is pos-
sible to examine and analyze the systematic variations
in the melting characteristic and correlate them with the
equilibrium geometries across the entire range of sizes
from 40 to 55. The heat capacity and δrms for Na43 are
shown in Figs. 7(a) and 7(b), respectively. We also show
typical low energy geometries (isomers) for Na43 in Fig.
8. The first isomer shown in Fig. 8(a) has two atoms clos-
est to each other capping the surface and second isomer
shows a distorted shape and no caps. The heat capacity
shows a weak peak around 160 K while the main peak
occurs at 270 K. An examination of the motion of ionic
trajectory seen as a movie indicates that isomerization
(Fig. 8(a)) is responsible for the weak peak.
It is interesting to observe the changes of radial dis-
tribution function (RDF) as a function of temperature
which is shown in Fig. 9. At low temperatures the shell
structure is clearly evident. The pattern seen at 180 K
and 210 K are mainly due to the fluctuation of the cluster
between the ground-state and low-lying states. At 300 K
and above the RDF shows the typical melting behavior
of a cluster. The δrms in Fig. 7(b) shows the effect of iso-
merization around 160 K. It can be seen that the melting
420340260180100
Temperature (K)
420340260180100
Temperature (K)
FIG. 10: (a) The normalized heat capacity and (b) the δrms
for Na45. The heat capacity curve shows a two stages melting
process.
150120906030
Time (ps)
150120906030
Time (ps)
FIG. 11: The MSDs of individual atoms calculated for Na45
(a) at 210 K and (b) at 225 K over the last 150 ps.
region is of the order of 60 K.
Let us recall that Na45, Na48 and Na52 belong to the
second class of disordered clusters. Among these Na45
shows some signature of partial shells. The heat capacity
and δrms for Na45 are shown in Figs. 10(a) and 10(b),
respectively. The heat capacity of Na45 shows a first
420340260180100
Temperature (K)
420340260180100
Temperature (K)
420340260180100
Temperature (K)
420340260180100
Temperature (K)
FIG. 12: (a) The normalized heat capacity and (b) the δrms
for Na48, (c) The normalized heat capacity and (d) the δrms
for Na52.
peak around 230 K and a second peak around 300 K. We
also show the MSDs of individual atoms at 210 and 225 K
in Figs. 11(a) and 11(b), respectively. We have observed
the ionic trajectories as a movie in the temperature at
225 K. It turns out that about one third of atoms in the
cluster “melt” at this temperature. This is brought out
by the contrasting behavior of the MSDs as shown in
Figs. 11(a) and 11(b). Interestingly, all these 15 atoms
are on the surface, indicating that surface melting takes
place first. The peak around 225 K in the heat capacity
400200
Temperature (K)
FIG. 13: The normalized heat capacity as a function of tem-
perature for Nan, n=40, 43, 45, 48, 50, 52, and 55.
is due to partial melting of these surface atoms. We show
the δrms as a function of the temperature in Fig. 10(b).
There is a sharp increases in δrms around 210 K and a
slow rise after 240 K consistent with the behavior of the
heat capacity. Thus in Na45 melting takes place in two
stages over the range of 120 K.
The heat capacity and δrms for Na48 and Na52 are
shown in Fig. 12. The heat capacity of Na48 (Fig. 12(a))
is very broad indicating almost continuous phase change
starting around 150 k. Thus it is difficult to identify a
definite melting temperature. A similar behavior is also
seen in that of Na52 except for the small peak seen 180 K
due to isomerization (Fig. 12(c)). It is interesting to note
that δrms for both clusters also show a gradual rise from
about 150 K to 350 K indicating a continuous melting
transition as shown in Figs. 12(b) and 12(d).
The systematic evolution of heat capacities can be bet-
ter appreciated by examination of all the available data
(calculated from density-functional simulation). In Fig.
13 we show the specific heats for all the available clusters
between Na40 and Na55. The most symmetric cluster
Na55 shows the sharpest peak in the heat capacity. The
heat capacity of Na40 and Na43 (partial icosahedral struc-
tures) have well recognizable peaks which are broader
than that of Na55. The disordered phase of the growth
is clearly reflected in the very broad heat capacities seen
around n=50.
V. SUMMARY AND CONCLUSION
The ab initio density-functional method has been ap-
plied to investigate systematic evolutionary trends in the
ground state geometries of the sodium clusters in the size
range of n=39-55. The DFMD finite temperature simula-
tions have been carried out for representative clusters. A
detailed comparison between the heat capacities and the
geometries firmly establishes a direct influence of the ge-
ometries on the shapes of the heat capacity curves. The
heat capacities show size sensitivity. The growth pattern
shows a transition from ordered −→ disordered −→ or-
dered sequence. The corresponding heat capacities show
a transition from peaked to a very broad to peaked se-
quence. It is seen that addition of a few atoms changes
the shape of heat capacity very significantly. We believe
that the size sensitive feature seen our simulation is uni-
versal. It may be noted that such a feature has been
observed experimentally in the case of gallium and alu-
minum and in the case of DFMD simulations for gold.
We await the experimental measurements of the heat ca-
pacities on the sodium clusters in these range showing
the size sensitivity.
VI. ACKNOWLEDGMENT
We acknowledge partial assistance from the Indo-
French center for Promotion of Advance Research (In-
dia)/ Center Franco-Indian pour la promotion de la
Recherche Avancée (France) (IFC-PAR, project No;
3104-2). We would like to thank Kavita Joshi and Sailaja
Krishnamurty for a number of useful discussions.
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http://arxiv.org/abs/cond-mat/0612287
|
0704.1551 | Quantum Zeno Effect in the Decoherent Histories | arXiv:0704.1551v1 [quant-ph] 12 Apr 2007
Quantum Zeno Effect in the Decoherent Histories
Petros Wallden∗
Abstract
The quantum Zeno effect arises due to frequent observation. That implies
the existence of some experimenter and its interaction with the system. In this
contribution, we examine what happens for a closed system if one considers
a quantum Zeno type of question, namely what is the probability of a system,
remaining always in a particular subspace. This has implications to the arrival
time problem that is also discussed. We employ the decoherent histories approach
to quantum theory, as this is the better developed formulation of closed system
quantum mechanics, and in particular, dealing with questions that involve time
in a non-trivial way. We get a very restrictive decoherence condition, that implies
that even if we do introduce an environment, there will be very few cases that
we can assign probabilities to these histories, but in those cases, the quantum
Zeno effect is still present.
1 Motivation
A remarkable property of quantum mechanics, is the so called quantum Zeno effect
[1]. This effect, is that frequent observation slow down the evolution of the state,
with the limit of continuous observation leading to “freezing” of the state1. This
has been experimentally verified. The intuitive explanation, is that the interaction of
the observer with the system leads to this apparent paradox. It would therefore be
interesting to see whether this effect persists if we consider a closed system. We would
try to see what is the probability of a closed system remaining in a particular subspace
of its Hilbert space with no external observer. This directly relates to the arrival time
problem as well (e.g. [2, 3]). Having said that, we should emphasize that in closed
systems, we cannot in general assign probabilities to histories, unless they decohere
and it is this property that resolves the apparent paradox that arises.
2 This paper
This contribution is largely based on Ref.[3]. In Section 3 we revise the quantum Zeno
effect and the decoherent histories, and introduce a new formula for the restricted
propagator that will be of use further. In Section 4.1 we see what probabilities we
would get if we had decoherence, that highlights the persistence of the quantum Zeno
effect. In Section 4.2 we get the decoherence condition that in Section 5 is stressed
how restrictive is by considering the arrival time problem. We conclude in Section 6.
∗Raman Research Institute, Theoretical Physics Group; Sadashivanagar, Bangalore - 560 080,
India; on leave from: Imperial College, Theoretical Physics Group; Blackett Laboratory, London
SW7 2BZ, UK; email: [email protected]
1To be more precise, restriction to a subspace.
http://arxiv.org/abs/0704.1551v1
3 Introductory material
3.1 Quantum Zeno effect
In standard Copenhagen quantum mechanics, the measurement is represented by pro-
jecting the state to a subspace defined by the eigenstates that correspond to the range
of eigenvalues of the measured physical quantity. The latter is represented by a self-
adjoint operator. The state, otherwise evolves unitarily: Û(t) = exp(−iĤt), where
Ĥ is the Hamiltonian. It is then a mathematical fact, that frequent measurement, of
the same quantity (subspace) leads to slow down of the evolution, i.e. decreases the
probability that the state evolves outside the subspace in question. This resembles the
ancient Greek, Zeno paradox (Zήνων), and thus the name.
The continuum measurement limit, leads to zero probability of leaving the observed
subspace. The state continues to evolve (unitarily), but restricted in the subspace of
observation [4]. This implies that if we project to a one-dimensional subspace, the state
stops evolving. In most literature, the question is of a particle decaying or not, so the
last comment applies. In particular, the above phenomenon is still present for infinite
dimensional Hilbert spaces, but provided that the restricted Hamiltonian (Hr = PHP )
is self-adjoint, as we will see later.
3.2 Decoherent histories
Decoherent histories approach to quantum theory is an alternative formulation de-
signed to deal with closed systems and it was developed by Griffiths [5], Omnès [6],
and Gell-Mann and Hartle [7]. There is no external observer, no a-priori environment-
system split. The main mathematical aim of this approach, is to see when is it mean-
ingful to assign probabilities to a history of a closed quantum system and of course to
determine this probability.
Here we will revise the standard non-relativistic quantum mechanics in decoherent
histories formulation. To each history (α) corresponds a particular class operator Cα,
Cα = Pαne
−iH(tn−tn−1)Pαn−1 · · · e
−iH(t2−t1)Pα1 (1)
Where Pα1 etc are projection operators corresponding to some observable, H is the
Hamiltonian, and tn is the total time interval we consider. This class operator corre-
sponds to the history, the system is at the subspace spanned by Pα1 at time t1 at Pα2
at time t2 and so on. The probability for this history, provided we had some external
observer making the measurement at each time tk would be
p(α) = D(α, α) = Tr(CαρC
α) (2)
where ρ is the initial state. In the case of a closed system, Eq.(2) fails in general to be
probability due to interference2.
There are, however, certain cases where we can assign probabilities. This happens
if for a complete set of histories, they pairwise obey
D(α, β) = Tr(CαρC
) = 0 ∀ α 6= β (3)
In that case, the complete set of histories is called decoherent set of histories and we
can assign to each history of this set the probability of Eq.(2). In order to achieve a set
2The additivity of disjoint regions of the sample space is not satisfied by Eq.(2)
of histories that satisfy Eq.(3) in general we need to consider coarse grained histories,
or/and very specific initial state ρ3.
To sum up, in decoherent histories we need to first construct a class operators that
corresponds to the histories of interest4, and then confirm that these histories satisfy
Eq.(3). Only then we can give an answer.
3.3 The restricted propagator
A mathematical object that will be needed for computing the suitable class operators,
is the restricted propagator. This is the propagator restricted to some particular region
∆ (of the configuration space) that corresponds to a subspace of the total Hilbert space
denoted by H∆. The most common (but not the most general) is the path integral
definition:
gr(x, t | x0, t0) =
Dx exp(iS[x(t)]) = 〈x|gr(t, t0)|x0〉 (4)
The integration is done over paths that remain in the region ∆ during the time interval
[t, t0]. The S[x(t)] is as usual the action. The operator form of the above is given by
[8, 9]:
gr(t, t0) = lim
Pe−iH(tn−tn−1)P · · ·Pe−iH(t1−t0)P (5)
With tn = t, δt → 0 and n → ∞ simultaneously keeping δt × n = (t − t0). H is
the Hamiltonian operator. P is a projection operator on the restricted region ∆. We
therefore have
gr(x, t | x0, t0) = 〈x|gr(t, t0)|x0〉 (6)
Note here that the expression Eq.(5) is the defining one for cases that the restricted
region is not a region of the configuration space, but some other subspace of the total
Hilbert space H. The differential equation obeyed by the restricted propagator is:
−H)gr(t, t0) = [P,H ]gr(t, t0) (7)
Which is almost the Schrödinger equation, differing by the commutator of the projec-
tion to the restricted region with the Hamiltonian.
The most useful form, for our discussion was derived in Ref. [3]
gr(t, t0) = P exp (−i(t − t0)PHP )P (8)
Note that PHP is the Hamiltonian projected in the subspace H∆. To prove Eq.(8) we
multiply Eq. (7) with P we will then get
− PHP )gr(t, t0) = 0 (9)
using the fact that P [H,P ]P = 0 and that the propagator has a projection P at the
final time. This is Schrödinger equation with Hamiltonian PHP . It is evident that
3Note that the interaction of a system with an environment that brings decoherence, in the histories
vocabulary, is just a particular type of coarse graining where we ignore the environments degrees of
freedom.
4Note that the same classical question can be turned to quantum with several, possibly inequivalent
ways. Due to this property, the construction of the suitable class operator is important for questions
such as for example, the arrival time or reparametrization invariant questions.
this leads to the full propagator in H∆ provided that the operator PHP is self-adjoint
in this subspace [4]5.
4 Quantum Zeno histories
In this section we will examine the question what is the probability for a system to
remain in a particular subspace, during a time interval ∆t = t − t0. We will see the
probabilities and decoherence conditions for the general case, and then see what this
implies for the arrival time problem, which is just a particular example.
4.1 The class operator and probabilities
There are several ways of turning the above classical proposition to a quantum me-
chanical one. The most straight forward is the following. We consider a system being
in one subspace by projecting to that, and the history of always remaining in that
subspace corresponds to the limit of projecting to the region evolving unitarily but for
infinitesimal time and then projecting again, i.e. taking the δt between the proposi-
tions going to zero. The class operator for remaining always in that subspace follows
from Eq.(1) by taking each Pαk being the same (P ) and taking the limit of (tk − tk−1)
going to zero for each k. We then have
Cα(t, t0) = gr(t, t0) (10)
and the class operator for not remaining at this subspace during all the interval is
naturally
Cβ(t, t0) = g(t, t0)− gr(t, t0) (11)
with g(t, t0) = exp(−iH(t− t0)) the full propagator.
Let us, for the moment, assume that the initial state |ψ〉 is such, that we do
have decoherence. We will return later to see when this is the case. The (candidate)
probability is
p(α) = 〈ψ|g†r(t, t0)gr(t, t0)|ψ〉 (12)
Following Eq.(8) it is clear6 that
g†r(t, t0)gr(t, t0) = P (13)
which then implies
p(α) = 〈ψ|P |ψ〉 (14)
For an initial state that is in the subspace defined by P , the probability to remain in this
subspace is one. This is the usual account of the quantum Zeno effect. As it is stressed
in other literature, to have the quantum Zeno is crucial that the restricted Hamiltonian
Hr = PHP to be self-adjoint operator in the subspace. Note, that this only states
that the system remain in the subspace, but it does not “freeze” completely and in
particular follows unitary evolution in the subspace with Hamiltonian, the restricted
5A detailed proof from Eq.(5) can be found in [3].
6Provided PHP is self-adjoint in the subspace. This is true for finite dimensional Hilbert spaces
and has been shown to be true for regions of the configuration space in a Hamiltonian with at most
quadratic momenta [4].
one Hr. The form of Eq.(8) of the restricted propagator makes the latter comment
more transparent.
4.2 Decoherence condition
All this is well understood for open systems with external observers. To assign the
candidate probability of Eq.(12) as a proper probability of a closed system, we need
the system to obey the decoherence condition, i.e.
D(α, β) = 〈ψ|C
βCα|ψ〉 = 0 (15)
and this implies that
〈ψ|g†r(t, t0)g(t, t0)|ψ〉 = 〈ψ|P |ψ〉 (16)
which is a very restrictive condition and only very few states satisfy this, as we will
see in the arrival time example. The condition, essentially states that the overlap of
the time evolved state (g(t, t0)|ψ〉) with the state evolved in the subspace (gr(t, t0)|ψ〉)
should be the same at the times t0 and t. Given that the restricted Hamiltonian leads,
in general, to different evolution, the condition refers only to very special initial states
with symmetries, or for particular time intervals ∆t.
5 Arrival time problem
The arrival time problem is the following: What is the probability that the system
crosses a particular region ∆ of the configuration space, at any time during the time
interval ∆t = (t − t0). One can attempt to answer this, by considering what is the
probability that the system remains always in the complementary region ∆̄. So if Q
is the total configuration space, we have ∆ ∪ ∆̄ = Q and ∆ ∩ ∆̄ = ∅. Taking this
approach to the arrival time problem, the relation with the quantum Zeno histories is
apparent, since it is just the special case, where the subspace of projection is a region
of the configuration space (∆̄) and the Hamiltonian is quadratic in momenta, i.e.
|x〉〈x|dx
Ĥ = p̂2/2m+ V (x̂) (17)
This particular case is infinite dimensional, but as shown in Ref. [4] the restricted
Hamiltonian is indeed self-adjoint and the arguments of the previous section apply.
Before proceeding further, we should point out that one could construct different
class operators that would also correspond to the (classical) arrival time question. For
example, one could consider having POVM’s7 instead of projections at each moment
of time, or could have a finite (but frequent) number of projections (not taking the
limit where δt→ 0). These and other approaches are not discussed here.
Let us see now, what the quantum Zeno effect implies about the arrival time. It
states that a system initially localized outside ∆ will always remain outside ∆ (if it
decoheres) and therefore we can only get zero crossing probabilities. This is definitely
surprising, since for a wave packet that is initially localized in ∆̄ and its classical
trajectory crosses region ∆, we would expect to get crossing probability one. The
resolution comes due to the decoherence condition as will be argued later.
7Positive Operator Valued Measure
Returning to the decoherence condition Eq.(16) we see that there is the overlap of
the time evolved state with the restricted time evolved state. In the arrival time case,
the restricted Hamiltonian corresponds to the Hamiltonian in the restricted region (∆̄)
but with infinite potential walls on the boundary (i.e. perfectly reflecting). We then
get decoherence in the following four cases.
(a) The initial state |ψ〉 is in an energy eigenstate, and it also vanishes on the bound-
ary of the region.
(b) The restricted propagator can be expressed by the method of images8 and the
initial state shares the same symmetry.
(c) The full unitary evolution in the time interval ∆t remains in the region ∆̄.
(d) Recurrence: Due to the period of the Hamiltonian and the restricted Hamiltonian
their overlap happens to be the same after some time t as it was in time t0. This
depends sensitively on the time interval and it is thus of less physical significance.
It is now apparent that most initial states do not satisfy any of those conditions. In
particular, the wavepacket that classically would cross the region ∆, will not satisfy
any of these conditions, and we would not be able to assign the candidate probability
as a proper one, and thus we avoid the paradox. The introduction of an interacting
environment to our system, (that usually produces decoherence by coarse-graining the
environment) does not change the probabilities and contrary to the intuitive feeling, it
does not provide decoherence for the particular type of question we consider. This still
leave us with no answer for any of the cases that the system would classically cross the
region. The latter implies, that the straight forward coarse grainings we used, were
not general enough to answer fully the arrival time question 9.
As a final note, we should point out that the quantum Zeno effect in the decoherent
histories, has implications for the decoherent histories approach to the problem of time
(e.g. Refs. [9, 3]).
6 Conclusions
We examined the quantum Zeno type of histories of a closed system, using the deco-
herent histories approach. We show that the quantum Zeno effect is still present, but
only for the very few cases that we have decoherence. The situation does not change
with the introduction of interacting environment. We see that while in the open sys-
tem quantum Zeno, the delay of the evolution arises as interaction with the observer,
in the closed system we have the decoherence condition “replacing” the observer and
resolving the apparent paradox.
Acknowledgments: The author is very grateful to Jonathan J. Halliwell for many
useful discussions and suggestions, and would like to thank the organizers for giving
the opportunity to give this talk and hosting this very interesting and nice conference.
8Note that the restricted propagator can be expressed using the method of images, if and only if
there exist a set of energy eigenstates, vanishing on the boundary, that when projected on the region
∆̄ forms a dense subset of the subspace H
, i.e. span H
. This is equivalent with requiring that
the restricted energy spectrum (i.e. spectrum of the restricted Hamiltonian H
) is a subset of the
(unrestricted) energy spectrum, which is not in general the case.
9For more details, examples and discussion see Ref. [3].
References
[1] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18, 756 (1977).
[2] J.J.Halliwell and E.Zafiris, Phys.Rev. D57, 3351 (1998).
[3] P. Wallden, gr-qc/0607072.
[4] P. Facchi, S. Pascazio, A. Scardicchio, and L. S. Schulman, Phys. Rev. A 65,
012108 (2002).
[5] R.B. Griffiths, J. Stat. Phys., 36 219, (1984).
[6] R. Omnès. J. Stat. Phys. 53, 893 (1988).
[7] M. Gell-Mann and J. Hartle, in Complexity, Entropy and the Physics of Infor-
mation, SFI Studies in the Science of Complexity, Vol. VIII, edited by W. Zurek,
(Addison-Wesley, Reading, 1990).
[8] J. J. Halliwell, quant-ph/9506021.
[9] J. J. Halliwell and P. Wallden, Phys. Rev. D 73 024011 (2006), gr-qc/0509013.
|
0704.1552 | Green function theory versus Quantum Monte Carlo calculations for thin
magnetic films | Green Function theory vs. Quantum Monte Carlo Calculation for thin magnetic films
S. Henning,∗ F. Körmann, J. Kienert, and W. Nolting
Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
S. Schwieger
Technische Universität Ilmenau, Theoretische Physik I, Postfach 10 05 65, 98684 Ilmenau, Germany
(Dated: October 31, 2018)
In this work we compare numerically exact Quantum Monte Carlo (QMC) calculations and Green
function theory (GFT) calculations of thin ferromagnetic films including second order anisotropies.
Thereby we concentrate on easy plane systems, i.e. systems for which the anisotropy favors a
magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e.
B parallel to the film normal. GFT results are in good agreement with QMC for high enough fields
and temperatures. Below a critical field or a critical temperature no collinear stable magnetization
exists in GFT. On the other hand QMC gives finite magnetization even below those critical values.
This indicates that there occurs a transition from non-collinear to collinear configurations with
increasing field or temperature. For slightly tilted external fields a rotation of magnetization from
out-of-plane to in-plane orientation is found with decreasing temperature.
PACS numbers: 75.10.Jm, 75.40.Mg, 75.70.Ak, 75.30.Gw
I. INTRODUCTION
The fast development of technological applications
based on magnetic systems in the last years, e.g.
magnetic data storage devices, causes a high interest
in thin magnetic films. One precondition for the tech-
nological development is the investigation of magnetic
anisotropies and spin reorientation transitions connected
therewith. Those reorientation transitions can occur
from out-of-plane to in-plane or vice versa for increasing
film thickness d1, temperature T 2,3,4,5,6,7,8, or external
field B0.
Quantum Monte Carlo (QMC) calculations give the
possibility to compare numerically exact results with
analytical approximations. In Ref. 9 the authors inves-
tigated a ferromagnetic monolayer including positive
second order anisotropy (easy axis perpendicular to the
film plane). They discuss the temperature dependence
of the magnetization 〈Sz〉(T ) as well as field induced
reorientation transitions from out-of-plane to in-plane
and compare the QMC results with Green function
theory (GFT). They found good agreement in the case
of applied external field in the easy direction (here
z-axis). However, their GFT fails for external field
applied in arbitrary direction, especially in the hard
direction (within the film plane). As shown in Ref. 10
for getting closer to the QMC results for magnetic field
induced reorientation from out-of-plane to in-plane a
more careful treatment of the local anisotropy terms is
needed. In Refs. 10,11,12,13 a decoupling scheme was
presented which yields excellent agreement with QMC
results for out-of-plane systems.
The availability of theories such as GFT and their
check against state-of-the-art numerical algorithms
is highly desirable because of the size limitations of
systems where QMC can be performed. On the other
hand the extension of GFT from a monolayer (where it
can be compared to QMC as in the the present work)
to multilayer systems is a straightforward task without
further approximations11.
Up to now, to our knowledge, there is no comparison
between QMC and approximative theories for easy-plane
systems and it is not obvious that the theory presented
in Refs. 10,11,12,13 can reproduce the QMC results for
in-plane systems as accurately as for the out-of-plane
case. In contrast to the easy-axis case where a certain
direction is preferred by the single ion anisotropy in
easy-plane systems the full xy-plane is favored and no
particular direction is distinguished within the plane. A
magnetic field applied perpendicular to the plane does
not destroy the xy-symmetry.
For systems exhibiting this kind of symmetry it was
shown in a classical treatment that for external fields
smaller than a critical field 0 ≤ B < Bcrit (B || z)
stable vortices, i.e. a non-collinear arrangement of
spins, can exist15,16,17,18,19. These vortices can undergo
a Berezinskii-Kosterlitz-Thouless (BKT) transition14.
Depending on the strength of the anisotropy K2 there
might be vortices with or without a finite z-component
of magnetization15. In the small anisotropy case (which
is considered in this work, |K2| < 0.1J) there is a
finite out-of-plane component and for zero field the
two possible directions of magnetization (±z) are en-
ergetically degenerate. For increasing magnetic field in
z-direction the vortices antiparallel to the field become
more and more unstable (heavy vortices). However the
so called light vortices (parallel to the field) are stable
up to a critical field Bz = Bcrit and contribute a finite
z-component to the net-magnetization of the considered
system19.
The vortices in connection with a finite z-component
of the net-magnetization emerge because of two reasons:
first the competition between the anisotropy (favoring
a orientation of the magnetization within the xy-plane)
http://arxiv.org/abs/0704.1552v1
and the external field (favoring a perpendicular magne-
tization), and second: the xy-symmetry of the system,
which does not allow for a rotated homogeneous phase.
In this paper we investigate both aspects, i.e. the
field vs. anisotropy competition as well as the symmetry
properties in detail for a quantum mechanical system.
We will compare the results of QMC and GFT calcula-
tions.
As explained in more detail below, the QMC al-
gorithm used here allows only for an external field
applied in z-direction. Thus the xy-symmetry can not
be broken and no comparison between xy-symmetric
and asymmetric systems is possible. We will use GFT
to clarify the influence of this symmetry breaking on
the homogeneous phase. On the other hand, the GFT
used here is by ansatz limited to the homogeneous
phase. Therefore it can not describe a non-collinear
(e.g, vortex-) magnetic phase, which is expected for
B || z and small field strengths. The breakdown of
magnetization in GFT as well as an exposed maximum
in the magnetization in QMC at certain critical values
of the external field or temperature gives, however, a
clear fingerprint of non-collinear configurations, at least
if there is no meta-stable homogeneous phase. Below
these critical values there will be a finite z-component
in QMC and a vanishing magnetization in GFT.
For parameters, where both theories are applicable,
QMC serves as a test for the approximations needed in
In this work we find indications for non-collinear
spin configurations below a critical field or temperature
for B || z by comparing results of QMC and GFT as
explained in the last clause. Above the critical field
we obtain good agreement between QMC and GFT
results. Breaking the xy-symmetry by adding a small
x-component to the external field yields a stable collinear
solution in GFT. The z-component of the magnetization
in this case is in good agreement with the QMC results
calculated with untilted field. Thus we can conclude
that except for the restriction to collinear magnetic
states GFT describes the competition between external
field and anisotropy quite well.
The paper is organized as follows: First we ex-
plain the basics of the GFT and the QMC calculations.
Then we apply both approaches to easy-plane systems
in external magnetic fields and report the results of our
calculations.
II. THEORY
A. Green Function Theory
In the following we present our theoretical approach us-
ing Green function theory. The focus of this work lies on
the translational invariant system of a two-dimensional
monolayer. Therefore the following Hamiltonian is used:
H = −
JijSiSj −B
Si −K2
(Sz,i)
2. (1)
The first term describes the Heisenberg coupling Jij be-
tween spins Si and Sj located at sites i and j. The second
term contains an external magnetic field B in arbitrary
direction (the Landé factor gJ and the Bohr magneton
µB are absorbed in B). The third term represents second
order lattice anisotropy due to spin-orbit coupling. Sz,i
is the z-component of Si (the z-axis of the coordinate
system is oriented perpendicular to the film-plane). The
lattice anisotropy favours in-plane (K2 < 0) or out-of-
plane (K2 > 0) orientation. Our Hamiltonian is similar
to that used in Refs. 10,11,13,22,23 for the investigation
of the magnetic anisotropy and the field induced reori-
entation transition. To simplify calculations we consider
nearest neighbor coupling only
Jij =
J (i), (j) n.n.
0 otherwise.
The main idea of the special treatment presented in
Refs. 10,11,12,13 is that, before any decoupling is applied,
the coordinate system Σ is rotated to a new system Σ′
where the new z′-axis is parallel to the magnetization im-
plying a collinear alignment of all spins within the layer.
Then a combination of Random Phase approximation
(RPA)24 for the nonlocal terms in Eq. (1) (Heisenberg
exchange interaction term) and Anderson-Callen approx-
imation (AC)25 for the local lattice anisotropy term is
applied in the rotated system. After application of the
approximation one gets an effective anisotropy
Keff (T ) = 2K2
S(S + 1)− 〈S2z′〉
〈Sz′〉 (3)
where 〈Sz′〉 is the norm of the magnetization and S is the
spin quantum number, that we have chosen to be S = 1
in all our calculations.
As shown in comparison with an exact treatment of the
local anisotropy term in Ref. 26 this approximation still
holds up to anisotropy strengths K2 ∼ 1/2J . Therefore
we restrict ourselves in the following to small anisotropies
(K2 ≤ 0.1J) as found in most real materials
33. For a
magnetic field applied in the xz-plane (B = (Bx, 0, Bz))
our theory gives a condition for the polar angle θ of the
magnetization:
sin θBz − cos θBx +Keff sin θ cos θ = 0 (4)
The uniform magnon energies (q = 0) which dominate
the physical behavior of the magnetic system can easily
be extracted from the theory12,13:
E2q=0 =
cos θBz + sin θBx +Keff(cos
2 θ − sin2 θ)
cos θBz + sin θBx +Keff cos
This result coincides with the spin-wave result13 if one
replaces 〈Sz′〉 by the spin quantum number S and Keff
by the bare anisotropy constant K2 in Eq. (5). For an
easy-plane system (Keff < 0) with external field B in
z-direction the polar angle θ of the magnetization34 is
given by:
cos θ =
−B/Keff(T ) for B < |Keff (T )|
1 otherwise
By inserting Eq.(6) into Eq.(5) one immediately gets:
Keff<0
q=0 (B) =
0 B < |Keff (T )|
B +Keff (T ) otherwise.
For gapless magnon energies Eq=0 = 0 the magnon oc-
cupation number φ diverges (φ → ∞) in film systems
with ferromagnetic coupling J > 0 and the magnetiza-
tion becomes zero 〈Sz′〉 = 0 in the collinear phase. This
can be seen by following an argument of Bloch20 already
given in 1930. Since the spin wave dispersion is E ≈ q2
in the vicinity of q = 0 the spin-wave density of states
N(E) is independent of E for a two-dimensional system
for E close to zero. The excitation of spin-waves at finite
temperature leads to a variation of the magnetization of
the order:
∆m(T ) ∼
N(E)dE
exp(E/kBT )− 1)
∼ kBT
exp(x)− 1
. (8)
Since the integral in Eq. (8) diverges for T 6= 0 and exited
spin-waves lead to a reduction of the magnetization one
can conclude that the magnetization should be zero at
finite temperature. However for an infinitesimally small
contribution of the external field parallel to the plane,
i.e. Bx 6= 0, a finite gap in the excitation spectrum
at q = 0 opens. This can be seen in Fig.1 where the
uniform magnon modes Eq=0(B) are shown for different
orientations θB, where θB is the polar angle of the ex-
ternal field. The integral (8) is now finite and a stable
finite magnetization in the collinear phase having a well
defined orientation in the xz-plane is possible.
Let us now come back to the case where the applied
field is aligned in z-direction. It can be seen from Eq.
(7) that for external field B (B || z) larger than a critical
field B > Bcrit given by:
Bcrit
= |Keff (T,B)| (9)
a stable collinear solution exists. Since Keff (T ) is a de-
creasing function of temperature T a transition from non-
collinear to collinear phase with increasing temperature
is possible. In Fig. 2 we show the normalized critical
field (9) Bcrit/K2 as a function of temperature T . For
a constant magnetic field B (B || z) at a temperature
T1 with B < Keff (T1, B) no stable collinear phase ex-
ist. Then by increasing the temperature up to T2 the ef-
fective anisotropy Keff is sufficiently reduced such that
0.01 0.02 0.03 0.04
B / J
=0.2°
FIG. 1: The energies of the uniform magnon mode Eq=0(B)
for different polar angles θB of the external field. Eq=0 is
zero below B/J ≈ 0.03 for θB = 0
◦. The prefactors gJµB
and kB are absorbed in B and T respectively. The latter are
given in units of the nearest neighbor Heisenberg coupling J .
Parameters: S = 1, K/J = −0.03 and T/J = 10−4.
FIG. 2: The normalized critical field Bcrit/K2 as a function
of temperature. Parameters: S = 1.
B > Keff (T2, B), and the collinear phase becomes sta-
ble. Before we come to the results let us briefly sketch
the main aspects of the QMC.
B. QMC
In the last section we gave a short description of the
theory used to treat a system described by a Hamilto-
nian of form (1). This theory applies to the thermody-
namic limit (films of infinte size) but contains certain
approximations. Additionally the GFT is restricted to
ordered phases with a collinear alignment of all spins.
Therefore it would be very useful to have exact results at
hand to crosscheck the predictions of GFT. A Quantum
Monte Carlo method, particularly well suited for spin
systems, is the stochastic series expansion (SSE) with
directed loop update. We will sketch this method here
only briefly as detailed descriptions can be already found
elsewhere28,29,30.
Our starting point is the series expansion of the parti-
tion function
Z = Tre−βH =
〈α|(−H)n|α〉 (10)
whereH denotes the Hamiltonian, {|α〉} are basis vectors
of a proper Hilbert space and β is the inverse tempera-
ture. The Hamiltonian is then rewritten in terms of bond
Hamiltonians:
H = −J
Hb (11)
where Hb can be further decomposed into a diagonal and
an off-diagonal part:
HD,b = C + S
i(b)S
j(b) + bb[S
i(b) + S
j(b)] (12)
+k2b[(S
i(b))
2 + (Szj(b))
HO,b =
] (13)
Here we have renormalized the anisotropy constant k2b
and the magnetic field bb in such a way that (11) coincides
with (1). i(b) and j(b) denotes the lattice sites connected
by the bond b and the additional constant C in HD,b will
be chosen such that all matrix elements of this term be-
come positive, a condition necessary to interpret them as
probabilities. Note that for a finite system at finite tem-
perature the power series of the partition function can
be truncated at a finite cutoff length Λ without intro-
ducing any systematic error in practical computations29.
Therefore reinserting (11) into (10) and rewriting the re-
sult yields:
βn(Λ− n)!
〈α|SCΛ |α〉. (14)
Here SCΛ denotes a product of operators (operator
string) consisting of n non-unity operators and (Λ − n)
unity operators H0 = Id which were inserted to get op-
erator strings of equal length Λ.
In fact it is impossible to evaluate all operator strings in
(14). The SSE-QMC replaces such an evaluation there-
fore by importance sampling over the strings according to
their relative weight. Hence an efficient scheme for gen-
erating new operator strings is needed. In the directed
loop version of the SSE this is done by dividing the up-
date into two parts. In a first step a diagonal update is
performed by traversing the operator string and replac-
ing some unity operators by diagonal bond operators and
vice versa (the probabilities for both substitutions have
to fulfill the detailed balance criterion). Then the loop
update follows in which new non-diagonal bond opera-
tors can appear in the operator string. For details of the
update procedure we refer the interested reader to the
0 0.5 1 1.5 2
GFT (RPA+AC)
QMC (N=16)
QMC (N=32)
QMC (N=64)
FIG. 3: Magnetization vs. temperature for an out-of-plane
easy-axis system (K2 > 0). Straight line: GFT (RPA+AC)
result; symbols: QMC results for different system sizes N2.
Parameters: S = 1, B/J = 0.01 (B || z) and K2/J = 0.01.
according literature28,29,30.
A full implementation of the SSE with directed loop
update which we have used for all QMC calculations
in this work can be found in the ALPS project30,31.
Since the SSE-QMC used by us is implemented in z-
representation (spin quantization axis along z-axis) in-
plane correlation functions e.g. the in-plane magnetiza-
tion are not accessible. Further B || z is the only possible
field direction in the used QMC implementation because
a traverse field (in-plane field component) would lead to
non-closing loops (see Ref. 9).
III. RESULTS
As mentioned in Sec. II A the results for the in-plane
systems are very sensitive to the effective anisotropy
Keff (T ). This sensitivity of the anisotropy is less pro-
nounced for out-of-plane systems (K2 > 0) since the ap-
plied field B (B || z) and the intrinsic easy axis are par-
allel. In order to test our decoupling scheme (RPA+AC)
we first compare GFT and QMC for an out-of-plane
system.35
In Fig. 3 the magnetization 〈Sz〉 as a function of tem-
perature T is shown. The straight line belongs to the
GFT whereas the symbols show the result of the QMC
for different system sizes. Let us first comment on finite
size effects in the QMC results.
It can be seen in Fig. 3 that the QMC results converge
for increasing system size N2 (for N ×N square lattice).
Indeed forN ≥ 32 the QMC results are unbiased by finite
size effects and resulting magnetization curves are almost
equal for increasing N ≥ 32. Note that we have omitted
error bars in the figures showing QMC results because
the relative errors are of the order 10−4.
We now compare the GFT with the QMC results
(N = 64). For low temperatures (T/J ≤ 0.5) we ob-
tain excellent quantitative agreement. This is plausible
0 0.05 0.1 0.15 0.2
QMC (16)
QMC (32)
QMC (64)
QMC (128)
FIG. 4: z-component of magnetization as a function of ex-
ternal magnetic field for fixed temperature T/J = 0.4. In
contrary to the GFT the magnetization obtained by QMC
remains finite for all fields. The QMC results are converged
for N ≥ 64. Parameters: S = 1, K2/J = −0.06 and θB = 0
0 0.05 0.1 0.15 0.2
GFT (θ
= 0°)
GFT (θ
= 0.5°)
QMC (128)
FIG. 5: z-component of magnetization vs. external field for
T/J = 0.4 with slightly tilted field (θB = 0.5
◦)in the GFT
result (solid line). The dotted line shows GFT result for (θB =
0◦). Other parameters as in Fig. 4.
because in this region the GFT result coincides with the
result of the spin-wave theory which is known to be re-
liable (exact for T = 0) for low temperatures. For the
intermediate region T/J = 0.5..1 the RPA slightly un-
derestimates the magnetization which was also found in
Ref. 9. The opposite is the case in the region near the
extrapolated Curie temperature TC
36, where the magne-
tization is overestimated. The reason is the presence of
longitudinal fluctuations, which play an important role
in this region and it is well known that the RPA fails to
treat them properly.
We consider now the case of in-plane systems (K2 < 0)
and applied field in the hard direction (B || z). As al-
ready mentioned there is no ’collinear’ magnetization in
the GFT for Bz < |Keff (T )|. In Fig. 4 the z-component
of the magnetization is shown as a function of the ex-
ternal field B for a constant temperature T/J = 0.4.
0 0.05 0.1 0.15
T/J = 0.4
T/J = 0.6
T/J = 0.9
T/J = 1.2
T/J = 1.4
FIG. 6: z-component of magnetization vs. external field for
different temperatures T/J and fixed system size N2 (N =
128). Solid lines: GFT (θB = 0.5
◦), dashed lines: GFT (θB =
0◦) other parameters as in Fig. 4.
As in Fig. 3 we see that the QMC results for N ≥ 64
are almost converged and the finite size of the calculated
system in QMC should not influence the results anymore.
The dotted line marks a critical field Bcrit. For magnetic
fields larger than the critical one B > Bcrit we obtain
good agreement between QMC and GFT results. Below
the critical field B < Bcrit GFT does not yield a stable
homogeneous magnetization. However the QMC results
show that there is a finite z-component of the magneti-
zation in the considered system for 0 ≤ B ≤ Bcrit.
In order to compare QMC with GFT results we have
tilted the magnetic field by θB = 0.5
◦ which corresponds
to Bx < 10
−2Bz in the GFT. As explained before any
symmetry breaking field Bx 6= 0 leads to a stable homo-
geneous magnetization with well-defined orientation in
the xz-plane. However such a small contribution of the
external field within the plane should hardly influence the
z-component of the magnetization. This is confirmed by
Fig. 5 where we show QMC results (N = 128, θB = 0
as well as the corresponding GFT results with θB = 0
and θB = 0.5
◦. As expected for |B| > Bcrit the two so-
lutions in the GFT are nearly the same and agree well
with QMC. Below the critical field only the solution with
the slightly tilted field yields a stable homogeneous mag-
netization and its z-component compares well with the
QMC result in the untilted case.
The above results can be interpreted within a semi-
classical picture of non-collinear vortex configurations
which are stable below a critical field Bcrit in z-direction
and contribute a finite z-component to the magnetiza-
tion in case of an applied field.19 Despite the lack of di-
rect, quantitative access to such states (or correspond-
ing physical in-plane observables) within the QMC al-
gorithm they are included in principle and one can ob-
serve their consequences, namely a finite z-component of
the magnetization below the critical GFT field. On the
other hand GFT can only describe homogeneous collinear
0 0.5 1 1.5 2
GFT (θ
=0.5°)
GFT (AC)
GFT (MF)
QMC (16)
QMC (64)
QMC (128)
QMC (256)
FIG. 7: The z-component of magnetization as function of
temperature for a fixed external field. Below a critical tem-
perature Tcrit there is a breakdown of magnetization in GFT
where is no in QMC. Parameters: B/J = 0.03, S = 1,
K2/J = −0.06.
configurations of spins therefore showing a breakdown of
magnetization. However by applying a small field in x-
direction the xy-symmetry is broken and the spins ro-
tate in the field direction (the vortices vanish) and the
collinear phase is retrieved. Our results corroborate this
interpretation based on the classical picture. Let us em-
phasize that both, GFT for slightly tilted field and QMC
for B || z, describe the competition between the external
field (which favors magnetization parallel to z) and the
anisotropy favoring in-plane magnetization. Comparing
the z-components of the magnetization for both cases,
one can conclude that the ratio of the competing forces
are comparable for QMC and GFT. This indicates that
this competition is correctly taken into account in GFT.
In Fig. 6 the same field dependence of the z-component
of the magnetization is shown for different temperatures.
We have plotted the result for the tilted field in case of
GFT, the point of breakdown in the untilted case is in-
dicated by the dotted line. It can be seen that for higher
temperatures no breakdown of collinear magnetization
occurs, meaning that the condition for the critical field
(B ≤ |Keff (T,B)|) is never fulfilled in this case. The dis-
crepancies at intermediate temperatures (T = 0.9..1.2)
are due to the RPA decoupling in the GFT as was dis-
cussed already.
In Figs. 7, 8 and 9 the z-component of the magnetiza-
tion is plotted as a function of temperature obtained by
GFT (straight line RPA+AC) as well as QMC (symbols)
for different system sizes and a constant applied magnetic
field.
Let us first discuss the qualitative behavior of the mag-
netization as a function of temperature which is found in
all three figures. For high T (T ≫ Tcrit) the magneti-
zation is reduced by thermal fluctuations (where the tail
of the curve above T/J ≈ 1.5 is due to the applied ex-
ternal field). In the vicinity of Tcrit, T − Tcrit → 0
competition between two effects sets in and has a pro-
0 0.5 1 1.5 2
GFT (θ
GFT (θ
=0.5°)
GFT (two layers)
QMC (128)
FIG. 8: Same situation as in Fig. 7 for K2/J = −0.04 (other
parameters as in Fig.7). The result for a two layer film treated
by GFT is plotted also (dashed-dotted line).
nounced influence on the magnetization. On the one
side the effective anisotropy acts against the external
field (Beff = B − |Keff (T )|, (B || z)). The effective
anisotropy Keff (T ) is reduced with increasing temper-
ature T and thus the effective field Beff increases with
T . This effect tends to enhance the magnetization with
T . On the other side thermal fluctuations suppress the
magnetization with increasing T . The flattening of the
magnetization curve near Tcrit is a result of this com-
petition. For low temperatures T < Tcrit the effective
anisotropy in the GFT cannot be overcome by the exter-
nal field (B < |Keff (T )|, (B ||z)). Therefore the collinear
magnetization in our approximation vanishes due to the
mentioned gapless excitations, in contrast to QMC which
yields again a finite magnetization because non-collinear
states are taken into account as discussed above. The
reduction of the z-component of magnetization in QMC
below Tcrit can be pictured classically as the spins being
in a non-collinear phase with an angle θ with respect to
the z-axis. Since in general anisotropy effects (which fa-
vor in-plane magnetization) increase when temperature is
lowered the z-component of the magnetization decreases.
Now we discuss the three figures in detail. In Fig. 7
we have plotted QMC results for different system size
showing again that these are well converged for N ≥ 64.
Thus we conclude that the striking difference between
GFT and QMC is not a mere finite size effect. The
breakdown of magnetization in GFT occurs at a critical
temperature Tcrit/J = 0.5 whereas no such breakdown
exists in QMC. However the exposed maximum of the
magnetization in QMC lies near the breakdown point.
The differences between QMC and GFT in the tempera-
ture range T/J ≈ 0.3 . . . 1.3 are due to the decoupling of
the exchange and anisotropy term in GFT as also seen
in Fig. 3. It is worth mentioning that the value of the
z-component of the magnetization is nearly the same at
the breakdown point in GFT and the maximum in the
QMC. Thus we have the result that although GFT can-
0 0.5 1 1.5 2
GFT (<S
>; θ
=0.5°)
GFT (<S
>; θ
=0.5°)
QMC (θ
FIG. 9: Same situation as in Fig. 7 for K2/J = −0.01, B/J =
0.005 and slightly tilted field (θB = 0.5
◦) for the GFT results.
not describe the non-collinear phase by ansatz its break-
down coincides rather well with the onset of this phase,
which we attribute to the maximum of the QMC curve.
Fig.8 shows the same situation for a different anisotropy
constant K2 = −0.04. The critical temperature is lower
than in Fig.7 since the ratio Bz/K2 becomes larger. The
tilted field case is also shown for the GFT results. Again
the qualitative agreement of the z-component of magne-
tization with QMC is good. To confirm this point we
have plotted the temperature dependence for an other
set of parameters in Fig. 9. There is as good qualitative
agreement of the two approaches. Additionally one gets
a finite component in x-direction in GFT which is also
plotted in the figure. The two effects of the external field
vs. anisotropy competition are nicely to be seen: a non-
collinear state for B || z (z-component only in QMC but
not in GFT) and rotation of magnetization for slightly
tilted external field (seen only in GFT). The ratio of the
competing forces agree well again in both treatments.
In Fig. 7 we have plotted the results of a different
decoupling scheme of the anisotropy terms (namely a
mean field decoupling, dashed line in Fig. 7). Although
the overall characteristic resembles the RPA+AC result
(breakdown of magnetization) the mean field results dif-
fer extremely from the QMC for a large range of tempera-
ture and underestimates the magnetization. This demon-
strates the reliability of the Anderson-Callen treatment of
the local anisotropy terms presented in Refs. 10,11,12,13.
The extension of the GFT method to multi-layer films
is straightforward.11 We have also included results for a
two-layer film in Fig. 8 for the same parameters as in
the monolayer case. One finds that for a double layer
magnetism is stabilized, which can be attributed to the
increased coordination number and thus higher exchange
energy. Just like for a monolayer, one observes a break-
down of collinear magnetization at some critical temper-
ature. This is due to the fact that the same reasoning
regarding the vanishing excitation gap also applies for
multilayer (slab) systems32. The effective anisotropy per
layer is essentially the same as for a single layer, thus the
critical 〈Sz〉-value (magnetization at critical field Bcrit) is
practically the same. The critical temperature is higher
than that of a monolayer due to the increased magnetic
stiffness of the double layer.
IV. SUMMARY AND CONCLUSIONS
Using GFT and QMC calculations we studied easy-
plane systems as well as easy-axis systems with an exter-
nal field applied perpendicularly to the film. The GFT
treatment of the Hamiltonian Eq. (1) consists of a RPA-
decoupling for the nonlocal terms and an AC-decoupling
for the local terms performed in a rotated frame, where
the new z′-axis is parallel to the magnetization. For the
QMC calculations we have used the stochastic series ex-
pansion (SSE) with directed loop updates, which is well
suited for spin-systems.
We have calculated the magnetization as a function of
the external field as well as temperature. We found a
critical field and critical temperature respectively below
which is no magnetization in GFT whereas there is one in
QMC. By tilting the field slightly in GFT so that it has
a small component in x-direction we get a stable magne-
tization even below the critical field or temperature. The
z-component of the magnetization in this case coincides
well with the z-component obtained by QMC for the un-
tilted field confirming that GFT and QMC agree well in
the description of the external field vs. anisotropy com-
petition. However, this comparison can be only some-
what indirect, since QMC has access to the non-collinear
(B || z) state only, while GFT is limited to collinear fer-
romagnetic states (rotated homogeneous magnetization)
found for slightly tilted external fields.
For parameters that are accessible by both QMC and
GFT (B || z; B > Bcrit(T )) QMC and GFT are in good
agreement. Thus one can conclude that the GFT is ap-
plicable to the homogeneous phases of systems described
by Eq. (1) and can be used also for system configurations
not accessible by QMC due to too large system size as
e.g. multilayer systems.
It would be an interesting task for a succeeding work
to extend the GFT in order to get a deeper insight into
the non-collinear configurations also.
APPENDIX A: MAGNETIZATION ANGLE
Here we will discuss the second mathematical solution
which occurs besides Eq. 6. For an external field in
the z-direction the angle dependent part of the free en-
ergy including second order anisotropy can be expanded
as1,27:
F = −MzBz cos θ − K̃2 cos
whereMz is the z-component of the magnetization and
K̃2 is the first nonvanishing coefficient in an expansion of
the free energy for a system with second order anisotropy.
For the equilibrium angle one gets:
∂F (θ)
= MzBz sin θ + 2K̃2 cos θ sin θ
= 0. (A1)
Therefore one gets two solutions for in-plane systems
(K̃2 < 0). For sin θ 6= 0 one gets immediately the so-
lution of Eq. 6 if 2K̃2/Mz ≡ Keff holds. This is the
stable solution. The trivial (second) solution sin θ = 0 is
unstable for Bz < |Keff | because
∂2F (θ)
|sin θ=0 =
< 0 for Bz < |Keff |
> 0 otherwise
holds. For a detailed discussion of stability conditions in
film systems we refer to Refs. 1,27.
∗ Electronic address: [email protected]
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33 Besides some rare earth materials where the anisotropy can
be of the order of J .
34 For B < |Keff | there is another mathematical solution
(sin θ = 0) which however is unstable (see appendix A).
35 Note that a similar result has already been published in
Ref. 9.
36 Strictly speaking there is no phase transition because of
the applied magnetic field as can be seen from the large
tail of the magnetization curve. However one can extract
a TC from the curves by extrapolating to the zero field
case and additionally to an infinte system size in the QMC
calculations.
mailto:[email protected]
http://arxiv.org/pdf/cond-mat/0607675
http://alps.comp-phys.org/
|
0704.1553 | Matrix Ordered Operator Algebras | 7 Matrix Ordered Operator Algebras.
Ekaterina Juschenko, Stanislav Popovych
Department of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
[email protected]
[email protected]
Abstract
We study the question when for a given ∗-algebra A a sequence of
cones Cn ∈ Mn(A) can be realized as cones of positive operators in a
faithful ∗-representation of A on a Hilbert space. A characterization
of operator algebras which are completely boundedly isomorphic to
∗-algebras is presented.
KEYWORDS: ∗-algebra, operator algebra, C∗-algebra, completely
bounded homomorphism, Kadison’s problem.
1 Introduction
Effros and Choi gave in [2] an abstract characterization of the self-adjoint sub-
spaces S in C∗-algebras with hierarchy of cones of positive elements inMn(S).
In Section 2 of the present paper we are concerned with the same question
for ∗-subalgebras of C∗-algebras. More precisely, let A be an associative
∗-algebra with unit. In Theorem 2 we present a characterization of the col-
lections of cones Cn ⊆Mn(A) for which there exist faithful ∗-representation
π of A on a Hilbert space H such that Cn coincides with the cone of positive
operators contained in π(n)(Mn(A)). Here π
(n)((xi,j)) = (π(xi,j)) for every
matrix (xi,j) ∈Mn(A). Note that we do not assume that A has any faithful
0 2000 Mathematics Subject Classification: 46L05, 46L07 (Primary) 47L55, 47L07,
47L30 (Secondary)
http://arxiv.org/abs/0704.1553v2
∗-representation. This follows from the requirements imposed on the cones.
In terms close to Effros and Choi we give an abstract characterizations of
matrix ordered (not necessary closed) operator ∗-algebras up to complete
order ∗-isomorphism.
Based on this characterization we study the question when an operator
algebra is similar to a C∗-algebra.
Let B be a unital (closed) operator algebra in B(H). In [8] C. Le Merdy
presented necessary and sufficient conditions for B to be self-adjoint. These
conditions involve all completely isometric representations of B on Hilbert
spaces. Our characterization is different in the following respect. If S is a
bounded invertible operator in B(H) and A is a C∗-algebra in B(H) then the
operator algebra S−1AS is not necessarily self-adjoint but only isomorphic to
a C∗-algebra via completely bounded isomorphism with completely bounded
inverse. By Haagerup’s theorem every completely bounded isomorphism π
from a C∗-algebra A to an operator algebra B has the form π(a) = S−1ρ(a)S,
a ∈ A, for some ∗-isomorphism ρ : A → B(H) and invertible S ∈ B(H).
Thus the question whether an operator algebra B is completely boundedly
isomorphic to a C∗-algebra via isomorphism which has completely bounded
inverse, is equivalent to the question if there is bounded invertible operator
S such that SBS−1 is a C∗-algebra.
We will present a criterion for an operator algebra B to be completely
boundedly isomorphic to a C∗-algebra in terms of the existence of a collection
of cones Cn ∈ Mn(B) satisfying certain axioms (see def. 3). The axioms are
derived from the properties of the cones of positive elements of a C∗-algebra
preserved under completely bounded isomorphisms.
The main results are contained in section 2. We define a ∗-admissible
sequence of cones in an operator algebra and present a criterion in Theorem 4
for an operator algebra to be completely boundedly isomorphic to a C∗-
algebra.
In the last section we consider the operator algebras and collections of
cones associated with Kadison similarity problem.
2 Operator realizations of matrix-ordered ∗-
algebras.
The aim of this section is to give necessary and sufficient conditions on a
sequences of cones Cn ⊆ Mn(A)sa for a unital ∗-algebra A such that Cn
coincides with the coneMn(A)∩Mn(B(H))
+ for some realization of A as a ∗-
subalgebra of B(H), where Mn(B(H))
+ denotes the set of positive operators
acting on Hn = H ⊕ . . .⊕H .
In [11] it was proved that a ∗-algebra A with unit e is a ∗-subalgebra of
B(H) if and only if there is an algebraically admissible cone on A such that
e is an Archimedean order unit. Applying this result to some inductive limit
of M2n(A) we obtain the desired characterization in Theorem 2.
First we give necessary definitions and fix notations. Let Asa denote the
set of self-adjoint elements in A. A subset C ⊂ Asa containing unit e of A
is algebraically admissible cone (see [12]) provided that
(i) C is a cone in Asa, i.e. λx+ βy ∈ C for all x, y ∈ C and λ ≥ 0, β ≥ 0,
λ, β ∈ R;
(ii) C ∩ (−C) = {0};
(iii) xCx∗ ⊆ C for every x ∈ A;
We call e ∈ Asa an order unit if for every x ∈ Asa there exists r > 0 such
that re+ x ∈ C. An order unit e is Archimedean if re+ x ∈ C for all r > 0
implies that x ∈ C
In what follows we will need the following.
Theorem 1. Let A be a ∗-algebra with unit e and C ⊆ Asa be a cone
containing e. If xCx∗ ⊆ C for every x ∈ A and e is an Archimedean
order unit then there is a unital ∗-representation π : A → B(H) such that
π(C) = π(Asa) ∩ B(H)
+. Moreover
1. ‖π(x)‖ = inf{r > 0 : r2 ± x∗x ∈ C}.
2. ker π = {x : x∗x ∈ C ∩ (−C)}.
3. If C ∩ (−C) = {0} then ker π = {0}, ‖π(a)‖ = inf{r > 0 : r ± a ∈ C}
for all a = a∗ ∈ A and π(C) = π(A) ∩B(H)+
Proof. Following the same lines as in [11] one obtains that the function ‖ · ‖ :
Asa → R+ defined as
‖a‖ = inf{r > 0 : re± a ∈ C}
is a seminorm on R-space Asa and |x| =
‖x∗x‖ for x ∈ A defines a pre-
C∗-norm on A. If N denote the null-space of | · | then the completion B =
A/N with respect to this norm is a C∗-algebra and canonical epimorphism
π : A → A/N extends to a unital ∗-homomorphism π : A → B. We can
assume without loss of generality that B is a concrete C∗-algebra in B(H)
for some Hilbert space H . Thus π : A → B(H) can be regarded as a unital
∗-representation. Clearly,
‖π(x)‖ = |x| for all x ∈ A.
This implies 1.
To show 2 take x ∈ ker π then ‖π(x)‖ = 0 and re ± x∗x ∈ C for all
r > 0. Since e is an Archimedean unit we have x∗x ∈ C ∩ (−C). Conversely
if x∗x ∈ C ∩ (−C) then re± x∗x ∈ C, for all r > 0, hence ‖π(x)‖ = 0 and 2
holds.
Let us prove that π(C) = π(Asa) ∩ B(H)
+. Let x ∈ Asa and π(x) ≥ 0.
Then there exists a constant λ > 0 such that ‖λIH − π(x)‖ ≤ λ, hence
|λe − x| ≤ λ. Since ‖a‖ ≤ |a| for all self-adjoint a ∈ A, see Lemma 3.3 of
[11], we have ‖λe−x‖ ≤ λ. Thus given ε > 0 we have (λ+ε)e±(λe−x) ∈ C.
Hence εe+ x ∈ C. Since e is Archimedean x ∈ C.
Conversely, let x ∈ C. To show that π(x) ≥ 0 it is sufficient to find λ > 0
such that ‖λIH − π(x)‖ ≤ λ. Since ‖λIH − π(x)‖ = |λe − x| we will prove
that |λe − x| ≤ λ for some λ > 0. From the definition of norm | · | we have
the following equivalences:
|λe− x| ≤ λ ⇔ (λ+ ε)2e− (λe− x)2 ∈ C for all ε > 0 (1)
⇔ ε1e + x(2λe− x) ≥ 0, for all ε1 > 0. (2)
By condition (iii) in the definition of algebraically admissible cone we
have that xyx ∈ C and yxy ∈ C for every x, y ∈ C. If xy = yx then
xy(x + y) ∈ C. Since e is an order unit we can choose r > 0 such that
re − x ∈ C. Put y = re − x to obtain rx(re − x) ∈ C. Hence (2) is
satisfied with λ = r
. Thus ‖λe − π(x)‖ ≤ λ and π(x) ≥ 0, which proves
π(C) = π(Asa) ∩ B(H)
In particular, for a = a∗ we have
‖π(a)‖ = inf{r > 0 : rIH ± π(a) ∈ π(C)}. (3)
We now in a position to prove 3. Suppose that C∩ (−C) = 0. Then ker π
is a ∗-ideal and ker π 6= 0 implies that there exists a self-adjoint 0 6= a ∈ ker π,
i.e. |a| = 0. Inequality ‖a‖ ≤ |a| implies re± a ∈ C for all r > 0. Since e is
Archimedean, ±a ∈ C, i.e. a ∈ C ∩ (−C) and, consequently, a = 0.
Since ker π = 0 the inclusion rIH±π(a) ∈ π(C) is equivalent to re±a ∈ C,
and by (3), ‖π(a)‖ = inf{r > 0 : re± a ∈ C}. Moreover if π(a) = π(a)∗ then
a = a∗. Thus we have π(C) = π(A) ∩ B(H)+.
We say that a ∗-algebra A with unit e is a matrix ordered if the following
conditions hold:
(a) for each n ≥ 1 we are given a cone Cn in Mn(A)sa and e ∈ C1,
(b) Cn ∩ (−Cn) = {0} for all n,
(c) for all n and m and all A ∈Mn×m(A), we have that A
∗CnA ⊆ Cm,
We call e ∈ Asa a matrix order unit provided that for every n ∈ N
and every x ∈ Mn(A)sa there exists r > 0 such that ren + x ∈ Cn, where
en = e ⊗ In. A matrix order unit is called Archimedean matrix order unit
provided that for all n ∈ N inclusion ren + x ∈ Cn for all r > 0 implies that
x ∈ Cn.
Let π : A → B(H) be a ∗-representation. Define π(n) : Mn(A) →
Mn(B(H)) by π
(n)((aij)) = (π(aij)).
Theorem 2. If A is a matrix-ordered ∗-algebra with a unit e which is
Archimedean matrix order unit then there exists a Hilbert spaceH and a faith-
ful unital ∗-representation τ : A → B(H), such that τ (n)(Cn) = Mn(τ(A))
for all n. Conversely, every unital ∗-subalgebra D of B(H) is matrix-ordered
by cones Mn(D)
+ = Mn(D) ∩ B(H)
+ and the unit of this algebra is an
Archimedean order unit.
Proof. Consider an inductive system of ∗-algebras and unital injective ∗-
homomorphisms:
φn :M2n(A) →M2n+1(A), φn(a) =
for all n ≥ 0, a ∈M2n(A).
Let B = lim
M2n(A) be the inductive limit of this system. By (c) in the
definition of the matrix ordered algebra we have φn(C2n) ⊆ C2n+1. We will
identify M2n(A) with a subalgebra of B via canonical inclusions. Let C =⋃
C2n ⊆ Bsa and let e∞ be the unit of B.
Let us prove that C is an algebraically admissible cone. Clearly, C satisfies
conditions (i) and (ii) of definition of algebraically admissible cone. To prove
(iii) suppose that x ∈ B and a ∈ C, then for sufficiently large n we have
a ∈ C2n and x ∈ M2n(A). Therefore, by (c), x
∗ax ∈ C. Thus (iii) is proved.
Since e is an Archimedean matrix order unit we obviously have that e∞ is
also an Archimedean order unit. Thus ∗-algebra B satisfies assumptions of
Theorem 1 and there is a faithful ∗-representation π : B → B(H) such that
π(C) = π(B) ∩ B(H)+.
Let ξn : M2n(A) → B be canonical injections (n ≥ 0). Then τ = π ◦ ξ0 :
A → B(H) is an injective ∗-homomorphism.
We claim that τ (2
n) is unitary equivalent to π◦ξn. By replacing π with π
where α is an infinite cardinal, we can assume that πα is unitary equivalent
to π. Since π◦ξn :M2n(A) → B(H) is a ∗-homomorphism there exist unique
Hilbert space Kn, ∗-homomorphism ρn : A → B(Kn) and unitary operator
Un : Kn ⊗ C
2n → H such that
π ◦ ξn = Un(ρn ⊗ idM2n )U
For a ∈ A, we have
π ◦ ξ0(a) = π ◦ ξn(a⊗ E2n)
= Un(ρn(a)⊗ E2n)U
where E2n is the identity matrix in M2n(C). Thus τ(a) = U0ρ0(a)U
Un(ρn(a) ⊗ E2n)U
n. Let ∼ stands for the unitary equivalence of represen-
tations. Since π ◦ ξn ∼ ρn ⊗ idM2n and π
α ∼ π we have that ραn ⊗ idM2n ∼
πα ◦ ξn ∼ ρn ⊗ idM2n . Hence ρ
n ∼ ρn. Thus ρn ⊗ E2n ∼ ρ
n ∼ ρn.
Consequently ρ0 ∼ ρn and π ◦ ξn ∼ ρ0 ⊗ idM2n ∼ τ ⊗ idM2n . Therefore
n) = τ ⊗ idM2n is unitary equivalent to π ◦ ξn.
What is left to show is that τ (n)(Cn) = Mn(τ(A))
+. Note that π ◦
ξn(M2n(A))∩B(H)
+ = π(C2n). Indeed, the inclusion π ◦ ξ(C2n) ⊆M2n(A)∩
B(H)+ is obvious. To show the converse take x ∈M2n(A) such that π(x) ≥ 0.
Then x ∈ C∩M2n(A). Using (c) one can easily show that C∩M2n(A) = C2n .
Hence π ◦ ξn(M2n(A)) ∩ B(H)
+ = π(C2n). Since τ
(2n) is unitary equivalent
to π ◦ ξn we have that τ
(2n)(C2n) =M2n(τ(A)) ∩ B(H
2n)+.
Let now show that τ (n)(Cn) =Mn(τ(A))
+. For X ∈Mn(A) denote
X 0n×(2n−n)
0(2n−n)×n 0(2n−n)×(2n−n)
∈M2n(A).
Then, clearly, τ (n)(X) ≥ 0 if and only if τ (2
n)(X̃) ≥ 0. Thus τ (n)(X) ≥ 0 is
equivalent to X̃ ∈ C2n which in turn is equivalent to X ∈ Cn by (c).
3 Operator Algebras completely boundedly
isomorphic to C∗-algebras.
The algebra Mn(B(H)) of n× n matrices with entries in B(H) has a norm
‖ · ‖n via the identification ofMn(B(H)) with B(H
n), where Hn is the direct
sum of n copies of a Hilbert space H . If A is a subalgebra of B(H) then
Mn(A) inherits a norm ‖·‖n via natural inclusion intoMn(B(H)). The norms
‖ · ‖n are called matrix norms on the operator algebra A. In the sequel all
operator algebras will be assumed to be norm closed.
Operator algebras A and B are called completely boundedly isomorphic
if there is a completely bounded isomorphism τ : A → B with completely
bounded inverse. The aim of this section is to give necessary and sufficient
conditions for an operator algebra to be completely boundedly isomorphic
to a C∗-algebra. To do this we introduce a concept of ∗-admissible cones
which reflect the properties of the cones of positive elements of a C∗-algebra
preserved under completely bounded isomorphism.
Definition 3. Let B be an operator algebra with unit e. A sequence Cn ⊆
Mn(B) of closed (in the norm ‖ · ‖n) cones will be called ∗-admissible if it
satisfies the following conditions:
1. e ∈ C1;
2. (i) Mn(B) = (Cn − Cn) + i(Cn − Cn), for all n ∈ N,
(ii) Cn ∩ (−Cn) = {0}, for all n ∈ N,
(iii) (Cn − Cn) ∩ i(Cn − Cn) = {0}, for all n ∈ N;
3. (i) for all c1, c2 ∈ Cn and c ∈ Cn, we have that (c1−c2)c(c1−c2) ∈ Cn,
(ii) for all n, m and B ∈Mn×m(C) we have that B
∗CnB ⊆ Cm;
4. there is r > 0 such that for every positive integer n and c ∈ Cn − Cn
we have r‖c‖en + c ∈ Cn,
5. there exists a constant K > 0 such that for all n ∈ N and a, b ∈ Cn−Cn
we have ‖a‖n ≤ K · ‖a+ ib‖n.
Theorem 4. If an operator algebra B has a ∗-admissible sequence of cones
then there is a completely bounded isomorphism τ from B onto a C∗-algebra
A. If, in addition, one of the following conditions holds
(1) there exists r > 0 such that for every n ≥ 1 and c, d ∈ Cn we have
‖c+ d‖ ≥ r‖c‖.
(2) ‖(x− iy)(x+ iy)‖ ≥ α‖x− iy‖‖x+ iy‖ for all x, y ∈ Cn − Cn
then the inverse τ−1 : A → B is also completely bounded.
Conversely, if such isomorphism τ exists then B possesses a ∗-admissible
sequence of cones and conditions (1) and (2) are satisfied.
The proof will be divided into 4 lemmas.
Let {Cn}n≥1 be a ∗-admissible sequence of cones of B. Let B2n =M2n(B),
φn : B2n → B2n+1 be unital homomorphisms given by φn(x) =
x ∈ B2n . Denote by B∞ = lim−→
B2n the inductive limit of the system (B2n , φn).
As all inclusions φn are unital B∞ has a unit, denoted by e∞. Since B∞ can
be considered as a subalgebra of a C∗-algebra of the corresponding induc-
tive limit of M2n(B(H)) we can define the closure of B∞ in this C
∗-algebra
denoted by B∞.
Now we will define an involution on B∞. Let ξn : M2n(B) → B∞ be the
canonical morphisms. By (3ii), φn(C2n) ⊆ C2n+1 . Hence C =
ξn(C2n) is a
well defined cone in B∞. Denote by C its completion. By (2i) and (2iii), for
every x ∈ B2n , we have x = x1+ ix2 with unique x1, x2 ∈ C2n −C2n . By (3ii)
we have
∈ C2n+1 − C2n+1, i = 1, 2. Thus for every x ∈ B∞ we
have unique decomposition x = x1+ ix2, x1 ∈ C−C, x2 ∈ C−C. Hence the
mapping x 7→ x♯ = x1− ix2 is a well defined involution on B∞. In particular,
we have an involution on B which depends only on the cone C1.
Lemma 5. Involution on B∞ is defined by the involution on B, i.e. for all
A = (aij)i,j ∈M2n(B)
A♯ = (a
ji)i,j.
Proof. Assignment A◦ = (a
ji)i,j, clearly, defines an involution on M2n(B).
We need to prove that A♯ = A◦.
Let A = (aij)i,j ∈ M2n(B) be self-adjoint A
◦ = A. Then A =
aii ⊗
Eii +
(aij ⊗ Eij + a
ij ⊗ Eji) and a
ii = aii, for all i. By (3ii) we have
aii ⊗ Eii ∈ C2n − C2n . Since aij = a
ij + ia
ij for some a
ij , a
ij ∈ C2n − C2n
we have
aij ⊗Eij + a
ij ⊗ Eji = (a
ij + ia
ij)⊗ Eij + (a
ij − ia
ij)⊗ Eji
= (a′ij ⊗ Eij + a
ij ⊗ Eji) + (ia
ij ⊗ Eij − ia
ij ⊗ Eji)
= (Eii + Eji)(a
ij ⊗ Eii + a
ij ⊗ Ejj)(Eii + Eij)
− (a′ij ⊗ Eii + a
ij ⊗ Ejj)
+ (Eii − iEji)(a
ij ⊗ Eii + a
ij ⊗ Ejj)(Eii + iEij)
− (a′′ij ⊗ Eii + a
ij ⊗ Ejj) ∈ C2n − C2n .
Thus A ∈ C2n − C2n and A
♯ = A. Since for every x ∈ M2n(B) there exist
unique x1 = x
1 and x2 = x
2 in M2n(B), such that x = x1 + ix2, and unique
x′1 = x
1 and x
2 = x
2 , such that x = x
1 + ix
2, we have that x1 = x
1 = x
x2 = x
2 = x
2 and involutions ♯ and ◦ coincide.
Lemma 6. Involution x → x♯ is continuous on B∞ and extends to the in-
volution on B∞. With respect to this involution C ⊆ (B∞)sa and x
♯Cx ⊆ C
for every x ∈ B∞.
Proof. Consider a convergent net {xi} ⊆ B∞ with the limit x ∈ B∞. Decom-
pose xi = x
i with x
i ∈ C−C. By (5), the nets {x
i} and {x
i } are also
convergent. Thus x = a+ ib, where a = lim x′i ∈ C − C, b = lim x
i ∈ C − C
and lim x
i = a− ib. Therefore the involution defined on B∞ can be extended
by continuity to B∞ by setting x
♯ = a− ib.
Under this involution C ⊆ (B∞)sa = {x ∈ B∞ : x = x
Let us show that x♯cx ∈ C for every x ∈ B∞ and c ∈ C. Take firstly
c ∈ C2n and x ∈ B2n . Then x = x1 + ix2 for some x1, x2 ∈ C2n − C2n and
(x1 + ix2)
♯c(x1 + ix2) = (x1 − ix2)c(x1 + ix2)
)( −x1 −ix2
ix2 x1
−x1 −ix2
ix2 x1
By (3i), Lemma 5 and (3ii) x♯cx ∈ C2n .
Let now c ∈ C and x ∈ B∞. Suppose that ci → c and xi → x, where
ci ∈ C, xi ∈ B∞. We can assume that ci, xi ∈ B2ni . Then x
icixi ∈ C2ni for
all i and since it is convergent we have x♯cx ∈ C.
Lemma 7. The unit of B∞ is an Archimedean order unit and (B∞)sa =
C − C.
Proof. Firstly let us show that e∞ is an order unit. Clearly, (B∞)sa = C − C.
For every a ∈ C − C, there is a net ai ∈ C2ni − C2ni convergent to a. Since
‖ai‖ <∞ there exists r1 > 0 such that r1eni − ai ∈ C2ni , i.e. r1e∞− ai ∈
C. Passing to the limit we get r1e∞ − a ∈ C. Replacing a by −a we can
find r2 > 0 such that r2e∞ + a ∈ C. If r = max(r1, r2) then re∞ ± a ∈ C.
This proves that e∞ is an order unit and that for all a ∈ C − C we have
a = re∞ − c for some c ∈ C. Thus C − C ∈ C − C. The converse inclusion,
clearly, holds. Thus C − C = C − C.
If x ∈ (B∞)sa such that for every r > 0 we have r + x ∈ C then x ∈ C
since C is closed. Hence e∞ is an Archimedean order unit.
Lemma 8. B∞ ∩ C = C.
Proof. Denote by D = lim
M2n(B(H)) the C
∗-algebra inductive limit corre-
sponding to the inductive system φn and denote φn,m = φm−1 ◦ . . . ◦ φn :
M2n(B(H)) → M2m(B(H)). For n < m we identify M2m−n(M2n(B(H)))
with M2m(B(H)) by omitting superfluous parentheses in a block matrix
B = [Bij ]ij with Bij ∈M2n(B(H)).
Denote by Pn,m the operator diag(I, 0, . . . , 0) ∈M2m−n(M2n(B(H))) and
set Vn,m =
∑2m−n
k=1 Ek,k−1. Here I is the identity matrix in M2n(B(H)) and
Ek,k−1 is 2
n×2n block matrix with identity operator at (k, k−1)-entry and all
other entries being zero. Define an operator ψn,m([Bij ]) = diag(B11, . . . , B11).
It is easy to see that
ψn,m([Bij ]) =
2m−n−1∑
(V kn,mPn,m)B(V
n,mPn,m)
Hence by (3ii)
ψn,m(C2m) ⊆ φ(C2n) ⊆ C2m . (4)
Clearly, ψn,m is a linear contraction and
ψn,m+k ◦ φm,m+k = φm,m+k ◦ ψn,m
Hence there is a well defined contraction ψn = lim
ψn,m : D → D such that
ψn|M2n(B(H)) = idM2n (B(H)),
whereM2n(B(H)) is considered as a subalgebra in D. Clearly, ψn(B∞) ⊆ B∞
and ψn|B2n = id. Consider C and C2n as subalgebras in B∞, by (4) we have
ψn : C → C2n.
To prove that B∞ ∩C = C take c ∈ B∞ ∩C. Then there is a net cj in C
such that ‖cj − c‖ → 0. Since c ∈ B∞, c ∈ B2n for some n, and consequently
ψn(c) = c. Thus
‖ψn(cj)− c‖ = ‖ψn(cj − c)‖ ≤ ‖cj − c‖.
Hence ψn(cj) → c. But ψn(cj) ∈ C2n and the latter is closed. Thus c ∈ C.
The converse inclusion is obvious.
Remark 9. Note that for every x ∈ D
ψn(x) = x. (5)
Indeed, for every ε > 0 there is x ∈ M2n(B(H)) such that ‖x − xn‖ < ε.
Since ψn is a contraction and ψn(xn) = xn we have
‖ψn(x)− x‖ ≤ ‖ψn(x)− xn‖+ ‖xn − x‖
= ‖ψn(x− xn)‖+ ‖xn − x‖ ≤ 2ε.
Since xn ∈ M2n(B(H)) also belong to M2m(B(H)) for all m ≥ n, we have
that ‖ψm(x)− x‖ ≤ 2ε. Thus lim
ψn(x) = x.
Proof of Theorem 4. By Lemma 6 and 7 the cone C and the unit
e∞ satisfies all assumptions of Theorem 1. Thus there is a homomorphism
τ : B∞ → B(H̃) such that τ(a
♯) = τ(a)∗ for all a ∈ B∞. Since the image
of τ is a ∗-subalgebra of B(H̃) we have that τ is bounded by [3, (23.11),
p. 81]. The arguments at the end of the proof of Theorem 2 show that the
restriction of τ to B2n is unitary equivalent to the 2
n-amplification of τ |B.
Thus τ |B is completely bounded.
Let us prove that ker(τ) = {0}. By Theorem 2.3 it is sufficient to show
that C ∩ (−C) = 0. If c, d ∈ C such that c + d = 0 then c = d = 0. Indeed,
for every n ≥ 1, ψn(c) + ψn(d) = 0. By Lemma 8, we have
ψn(C) ⊆ C ∩ B2n = C2n .
Therefore ψn(c), ψn(d) ∈ C2n . Hence ψn(c) = −ψn(d) ∈ C2n ∩ (−C2n) and,
consequently, ψn(c) = ψn(d) = 0. Since ‖ψn(c)−c‖ → 0 and ‖ψn(d)−d‖ → 0
by Remark 9, we have that c = d = 0. If x ∈ C ∩ (−C) then x+ (−x) = 0,
x,−x ∈ C and x = 0. Thus τ is injective.
We will show that the image of τ is closed if one of the conditions (1) or
(2) of the statement holds.
Assume firstly that operator algebra B satisfies the first condition. Since
τ(B∞) = τ(C)−τ(C)+ i(τ(C)−τ(C)) and τ(C) is exactly the set of positive
operators in the image of τ , it is suffices to prove that τ(C) is closed. By
Theorem 1.3, for self-adjoint (under involution ♯) x ∈ B∞ we have
‖τ(x)‖
B( eH)
= inf{r > 0 : re∞ ± x ∈ C}.
If τ(cα) ∈ τ(C) is a Cauchy net in B(H̃) then for every ε > 0 there is γ
such that ε ± (cα − cβ) ∈ C when α ≥ γ and β ≥ γ. Since C ∩ B∞ = C,
ε ± (cα − cβ) ∈ C. Denote cαβ = ε + (cα − cβ) and dαβ = ε − (cα − cβ).
The set of pairs (α, β) is directed if (α, β) ≥ (α1, β1) iff α ≥ α1 and β ≥ β1.
Since cαβ + dαβ = 2ε this net converges to zero in the norm of B∞. Thus by
assumption 4 in the definition of ∗-admissible sequence of cones, ‖cαβ‖B∞ →
0. This implies that cα is a Cauchy net in B∞. Let c = lim cα. Clearly,
c ∈ C. Since τ is continuous ‖τ(cα) − τ(c)‖B∞ → 0. Hence the closure
τ(C) is contained in τ(C). By continuity of τ we have τ(C) ⊆ τ(C). Hence
τ(C) = τ(C), τ(C) is closed.
Let now B satisfy condition (2) of the Theorem. Then for every x ∈ B∞
we have ‖x♯x‖ ≥ α‖x‖‖x♯‖. By [3, theorem 34.3] B∞ admits an equivalent
C∗-norm |·|. Since τ is a faithful ∗-representation of the C∗-algebra (B∞, |·|)
it is isometric. Therefore τ(B∞) is closed.
Let us show that (τ |B)
−1 : τ(B) → B is completely bounded. The image
A = τ(B∞) is a C
∗-algebra inB(H̃) isomorphic to B∞. By Johnson’s theorem
(see [6]), two Banach algebra norms on a semi-simple algebra are equivalent,
hence, τ−1 : A → B∞ is bounded homomorphism, say ‖τ
−1‖ = R. Let us
show that ‖(τ |B)
−1‖cb = R. Since
τ |B2n = Un(τ |B ⊗ idM2n )U
for some unitary Un : K ⊗ C
2n → H̃ we have for any B = [bij ] ∈M2n(B)
bij ⊗ Eij‖ ≤ R‖τ(
bij ⊗ Eij)‖
= R‖Un(
τ(bij)⊗ Eij)U
τ(bij)⊗ Eij‖.
This is equivalent to
τ−1(bij)⊗ eij‖ ≤ R‖
bij ⊗Eij‖,
hence ‖(τ−1)2
(B)‖ ≤ R‖B‖. This proves that ‖(τ |B)
−1‖cb = R.
The converse statement evidently holds with ∗-admissible sequence of
cones given by (τ (n))−1(Mn(A)
Conditions (1) and (2) were used to prove that the image of isomorphism τ
is closed. The natural question one can ask is wether there exists an operator
algebra B and isomorphism ρ : B → B(H) with non-closed self-adjoint image.
The following example gives the affirmative answer.
Example 10. Consider the algebra B = C1([0, 1]) as an operator algebra in
C∗-algebra
M2(C([0, 1])) via inclusion
f(·) 7→ ⊕q∈Q
f(q) f ′(q)
0 f(q)
The induced norm
‖f‖ = sup
(2|f(q)|2 + |f ′(q)|2 + |f ′(q)|
4|f(q)|2 + |f ′(q)|2)
satisfies the inequality ‖f‖ ≥ 1√
max{‖f‖∞, ‖f
′‖∞} ≥
‖f‖1 where ‖f‖1 =
‖f‖∞+‖f
′‖∞ is the standard Banach norm on C
1([0, 1]). Thus B is a closed
operator algebra with isometric involution f ♯(x) = f(x), (x ∈ [0, 1]). The
identity map C1([0, 1]) → C([0, 1]), f 7→ f is a ∗-isomorphism of B into
C∗-algebra with non-closed self-adjoint image.
4 Operator Algebra associated with Kadison’s
similarity problem.
In 1955 R. Kadison raised the following problem. Is any bounded homomor-
phism π of a C∗-algebra A into B(H) similar to a ∗-representation? The
similarity above means that there exists invertible operator S ∈ B(H) such
that x→ S−1π(x)S is a ∗-representation of A.
The following criterion due to Haagerup (see [4]) is widely used in refor-
mulations of Kadison’s problem: non-degenerate homomorphism π is similar
to a ∗-representation iff π is completely bounded. Moreover the similarity S
can be chosen in such a way that ‖S−1‖‖S‖ = ‖π‖cb.
The affirmative answer to the Kadison’s problem is obtained in many
important cases. In particular, for nuclear A, π is automatically completely
bounded with ‖π‖cb ≤ ‖π‖
2 (see [1]).
About recent state of the problem we refer the reader to [9, 5].
We can associate an operator algebra π(B) to every bounded injective
homomorphism π of a C∗-algebra A. The fact that π(B) is closed can be seen
by restricting π to a nuclear C∗-algebra C∗(x∗x). This restriction is similar to
∗-homomorphism for every x ∈ A which gives the estimate ‖x‖ ≤ ‖π‖3‖π(x)‖
(for details see [10, p. 4]). Denote Cn = π
(n)(Mn(A)
Let J be an involution in B(H), i.e. self-adjoint operator such that J2 =
I. Clearly, J is also a unitary operator. A representation π : A → B(H) of a
∗-algebra A is called J-symmetric if π(a∗) = Jπ(a)∗J . Such representations
are natural analogs of ∗-representations for Krein space with indefinite metric
[x, y] = 〈Jx, y〉.
We will need the following observation due to V. Shulman [13] (see also
[7, lemma 9.3, p.131]). If π is an arbitrary representation of A in B(H) then
the representation ρ : A → B(H⊕H), a 7→ π(a)⊕π(a∗)∗ is J-symmetric with
J(x⊕ y) = y⊕x and representation π is a restriction ρ|K⊕{0}. Moreover, if ρ
is similar to ∗-representation then so is π. Clearly the converse is also true,
thus π and ρ are simultaneously similar to ∗-representations or not. In sequel
for an operator algebra D ∈ B(H) we denote by lim
M2n(D) the closure of the
algebraic direct limit of ofM2n(D) in the C
∗-algebra direct limit of inductive
system M2n(B(H)) with standard inclusions x→
Theorem 11. Let π : A → B(H) be a bounded unital J-symmmetric in-
jective homomorphism of a C∗-algebra A and let B = π(A). Then π−1 is
a completely bounded homomorphism. Its extension π̃−1 to the homomor-
phism between the inductive limits B∞ = lim−→
M2n(B) and A∞ = lim−→
M2n(A)
is injective.
Proof. Let us show that {Cn}n≥1 is a ∗-admissible sequence of cones. It
is routine to verify that conditions (1)-(3) in the definition of ∗-admissible
cones are satisfied for {Cn}. To see that condition (4) also holds take B ∈
Cn − Cn and denote r = ‖B‖. Let D ∈ Mn(A)sa be such that B = π
(n)(D).
Since π(n) : Mn(A) → Mn(B) is algebraic isomorphism it preserves spectra
σMn(A)(x) = σMn(B)(π
(n)(x)). Since the spectral radius spr(B) ≤ r we have
spr(D) ≤ r. Hence ren +D ∈ Mn(A)
+ because D is self-adjoint. Applying
π(n) we get ren +B ∈ Cn which proves condition (4).
Since π is J-symmetric
‖π(n)(a)‖ = ‖(J ⊗En)π
(n)(a)∗(J ⊗En)‖ = ‖π
(n)(a∗)‖
for every a ∈Mn(A), and
‖π(n)(h1)‖ ≤ 1/2(‖π
(n)(h1) + iπ
(n)(h2)‖+ ‖π
(n)(h1)− iπ
(n)(h2)‖)
= ‖π(n)(h1) + iπ
(n)(h2)‖
for all h1, h2 ∈ Cn − Cn. Thus condition (5) is satisfied and {Cn} is ∗-
admissible. By Theorem 4, there is an injective bounded homomorphism
τ : B∞ → B(H̃) such that its restriction to B is completely bounded, τ(b
τ(b)∗ and τn(Cn) = τn(Mn(B))
Denote ρ = τ ◦ π : A → B(H̃). Since ρ is a positive homomorphism, it is
a ∗-representation. Moreover, ker ρ = {0} because both π and τ are injective.
Therefore ρ−1 is ∗-isomorphism. Since τ : B → B(H̃) extends to an injective
homomorphism of inductive limit B∞ and ρ
−1 is completely isometric, we
have that π−1 = ρ−1 ◦ τ extends to injective homomorphism of B∞. It is also
clear that π−1 is completely bounded as a superposition of two completely
bounded maps.
Remark 12. The first statement of Theorem 11 can be deduced also from [10,
Theorem 2.6].
Remark 13. Note that condition (1) and (2) in Theorem 4 for cones Cn
from the proof of Theorem 11 is obviously equivalent to π being completely
bounded.
Acknowledgments.
The authors wish to express their thanks to Victor Shulman for helpful
comments and providing the reference [13].
The work was written when the second author was visiting Chalmers
University of Technology in Göteborg, Sweden. The second author was sup-
ported by the Swedish Institute.
References
[1] J. Bunce, The similarity problem for representations of C∗-algebras,
Proc. Amer. Math. Soc. 81 (1981), p. 409-414.
[2] M.D. Choi, E.G. Effros, Injectivity and operator spaces. J. Functional
Analysis 24 (1977), no. 2, 156–209.
[3] R.S. Doran, V.A. Belfi, Characterizations of C∗-algebras. The Gelfand-
Năımark theorems. Monographs and Textbooks in Pure and Applied
Mathematics, 101. Marcel Dekker, Inc., New York, 1986. xi+426 pp.
[4] U. Haagerup, Solution of the similarity problem for cyclic representa-
tions of C∗-algebras, Annals of Math. 118 (1983), p. 215-240
[5] D. Hadwin, V. Paulsen, Two reformulations of Kadison’s similarity
problem, J. Oper. Theory, Vol. 55, No. 1, (2006), 3-16.
[6] B. Johnson The uniqueness of the (complete) norm topology, Bull.
Amer. Math. Soc. 73 (1967), 537-539
[7] E. Kissin, V. Shulman, Representations on Krein spaces and deriva-
tions of C∗-algebras, Pitman Monographs and Surveys in Pure and Ap-
plied Mathematics 89, 1997
[8] C. Le Merdy, Self adjointness criteria for operator algebras, Arch.
Math. 74 (2000), p. 212- 220.
[9] G. Pisier, Similarity Problems and Completely Bounded Maps,
Springer-Verlag Lecture Notes in Math 1618, 1996
[10] D. Pitts, Norming algebras and automatic complete boundedness of
isomorphism of operator algebras, arXiv: math.OA/0609604, 2006
http://arxiv.org/abs/math/0609604
[11] S. Popovych, On O∗-representability and C∗-representability of ∗-
algebras, Chalmers & Göteborg University math. preprint 2006:35.
[12] R. Powers, Selfadjoint algebras of unbounded operators II, Trans.
Amer. Math. Soc. 187 (1974), 261–293.
[13] V.S. Shulman, On representations of C∗-algebras on indefinite metric
spaces, Mat. Zametki, 22(1977), 583-592 = Math Notes 22(1977)
Introduction
Operator realizations of matrix-ordered *-algebras.
Operator Algebras completely boundedly isomorphic to C*-algebras.
Operator Algebra associated with Kadison's similarity problem.
|
0704.1554 | Properly infinite C(X)-algebras and K_1-injectivity | Properly infinite C(X)-algebras and K1-injectivity
Etienne Blanchard, Randi Rohde and Mikael Rørdam
Abstract
We investigate if a unital C(X)-algebra is properly infinite when all its fibres are prop-
erly infinite. We show that this question can be rephrased in several different ways,
including the question if every unital properly infinite C∗-algebra is K1-injective.
We provide partial answers to these questions, and we show that the general ques-
tion on proper infiniteness of C(X)-algebras can be reduced to establishing proper
infiniteness of a specific C([0, 1])-algebra with properly infinite fibres.
1 Introduction
The problem that we mainly are concerned with in this paper is if any unital C(X)-algebra
with properly infinite fibres is itself properly infinite (see Section 2 for a brief introduction
to C(X)-algebras). An analogous study was carried out in the recent paper [8] where it
was decided when C(X)-algebras, whose fibres are either stable or absorb tensorially a
given strongly self-absorbing C∗-algebra, itself has the same property. This was answered
in the affirmative in [8] under the crucial assumption that the dimension of the space X is
finite, and counterexamples were given in the infinite dimensional case.
Along similar lines, Dadarlat, [5], recently proved that C(X)-algebras, whose fibres are
Cuntz algebras, are trivial under some K-theoretical conditions provided that the space X
is finite dimensional.
The property of being properly infinite turns out to behave very differently than the
property of being stable or of absorbing a strongly self-absorbing C∗-algebra. It is relative
easy to see (Lemma 2.10) that if a fibre Ax of a C(X)-algebra A is properly infinite, then
AF is properly infinite for some closed neighborhood F of x. The (possible) obstruction
to proper infiniteness of the C(X)-algebra is hence not local. Such an obstruction is also
not related to the possible complicated structure of the space X , as we can show that a
counterexample, if it exists, can be taken to be a (specific) C([0, 1])-algebra (Example 4.1
and Theorem 5.5). The problem appears to be related with some rather subtle internal
structure properties of properly infinite C∗-algebras.
Cuntz studied purely infinite—and in the process also properly infinite—C∗-algebras,
[4], where he among many other things (he was primarily interested in calculating the
http://arxiv.org/abs/0704.1554v1
K-theory of his algebras On) showed that any unital properly infinite C
∗-algebra A is K1-
surjective, i.e., the mapping U(A) → K1(A) is onto; and that any purely infinite simple
C∗-algebra A is K1-injective, i.e., the mapping U(A)/U
0(A) → K1(A) is injective (and
hence an isomorphism). He did not address the question if any properly infinite C∗-al-
gebra is K1-injective. That question has not been raised formally to our knowledge—we
do so here—but it does appear implicitly, eg. in [10] and in [14], where K1-injectivity of
properly infinite C∗-algebras has to be assumed.
Proper infiniteness of C∗-algebras has relevance for existence (or rather non-existence)
of traces and quasitraces. Indeed, a unital C∗-algebra admits a 2-quasitrace if and only if
no matrix algebra over the C∗-algebra is properly infinite, and a unital exact C∗-algebra
admits tracial state again if and only if no matrix algebra over the C∗-algebra is properly
infinite.
In this paper we show that every properly infinite C∗-algebra is K1-injective if and only
if every C(X)-algebra with properly infinite fibres itself is properly infinite. We also show
that a matrix algebra over any such C(X)-algebra is properly infinite. Examples of unital
C∗-algebras A, where Mn(A) is properly infinite for some natural number n ≥ 2 but where
Mn−1(A) is not properly infinite, are known, see [12] and [11], but still quite exotic.
We relate the question if a given properly infinite C∗-algebra isK1-injective to questions
regarding homotopy of projections (Proposition 5.1). In particular we show that our main
questions are equivalent to the following question: is any non-trivial projection in the first
copy of O∞ in the full unital universal free product O∞∗O∞ homotopic to any (non-trivial)
projection in the second copy of O∞? The specific C([0, 1])-algebra, mentioned above, is
perhaps not surprisingly a sub-algebra of C([0, 1],O∞ ∗ O∞).
Using ideas implicit in Rieffel’s paper, [9], we construct in Section 4 a C(T)-algebra
B for each C∗-algebra A and for each unitary u ∈ A for which diag(u, 1) is homotopic to
1M2(A); and B is non-trivial if u is not homotopic to 1A. In this way we relate our question
about proper infiniteness of C(X)-algebras to a question about K1-injectivity.
The last mentioned author thanks Bruce Blackadar for many inspiring conversations
on topics related to this paper.
2 C(X)-algebras with properly infinite fibres
A powerful tool in the classification of C∗-algebras is the study of their projections. A
projection in a C∗-algebra is said to be infinite if it is equivalent to a proper subprojection
of itself, and it is said to be properly infinite if it is equivalent to two mutually orthogonal
subprojections of itself.
A projection which is not infinite is said to be finite. A unital C∗-algebra is said
to be finite, infinite, or properly infinite if its unit is finite, infinite, or properly infinite,
respectively. If A is a C∗-algebra for which Mn(A) is finite for all positive integers n, then
A is stably finite.
In this section we will study stability properties of proper infiniteness under (upper-
semi-)continuous deformations using the Cuntz-Toeplitz algebra which is defined as follows.
For all integers n ≥ 2 the Cuntz-Toeplitz algebra Tn is the universal C
∗-algebra generated
by n isometries s1, . . . , sn satisfying the relation
1 + · · ·+ sns
n ≤ 1.
Remark 2.1 A unital C∗-algebra A is properly infinite if and only if Tn embeds unitally
into A for some n ≥ 2, in which case Tn embeds unitally into A for all n ≥ 2.
In order to study deformations of such algebras, let us recall a few notions from the theory
of C(X)-algebras.
Let X be a compact Hausdorff space and C(X) be the C∗-algebra of continuous func-
tions on X with values in the complex field C.
Definition 2.2 A C(X)-algebra is a C∗-algebra A endowed with a unital ∗-homomorphism
from C(X) to the center of the multiplier C∗-algebra M(A) of A.
If A is as above and Y ⊆ X is a closed subset, then we put IY = C0(X \ Y )A, which is a
closed two-sided ideal in A. We set AY = A/IY and denote the quotient map by πY .
For an element a ∈ A we put aY = πY (a), and if Y consists of a single point x, we will
write Ax, Ix, πx and ax in the place of A{x}, I{x}, π{x} and a{x}, respectively. We say that
Ax is the fibre of A at x.
The function
x 7→ ‖ax‖ = inf{‖ [1− f + f(x)]a‖ : f ∈ C(X)}
is upper semi-continuous for all a ∈ A (as one can see using the right-hand side identity
above). A C(X)-algebra A is said to be continuous (or to be a continuous C∗-bundle over
X) if the function x 7→ ‖ax‖ is actually continuous for all element a in A.
For any unital C∗-algebra A we let U(A) denote the group of unitary elements in A,
U0(A) denotes its connected component containing the unit of A, and Un(A) and U
are equal to U(Mn(A)) and U
0(Mn(A)), respectively.
An element in a C∗-algebra A is said to be full if it is not contained in any proper
closed two-sided ideal in A.
It is well-known (see for example [13, Exercise 4.9]) that if p is a properly infinite, full
projection in a C∗-algebra A, then e - p, i.e., e is equivalent to a subprojection of p, for
every projection e ∈ A.
We state below more formally three more or less well-known results that will be used
frequently throughout this paper, the first of which is due to Cuntz, [4].
Proposition 2.3 (Cuntz) Let A be a C∗-algebra which contains at least one properly
infinite, full projection.
(i) Let p and q be properly infinite, full projections in A. Then [p] = [q] in K0(A) if and
only if p ∼ q.
(ii) For each element g ∈ K0(A) there is a properly infinite, full projection p ∈ A such
that g = [p].
The second statement is a variation of the Whitehead lemma.
Lemma 2.4 Let A be a unital C∗-algebra.
(i) Let v be a partial isometry in A such that 1 − vv∗ and 1 − v∗v are properly infinite
and full projections. Then there is a unitary element u in A such that [u] = 0 in
K1(A) and v = uv
∗v, i.e., u extends v.
(ii) Let u be a unitary element A such that [u] = 0 in K1(A). Suppose there exists a
projection p ∈ A such that ‖up− pu‖ < 1 and p and 1 − p are properly infinite and
full. Then u belongs to U0(A).
Proof: (i). It follows from Proposition 2.3 (i) that 1 − v∗v ∼ 1 − vv∗, so there is a
partial isometry w such that 1 − v∗v = w∗w and 1 − vv∗ = ww∗. Now, z = v + w is
a unitary element in A with zv∗v = v. The projection 1 − v∗v is properly infinite and
full, so 1 - 1 − v∗v, which implies that there is an isometry s in A with ss∗ ≤ 1 − v∗v.
As −[z] = [z∗] = [sz∗s∗ + (1 − ss∗)] in K1(A) (see eg. [13, Exercise 8.9 (i)]), we see that
u = z(sz∗s∗ + (1− ss∗)) is as desired.
(ii). Put x = pup + (1 − p)u(1 − p) and note that ‖u − x‖ < 1. It follows that x is
invertible in A and that u ∼h x in GL(A). Let x = v|x| be the polar decomposition of
x, where |x| = (x∗x)1/2 and v = x|x|−1 is unitary. Then u ∼h v in U(A) (see eg. [13,
Proposition 2.1.8]), and pv = vp. We proceed to show that v belongs to U0(A) (which will
entail that u belongs to U0(A)).
Write v = v1v2, where
v1 = pvp+ (1− p), v2 = p+ (1− p)v(1− p).
As 1−p - p we can find a symmetry t in A such that t(1−p)t ≤ p. As t belongs to U0(A)
(being a symmetry), we conclude that v2 ∼h tv2t, and one checks that tv2t is of the form
w + (1− p) for some unitary w in pAp. It follows that v is homotopic to a unitary of the
form v0 + (1− p), where v0 is a unitary in pAp. We can now apply eg. [13, Exercise 8.11]
to conclude that v ∼h 1 in U(A). �
We remind the reader that if p, q are projections in a unital C∗-algebra A, then p and q are
homotopic, in symbols p ∼h q, (meaning that they can be connected by a continuous path
of projections in A) if and only if q = upu∗ for some u ∈ U0(A), eg. cf. [13, Proposition
2.2.6].
Proposition 2.5 Let A be a unital C∗-algebra. Let p and q be two properly infinite, full
projections in A such that p ∼ q. Suppose that there exists a properly infinite, full projection
r ∈ A such that p ⊥ r and q ⊥ r. Then p ∼h q.
Proof: Take a partial isometry v0 ∈ A such that v
0v0 = p and v0v
0 = q. Take a subpro-
jection r0 of r such that r0 and r− r0 both are properly infinite and full. Put v = v0 + r0.
Then vpv∗ = q and vr0 = r0 = r0v. Note that 1 − v
∗v and 1 − vv∗ are properly infi-
nite and full (because they dominate the properly infinite, full projection r − r0). Use
Lemma 2.4 (i) to extend v to a unitary u ∈ A with [u] = 0 in K1(A). Now, upu
∗ = q and
ur0 = vr0 = r0 = r0v = r0u. Hence u ∈ U
0(A) by Lemma 2.4 (ii), and so p ∼h q as desired.
Definition 2.6 A unital C∗-algebra A is said to be K1-injective if the natural mapping
U(A)/U0(A) → K1(A)
is injective. In other words, if A is K1-injective, and if u is a unitary element in A, then
u ∼h 1 in U(A) if (and only if) [u] = 0 in K1(A).
One could argue thatK1-injectivity should entail that the natural mappings Un(A)/U
n(A) →
K1(A) be injective for every natural number n. However there seem to be an agreement for
defining K1-injectivity as above. As we shall see later, in Proposition 5.2, if A is properly
infinite, then the two definitions agree.
Proposition 2.7 Let A be a unital C∗-algebra that is the pull-back of two unital, properly
infinite C∗-algebras A1 and A2 along the
∗-epimorphisms π1 : A1 → B and π2 : A2 → B:
}} ϕ2
π1 A
π2~~}}
Then M2(A) is properly infinite. Moreover, if B is K1-injective, then A itself is properly
infinite.
Proof: Take unital embeddings σi : T3 → Ai for i = 1, 2, where T3 is the Cuntz-Toeplitz
algebra (defined earlier), and put
(π1 ◦ σ1)(tj)(π2 ◦ σ2)(t
where t1, t2, t3 are the canonical generators of T3. Note that v is a partial isometry with
(π1 ◦ σ1)(tj) = v(π2 ◦ σ2)(tj) for j = 1, 2. As (π1 ◦ σ1)(t3t
3) ≤ 1− vv
∗ and (π2 ◦ σ2)(t3t
1−v∗v, Lemma 2.4 (i) yields a unitary u ∈ B with [u] = 0 in K1(B) and with (π1◦σ1)(tj) =
u(π2 ◦ σ2)(tj) for j = 1, 2.
If B is K1-injective, then u belongs to U
0(B), whence u lifts to a unitary v ∈ A2.
Define σ̃2 : T2 → A2 by σ̃2(tj) = vσ2(tj) for j = 1, 2 (observing that t1, t2 generate T2).
Then π1 ◦ σ1 = π2 ◦ σ̃2, which by the universal property of the pull-back implies that σ1
and σ̃2 lift to a (necessarily unital) embedding σ : T2 → A, thus forcing A to be properly
infinite.
In the general case (where B is not necessarily K1-injective) u may not lift to a unitary
element in A2, but diag(u, u) does lift to a unitary element v in M2(A2) by Lemma 2.4 (ii)
(applied with p = diag(1, 0)). Define unital embeddings σ̃i : T2 → M2(Ai), i = 1, 2, by
σ̃1(tj) =
σ1(tj) 0
0 σ1(tj)
, σ̃2(tj) = v
σ2(tj) 0
0 σ2(tj)
for j = 1, 2. As (π1 ⊗ idM2) ◦ σ̃1 = (π2 ⊗ idM2) ◦ σ̃2, the unital embeddings σ̃1 and σ̃2 lift
to a (necessarily unital) embedding of T2 into M2(A), thus completing the proof. �
Question 2.8 Is the pull-back of any two properly infinite unital C∗-algebras again prop-
erly infinite?
As mentioned in the introduction, one cannot in general conclude that A is properly infinite
if one knows that Mn(A) is properly infinite for some n ≥ 2.
One obvious way of obtaining an answer to Question 2.8, in the light of the last state-
ment in Proposition 2.7, is to answer the question below in the affirmative:
Question 2.9 Is every properly infinite unital C∗-algebra K1-injective?
We shall see later, in Section 5, that the two questions above in fact are equivalent.
The lemma below, which shall be used several times in this paper, shows that one can
lift proper infiniteness from a fibre of a C(X)-algebra to a whole neighborhood of that
fibre.
Lemma 2.10 Let X be a compact Hausdorff space, let A be a unital C(X)-algebra, let
x ∈ X, and suppose that the fibre Ax is properly infinite. Then AF is properly infinite for
some closed neighborhood F of x.
Proof: Let {Fλ}λ∈Λ be a decreasing net of closed neighborhoods of x ∈ X , fulfilling that⋂
λ∈Λ Fλ = {x}, and set Iλ = C0(X \Fλ)A. Then {Iλ}λ∈Λ is an increasing net of ideals in
A, AFλ = A/Iλ, I :=
λ∈Λ Iλ = C0(X\{x}), and Ax = A/I.
By the assumption that Ax is properly infinite there is a unital
∗-homomorphism
ψ : T2 → Ax, and since T2 is semi-projective there is a λ0 ∈ Λ and a unital
∗-homomorphism
ϕ : T2 → AFλ0 making the diagram
// Ax
commutative. We can thus take F to be Fλ0 . �
Theorem 2.11 Let A be a unital C(X)-algebra where X is a compact Hausdorff space.
If all fibres Ax, x ∈ X, are properly infinite, then some matrix algebra over A is properly
infinite.
Proof: By Lemma 2.10, X can be covered by finitely many closed sets F1, F2, . . . , Fn
such that AFj is properly infinite for each j. Put Gj = F1 ∪ F2 ∪ · · · ∪ Fj . For each
j = 1, 2, . . . , n− 1 we have a pull-back diagram
AGj+1
yyrrr
AFj+1
AGj∩Fj+1
We know that M2j−1(AGj ) is properly infinite when j = 1. Proposition 2.7 (applied to
the diagram above tensored with M2j−1(C)) tells us that M2j (AGj+1) is properly infinite if
M2j−1(AGj) is properly infinite. Hence M2n−1(A) is properly infinite. �
Remark 2.12 Uffe Haagerup has suggested another way to prove Theorem 2.11: If no
matrix-algebra over A is properly infinite, then there exists a bounded non-zero lower
semi-continuous 2-quasi-trace on A, see [7] and [1, page 327], and hence also an extremal
2-quasi-trace. Now, if A is also a C(X)-algebra for some compact Hausdorff space X , this
implies that there is a bounded non-zero lower semi-continuous 2-quasitrace on Ax for (at
least) one point x ∈ X (see eg. [8, Proposition 3.7]). But then the fibre Ax cannot be
properly infinite.
Question 2.13 Is any unital C(X)-algebra A properly infinite if all its fibres Ax, x ∈ X ,
are properly infinite?
We shall show in Section 5 that the question above is equivalent to Question 2.8 which
again is equivalent to Question 2.9.
3 Lower semi-continuous fields of properly infinite C∗-
algebras
Let us briefly discuss whether the results from Section 2 can be extended to lower semi-
continuous C∗-bundles (A, {σx}) over a compact Hausdorff space X . Recall that any such
separable lower semi-continuous C∗-bundle admits a faithful C(X)-linear representation on
a Hilbert C(X)-module E such that, for all x ∈ X , the fibre σx(A) is isomorphic to the
induced image of A in L(Ex), [2]. Thus, the problem boils down to the following: Given a
separable Hilbert C(X)-module E with infinite dimensional fibres Ex, such that the unit
p of the C∗-algebra LC(X)(E) of bounded adjointable C(X)-linear operators acting on E
has a properly infinite image in L(Ex) for all x ∈ X . Is the projection p itself properly
infinite in LC(X)(E)?
Dixmier and Douady proved that this is always the case if the space X has finite
topological dimension, [6]. But it does not hold anymore in the infinite dimensional case,
see [6, §16, Corollaire 1] and [11], not even if X is contractible, [3, Corollary 3.7].
4 Two examples
We describe here two examples of continuous fields; the first is over the interval and the
second (which really is a class of examples) is over the circle.
Example 4.1 Let (O∞ ∗O∞, (ι1, ι2)) be the universal unital free product of two copies of
O∞, and let A be the unital sub-C
∗-algebra of C([0, 1],O∞ ∗ O∞) given by
A = {f ∈ C([0, 1],O∞ ∗ O∞) : f(0) ∈ ι1(O∞), f(1) ∈ ι2(O∞)}.
Observe that A (in a canonical way) is a C([0, 1])-algebra with fibres
ι1(O∞), t = 0,
O∞ ∗ O∞, 0 < t < 1,
ι2(O∞), t = 1
O∞, t = 0,
O∞ ∗ O∞, 0 < t < 1,
O∞, t = 1.
In particular, all fibres of A are properly infinite.
One claim to fame of the example above is that the question below is equivalent to Ques-
tion 2.13 above. Hence, to answer Question 2.13 in the affirmative (or in the negative)
we need only consider the case where X = [0, 1], and we need only worry about this one
particular C([0, 1])-algebra (which of course is bad enough!).
Question 4.2 Is the C([0, 1])-algebra A from Example 4.1 above properly infinite?
The three equivalent statements in the proposition below will in Section 5 be shown to be
equivalent to Question 4.2.
Proposition 4.3 The following three statements concerning the C([0, 1])-algebra A and
the C∗-algebra (O∞ ∗ O∞, (ι1, ι2)) defined above are equivalent:
(i) A contains a non-trivial projection (i.e., a projection other than 0 and 1).
(ii) There are non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(q).
(iii) Let s be any isometry in O∞. Then ι1(ss
∗) ∼h ι2(ss
∗) in O∞ ∗ O∞.
We warn the reader that all three statements above could be false.
Proof: (i) ⇒ (ii). Let e be a non-trivial projection in A. Let πt : A → At, t ∈ [0, 1],
denote the fibre map. As A ⊆ C([0, 1],O∞ ∗ O∞), the mapping t 7→ πt(e) ∈ O∞ ∗ O∞
is continuous, so in particular, π0(e) ∼h π1(e) in O∞ ∗ O∞. The mappings ι1 and ι2 are
injective, so there are projections p, q ∈ O∞ such that π0(e) = ι1(p) and π1(e) = ι2(q). The
projections p and q are non-zero because the mapping t 7→ ‖πt(e)‖ is continuous and not
constant equal to 0. Similarly, and 1− p and 1− q are non-zero because 1− e is non-zero.
(ii) ⇒ (iii). Take non-trivial projections p, q ∈ O∞ such that ι1(p) ∼h ι2(q). Take a
unitary v in U0(O∞∗O∞) with ι2(q) = vι1(p)v
∗. Let s ∈ O∞ be an isometry. If s is unitary,
then ι1(ss
∗) = 1 = ι2(ss
∗) and there is nothing to prove. Suppose that s is non-unitary.
Then ss∗ is homotopic to a subprojection p0 of p and to a subprojection q0 of q (use that p
and q are properly infinite and full, then Lemma 2.4 (i), and last the fact that the unitary
group of O∞ is connected). Hence ι1(ss
∗) ∼h ι1(p0) ∼h vι1(p0)v
∗ and ι2(ss
∗) ∼h ι2(q0),
so we need only show that vι1(p0)v
∗ ∼h ι2(q0). But this follows from Proposition 2.5 with
r = 1− ι2(q) = ι2(1− q), as we note that p0 ∼ 1 ∼ q0 in O∞, whence
ι2(q0) ∼ ι2(1) = 1 = ι1(1) ∼ ι1(p0) ∼ vι1(p0)v
(iii) ⇒ (i). Take a non-unitary isometry s ∈ O∞. Then ι1(ss
∗) ∼h ι2(ss
∗), and so there
is a continuous function e : [0, 1] → O∞ ∗O∞ such that e(t) is a projection for all t ∈ [0, 1],
e(0) = ι1(ss
∗) and e(1) = ι2(ss
∗). But then e is a non-trivial projection in A. �
It follows from Theorem 2.11 that some matrix algebra over A (from Example 4.1) is
properly infinite. We can sharpen that statement as follows:
Proposition 4.4 M2(A) is properly infinite; and if O∞ ∗O∞ is K1-injective, then A itself
is properly infinite.
It follows from Theorem 5.5 below that A is properly infinite if and only if O∞ ∗ O∞ is
K1-injective.
Proof: We have a pull-back diagram
A[0, 1
π1/2 %%KK
π1/2yysss
O∞ ∗ O∞
One can unitally embed O∞ into A[0, 1
] via ι1, so A[0, 1
] is properly infinite, and a similar
argument shows that A[ 1
,1] is properly infinite. The two statements now follow from
Proposition 2.7. �
The example below, which will be the focus of the rest of this section, and in parts also of
Section 5, is inspired by arguments from Rieffel’s paper [9].
Example 4.5 Let A be a unital C∗-algebra, and let v be a unitary element in A such that
in U2(A).
Let t 7→ ut be a continuous path of unitaries in U2(A) such that u0 = 1 and u1 = diag(v, 1).
p(t) = ut
u∗t ∈M2(A),
and note that p(0) = p(1). Identifying, for each C∗-algebra D, C(T, D) with the algebra
of all continuous functions f : [0, 1] → D such that f(1) = f(0), we see that p belongs to
C(T,M2(A)). Put
B = pC(T,M2(A))p,
and note that B is a unital (sub-trivial) C(T)-algebra, being a corner of the trivial C(T)-
algebra C(T,M2(A)). The fibres of B are
Bt = p(t)M2(A)p(t) ∼= A
for all t ∈ T.
Summing up, for each unital C∗-algebraA, for each unitary v inA for which diag(v, 1) ∼h
1 in U2(A), and for each path t 7→ ut ∈ U2(A) implementing this homotopy we get a C(T)-
algebra B with fibres Bt ∼= A. We shall investigate this class of C(T)-algebras below.
Lemma 4.6 In the notation of Example 4.5,
− p ∼
in C(T,M2(A)).
In particular, p is stably equivalent to diag(1, 0).
Proof: Put
vt = ut
, t ∈ [0, 1].
v0 = u0
, v1 = u1
so v belongs to C(T,M2(A)). It is easy to see that v
t vt = diag(0, 1) and vtv
t = 1 − p(t),
and so the lemma is proved. �
Proposition 4.7 Let A, v ∈ U(A), and B be as in Example 4.5. Conditions (i) and (ii)
below are equivalent for any unital C∗-algebra A, and all three conditions are equivalent if
A in addition is assumed to be properly infinite.
(i) v ∼h 1 in U(A).
(ii) p ∼ diag(1A, 0) in C(T,M2(A)).
(iii) The C(T)-algebra B is properly infinite.
Proof: (ii) ⇒ (i). Suppose that p ∼ diag(1, 0) in C(T,M2(A)). Then there is a w ∈
C(T,M2(A)) such that
and w∗twt = pt
for all t ∈ [0, 1] and w1 = w0 (as we identify C(T,M2(A)) with the set of continuous
functions f : [0, 1] → M2(A) with f(1) = f(0)). Upon replacing wt with w
0wt we can
assume that w1 = w0 = diag(1, 0). Now, with t 7→ ut as in Example 4.5,
where t 7→ at is a continuous path of unitaries in A. Because u0 = diag(1, 1) and u1 =
diag(v, 1) we see that a0 = 1 and a1 = v, whence v ∼h 1 in U(A).
(i) ⇒ (ii). Suppose conversely that v ∼h 1 in U(A). Then we can find a continuous
path t 7→ vt ∈ U(A), t ∈ [1 − ε, 1], such that v1−ε = v and v1 = 1 for an ε > 0 (to be
determined below). Again with t 7→ ut as in Example 4.5, define
ũt =
u(1−ε)−1t, 0 ≤ t ≤ 1− ε,
diag(vt, 1), 1− ε ≤ t ≤ 1.
Then t 7→ ũt is a continuous path of unitaries in U2(A) such that ũ1−ε = u1 = diag(v, 1)
and ũ0 = ũ1 = 1. It follows that ũ belongs to C(T,M2(A)). Provided that ε > 0 is chosen
small enough we obtain the following inequality:
∥∥∥∥ũt
ũ∗t − p(t)
∥∥∥∥ =
∥∥∥∥ũt
ũ∗t − ut
∥∥∥∥ < 1
for all t ∈ [0, 1], whence p ∼ ũ diag(1, 0) ũ∗ ∼ diag(1, 0) as desired.
(iii) ⇒ (ii). Suppose that B is properly infinite. From Lemma 4.6 we know that
[p] = [diag(1A, 0)] in K0(C(T, A)). Because B and A are properly infinite, it follows that p
and diag(1A, 0) are properly infinite (and full) projections, and hence they are equivalent
by Proposition 2.3 (i).
(ii) ⇒ (iii). Since A is properly infinite, diag(1A, 0) and hence p (being equivalent to
diag(1A, 0)) are properly infinite (and full) projections, whence B is properly infinite. �
We will now use (the ideas behind) Lemma 4.6 and Proposition 4.7 to prove the following
general statement about C∗-algebras.
Corollary 4.8 Let A be a unital C∗-algebra such that C(T, A) has the cancellation prop-
erty. Then A is K1-injective.
Proof: It suffices to show that the natural maps Un−1(A)/U
n−1(A) → Un(A)/U
n(A) are
injective for all n ≥ 2. Let v ∈ Un−1(A) be such that diag(v, 1A) ∈ U
n(A) and find a
continuous path of unitaries t 7→ ut in Un(A) such that
u0 = 1Mn(A) =
1Mn−1(A) 0
and u1 =
pt = ut
1Mn−1(A) 0
u∗t , t ∈ [0, 1],
and note that p0 = p1 so that p defines a projection in C(T,Mn(A)). Repeating the
proof of Lemma 4.6 we find that 1Mn(A) − p ∼ diag(0, 1A) in C(T,Mn(A)), whence p ∼
diag(1Mn−1(A), 0) by the cancellation property of C(T, A), where we identify projections in
Mn(A) with constant projections in C(T,Mn(A)). The arguments going into the proof of
Proposition 4.7 show that v ∼h 1Mn−1(A) in Un−1(A) if (and only if) p ∼ diag(1Mn−1(A), 0).
Hence v belongs to U0n−1(A) as desired. �
5 K1-injectivity of properly infinite C
∗-algebras
In this section we prove our main result that relate K1-injectivity of arbitrary unital prop-
erly infinite C∗-algebras to proper infiniteness of C(X)-algebras and pull-back C∗-algebras.
More specifically we shall show that Question 2.9, Question 2.13, Question 2.8, and Ques-
tion 4.2 are equivalent.
First we reformulate in two different ways the question if a given properly infinite unital
C∗-algebra is K1-injective.
Proposition 5.1 The following conditions are equivalent for any unital properly infinite
C∗-algebra A:
(i) A is K1-injective.
(ii) Let p, q be projections in A such that p ∼ q and p, q, 1−p, 1−q are properly infinite
and full. Then p ∼h q.
(iii) Let p and q be properly infinite, full projections in A. There exist properly infinite,
full projections p0, q0 ∈ A such that p0 ≤ p, q0 ≤ q, and p0 ∼h q0.
Proof: (i) ⇒ (ii). Let p, q be properly infinite, full projections in A with p ∼ q such that
1− p, 1− q are properly infinite and full. Then by Lemma 2.4 (i) there is a unitary v ∈ A
such that vpv∗ = q and [v] = 0 in K1(A). By the assumption in (i), v ∈ U
0(A), whence
p ∼h q.
(ii) ⇒ (i). Let u ∈ U(A) be such that [u] = 0 in K1(A). Take, as we can, a projection
p in A such that p and 1 − p are properly infinite and full. Set q = upu∗. Then p ∼h q by
(ii), and so there exists a unitary v ∈ U0(A) with p = vqv∗. It follows that
pvu = vqv∗vu = v(upu∗)v∗vu = vup.
Therefore vu ∈ U0(A) by Lemma 2.4 (ii), which in turn implies that u ∈ U0(A).
(ii) ⇒ (iii). Let p, q be properly infinite and full projections in A. There exist mutually
orthogonal projections e1, f1 such that e1 ≤ p, f1 ≤ p and e1 ∼ p ∼ f1, and mutually
orthogonal projections e2, f2 such that e2 ≤ q, f2 ≤ q and e2 ∼ q ∼ f2. Being equivalent
to either p or q, the projections e1, e2, f1 and f2 are properly infinite and full. There are
properly infinite, full projections p0 ≤ e1 and q0 ≤ e2 such that [p0] = [q0] = 0 in K0(A)
and p0 ∼ q0 (cf. Proposition 2.3). As f1 ≤ 1 − p0 and f2 ≤ 1− q0, we see that 1 − p0 and
1− q0 are properly infinite and full, and so we get p0 ∼h q0 by (ii).
(iii) ⇒ (ii). Let p, q be equivalent properly infinite, full projections in A such that
1 − p, 1 − q are properly infinite and full. From (iii) we get properly infinite and full
projections p0 ≤ p, q0 ≤ q which satisfy p0 ∼h q0. Thus there is a unitary v ∈ U0(A)
such that vp0v
∗ = q0. Upon replacing p by vpv
∗ (as we may do because p ∼h vpv
∗) we
can assume that q0 ≤ p and q0 ≤ q. Now, q0 is orthogonal to 1 − p and to 1 − q, and so
1− p ∼h 1− q by Proposition 2.5, whence p ∼h q. �
Proposition 5.2 Let A be a unital properly infinite C∗-algebra. The following conditions
are equivalent:
(i) A is K1-injective, ie., the natural map U(A)/U
0(A) → K1(A) is injective.
(ii) The natural map U(A)/U0(A) → U2(A)/U
2 (A) is injective.
(iii) The natural maps Un(A)/U
n(A) → K1(A) are injective for each natural number n.
Proof: (i) ⇒ (ii) holds because the map U(A)/U0(A) → K1(A) factors through the map
U(A)/U0(A) → U2(A)/U
2 (A).
(ii)⇒ (i). Take u ∈ U(A) and suppose that [u] = 0 inK1(A). Then diag(u, 1A) ∈ U
2 (A)
by Lemma 2.4 (ii) (with p = diag(1A, 0)). Hence u ∈ U0(A) by injectivity of the map
U(A)/U0(A) → U2(A)/U
2 (A).
(i) ⇒ (iii). Let n ≥ 1 be given and consider the natural maps
U(A)/U0(A) → Un(A)/U
n(A) → K1(A).
The first map is onto, as proved by Cuntz in [4], see also [13, Exercise 8.9], and the
composition of the two maps is injective by assumption, hence the second map is injective.
(iii) ⇒ (i) is trivial. �
We give below another application of K1-injectivity for properly infinite C
∗-algebras. First
we need a lemma:
Lemma 5.3 Let A be a unital, properly infinite C∗-algebra, and let ϕ, ψ : O∞ → A be
unital embeddings. Then ψ is homotopic to a unital embedding ψ′ : O∞ → A for which
there is a unitary u ∈ A with [u] = 0 in K1(A) and for which ψ
′(sj) = uϕ(sj) for all j
(where s1, s2, . . . are the canonical generators of O∞).
Proof: For each n set
ψ(sj)ϕ(sj)
∗ ∈ A, en =
j ∈ O∞.
Then vn is a partial isometry in A with vnv
n = ψ(en), v
nvn = ϕ(en), and ψ(sj) = vnϕ(sj)
for j = 1, 2, . . . , n. Since 1− en is full and properly infinite it follows from Lemma 2.4 that
each vn extends to a unitary un ∈ A with [un] = 0 in K1(A). In particular, ψ(sj) = unϕ(sj)
for j = 1, 2, . . . , n.
We proceed to show that n 7→ un extends to a continuous path of unitaries t 7→ ut, for
t ∈ [2,∞), such that utϕ(en) = unϕ(en) for t ≥ n + 1. Fix n ≥ 2. To this end it suffices
to show that we can find a continuous path t 7→ zt, t ∈ [0, 1], of unitaries in A such that
z0 = 1, z1 = u
nun+1, and ztϕ(en−1) = ϕ(en−1) (as we then can set ut to be unzt−n for
t ∈ [n, n+ 1]).
Observe that
un+1ϕ(en) = vn+1ϕ(en) = vn = unϕ(en).
Set A0 = (1 − ϕ(en−1))A(1 − ϕ(en−1)), and set y = u
nun+1(1 − ϕ(en−1)). Then y is a
unitary element in A0 and [y] = 0 in K1(A0). Moreover, y commutes with the properly
infinite full projection ϕ(en) − ϕ(en−1) ∈ A0. We can therefore use Lemma 2.4 to find a
continuous path t 7→ yt of unitaries in A0 such that y0 = 1A0 = 1 − ϕ(en−1) and y1 = y.
The continuous path t 7→ zt = yt + ϕ(en−1) is then as desired.
For each t ≥ 2 let ψt : O∞ → A be the
∗-homomorphism given by ψt(sj) = utϕ(sj).
Then ψt(sj) = ψ(sj) for all t ≥ j + 1, and so it follows that
ψt(x) = ψ(x)
for all x ∈ O∞. Hence ψ2 is homotopic to ψ, and so we can take ψ
′ to be ψ2. �
Proposition 5.4 Any two unital ∗-homomorphisms from O∞ into a unital K1-injective
(properly infinite) C∗-algebra are homotopic.
Proof: In the light of Lemma 5.3 it suffices to show that if ϕ, ψ : O∞ → A are unital
homomorphisms such that, for some unitary u ∈ A with [u] = 0 in K1(A), ψ(sj) = uϕ(sj)
for all j, then ψ ∼h ϕ. By assumption, u ∼h 1, so there is a continuous path t 7→ ut of
unitaries in A such that u0 = 1 and u1 = u. Letting ϕt : O∞ → A be the
∗-homomorphism
given by ϕt(sj) = utϕ(sj) for all j, we get t 7→ ϕt is a continuous path of
∗-homomorphisms
connecting ϕ0 = ϕ to ϕ1 = ψ. �
Our main theorem below, which in particular implies that Question 2.9, Question 2.13,
Question 2.8 and Question 4.2 all are equivalent, also give a special converse to Proposi-
tion 5.4: Indeed, with ι1, ι2 : O∞ → O∞ ∗O∞ the two canonical inclusions, if ι1 ∼h ι2, then
condition (iv) below holds, whence O∞ ∗ O∞ is K1-injective, which again implies that all
unital properly infinite C∗-algebras are K1-injective. Below we retain the convention that
O∞ ∗ O∞ is the universal unital free product of two copies of O∞ and that ι1 and ι2 are
the two natural inclusions of O∞ into O∞ ∗ O∞.
Theorem 5.5 The following statements are equivalent:
(i) Every unital, properly infinite C∗-algebra is K1-injective.
(ii) For every compact Hausdorff space X, every unital C(X)-algebra A, for which Ax is
properly infinite for all x ∈ X, is properly infinite.
(iii) Every unital C∗-algebra A, that is the pull-back of two unital, properly infinite C∗-
algebras A1 and A2 along
∗-epimorphisms π1 : A1 → B, π2 : A2 → B:
}} ϕ2
π1 A
π2~~}}
is properly infinite.
(iv) There exist non-zero projections p, q ∈ O∞ such that p 6= 1, q 6= 1, and ι1(p) ∼h ι2(p)
in O∞ ∗ O∞.
(v) The specific C([0, 1])-algebra A considered in Example 4.1 (and whose fibres are prop-
erly infinite) is properly infinite.
(vi) O∞ ∗ O∞ is K1-injective.
Note that statement (i) is reformulated in Propositions 5.1, 5.2, and 5.4; and that statement
(iv) is reformulated in Proposition 4.3. We warn the reader that all these statements may
turn out to be false (in which case, of course, there will be counterexamples to all of them).
Proof: (i) ⇒ (iii) follows from Proposition 2.7.
(iii) ⇒ (ii). This follows from Lemma 2.10 as in the proof of Theorem 2.11, except that
one does not need to pass to matrix algebras.
(ii) ⇒ (i). Suppose that A is unital and properly infinite. Take a unitary v ∈ U(A)
such that diag(v, 1) ∈ U02 (A). Let B be the C(T)-algebra constructed in Example 4.5 from
A, v, and a path of unitaries t 7→ ut connecting 1M2(A) to diag(v, 1). Then Bt
∼= A for all
t ∈ T, so all fibres of B are properly infinite. Assuming (ii), we can conclude that B is
properly infinite. Proposition 4.7 then yields that v ∈ U0(A). It follows that the natural
map U(A)/U0(A) → U2(A)/U
2 (A) is injective, whence A is K1-injective by Proposition 5.2.
(ii) ⇒ (v) is trivial (because A is a C([0, 1])-algebra with properly infinite fibres).
(v) ⇒ (iv) follows from Proposition 4.3.
(iv) ⇒ (i). We show that Condition (iii) of Proposition 4.3 implies Condition (iii) of
Proposition 5.1.
Let A be a properly infinite C∗-algebra and let p, q be properly infinite, full projections
in A. Then there exist (properly infinite, full) projections p0 ≤ p and q0 ≤ q such that
p0 ∼ 1 ∼ q0 and such that 1−p0 and 1−q0 are properly infinite and full, cf. Propositions 2.3.
Take isometries t1, r1 ∈ A with t1t
1 = p0 and r1r
1 = q0; use the fact that 1 - 1 − p0 and
1 - 1− q0 to find sequences of isometries t2, t3, t4, . . . and r2, r3, r4, . . . in A such that each
of the two sequences {tjt
j=1 and {rjr
j=1 consist of pairwise orthogonal projections.
By the universal property of O∞ there are unital
∗-homomorphisms ϕj : O∞ → A,
j = 1, 2, such that ϕ1(sj) = tj and ϕ2(sj) = rj, where s1, s2, s3, . . . are the canonical
generators of O∞. In particular,
ϕ1(s1s
1) = p0 and ϕ2(s1s
1) = q0.
By the property of the universal unital free products of C∗-algebras, there is a unique
unital ∗-homomorphism ϕ : O∞ ∗ O∞ → A making the diagram
O∞ ∗ O∞
ϕ1 %%KK
99ssssssssss
ϕ2yysss
eeKKKKKKKKKK
commutative. It follows that p0 = ϕ(ι1(s1s
1)) and q0 = ϕ(ι2(s1s
1)). By Condition (iii) of
Proposition 4.3, ι1(s1s
1) ∼h ι2(s1s
1) in O∞ ∗ O∞, whence p0 ∼h q0 as desired.
(i) ⇒ (vi) is trivial.
(vi) ⇒ (v) follows from Proposition 4.4. �
6 Concluding remarks
We do not know if all unital properly infinite C∗-algebras are K1-injective, but we observe
that K1-injectivity is assured in the presence of certain central sequences:
Proposition 6.1 Let A be a unital properly infinite C∗-algebras that contains an asymp-
totically central sequence {pn}
n=1, where pn and 1−pn are properly infinite, full projections
for all n. Then A is K1-injective
Proof: This follows immediately from Lemma 2.4 (ii). �
It remains open if arbitrary C(X)-algebras with properly infinite fibres must be properly
infinite. If this fails, then we already have a counterexample of the form B = pC(T, A)p,
cf. Example 4.5, for some unital properly infinite C∗-algebra A and for some projection
p ∈ C(T, A). (The C∗-algebra B is a C(T)-algebra with fibres Bt ∼= A.)
On the other hand, any trivial C(X)-algebra C(X,D) with constant fibre D is clearly
properly infinite if its fibre(s) D is unital and properly infinite (because C(X,D) ∼= C(X)⊗
D). We extend this observation in the following easy:
Proposition 6.2 Let X be a compact Hausdorff space, let p ∈ C(X,D) be a projection,
and consider the sub-trivial C(X)-algebra pC(X,D)p whose fibre at x is equal to p(x)Dp(x).
If p is Murray-von Neumann equivalent to a constant projection q, then pC(X,D)p is
C(X)-isomorphic to the trivial C(X)-algebra C(X,D0), where D0 = qDq. In this case,
pC(X,D)p is properly infinite if and only if D0 is properly infinite.
In particular, if X is contractible, then pC(X,D)p is C(X)-isomorphic to a trivial
C(X)-algebra for any projection p ∈ C(X,D) and for any C∗-algebra D.
Proof: Suppose that p = v∗v and q = vv∗ for some partial isometry v ∈ C(X,D).
The map f 7→ vfv∗ defines a C(X)-isomorphism from pC(X,D)p onto qC(X,D)q, and
qC(X,D)q = C(X,D0).
If X is contractible, then any projection p ∈ C(X,D) is homotopic, and hence equiva-
lent, to the constant projection x 7→ p(x0) for any fixed x0 ∈ X . �
Remark 6.3 One can elaborate a little more on the construction considered above. Take
a unital C∗-algebra D such that for some natural number n ≥ 2, Mn(D) is properly
infinite, butMn−1(D) is not properly infinite (see [12] or [11] for such examples). Take any
space X , preferably one with highly non-trivial topology, eg. X = Sn, and take, for some
k ≥ n, a sufficiently non-trivial n-dimensional projection p in C(X,Mk(D)) such that p(x)
is equivalent to the trivial n dimensional projection 1Mn(D) for all x (if X is connected we
need only assume that this holds for one x ∈ X). The C(X)-algebra
A = pC(X,Mk(D)) p,
then has properly infinite fibres Ax = p(x)Dp(x) ∼= Mn(D). Is A always properly infinite?
We guess that a possible counterexample to the questions posed in this paper could be of
this form (for suitable D, X , and p).
Let us end this paper by remarking that the answer to Question 2.13, which asks if any
C(X)-algebra with properly infinite fibres is itself properly infinite, does not depend (very
much) on X . If it fails, then it fails already for X = [0, 1] (cf. Theorem 5.5), and [0, 1] is a
contractible space of low dimension. However, if we make the dimension of X even lower
than the dimension of [0, 1], then we do get a positive anwer to our question:
Proposition 6.4 Let X be a totally disconnected space, and let A be a C(X)-algebra such
that all fibres Ax, x ∈ X, of A are properly infinite. Then A is properly infinite.
Proof: Using Lemma 2.10 and the fact that X is totally disconnected we can write X as
the disjoint union of clopen sets F1, F2, . . . , Fn such that AFj is properly infinite for all j.
A = AF1 ⊕ AF2 ⊕ · · · ⊕AFn ,
the claim is proved. �
References
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Projet Algbres d’oprateurs, Institut de Mathmatiques de Jussieu, 175, rue
du Chevaleret, F-75013 PARIS, France
E-mail address: [email protected]
Internet home page: www.math.jussieu.fr/∼blanchar
Department of Mathematics, University of Southern Denmark, Odense,
Campusvej 55, 5230 Odense M, Denmark
E-mail address: [email protected]
Department of Mathematics, University of Southern Denmark, Odense,
Campusvej 55, 5230 Odense M, Denmark
E-mail address: [email protected]
Internet home page: www.imada.sdu.dk/∼mikael/welcome
Introduction
C(X)-algebras with properly infinite fibres
Lower semi-continuous fields of properly infinite C-algebras
Two examples
K1-injectivity of properly infinite C*-algebras
Concluding remarks
|
0704.1555 | An information-based traffic control in a public conveyance system:
reduced clustering and enhanced efficiency | An information-based traffic control in a public conveyance system:
reduced clustering and enhanced efficiency
Akiyasu Tomoeda and Katsuhiro Nishinari
Department of Aeronautics and Astronautics, Faculty of Engineering,
University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
Debashish Chowdhury
Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
Andreas Schadschneider
Institut für Theoretische Physik, Universität zu Köln D-50937 Köln, Germany
(Dated: October 31, 2018)
A new public conveyance model applicable to buses and trains is proposed in this paper by
using stochastic cellular automaton. We have found the optimal density of vehicles, at which the
average velocity becomes maximum, significantly depends on the number of stops and passengers
behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed
by comparing the different behavior of passengers. Moreover, we have found that a big cluster
of vehicles is divided into small clusters, by incorporating information of the number of vehicles
between successive stops.
I. INTRODUCTION
The totally asymmetric simple exclusion process [1, 2,
3] is the simplest model of non-equilibrium systems of in-
teracting self-driven particles. Various extensions of this
model have been reported in the last few years for captur-
ing the essential features of the collective spatio-temporal
organizations in wide varieties of systems, including those
in vehicular traffic [4, 5, 6, 7, 8]. Traffic of buses and bicy-
cles have also been modeled following similar approaches
[9, 10]. A simple bus route model [10] exhibits clustering
of the buses along the route and the quantitative features
of the coarsening of the clusters have strong similarities
with coarsening phenomena in many other physical sys-
tems. Under normal circumstances, such clustering of
buses is undesirable in any real bus route as the effi-
ciency of the transport system is adversely affected by
clustering. The main aim of this paper is to introduce
a traffic control system into the bus route model in such
a way that helps in suppressing this tendency of cluster-
ing of the buses. This new model exhibits a competition
between the two opposing tendencies of clustering and
de-clustering which is interesting from the point of view
of fundamental physical principles. However, the model
may also find application in developing adaptive traffic
control systems for public conveyance systems.
In some of earlier bus-route models, movement of the
buses was monitored on coarse time intervals so that the
details of the dynamics of the buses in between two suc-
cessive bus stops was not described explicitly. Instead,
the movement of the bus from one stop to the next was
captured only through probabilities of hopping from one
stop to the next; hopping takes place with the lower prob-
ability if passengers are waiting at the approaching bus
stop [10]. An alternative interpretation of the model is
as follows: the passengers could board the bus whenever
and wherever they stopped a bus by raising their hand,
this is called the hail-and-ride system.
Several possible extensions of the bus route model have
been reported in the past [11, 12, 13]. For example, in
[11], in order to elucidate the connection between the
bus route model with parallel updating and the Nagel-
Schreckenberg model, two alternative extensions of the
latter model with space-/time-dependent hopping rates
are proposed. If a bus does not stop at a bus stop, the
waiting passengers have to wait further for the next bus;
such scenarios were captured in one of the earlier bus
route models [12], using modified car-following model.
In [13], the bus capacity, as well as the number of pas-
sengers getting on and off at each stop, were introduced
to make the model more realistic. Interestingly, it has
been claimed that the distribution of the time gaps be-
tween the arrival of successive buses is described well by
the Gaussian Unitary Ensemble of random matrices [14].
In this paper, by extending the model in [10], we sug-
gest a new public conveyance model (PCM). Although
we refer to each of the public vehicles in this model as
a “bus”, the model is equally applicable to train traffic
on a given route. In this PCM we can set up arbitrary
number of bus stops on the given route. The hail-and-
ride system turns out to be a special case of the general
PCM. Moreover, in the PCM the duration of the halt
of a bus at any arbitrary bus stop depends on the num-
ber of waiting passengers. As we shall demonstrate in
this paper, the delay in the departure of the buses from
crowded bus stops leads to the tendency of the buses to
cluster on the route. Furthermore, in the PCM, we also
introduce a traffic control system that exploits the in-
formation on the number of buses in the “segments” in
between successive bus stops; this traffic control system
helps in reducing the undesirable tendency of clustering
by dispersing the buses more or less uniformly along the
http://arxiv.org/abs/0704.1555v2
route.
In this study we introduce two different quantitative
measures of the efficiency of the bus transport system,
and calculate these quantities, both numerically and an-
alytically, to determine the conditions under which the
system would operate optimally.
This paper is organized as follows, in Sec. 2 PCM is in-
troduced and we show several simulation results in Sec. 3.
The average speed and the number of waiting passengers
are studied by mean field analysis in Sec. 4, and conclu-
sions are given in Sec. 5.
II. A STOCHASTIC CA MODEL FOR PUBLIC
CONVEYANCE
In this section, we explain the PCM in detail. For the
sake of simplicity, we impose periodic boundary condi-
tions. Let us imagine that the road is partitioned into
L identical cells such that each cell can accommodate
at most one bus at a time. Moreover, a total of S
(0 ≤ S ≤ L) equispaced cells are identified in the begin-
ning as bus stops. Note that, the special case S = L cor-
responds to the hail-and-ride system. At any given time
step, a passenger arrives with probability f to the sys-
tem. Here, we assume that a given passenger is equally
likely to arrive at any one of the bus stops with a proba-
bility 1/S. Thus, the average number of passengers that
arrive at each bus stop per unit time is given by f/S.
In contrast to this model, in ref. [15, 16] the passengers
were assumed to arrive with probability f at all the bus
stops in every time step.
Model AModel A
Model BModel B
BUS BUS BUS
BUSBUS BUS
)0( =iNQ)0( =iNQ 1H 2H
)( qQ >
1 2( )Q H H> >
FIG. 1: Schematic illustration of the PCM. In the model A,
the hopping probability to the bus stop does not depend on
the number of waiting passengers. In contrast, in the model
B the hopping probability to the bus stop depends on the
number of waiting passengers. Thus if the waiting passengers
increase, the hopping probability to the bus stop is decreased.
The model A corresponds to those situations where,
because of sufficiently large number of broad doors, the
time interval during which the doors of the bus remain
open after halting at a stop, is independent of the size
of waiting crowd of passengers. In contrast, the model
B captures those situations where a bus has to halt for
a longer period to pick up a larger crowd of waiting pas-
sengers.
The symbol H is used to denote the hopping probabil-
ity of a bus entering into a cell that has been designated
as a bus stop. We consider two different forms of H in
the two versions of our model which are named as model
A and model B. In the model A we assume the form
Q no waiting passengers
q waiting passengers exist
where both Q and q (Q > q) are constants independent
of the number of waiting passengers. The form (1) was
used in the original formulation of the bus route model
by O’Loan et al. [10].
In contrast to most of all the earlier bus route models,
we assume in the model B that the maximum number
of passengers that can get into one bus at a bus stop is
Nmax. Suppose, Ni denotes the number of passengers
waiting at the bus stop i (i = 1, · · · , S) at the instant of
time when a bus arrives there. In contrast to the form
(1) for H in model A, we assume in model B the form
min(Ni, Nmax) + 1
where min(Ni, Nmax) is the number of passengers who
can get into a bus which arrives at the bus stop i at
the instant of time when the number of passengers wait-
ing there is Ni. The form (2) is motivated by the com-
mon expectation that the time needed for the passengers
boarding a bus is proportional to their number. FIG. 1
depicts the hopping probabilities in the two models A
and B schematically.
The hopping probability of a bus to the cells that are
not designated as bus stops is Q; this is already captured
by the expressions (1) and (2) since no passenger ever
waits at those locations.
In principle, the hopping probability H for a real bus
would depend also on the number of passengers who get
off at the bus stop; in the extreme situations where no
passenger waits at a bus stop the hopping probability H
would be solely decided by the disembarking passengers.
However, in order to keep the model theoretically simple
and tractable, we ignore the latter situation and assume
that passengers get off only at those stops where waiting
passengers get into the bus and that the time taken by the
waiting passengers to get into the bus is always adequate
for the disembarking passengers to get off the bus.
Note that Nmax is the maximum boarding capacity at
each bus stop rather than themaximum carrying capacity
of each bus. The PCM model reported here can be easily
extended to incorporate an additional dynamical variable
associated with each bus to account for the instantaneous
number of passengers in it. But, for the sake of simplic-
ity, such an extension of the model is not reported here.
Instead, in the simple version of the PCMmodel reported
here, Nmax can be interpreted as the maximum carrying
capacity of each bus if we assume that all of the passen-
gers on the bus get off whenever it stops.
The model is updated according to the following rules.
In step 2 − 4, these rules are applied in parallel to all
buses and passengers, respectively:
1. Arrival of a passenger
A bus stop i (i = 1, · · · , S) is picked up randomly,
with probability 1/S, and then the corresponding
number of waiting passengers in increased by unity,
i.e. Ni → Ni+1, with probability f to account for
the arrival of a passenger at the selected bus stop.
2. Bus motion
If the cell in front of a bus is not occupied by an-
other bus, each bus hops to the next cell with the
probability H . Specifically, if passengers do not ex-
ist in the next cell in both model A and model B
hopping probability equals to Q because Ni equals
to 0. Else, if passengers exist in the next cell,
the hopping probability equals to q in the model
A, whereas in the model B the corresponding hop-
ping probability equals to Q/(min(Ni, Nmax) + 1).
Note that, when a bus is loaded with passengers to
its maximum boarding capacity Nmax, the hopping
probability in the model B equals to Q/(Nmax+1),
the smallest allowed hopping probability.
3. Boarding a bus
When a bus arrives at the i-th (i = 1, · · · , S) bus
stop cell, the corresponding number Ni of waiting
passengers is updated to max(Ni −Nmax, 0) to ac-
count for the passengers boarding the bus. Once
the door is closed, no more waiting passenger can
get into the bus at the same bus stop although the
bus may remain stranded at the same stop for a
longer period of time either because of the unavail-
ability of the next bus stop or because of the traffic
control rule explained next.
4. Bus information update
Every bus stop has information Ij (j = 1, · · · , S)
which is the number of buses in the segment of the
route between the stop j and the next stop j+1 at
that instant of time. This information is updated at
each time steps. When one bus leaves the j-th bus
stop, Ij is increased to Ij + 1. On the other hand,
when a bus leaves (j+1)-th bus stop, Ij is reduced
to Ij − 1. The desirable value of Ij is I0 = m/S,
where m is the total number of buses, for all j so
that buses are not clustered in any segment of the
route. We implement a traffic control rule based
on the information Ij : a bus remains stranded at a
stop j as long as Ij exceeds I0.
We use the average speed 〈V 〉 of the buses and the
number of the waiting passengers 〈N〉 at a bus stop as
two quantitative measures of the efficiency of the pub-
lic conveyance system under consideration; a higher 〈V 〉
and smaller 〈N〉 correspond to an efficient transportation
system.
III. COMPUTER SIMULATIONS OF PCM
In the simulations we set L = 500, Q = 0.9, q = 0.5
and Nmax = 60. The main parameters of this model,
which we varied, are the number of buses (m), the num-
ber of bus stops (S) and the probability (f) of arrival of
passengers. The number density of buses is defined by
ρ = m/L.
SPACE
SPACE
Initial stage late stage
FIG. 2: Space-time plots in the model B for the parameter
values f = 0.6, S = 5, m = 30. The upper two figures cor-
respond to the case where no traffic control system based on
the information {I} is operational. The upper left figure cor-
responds to the initial stage (from t = 1000 to t = 1500)
whereas the upper right plot corresponds to the late stages
(from t = 4000 to t = 4500). The lower figures correspond to
the case where the information ({I}) based bus-traffic control
system is operational (left figure shows data from t = 1000
to t = 1500 while the right figure corresponds to t = 4000
to t = 4500). Clearly, information-based traffic control sys-
tem disperses the buses which, in the absence of this control
system, would have a tendency to cluster.
Typical space-time plots of the model B are given in
FIG. 2. If no information-based traffic control system
exits, the buses have a tendency to cluster; this phe-
nomenon is very simular to that observed in the ant-
trail model [15, 16]. However, implementation of the
information-based traffic control system restricts the size
of such clusters to a maximum of I0 buses in a segment
of the route in between two successive bus stops. We
study the effects of this control system below by compar-
ing the characteristics of two traffic systems one of which
includes the information-based control system while the
other does not.
A. PCM without information-based traffic control
In the FIG. 3 - FIG. 8, we plot 〈V 〉 and 〈N〉 against
the density of buses for several different values of f . Note
that, the FIG. 5 and FIG. 8 corresponds to the hail-and-
ride system for models A and B, respectively.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 3: The average speed and the average number of waiting
passengers in the model A are plotted against the density for
the parameters S = 5 and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 4: The plot of 〈V 〉 and 〈N〉 of the model A for S = 50
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 5: The plot of 〈V 〉 and 〈N〉 of the model A for S =
500(= L) and f = 0.3, 0.6 and 0.9.
These figures demonstrate that the average speed 〈V 〉,
which is a measure of the efficiency of the bus traffic
system, exhibits a maximum at around ρ = 0.2 ∼ 0.3
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 6: The plot of 〈V 〉 and 〈N〉 of the model B for S = 5
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 7: The plot of 〈V 〉 and 〈N〉 of the model B for S = 50
and f = 0.3, 0.6 and 0.9.
especially in the model B (comparing FIG. 3 with FIG. 6,
it shows the model B (FIG. 6) reflects the bus bunching
more clearly than the model A (FIG. 3) especially at large
f and small ρ). The average number of waiting passengers
〈N〉, whose inverse is another measure of the efficiency of
the bus traffic system, is vanishingly small in the region
0.3 < ρ < 0.7; 〈N〉 increases with decreasing (increasing)
ρ in the regime ρ < 0.3 (ρ > 0.7).
The average velocity of the model A becomes smaller as
S increases in the low density region (see FIG. 3, FIG. 4
and FIG. 5). In contrast, in the model B (FIG. 7 and
FIG. 8) we observe that there is no significant differ-
ence in the average velocity. Note that the number of
waiting passengers is calculated by (total waiting pas-
sengers)/(number of bus stops). The total number of
waiting passengers in this system is almost the same un-
der the case S = 50 and hail-and-ride system S = L in
both models. When the parameter S is small (comparing
FIG. 3 and FIG. 6), in the model B the waiting passen-
gers are larger and the average velocity is smaller than
in the model A, since the effect of the delay in getting on
a bus is taken into account. In the model B (comparing
FIG. 6, FIG. 7 and FIG. 8), the case S = 50 is more effi-
cient than S = 5, i.e. the system is likely to become more
efficient, as S increases. However, we do not find any sig-
nificant variation between S = 50 and S = 500. When S
is small, the system becomes more efficient by increasing
the number of bus stops. If the number of bus stops in-
crease beyond 50, then there is little further variation of
the efficiency as S is increased up to the maximum value
From FIG. 9, the distribution of 〈N〉 over all the bus
stops in the system is shown. We see that the distribution
does not show the Zipf’s law, which is sometimes seen in
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 8: The plot of 〈V 〉 and 〈N〉 of the model B for S =
500(= L) and f = 0.3, 0.6 and 0.9.
0 10 20 30 40 50
Ranking
FIG. 9: The distribution of waiting passengers is plotted
against all bus stops for the parameters f = 0.6, B = 50,
S = 50. The horizontal line means the ranking, where we
arrange the bus stops according to the descending order of
natural and social phenomena; frequency of used words
[17], population of a city [18], the number of the access to
a web site [19], and intervals between successive transit
times of the cars of traffic flow [20].
Next, we investigate the optimal density of buses at
which the average velocity becomes maximum. The op-
timal density depends on Q and is ρ = 0.3 for Q = 0.8
(FIG. 10, see also FIG. 11). In FIG. 10, it is shown
that the density corresponding to the maximum veloc-
ity shifts to higher values as Q becomes larger. FIG. 11
shows the optimal density of buses in the model B with-
out information-based control system. From this figure,
we find that the optimal density, for case S = 50, is
smaller than that for S = 5. Moreover, for given S, the
optimal density decreases with decreasing f . However,
for both S = 5 and S = 50, the optimal density corre-
sponding to Q = 1.0 is higher for f = 0.6 than that for
f = 0.9.
What is more effective way of increasing the efficiency
of the public conveyance system on a given route by in-
creasing the number of buses without increasing the car-
rying capacity of each bus, or by increasing the carrying
capacity of each bus without recruiting more buses? Or,
are these two prescriptions for enhancing efficiency of the
public conveyance system equally effective? In order to
address these questions, we make a comparative study
of two situations on the same route: for example, in the
0 0.2 0.4 0.6 0.8 1
Density
Q = 0.8
Q = 1.0
0 0.2 0.4 0.6 0.8 1
Density
rs Q = 0.8
Q = 1.0
FIG. 10: The average speed and the average number of wait-
ing passengers in the model B are plotted against the density
for the parameters f = 0.9, S = 50; the hopping parameters
are Q = 0.8 and Q = 1.0.
0.6 0.7 0.8 0.9 1
f = 0.9, S = 5
f = 0.6, S = 5
f = 0.9, S = 50
f = 0.6, S = 50
FIG. 11: The optimal density of buses in the model B is
plotted against Q. The parameters are f = 0.9, S = 5 (normal
line),f = 0.6, S = 5 (finer broken line), f = 0.9, S = 50 (bold
broken line), f = 0.6, S = 50 (longer broken line).
first situation the number of buses is 10 and each has a
capacity of 60, whereas in the second the number of buses
is 5 and each has a capacity of 120. Note that the total
carrying capacity of all the buses together is 600 (60×10
and 120× 5 in the two situations), i.e., same in both the
situations. But, the number density of the buses in the
second situation is just half of that in the first as the
length of the bus route is same in both the situations. In
FIG. 12, the results for these two cases are plotted; the
different scales of density used along the X-axis arises
from the differences in the number densities mentioned
above.
From FIG. 12, we conclude that, at sufficiently low
densities, the average velocity is higher for Nmax = 60
compared to those for Nmax = 120. But, in the same
regime of the number density of buses, larger number of
passengers wait at bus stops when the bus capacity is
smaller. Thus, in the region ρ < 0.05, system adminis-
trators face a dilemma: if they give priority to the aver-
age velocity and decide to choose buses with Nmax = 60,
the number of passengers waiting at the bus stops in-
creases. On the other hand if they decide to make the
passengers happy by reducing their waiting time at the
bus stops and, therefore, choose buses with Nmax = 120,
the travel time of the passengers after boarding a bus
becomes longer.
However, at densities ρ > 0.05, the system administra-
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
FIG. 12: Comparison between the case of bus capacity 60
with bus capacity 120. The parameters are Q = 0.9, S = 10,
f = 0.6 in the model B without information. The top figure
shows the average velocity, the center figure shows waiting
passengers and the bottom figure shows the number of con-
veyed passengers per unit bus, i.e. this number is calculated
by (total number of on-boarding passengers on all buses)/(the
number of buses), against the bus density up to 0.5. In each
figure, the horizontal axis shows the density; the numbers
without parentheses denote the number densities in the case
Nmax = 60, whereas the numbers in the parentheses denote
the number densities in the case Nmax = 120.
tors can satisfy both the criteria, namely, fewer wait-
ing passengers and shorter travel times, by one sin-
gle choice. In this region of density, the public con-
veyance system with Nmax = 60 is more efficient than
that with Nmax = 120 because the average velocity is
higher and the number of waiting passengers is smaller
for Nmax = 60 than for Nmax = 120. Thus, in this regime
of bus density, efficiency of the system is enhanced by re-
ducing the capacity of individual buses and increasing
their number on the same bus route.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 13: The plot of 〈V 〉 and 〈N〉 of the model B with infor-
mation (S = 5 and f = 0.3, 0.6 and 0.9)
0 0.2 0.4 0.6 0.8 1
Density
without info
with info
0 0.2 0.4 0.6 0.8 1
Density
rs without info
with info
FIG. 14: The model B with S = 5 and f = 0.9. The left
vertical dash line is ρ = 0.28 and the right is ρ = 0.73 in the
two figures.
B. PCM with information-based traffic control
The results for the PCM with information-based traf-
fic control system is shown in FIG. 13 and FIG. 14. In
the FIG. 13 we plot 〈V 〉 and 〈N〉 against the density of
buses for the parameter S = 5. The density correspond-
ing to the peak of the average velocity shifts to lower
values when the information-based traffic control system
is switched on.
The data shown in FIG. 14 establish that implemen-
tation of the information-based traffic control system
does not necessarily always improve the efficiency of
the public conveyance system. In fact, in the region
0.3 < ρ < 0.7, the average velocity of the buses is
higher if the information-based control system is switched
off. Comparing 〈V 〉 and 〈N〉 in FIG. 14, we find that
information-based traffic control system can improves the
efficiency by reducing the crowd of waiting passengers.
But, in the absence of waiting passengers, introduction
of the information-based control system adversely affects
the efficiency of the public conveyance system by holding
up the buses at bus stops when the number of buses in
the next segment of the route exceeds I0.
0 10 20 30 40 50
Ranking
without information
with information
FIG. 15: Distribution of headway distance for S = 10, m =
50, f = 0.9 in model B. This figure shows the plot of headway
distance against the ranking.
Finally, FIG. 15 shows the distribution of headway dis-
tance against the ranking, where we arrange the order of
magnitude according to the headway distance of buses
in descending order. From this figure it is found that
the headway distribution is dispersed by the effect of
the information. The average headway distance with the
information-based traffic control is equal to 8.34, in con-
trast to a much shorter value of 0.66 when that control
system is switched off. Thus we confirm that the avail-
ability of the information Ij and implementation of the
traffic control system based on this information, signifi-
cantly reduces the undesirable clustering of buses.
IV. MEAN FIELD ANALYSIS
Let us estimate 〈V 〉 theoretically in the low density
limit ρ → 0. Suppose, T is the average time taken by a
bus to complete one circuit of the route. In the model
A, the number of hops made by a bus with probability
q during the time T is S, i.e. the total number of bus
stops. Therefore the average period T for a bus in the
model A is well approximated by
and hence,
〈V 〉 =
q(L− S) +QS
. (4)
In model B, in the low density limit where m buses run
practically unhindered and are distributed uniformly in
the system without correlations, the average number of
passengersN waiting at a bus stop, just before the arrival
of the next bus, is
. (5)
The first factor f/S on the right hand side of the equation
(5) is the probability of arrival of passengers per unit
time. The second factor on the right hand side of (5)
is an estimate of the average time taken by a bus to
traverse one segment of the route, i.e. the part of the
route between successive bus stops. The last factor in the
same equation is the average number of segments of the
route in between two successive buses on the same route.
Instead of the constant q used in (4) for the evaluation
of 〈V 〉 in the model A, we use
N + 1
in eq. (4) and eq. (5) for the model B. Then, for the
model B, the hopping probability Q is estimated self-
consistently solving
〈V 〉 = Q−
, (7)
(4) and (6) simultaneously.
We also obtain, for the model B, the average number
of passengers 〈N〉 waiting at a bus stop in the ρ → 0
limit. The average time for moving from one bus stop to
the next is ∆t = (L/S − 1)/Q + 1/q̄ and, therefore, we
〈N〉 = (f/S) · (∆t+ 2∆t+ · · ·+ (S − 1)∆t)/S
f(S − 1)(q̄(L− S) + SQ)
2S2Qq̄
. (8)
As long as the number of waiting passengers does not
exceed Nmax, we have observed reasonably good agree-
ment between the analytical estimates (4), (8) and the
corresponding numerical data obtained from computer
simulations. For example, in the model A, we get the es-
timates 〈V 〉 = 0.85 and 〈N〉 = 1.71 from the approximate
mean field theory for the parameter set S = 50, m = 1,
Q = 0.9, q = 0.5, f = 0.3. The corresponding numbers
obtained from direct computer simulations of the model
A version of PCM are 0.84 and 1.78, respectively. Simi-
larly, in the model B under the same conditions, we get
〈V 〉 = 0.60 and 〈N〉 = 2.45 from the mean field theory,
while the corresponding numerical values are 0.60 and
2.51, respectively. If we take sufficiently small f ’s, then
the mean-field estimates agree almost perfectly with the
corresponding simulation data. However, our mean field
analysis breaks down when a bus can not pick up all the
passengers waiting at a bus stop.
V. CONCLUDING DISCUSSIONS
In this paper, we have proposed a public conveyance
model (PCM) by using stochastic CA. In our PCM, some
realistic elements are introduced: e.g., the carrying ca-
pacity of a bus, the arbitrary number of bus stops, the
halt time of a bus that depends on the number of waiting
passengers, and an information-based bus traffic control
system which reduces clustering of the buses on the given
route.
We have obtained quantitative results by using both
computer simulations and analytical calculations. In par-
ticular, we have introduced two different quantitative
measures of the efficiency of the public conveyance sys-
tem. We have found that the bus system works efficiently
in a region of moderate number density of buses; too
many or too few buses drastically reduce the efficiency
of the bus-transport system. If the density of the buses
is lower than optimal, not only large number of passen-
gers are kept waiting at the stops for longer duration, but
also the passengers in the buses get a slow ride as buses
run slowly because they are slowed down at each stop to
pick up the waiting passengers. On the other hand, if
the density of the buses is higher than optimal, the mu-
tual hindrance created by the buses in the overcrowded
route also lowers the efficiency of the transport system.
Moreover, we have found that the average velocity in-
creases, and the number of waiting passengers decreases,
when the information-based bus traffic control system is
switched on. However, this enhancement of efficiency of
the conveyance system takes place only over a particular
range of density; the information-based bus traffic con-
trol system does not necessarily improve the efficiency of
the system in all possible situations.
We have compared two situations where the second
situation is obtained from the first one by doubling the
carrying capacity of each bus and reducing their number
to half the original number on the same route. In the
density region ρ > 0.05 the system of Nmax = 60 is more
efficient than that with Nmax = 120. However, at small
densities (ρ < 0.05), although the average velocity in-
creases, the number of waiting passengers also increases,
by doubling the carrying capacity from Nmax = 60 to
Nmax = 120. Hence, bus-transport system administra-
tors would face a dilemma in this region of small density.
Finally, in our PCM, the effect of the disembarking
passengers on the halt time of the buses has not been
captured explicitly. Moreover, this study is restricted to
periodic boundary conditions. The clustering of particles
occurs not only in a ring-like bus route, but also in shuttle
services of buses and trains. Thus it would be interesting
to investigate the effects of the information-based traffic
control system also on such public transport systems. In
a future work, we intend to report the results of our in-
vestigations of the model under non-periodic boundary
conditions. We hope our model will help in understand-
ing the mechanism of congestion in public conveyance
system and will provide insight as to the possible ways
to reduce undesirable clustering of the vehicles.
Acknowledgments: Work of one of the authors (DC)
has been supported, in part, by the Council of Scientific
and Industrial Research (CSIR), government of India.
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|
0704.1556 | A separable deformation of the quaternion group algebra | A SEPARABLE DEFORMATION OF THE QUATERNION
GROUP ALGEBRA
NURIT BARNEA AND YUVAL GINOSAR
Abstract. The Donald-Flanigan conjecture asserts that for any finite group
G and any field k, the group algebra kG can be deformed to a separable algebra.
The minimal unsolved instance, namely the quaternion group Q8 over a field
k of characteristic 2 was considered as a counterexample. We present here a
separable deformation of kQ8. In a sense, the conjecture for any finite group
is open again.
1. Introduction
In their paper [1], J.D. Donald and F.J. Flanigan conjectured that any group
algebra kG of a finite group G over a field k can be deformed to a semisimple
algebra even in the modular case, namely where the order of G is not invertible
in k. A more customary formulation of the Donald-Flanigan (DF) conjecture is
by demanding that the deformed algebra [kG]t should be separable, i.e. it remains
semisimple when tensored with the algebraic closure of its base field. If, additionally,
the dimensions of the simple components of [kG]t are in one-to-one correspondence
with those of the complex group algebra CG, then [kG]t is called a strong solution
to the problem.
The DF conjecture was solved for groups G which have either a cyclic p-Sylow
subgroup over an algebraically closed field [11] or a normal abelian p-Sylow sub-
group [5] where p =char(k), and for all but six reflection groups in any characteristic
[6, 7, 10]. In [4], it is claimed that the group algebra kQ8, where
Q8 = 〈σ, τ |σ
4 = 1, τσ = σ3τ, σ2 = τ2 〉
is the quaternion group of order 8 and k a field of characteristic 2, does not admit
a separable deformation. This result allegedly gives a counterexample to the DF
conjecture. However, as observed by M. Schaps, the proof apparently contains an
error (see §7).
The aim of this note is to present a separable deformation of kQ8, where k is
any field of characteristic 2, reopening the DF conjecture.
2. Preliminaries
Let k[[t]] be the ring of formal power series over k, and let k((t)) be its field of
fractions. Recall that the deformed algebra [kG]t has the same underlying k((t))-
vector space as k((t))⊗k kG, with multiplication defined on basis elements
(2.1) g1 ∗ g2 := g1g2 +
Ψi(g1, g2)t
i, g1, g2 ∈ G
Date: November 13, 2018.
http://arxiv.org/abs/0704.1556v1
2 NURIT BARNEA AND YUVAL GINOSAR
and extended k((t))-linearly (such that t is central). Here g1g2 is the group multi-
plication. The functions Ψi : G×G → kG satisfy certain cohomological conditions
induced by the associativity of [kG]t [3, §1 ; §2].
Note that the set of equations (2.1) determines a multiplication on the free
k[[t]]-module Λt spanned by the elements {g}g∈G such that kG ≃ Λt/〈tΛt〉 and
[kG]t ≃ Λt ⊗k[[t]] k((t)). In a more general context, namely over a domain R which
is not necessarily local, the R-module Λt which determines the deformation, is
required only to be flat rather than free [2, §1].
In what follows, we shall define the deformed algebra [kG]t by using generators
and relations. These will implicitly determine the set of equations (2.1).
3. Sketch of the construction
Consider the extension
(3.1) [β] : 1 → C4 → Q8 → C2 → 1,
where C2 = 〈 τ̄ 〉 acts on C4 = 〈σ 〉 by
η : C2 → Aut(C4)
η(τ̄ ) : σ 7→ σ3(= σ−1),
and the associated 2-cocycle β : C2 × C2 → C4 is given by
β(1, 1) = β(1, τ̄) = β(τ̄ , 1) = 1, β(τ̄ , τ̄) = σ2.
The group algebra kQ8 (k any field) is isomorphic to the quotient kC4[y; η]/〈 q(y) 〉,
where kC4[y; η] is a skew polynomial ring [9, §1.2], whose indeterminate y acts
on the ring of coefficients kC4 via the automorphism η(τ̄ ) (extended linearly) and
where
(3.2) q(y) := y2 − σ2 ∈ kC4[y; η]
is central. The above isomorphism is established by identifying τ with the indeter-
minate y.
Suppose now that Char(k) = 2. The deformed algebra [kQ8]t is constructed as
follows.
In §4.1 the subgroup algebra kC4 is deformed to a separable algebra [kC4]t which
is isomorphic to K⊕k((t))⊕k((t)), where K is a separable field extension of k((t))
of degree 2.
The next step (§4.2) is to construct an automorphism ηt of [kC4]t which agrees
with the action of C2 on kC4 when specializing t = 0. This action fixes all three
primitive idempotents of [kC4]t. By that we obtain the skew polynomial ring
[kC4]t[y; ηt].
In §5 we deform q(y) = y2 + σ2 to qt(y), a separable polynomial of degree 2 in
the center of [kC4]t[y; ηt].
By factoring out the two-sided ideal generated by qt(y), we establish the defor-
mation
[kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉.
In §6 we show that [kQ8]t as above is separable. Moreover, passing to the
algebraic closure k((t)) we have
[kQ8]t ⊗k((t)) k((t)) ≃
k((t))⊕M2(k((t))).
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 3
This is a strong solution to the DF conjecture since its decomposition to simple
components is the same as
CQ8 ≃
C⊕M2(C).
4. A Deformation of kC4[y; η]
4.1. We begin by constructing [kC4]t, C4 = 〈σ 〉. Recall that
kC4 ≃ k[x]/〈x
4 + 1 〉
by identifying σ with x+ 〈x4+1 〉. We deform the polynomial x4+1 to a separable
polynomial pt(x) as follows.
Let k[[t]]∗ be the group of invertible elements of k[[t]] and denote by
U := {1 + zt|z ∈ k[[t]]∗}
its subgroup of 1-units (when k = F2, U is equal to k[[t]]
a ∈ k[[t]] \ k[[t]]∗
be a non-zero element, and let
b, c, d ∈ U, (c 6= d),
such that
π(x) := x2 + ax+ b
is an irreducible (separable) polynomial in k((t))[x]. Let
pt(x) := π(x)(x + c)(x + d) ∈ k((t))[x].
Then the quotient k((t))[x]/〈 pt(x) 〉 is isomorphic to the direct sum K ⊕ k((t)) ⊕
k((t)), where K := k((t))[x]/〈π(x) 〉. The field extension K/k((t)) is separable and
of dimension 2.
Note that pt=0(x) = x
4+1 and that only lower order terms of the polynomial were
deformed. Hence, the quotient k[[t]][x]/〈 pt(x) 〉 is k[[t]]-free and k((t))[x]/〈 pt(x) 〉
indeed defines a deformation [kC4]t of kC4 ≃ k[x]/〈x
4+1 〉. The new multiplication
σi∗σj of basis elements (2.1) is determined by identifying σi with x̄i := xi+〈 pt(x) 〉.
We shall continue to use the term x̄ in [kC4]t rather than σ.
Assume further that there exists w ∈ k[[t]] such that
(4.1) (x + w)(x + c)(x+ d) = xπ(x) + a
(see example 4.3). Then K ≃ ([kC4]t)e1, where
(4.2) e1 =
(x̄+ w)(x̄ + c)(x̄ + d)
The two other primitive idempotents of [kC4]t are
(4.3) e2 =
c(x̄ + d)π(x̄)
a(c+ d)
, e3 =
d(x̄ + c)π(x̄)
a(c+ d)
4 NURIT BARNEA AND YUVAL GINOSAR
4.2. Let
ηt : k((t))[x] → k((t))[x]
be an algebra endomorphism determined by its value on the generator x as follows.
(4.4) ηt(x) := xπ(x) + x+ a.
We compute ηt(π(x)), ηt(x+ c) and ηt(x+ d):
ηt(π(x)) = ηt(x)
2 + aηt(x) + b = x
2π(x)2 + x2 + a2 + axπ(x) + ax+ a2 + b
= π(x)(x2π(x) + ax+ 1).
By (4.1),
(4.5) ηt(π(x)) = π(x) + x(x+ w)pt(x) ∈ 〈π(x) 〉.
Next,
ηt(x+ c) = xπ(x) + x+ a+ c.
By (4.1),
(4.6) ηt(x+ c) = (x+ c)[(x + w)(x + d) + 1] ∈ 〈x+ c 〉.
Similarly,
(4.7) ηt(x+ d) = (x+ d)[(x + w)(x + c) + 1] ∈ 〈x+ d 〉.
By (4.5), (4.6) and (4.7), we obtain that ηt(pt(x)) ∈ 〈 pt(x) 〉, and hence ηt induces
an endomorphism of k((t))[x]/〈 pt(x) 〉 which we continue to denote by ηt. As can
easily be verified, the primitive idempotents given in (4.2) and (4.3) are fixed under
(4.8) ηt(ei) = ei, i = 1, 2, 3,
whereas
(4.9) ηt(x̄e1) = ηt(x̄)e1 = (x̄π(x̄) + x̄+ a)e1 = (x̄+ a)e1.
Hence, ηt induces an automorphism of K of order 2 while fixing the two copies of
k((t)) pointwise. Furthermore, one can easily verify that
ηt=0(x̄) = x̄
Consequently, the automorphism ηt of [kC4]t agrees with the automorphism η(τ̄ )
of kC4 when t = 0. The skew polynomial ring
[kC4]t[y; ηt] = (k((t))[x]/〈 pt(x) 〉)[y; ηt]
is therefore a deformation of kC4[y; η].
Note that by (4.8), the idempotents ei, i = 1, 2, 3 are central in [kC4]t[y; ηt] and
hence
(4.10) [kC4]t[y; ηt] =
[kC4]t[y; ηt]ei.
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 5
4.3. Example. The following is an example for the above construction.
t+ t2 + t3
1 + t
, b := 1 + t2 + t3, c :=
1 + t
, d := 1 + t+ t2, w := t.
These elements satisfy equation (4.1):
(x+ w)(x+ c)(x + d) = (x+ t)(x +
1 + t
)(x+ 1 + t+ t2)
= x3 +
t+ t2 + t3
1 + t
x2 + (1 + t2 + t3)x +
t+ t2 + t3
1 + t
= xπ(x) + a.
The polynomial
π(x) = x2 +
t+ t2 + t3
1 + t
x+ 1 + t2 + t3
does not admit roots in k[[t]]/〈 t2 〉, thus it is irreducible over k((t)).
5. A Deformation of q(y)
The construction of [kQ8]t will be completed once the product τ̄ ∗ τ̄ is defined.
For this purpose the polynomial q(y) (3.2), which determined the ordinary multi-
plication τ2, will now be developed in powers of t.
For any non-zero element z ∈ k[[t]] \ k[[t]]∗, let
(5.1) qt(y) := y
2 + zx̄π(x̄)y + x̄2 + ax̄ ∈ [kC4]t[y; ηt].
Decomposition of (5.1) with respect to the idempotents e1, e2, e3 yields
(5.2) qt(y) = (y
2 + b)e1 + [y
2 + zay + c(c+ a)]e2 + [y
2 + zay + d(d+ a)]e3.
We now show that qt(y) is in the center of [kC4]t[y; ηt] :
First, the leading term y2 is central since the automorphism ηt is of order 2.
Next, by (4.8), the free term be1 + c(c + a)e2 + d(d + a)e3 is invariant under the
action of ηt and hence central. It is left to check that the term za(e2 + e3)y is
central. Indeed, since e2 and e3 are ηt-invariant, then za(e2 + e3)y commutes both
with [kC4]t[y; ηt]e2 and [kC4]t[y; ηt]e3. Furthermore, by orthogonality
za(e2 + e3)y · [kC4]t[y; ηt]e1 = [kC4]t[y; ηt]e1 · za(e2 + e3)y = 0,
and hence za(e2 + e3)y commutes with [kC4]t[y; ηt].
Consequently, 〈 qt(y) 〉 = qt(y)[kC4]t[y; ηt] is a two-sided ideal.
Now, as can easily be deduced from (5.1),
(5.3) qt=0(y) = y
2 + x̄2 = q(y),
where the leading term y2 remains unchanged. Then
[kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉
is a deformation of kQ8, identifying τ̄ with ȳ := y + 〈 qt(y) 〉.
6 NURIT BARNEA AND YUVAL GINOSAR
6. Separability of [kQ8]t
Finally, we need to prove that the deformed algebra [kQ8]t is separable. More-
over, we prove that its decomposition to simple components over the algebraic
closure of k((t)) resembles that of CQ8. By (4.10), we obtain
(6.1) [kQ8]t =
[kC4]t[y; ηt]ei/〈 qt(y)ei 〉.
We handle the three summands in (6.1) separately:
By (5.2),
[kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ≃ K[y; ηt]/〈 y
2 + b 〉 ≃ Kf ∗ C2.
The rightmost term is the crossed product of the group C2 := 〈 τ̄ 〉 acting faithfully
on the field K = [kC4]te1 via ηt (4.9), with a twisting determined by the 2-cocycle
f : C2 × C2 → K
f(1, 1) = f(1, τ̄) = f(τ̄ , 1) = 1, f(τ̄ , τ̄ ) = b.
This is a central simple algebra over the subfield of invariants k((t)) [8, Theorem
4.4.1]. Evidently, this simple algebra is split by k((t)), i.e.
(6.2) [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ⊗k((t)) k((t)) ≃ M2(k((t))).
Next, since ηt is trivial on [kC4]te2, the skew polynomial ring [kC4]te2[y; ηt] is
actually an ordinary polynomial ring k((t))[y]. Again by (5.2),
[kC4]t[y; ηt]e2/〈 qt(y)e2 〉 ≃ k((t))[y]/〈 y
2 + zay + c(c+ a) 〉.
Similarly,
[kC4]t[y; ηt]e3/〈 qt(y)e3 〉 ≃ k((t))[y]/〈 y
2 + zay + d(d+ a) 〉.
The polynomials y2 + zay + c(c+ a) and y2 + zay + d(d + a) are separable (since
za is non-zero). Thus, both [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 and [kC4]t[y; ηt]e3/〈 qt(y)e3 〉
are separable k((t))-algebras, and for i = 2, 3
(6.3) [kC4]t[y; ηt]ei/〈 qt(y)ei 〉 ⊗k((t)) k((t)) ≃ k((t)) ⊕ k((t)).
Equations (6.1), (6.2) and (6.3) yield
[kQ8]t ⊗k((t)) k((t)) ≃
k((t)) ⊕M2(k((t)))
as required.
7. Acknowledgement
We wish to thank M. Schaps for pointing out to us that there is an error in the
attempted proof in [4] that the quaternion group is a counterexample to the DF
conjecture. Here is her explanation: The given relations for the group algebra are
incorrect. Using the notation in pages 166-7 of [4], if a = 1 + i, b = 1 + j and
z = i2 = j2, then ab + ba = ij(1 + z) while a2 = b2 = 1 + z. There is a further
error later on when the matrix algebra is deformed to four copies of the field, since
a non-commutative algebra can never have a flat deformation to a commutative
algebra.
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 7
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deformations of algebras and their representations, Israel Math. Conf. Proc., 7, (1993), 25–44.
[3] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964), 59–103.
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[6] M. Gerstenhaber and M.E. Schaps, Hecke algebras, Uqsln, and the Donald-Flanigan conjecture
for Sn, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3353–3371.
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Department of Mathematics, University of Haifa, Haifa 31905, Israel
E-mail address: [email protected]
1. Introduction
2. Preliminaries
3. Sketch of the construction
4. A Deformation of kC4[y;]
4.1.
4.2.
4.3. Example
5. A Deformation of q(y)
6. Separability of [kQ8]t
7. Acknowledgement
References
|
0704.1557 | On the residue fields of Henselian valued stable fields, II | arXiv:0704.1557v1 [math.RA] 12 Apr 2007
On the residue fields of Henselian valued stable fields, II
I.D. Chipchakov ∗
Introduction
This paper is a continuation of [Ch2]. Let E be a field, Esep a separable closure of
E , E∗ the multiplicative group of E , d(E) the class of finite-dimensional central
division E -algebras, Br (E) the Brauer group of E , Gal (E) the set of finite Galois
extensions of E in Esep , and NG (E) the set of norm groups N(M/E): M ∈ Gal(E) .
Let also P be the set of prime numbers, Br (E)p the p -component of Br (E) , for each
p ∈ P , P(E) the set of those p ∈ P , for which E is properly included in its maximal
p -extension E(p) (in Esep ), and Π(E) the set of all p
′ ∈ P , for which the absolute
Galois group GE = G(Esep/E) is of nonzero cohomological p
′ -dimension cd p′(GE) .
Recall that E is p -quasilocal, for a given p ∈ P , if p 6∈ P(E) or the relative
Brauer group Br (F/E) equals the subgroup pBr(E) = {bp ∈ Br(E): pbp = 0} , for
every cyclic extension F of E of degree p . The field E is said to be primarily
quasilocal (PQL), if it is p -quasilocal, for all p ∈ P ; E is called quasilocal, if its
finite extensions are PQL. We say that E is a strictly PQL-field, if it is PQL and
Br (E)p 6= {0} , for every p ∈ P(E) (see [Ch4, Corollary 3.7], for a characterization
of the SQL-property). The purpose of this note is to complement the main result of
[Ch2] and to shed light on the structure of Br (K) , where K is an absolutely stable
field (in the sense of E. Brussel) with a Henselian valuation whose value group is
totally indivisible.
1. Subfields of central division algebras over PQL-fields
Assume that E is a PQL-field and M ∈ Gal(E) , such that G(M/E) is nilpotent, and
let R be an intermediate field of M/E . It follows from Galois theory, [Ch2, Theorem
4.1 (iii)], the Burnside-Wielandt characterization of nilpotent finite groups, and the
general properties of central division algebras and their Schur indices (see [KM, Ch.
6, Sect. 2] and [P, Sects. 13.4 and 14.4]) that R embeds as an E -subalgebra in each
∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scien-
tific Research.
http://arxiv.org/abs/0704.1557v1
algebra D ∈ d(E) of index divisible by [R: E] . In this Section, we demonstrate the
optimality of this result in the class of strictly PQL-fields.
Theorem 1.1. For each nonnilpotent finite group G , there exists a strictly PQL-
field E = E(G) with Br (E) ∼= Q/Z and Gal (E) containing an element M such that
G(M/E) ∼= G and M does not embed as an E -subalgebra in any ∆ ∈ d(E) of index
equal to [M:E] .
Proof. Let P(G) be the set of prime divisors of the order of G . Our argument relies
on the existence (cf. [Ch1, Sect. 4]) of an algebraic number field E0 , such that there
is M0 ∈ Gal(E0) with G(M0/E0) ∼= G , and E0 possesses a system of (real-valued)
valuations {w(p): p ∈ P(G)} satisfying the following conditions:
(1.1) (i) w(p) extends the normalized p -adic valuation (in the sense of [CF, Ch.
VII]) of the field Q of rational numbers, for each p ∈ P(G) ;
(ii) The completion M0,w(p)′ lies in Gal (E0,w(p)) and G(M0,w(p)′/E0,w(p)) is isomor-
phic to a Sylow p -subgroup of G(M0/E0) , whenever p ∈ P(G) and w(p)′ extends
w(p) on M0 .
Let Ap be the maximal abelian p -extension of E0 in M0 , for each p ∈ P(G) . It
follows from the nonnilpotency of G , the Burnside-Wielandt theorem and Galois
theory that M0/E0 has an intermediate field F 6= E0 such that F ∩ Ap = E , for
all p ∈ P(G) (see [Ch1, Sect. 4]). Fix a prime divisor π of [F: E0] as well as Sylow
π -subgroups Hπ ∈ SylπG(M0/F) and Gπ ∈ SylπG(M0/E0) with Hπ ⊂ Gπ , and
denote by Fπ and Eπ the maximal extensions of E0 in M0 fixed by Hπ and
Gπ , respectively. It is clear from (1.1) (ii) and [CF, Ch. II, Theorem 10.2] that Eπ
has a valuation t(π) extending w(π) so that M0 ⊗Eπ Eπ,t(π) is a field. This means
that t(π) is uniquely (up-to an equivalence) extendable to a valuation µ(π) of M0 ,
and allows us to identify E0,w(π) with Eπ,t(π) . Using repeatedly the Grunwald-Wang
theorem [W] and the normality of maximal subgroups of finite π -groups (cf. [L, Ch. I,
Sect. 6]), one also obtains that there is a finite extension K of E0 in E0(π) , such that
K⊗E0 E0,w(π) is a field isomorphic to Fπ,ν(π) as an E0 -algebra, where ν(π) is the
valuation of Fπ induced by µ(π) . These observations indicate that K ∩M0 = E0 ,
i.e. the compositum M0K lies in Gal (K) and G((M0K)/K) ∼= G(M0/E0) . At the
same time, our argument proves that F is dense in Fπ,ν(p) with respect to the
topology induced by ν(p) , K has a unique valuation κ(π) extending w(π) ,
and (M0K)t(π)′ ∈ Gal(Kκ(π)) with G((M0K)t(π)′/Kκ(π)) ∼= Hπ , for any prolongation
t(π)′ of t(π) on M0K . Let now M(K) = {κ(p): p ∈ P} be a system of valuations
of K complementing κ(π) so that κ(p) extends w(p) or the normalized p -adic
valuation of Q , depending on whether or not p ∈ (P(G) \ {π}) . Applying [Ch3,
Theorem 2.2] to K , P and M(K) , and using [Ch3, Lemmas 3.1 and 3.2], one proves
the existence of an extension E of K in E0,sep , such that P(E) = P , M0E := M lies
in Gal (E) , G(M/E) ∼= G , and E possesses a system {v(p): p ∈ P} of valuations
with the following properties:
(1.2) For each p ∈ P , v(p) extends κ(p) , Ev(p) is a PQL-field, Ev(p)(p) is E -
isomorphic to E(p)⊗E Ev(p) , Br (Ev(p))p 6= {0} and Br (Ev(p′))p = {0} , for every
p′ ∈ (P \ {p}) . Moreover, if p ∈ P(G) and v(p)′ is a valuation of M extending
v(p) , then Mv(p)′ ∈ Gal(Ev(p)) , G(Mv(p)′/Ev(p)) ∼= G(M0,w(p)′/E0,w(p)): p 6= π , and
G(Mv(π)′/Ev(π))
∼= Hπ .
Hence, by [Ch3, Theorem 2.1], E is a nonreal strictly PQL-field. As Ev(π) is PQL,
it follows from [Ch2, Theorem 4.1], statements (1.2) and the Brauer-Hasse-Noether
and Albert theorem (in the form of [Ch3, Proposition 1.2]) that M splits an algebra
∆ ∈ d(E) of π -primary dimension if and only if ind (∆) divides the order of Hπ .
In view of the general theory of simple algebras (see [P, Sects. 13.4 and 14.4]), this
proves Theorem 1.1.
2. Divisible and reduced parts of the multiplicative group of an SQL-field
A field E is said to be strictly quasilocal (SQL), if its finite extensions are strictly
PQL. When this holds and E is almost perfect (in the sense of [Ch2, I, (1.8)]), this
Section gives a Galois-theoretic characterization of the maximal divisible subgroup
D(E) of E∗ .
Theorem 2.1. Let E be an almost perfect SQL-field and N1(E) the intersection of
the groups from NG (E) . Then N1(E) = D(E
To prove Theorem 2.1 we need the following lemma.
Lemma 2.2. Let E and M be fields, M ∈ Gal(E) , and let P(M/E) be the set of
prime divisors of [M:E] . Then N(M/E) ⊆ N(M/F) , for every intermediate field F
of M/E . Moreover, if Ep is the fixed field of some Sylow p -subgroup of G(M/E) ,
then N(M/E) = ∩p∈P(M/E)N(M/Ep) .
Proof. It is easily verified that if [F: E] = m , {τj : j = 1, . . . ,m} is the set of E -
embeddings of F into M , and σu is an automorphism of M extending τu , for each
index j , then G(M/E) = ∪mj=1G(M/F)σ
j . This yields N
E (γ) = N
j=1 σ
j (γ)) ,
for any γ ∈ M∗ , which proves that N(M/F) ⊆ N(M/E) . We show that ∩p∈P(M/E)
N(M/Ep) := N0(M/E) ⊆ N(M/E) . Take an element α ∈ N0(M/E) and put [Ep: E]
= mp , for each p ∈ P(M/E) . It is known that p 6 |mp , for any p ∈ P(M/E) , which
implies consecutively that g.c.d.(mp: p ∈ P(M/E)) = 1 , α ∈ E∗ and αmp ∈ N(M/E) ,
for every p ∈ P(M/E) . Thus it turns out that α ∈ N(M/E) , so Lemma 2.2 is proved.
Proof of Theorem 2.1. The inclusion D(E∗) ⊆ N1(E) is obvious, so our objective is
to prove the converse. We show that N1(E) ⊆ N1(E)p
, for every p ∈ P and n ∈ N .
It is clearly sufficient to consider the special case where p ∈ Π(E) . Then there is a
finite extension Fp of E in Esep , such that Br (Fp)p 6= {0} and p 6 |[Fp: E] . Note
also that each finite extension of Fp in E
∗ is included in a field F̃p ∈ Gal(E) , so it
follows from Lemma 2.2 that N1(E) ⊆ N1(Fp) . At the same time, the norm mapping
E clearly induces homomorphisms D(F
p) → D(E∗) and N1(Fp) → Np(E) (for
the latter, see [L, Ch. VIII, Sect. 5]) as well as an automorphism of the maximal
p -divisible subgroup of E∗ . These observations allow one to assume additionally
that p ∈ P(E) (and Br (E)p 6= {0} ). If p = char(E) , our assertion is implied by
[Ch2, Lemma 8.4], so we turn to the case of p 6= char(E) . Let εp be a primitive
p -th root of unity in Esep . Then [E(εp): E] divides p− 1 (cf. [L, Ch. VIII, Sect.
3]), whence our considerations reduce further to the case in which εp ∈ E . Now
the proof of Theorem 2.1 is completed by applying the following two lemmas. Before
stating them, let us recall that the character group C(Y/E) of G(Y/E) is an abelian
torsion group, for every Galois extension Y/E (see [K, Ch. 7, Sect. 5]). Therefore,
by [F, Theorem 24.5], the divisible part D(Y/E) of C(Y/E) is a direct summand
in C(Y/E) , i.e. C(Y/E) is isomorphic to the direct sum D(Y/E)⊕ R(Y/E) , where
R(Y/E) ∼= C(Y/E)/D(Y/E) is a maximal reduced subgroup of C(Y/E) .
Lemma 2.3. Let E be a p -quasilocal field containing a primitive p -th root of unity
ε , for some prime p ∈ P(E) . Assume also that rp(E) is the group of roots of unity
in E of p -primary degrees. Then:
(i) R(E(p)/E) = {0} if and only if Br (E)p = {0} or rp(E) is infinite;
(ii) If Br (E)p 6= {0} , d is the dimension of pBr(E) as a vector space over the
field Fp with p elements, and Rp(E) is of order p
µ , for some µ ∈ N , then
D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is presentable as a direct sum of cyclic
groups of order pµ , indexed by a set of cardinality d .
Proof. Statement (i) is a well-known consequence of Kummer theory, the Merkurjev-
Suslin theorem and Galois cohomology (see [MS, (11.5), (16.1)] and [S, Ch. I, 3.4
and 4.2]), so we assume further that d > 0 and rp(E) is of order p
µ , for some
µ ∈ N . We first show that pµC(E(p)/E) is divisible. Fix a primitive pµ -th root of
unity εµ in E and a subset S = {εn: n ∈ N, n ≥ µ+ 1} of E(p) so that εpn+1 = εn ,
for each index n . Also, let E∞ = E(S) and En = E(εn) , for every integer n > µ .
Suppose first that p > 2 or µ ≥ 2 . Then E∞/E is a Zp -extension and En is the
unique subextension of E in E∞ of degree p
n−µ . Since Zp is a projective profinite
group (see [G, Theorem 1]), this enables one to deduce from Galois theory that E(p)
possesses a subfield E′ such that E′ ∩ E∞ = E and E∞E′ = E(p) . In addition, it
becomes clear that C(E(p)/E) is isomorphic to the direct sum Z(p∞)⊕ C(F/E) ,
where Z(p∞) is the quasicyclic p -group (identified with the character group of Zp )
and F is the maximal abelian extension of E in E′ . Let now Φ be a cyclic extension
of E in F . As E is p -quasilocal, an element βn ∈ E∗n lies in N((ΦEn)/En) , for an
arbitrary integer n ≥ µ , if and only if the norm NEnE (βn) ∈ N(Φ/E) . In particular,
this proves that εµ ∈ N(Φ/E) if and only if εn ∈ N((ΦEn)/En) . In view of the
results of [FSS, Sect. 2], this means that pµC(F/E) = pnC(F/E) , for every index
n ≥ µ . In other words, C(F/E) is divisible, so pµC(E(p)/E) has the same property,
as claimed.
Our objective now is to prove the divisibility of 2C(E(2)/E) , under the hypothesis
that p = 2 and µ = 1 . If E∞ = E(
−1) = E2 , this is contained in [FSS, Proposi-
tion 2], so we assume further that E∞ 6= E2 , i.e. E2 = Eν 6= Eν+1 , for some in-
teger ν ≥ 2 . Let R be a cyclic extension of E in E(2) . By Albert’s theorem
(see [FSS, Sect. 2]), we have C(R/E) ⊂ 2C(E(2)/E) if and only if −1 ∈ N(R/E) .
We show that if C(R/E) ⊂ 2C(E(2)/E) , then εν ∈ N((RE2)/E2) . Since NE2E (εν)
equals 1 or −1 , this follows from [Ch2, Lemma 4.2 (iii)] in case R ∩ E2 = E .
Suppose further that E2 ⊆ R and fix a generator σ of G(F/E) . Applying [Ch2,
Theorem 4.1] one obtains that the cyclic E2 -algebra (R/E2, σ
2, εν) is similar
to (R/E, σ, c)⊗E E2 , for some c ∈ E∗ . In view of [P, Sect. 14.7, Proposition
b], this ensures that (R/E2, σ
2, εν) ∼= (R/E2, σ2, c) as E2 -algebras. Therefore, [P,
Sect. 15.1, Proposition b] yields cε−1ν ∈ N(R/E2) and c2N
E (εν) ∈ N(R/E) . Since
−1 ∈ N(R/E) , this means that c2 ∈ N(R/E) as well. Observing now that the core-
striction homomorphism of Br (E2) into Br (E) maps the similarity class Ac of
(R/E2, σ
2, c) into the similarity class of (R/E, σ, c2) (cf. [T, Theorem 2.5]), one con-
cludes that Ac lies in the kernel Ker E2/E of this homomorphism. At the same
time, by [Ch2, Lemma 4.2 (i)], we have Br (E2)2 ∩KerE2/E = {0} . In particu-
lar, Ac = 0 , so it follows from [P, Sect. 15.1, Proposition b] that εν ∈ N(R/E2) ,
as claimed. The obtained result, combined with [AFSS, Theorem 3], implies
that 2C(E(2)/E) = 2νC(E(2)/E) = 4C(E(2)/E) , which proves the divisibility of
2C(E(2)/E) .
In order to complete the proof of Lemma 2.3, it remains to be seen that the quotient
group C(E(p)/E)/pµC(E(p)/E) is isomorphic to the direct sum of cyclic groups of
order pµ , indexed by a set I of cardinality d (see [F, Theorem 24.5]). Since, by Kum-
mer’s theory, pµ−1C(E(p)/E) includes the group Xp(E) = {χ ∈ C(E(p)/E): pχ = 0} ,
it is sufficient to show that (Xp(E) + p
µC(E(p)/E))/pµC(E(p)/E) := Xp(E) is iso-
morphic to pBr(E) . Applying Albert’s theorem and elementary properties of symbol
E -algebras, and taking into account that Xp(E) ∼= Xp(E)/(Xp(E) ∩ pµC(E(p)/E)) ,
one obtains that Xp(E) ∼= E∗/N(Eµ+1/E) . At the same time, the cyclicity of Eµ+1/E
ensures that E∗/N(Eµ+1/E) ∼= Br(Eµ+1/E) (cf. [P, Sect. 15.1, Proposition b]), and
by the p -quasilocal property of E , we have Br (Eµ+1/E) = pBr(E) , so our proof is
complete.
Lemma 2.4. Let E be a p -quasilocal field containing a primitive p -th root
of unity ε , and such that Br (E)p 6= {0} , and let Ωp(E) be the set of finite
abelian extensions of E in E(p) . Then the intersection Np(E) of the norm groups
N(M/E): M ∈ Ωp(E) , coincides with the maximal p -divisible subgroup of E∗ .
Proof. It is clearly sufficient to show that Np(E) ⊆ N1(E)p
, for each n ∈ N . Denote
by rp(E) the group of roots of unity in E of p -primary degrees and fix an
algebra D ∈ d(E) of index p (the existence of D is guaranteed by the assumption
that Br (E)p 6= {0} and the Merkurjev-Suslin theorem [MS, (16.1)]). As E is p -
quasilocal, one obtains from Kummer theory that, for each c ∈ E∗ \ E∗p , there is
c′ ∈ E∗ \ E∗p , such that D is E -isomorphic to the symbol E -algebra Aε(c, c′; E) .
Hence, c 6∈ N(Ec′/E) , where Ec′ is the extension of E in E(p) obtained by
adjunction of a p -th root of c′ (see [P, Sect. 15.1, Proposition b]). The obtained
result implies Np(E) ⊆ E∗p . Arguing in a similar manner and applying general
properties of cyclic E -algebras (cf. [P, Sect. 15.1, Corollary b]), one deduces that
Np(E) ⊆ E∗p
, if E contains a primitive pm -th root of unity (and proves the
lemma in case rp(E) is infinite). Suppose further that rp(E) is of order p
µ , for
some µ ∈ N , and take C(E(p)/E) , D(E(p)/E) and R(E(p)/E) as in Lemma 2.3.
Also, let c ∈ (E∗pm ∩Np(E)) , for some integer m ≥ µ . We prove Lemma 2.4 by
showing that c ∈ (E∗pm+µ ∩Np(E)p
) . Fix a primitive pµ -th root of unity δµ ∈ E ,
take an element cm ∈ E∗ so that cp
m = c , and for any λ ∈ E∗ , denote by Eλ
the extension of E in E(p) obtained by adjunction of a pµ -th root of λ . Using
again [P, Sect. 15.1, Corollary b], one obtains that cm ∈ N(F/E) whenever F is an
intermediate field of a Zp -extension of E . Hence, by Albert’s theorem (see [FSS,
Sect. 2]), Kummer theory and the basic properties of symbol E -algebras of dimension
p2µ , N(Eδµ)/E) ⊆ N(Ecm)/E) . Since Br (E)p 6= {0} , E admits (one-dimensional)
local p -class field theory [Ch4, Theorem 3.1], which means that Ecm ⊆ Eδµ . In
view of Kummer theory, the obtained result yields cm = δ
m , for some k ∈ N ,
tm ∈ E∗ . Therefore, we have tp
m = (t
pm = c . Let now M be an arbitrary
finite abelian extension of E in E(p) . Then it follows from Galois theory and
Lemma 2.3 that M is a subfield of the compositum of finitely many cyclic extensions
Di, i ∈ I , and Rj , j ∈ J , of E in E(p) , such that [Rj : E] = pµ , for every j ∈ J , and
Di ⊆ ∆i , where ∆i is a Zp -extension of E in E(p) , for each index i . Hence, by
[Ch4, Theorem 3.1], cp
m ∈ N(M̃/E) ⊆ N(M/E) , i.e. cp
m ∈ Np(E) . Replacing cm by
tm and arguing as above, one obtains that t
m ∈ Np(E) and c ∈ Np(E)p
. Thus
the assertion that c ∈ (E∗pm+µ ∩Np(E)p
) is proved. It is now easy to see that
Np(E) = Np(E)
pn , for any n ∈ N , as required by Lemma 2.4.
Remark 2.5. Analyzing the proof of Theorem 2.1, one obtains that its conclu-
sion remains valid, if E is a quasilocal perfect field and cd p(GE) 6= 1 , for all
p ∈ (P \ char(E)) . One also proves that R(E∗) ∼= Φ(E)⊕ R0(E) , where Φ(E) is a
torsion-free group and R0(E) is the direct sum of the groups rp(E) , indexed by
those p ∈ Π(E) , for which rp(E) is nontrivial and finite. Moreover, it turns out that
if Br (E)p 6= {0} , for every p ∈ (Π(E) \ char(E)) , then D(E∗) equals the intersec-
tion of the groups N(M/E) , defined over all M ∈ Gal(E) , such that G(M/E) lies
in any fixed class χ of finite groups, which is closed under the formation of sub-
groups, homomorphic images and finite direct products, and which contains all finite
metabelian groups of orders not divisible by any p ∈ (P \Π(E)) .
3. The reduced part of the Brauer group of an equicharacteristic Henselian
valued absolutely stable field with a totally indivisible value group
In what follows, our notation agrees with that of Lemma 2.3 and its proof. For
each Henselian valued field (K, v) , K̂ and v(K) denote the residue field and
the value group of (K, v) , respectively. In this Section we announce the following
characterization of the reduced components of the Brauer groups of the fields pointed
out in its title:
Theorem 3.1. An abelian torsion group T is isomorphic to the reduced part of
Br (K) , for some absolutely stable field K = K(T) with a Henselian valuation v
such that char (K) = char(K̂) and the value group v(K) is totally indivisible (i.e.
v(K) 6= pv(K) , for every p ∈ P ), if and only if the p -component Tp of T is
presentable as a direct sum of cyclic groups of one and the same order pnp , for
each p ∈ P .
The rest of this Section is devoted to the proof of the necessity in Theorem 3.1
(the sufficiency will be proved elsewhere in a more general situation covering the
case where char (K) 6= char(K̂) , v is Henselian and discrete, and K̂ is perfect). We
begin with a description of some known basic relations between Br (K)p , Br (K̂)p
and C(K̂(p)/K̂) (proved for convenience of the reader).
Lemma 3.2. Let (K, v) be a Henselian valued field with v(K) 6= pv(K) , for
some p ∈ P , and let π be an element of K∗ of value v(π) 6∈ pv(K) . For each
χ ∈ C(K̂(p)/K̂) , denote by L̃χ the extension of K̂ in K̂sep corresponding by Ga-
lois theory to the kernel Ker (χ) , and by σ̃χ the generator of G(L̃χ/K̂) satis-
fying the equality χ(σ̃χ) = (1/[Lχ: K̂]) +Q/Z . Assume also that Lχ is the iner-
tial lift of L̃χ (over K̂ ) in Ksep , ηχ is the canonical isomorphism of G(Lχ/K)
on G(L̃χ/K̂) , and σχ is the preimage of σ̃χ in G(Lχ/K) with respect to
ηχ . Then the mapping Wπ of C(K̂(p)/K̂) into Br (K)p defined by the rule
Wπ(χ) = [(Lχ/K, σχ, π)]: χ ∈ C(K̂(p)/K̂) , is an injective group homomorphism.
Proof. It is clearly sufficient to show that [(Lχ1+χ2/K, σχ1+χ2 , π)] = [(Lχ1/L, σχ1 , π)]
+[(Lχ2/K, σχ2 , π)] , where χ1, χ2 ∈ C(K̂(p)/K̂) , in each of the following special
cases:
(3.1) (i) χ2 ∈ 〈χ1〉 , i.e. Lχ2 ⊆ Lχ1 ;
(ii) 〈χ1〉 ∩ 〈χ2〉 = {0} , i.e. Lχ1 ∩ Lχ2 = K .
In case (3.1) (i), this follows from the general properties of cyclic algebras (cf. [P,
Sect. 15.1, Corollary b and Proposition b]). Assuming that Lχ1 ∩ Lχ2 = K and
[Lχ1 : K] ≥ [Lχ2 : K] , one obtains that the K -algebras (Lχ1/K, σχ2 , π)⊗K (Lχ2/K, σχ2 ,
π) and (Lχ1+χ2/K, σχ1+χ2 , π)⊗K (Lχ2/K, σχ2 , 1) are isomorphic, which completes
our proof.
With assumptions being as in the lemma, let v(K) 6= pv(K) , for some prime
p 6= char(K̂) , and let Bp be a basis and n(p) the dimension of v(K)/pv(K)
as a vector space over Fp . Denote by C(K̂(p)/K̂)
n(p) the direct sum of some
isomorphic copies of C(K̂(p)/K̂) , indexed by Bp . Fix a linear ordering ≤ on Bp
and put Jp = {(cp, dp) ∈ (Bp × Bp): cp < dp} , and in case n(p) ≥ 2 , take a direct
sum Rp(K̂)
Jp of isomorphic copies of Rp(K̂) , indexed by Jp . Then the Ostrowski-
Draxl theorem, the Jacob-Wadsworth decomposition lemmas (see [JW]) and Lemma
3.2 imply the following variant of the Scharlau-Witt theorem [Sch]:
(3.2) (i) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) unless n(p) ≥ 2 and Rp(K̂) 6= {1} ;
(ii) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) ⊕ Rp(K̂)Jp , if n(p) ≥ 2 and Rp(K̂) 6= {1} .
Suppose now that K is a Henselian valued absolutely stable field such that v(K)
is totally indivisible. By [Ch2, Proposition 2.3], K̂ is quasilocal, so the necessity in
Theorem 3.1 can be deduced from (3.2), the divisibility of Br (F)q , for any field F of
characteristic q > 0 (Witt’s theorem, see [Dr, Sect. 15]), and the following lemma.
Before stating it, note that the extension of E in Esep obtained by adjunction of a
primitive p -th root of unity, for some p ∈ P , is cyclic of degree dividing p− 1 (cf.
[L, Ch. VIII, Sect. 3]).
Lemma 3.3. Let E be a quasilocal field not containing a primitive p -th root of unity,
for a given prime p 6= char(E) , and let ε be such a root in Esep . Fix a generator
ϕ of G(E(ε)/E) , take an integer s so as to satisfy the equality ϕ(ε) = εs , and put
Bs = {b ∈ pBr(E(ε)): ϕ(b) = sb} . Then:
(i) R(E(p)/E) = {0} if and only if rp(E(ε)) is infinite or the group Bs is trivial;
(ii) If Bs is nontrivial and of dimension d as an Fp -vector space, and if rp(E(ε))
is of order pµ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E)
is a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d .
Proof. Let Λ = {λ ∈ E(ε)∗: ϕ(λ)λ−s ∈ E(ε)∗p} . It is known (Albert, see [A, Ch. IX,
Theorem 15]) that an extension L of E in Esep is cyclic of degree p if and only if
L(ε) is generated over E(ε) by a p -th root of an element λ0 ∈ (Λ \ E(ε)∗p) . Hence,
the Kummer isomorphism of Xp(E(ε)) on E(ε)
∗/E(ε)∗p induces an isomorphism
of Xp(E) on Λ/E(ε)
∗p . Observe also that the similarity class of the symbol E(ε) -
algebra Aε(λ1, λ2; E(ε)) lies in Bi whenever λj ∈ Λ , j = 1, 2 . Conversely, by the
p -quasilocal property of E(ε) , each element of Bs is presented by a symbol E(ε) -
algebra determined by a pair of elements of Λ . The noted results enable one to prove
the lemma arguing as in the proof of Lemma 2.3.
REFERENCES
[A] A.A. ALBERT, Modern Higher Algebra. Univ. of Chicago Press, XIV,
Chicago, Ill., 1937.
[AFSS] J.K. ARASON, B. FEIN, M. SCHCHER, J. SONN, Cyclic extensions of K(
−1) .
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[CF] J.W.S. CASSELS, A. FRöHLICH (Eds.), Algebraic Number Theory. Aca-
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[Ch1] I.D. CHIPCHAKOV, On nilpotent Galois groups and the scope of the norm
limitation theorem in one-dimensional abstract local class field theory. In:
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Apulensis, No 10 (2005), 149-167.
[Ch2] I.D. CHIPCHAKOV, On the residue fields of Henselian valued stable fields.
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dimensional local class field theory. Preprint (available at www.arXiv.org/
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[Ch4] I.D. CHIPCHAKOV, One-dimensional abstract local class field theory. Preprint,
v. 3 (available at www.arXiv.org/math.RA/0506515).
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bridge etc., Cambridge Univ. Press, 1983.
[FSS] B. FEIN, D. SALTMAN, M. SCHACHER, Heights of cyclic field extensions. Bull.
Soc. Math. Belg., Ser. A, 40 (1988), 213-223.
[F] L. FUCHS, Infinite Abelian Groups. Academic Press, New York-London,
1970.
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[G] K.W. GRUENBERG, Projective profinite groups. J. Lond. Math. Soc. 42
(1967), 155-165.
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Groups, 2nd Ed. Nauka, Moscow, 1977 (Russian: Engl. Transl. in Graduate
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Amsterdam, 1989.
[JW] B. JACOB, A.R. WADSWORTH, Division algebras over Henselian fields. J.
Algebra 128 (1990), 528-579.
[L] S. LANG, Algebra, Addison-Wesley, Reading, MA, 1965.
[MS] A.S. MERKURJEV, A.A. SUSLIN, K -cohomology of Brauer-Severi varieties
and norm residue homomorphisms. Izv. Akad. Nauk SSSR 46 (1982), 1011-
1046 (Russian: Engl. transl. in Math. USSR Izv. 21 (1983), 307-340).
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Verlag, New York-Heidelberg-Berlin, 1982.
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Semin. Univ. Hamb. 33 (1969), 243-249.
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Verlag, Berlin-Heidelberg-New York, 1965.
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Ivan CHIPCHAKOV
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., bl. 8
1113 SOFIA, Bulgaria
|
0704.1558 | Chromospheric Cloud-Model Inversion Techniques | The Physics of Chromospheric Plasmas
ASP Conference Series, Vol. 368, 2007
Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds.
Chromospheric Cloud-Model Inversion Techniques
Kostas Tziotziou
National Observatory of Athens, Institute for Space Applications and
Remote Sensing, Greece
Abstract. Spectral inversion techniques based on the cloud model are ex-
tremely useful for the study of properties and dynamics of various chromospheric
cloud-like structures. Several inversion techniques are reviewed based on simple
(constant source function) and more elaborated cloud models, as well as on grids
of synthetic line profiles produced for a wide range of physical parameters by
different NLTE codes. Several examples are shown of how such techniques can
be used in different chromospheric lines, for the study of structures of the quiet
chromosphere, such as mottles/spicules, as well as for active region structures
such as fibrils, arch filament systems (AFS), filaments and flares.
1. Introduction
Observed intensity line profiles are a function of several parameters describing
the three-dimensional solar atmosphere, such as chemical abundance, density,
temperature, velocity, magnetic field, microturbulence etc (which one would like
to determine), as well as of wavelength, space (solar coordinates) and time. How-
ever, due to the large number of parameters that an observed profile depends
on, as well as data noise, model atmospheres have to be assumed in order to
restrict the number of these unknown parameters. The term “inversion tech-
niques” refers to the procedures used for inferring these model parameters from
observed profiles. We refer the reader to Mein (2000) for an extended overview of
inversion techniques. In this paper, we will review only a class of such inversion
techniques known in the solar community as “cloud models”.
Cloud models refer to models describing the transfer of radiation through
structures located higher up from the solar photosphere, which represents the
solar surface, resembling clouds on earth’s sky (see Fig. 1). Such cloud-like
structures, when observed from above, would seem to mostly absorb the radi-
ation coming from below, an absorption which mostly depends on the optical
thickness of the cloud, that is the “transparency” of the cloud to the incident
radiation and also on the physical parameters that describe it. The possibility of
observed emission from such structures cannot, of course, be excluded when the
radiation produced by the cloud-like structure is higher than the absorbed one.
The aforementioned processes are described by the radiative transfer equation
I(∆λ) = I0(∆λ) e
−τ(∆λ) +
∫ τ(∆λ)
−t(∆λ) dt , (1)
where I(∆λ) is the observed intensity, I0(∆λ) is the reference profile emitted by
the background (the incident radiation to the cloud from below), τ(∆λ) is the
http://arxiv.org/abs/0704.1558v1
218 Tziotziou
Figure 1. Geometry of the cloud model. D is the geometrical thickness
of the cloud at height H above the solar surface and V its velocity. From
Heinzel et al. (1999).
optical thickness and S the source function which is a function of optical depth
along the cloud. The first term of the right hand part of the equation represents
the absorption of the incident radiation by the cloud, while the second term
represents emission by the cloud itself.
The simple cloud model method introduced by Beckers (1964) arose from
the need to solve fast the radiation transfer equation and deduce the physical
parameters that describe the observed structure. Beckers assumed that a) the
structure is fully separated from the underlying chromosphere, b) the source
function, radial velocity, Doppler width and the absorption coefficient are con-
stant along the line-of-sight (hereafter LOS) and c) the background intensity is
the same below the structure and the surrounding atmosphere; hence it can be
extrapolated from a neighboring to the structure under study region. Under the
above assumptions the radiative transfer equation is simplified to
I(∆λ) = I0(∆λ) e
−τ(∆λ) + S(1− e−τ(∆λ)) (2)
and can be rewritten as
C(∆λ) =
I(∆λ)− I0(∆λ)
I0(∆λ)
I0(∆λ)
(1− e−τ(∆λ)) , (3)
where C(∆λ) defines the contrast profile. A Gaussian wavelength dependence
is usually assumed for the optical depth as follows
τ(∆λ) = τ0 e
∆λ−∆λI
, (4)
Cloud-model Inversion Techniques 219
where τ0 is the line center optical thickness, ∆λI = λ0v/c is the Doppler shift
with λ0 being the line center wavelength, c the speed of light and ∆λD is the
Doppler width. The latter depends on temperature T and microturbulent ve-
locity ξt through the relationship
∆λD =
ξ2t +
, (5)
where m is the atom rest mass. Other wavelength dependent profiles than the
Gaussian one can also be assumed for the optical depth, e.g., a Voigt profile
(Tsiropoula et al. 1999).
The four adjustable parameters of the model are the source function S, the
Doppler width ∆λD, the optical thickness τ0 and the LOS velocity v. All these
parameters are assumed to be constant through the structure. There are some
crucial assumptions concerning Beckers’ cloud model (hereafter BCM):
– the uniform background radiation assumption, which is not always true es-
pecially for cloud-like structures that do not reside above quiet Sun regions.
Moreover, the background radiation plays an important role in the correct
quantitative determination of the physical parameters.
– the neglect of incident radiation, the effects of which are of course not directly
considered in BCM, but does play an important role in non-Local Thermody-
namic Equilibrium (hereafter NLTE) modeling, since it determines the radia-
tion field within the structure, that is the excitation and ionization conditions
and hence the source function.
– the constant source function assumption which is not realistic especially in
the optically thick case or not valid in the presence of large velocity gradients.
However, the cloud model works quite well for a large number of optically thin
structures and can provide useful, reasonable estimates for the physical param-
eters that describe them. We refer the reader to Alissandrakis et al. (1990) for
a detailed discussion on the validity conditions of BCM for different types of
contrast profiles.
2. Cloud Model Variants
Since the introduction of the BCM method several improvements have been
suggested in the literature. When looking at the radiative transfer equation
(Eq. 1), it is obvious that all efforts concentrate on a better description of the
source function S which in BCM is considered to be constant. In the following
subsections some of these suggested improvements are described.
2.1. Variable source function
Mein et al. (1996a) considered a source function that is a function of optical
depth and is approximated by a second-order polynomial
St = S0 + S1
τ0,max
τ0,max
220 Tziotziou
Figure 2. Left: Geometry of the cloud model in the case of first-order differ-
ential cloud model. From Heinzel et al. (1992). Right: The “3-optical depths”
procedure for solving the differential cloud model case which is described in
Sect. 3.5 (from Mein & Mein (1988)).
with the optical depth at the center of the line τ0 taking values between 0 and
the total optical thickness at line center τ0,max, while S0, S1 and S2 are functions
of τ0,max. This formulation was further improved by Heinzel et al. (1999), who
included also the effect of cloud motion by assuming that S0, S1 and S2 are now
not only functions of τ0,max, but also depend on the velocity v of the structure.
Tsiropoula et al. (1999) assumed the parabolic formula
St = S0
1 + α
as an initial condition for the variation of the source function with optical depth,
where S0 is the source function at the middle of the structure, τ0 the optical
depth at line center, and α a constant expressing the variation of the source
function. However, their final results on the dependence of the source on optical
depth were in good agreement with the results of Mein et al. (1996a).
2.2. Differential cloud models
First and second order differential cloud models (hereafter DCM1 and DCM2)
were introduced by Mein & Mein (1988) to account for fast mass flows observed
on the disc, where BCM is not valid due to fluctuations of the background and
strong velocity gradients along the LOS. DCM1 assumes that the source function
S, temperature T and velocity v are constant within a small volume contained
between two close lines of sight P and R (see Fig. 2, left panel). If we assume
that the variation of the background profile is negligible (I0P ≃ I0R) for such
close points then the differential cloud contrast profile can be written as
C(P,R, λ) =
IP (λ)− IR(λ)
IR(λ)
IR(λ)
(1− e−δτ(λ)) (8)
Cloud-model Inversion Techniques 221
δτ(λ) = δτ0 e
λ− λ0 − vλ0/c
, (9)
where ∆λD is the Doppler width. The zero velocity reference wavelength λ0
is obtained by averaging over the whole field of view. DCM1 is a method for
suppressing the use of the background radiation. If velocity shears are present
between neighboring LOS then DCM2 can be used instead which requires the
use of three neighboring LOS. We refer the reader to Mein & Mein (1988) for the
precise formulation of DCM2 and to Table 1 of the same paper which summarizes
the validity conditions, constrains and results of the two models in comparison
to the classical BCM.
2.3. Multi-cloud models
The multi-cloud model (Gu et al. 1992, 1996) – hereafter MCM – was intro-
duced for the study of asymmetric, non-Gaussian profiles, such as line profiles
of post-flare loops, prominences and surges and was based on the BCM and
DCMs models. These asymmetric line profiles are assumed to be the result of
overlapping of several symmetric Gaussian profiles along the LOS, formed in
small radiative elements (clouds) which have a) different or identical physical
properties and b) a source function and velocity independent of depth. The
profile asymmetry mostly results from the relative Doppler shifts of the different
clouds. The total intensity Iλ emitted by m clouds is then given by the relation
Iλ = I0,λ e
−τλ +
Sj(1− e
−τλ,j ) exp
, (10)
where τλ,0 = 0, I0,λ is the background intensity, τλ =
j=1 τλ,j is the total
optical depth of the m clouds and
τλ,j = τ0,j e
λ− λ0 −∆λ0,j
∆λD,j
∆λ0,j = λ0vj/c, Sj, τ0,j, vj , ∆λD,j are respectively the optical depth, Doppler
shift, source function, line-center thickness, velocity and Doppler width of the
jth cloud.
A somewhat similar in philosophy, two-cloud model method was used by
Heinzel & Schmieder (1994) in their study of black and white mottles. It was
assumed that the LOS intersects two mottles treated as two different clouds c1
and c2 with optical depths τ1 and τ2 respectively. Hence the emerging intensity
from the lower mottle I1 is assumed to be the background incident intensity for
the second upper mottle. Then, the equations describing the radiation transfer
through the two mottles are
I2(∆λ) = I1 e
−τ2(∆λ) + Ic2(∆λ)
I1(∆λ) = I0 e
−τ1(∆λ) + Ic1(∆λ) , (12)
222 Tziotziou
where I0 is the background chromospheric intensity and Ic1, Ic2 the intensity
emitted by the two clouds respectively. The novelty of the method is that for
the emitted by the clouds intensity, a grid of 140 NLTE models was used which
was computed for prominence-like structures by Gouttebroze et al. (1993). So
this method is a combination of MCM with NLTE source function calculations
which will be further discussed in Section 2.6.
2.4. The Doppler signal method
The Doppler signal method (Georgakilas et al. 1990; Tsiropoula 2000) can be
used when filtergrams at two wavelengths −∆λ and +∆λ (blue and red side of
the line) are available and a fast determination of mass motions is needed. Then
the Doppler signal DS can be defined from the BCM equations as
I − 2I0
− e−τ
2− e−τ
− e−τ
, (13)
where ∆I = I(−∆λ) − I(+∆λ),
I = I(−∆λ) + I(+∆λ) and τ± = τ(±∆λ).
The Doppler signal DS has the same sign as velocity and can be used for a
qualitative description of the velocity field. The left hand side of the above
equation can be determined by the observations while the right hand clearly
does not depend on the source function. Quantitative values for the velocity can
be obtained when τ0 < 1; then the Doppler signal equation reduces to
τ− − τ+
τ− + τ+
and the velocity v – once DS is calculated from the observations and a value of
the Doppler width ∆λD is obtained from the literature or assumed – is given by
the equation
1 +DS
. (15)
2.5. Avoiding the background profile
Liu & Ding (2001) in order to avoid the use of the background profile needed
in BCM assumed that it is symmetric, that is I0(∆λ) = I0(−∆λ). Then it can
easily be shown that we can obtain the relationship
∆I(∆λ) = I(∆λ)− I(−∆λ) = [I(∆λ)− S][1− eτ(∆λ)−τ(−∆λ)] , (16)
which does not require the use of the background for the derivation of the phys-
ical parameters.
2.6. NLTE methods
As Eq. 1 shows, in the general case, the source function S within a cloud-like
structure is not constant, but usually depends on optical depth. In order to cal-
culate this dependence, the NLTE radiative transfer problem within the struc-
ture has to be solved, taking into account all excitation and ionization conditions
within the structure. Several efforts have been undertaken in the past for such
Cloud-model Inversion Techniques 223
Figure 3. Geometry of a two-dimensional cloud model slab. The incident
radiation comes not only from below, but also from the sides of the structure.
From Vial (1982).
NLTE calculations, usually for the case of filaments or prominences. Such NLTE
calculations started from the one-dimensional regime, where the cloud-like struc-
ture is approximated by an infinite one-dimensional slab (see Fig. 1) or a cylin-
der. We refer the reader to the works of Heasley et al. (1974), Heasley & Mihalas
(1976), Heasley & Milkey (1976), Mozozhenko (1978), Fontenla & Rovira (1985),
Heinzel et al. (1987), Gouttebroze et al. (1993), Heinzel (1995), Gouttebroze
(2004) for an overview of such one-dimensional NLTE models. The philosophy of
two-dimensional NLTE models is similar to the one-dimensional models, but now
the cloud-like structure is replaced by a two-dimensional slab or cylinder which
is infinite in the third dimension, allowing both vertical, as well as horizontal
radiation transport (see Fig. 3). Furthermore, the incident radiation is treated
as anisotropic and comes now not only from below, but also from the sides of
the structure. We refer the reader to the works of Mihalas et al. (1978), Vial
(1982), Paletou et al. (1993), Auer & Paletou (1994), Heinzel & Anzer (2001),
Gouttebroze (2005) for an overview of such two-dimensional NLTE models.
A general recipe for such NLTE models, which is modified according to the
specific needs, i.e. the line profile used and the structure observed, has as follows:
– The cloud-like structure is assumed to be a 1-D or 2-D slab or cylinder at
a height H above the photosphere. This slab/cylinder can be considered
to be either isothermal (e.g., Heinzel 1995) or isothermal and isobaric (e.g.,
Paletou et al. 1993).
– The incident radiation comes in the case of 1-D models only from below and
in the case of 2-D models also from the sides and determines the radiation
field within the structure, that is all excitation and ionization conditions.
– A multi-level atom plus continuum is assumed. The larger the number of
atomic levels used, the more computationally demanding the method is.
Complete or partial redistribution effects (CRD or PRD) are also assumed
224 Tziotziou
depending on the formation properties of the line. Methods with CRD are
computationally much faster so sometimes CRD is used but with simulated
PRD effects taken into account (e.g., Heinzel 1995).
– Some physical parameters are assigned to the slab/cylinder, like temperature
T , bulk velocity v, geometrical thickness Z, electronic density Ne or pressure
p. Calculations with electronic density are usually faster than calculations
with pressure.
– The radiative transfer statistical equilibrium equations are numerically solved
and the population levels are found and hence the source function as a func-
tion of optical depth for a set of selected physical parameters.
Once the source function S is obtained as a function of optical depth, Eq. 1 can
be solved in order to calculate the emerging observed profile from the structure
which is going to be compared to the observed one.
3. Solving the Cloud Model Equations
In the following subsections some of the methods used to solve the cloud model
equations are reviewed. We remind the reader that whenever the background
profile is needed, either the average profile of a quiet Sun region is taken or the
average profile of a region close to the structure under study.
3.1. Solving the constant-S case with the “5-point” method
Mein et al. (1996a) introduced the “5-point method” for solving the BCM equa-
tion with constant S. According to this method five intensities of the observed
and the background profile at wavelengths λ1, λ2 (blue wing of the observed
profile), λ3, λ4 (red wing of the the observed profile) and the line-center wave-
length λ0 are used for solving Eqs. 3 and 4. It is an iterative method that works
as follows:
– The line-center wavelength λ0 profile and background intensities are used for
calculating S, where τ0, ∆λD and v are determined in a previous iteration.
At the first step of the iteration some values can be assumed and S can be
taken as equal to zero.
– Profile and background intensities at wavelengths λ1 and λ3 are used for
calculating a new τ0.
– Afterwards a new ∆λD is calculated using the other two remaining wave-
lengths λ2 and λ4.
– Finally a new velocity is calculated from wavelengths λ1, λ2, λ3, and λ4 and
then a reconstructed profile obtained using the derived parameters which is
compared to the observed one. If any of the departures between the recon-
structed and the observed profile is higher than an assumed small threshold
value (i.e. 10−4) then the aforementioned procedure is repeated until con-
vergence is achieved. If no convergence is gained after a certain number of
iterations then it is assumed that no solution exists.
We refer the reader to Mein et al. (1996a) for a detailed description of the ana-
lytical equations described above.
Cloud-model Inversion Techniques 225
3.2. Solving the constant-S case with an iterative least-square fit
This method which was used by Alissandrakis et al. (1990) and further described
in Tsiropoula et al. (1999) and Tziotziou et al. (2003) fits the observed contrast
profile with a curve that results from an iterative least-square procedure for
non-linear functions which is repeated until the departures between computed
and observed profiles are minimized. The coefficients of the fitted curve are
functions of the free parameters of the cloud model. At the beginning of the
iteration procedure initial values have to be assumed for the free parameters
and especially for the source function S which is usually estimated from some
empirical approximate expressions that relate it to the line-center contrast. This
method is very accurate and usually converges within a few iterations. The more
observed wavelengths used within the profile, the better the determination of
the ohysical parameters is. However, as Tziotziou et al. (2003) have reported,
the velocity calculation can overshoot producing very high values, if the wings of
the profile are not sufficiently covered by observed wavelengths. The suggested
way to overcome the problem is to artificially add two extra contrast points near
the continuum of the observed profile where the contrast should be in theory
equal to zero.
This iterative method can also be successfully used not only in the case
of a constant source function S, but also for cases with a prescribed expres-
sion for the source function, such as the parabolic expression of Eq. 7 used by
Tsiropoula et al. (1999).
3.3. Solving the constant-S case with a constrained nonlinear least-
square fitting technique
The constrained nonlinear least-square fitting technique, used by Chae et al.
(2006) for the inversion of a filament with BCM, was introduced by Chae et al.
(1998). According to the method a) expectation values pei of the ith free param-
eters, b) their uncertainties εi, as well as c) the data to fit are provided (M wave-
lengths along the profile) and then a set ofN free parameters p = (p0, p1, ...pN−1)
are sought, i.e. p = (S, τ0, λ0,∆λD), that minimize the function
H(p) =
Cobsj − C
j (p)
pi − p
, (17)
where Cobsj and C
j are respectively the observed and calculated with the
expectation values contrasts and σj the noise in the data. The first term of the
sum H represents the data χ2, while the second term the expectation χ2 which
regularizes the solution by constraining the probable range of free parameters.
For very small values of εi the solution will not be much constrained by the
data and will be close to the chosen set of expectation values pei , while for large
values of εi it will be mostly constrained by the data and not by the expectation
values. We refer the reader to Chae et al. (2006) for a detailed discussion of the
effects of constrained fitting.
3.4. Solving the variable-S case
Apart from the iterative least-square procedure described above which can be
used when the source function varies in a prescribed way, Mein et al. (1996a)
226 Tziotziou
have introduced also the “4-point method” for solving the case of a source func-
tion that is described by the second order polynomial of Eq. 6. According to
the method an intensity I ′(∆λ) can be defined as follows
I ′(∆λ) = I(∆λ)−
1− [τ(∆λ) + 1] e−τ(∆λ)
τ(∆λ)
2− [τ2(∆λ) + 2τ(∆λ) + 2] e−τ(∆λ)
τ2(∆λ)
S2 (18)
and then the radiative transfer equation reduces to
I ′(∆λ) = S0 + (I0 − S0) e
−τ(∆λ) . (19)
This equation can be solved now using the iteration procedure described in
Sect. 3.1, with the modification that I(∆λ) is now replaced by I ′(∆λ) and
that the source function calculation in the first step is replaced by the assumed
theoretical relation for S given by Eq. 6.
3.5. Solving the DCM cases
A method for solving the differential cloud model cases is the “3-optical depths”
procedure introduced by Mein & Mein (1988). According to this procedure:
– the zero velocity reference is obtained from the average profile over the whole
field of view;
– a value S is assumed between zero and the line-center intensity (in principle
it could even work also for emitting clouds) and a function δτ(λ) is derived
from Eq. 8. The latter is characterized by the maximum value δτ0 and δτ1,
and the values δτ2 (see right panel of Fig. 2) which correspond to the half
widths ∆λ1 and ∆λ2 respectively and are given by the following relations
δτ1 = δτ0 e
−(∆λ1/∆λD)
δτ2 = δτ0 e
−(∆λ2/∆λD)
; (20)
– the code fits S and ∆λD by the conditions of Eq. 20 coupled with Eq. 8 and
the solutions are assumed to be acceptable when the radial velocities v1 and
v2, which correspond to widths ∆λ1 and ∆λ2 respectively and are defined
as the displacement of the middle of these chords compared with the zero
reference position, are not that different. When convergence is achieved the
δτ(λ) curve is well represented by a Gaussian and the Doppler width ∆λD is
independent of the chord ∆λ.
3.6. Solving the MCM case
We refer the reader to the papers by Li & Ding (1992) and Li et al. (1993, 1994)
for a detailed description of the methods and mathematical manipulations used
for fitting observed profiles with the multi-cloud method, which unfortunately
are not easy to concisely describe within a few lines.
Cloud-model Inversion Techniques 227
3.7. Using NLTE Methods
The most straightforward method for deriving the parameters of an observed
structure with NLTE calculations would be the calculation of a grid of models
for a wide range of the physical parameters used to describe the structure. How-
ever, the calculation of such a grid is computationally demanding, especially in
the case when a) a large number of atomic levels is assumed and/or b) partial
redistribution effects (PRD) are taken into account and/or c) a two-dimensional
geometry is considered. For such cases, either a very small grid of models is con-
structed and thus only approximate values for the observed structure are derived
or “test and try” methods are used where the user makes a “good guess” for the
physical parameter values, proceeds to the respective NLTE calculations, com-
pares the derived profile(s) with the observed one(s) and applies the necessary
adjustments to the model parameters according to the derived results.
However, nowadays the construction of a large grid of models, although
time-demanding, becomes more of a common practice with the extended ca-
pabilities of modern computers. We refer the reader to Molowny-Horas et al.
(2001) and Tziotziou et al. (2001) for two such examples, both considering a
one-dimensional isothermal slab for a cloud-like structure, which is the same
filament observed and studied in the Hα in Ca II 8542 Å lines respectively. The
general methodology used in the case of grids of models is the following:
– a grid of synthetic line profiles for a wide range of model parameters is
computed using NLTE calculations for the source function, as described in
Sect. 2.6;
– these synthetic profiles are convolved with the characteristics of the instru-
ment used for the observations in order to simulate its effects on the observed
profiles;
– each observed profile is compared with the whole library of convolved syn-
thetic profiles and the best fit is derived, that is the synthetic profile with the
smallest departure, and hence the physical parameters that describe it;
– an interpolation (linear or parabolic) between neighboring points in the pa-
rameter space can also be used, for a more accurate quantitative determina-
tion of the physical parameters that best describe the observed profile.
Grid models based on NLTE calculations have many advantages since pre-
ferred geometries, temperature structures, etc can be used, no iterations are
required, errors can be easily defined from the parameter space and inversions
are nowadays becoming faster with modern computers.
4. Validity of the Cloud Models
The validity of the cloud model used for an inversion obviously strongly de-
pends on a) the method used, b) the assumptions that were made for the model
atmosphere describing the structure and c) the specific characteristics of the
structure under study. Most of the reviewed papers in Sect. 5, concerning ap-
plications of different cloud models, have extended discussions on the validity
of the cloud model method and the results obtained, as well as the limitations
228 Tziotziou
Figure 4. Two left panels: The calculated optical depth τ0,max and velocity
v with BCM (constant source function) versus the assumed optical thickness.
The dashed curve is the model, the solid curve the inversion. Two right panels:
Same plots but with added Gaussian noise. From Mein et al. (1996a).
Figure 5. Two left panels: The calculated optical depth τ0,max and velocity
v using a cloud model with variable source function (see Eq. 6) depending
only on line-center optical thickness versus the assumed optical thickness.
Gaussian noise has also been taken into account. Two right panels: Same plots
but for an over-estimated chromospheric background profile. FromMein et al.
(1996a).
of the method for the specific structure. However, below, some studies found in
literature about the validity of cloud models are presented.
Mein et al. (1996a) presented a rather detailed study about the validity of
BCM (constant source function), as well as of cloud models with a variable source
function as described in Eq. 6 (depending only on line-center optical thickness)
by inverting theoretical profiles produced with a NLTE code and comparing
the resulting model parameters from the inversion with the assumed ones. Fig-
ure 4 (two left panels) shows the results of the inversion versus the assumed
model optical thickness for the BCM inversion (constant source function). The
calculated optical thickness is smaller, with the difference increasing with the
thickness of the cloud, while the difference in velocity is no more than 20% and
only for high values of the thickness. The figure shows that for optically thin
structures there is practically no difference in the obtained results. When noise
is included (Fig. 4, two right panels) the error increases for increasing thickness
but the mean values stay almost the same. Again for optically thin structures
the difference in the results is very small.
Figure 5 (two left panels) shows the results of the inversion versus the
assumed model optical thickness for a cloud model with variable source function
Cloud-model Inversion Techniques 229
Figure 6. Comparison of the results obtained with method (a) represented
by dots and with method (b) represented by asterisks (see text for de-
tails of the methods) with the assumed model values (solid curve). From
Heinzel et al. (1999).
according to Eq. 6 depending only on line-center optical thickness with an added
Gaussian noise; without noise the results are perfectly reproduced. We see
that the differences are now almost negligible for a large range of the assumed
optical thickness and the parameters are better determined. However, when
taking a slightly brighter background (Fig. 5, two right panels) we see that the
calculated values of the optical thickness are larger than the assumed ones, while
the estimation of velocity is still rather good. This shows the importance of a
correct background profile choice in cloud model calculations.
Heinzel et al. (1999) has repeated the same exercise (inversion of NLTE
synthetic profiles) for a cloud model with a variable source function according
to Eq. 6 depending a) only on line-center optical thickness (method a) and b) on
line-center optical thickness and velocity (method b). Some of their results are
shown in Fig. 6. We see that although with method (a) there are some differences
in the calculation of optical thickness, similarly to Mein et al. (1996a), method
(b) gives exact solutions. Heinzel et al. (1999) have also applied the two methods
in observed profiles of a dark arch filament. Figure 7 shows the comparison of
the results obtained with the two methods.
We refer also the reader selectively to a) Molowny-Horas et al. (2001) (Fig. 12
of their paper) for a comparison of inversion results for a filament with a NLTE
method and a cloud model with a parabolic S, b) Schmieder et al. (2003) (Fig. 16
of their paper) for a comparison of inversion results for a filament with a NLTE
230 Tziotziou
Figure 7. Comparison of the results obtained with the two methods (a)
and (b) (see text for details) from the inversion of observed profiles of a dark
arch filament. Scatter plots are shown for (1) optical thickness, (2) velocity
(in km s−1), and (3) Doppler width (in Å). From Heinzel et al. (1999).
method and a constant source function cloud model, c) Tsiropoula et al. (1999)
(Fig. 5 of their paper) for a comparison of inversion results for mottles for
cloud models with a constant and parabolic S, and d) Alissandrakis et al. (1990)
(Fig. 8 to 11 of their paper) for a comparison of inversion results for an arch
filament system with Beckers’ cloud model, the Doppler signal method and the
differential cloud model.
5. Examples of Cloud Model Inversions
Cloud models have been so far successfully applied for the derivation of the
parameters of several cloud-like solar structures of the quiet Sun, such as mot-
tles/spicules, as well of active region structures, such as arch filament systems
(AFS), filaments, fibrils, flaring regions, surges etc. Below, some examples of
such cloud model inversions are presented.
5.1. Application to filaments
Filaments are commonly observed features that appear on the solar disc as dark
long structures, lying along longitudinal magnetic field inversion lines. When
observed on the limb they are bright and are called prominences. Filaments
were some of the first solar structures to be studied with cloud models (see
for example Maltby 1976, and references therein). Since then several authors
used different cloud models to infer the dynamics and physical parameters of
filaments. Mein, Mein & Wiik (1994), for example, studied the dynamical fine
structure (threads) of a quiescent filament assuming a number of identical –
except for the velocity – threads seen over the chromosphere and using a variant
of BCM, while Schmieder et al. (1991) performed a similar study for threads by
using the DCM. Morimoto & Kurokawa (2003) developed an interesting method
applying BCM to determine the three-dimensional velocity fields of disappearing
filaments.
Molowny-Horas et al. (2001) and Tziotziou et al. (2001) studied the same
filament observed in Hα and Ca II 8542 Å respectively with the Multichannel
Subtractive Double Pass (MSDP) spectrograph (Mein 1991, 2002) mounted on
the German solar telescope VTT in Tenerife. The filament was studied by
using two very large grids of models in Hα and Ca II 8542 Å respectively which
were constructed with the NLTE one-dimensional code MALI (Heinzel 1995), as
Cloud-model Inversion Techniques 231
Figure 8. Top row: A filament observed in Hα and the two-dimensional
parameter distributions derived with a Hα NLTE inversion using a grid of
models. From Molowny-Horas et al. (2001). Bottom row: Same filament
observed in Ca II 8542 Å and the two-dimensional parameter distributions
derived with a Ca II 8542 Å NLTE grid model inversion. From Tziotziou et al.
(2001).
described in Sect. 2.6 Two-dimensional distributions of the physical parameters
were obtained (see Fig. 8) which are not that similar due to the different physical
formation properties and formation heights of the two lines. Schmieder et al.
(2003) performed a similar NLTE grid inversion of a filament combined with a
classical BCM inversion in a multi-wavelength study of filament channels. More
recently, Chae et al. (2006) used Hα images obtained with a tunable filter and a
BCM inversion to obtain detailed two-dimensional distributions of the physical
parameters describing a quiescent filament.
5.2. Application to arch filaments (AFS)
Arch filaments systems (AFSs) are low-lying dark loop-like structures formed
during the emergence of solar magnetic flux in active regions. Georgakilas et al.
(1990) have used the Doppler signal method described in Sect. 2.4 to study
mass motions in AFSs observed in Hα, while Alissandrakis et al. (1990) and
Tsiropoula et al. (1992) used the standard BCM to obtain the physical param-
eters describing arch filament regions observed in the same line (see Fig. 9). An
example of the use of the differential cloud model described in Sect. 2.2 for the
232 Tziotziou
Figure 9. Contours maps of source function (top right panel) and the ve-
locity (bottom right panel) derived with the cloud model for the AFS shown
in Hα in the left panel of the figure. From Alissandrakis et al. (1990).
study of the dynamics of AFSs can be found in Mein et al. (1996b) who applied
the method to Hα observations from a two-telescope coordinated campaign. Fi-
nally Mein et al. (2000) present a study of AFSs in Ca II 8542 Å using a fitting
done with NLTE synthetic profile calculations – as described in Sect. 2.6 – with
the one-dimensional MALI code (Heinzel 1995).
5.3. Application to fibrils
Fibrils are small dark structures, belonging to the family of “chromospheric fine
structures”, found in active regions surrounding plages or sunspots (penum-
bral fibrils). One of the first studies of fibrils was conducted by Bray (1974)
who compared observed profiles of fibrils with profiles calculated with BCM.
Alissandrakis et al. (1990) used the standard BCM to obtain two-dimensional
maps of several physical parameters distributions describing fibrils using Hα ob-
servations obtained at Pic du Midi Observatory. Georgakilas et al. (2003) used
filtergrams obtained at nine wavelengths along the Hα to study the Evershed
flow in sunspots and reconstruct the three-dimensional velocity vector using
the Doppler signal method (see Fig. 10), while Tsiropoula (2000) used also the
Doppler signal method to determine LOS velocities of dark penumbral fibrils.
5.4. Application to mottles
Mottles are small-scale structures (appearing both bright and dark) belonging
also to the family of “chromospheric fine structures” and occurring at quiet
Sun regions at the boundaries of supergranular cells. Mottles are believed to
be the counterparts of limb spicules. They form groups called chains (when
they are almost parallel to each other) or rosettes (when they are more or less
Cloud-model Inversion Techniques 233
Figure 10. Image of a sunspot observed in Hα (a) and Doppler veloc-
ity maps computed with the Doppler signal method from filtergrams in
Hα±0.35Å (b), in Hα±0.5Å (c) and in Hα±0.75Å (d). The intensity gray
scale bar corresponds to normalized intensities while the Doppler velocity gray
scale bars to velocities in km s−1. From Georgakilas et al. (2003).
circularly aligned, pointing radially outwards from a central core) depending on
their location at the chromospheric network.
First cloud studies of mottles started with a controversy about the ability of
BCM to explain their contrast profiles. Bray (1973) and Loughhead (1973) who
studied bright and dark mottles found that their contrast profiles are in good
agreement with BCM. However, Loughhead (1973) used also BCM to deduce
that it could not explain the contrast of individual bright and dark mottles
observed in Hα near the limb, while Cram (1975) claimed that the parameters
inferred from an application of BCM to contrast profiles of chromospheric fine
structures are unreliable.
Since then cloud models have been established as a reliable method for the
study of physical parameters of mottles. Tsiropoula et al. (1999) studied several
bright and dark mottles to derive physical parameters assuming a constant as
well as a varying source function according to Eq. 7. Tsiropoula & Schmieder
(1997) applied Beckers’ cloud model to determine physical parameters in Hα
dark mottles of a rosette region, while Tsiropoula et al. (1993, 1994) studied
the time evolution and fine structure of a rosette with BCM and first showed
an alternating behaviour with time for velocity along mottles (see Fig. 11, left
panel). A similar behaviour has also been found by Tziotziou et al. (2003) using
BCM for a chain of mottles (see Fig. 11, right panel), while the dynamics of an
enhanced network region were also explored in high resolution Hα images by
Al et al. (2004).
234 Tziotziou
Figure 11. Left panel: Cloud velocity as a function of position and time
along the axis of a dark mottle belonging to a rosette. From Tsiropoula et al.
(1994). Right panel: Cloud velocity as a function of position and time along
the axis of a dark mottle belonging to a chain of mottles. White contours
denote downward velocities, black upward velocities, while the thick gray
curve is the zero velocity contour. From Tziotziou et al. (2003).
5.5. Application to post-flare loops
Post-flare loops are loops generally observed between two-ribbon flares. We
refer the reader to Bray & Loughhead (1983) for one of the first post-flare loop
studies, who constructed theoretical curves based on the cloud model to fit
observed contrast profiles of active region loops. Later Schmieder et al. (1988)
and Heinzel et al. (1992) used a differential cloud model to study the structure
and dynamics of post-flare loops. Heinzel et al. (1992) also constructed several
isobaric and isothermal NLTE models of post-flare loops. Their results were
compared by Gu et al. (1997) with two-dimensional maps of Hα post-flare loop
cloud parameters obtained using a two-cloud model. Multi-cloud models like the
ones described in Sect. 2.3 were used by Liang et al. (2004) to study Hα post-
flare loops at the limb (see Fig. 12), by Gu & Ding (2002) for the study of Hα
and Ca II 8542 Å post-flare loops and by Dun et al. (2000) for the study of Hβ
post-flare loops. Liu & Ding (2001) obtained parameters of Hα post-flare loops
using the modified cloud model method presented in Sect. 2.5 that eliminates
the use of the background profile while Gu et al. (1992) presented an extensive
study using BCM, the differential cloud model and a two-cloud model to study
the time evolution of post-flare loops in two-ribbon flares. Finally, we refer the
reader to Berlicki et al. (2005) who studied Hα ribbons during the gradual phase
of a flare by comparing observed Hα profiles with a grid of synthetic Hα profiles
calculated with the NLTE code MALI (Heinzel 1995) which was modified to
account for flare conditions.
5.6. Application to surges
Surges are large jet-like structures observed in opposite polarity flux emer-
gence areas in active regions believed to be supported by magnetic reconnection.
Gu et al. (1994) studied a surge on the limb observed in Hα, using a two-cloud
model inversion as described in Sect. 2.3 (see Fig 13). The inversion result was
Cloud-model Inversion Techniques 235
Figure 12. The distributions of Doppler velocity (in km s−1) derived with
a multi-cloud method for Hα limb post-flare loops. Coordinates are in units
of arcsec, dashed curves show red-shifted mass motions, while solid curves
indicate blue-shifted ones. From Liang et al. (2004).
Figure 13. An Hα filtergram of a surge (left panle) and the two-dimensional
isocontours of Doppler velocity derived with a two-cloud model. Dashed
curves refer to blue-shifted velocities (middle panel), solid curves red-shifted
ones (right panel), while the unit of velocity is in km s−1. From Gu et al.
(1994).
detailed two-dimensional maps of the blue-shifted and red-shifted LOS velocity
distributions.
6. Conclusions
Several inversion techniques for chromospheric structures based on the cloud
model have been reviewed. Cloud models are fast, quite reliable tools for in-
ferring the physical parameters describing cloud-like chromospheric structures
located above the solar photosphere and being illuminated by a background radi-
ation. Cloud model techniques usually provide unique solutions and the results
do not differ – in principle – qualitatively, especially for velocity, when using
different cloud model techniques. However there can be quantitative differences
arising from a) the selection of the background intensity, b) the physical condi-
tions and especially the behaviour of the source function within the structure
under study, and c) the particular model assumptions. Cloud models are mainly
used for absorbing structures, however most of the techniques do work also for
236 Tziotziou
line-center contrasts that are slightly higher than zero, indicating an important
emission by the structure itself.
Several different variants for cloud modeling have been proposed in litera-
ture so far that mainly deal with different assumptions or calculations for the
source functions and span from the simple BCM that assumes a constant source
function to more complicated NLTE calculations of the radiation transfer and
hence the source function within the structure. Accordingly, several different
techniques – most of them iterative – have been proposed for solving cloud
model equations. The latest and more accurate inversion techniques involve
the construction of large grids of synthetic profiles, for different geometries and
physical conditions, which are used for comparison with observed profiles.
Cloud models can be applied with success to several, different in geometry
and physical conditions, solar structures both of the quiet Sun, as well as of
active regions. The resulting parameter inversions has shed light to several
problems involving the physics and dynamics of chromospheric structures.
The future of cloud modeling looks even more brighter. New high resolution
data from telescopes combined with an always increasing computer power and
the continuous development of new, state of the art, NLTE one-dimensional and
two-dimensional cloud model codes will provide further detailed insights to the
physics and dynamics that govern chromospheric structures.
Acknowledgments. KT thanks G. Tsiropoula for constructive comments on the
manuscript and acknowledges support by the organizers of the meeting and by Marie
Curie European Reintegration Grant MERG-CT-2004-021626.
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|
0704.1559 | Dust covering factor, silicate emission and star formation in luminous
QSOs | Astronomy & Astrophysics manuscript no. paper˙fin c© ESO 2018
October 30, 2018
Dust covering factor, silicate emission and star formation in
luminous QSOs
R. Maiolino1, O. Shemmer2, M. Imanishi3, Hagai Netzer4, E. Oliva5, D. Lutz6, and E. Sturm6
1 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
2 Department of Astronomy and Astrophysics, 525 Davey Laboratory, Pennsylvania State University, University Park, PA 16802
3 National Astronomical Observatory, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan
4 School of Physics and Astronomy and the Wise Observatory , Tel-Aviv University, Tel-Aviv 69978, Israel
5 INAF - Telescopio Nazionale Galileo, PO Box 565, 38700 Santa Cruz de La Palma, Tenerife, Spain
6 Max-Planck-Institut für Extraterrestrische Physik, D-85741 Garching, Germany
Received ; accepted
ABSTRACT
We present Spitzer IRS low resolution, mid-IR spectra of a sample of 25 high luminosity QSOs at 2<z<3.5. When combined with
archival IRS observations of local, low luminosity type-I active galactic nuclei (AGNs), the sample spans five orders of magnitude
in luminosity. We find that the continuum dust thermal emission at λrest = 6.7µm is correlated with the optical luminosity, following
the non-linear relation λLλ(6.7µm) ∝ λLλ(5100Å)
0.82 . We also find an anti correlation between λLλ(6.7µm)/λLλ(5100Å) and the
[Oiii]λ5007 line luminosity. These effects are interpreted as a decreasing covering factor of the circumnuclear dust as a function of
luminosity. Such a result is in agreement with the decreasing fraction of absorbed AGNs as a function of luminosity recently found in
various surveys. In particular, while X-ray surveys find a decreasing covering factor of the absorbing gas as a function of luminosity,
our data provides an independent and complementary confirmation by finding a decreasing covering factor of dust. We clearly detect
the silicate emission feature in the average spectrum, but also in four individual objects. These are the Silicate emission in the most
luminous objects obtained so far. When combined with the silicate emission observed in local, low luminosity type-I AGNs, we
find that the silicate emission strength is correlated with luminosity. The silicate strength of all type-I AGNs also follows a positive
correlation with the black hole mass and with the accretion rate. The Polycyclic Aromatic Hydrocarbon (PAH) emission features,
expected from starburst activity, are not detected in the average spectrum of luminous, high-z QSOs. The upper limit inferred from the
average spectrum points to a ratio between PAH luminosity and QSO optical luminosity significantly lower than observed in lower
luminosity AGNs, implying that the correlation between star formation rate and AGN power saturates at high luminosities.
Key words. infrared: galaxies – galaxies: nuclei – galaxies: active – galaxies: Seyfert – galaxies: starburst – quasars: general
1. Introduction
The mid-IR (MIR) spectrum of AGNs contains a wealth of in-
formation which is crucial to the understanding of their inner
region. The observed prominent continuum emission is due to
circumnuclear dust heated to a temperature of several hundred
degrees by the nuclear, primary optical/UV/X-ray source (pri-
marily the central accretion disk); therefore, the MIR contin-
uum provides information on the amount and/or covering factor
of the circumnuclear dust. The MIR region is also rich of sev-
eral emission features which are important tracers of the ISM.
Among the dust features, the Polycyclic-Aromatic-Hydrocarbon
bands (PAH, whose most prominent feature is at ∼ 7.7µm)
are emitted by very small carbon grains excited in the Photo
Dissociation Regions, that are tracers of star forming activity
(although PAHs may not be reliable SF tracers for compact
HII regions or heavily embedded starbursts, Peeters et al., 2004;
Förster Schreiber et al., 2004). Additional MIR dust features are
the Silicate bands at ∼ 10µm and at ∼ 18µm, often seen in ab-
sorption in obscured AGNs and in luminous IR galaxies.
Major steps forward in this field were achieved thanks to
the Spitzer Space Observatory, and to its infrared spectrome-
ter, IRS, which allows a detailed investigation of the MIR spec-
tral features in a large number of sources. In particular, IRS
Send offprint requests to: R. Maiolino
allowed the detection of MIR emission lines in several AGNs
(e.g. Armus et al., 2004; Haas et al., 2005; Sturm et al., 2006a;
Weedman et al., 2005), the detection of PAHs in local PG QSOs
(Schweitzer et al., 2006), the first detection of the Silicate fea-
ture in emission (Siebenmorgen et al., 2005; Hao et al., 2005), as
well as detailed studies of the silicate strength in various classes
of sources (Spoon et al., 2007; Hao et al., 2007; Shi et al., 2006;
Imanishi et al., 2007).
However, most of the current Spitzer IRS studies have fo-
cused on local and modest luminosity AGNs (including low
luminosity QSOs), with the exception of a few bright, lensed
objects at high redshift (Soifer et al., 2004; Teplitz et al., 2006;
Lutz et al., 2007). We have obtained short IRS integrations of a
sample of 25 luminous AGNs (hereafter QSOs) at high redshifts
with the goal of extending the investigation of the MIR prop-
erties to the high luminosity range. The primary goals were to
investigate the covering factor of the circumnuclear dust and the
dependence of the star formation rate (SFR), as traced by the
PAH features, on various quantities such as metallicity, narrow
line luminosity, accretion rate and black hole mass. In combi-
nation with lower luminosity AGNs obtained by previous IRS
studies, our sample spans about 5 orders of magnitude in lumi-
nosity. This allows us to look for the dependence of the covering
factor on luminosity and black hole mass. We also search for the
silicate emission and PAH-related properties although the inte-
http://arxiv.org/abs/0704.1559v1
2 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
gration times were too short, in most cases, to unveil the proper-
ties of individual sources.
In Sect. 2 we discuss the sample selection, the observations
and the data reduction. In Sect. 3.1 we describe the spectral
analysis and the main observational results, and in Sect. 3.2 we
include additional data on local, low luminosity sources from
the literature and from the Spitzer archive. The dust covering
factor is discussed in Sect. 4.1, the properties of the Silicate
emission feature in Sect. 4.2 and the constraints on the star
formation in Sect. 4.3. The conclusion are outlined in Sect. 5.
Throughout the paper we assume a concordance Λ-cosmology
with H0 = 71 km s
−1 Mpc−1, Ωm = 0.27 and ΩΛ = 0.73
(Spergel et al., 2003).
2. Sample selection, observations and data
reduction
High redshift, high luminosity QSOs in our sample were
mostly drawn from Shemmer et al. (2004) and from Netzer et al.
(2004). The latter papers presented near-IR spectra (optical rest-
frame) of a large sample of QSOs at 2<z<3.5, which were used
to obtain detailed information on the black hole (BH) mass
(by means of the width of the Hβ line), on the accretion rate
and on the strength of the narrow emission line [OIII]λ5007.
The sample contains also infrared data on two sources from
Dietrich et al. (2002) and a few additional QSOs in the same
redshift range, for which near-IR spectra where obtained after
Shemmer et al. (2004), but unpublished yet. This sample allows
us not only to extend the investigation of the MIR properties
as a function of luminosity, but also to relate those properties
to other physical quantities such as BH mass, accretion rate
and luminosity of the narrow line region. In total our sample
includes 25 sources which are listed in Table 1. Note that the
QSOs in Shemmer et al. (2004) and in Netzer et al. (2004) were
extracted from optically or radio selected catalogs, without any
pre-selection in terms of mid- or far-IR brightness. Therefore,
the sample is not biased in terms of star formation or dust con-
tent in the host galaxy.
We observed these QSOs with the Long-Low resolution
module of the Spitzer Infrared Spectrograph IRS (Houck et al.,
2004), covering the wavelength range 22–35µm, in staring
mode. Objects were acquired by a blind offset from a nearby,
bright 2MASS star, whose location and proper motion were
known accurately from the Hipparcos catalog. We adopted the
“high accuracy” acquisition procedure, which provides a slit
centering good enough to deliver a flux calibration accuracy bet-
ter than 5%. The integration time was of 12 minutes on source,
with the exception of seven which were observed only 4 minutes
each1
We started our reduction from the Basic Calibrated Data
(BCD). For each observation, we combined all images with the
same position on the slit. Then the sky background was sub-
tracted by using pairs of frames where the sources appears at
two different positions along the slit. The spectra were cleaned
for bad, hot and rogue pixels by using the IRSCLEAN algorithm.
The monodimensional spectra were then extracted by means of
the SPICE software.
1 More specifically: LBQS0109+0213, [HB89]1318-113,
[HB89]1346-036,SBS1425+606,[HB89]2126-158, 2QZJ222006.7-
280324,VV0017.
Fig. 1. Average spectrum of all high-z, luminous QSOs in our
sample, normalized to the flux at 6.7µm (black solid line). The
blue dashed line indicates the power-law fitted to the data at
λ < 8µm; the green solid line is the fitted silicate emission and
the red, dot-dashed line is the resulting fit to the stacked spec-
trum (sum of the power-law and silicate emission). The bottom
panel indicates the number of objects contributing to the stacked
spectrum at each wavelength.
3. Analysis
3.1. Main observational results
All of the objects were clearly detected. In Tab. 2 we list the
observed continuum flux densities at the observed wavelength
corresponding to λrest = 6.7µm. This wavelength was chosen
both because it is directly observed in the spectra of all objects
and because it is far from the Silicate feature and in-between
PAH features. Thus the determination of L(6.7µm) should be
little affected by uncertainties in the subtraction of the star-
burst component (see below). For two of the radio-loud objects
([HB89]0123+257 and TON618) the MIR flux lies on the ex-
trapolation of the synchrotron radio emission and therefore the
former is also probably non-thermal. Since in this paper we are
mostly interested in the thermal emission by dust, the latter two
objects are not used in the statistical analysis. For the other two
radio loud QSOs, the extrapolation of the radio spectrum falls
below the observed MIR emission and the latter is little affected
by synchrotron contamination.
Fig. 1 shows the mean spectrum of all sources in the sample,
except for the two which are likely dominated by synchrotron
emission. Each spectrum has been normalized to 6.7µm prior
to averaging. The bottom panel shows the number of sources
contributing to the mean spectrum in different spectral regions.
We only consider the rest frame spectral range where at least
5 objects contribute to the mean spectrum. The spectrum at
λ < 8µm has been fitted with a simple power-law. While other
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 3
Fig. 2. IRS spectra of four individual high-z luminous QSOs showing evidence for silicate emission. The black solid lines indicate
the IRS spectra smoothed with a 5 pixels boxcar. The shaded areas indicate the flux uncertainty. The blue dashed line and the green
solid line are the power-law and the silicate emission components of the fits. The black dotted line shows the starburst component,
which is formally required by the fit, but statistically not significant. The red dot-dashed lines are the global fits to the observed
spectra.
workers in this field assumed more complicated continuum (e.g.
spline, polynomial) we do not consider it justified given the
limited wavelength range of our spectra. The extrapolation of
the continuum to 10µm clearly reveals an excess identified with
Silicate emission. Fitting and measuring the strength of this fea-
ture is not easy given the limited rest-frame spectral coverage
of our spectra. Therefore, we resort to the use of templates.
In particular, we fit the Silicate feature by using as a template
the (continuum-subtracted) silicate feature observed in the aver-
age spectrum of local QSOs as obtained by the QUEST project
(Schweitzer et al., 2006) and kindly provided by M. Schweitzer.
The template Silicate spectrum, with the best fitting scaling fac-
tor is shown in green in Fig. 1, while the red dot-dashed line
shows the resulting fit including the power-law. We adopt the
definition of “silicate strength” given in (Shi et al., 2006) which
is the ratio between the maximum of the silicate feature and the
interpolated featureless continuum at the same wavelength. In
the QSO-QUEST template the maximum of the Silicate feature
is at 10.5µm. This wavelength is slightly outside the band cov-
ered by our spectra but the uncertainty on the extrapolation is
not large (the latter is included in the error estimate of the sil-
icate strength). The silicate strength in the mean spectrum is
0.58±0.10 (Tab. 2).
We note that the average spectrum does not show evidence
for PAH features at 7.7µm and 6.2µm. Such features are ob-
served in lower luminosity AGNs. More specifically, a starburst
4 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
template (Sturm et al., 2000) is not required by the fit shown in
Fig. 1. In Sect. 4.3 we will infer an upper limit on the PAH lu-
minosity and discuss its implication.
We clearly detect the blue wing of the silicate feature in four
individual spectra, which are shown in Fig. 2. These spectra were
fitted with a power-law and a silicate template exactly as the
stacked spectrum. The resulting values for the Silicate strength
are given in Tab. 2. The presence of silicate emission in all other
cases is poorly constrained (or totally unconstrained) either be-
cause of low signal-to-noise (S/N) or because of a lack of spec-
tral coverage. The one exception is Ton 618 which has a high
S/N spectrum and a redshift (z=2.22) appropriate to observe the
Silicate 10.5 µm feature. No silicate emission is detected in this
case, but note that this is not expected since the MIR radiation
of this source is probably dominated by synchrotron emission.
Tables 1 and 2 list the more important MIR information on
the sources and physical properties deduced from the rest-frame
optical spectra and obtained from Shemmer et al. (2004): opti-
cal continuum luminosity λLλ(5100Å), [OIII]λ5007 line lumi-
nosity, BH mass and Eddington accretion rate L/LEdd.
3.2. A comparison with MIR properties of lower luminosity
To compare the MIR properties of our luminous QSOs with
those of lower luminosity sources we have included in our study
the IRS/MIR spectra of various low redshift, lower luminosity
type-I AGNs. We purposely avoid type-II sources because of the
additional complication due to absorption along the line of sight.
We have used data from Shi et al. (2006) who analyze the in-
tensity of the silicate features in several, local AGNs with lumi-
nosities ranging from those of nearby Seyfert 1s to intermediate
luminosity QSOs. We discarded BAL QSOs (which are known
to have intervening gas and dust absorption) as well as dust red-
dened type-I nuclei (e.g. 2MASS red QSOs). We also discard
those cases (e.g. 3C273) where the optical and MIR continuum
is likely dominated by synchrotron radiation. Note that Shi et al.
(2006) selected type-I objects with “high brightness” and, there-
fore, low-luminosity AGNs tend to be excluded from their sam-
Shi et al. (2006) provide a measure of the silicate feature
strength (whose definition was adopted also by us). The con-
tinuum emission at 6.7µm was measured by us from the archival
spectra. We also subtracted from the 6.7µm emission the pos-
sible contribution of a starburst component by using the M82
template. We estimate the host galaxy contribution (stellar pho-
tospheres) in all sources to be negligible.
We include in the sample of local Sy1s also some IRS spectra
taken from the sample of Buchanan et al. (2006), whose spectral
parameters were determined by us from the archival spectral,
in the same manner as for the Shi et al. (2006) spectra. As for
the previous sample, we discarded reddened/absorbed sources as
well as those affected by synchrotron emission. As discussed in
Buchanan et al. (2006), these spectra are affected by significant
flux calibration uncertainties, due to the adopted mapping tech-
nique. Therefore, the spectra were re-calibrated by using IRAC
photometric images. We discarded objects for which IRAC data
are not available or not usable (e.g. because saturated). Finally,
we also discarded data for which optical spectroscopic data are
not available (see below).
The mid-IR parameters of the sources in both samples are
listed in Tab. 2.
Optical data were mostly taken from Marziani et al. (2003)
and BH masses and Eddington accretion rates inferred as in
Shemmer et al. (2004). The resulting parameters are listed in
Table 1.
The type-I sources in Shi et al. (2006) and Buchanan et al.
(2006) are only used for the investigation of the covering fac-
tor and silicate strength, which are the main aims of our work.
The Shi et al. (2006) and Buchanan et al. (2006) samples are not
suitable for investigating the PAH features because most of these
objects are at small distances and the IRS slit misses most of the
star formation regions in the host galaxy. For what concerns the
the PAH emission, we use the data in Schweitzer et al. (2006)
who performed a detailed analysis of the PAH features in their
local QSOs sample. The slit losses in those sources are minor.
The Schweitzer et al. (2006) sample is also used for the investi-
gation of the MIR-to-optical luminosity ratio. The mid-IR data
of this sample are not listed in Tab. 2, since such data are already
reported in Schweitzer et al. (2006) and in Netzer et al. (2007).
4. Discussion
4.1. Dust covering factor
4.1.1. Covering factor as a function of source luminosity and
BH mass
The main assumption used here is that the covering factor of the
circumnuclear dust is given by the ratio of the thermal infrared
emission to the primary AGN radiation. The latter is mostly the
“big blue bump” radiation with additional contribution from the
optical and X-ray wavelength ranges (Blandford et al., 1990).
Determining the integral of the AGN-heated dust emission,
and disentangling it from other spectral components is not sim-
ple. The FIR emission in type-I AGNs is generally dominated by
a starburst component, even in QSOs (Schweitzer et al., 2006).
In lower luminosity AGNs, the near-IR emission may be affected
by stellar emission in the host galaxy, while in QSOs the near-
IR light is often contributed also by the direct primary radia-
tion. The MIR range (∼ 4 − 10µm) is where the contrast be-
tween AGN-heated dust emission and other components is max-
imal. This spectral region contains various spectral features, like
PAHs and silicate emission, yet MIR spectra allow us to disen-
tangle and remove these components, and determine the hot dust
continuum. In particular, by focusing on the continuum emis-
sion at 6.7µm, the uncertainty in the removal of PAH emission
is minimized, while the contribution from the Silicate emission
is totally negligible at this wavelength (note that such a spectral
decomposition is unfeasible with photometric data). If the spec-
tral shape of the AGN-heated dust does not change from object
to object (and in particular it does not change significantly with
luminosity), then the 6.7µm emission is a proxy of the global
circumnuclear hot dust emission. It is possible to infer a quanti-
tative relation between L(6.7µm) and the total AGN-heated hot
dust emission through the work of Silva et al. (2004), who use
observations of various nearby AGNs to determine their average,
nuclear IR SED (divided into absorption classes). From their
type I AGNs SED, we find that the integrated nuclear, thermal
IR bump is about ∼ 2.7 λLλ(6.7µm). This ratio is also consistent
with that found in the QUEST QSO sample, once the contribu-
tion by the silicate features is subtracted.
Regarding the primary optical-UV radiation, determining its
integrated flux would require observations of the entire intrin-
sic spectral energy distribution (SED) from the far-UV to the
near-IR. This is not available for most sources in our sample.
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 5
Fig. 3. MIR continuum luminosity at 6.7µm versus optical con-
tinuum luminosity at 5100Å. Red triangles mark high-z, lumi-
nous QSOs and blue diamonds mark low redshift type-I AGNs.
The black solid line is a fit to the data, which has a slope with a
power index of 0.82.
Moreover, there are indications that the SED is luminosity de-
pendent (e.g. Scott et al., 2004; Shang et al., 2005) and thus an
estimate of the bolometric luminosity of the primary continuum
based on the observed luminosity in a certain band is somewhat
uncertain. Notwithstanding these limitations, we assume in this
work that the optical continuum luminosity can be used as a
proxy of the bolometric luminosity of the primary continuum.
We use the continuum luminosity at the rest frame wavelength
of 5100Å, (λLλ, hereafter L5100) because it is directly measured
for all sources in our sample. We can infer the ratio between
the bolometric luminosity and L5100 from the mean spectrum of
QSOs obtained in recent studies, mostly the results of Scott et al.
(2004) and Richards et al. (2006). These studies indicate a bolo-
metric correction in the range of 5–9. Here we chose, rather ar-
bitrarily, the mean value of 7.
The comparison of L(6.7µm) and L5100 is our way of deduc-
ing the hot dust covering factor. Fig. 3 shows the λLλ(6.7µm)
versus L5100 for our sample. Red triangles are the high-z, lumi-
nous QSOs and blue diamonds are local type-I AGNs. Not sur-
prisingly, the two quantities show a good correlation. However,
the very large luminosity range spanned by our sample allows us
to clearly state that the correlation is not linear, but has a slope
α = 0.82 ± 0.02 defined by
log[λLλ(6.7µm)] = K + α log[λLλ(5100Å)] (1)
where K = 8.36±0.80 and luminosities are expressed in erg s−1.
This indicating that the MIR, reprocessed emission increases
more slowly than the primary luminosity.
The same phenomenon is observed in a cleaner way in
Fig. 4a, where the ratio between the two continuum luminosi-
ties is plotted as a function of L5100. There is a clear anti-
correlation between the MIR–to–optical ratio and optical lumi-
nosity. Fig. 4b shows the same MIR–to–optical ratio as a func-
tion of L([OIII]λ5007) as an alternative tracer of the global AGN
luminosity (although the latter is not a linear tracer of the nu-
clear luminosity, as discussed in Netzer et al., 2006), which dis-
plays the same anti-correlation as for the continuum optical flux.
Spearman-rank coefficients and probabilities for these correla-
tions are given in Tab. 3.
According to the above discussion, an obvious interpretation
of the decreasing MIR–to–optical ratio is that the covering factor
of the dust surrounding the AGN decreases with luminosity. In
particular, if the covering factor is proportional to the MIR–to–
optical ratio, then Figs. 4a-b indicate that the dust covering factor
decreases by about a factor 10 over the luminosity range probed
by us.
It is possible to convert the MIR–to–optical luminosity ra-
tio into absolute dust covering factor by assuming ratios of
broad band to monochromatic continuum luminosities observed
in AGNs, as discussed above. In particular, by using the 6.7µm–
to–MIR and 5100Å–to–bolometric luminosity ratios reported
above, we obtain that the absolute value of the dust covering
factor (CF) can be written as:
CF(dust) ≈ 0.39 ·
λLλ(6.7µm)
λLλ(5100Å)
. (2)
In Fig. 4 the axes on the right hand side provide the dust
covering factor inferred from the the equation above. A fraction
of objects have covering factor formally larger than one, these
could be due to uncertainties in the observational data, or nu-
clear SED differing from the ones assumed above, or to optical
variability, as discussed in Sect. 4.1.2. The dust covering fac-
tor ranges from about unity in low luminosity AGNs to about
10% in high luminosity QSOs. As it will be discussed in detail
in Sect. 4.1.3, the dust covering factor is expected to be equal
to the fraction of type 2 (obscured) AGNs relative to the total
AGN population. The finding of a large covering factor in low
luminosity AGNs is in agreement with the large fraction of type
2 nuclei observed in local Seyferts (∼ 0.8, Maiolino & Rieke,
1995).
A similar result has been obtained, in an independent way,
through the finding of a systemic decrease of the the obscured
to unobscured AGN ratio as a function of luminosity in various
surveys. The comparison with these results will be discussed in
more detail in the next section.
The physical origin of the decreasing covering factor is still
unknown. One possibility is that higher luminosities imply a
larger dust sublimation radius: if the obscuring medium is dis-
tributed in a disk with constant height, then a larger dust subli-
mation radius would automatically give a lower covering factor
of dust at higher luminosities (Lawrence, 1991). However, this
effect can only explain the decreasing covering factor with lu-
minosity for the dusty medium, and not for the gaseous X-ray
absorbing medium. Moreover, Simpson (2005) showed that the
simple scenario of such a “receding torus” is unable to account
for the observed dependence of the type 2 to type 1 AGN ratio
as a function of luminosity.
Another possibility is that the radiation pressure on dust is
stronger, relative to the BH gravitational potential, in luminous
AGNs (e.g. Laor & Draine, 1993; Scoville & Norman, 1995),
thus sweeping away circumnuclear dust more effectively. In this
scenario a more direct relation of the covering factor should be
with L/LEdd, rather than with luminosity. Our sample does not
show such a relation, as illustrated in Fig. 4c. However, the un-
certainties on the accretion rates (horizontal error bar in Fig. 4c)
may hamper the identification of such a correlation.
Lamastra et al. (2006) proposed that, independently of lumi-
nosity, the BH gravitational potential is responsible for flatten-
6 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
Fig. 4. MIR–to–optical continuum luminosity [λLλ(6.7µm)]/[λLλ(5100Å)] versus (a) continuum optical luminosity, (b) [OIII]λ5007
line luminosity, (c) accretion rate L/LEdd and (d) black hole mass. Symbols are the same as in Fig. 3. The magenta square indicates
the location of the mean SDSS QSO SED in Richards et al. (2006). The horizontal error bars in panels (c) and (d) indicate con-
servative uncertainties on the accretion rates and BH masses. The right hand side axis on each panel shows the circumnuclear dust
covering factor inferred from Eq. 2. The dashed lines in panels (a) and (b) shows the fit resulting from the analytical forms in Eqs. 3
and 6, respectively.
ing the circumnuclear medium, so that larger BH masses effec-
tively produce a lower covering factor. According to this sce-
nario, the relation between covering factor and luminosity is
only an indirect one, in the sense that more luminous AGN tend
to have larger BH masses (if the Eddington accretion rate does
not change strongly on average). Fig. 4d shows the MIR–to–
optical ratio (and dust covering factor) as a function of BH mass,
indicating a clear (anti-)correlation between these two quanti-
ties. However, the correlation is not any tighter than the relation
with luminosity in Fig. 4a-b, as quantified by the comparison of
the Spearman-rank coefficients and probabilities in Tab. 3. The
degeneracy between luminosity and BH mass prevents us to dis-
criminate which of the two is the physical quantity driving the
relation.
4.1.2. Model uncertainties
In this section we discuss some possible caveats in our interpre-
tation of the MIR–to–optical ratio as an indicator of the hot dust
covering factor.
Our analysis assumes that the shape of the hot dust IR spec-
trum SED is not luminosity dependent. However, an alternative
interpretation of the trends observed in Fig. 4 could be that the
dust temperature distribution changes with luminosity. Yet, to
explain the decreasing 6.7µm to 5100Å flux ratio in terms of dust
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 7
Fig. 5. Mid-IR spectral slope (5–8µm) versus
λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3.
No clear correlation is observed between these two quan-
tities (see also Tab. 3). Note that the exceptional object
SDSSJ173352.22+540030.5 is out of scale, and it is discussed
in the text.
Fig. 6. Optical-to-UV spectral slope (1450–5100Å) versus
λLλ(6.7µm)/λLλ(5100Å) ratio. Symbols are as in Fig. 3. The
arrow indicates the effect of dust reddening with AV = 2 mag.
The data do not show any evidence for dust reddening effects.
temperature would require that the circumnuclear dust is cooler
at higher luminosities. This, besides being contrary to expecta-
tions, is ruled out by the observations which show no clear corre-
lation between the mid-IR continuum slope (which is associated
with the average dust temperature) and the MIR–to–optical ratio,
as illustrated in Fig. 5 and in Tab. 3. The one remarkable excep-
tion is SDSSJ173352.22+540030.5,which has the most negative
mid-IR continuum slope (αMIR = −2.88, i.e. an inverted spec-
trum in λLλ) and the lowest MIR-to-optical ratio of the whole
sample (λLλ(6.7µm)/λLλ(5100Å) = 0.22), which is out of scale
in Fig. 5. This QSO may be totally devoid of circumnuclear hot
dust, and its MIR emission may simply be the continuation of the
optical “blue-bump”. Similar high-z QSOs, with an exceptional
deficiency of mir-MIR flux, have been reported by Jiang et al.
(2006).
As explained earlier, there are indications that the UV-optical
SED, and hence the bolometric correction based to the observed
L5100, are luminosity dependent. If correct, this would mean a
smaller bolometric correction for higher L5100 sources which
would flatten the relationship found here (i.e. will result in a
slower decrease of the covering factor with increasing L5100).
However, the expected range (a factor of at most 2 in bolometric
correction) is much smaller than the deduced change in covering
factor.
Variability is an additional potential caveat because of the
time delay between the original L5100 “input” and the response
of the dusty absorbing “torus”. While we do not have the obser-
vations to test this effect (multi-epoch, high-quality optical spec-
troscopic data are available only for a few sources in our sam-
ple), we note that the location of the 6.7µm emitting gas from
the central accretion disk is at least several light years and thus
L(MIR) used here reflects the mean L5100 in most sources. We
expect that the average luminosity of a large sample will not be
affected much by individual source variations. Moreover, in the
high luminosity sources of our sample we do not expect much
variability (since luminous QSOs are known to show little or no
variability).
An alternative, possible explanation of the variation of the
optical-to-MIR luminosity ratio could be dust extinction affect-
ing the observed optical flux. Optical dust absorption increasing
towards low luminosities may in principle explain the trends ob-
served in Fig. 4. However, we have pre-selected the sample of
local QSOs and Sy1s to avoid objects showing any indication
of absorption, thus probably shielding us from such spurious ef-
fects. Yet, we have further investigated the extinction scenario
by analyzing the optical-to-UV continuum shape of our sample.
The optical-UV continuum slope does not necessarily trace dust
reddening, since intrinsic variations of the continuum shape are
known to occur, as discussed above. Variability introduce ad-
ditional uncertainties, since optical and UV data are not simul-
taneous. However, if the variations of λLλ(6.7µm)/λLλ(5100Å)
are mostly due to dust reddening, one would expect the MIR-
to-optical ratio to correlate with the optical-UV slope, at least
on average. We have compiled UV rest-frame continuum fluxes
(at λrest ∼ 1450Å) from spectra in the literature or in the HST
archive. By combining such data with the continuum luminosi-
ties at λrest = 5100Å we inferred the optical-UV continuum
slope αopt−UV defined as
2 Lλ ∝ λ
αopt−UV , as listed in Tab. 1.
Fig. 6 shows αopt−UV versus λLλ(6.7µm)/λLλ(5100Å). The ar-
row indicates the effect of dust reddening with AV = 2 mag
(by assuming a SMC extinction curve, as appropriate for type
1 AGNs, Hopkins et al., 2004), which would be required to ac-
count for the observed variations of λLλ(6.7µm)/λLλ(5100Å).
Fig. 6 does not show evidence for any (positive) correlation be-
tween αopt−UV and λLλ(6.7µm)/λLλ(5100Å) (see also Tab 3).
In particular, if the variation of MIR-to-optical ratio (spanning
more than a factor of ten) was due to dust reddening, we would
expect to find ∆αopt−UV > 5 (as indicated by the arrow in Fig. 6),
which is clearly not observed. If any, the data show a marginal
2 Our definition of power law index is linked to the αν given in
Vanden Berk et al. (2001) by the relation αopt−UV = −(αν + 2). Our dis-
tribution of αopt−UV is roughly consistent (within the uncertainties and
the scatter) with αν = −0.44 obtained by Vanden Berk et al. (2001) for
the SDSS QSO composite spectrum (see also Shemmer et al., 2004).
8 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
anti-correlation between αopt−UV and λLλ(6.7µm)/λLλ(5100Å)
(Tab. 3), i.e. opposite to that expected from dust reddening.
Finally, possible evolutionary effects on the dust covering
factor have not been considered. We have been comparing lo-
cal objects with QSOs at z∼2–3 yet assumed that only lumi-
nosity or BH mass plays a role. La Franca et al. (2005) and
Akylas et al. (2006) find evidence for an increasing fraction of
obscured AGNs as a function of redshift, a result which is still
debated (see Ueda et al., 2003; Gilli et al., 2007). If the ab-
sorbing medium covering factor really increases with redshift,
then the actual dependence of the covering factor on luminos-
ity would be stronger than shown in Fig. 4. Indeed, according to
La Franca et al. (2005) and Akylas et al. (2006), putative low-z
counterparts of our high-z QSOs (matching the same luminosi-
ties) should be affected by an even lower covering factor. As a
consequence, the diagrams in Figs. 4 and 7 should have even
steeper trends once the data are corrected for such putative evo-
lutionary effects, thus strengthening our conclusions.
4.1.3. Comparison with previous works
Wang et al. (2005) used IRAS and ISO mid-IR data of (mostly
local) PG quasars to infer their dust covering factor. They find
that the covering factor decreases as a function of X-ray lumi-
nosity. They probe a narrower luminosity range with respect to
our work, nonetheless their results are generally consistent with
ours, although with significant scatter.
More recently Richards et al. (2006) derived the broad band
SED of a large sample of SDSS QSOs by including Spitzer pho-
tometric data. Although such data do not have spectroscopic in-
formation, they can be used to obtain a rough indication of the
dust covering factor in their QSO sample, to be compared with
our result. The average optical luminosity of the Richards et al.
(2006) sample is 〈log[λLλ(5100Å)]〉 ∼ 45.5 erg s
−1. From their
mean SED we derive λLλ(6.7µm)/λLλ(5100Å) = 1.17. The cor-
responding location on the diagram of Fig. 4a is marked with a
magenta square, and it is in agreement with the general trend of
our data.
In a companion paper, Gallagher et al. (2007) use the same
set of data to investigate the MIR-to-optical properties as a func-
tion of luminosity. They find a result similar to ours, i.e. the
MIR-to-optical ratio decreases with luminosity. However, they
interpret such a result as a consequence of dust reddening in the
optical, since the effect is stronger in QSOs with redder opti-
cal slope. As discussed in the previous section, our data do no
support this scenario, at least for our sample. In particular, the
analysis of the optical-UV slope indicates that dust reddening
does not play a significant role in the variations of the MIR-
to-optical luminosity ratio. The discrepancy between our and
Gallagher et al. (2007) results may have various explanations.
The QSOs in the Gallagher et al. (2007) sample span about two
orders of magnitudes in luminosity, while we have seen that
to properly quantify the effect a wider luminosity range is re-
quired. Moreover, the majority of their sources are clustered
around the mean luminosity of 1045.5 erg s−1. In addition, the
the lack of spectroscopic information makes it difficult to allow
for the presence of other spectral features such as the silicate
emission which, as we show later, is luminosity dependent. The
lack of spectroscopic information may be an issue specially for
samples spanning a wide redshift range (as in Gallagher et al. ,
2007), where the photometric bands probe different rest-frame
bands. The same concerns applies for the optical luminosities.
Our rest-frame continuum optical luminosities are always in-
ferred through rest-frame optical spectra, even at high-z (through
near-IR spectra). Gallagher et al. (2007) do not probe directly
the optical continuum luminosity of high-z sources (at high-z
they only have optical and Spitzer data, which probe UV and
near-IR rest-frame, respectively). Finally, differences between
our and Gallagher et al. (2007) results may be simply due to
the different samples. As discussed in the previous section, we
avoided dust reddened targets, thus making us little sensitive to
extinction effects, while Gallagher et al. (2007) sample may in-
clude a larger fraction of reddened objects.
A decreasing dust covering factor as a function of lumi-
nosity must translate into a decreasing fraction of obscured
AGNs as a function of luminosity. The effect has been noted
in various X-ray surveys (Ueda et al., 2003; Steffen et al., 2003;
La Franca et al., 2005; Akylas et al., 2006; Barger et al., 2005;
Tozzi et al., 2006; Simpson, 2005), although the results have
been questioned by other authors (e.g. Dwelly & Page, 2006;
Treister & Urry, 2005; Wang et al., 2007). The X-ray based
studies do not distinguish between dust and gas and thus probe
mostly trends of the gaseous absorption. Our result provides an
independent confirmation of these trends. Moreover, our find-
ings are complementary to those obtained in the X-rays since,
instead of the covering factor of gas, we probe the covering fac-
tor of dust.
In order to compare our findings with the results obtained
from X-ray surveys, we have derived the expected fraction of
obscured AGNs by fitting the dust covering factor versus lumi-
nosity relation with an analytical function. Instead of using the
simple power-law illustrated in Fig. 3 (Eq. 1) we fit the depen-
dence of the covering factor on luminosity with a broken power-
law. The latter analytical function is preferred both because it
provides a statistically better fit and because a simple power-law
would yield a covering factor larger than unity at low luminosi-
ties. As a result we obtain the following best fit for the fraction
of obscured AGNs as a function of luminosity:
fobsc =
1 +Lopt
0.414
where fobsc is the fraction of obscured AGNs relative to the total
Lopt =
λLλ(5100Å) [erg s
1045.63
The resulting fit is shown with a dashed line in Fig. 4a. The frac-
tion of obscured AGNs as a function of luminosity is also re-
ported with a blue, solid line in Fig. 7a. The shaded area reflects
the uncertainty in the bolometric correction discussed above.
The most recent and most complete investigation on the frac-
tion of X-ray obscured AGNs as a function of luminosity has
been obtained by Hasinger (2007, in prep., see also Hasinger,
2004) who combined the data from surveys of different areas
and limiting fluxes to get a sample of ∼700 objects. We convert
from X-ray to optical luminosity by using the non-linear relation
obtained by Steffen et al. (2006). The latter use flux densities at
2 keV and 2500Å; we adapt their relation to our reference optical
wavelength (5100Å) by assuming the optical-UV spectral slope
obtained by Vanden Berk et al. (2001), and to the 2–10 keV inte-
grated luminosity (adopted in most X-ray surveys) by assuming
a photon index of –1.7, yielding the relation
log[L(2 − 10 keV)] = 0.721 · log[λLλ(5100Å)]+ 11.78 (5)
(where luminosities are in units of erg s−1). Fig. 7a compares the
fraction of obscured AGN obtained by Hasinger (2007) through
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 9
Fig. 7. a) Fraction of obscured AGNs (relative to total) as a function of optical continuum luminosity. The blue line shows the
fraction of obscured AGNs inferred from the hot dust covering factor with the analytical form of Eq. 3. The shaded area is the
uncertainty resulting from the plausible range of the bolometric correction (see text). Points with error bars are the fraction of X-ray
obscured AGNs as a function of X-ray luminosity inferred by Hasinger (2007). The upper scale of the diagram gives the intrinsic
hard X-ray luminosity; the (non-linear) correspondence between X-ray and optical luminosity is obtained from Eq. 5. b) Fraction of
obscured AGNs as a function of L([OIII]λ5007). The blue line is the fraction of obscured AGNs inferred from the hot dust covering
factor with the analytical form of Eq. 6. Points with error bars show the fraction of type 2 AGNs as a function of L([OIII]λ5007)
inferred by Simpson (2005).
X-ray surveys with our result based on the covering factor of
hot dust. Both have the same trends with luminosity, but the
fraction of obscured AGN expected from the hot dust cover-
ing factor is systematically higher. Such an offset is however ex-
pected. Indeed, current high redshift X-ray surveys do not probe
the Compton thick population of obscured AGNs since these
are heavily absorbed even in the hard X-rays. In local AGNs,
Compton thick nuclei are about as numerous as Compton thin
ones (Risaliti et al., 1999; Guainazzi et al., 2005; Cappi et al.,
2006). The fraction of Compton thick, high luminosity, high
redshift AGNs is still debated, but their contribution certainly
makes the fraction of X-ray obscured AGNs higher than in-
ferred by Hasinger (2007), who can only account for Compton
thin sources. The ratio between the dust covering factor curve in
Fig. 7a and the X-ray data from Hasinger (2007), indicates that
the ratio between the total number of obscured AGNs (includ-
ing Compton thick ones) and Compton thin ones is about 2 even
at high luminosities, i.e. consistent (within uncertainties) with
local, low-luminosity AGNs.
An analogous result on the decreasing fraction of obscured
AGNs as a function of luminosity was obtained by Simpson
(2005) who compared the numbers of type 2 and type 1 AGNs
at a given L([OIII]λ5007). To compare with Simpson (2005), we
used our sample to derive the following analytical description for
the fraction of obscured AGN as a function of L([OIII]λ5007):
fobsc =
1 +L[OIII]
0.409
where fobsc is the fraction of obscured AGNs relative to the total
L[OIII] =
L([OIII]) [erg s−1]
1043.21
The corresponding fit is shown with a dashed line in Fig. 4b, and
the fraction of obscured AGNs as a function of L([OIII]) is also
shown with a blue line in Fig. 7b. In the latter figure we also
compare the fraction of obscured AGNs obtained by Simpson
(2005) with our result based on the covering factor of the hot
dust. There is a good agreement (within uncertainties) between
the fraction of obscured AGNs inferred through the two meth-
ods, as expected since both optical surveys and our method probe
the covering factor of dust. However, we shall also mention that
the results obtained by Simpson (2005) have been questioned by
Haas et al. (2005), by arguing that at high luminosities, the de-
rived L([OIII]λ5007) may be affected by a large scale absorber.
4.2. Silicate emission
In this paper we have presented the most luminous (type 1)
QSOs where Silicate emission has been detected so far. When
combined with MIR spectra of lower luminosity sources, it is
possible to investigate the properties and behavior of this feature
over a wide luminosity range.
The discovery of silicate emission in the spectrum of (mostly
type 1) AGNs obtained by Spitzer was regarded as the so-
lution of a long standing puzzle on the properties of the cir-
10 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
Fig. 8. Silicate strength versus (a) L5100, (b) L([OIII]λ5007), (c) normalized accretion rate (L/LEdd) and (d) black hole mass. Symbols
are the same as in Fig. 3. The red square indicates the silicate strength measured in the average spectrum. The horizontal, black error
bars in panels (c) and (d) indicate conservative uncertainties on the accretion rates and BH masses.
cumnuclear dusty medium. Indeed, silicate emission was ex-
pected by various models of the dusty torus. However, the ab-
sence of clear detections prior to the Spitzer epoch induced
various authors to either postulate a very compact and dense
torus (e.g. Pier & Krolik, 1993) or different dust compositions
(Laor & Draine, 1993; Maiolino et al., 2001a,b). Initial Spitzer
detections of silicate emission relaxed the torus model assump-
tions (Fritz et al., 2006), but more detailed investigations re-
vealed a complex scenario. The detection of silicate emission
even in type 2 AGNs (Sturm et al., 2006b; Teplitz et al., 2006;
Shi et al., 2006) suggested that part of the silicate emission may
originate in the Narrow Line Region (NLR) (Efstathiou, 2006).
Further support for a NLR origin of the silicate emission comes
from the temperature inferred for the Silicate features, which is
much lower (<200 K) than for the circumnuclear dust emitting
the featureless MIR continuum (>500 K), as well as from MIR
high resolution maps spatially resolving the silicate emission on
scales of 100 pc (Schweitzer et al. in prep.).
If most of the observed silicate emission originates in the
NLR, then the effects of circumnuclear hot dust covering fac-
tor should be amplified when looking at the “silicate strength”
(which we recall is defined as the ratio of the silicate maximum
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 11
intensity and the featureless hot dust continuum). Indeed, if the
covering factor of the circumnuclear dusty torus decreases, it im-
plies that the MIR hot dust continuum decreases and the silicate
emission increases because a larger volume of the NLR is il-
luminated. Both effects go in the same direction of increasing
the “silicate strength”. This scenario is made more complex by
the tendency of the NLR to disappear at very high luminosities,
or to get very dense and not to scale linearly with the nuclear
luminosity (Netzer et al., 2004, 2006). Moreover, the schematic
division of a silicate feature totally emitted by the NLR and a
MIR featureless continuum totally emitted by the inner side of
the obscuring torus is probably too simplistic. There must be
at least a small contribution to the featureless MIR continuum
from dust in the NLR, while some silicate emission is probably
also coming from the obscuring torus. However, from a general
qualitative point of view we expect a monotonic behavior of the
“silicate strength” with the physical quantity responsible for the
changes in the hot dust covering factor.
Before investigating the various trends of the Silicate
strength, we mention that by using the four silicate detections
shown in Fig. 2 and listed in Tab. 2, we may in principle intro-
duce a bias against weak silicate emitters. Indeed, although we
cannot set useful upper limits on the silicate strength in most of
the other objects, we have likely missed objects with low sili-
cate strength. However, the mean spectrum in Fig. 1 includes all
QSOs in our sample, and therefore its silicate strength should
be representative of the average Silicate emission in the sample
(at least for the objects at z<2.5, which are the ones where the
observed band includes the silicate feature, and which are the
majority).
Figs. 8a-b show the silicate strength of the objects in our
combined sample as functions of L5100 and L([OIII]λ5007). The
red square indicates the silicate strength in the high-z QSO mean
spectrum, while its horizontal bar indicates the range of lumi-
nosities spanned by the subsample of objects at z<2.5 (i.e. those
contributing to the silicate feature in the mean spectrum). Low
redshift AGNs and high redshift QSOs show an apparently clear
correlation between silicate strength and luminosity. Although
with a significant spread, the Silicate strength is observed to pos-
itively correlate also with the accretion rate L/LEdd and with the
BH mass, as shown in Figs. 8c-d. Essentially, the correlations
observed for the Silicate strength reflects the same correlation
observed for the L(6.7µm)/L(5100Å) (with the exception of the
accretion rate), in agreement with the idea that also the Silicate
strength is a proxy of the covering factor of the circumnuclear
hot dust, for the reasons discussed above.
Unfortunately, the correlations observed for the Silicate
strength do not improve our understanding on the origin of the
decreasing covering factor with luminosity, i.e. whether the driv-
ing physical quantity is the luminosity itself, the accretion rate or
the black hole mass. Formally, the correlation between Silicate
strength and optical continuum luminosity is tighter than the oth-
ers (Tab. 3), possibly hinting at the luminosity itself as the quan-
tity driving the dust covering factor. However, there are a few
low luminosity objects, such as a few LINERs, which have large
silicate strengths (Sturm et al., 2005) and which clearly deviate
from the correlation shown in Fig. 8a, thus questioning the role
of luminosity in determining the Silicate strength. In addition,
the apparently looser correlations of Silicate strength versus ac-
cretion rate and BH mass may simply be due to the additional
uncertainties affecting the latter two quantities (horizontal, black
error bars in Figs. 8c,d).
4.3. PAHs and star formation
The presence and intensity of star formation in QSOs has been
a hotly debated issue during the past few years. A major step
forward in this debate was achieved by Schweitzer et al. (2006)
through the Spitzer IRS detection of PAH features in a sample
of nearby QSOs, revealing vigorous star formation in these ob-
jects. The analysis also shows that the far-IR emission in these
QSOs is dominated by star formation and that the star forming
activity correlates with the nuclear AGN power. Here we show in
Fig. 9 the latter correlation in terms of PAH(7.7µm) luminosity
versus L5100 by using the PAH luminosities from the sample of
Schweitzer et al. (2006) and the corresponding optical data from
Marziani et al. (2003). Although the large fraction of upper lim-
its in the former sample prevents a careful statistical characteri-
zation, Fig. 9 shows a general correlation between QSO optical
luminosity and starburst activity in the host galaxy as traced by
the PAH luminosity.
The scale on the right hand side of Fig. 9 translates the 7.7µm
PAH luminosity into star formation rate (SFR). This was ob-
tained by combining the average L(PAH7.7µm)/L(FIR) ratio ob-
tained by Schweitzer et al. (2006) for the starburst dominated
QSOs in their sample with the SFR/L(FIR) given in Kennicutt
(1998), yielding
SFR [M⊙ yr
−1] = 3.46 10−42 L(PAH7.7µm) [erg s
−1] (8)
The average spectrum of high-z, luminous QSOs in Fig. 1
does not show evidence for the presence of PAH features and
can only provide an upper limit on the PAH flux relative to
the flux at 6.7µm (since all spectra were normalized to the lat-
ter wavelength prior to computing the average). However, we
can derive an upper limit on the PAH luminosity by assum-
ing the average distance of the sources in the sample. The in-
ferred upper limit on the PAH luminosity is reported with a red
square in Fig. 9a, and it is clearly below the extrapolation of
the L(PAH7.7µm) − λLλ(5100Å) relation found for local, low-
luminosity QSOs. This is shown more clearly in Fig. 9b which
shows the distribution of the ratio L(PAH7.7µm)/λLλ(5100Å) for
local QSOs (histogram) and the upper limit inferred from the av-
erage spectrum of high-z, luminous QSOs (red solid line). The
corresponding upper limit on the SFR is ∼ 700 M⊙ yr
Note that certainly there are luminous, high-z QSOs
with larger star formation rates (e.g. Bertoldi et al., 2003;
Beelen et al., 2006; Lutz et al., 2007). However, since our sam-
ple is not pre-selected in terms of MIR or FIR emission, our
result is not biased in terms of star formation and dust content,
and therefore it is representative of the general high-z, luminous
QSO population.
Our results indicate that the relation between star forma-
tion activity, as traced by the PAH features, and QSO power,
as traced by L5100, saturates at high luminosity. This result is
not surprising. Indeed, if high-z, luminous QSOs were char-
acterized by the same average L(PAH7.7µm)/λLλ(5100Å) ob-
served in local QSOs, this would imply huge star formation rates
of ∼ 7000 M⊙ yr
−1 at λLλ(5100Å) ∼ 10
47 erg s−1. On the
contrary, the few high-z QSOs detected at submm-mm wave-
lengths have far-IR luminosities corresponding to SFR of about
1000−3000 M⊙ yr
−1 (Omont et al., 2003). The majority of high-
z QSO (∼70%) are undetected at submm-mm wavelengths. The
mean mm-submm fluxes of QSOs in various surveys (includ-
ing both detections and non-detections) imply SFRs in the range
∼ 500−1500 M⊙ yr
−1 (Omont et al., 2003; Priddey et al., 2003),
in fair agreement with our finding, especially if we consider the
12 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
Fig. 9. a) PAH(7.7µm) luminosity as a function of the QSO optical luminosity. Blue diamonds are data from Schweitzer et al. (2006).
The red square is the upper limit obtained by the average spectrum of luminous, high-z QSOs. b) Distribution of the PAH(7.7µm)
to optical luminosity ratio in the local QSOs sample of Schweitzer et al. (2006). The hatched region indicates upper limits. The red
vertical line indicate the upper limit inferred from the average spectrum of luminous QSOs at high-z.
uncertainties involved in the two different observational methods
to infer the SFR.
We also note that a similar, independent result was obtained
by Haas et al. (2003), who found that in a large sample of QSOs
the ratio between LFIR (powered by star formation) and LB de-
creases at high luminosities.
The finding of a “saturation” of the relation between star for-
mation activity and QSO power may provide an explanation for
the evolution of the relation between BH mass and galaxy mass
at high redshift. Indeed, Peng et al. (2006) and McLure et al.
(2006) found that, for a given BH mass, QSO hosts at z∼2 are
characterized by a stellar mass lower than expected from the
local BH-galaxy mass relation. In other terms, the BH growth
is faster, relative to star formation, in high-z, luminous QSOs.
Our result supports this scenario by independently showing that
the correlation between star formation and AGN activity breaks
down at high luminosities.
5. Conclusions
We have presented low resolution, mid-IR Spitzer spectra of a
sample of 25 luminous QSOs at high redshifts (2 < z < 3.5).
We have combined our data with Spitzer spectra of lower lumi-
nosity, type-I AGNs, either published in the literature or in the
Spitzer archive. The combined sample spans five orders of mag-
nitude in luminosity, and allowed us to investigate the dust prop-
erties and star formation rate as a function of luminosity. The
spectroscopic information allowed us to disentangle the various
spectral components contributing to the MIR band (PAH and sil-
icate emission) and to sample the continuum at a specific λrest, in
contrast to photometric MIR observations. The main results are:
– The mid-IR continuum luminosity at 6.7µm correlates with
the optical continuum luminosity but the correlation is not
linear. In particular, the ratio λLλ(6.7µm)/λLλ(5100Å) de-
creases by about a factor of ten as a function of lumi-
nosity over the luminosity range 1042.5 < λLλ(5100Å) <
1047.5 erg s−1. This is interpreted as a reduction of the cov-
ering factor of the circumnuclear hot dust as a function of
luminosity. This result is in agreement and provides an in-
dependent confirmation of the recent findings of a decreas-
ing fraction of obscured AGN as a function of luminosity,
obtained in X-ray and optical surveys. We stress that while
X-ray surveys probe the covering factor of the gas, our result
provides an independent confirmation by probing the cover-
ing factor of the dust. We have also shown that the dust cov-
ering factor, as traced by the λLλ(6.7µm)/λLλ(5100Å) ratio,
decreases also as a function of the BH mass. Based on these
correlations alone it is not possible to determine whether the
physical quantity primarily driving the reduction of the cov-
ering factor is the AGN luminosity or the BH mass.
– The mean spectrum of the luminous, high-z QSOs in our
sample shows a clear silicate emission at λrest ∼ 10µm.
Silicate emission is also detected in the individual spectra
of four high redshift QSOs. When combined with the spec-
tra of local, lower luminosity AGNs we find that the sili-
cate strength (defined as the ratio between the maximum of
the silicate feature and the extrapolated featureless contin-
uum) tend to increase as a function of luminosity. The sili-
cate strength correlates positively also with the accretion rate
and with the BH mass, albeit with a large scatter.
– The mean MIR spectrum of the luminous, high-z QSOs in
our sample does not show evidence for PAH emission. Our
sample is not pre-selected by the FIR emission and therefore
it is not biased in terms star formation. As a consequence, the
upper limit on the PAH emission in the total mean spectrum
provides a useful, representative upper limit on the SFR in
luminous QSOs at high redshifts. We find that the ratio be-
tween PAH luminosity and QSO optical luminosity is signifi-
cantly lower than observed in local, lower luminosity AGNs,
implying that the correlation between star formation rate and
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 13
AGN power probably saturates at high luminosities. This re-
sult may explain the evolution of the correlation between BH
mass and galaxy stellar mass recently observed in luminous
QSOs at high redshift.
Acknowledgements. We are grateful to M. Salvati for useful comments. We are
grateful to G. Hasinger for providing us with some of his results prior to publica-
tion. This work is based on observations made with the Spitzer Space Telescope,
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology under a contract with NASA. Support for this work was provided
by NASA under contract 1276513 (O.S.). RM acknowledges partial support
from the Italian Space Agency (ASI). MI is supported by Grants-in-Aid for
Scientific Research (16740117). OS acknowledges support by the Israel Science
Foundation under grant 232/03.
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List of Objects
‘2QZJ002830.4-281706’ on page 16
‘LBQS0109+0213’ on page 16
http://arxiv.org/abs/astro-ph/0702272
http://arxiv.org/abs/astro-ph/0610939
http://arxiv.org/abs/astro-ph/0702136
http://arxiv.org/abs/astro-ph/0702701
14 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
‘[HB89]0123+257’ on page 16
‘HS0211+1858’ on page 16
‘2QZJ023805.8-274337’ on page 16
‘SDSSJ024933.42-083454.4’ on page 16
‘Q0256-0000’ on page 16
‘Q0302-0019’ on page 16
‘[HB89]0329-385’ on page 16
‘SDSSJ100428.43+001825.6’ on page 16
‘TON618’ on page 16
‘[HB89]1318-113’ on page 16
‘[HB89]1346-036’ on page 16
‘UM629’ on page 16
‘UM632’ on page 16
‘BS1425+606’ on page 16
‘[VCV01]J1649+5303’ on page 16
‘SDSSJ170102.18+612301.0’ on page 16
‘SDSSJ173352.22+540030.5’ on page 16
‘[HB89]2126-158’ on page 16
‘2QZJ221814.4-300306’ on page 16
‘2QZJ222006.7-280324’ on page 16
‘Q2227-3928’ on page 16
‘[HB89]2254+024’ on page 16
‘2QZJ234510.3-293155’ on page 16
‘Mrk335’ on page 16
‘IIIZw2’ on page 16
‘PG0050+124’ on page 16
‘PG0052+251’ on page 16
‘Fairall9’ on page 16
‘Mkr79’ on page 16
‘PG0804+761’ on page 16
‘Mrk704’ on page 16
‘PG0953+414’ on page 16
‘NGC3516’ on page 16
‘PG1116+215’ on page 16
‘NGC3783’ on page 16
‘PG1151+117’ on page 16
‘NGC4051’ on page 16
‘PG1211+143’ on page 16
‘NGC4593’ on page 16
‘PG1309+355’ on page 16
‘PG1351+640’ on page 16
‘IC4329a’ on page 16
‘NGC5548’ on page 16
‘Mrk817’ on page 16
‘Mrk509’ on page 16
‘Mrk926’ on page 16
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 15
Table 1. Combined sample of high-z luminous QSO, local QSO and Sy1, and physical properties inferred from optical-UV spectra.
Name RA(J2000) Dec(J2000) z log(λLλ(5100Å)) logL([OIII]) logM(BH)
d L/LdEdd αopt−UV Ref.
(erg/s) (erg/s) M⊙
High-z luminous QSOs
2QZJ002830.4-281706 00:12:21.18 −28:36:30.2 2.401 46.59 44.41 9.72 0.35 −1.42 1
LBQS0109+0213 01:12:16.91 +02:29:47.6 2.349 46.81 44.51 10.01 0.30 −1.75 1
[HB89]0123+257a 01:26:42.79 +25:59:01.3 2.369 46.58 44.32 9.10 1.40 −1.65 1
HS0211+1858 02:14:29.70 +19:12:37.0 2.470 46.63 44.38 10.11 0.16 −0.01 3,10
2QZJ023805.8-274337 02:38:05.80 −27:43:37.0 2.471 46.58 <43.71 9.41 0.69 −1.59 1
SDSSJ024933.42-083454.4 02:49:33.41 −08:34:54.4 2.491 46.39 44.12 9.67 0.25 −1.36 1
Q0256-0000 02:59:05.64 +00:11:21.9 3.377 46.99 44.55 10.11 0.19 −0.96 2
Q0302-0019 03:04:49.86 −00:08:13.4 3.286 46.83 45.01 10.11 0.30 −1.66 2
[HB89]0329-385 03:31:06.34 −38:24:04.8 2.435 46.72 44.31 10.11 0.18 −1.79 1
SDSSJ100428.43+001825.6 10:04:28.44 +00:18:25.6 3.040 46.45 44.47 9.34 0.70 −0.70 3,11
TON618a 12:28:24.97 +31:28:37.6 2.226 47.32 <44.12 10.81 0.14 −1.27 1
[HB89]1318-113 13:21:09.38 −11:39:31.6 2.306 46.90 44.32 9.76 0.62 −0.99 1
[HB89]1346-036 13:48:44.08 −03:53:24.9 2.370 46.89 43.73 9.95 0.41 −1.26 1
UM629 14:03:23.39 −00:06:06.9 2.460 46.57 44.41 9.17 1.16 −1.40 1
UM632b 14:04:45.89 −01:30:21.9 2.517 46.55 44.04 9.44 0.61 −1.27 1
SBS1425+606 14:26:56.10 +60:25:50.0 3.202 47.39 45.04 9.83 1.73 −1.45 1
[VCV01]J1649+5303 16:49:14.90 +53:03:16.0 2.260 46.70 44.19 9.99 0.24 −0.86 3,11
SDSSJ170102.18+612301.0 17:01:02.18 +61:23:01.0 2.301 46.35 <43.51 9.73 0.20 −1.48 1
SDSSJ173352.22+540030.5 17:33:52.23 +54:00:30.5 3.428 47.02 44.36 9.58 1.28 −1.51 1
[HB89]2126-158b 21:29:12.17 −15:38:41.0 3.282 47.27 44.66 9.73 1.60 0.72 1
2QZJ221814.4-300306 22:18:14.40 −30:03:06.0 2.389 46.55 43.95 9.28 0.89 −1.27 1
2QZJ222006.7-280324 22:20:06.70 −28:03:23.0 2.414 47.23 44.64 10.21 0.54 −1.28 1
Q2227-3928 22:30:32.95 −39:13:06.8 3.438 46.95 <44.02 10.31 0.19 −1.25 2
[HB89]2254+024 22:57:17.56 +02:43:17.5 2.083 46.46 43.95 9.10 1.08 −1.37 1
2QZJ234510.3-293155 23:45:10.36 −29:31:54.7 2.382 46.33 43.97 9.38 0.42 −1.26 1
High-z QSO aver. (z<2.5)c 46.63 44.07 9.68 0.51
Local QSOs and Sy1s
Mrk335 00:06:19.52 +20:12:10.4 0.025 43.62 41.29 7.10 0.28 −2.00 4,14
IIIZw2 00:10:30.80 +10:58:13.0 0.090 44.02 42.25 8.19 0.16 −1.56 4,13
PG0050+124 00:53:34.94 +12:41:36.2 0.058 44.36 41.87 7.09 0.96 −0.91 4,13
PG0052+251 00:54:52.10 +25:25:38.0 0.155 44.46 42.57 8.55 0.21 −2.27 4,13
Fairall9 01:23:45.78 −58:48:20.5 0.046 43.80 41.91 8.27 0.10 −1.99 4,13
Mrk79 07:42:32.79 +49:48:34.7 0.022 43.58 41.37 8.12 0.08 −0.83 9,14
PG0804+761 08:10:58.60 +76:02:42.0 0.100 44.42 42.03 8.08 0.33 −2.64 4,12
Mrk704 09:18:26.00 +16:18:19.2 0.029 43.44 41.18 7.97 0.08 – 4
PG0953+414 09:56:52.40 +41:15:22.0 0.234 44.96 42.69 8.39 0.56 −2.12 4,13
NGC3516 11:06:47.49 +72:34:06.8 0.009 42.81 40.52 7.39 0.06 −1.09 7,8,14
PG1116+215 11:19:08.60 +21:19:18.0 0.176 44.84 42.27 8.27 0.53 −2.58 4,13
NGC3783 11:39:01.72 −37:44:18.9 0.010 43.05 41.10 7.33 0.09 −1.46 4,13
PG1151+117 11:53:49.27 +11:28:30.4 0.176 44.48 42.09 8.31 0.28 −2.50 4,12
NGC4051 12:03:09.61 +44:31:52.8 0.002 41.39 39.64 5.32 0.06 −0.82 5,6,14
PG1211+143 12:14:17.70 +14:03:12.6 0.085 44.58 41.94 7.69 0.68 −1.32 4,13
NGC4593 12:39:39.42 −05:20:39.3 0.009 42.60 40.34 7.40 0.04 −1.26 4,14
PG1309+355 13:12:17.76 +35:15:21.2 0.184 44.50 42.18 8.29 0.30 −2.45 4,12
PG1351+640 13:53:15.80 +63:45:45.4 0.087 44.80 42.52 8.76 0.28 −0.86 4,13
IC4329a 13:49:19.26 −30:18:34.0 0.016 43.13 40.89 7.77 0.06 – 4
NGC5548 14:17:59.53 +25:08:12.4 0.017 43.10 41.15 7.78 0.06 −1.73 4,13
Mrk817 14:36:22.06 +58:47:39.3 0.033 43.96 41.65 8.11 0.16 −0.56 4,14
Mrk509 20:44:09.73 −10:43:24.5 0.034 44.01 42.13 7.87 0.22 −1.69 4,13
Mrk926 23:04:43.47 −08:41:08.6 0.047 43.83 42.29 8.55 0.08 – 4
The following quantities are reported in each column: column 1, object name; columns 2-3, coordinates (J2000); column 4, redshift; column 5, log
of the continuum luminosity λLλ (in units of erg/s) at the rest frame wavelength 5100Å; column 6, log of the [OIII]λ5007 emission line luminosity
(in units of erg/s); column 7, log of the black hole mass (in units of M⊙); column 8, Eddington ratio Lbol/LEdd; column 9, optical-to-UV (1450Å–
5100Å) continuum slope (Fλ ∝ λ
αopt−UV); column 11: reference for the optical and UV data: 1 - Shemmer et al. (2004), Netzer et al. (2004) and
therein references for UV data, 2 - Dietrich et al. (2002) and therein references for UV data, 3 - Juarez et al (in prep.), 4 - Marziani et al. (2003), 5 -
Suganuma et al. (2006), 6 - Peterson et al. (2000), 7 - Wanders et al. (1993), 8 - Ho & Ulvestad (2001), 9 - Peterson et al. (1998), 10 - Engels et al.
(1998), 11 - SDSS DR5 archive, 12- Baskin & Laor (2005), 13 - Evans & Koratkar (2004), 14 - Kaspi et al. (2005).
a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the
6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses.
b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm
flux is likely thermal.
c Optical luminosities, black hole mass and Eddington ratio for the stacked spectrum refer to the average values of only the objects at z<2.5, i.e.
those who contribute to the Silicate feature observed in the stacked spectrum.
d As discussed in Shemmer et al. (2004), the uncertainties on the BH masses and accretion rate are no larger than a factor of two.
16 R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs
Table 2. Infrared properties of the combined sample of high-z luminous QSO, local QSO and Sy1
Name FMIR αMIR
λLλ (6.7µm)
λLλ (5100Å)
Si strength Ref.
(mJy)
High-z luminous QSOs
2QZJ002830.4-281706 4.8 -0.59 0.75 1
LBQS0109+0213 8.3 -0.91 0.75 1
[HB89]0123+257a 4.9 -1.69 0.76 1
HS0211+1858 7.3 -1.36 1.09 1
2QZJ023805.8-274337 3.7 -1.04 0.62 0.81±0.16 1
SDSSJ024933.42-083454.4 2.2 -1.26 0.57 1
Q0256-0000 2.7 -1.34 0.30 1
Q0302-0019 3.6 -1.67 0.56 1
[HB89]0329-385 5.5 -1.74 0.65 1
SDSSJ100428.43+001825.6 2.3 -1.51 0.73 1
TON618a 20.1 -1.16 0.51 0.04±0.01 1
[HB89]1318-113 6.6 -1.70 0.47 1
[HB89]1346-036 12.9 -1.68 0.99 1
UM629 3.8 -0.68 0.65 1
UM632b 2.1 -1.77 0.40 1
BS1425+606 23.9 -1.56 0.96 1
[VCV01]J1649+5303 9.4 -1.28 1.03 1
SDSSJ170102.18+612301.0 3.3 -0.66 0.83 1
SDSSJ173352.22+540030.5 2.0 -2.88 0.21 1
[HB89]2126-158b 19.1 -1.18 1.08 1
2QZJ221814.4-300306 4.6 -1.03 0.78 1
2QZJ222006.7-280324 16.0 -1.52 0.58 0.65±0.05 1
Q2227-3928 3.0 -1.70 0.38 1
[HB89]2254+024 3.6 -1.12 0.60 0.63±0.15 1
2QZJ234510.3-293155 4.2 -1.33 1.16 0.92±0.15 1
High-z QSO aver. (z<2.5)c -1.57 0.58±0.10
Local QSOs and Sy1s
Mrk335 130. -1.38 1.90 0.25±0.06 3
IIIZw2 52. -1.09 4.02 0.05±0.03 2,3
PG0050+124 245. -0.74 3.55 0.38±0.05 2,3
PG0052+251 28. -1.35 2.40 0.33±0.06 2,3
Fairall9 146. -0.90 4.83 0.21±0.07 2,3
Mkr79 200. -1.03 2.44 0.10±0.06 3
PG0804+761 88. -1.81 3.38 0.60±0.05 2,3
Mrk704 190. -0.95 5.64 0.09±0.06 3
PG0953+414 26. -1.88 1.68 0.40±0.08 2,3
NGC3516 210. -1.05 2.40 0.06±0.05 3
PG1116+215 66. -1.71 3.08 0.22±0.05 2,3
NGC3783 315. -1.08 2.69 -0.01±0.03 2,3
PG1151+117 10. -2.19 1.08 0.36±0.13 2,3
NGC4051 230. -0.43 4.71 0.06±0.05 3
PG1211+143 100. -1.26 1.89 0.55±0.05 2,3
NGC4593 184. -1.14 3.61 0.08±0.05 2,3
PG1309+355 25. -1.24 2.82 0.41±0.07 2,3
PG1351+640 53. -0.89 0.64 1.25±0.05 2,3
IC4329a 487. -0.67 9.00 0.01±0.03 2,3
NGC5548 69. -1.06 1.53 0.27±0.05 2,3
Mrk817 140. -0.65 1.61 0.16±0.06 3
Mrk509 179. -1.29 1.98 0.11±0.04 2,3
Mrk926 55. -1.40 1.77 0.26±0.04 2,3
The following quantities are reported in each column: column 1, object name; column 2, continuum flux density at the observed wavelength
corresponding to λrest = 6.7µm (after removing starburst and stellar components, in units of mJy); column 3, power-law index (Fλ ∝ λ
α) fitted
to the continuum in the 5–8µm range (starburst component–subtracted); column 4, ratio of the continuum emission at 5100Å and at 6.7µm,
λLλ(5100Å)/λLλ(6.7µm); column 5, Silicate strength; column 6: reference for the infrared data: 1 - this work (from Spitzer program 20493) ; 2 -
Shi et al. (2006); 3 - this work (from Spitzer archival data).
a Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is near or above the observed value, hence the
6.7µm flux is probably dominated by synchrotron emission; these objects will be excluded from statistical analyses.
b Radio loud QSOs for which the extrapolation of the radio synchrotron emission to the MIR is well below the observed value, hence the 6.7µm
flux is likely thermal.
c The ratio λLλ(5100Å)/λLλ(6.7µm) is not defined for the stacked spectrum, since all spectra were normalized to the 6.7µm flux before stacking.
As a consequence, only the Silicate strength (and more generally the continuum shape) has a physical meaning for the stacked spectrum.
R. Maiolino et al.: Dust covering factor, silicate emission and star formation in luminous QSOs 17
Table 3. Spearman-rank coefficients for the correlations in Figs.4, 8, 5 and 6.
log(λLλ(5100Å)) log(L[OIII]) log(L/LEdd) log(MBH) αMIR αopt−UV
λLλ (6.7µm)
λLλ (5100Å)
−0.76 (< 10−6) −0.72 (< 10−6) −0.44 (3 10−4) −0.70 (< 10−6) 0.29 (0.05) −0.20 (0.19)
log(Si str.) 0.83 (2 10−5) 0.75 (6 10−6) 0.75 (8 10−6) 0.66 (8 10−4)
Numbers in parenthesis give the probability for the correlation coefficient to deviate from zero.
Introduction
Sample selection, observations and data reduction
Analysis
Main observational results
A comparison with MIR properties of lower luminosity AGNs
Discussion
Dust covering factor
Covering factor as a function of source luminosity and BH mass
Model uncertainties
Comparison with previous works
Silicate emission
PAHs and star formation
Conclusions
|
0704.1560 | Generating entanglement of photon-number states with coherent light via
cross-Kerr nonlinearity | Generating entanglement of photon-number states with
coherent light via cross-Kerr nonlinearity
Zhi-Ming Zhang∗, Jian Yang, and Yafei Yu
Laboratory of Photonic Information Technology,
School of Information and Photoelectronics,
South China Normal University, Guangzhou 510006, China
November 25, 2018
Abstract
We propose a scheme for generating entangled states of light fields. This scheme only requires
the cross-Kerr nonlinear interaction between coherent light-beams, followed by a homodyne de-
tection. Therefore, this scheme is within the reach of current technology. We study in detail the
generation of the entangled states between two modes, and that among three modes. In addition
to the Bell states between two modes and the W states among three modes, we find plentiful
new kinds of entangled states. Finally, the scheme can be extend to generate the entangled states
among more than three modes.
PACS: 03.67.Mn; 42.50.Dv; 42.50.Ct
1 Introduction
Entanglement is a characteristic feature of quantum states and has important applications in quantum
science and technology, for example, in quantum computation and quantum information [1]. There
are a lot of schemes for generating various kinds of entanglement, for example, the entanglement
between photons, the entanglement between atoms, the entanglement between trapped ions, and
the entanglement between different kinds of particles (for example, between photons and atoms).
In addition to the entanglement between two parties, there are also entanglement of multiparties.
Among these schemes many use single-photon sources and/or single-photon detectors. Although
there are great progresses in the study on these single-photon devices, how to obtain them is still a
challenging task. In this paper we propose a simple scheme for generating entangled states of light
fields. This scheme only requires the cross-Kerr nonlinear interaction between light fields in coherent
states, followed by a homodyne detection. Therefore, this scheme is within the reach of current
technology.The basic idea of this scheme is shown in Figure 1. Mode a is a bright beam which is in a
coherent state |α〉. Mode b is a weak or bright beam which is also in a coherent state. BS is a 50/50
beam splitter. KM1 and KM2 are Kerr media. HD means homodyne detection [2].
This paper is organized as follows: In section 2 we briefly introduce the cross-Kerr nonlinear inter-
action between two field-modes. In section 3 and section 4 we study the generation of entanglement
between two modes and that among three modes, respectively. Section 5 is a summary.
∗Corresponding author: [email protected]
http://arxiv.org/abs/0704.1560v1
2 Cross-Kerr nonlinear interaction
First, let us briefly review the cross-Kerr nonlinear interaction between a mode A and a mode B. The
interaction Hamiltonian has the form [3]
HCK = ~Kn̂An̂B, (1)
where n̂A and n̂B are the photon-number operator of mode A and mode B, respectively. The coupling
coefficient K is proportional to the third-order nonlinear susceptibility χ(3). The time-evolution
operator is
U (t) = exp
= exp {−iKn̂An̂Bt} = exp {−iτ n̂An̂B} = U (τ) , (2)
in which τ = Kt = K (l/v), it can be named as the scaled interaction time, or the nonlinear phase
shift. Here l is the length of the Kerr medium and v is the velocity of light in the Kerr medium. The
cross Kerr nonlinearity has following property
U (τ) |n〉B |α〉A = |n〉B
∣αe−inτ
, (3)
here |n〉 and |α〉 are the photon number state and the coherent state, respectively.
3 Entanglement between two modes
Now let us study the generation of the entangled states between two modes. The scheme is shown
in Figure 1. Assume that mode a is in a coherent state |α〉 [4]. Mode b is also in a coherent state
which is divided by the 50/50 beam splitter BS into two beams b1 and b2, and both b1 and b2 are in
coherent state |β〉.
We first consider the case of weak coherent state |β〉. In this case we have
|β〉 ≈ 1√
1 + |β|2
(|0〉+ β |1〉) , (4)
where |0〉 and |1〉 are the vacuum state and one-photon state, respectively. Let mode a interacts with
mode b1 and b2 successively. For simplicity, we assume that both the scaled interaction times are τ ,
thai is, τ1 = K1t1 = τ2 = K2t2 = τ.The interactions change the state as following way
|β〉2 |β〉1 |α〉a →
1 + |β|2
|0〉2 |0〉1 |α〉a + β (|1〉2 |0〉1 + |0〉2 |1〉1)
∣αe−iτ
+ β2 |1〉2 |1〉1
∣αe−i2τ
where the subscripts 1 and 2 denote modes b1 and b2, respectively. We note that the internal product
of coherent states satisfies [4]
αe−inτ |αe−i(n+1)τ
= e−4|α|
2 sin2(τ/2) ≈ e−|α|
2τ2 , (6)
in which we have taken into account the fact that in practice τ is small [3] and therefore sin (τ/2) ≈
τ/2. However, if mode a is bright enough so that |α|2 τ2 ≫ 1, then the two coherent states will
be approximately orthogonal. This condition can be easily satisfied in experiments and in following
discussions we assume that it is satisfied. In this case, a homodyne detection can distinguish different
coherent states [5]. Therefore, when we find that mode a is in the coherent state
∣αe−iτ
, then beam
b1 and beam b2 will be projected into the entangled state
(|1〉2 |0〉1 + |0〉2 |1〉1) , (7)
and the probability for getting this entangled state is 2 |β|2 /
1 + |β|2
.This state is one of Bell
states [1] and a special case of the NOON states [6].
Now let us consider the general situation in which beam b1 and beam b2 are normal coherent states
[4]. In this situation,
|β〉 = exp
|n〉 . (8)
The cross-Kerr interactions transform the state as follows
|β〉2 |β〉1 |α〉a = e
−|β|2
βm+n√
|m〉2 |n〉1 |α〉a
→ e−|β|
βm+n√
|m〉2 |n〉1
αe−i(m+n)τ
. (9)
If the homedyne detection finds mode a in the state
∣αe−i(m+n)τ
∣αe−ikτ
(k = m+n = 1, 2, ...),
then mode b1 and mode b2 will be collapse into the entangled state
n! (k − n)!
|k − n〉2 |n〉1 (k = 1, 2, ...). (10)
Since in this state the sum of photon numbers of the two modes is equal to k, we name this state as
the 2-mode k-photon entangled state. The probability for getting this state is exp(−2 |β|2)2
|β|2k .
The entanglement property of the states expressed by Eq.(10) can be proved by using following
entanglement criteria [7]
b+1 b2
> 〈Nb1Nb2〉 , (11)
where Nb1(Nb2), b1(b2) and b
2 ) are the photon-number operator, the photon annihilation operator
and the photon creation operator of mode b1(b2), respectively. For the states of equation (10), we
can find
b+1 b2
k2 , and 〈Nb1Nb2〉 = 14k (k − 1) . Therefore the entanglement condition (11) is
satisfied, and the states (10) are indeed entangled states. For k = 1, equation (10) reduces to equation
(7), and some other examples of the 2-mode k-photon entangled states are listed below.
(|2〉2 |0〉1 + |0〉2 |2〉1) +
2 |1〉2 |1〉1
(k = 2) (12)
(|3〉2 |0〉1 + |0〉2 |3〉1) +
3 (|2〉2 |1〉1 + |1〉2 |2〉1)
(k = 3) (13)
Equations (12) and (13) are new kinds of entangled states. Equation (12) can be understood as a
superposition of a NOON state (|2〉2 |0〉1 + |0〉2 |2〉1) and a product state |1〉2 |1〉1 ,while equation (13)
can be understood as a superposition of a NOON state (|3〉2 |0〉1 + |0〉2 |3〉1) and a NOON − like
state (|2〉2 |1〉1 + |1〉2 |2〉1). We also note that in the superposition (13) the probability of getting the
state (|2〉2 |1〉1 + |1〉2 |2〉1) is larger than that of getting the state (|3〉2 |0〉1 + |0〉2 |3〉1) .That is, the
photons trend to distribute between the two modes symmetrically. The properties and applications
of these new kinds of entangled states will be studied in the future.
4 Entanglement among three modes
We can extend the scheme above to generate the entanglement among three modes. For this purpose
we modify the scheme from Figure 1 to Figure 2, in which BS1 has the reflection/transmission = 1/2
and BS2 has the reflection/transmission = 1/1, so that the three beams b1, b2 and b3 have the same
strength, and we assume all of them are in the coherent state |β〉 . We let mode a, in a coherent state
|α〉, interacts with modes b1, b2 and b3 successively. And for simplicity, we assume that all of the
scaled interaction times are equal, thai is, τ1 = τ2 = τ3 = τ.
For the situation in which |β〉 is weak and can be expressed as in equation (4), the interactions
transform the states in the following way
|β〉3 |β〉2 |β〉1 |α〉a →
1 + |β|2
{|0〉3 |0〉2 |0〉1 |α〉a
+β (|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1)
∣αe−iτ
+β2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)
∣αe−i2τ
+β3 |1〉3 |1〉2 |1〉1
∣αe−i3τ
}. (14)
As discussed above, we assume that different coherent states in above equation are approximately
orthogonal, and we can use homodyne detection to distinguish them [5]. If we find that mode a is in
state
∣αe−iτ
then modes b1, b2 and b3 will be projected to the entangled state
(|1〉3 |0〉2 |0〉1 + |0〉3 |1〉2 |0〉1 + |0〉3 |0〉2 |1〉1) , (15)
and the probability for obtaining this state is 3 |β|2 /
1 + |β|2
. On the other hand, If we find that
mode a is in state
∣αe−i2τ
then modes b1, b2 and b3 will be projected to the entangled state
(|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1) , (16)
and the probability for getting this state is 3 |β|4 /
1 + |β|2
. Equations (15) and (16) can be named
as 1-photon W state [8] and 2-photon W state, respectively.
For the general case in which |β〉 is not very weak we use equation (8). In this case the interactions
transform the states as follows:
|β〉3 |β〉2 |β〉1 |α〉a = e
−3|β|2/2
l,m,n
βl+m+n√
l!m!n!
|l〉3 |m〉2 |n〉1 |α〉a
→ e−3|β|
l,m,n
βl+m+n√
l!m!n!
|l〉3 |m〉2 |n〉1
αe−i(l+m+n)τ
. (17)
If we find that mode a is in the state
∣αe−i(l+m+n)τ
∣αe−ikτ
(k = l+m+n = 1, 2, ...), then
modes b1, b2 and b3 will be projected to the entangled state
(k −m− n)!m!n!
|k −m− n〉3 |m〉2 |n〉1 (k = 1, 2, ...). (18)
We name this state as the 3-mode k-photon entangled state. The probability for getting this state
is exp(−3 |β|2)3
|β|2k .The entanglement property of the states of Eq.(18) can be proved by using
following entanglement criteria [7]
b+1 b2
> 〈Nb1Nb2〉 and
b+2 b3
> 〈Nb2Nb3〉 . (19)
For the states (18), we can find
b+1 b2
b+2 b3
k2, and 〈Nb1Nb2〉 = 〈Nb2Nb3〉 = 19k (k − 1) .Therefore
the entanglement condition (19) is satisfied, and the states (18) are indeed entangled states of three
modes. For k = 1, equation (18) reduces to equation (15), and some other examples of the 3-mode
k-photon entangled state are as follows:
{(|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1)
2 (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)} (k = 2) , (20)
{(|3〉3 |0〉2 |0〉1 + |0〉3 |3〉2 |0〉1 + |0〉3 |0〉2 |3〉1)
3 (|2〉3 |1〉2 |0〉1 + |2〉3 |0〉2 |1〉1 + |1〉3 |2〉2 |0〉1 + |1〉3 |0〉2 |2〉1 + |0〉3 |2〉2 |1〉1 + |0〉3 |1〉2 |2〉1)
6 |1〉3 |1〉2 |1〉1} (k = 3) . (21)
Equation (20) is a superposition of two 2-photon W states. While equation (21) is a superposition
of a 3-photon W state (the first line), a product state (the third line), and a state (the second line)
which can be expressed as
|1〉j |0〉k + |0〉j |1〉k
+ |1〉i
|2〉j |0〉k + |0〉j |2〉k
+ |0〉i
|2〉j |1〉k + |1〉j |2〉k
, (22)
where the subscripts i = 1,or 2, or 3,and j,k are the other two, respectively. We also note that in
the superposition (20) the probability of getting the state (|1〉3 |1〉2 |0〉1 + |1〉3 |0〉2 |1〉1 + |0〉3 |1〉2 |1〉1)
is larger than that of getting the state (|2〉3 |0〉2 |0〉1 + |0〉3 |2〉2 |0〉1 + |0〉3 |0〉2 |2〉1). This shows again
that the photons trend to distribute among different modes symmetrically.
5 Summary
In summary, we have proposed a scheme for generating entangled states of light fields. This scheme
has following advantages: First, the scheme only involves the cross-Kerr nonlinear interaction between
coherent light-beams, followed by a homodyne detection. It is not necessary that the cross-Kerr
nonlinearity is very large, as long as the coherent light is bright enough. Therefore, this scheme is
within the reach of current technology. Second, in addition to the Bell states between two modes and
the W states among three modes, plentiful new kinds of entangled states can be generated with this
scheme. We also found that in the generated entangled states, the photons have a trend to distribute
among different modes symmetrically. Finally, we would like to point out that the scheme can be
extend to generate the entangled states among more than three modes.
Acknowledgement This work was supported by the National Natural Science Foundation of
China under grant nos 60578055 and 10404007.
References
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Cambridge University Press, 2000 ); D.Bouwmeester, A.Ekert, and A.Zeilinger, The Physics of
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[2] U.Leonhardt,Measuring the Quantum state of light (Cambridge: Cambridge University
Press,1997); H.A.Bachor and T.C.Ralph,A Guide to Experiments in Quantum Optics(Weinheim:
Wiley-VCH Verlag GmbH & Co.KGaA, 2004).
[3] B.C.Sanders and G.J.Milburn, Phys. Rev. A 45, 1919 (1992); C.C.Gerry, Phys. Rev. A 59, 4095
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[4] C.C.Gerry and P.L.Knight, Introductory Quantum Optics (Cambridge: Cambridge University
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R.G.Beauoleil, and T.P.Spiller,Phys. Rev. A 71, 033819 (2005).
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Z. Y. Ou, Phys. Rev. A 55, 2598 (1997); F.Shafiei, P.Srinivasan, and Z. Y. Ou,Phys. Rev. A
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G.C.Guo, Phys. Rev. A 73, 023808 (2006).
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Rev. A 74, 062320 (2006); R.Lohmayer, A.Osterloh, J.Siewert, and A.Uhlmann, Phys. Rev. Lett.
97, 260502 (2006); M.Bourennane, M.Eibl, S.Gaertner, N.Kiesel, C.Kurtsiefer, and H.Weinfurter,
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Rev. A 75, 032317 (2007).
Figure captions
Figure 1. Scheme for generating entanglement between two modes. KM:cross-Kerr medium; BS:
beam splitter; M: mirror, HD: homodyne detection.
Figure 2. Scheme for generating entanglement among three modes. KM:cross-Kerr medium; BS:
beam splitter; M: mirror, HD: homodyne detection.
Introduction
Cross-Kerr nonlinear interaction
Entanglement between two modes
Entanglement among three modes
Summary
|
0704.1561 | A parachute for the degree of a polynomial in algebraically independent
ones | A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN
ALGEBRAICALLY INDEPENDENT ONES
S. VÉNÉREAU
Abstract. We give a simpler proof as well as a generalization of the main
result of an article of Shestakov and Umirbaev ([3]). This latter article being
the first of two that solve a long-standing conjecture about the non-tameness,
or ”wildness”, of Nagata’s automorphism. As corollaries we get interesting
informations about the leading terms of polynomials forming an automorphism
of K[x1, · · · , xn] and reprove the tameness of automorphisms of K[x1, x2].
The following notations are fixed throughout the article: K is a field of cha-
racteristic 0 and K[x1, · · · , xn] is the ring of polynomials in the n indetermi-
nates x1, · · · , xn with coefficients in K, endowed with the classical degree func-
tion: deg. We consider m algebraically independent polynomials in K[x1, · · · , xn]:
f1, · · · , fm of respective degrees d1, · · · , dm. There is also, for every polynomial G ∈
K[f1, · · · , fm] a unique one G(X1, · · · , Xm) ∈ K[X1, · · · , Xm], where X1, · · · , Xm
are new indeterminates, such that G = G(f1, · · · , fm). By abuse of notation we will
write ∂G
to denote ∂G
(f1, · · · , fm), ∀1 ≤ i ≤ m and degfi G to denote degXi G,
the degree of G in Xi.
The following definition and properties are only formally new, and come from
Definition. We call the parachute of f1, · · · , fm and denote ∇ = ∇(f1, · · · , fm)
the integer
∇ = d1 + · · ·+ dm −m− max
1≤i1,··· ,im≤n
deg jxi1 ,··· ,xim (f1, · · · , fm)
where jxi1 ,··· ,xim (f1, · · · , fm) is the jacobian determinant of f1, · · · , fm with respect
to xi1 , · · · , xim that is jxi1 ,··· ,xim (f1, · · · , fm) = det(∂fi/∂xij )i,j.
Properties. The parachute of f1, · · · , fm has the following estimate:
0 ≤ ∇ = ∇(f1, · · · , fm) ≤ d1 + · · ·+ dm −m.(1)
For any G ∈ K[f1, · · · , fm] and ∀1 ≤ i ≤ m, one has
degG ≥ deg ∂G
+ di −∇ and, inductively,
degG ≥ deg ∂
+ kdi − k∇, ∀k ≥ 0.
Proof. The left minoration 0 ≤ ∇ in (1) is an easy exercise. The right majoration
is a direct consequence of the following
Fact. The vectors gradf1 = (∂f1/∂x1, · · · , ∂f1/∂xn), · · · , gradfm = (∂fm/∂x1, · · · , ∂fm/∂xn)
are linearly independent over K[x1, · · · , xn] therefore the minors of order m of the
matrix
gradf1
gradfm
are not all 0 and the number max1≤i1,··· ,im≤n deg jxi1 ,··· ,xim (f1, · · · , fm)
is non-negative.
Proof. As mentioned in [3], it is a well-known fact (see e.g. [1] or [2] for a nice proof)
that n rational functions f1, · · · , fn ∈ K(x1, · · · , xn) are algebraically independant
http://arxiv.org/abs/0704.1561v2
2 S. VÉNÉREAU
if and only if their jacobian determinant is not zero. Our fact is then proved
by completing our m algebraically independant polynomials to get n algebraically
independant rational functions: the jacobian determinant is not zero and it follows
that gradf1, · · · , gradfm must be linearly independant. �
It is clearly sufficient to show (2) for i = m.
Take any m integers 1 ≤ i1, · · · , im ≤ n. From the definition of jxi1 ,··· ,xim it is
clear that
deg jxi1 ,··· ,xim (f1, · · · , fm−1, G) ≤ d1 − 1 + · · ·+ dm−1 − 1 + degG− 1
≤ d1 + · · ·+ dm−1 −m+ degG .
On the other hand the chain rule gives
jxi1 ,··· ,xim (f1, · · · , fm−1, G) = jxi1 ,··· ,xim (f1, · · · , fm−1, fm)
Hence we get
deg jxi1 ,··· ,xim (f1, · · · , fm−1, fm) + deg
≤ d1 + · · ·+ dm−1 −m+ degG
deg ∂G
+ dm − (d1 + · · ·+ dm−1 + dm −m) + deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG .
In particular, when the maximum is realized,
+dm−(d1+· · ·+dm−1+dm−m)+ max
1≤i1,··· ,im≤n
deg jxi1 ,··· ,xim (f1, · · · , fm) ≤ degG
+dm−(d1+· · ·+dm−1+dm−m− max
1≤i1,··· ,im≤
deg jxi1 ,··· ,xim (f1, · · · , fm)) ≤ degG
+ dm −∇ ≤ degG.
In order to state our main theorem one needs to fix some more notations: we
denote p̄ the leading term of a polynomial p ∈ K[x1, · · · , xn] and for any subalgebra
A ⊂ K[x1, · · · , xn], we denote gr(A) := K[Ā] the subalgebra generated by Ā =
{ā|a ∈ A}. We define si, ∀1 ≤ i ≤ m, as the degree of the minimal, if any,
polynomial of f̄i over Frac( gr(K[fj ]j 6=i)), the field of fractions of the subalgebra
generated by K[fj]j 6=i = K[f1, · · · , fi−1, fi+1, · · · , fm] and as +∞ otherwise. We
denote ⌊α⌋ the integral part of a real number α and agree that k/∞ = 0 when
0 ≤ k < ∞.
Theorem. Let G be a polynomial in K[f1, · · · , fm]. Then the following minoration
holds, ∀1 ≤ i ≤ m,
degG ≥ di · degfi G−∇ · ⌊
degfi G
Proof. It is of course sufficient to prove it for i = m. First remark that a polynomial
m ∈ A[fm], where gi ∈ A := K[f1, · · · , fm−1], has degree strictly smaller
than maxi deg gi + i · dm if and only if
Ĝ :=
deg gi+i·dm=max
ḡif̄
m = 0
so if sm = +∞, which means such an annihilation cannot occur, then the minoration
in the Theorem is clear. Let’s assume now that f̄m does have a minimal polynomial
p(f̄m) = 0 with p = p(X) ∈ F [X ] where F is the field of fractions of gr(A) and X a
new indeterminate (whence sm := degX p). The following easy lemma constitutes
the very improvement with respect to [3]: it simplifies the proof a lot, makes it
A PARACHUTE FOR THE DEGREE OF A POLYNOMIAL IN ALGEBRAICALLY INDEPENDENT ONES3
more general and even stronger in the sense that one does not need the estimate
(1) anymore.
Lemma. Let G =
m be in A[fm] and
h(X) :=
deg gi+i·dm=max
i ∈ gr(A)[X ] ( hence Ĝ = h(f̄m)).
If degG < degfm G ·dm then Ĝ = 0 or, equivalently, h(X) ∈ (p(X)) := p(X) ·F [X ].
Moreover if h′(X) 6= 0, where h′ is the derivative of h, then ∂̂G
= h′(f̄m) and more
generally, while h(k) 6= 0, one has ∂̂
= h(k)(f̄m).
Proof. If degG < degfm G · dm then degG < maxi deg gi + i · dm and Ĝ = 0 as
already remarked above.
Assume that h′ 6= 0. One has
ḡif̄
m = h(f̄m) where I := {i| deg gi + i · dm ≥ deg gj + j · dm ∀j} and
iḡif̄
m where I
′ := {i| deg igi + (i− 1) · dm ≥ deg jgj + (j − 1) · dm ∀j}.
It remains to notice that I ′ = I ∩ N∗ when this intersection is not empty, which
occurs exactly when h′ 6= 0. �
Let now k be the maximal number such that h(X) ∈ (p(X)k). Clearly degfm G ≥
deg h ≥ k · deg p = ksm hence k ≤ ⌊
degfm G
⌋. One has h(k) /∈ (p(X)) hence, by the
Lemma,
≥ dm · degfm
= dm · (degfm G− k)
and, by property (2),
degG ≥ dm · (degfm G− k) + k · dm − k · ∇ = dm · degfm G− k · ∇.
A straightforward computation gives the following
Corollary 1. Define, ∀i = 1, · · · ,m, Ni = Ni(f1, · · · , fm) := sidi − ∇. Let G be
a polynomial in K[f1, · · · , fm] and, ∀i = 1, · · · ,m, let degfi G = qisi + ri be the
euclidean division of degfi G by si. Then the following minoration holds
degG ≥ qi ·Ni + ridi.
The special case m = 2 corresponds to the main result of [3] (where s1, s2 are
easy to compute):
Corollary 2. If m = 2, σi :=
gcd(d1,d2)
with (i, j) = (1, 2) and (2, 1) and N :=
σ1d1 −∇ = σ2d2 −∇ then the following minoration holds, for i = 1, 2,
degG ≥ qi ·N + ridi
where degfi G = qisi + ri.
Proof. Let us prove it for i = 2. By corollary 1 it suffices to prove that s2 ≥ σ2
1: s2
is the degree of the minimal polynomial of f2 over Frac( gr(K[f1])) = Frac(K[f̄1]) =
K(f̄1):
p(f̄2) = f̄
2 + ps2−1(f̄1)f̄
2 + · · ·+ p1(f̄1)f̄2 + p0(f̄1) = 0(3)
1Actually equality holds, as proved in [3] using Zaks Lemma, it is however possible to show it
easily and without this result.
4 S. VÉNÉREAU
hence ∃0 ≤ i 6= j ≤ s2 such that deg pi(f̄1)f̄
2 = deg pj(f̄1)f̄
2 . It follows that
i · d2 ≡ jd2 mod d1 whence d1 | (i − j)d2 and i − j ∈ Z
gcd(d1,d2)
which gives
s2 ≥ |i− j| ≥
gcd(d1,d2)
= σ2. �
Corollary 3. Let G be a polynomial in K[f1, · · · , fm] such that degG = 1. Then,
∀i = 1, · · · ,m, degfi G = 0 or di = 1 or Ni = sidi −∇ ≤ 1.
Proof. Otherwise, by corollary 1, degG = 1 ≥ qiNi + ridi ≥ min{Ni, di} ≥ 2, a
contradiction. �
Corollary 4. Assume m = n and K[f1, · · · , fn] = K[x1, · · · , xn] i.e. f1, · · · , fn
define an automorphism (well-known fact). Then ∀i = 1, · · · , n, di = 1 or sidi ≤
d1 + · · · + dn − n + 1. In particular, if dmax ≥ dj , ∀j, and dmax ≥ 2 (i.e. the
automorphism is not affine) then smax ≤ n− 1.
Proof. One has ∇ = ∇(f1, · · · , fn) = d1+ · · ·+dn−n−deg jx1,··· ,xn(f1, · · · , fn) =
d1+ · · ·+dn−n. Moreover ∀j = 1, · · · , n, there exists Gj ∈ K[f1, · · · , fn] such that
xj = Gj and, ∀i = 1, · · · , n, degfi Gj ≤ 1 for at least one j = 1, · · · , n otherwise
K[f1, · · · , fj−1, fj+1, · · · , fm] = K[x1, · · · , xn] which is impossible. Whence, by
corollary 3, di = 1 or sidi ≤ ∇ + 1 = d1 + · · · + dn − n + 1. With dmax one gets
smaxdmax ≤ d1 + · · · + dn − n + 1 ≤ ndmax − n + 1 ≤ ndmax − 1 (n ≥ 2) and it
follows that smax ≤ n− 1. �
Corollary 5 (Tameness Theorem in dimension two). Every automorphism of K[x1, x2]
is tame i.e. a product of affine and elementary ones. Recall that an automorphism
τ : K[x1, x2] → K[x1, x2] is called elementary when, up to exchanging x1 and x2,
τ(x1) = x1 + p(x2) and τ(x2) = x2 for some p(X) ∈ K[X ].
Proof. Let α : K[x1, x2] → K[x1, x2] be an automorphism defined by α(xi) = fi
for i = 1, 2. We prove the corollary by induction on d1 + d2 = deg f1 + deg f2.
If d1 + d2 = 2 then d1 = d2 = 1 and α is affine.
Assume d1 + d2 ≥ 3. Without loss of generality d1 ≤ d2 and d2 ≥ 2 whence,
by corollary 4, s2 = 1 and the relation (3) in the proof of corollary 2 becomes:
f̄2 = p(f̄1) where p(X) must be of the form p(X) = ps1X
s1 ∈ K[X ]. Taking the
elementary automorphism τ defined τ(x1) = x1 and τ(x2) = x2 − p(X) one has a
new pair f ′1 := ατ(x1) = α(x1) = f1 and f
2 := ατ(x2) = α(x2 − p(X)) = f2− p(f1)
with degrees d′1 = d1 and d
2 < d2 hence d
2 < d1+d2. By induction ατ is tame
and so is α. �
References
[1] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant and applications,
preprint.
[2] J.T. Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl.
221, 1995, 19–29. MR 96c:14014
[3] Shestakov, Ivan P. and Umirbaev, Ualbai U., Poisson brackets and two-generated subalgebras
of rings of polynomials, J. Amer. Math. Soc. 17(1), 2004, 181–196 (electronic).
Stéphane Vénéreau
Mathematisches Institut
Universität Basel
Rheinsprung 21, CH-4051 Basel
Switzerland
[email protected]
References
|
0704.1562 | Galaxy evolution in the infra-red: comparison of a hierarchical galaxy
formation model with SPITZER data | Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 8 November 2021 (MN LATEX style file v2.2)
Galaxy evolution in the infra-red: comparison of a hierarchical
galaxy formation model with SPITZER data
C. G. Lacey ⋆,1 C. M. Baugh,1 C.S. Frenk,1 L. Silva,2 G.L. Granato,3 and A. Bressan,3
1Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK
2INAF, Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy
3INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 2, I-35122 Padova, Italy.
8 November 2021
ABSTRACT
We present predictions for the evolution of the galaxy luminosity function, number counts and
redshift distributions in the IR based on the ΛCDM cosmological model. We use the combined
GALFORM semi-analytical galaxy formation model and GRASIL spectrophotometric code to
compute galaxy SEDs including the reprocessing of radiation by dust. The model, which is
the same as that in Baugh et al. (2005), assumes two different IMFs: a normal solar neigh-
bourhood IMF for quiescent star formation in disks, and a very top-heavy IMF in starbursts
triggered by galaxy mergers. We have shown previously that the top-heavy IMF seems to be
necessary to explain the number counts of faint sub-mm galaxies. We compare the model
with observational data from the Spitzer Space Telescope, with the model parameters fixed
at values chosen before Spitzer data became available. We find that the model matches the
observed evolution in the IR remarkably well over the whole range of wavelengths probed by
Spitzer. In particular, the Spitzer data show that there is strong evolution in the mid-IR galaxy
luminosity function over the redshift range z ∼ 0 − 2, and this is reproduced by our model
without requiring any adjustment of parameters. On the other hand, a model with a normal
IMF in starbursts predicts far too little evolution in the mid-IR luminosity function, and is
therefore excluded.
Key words: galaxies: evolution – galaxies: formation – galaxies: high-redshift – infrared:
galaxies – ISM: dust, extinction
1 INTRODUCTION
In recent years, the evolution of galaxies at mid- and far-infrared
wavelengths has been opened up for direct observational study by
infrared telescopes in space. Already in the 1980s, the IRAS satel-
lite surveyed the local universe in the IR, showing that much of
present-day star formation is optically obscured, revealing a pop-
ulation of luminous and ultra-luminous infrared galaxies (LIRGs
with total IR luminosities LIR ∼ 10
− 1012L⊙ and ULIRGs
with LIR & 10
12L⊙), and providing the first hints of strong evo-
lution in the number density of ULIRGs at recent cosmic epochs
(e.g. Wright et al. 1984; Soifer et al. 1987a; Sanders & Mirabel
1996). The next major advance came with the discovery by COBE
of the cosmic far-IR background which has an energy density
comparable to that in the optical/near-IR background (Puget et al.
1996; Hauser et al. 1998). This implies that, over the history of the
universe, as much energy has been emitted by dust in galaxies as
reaches us directly in starlight, after dust extinction is taken into
account. This discovery made apparent the need to understand the
IR as much as the optical emission from galaxies in order to have
⋆ E-mail: [email protected] (CGL)
a complete picture of galaxy evolution. In particular, it is essen-
tial to understand IR emission from dust in order to understand the
cosmic history of star formation, since most of the radiation from
young stars must have been absorbed by dust over the history of
the universe, in order to account for the far-IR background (e.g.
Hauser et al. 1998).
Following these early discoveries, the ISO satellite enabled the
first deep surveys of galaxies in the mid- and far-IR. The deep-
est of these surveys were in the mid-IR at 15µm, and probed the
evolution of LIRGs and ULIRGs out to z ∼ 1, showing strong
evolution in these populations, and directly resolving most of the
cosmic infrared background at that wavelength (Elbaz et al. 1999,
2002; Gruppioni et al. 2002). Deep ISO surveys in the far-IR at
170µm (Dole et al. 2001; Patris et al. 2003) probed lower red-
shifts, z ∼ 0.5. Around the same time, sub-mm observations using
the SCUBA instrument on the JCMT revealed a huge population of
high-z ULIRGs (Smail, Ivison & Blain 1997; Hughes et al. 1998)
which were subsequently found to have a redshift distribution peak-
ing at z ∼ 2 (Chapman et al. 2005), confirming the dramatic evo-
lution in number density for this population seen at shorter wave-
lengths and lower redshifts. The sub-mm galaxies have been stud-
c© 0000 RAS
http://arxiv.org/abs/0704.1562v2
2 Lacey et al.
ied in more detail in subsequent SCUBA surveys (e.g. SHADES,
Mortier et al. 2005).
Now observations using the Spitzer satellite (Werner et al.
2004), with its hugely increased sensitivity and mapping speed
are revolutionizing our knowledge of galaxy evolution at IR wave-
lengths from 3.6 to 160 µm. Spitzer surveys have allowed direct
determinations of the evolution of the galaxy luminosity func-
tion out to z ∼ 1 in the rest-frame near-IR and to z ∼ 2 in
the mid-IR (Le Floc’h et al. 2005; Perez-Gonzalez et al. 2005;
Babbedge et al. 2006; Franceschini et al. 2006). Individual galax-
ies have been detected by Spitzer out to z ∼ 6 (Eyles et al. 2005).
In the near future, the Herschel satellite (Pilbratt 2003) should
make it possible to measure the far-IR luminosity function out to
z ∼ 2, and thus directly measure the total IR luminosities of galax-
ies over most of the history of the universe.
Accompanying these observational advances, various types of
theoretical models have been developed to interpret or explain the
observational data on galaxy evolution in the IR. We can distinguish
three main classes of model:
(a) Purely phenomenological models: In these models,
the galaxy luminosity function and its evolution are described
by a purely empirical expression, and this is combined with
observationally-based templates for the IR spectral energy dis-
tribution (SED). The free parameters in the expression for the
luminosity function are then chosen to obtain the best match
to some set of observational data, such as number counts
and redshift distributions in different IR bands. These pa-
rameters are purely descriptive and provide little insight into
the physical processes which control galaxy evolution. Exam-
ples of these models are Pearson & Rowan-Robinson (1996);
Xu et al. (1998); Blain et al. (1999); Franceschini et al. (2001);
Chary & Elbaz (2001); Rowan-Robinson (2001); Lagache et al.
(2003); Gruppioni et al. (2005).
(b) Hierarchical galaxy formation models with phenomeno-
logical SEDs: In these models, the evolution of the luminosity func-
tions of the stellar and total dust emission are calculated from a
detailed model of galaxy formation based on the cold dark mat-
ter (CDM) model of structure formation, including physical mod-
elling of processes such as gas cooling and galaxy mergers. The
stellar luminosity of a model galaxy is computed from its star
formation history, and the stellar luminosity absorbed by dust,
which equals the total IR luminosity emitted by dust, is calculated
from this based on some treatment of dust extinction. However,
the SED shapes required to calculate the distribution of the dust
emission over wavelength from the total IR dust emission are ei-
ther observationally-based templates (e.g. Guiderdoni et al. 1998;
Devriendt & Guiderdoni 2000) or are purely phenomenological,
e.g. a modified Planck function with an empirically chosen dust
temperature (e.g. Kaviani et al. 2003). In this approach, the shape
of the IR SED assumed for a model galaxy may be incompatible
with its other predicted properties, such as its dust mass and radius.
(c) Hierarchical galaxy formation models with theoretical
SEDs: These models are similar to those of type (b), in that the
evolution of the galaxy population is calculated from a detailed
physical model of galaxy formation based on CDM, but instead
of using phenomenological SEDs for the dust emission, the com-
plete SED of each model galaxy, from the far-UV to the radio, is
calculated by combining a theoretical stellar population synthesis
model for the stellar emission with a theoretical radiative transfer
and dust heating model to predict both the extinction of starlight
by dust and the IR/sub-mm SED of the dust emission. The ad-
vantages of this type of model are that it is completely ab initio,
with the maximum possible theoretical self-consistency, and all of
the model parameters relate directly to physical processes. For ex-
ample, the typical dust temperature and the shape of the SED of
dust emission depend on the stellar luminosity and the dust mass,
and evolution in all of these quantities is computed self-consistently
in this type of model. Following this modelling approach thus al-
lows more rigorous testing of the predictions of physical models
for galaxy formation against observational data at IR wavelengths,
as well as shrinking the parameter space of the predictions. Ex-
amples of such models are Granato et al. (2000) and Baugh et al.
(2005). (An alternative modelling approach also based on theoret-
ical IR SEDs but with a simplified treatment of the assembly of
galaxies and halos has been presented by Granato et al. (2004) and
Silva et al. (2005).)
In this paper, we follow the third approach, with physical mod-
elling both of galaxy formation and of the galaxy SEDs, includ-
ing the effects of dust. This paper is the third in a series, where
we combine the GALFORM semi-analytical model of galaxy forma-
tion (Cole et al. 2000) with the GRASILmodel for stellar and dust
emission from galaxies (Silva et al. 1998). The GALFORM model
computes the evolution of galaxies in the framework of the ΛCDM
model for structure formation, based on physical treatments of the
assembly of dark matter halos by merging, gas cooling in halos,
star formation and supernova feedback, galaxy mergers, and chem-
ical enrichment. The GRASILmodel computes the SED of a model
galaxy from the far-UV to the radio, based on theoretical models of
stellar evolution and stellar atmospheres, radiative transfer through
a two-phase dust medium to calculate both the dust extinction and
dust heating, and a distribution of dust temperatures in each galaxy
calculated from a detailed dust grain model. In the first paper in
the series (Granato et al. 2000), we modelled the IR properties of
galaxies in the local universe. While this model was very success-
ful in explaining observations of the local universe, we found sub-
sequently that it failed when confronted with observations of star-
forming galaxies at high redshifts, predicting far too few sub-mm
galaxies (SMGs) at z ∼ 2 and Lyman-break galaxies (LBGs) at
z ∼ 3. Therefore, in the second paper (Baugh et al. 2005), we
proposed a new version of the model which assumes a top-heavy
IMF in starbursts (with slope x = 0, compared to Salpeter slope
x = 1.35), but a normal solar neighbourhood IMF for quiescent
star formation. In this new model, the star formation parameters
were also changed to force more star formation to happen in bursts.
This revised model agreed well with both the number counts and
redshift distributions of SMGs detected at 850µm, and with the
rest-frame far-UV luminosity function of LBGs at z ∼ 3, while
still maintaining consistency with galaxy properties in the local uni-
verse such as the optical, near-IR and far-IR luminosity functions,
and gas fractions, metallicities, morphologies and sizes.
This same model of Baugh et al. (2005) was found by
Le Delliou et al. (2005a, 2006) to provide a good match to the ob-
served evolution of the population of Lyα-emitting galaxies over
the redshift range z ∼ 3−6. Support for the controversial assump-
tion of a top-heavy IMF in bursts came from the studies of chem-
ical enrichment in this model by Nagashima et al. (2005a,b), who
found that the metallicities of both the intergalactic gas in galaxy
clusters and the stars in elliptical galaxies were predicted to be sig-
nificantly lower than observed values if a normal IMF was assumed
for all star formation, but agreed much better if a top-heavy IMF in
bursts was assumed, as in Baugh et al. . In this third paper in the
series, we extend the Baugh et al. (2005) model to make predic-
tions for galaxy evolution in the IR, and compare these predictions
with observational data from Spitzer. We emphasize that all of the
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 3
model parameters for the predictions presented in this paper were
fixed by Baugh et al. prior to the publication of any results from
Spitzer, and we have not tried to obtain a better fit to any of the
Spitzer data by adjusting these parameters1.
Our goals in this paper are to test our model of galaxy evo-
lution with a top-heavy IMF in starbursts against observations of
dust-obscured star-forming galaxies over the redshift range z ∼
0−2, and also to test our predictions for the evolution of the stellar
populations of galaxies against observational data in the rest-frame
near- and mid-IR. The plan of the paper is as follows: In Section 2,
we give an overview of the GALFORM and GRASIL models, fo-
cusing on how the predictions we present later on are calculated.
In Section 3, we compare the galaxy number counts predicted by
our model with observational data in all 7 Spitzer bands, from 3.6
to 160 µm. In Section 4, we investigate galaxy evolution in the
IR in more detail, by comparing model predictions directly with
galaxy luminosity functions constructed from Spitzer data. In Sec-
tion 5, we present the predictions of our model for the evolution
of the galaxy stellar mass function and star formation rate distribu-
tion, and investigate the insight our model offers on how well stel-
lar masses and star formation rates can be estimated from Spitzer
data. We present our conclusions in Section 6. In the Appendix, we
present model predictions for galaxy redshift distributions in the
different Spitzer bands, to assist in interpreting data from different
surveys.
2 MODEL
In this paper use the GALFORM semi-analytical model to predict
the physical properties of the galaxy population at different red-
shifts, and combine it with the GRASIL spectrophotometric model
to predict the detailed SEDs of model galaxies. Both GALFORM
and GRASIL have been described in detail in various previous
papers, but since the descriptions of the different model compo-
nents, as well as of our particular choice of parameters, are spread
among different papers, we give an overview of both of these here.
GALFORM is described in §2.1, and GRASIL in §2.2. Particularly
important features of our model are the triggering of starbursts by
mergers (discussed in §2.1.4) and the assumption of a top-heavy
IMF in starbursts (discussed in §2.1.7). We further discuss the
choice of model parameters in §2.3. Readers who are already fa-
miliar with the Baugh et al. (2005) model can skip straight to the
results, starting in §3.
2.1 GALFORM galaxy formation model
We compute the formation and evolution of galaxies within the
framework of the ΛCDM model of structure formation using the
semi-analytical galaxy formation model GALFORM. The general
methodology and approximations behind the GALFORM model are
set out in detail in Cole et al. (2000) (see also the review by Baugh
(2006)). In summary, the GALFORM model follows the main pro-
cesses which shape the formation and evolution of galaxies. These
include: (i) the collapse and merging of dark matter halos; (ii) the
1 A closely related model of galaxy formation obtained by applying
GALFORM principles to the Millennium simulation of Springel et al. (2005)
has recently been published by Bower et al. (2006). This model differs
from the current one primarily in that it includes feedback from AGN activ-
ity, but does not have a top-heavy IMF in bursts. We plan to investigate the
IR predictions of this alternative model in a subsequent paper.
shock-heating and radiative cooling of gas inside dark halos, lead-
ing to the formation of galaxy disks; (iii) quiescent star formation
in galaxy disks; (iv) feedback both from supernova explosions and
from photo-ionization of the IGM; (v) chemical enrichment of the
stars and gas; (vi) galaxy mergers driven by dynamical friction
within common dark matter halos, leading to the formation of stel-
lar spheroids, and also triggering bursts of star formation. The end
product of the calculations is a prediction of the numbers and prop-
erties of galaxies that reside within dark matter haloes of different
masses. The model predicts the stellar and cold gas masses of the
galaxies, along with their star formation and merger histories, their
sizes and metallicities.
The prescriptions and parameters for the different processes
which we use in this paper are identical to those adopted by
Baugh et al. (2005), but differ in several important respects from
Cole et al. (2000). All of these parameters were chosen by com-
parison with pre-Spitzer observational data. The background cos-
mology is a spatially flat CDM universe with a cosmological con-
stant, with “concordance” parameters Ωm = 0.3, ΩΛ = 0.7,
Ωb = 0.04, and h ≡ H0/(100km s
−1Mpc−1) = 0.7. The am-
plitude of the initial spectrum of density fluctuations is set by the
r.m.s. linear fluctuation in a sphere of radius 8h−1Mpc, σ8 = 0.93.
For completeness, we now summarize the prescriptions and param-
eters used, but give details mainly where they differ from those in
Cole et al. (2000), or where they are particularly relevant to pre-
dicting IR emission from dust.
2.1.1 Halo assembly histories
As in Cole et al. (2000), we describe the assembly histories of dark
matter halos through halo merger trees which are calculated using
a Monte Carlo method based on the extended Press-Schechter ap-
proach (e.g. Lacey & Cole 1993). The process of galaxy forma-
tion is then calculated separately for each halo merger tree, follow-
ing the baryonic physics in all of the separate branches of the tree.
As has been shown by Helly et al. (2003), the statistical properties
of galaxies calculated in semi-analytical models using these Monte
Carlo merger trees are very similar to those computed using merger
trees extracted directly from N-body simulations.
2.1.2 Gas cooling in halos
The cooling of gas in halos is calculated using the same simple
spherical model as in Cole et al. (2000). The diffuse gas in halos
(consisting of all of the gas which has not previously condensed
into galaxies) is assumed to be shock-heated to the halo virial tem-
perature when the halo is assembled, and then to cool radiatively by
atomic processes. The cooling time depends on radius through the
gas density profile, which is assumed to have a core radius which
grows as gas is removed from the diffuse phase by condensing into
galaxies. The gas at some radius r in the halo then cools and col-
lapses to the halo centre on a timescale which is the larger of the
cooling time tcool and the free-fall time tff at that radius. Thus, for
tcool(r) > tff(r), we have hot accretion, and for tcool(r) < tff(r),
we have cold accretion 2. In our model, gas only accretes onto the
central galaxy in a halo, not onto any satellite galaxies which share
2 Note that contrary to claims by Birnboim & Dekel (2003), the process
of “cold accretion”, if not the name, has always been part of semi-analytical
models (see Croton et al. (2006) for a detailed discussion)
c© 0000 RAS, MNRAS 000, 000–000
4 Lacey et al.
that halo. We denote all of the diffuse gas in halos as “hot”, and all
of the gas which has condensed into galaxies as “cold”.
2.1.3 Star formation timescale in disks
The global rate of star formation ψ in galaxy disks is assumed to be
related to the cold gas mass, Mgas, by ψ = Mgas/τ∗,disk, where
the star formation timescale is taken to be
τ∗,disk = τ∗0
Vc/200 km s
, (1)
where Vc is the circular velocity of the galaxy disk (at its half-
mass radius) and τ∗0 is a constant. We adopt values τ∗0 = 8Gyr
and α∗ = −3, chosen to reproduce the observed relation between
gas mass and B-band luminosity for present-day disk galaxies. As
discussed in Baugh et al. (2005), this assumption means that the
disk star formation timescale is independent of redshift (at a given
Vc), resulting in disks at high redshift that are much more gas-rich
than at low redshift, and have more gas available for star formation
in bursts triggered by galaxy mergers at high redshift.
2.1.4 Galaxy mergers and triggering of starbursts
In the model, all galaxies originate as central galaxies in some halo,
but can then become satellite galaxies if their host halo merges into
another halo. Mergers can then occur between satellite and central
galaxies within the same halo, after dynamical friction has caused
the satellite galaxy to sink to the centre of the halo. Galaxy mergers
can produce changes in galaxy morphology and trigger bursts. We
classify galaxy mergers according to the ratio of masses (including
stars and gas) M2/M1 6 1 of the secondary to primary galaxy
involved. We define mergers to be major or minor according to
whether M2/M1 > fellip or M2/M1 < fellip (Kauffmann et al.
1993). In major mergers, any stellar disks in either the primary or
secondary are assumed to be disrupted, and the stars rearranged into
a spheroid. In minor mergers, the stellar disk in the primary galaxy
is assumed to remain intact, while all of the stars in the secondary
are assumed to be added to the spheroid of the primary. We adopt a
threshold fellip = 0.3 for major mergers, consistent with the results
of numerical simulations (e.g. Barnes 1998), which reproduces the
observed present-day fraction of spheroidal galaxies. We assume
that major mergers always trigger a starburst if any gas is present.
We also assume that minor mergers can trigger bursts, if they sat-
isfy both M2/M1 > fburst and the gas fraction in the disk of the
primary galaxy exceeds fgas,crit. Following Baugh et al. (2005),
we adopt fburst = 0.05 and fgas,crit = 0.75. The parameters for
bursts in minor mergers were motivated by trying to explain the
number of sub-mm galaxies. An important consequence of assum-
ing eqn.(1) for the star formation timescale in disks, combined with
the triggering of starbursts in minor mergers, is that the global star
formation rate at high redshifts is dominated by bursts, while that at
low redshifts it is dominated by quiescent disks (see Baugh et al.
for a detailed discussion of these points).
In either kind of starburst, we assume that the burst consumes
all of the cold gas in the two galaxies involved in the merger, and
that the stars produced are added to the spheroid of the merger rem-
nant. During the burst, we assume that star formation proceeds ac-
cording to the relation ψ =Mgas/τ∗,burst. For the burst timescale,
we assume
τ∗,burst = max [fdynτdyn,sph; τ∗,burst,min] , (2)
where τdyn,sph is the dynamical time in the newly-formed spheroid.
We adopt fdyn = 50 and τ∗,burst,min = 0.2Gyr (these parame-
ters were chosen by Baugh et al. (2005) to allow a simultaneous
match to the sub-mm number counts and to the local 60µm lu-
minosity function). The star formation rate in a burst thus decays
exponentially with time after the galaxy merger. It is assumed to be
truncated after 3 e-folding times (where the e-folding time takes ac-
count of stellar recycling and feedback - see Granato et al. (2000)
for details), with the remaining gas being ejected into the galaxy
halo at that time.
2.1.5 Feedback from photo-ionization
After the intergalactic medium (IGM) has been reionized at redshift
zreion, the formation of low-mass galaxies is inhibited, both by the
effect of the IGM pressure inhibiting collapse of gas into halos, and
by the reduction of gas cooling in halos due to the photo-ionizing
background. We model this in a simple way, by assuming that for
z < zreion, cooling of gas is completely suppressed in halos with
circular velocities Vc < Vcrit. We adopt Vcrit = 60 kms
−1, based
on the detailed modelling by Benson et al. (2002). We assume in
this paper that reionization occurs at zreion = 6, for consistency
with Baugh et al. (2005), but increasing this to zreion ∼ 10 in
line with the WMAP 3-year estimate of the polarization of the mi-
crowave background (Spergel et al. 2006) has no significant effect
on the model results presented in this paper.
2.1.6 Feedback from supernovae
Photo-ionization feedback only affects very low mass galaxies.
More important for most galaxies is feedback from supernova ex-
plosions. We assume that energy input from supernovae causes gas
to be ejected from galaxies at a rate
Ṁej = β(Vc)ψ = [βreh(Vc) + βsw(Vc)] ψ (3)
The supernova feedback is assumed to operate for both quiescent
star formation in disks and for starbursts triggered by galaxy merg-
ers, with the only difference being that we take Vc to be the circular
velocity at the half mass radius of the disk in the former case, and
at the half-mass radius of the spheroid in the latter case. For sim-
plicity, we keep the same feedback parameters for starbursts as for
quiescent star formation.
The supernova feedback has two components: the reheating
term βrehψ describes gas which is reheated and ejected into the
galaxy halo, from where it is allowed to cool again after the halo
mass has doubled through hierachical mass build-up. For this, we
use the parametrization of Cole et al. (2000):
βreh = (Vc/Vhot)
−αhot , (4)
where we adopt parameter values Vhot = 300 kms
−1 and αhot =
2. The reheating term has the largest effect on low-mass galaxies,
for which ejection of gas from galaxies flattens the faint-end slope
of the galaxy luminosity function.
The second term βswψ in eqn.(3) is the superwind term, which
describes ejection of gas out of the halo rather than just the galaxy.
Once ejected, this gas is assumed never to re-accrete onto any halo.
We model the superwind ejection efficiency as
βsw = fsw min
1, (Vc/Vsw)
based on Benson et al. (2003). We adopt parameter values fsw = 2
and Vsw = 200 km s
−1, as in Baugh et al. (2005). The superwind
term mainly affects higher mass galaxies, where the ejection of gas
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 5
from halos causes an increase in the cooling time of gas in halos
by reducing the gas densities. This brings the predicted break at
the bright end of the local galaxy luminosity function into agree-
ment with observations, as discussed in Benson et al. (2003). The
various parameters for supernova feedback are thus chosen in or-
der to match the observed present-day optical and near-IR galaxy
luminosity functions, as well as the galaxy metallicity-luminosity
relation.
We note that the galaxy formation model in this paper, un-
like some other recent semi-analytical models, does not include
AGN feedback. Instead, the role of AGN feedback in reducing the
amount of gas cooling to form massive galaxies is taken by su-
perwinds driven by supernova explosions. The first semi-analytical
model to include AGN feedback was that of Granato et al. (2004),
who introduced a detailed model of feedback from QSO winds
during the formation phase of supermassive black holes (SMBHs),
with the aim of explaining the co-evolution of the spheroidal com-
ponents of galaxies and their SMBHs. The predictions of the
Granato et al. model for number counts and redshift distributions
in the IR have been computed by Silva et al. (2005) using the
GRASIL spectrophotometric model, and compared to ISO and
Spitzer data. However, the Granato et al. (2004) model has the
limitations that it does not include the merging of galaxies or of
dark halos, and does not treat the formation and evolution of galac-
tic disks. More recently, several semi-analytical models have been
published which propose that heating of halo gas by relativistic
jets from an AGN in an optically inconspicuous or “radio” mode
can balance radiative cooling of gas in high-mass halos, thus sup-
pressing hot accretion of gas onto galaxies (Bower et al. 2006;
Croton et al. 2006; Cattaneo et al. 2006; Monaco et al. 2007).
However, these AGN feedback models differ in detail, and all are
fairly schematic. None of these models has been shown to repro-
duce the observed number counts and redshifts of the faint sub-mm
galaxies.
The effects of our superwind feedback are qualitatively quite
similar to those of the radio-mode AGN feedback. Both superwind
and AGN feedback models contain free parameters, which are ad-
justed in order to make the model fit the bright end of the ob-
served present-day galaxy luminosity function at optical and near-
IR wavelengths. However, since the physical mechanisms are dif-
ferent, they make different predictions for how the galaxy lumi-
nosity function should evolve with redshift. Current models for the
radio-mode AGN feedback are very schematic, but they have the
advantage over the superwind model that the energetic constraints
are greatly relaxed, since accretion onto black holes can convert
mass into energy with a much higher efficiency than can supernova
explosions. We will investigate the predictions of models with AGN
feedback for the IR and sub-mm evolution of galaxies in a future
paper.
2.1.7 The Stellar Initial Mass Function and Chemical Evolution
Stars in our model are assumed to form with different Initial Mass
Functions (IMFs), depending on whether they form in disks or in
bursts. Both IMFs are taken to be piecewise power laws, with slopes
x defined by dN/d lnm ∝ m−x, with N the number of stars and
m the stellar mass (so the Salpeter slope is x = 1.35), and covering
a stellar mass range 0.15 < m < 120M⊙. Quiescent star forma-
tion in galaxy disks is assumed to have a solar neighbourhood IMF,
for which we use the Kennicutt (1983) paramerization, with slope
x = 0.4 for m < M⊙ and x = 1.5 for m > M⊙. (The Kennicutt
(1983) IMF is similar to other popular parametrizations of the solar
neighbourhood IMF, such as that of Kroupa (2001).) Bursts of star
formation triggered by galaxy mergers are assumed to form stars
with a top-heavy IMF with slope x = 0. As discussed in detail
in Baugh et al. (2005), the top-heavy IMF in bursts was found to
be required in order to reproduce the observed number counts and
redshift distributions of the faint sub-mm galaxies. Furthermore,
as shown by Nagashima et al. (2005a,b), the predicted chemical
abundances of the X-ray emitting gas in galaxy clusters and of the
stars in elliptical galaxies also agree better with observational data
in a model with the top-heavy IMF in bursts, rather than a universal
solar neighbourhood IMF.
A variety of other observational evidence has accumulated
which suggests that the IMF in some environments may be top-
heavy compared to the solar neighbourhood IMF. Rieke et al.
(1993) argued for a top-heavy IMF in the nearby starburst M82,
based on modelling its integrated properties, while Parra et al.
(2007) found possible evidence for a top-heavy IMF in the ultra-
luminous starburst Arp220 from the relative numbers of super-
novae of different types observed at radio wavelengths. Evidence
has been found for a top-heavy IMF in some star clusters in in-
tensely star-forming regions, both in M82 (e.g. McCrady et al.
2003), and in our own Galaxy (e.g. Figer et al. 1999; Stolte et al.
2005; Harayama et al. 2007). Observations of both the old and
young stellar populations in the central 1 pc of our Galaxy
also favour a top-heavy IMF (Paumard et al. 2006; Maness et al.
2007). Fardal et al. (2006) found that reconciling measurements of
the optical and IR extragalactic background with measurements of
the cosmic star formation history also seemed to require an average
IMF that was somewhat top-heavy. Finally, van Dokkum (2007)
found that reconciling the colour and luminosity evolution of early-
type galaxies in clusters also favoured a top-heavy IMF. Larson
(1998) summarized other evidence for a top-heavy IMF during the
earlier phases of galaxy evolution, and argued that this could be
a natural consequence of the temperature-dependence of the Jeans
mass for gravitational instability in gas clouds. Larson (2005) ex-
tended this to argue that a top-heavy IMF might also be expected
in starburst regions, where there is strong heating of the dust by the
young stars.
In our model, the fraction of star formation occuring in the
burst mode increases with redshift (see Baugh et al. (2005)), so the
average IMF with which stars are being formed shifts from being
close to a solar neighbourhood IMF at the present-day to being very
top-heavy at high redshift. In this model, 30% of star formation
occured in the burst mode when integrated over the past history of
the universe, but only 7% of the current stellar mass was formed
in bursts, because of the much larger fraction of mass recycled by
dying stars for the top-heavy IMF. We note that our predictions for
the IR and sub-mm luminosities of starbursts are not sensitive to
the precise form of the top-heavy IMF, but simply require a larger
fraction of m ∼ 5− 20M⊙ stars relative to a solar neighbourhood
In this paper, we calculate chemical evolution using the instan-
taneous recycling approximation, which depends on the total frac-
tion of mass recycled from dying stars (R), and the total yield of
heavy elements (p). Both of these parameters depend on the IMF.
We use the results of stellar evolution computations to calculate
values of R and p consistent with each IMF (see Nagashima et al.
(2005a) for details of the stellar evolution data used). Thus, we use
R = 0.41 and p = 0.023 for the quiescent IMF, and R = 0.91 and
p = 0.15 for the burst IMF. Our chemical evolution model then
predicts the masses and total metallicities of the gas and stars in
each galaxy as a function of time.
c© 0000 RAS, MNRAS 000, 000–000
6 Lacey et al.
2.1.8 Galaxy sizes and dust masses
For calculating the extinction and emission by dust, it is essential to
have an accurate calculation of the dust optical depths in the model
galaxies, which in turn depends on the mass of dust and the size
of the galaxy. The dust mass is calculated from the gas mass and
metallicity predicted by the chemical enrichment model, assuming
that the dust-to-gas ratio is proportional to metallicity, normalized
to match the local ISM value at solar metallicity. The sizes of galax-
ies are computed exactly as in Cole et al. (2000): gas which cools
in a halo is assumed to conserve its angular momentum as it col-
lapses, forming a rotationally-supported galaxy disk; the radius of
this disk is then calculated from its angular momentum, includ-
ing the gravity of the disk, spheroid (if any) and dark halo. Galaxy
spheroids are built up both from pre-existing stars in galaxy merg-
ers, and from the stars formed in bursts triggered by these mergers;
the radii of spheroids formed in mergers are computed using an
energy conservation argument. In calculating the sizes of disks and
spheroids, we include the adiabatic contraction of the dark halo due
to the gravity of the baryonic components. This model was tested
for disks by Cole et al. (2000) and for spheroids by Almeida et al.
(2007) (see also Coenda et al. in preparation, and Gonzalez et al.
in preparation). During a burst, we assume that the gas and stars
involved in the burst have a distribution with the same half-mass
radius as the spheroid (i.e. η = 1 in the notation of Granato et al.
(2000), who used a value η = 0.1).
2.2 GRASIL model for stellar and dust emission
For each galaxy in our model, we compute the spectral energy dis-
tribution using the spectrophotometric model GRASIL (Silva et al.
1998; Granato et al. 2000). GRASIL computes the emission from
the stellar population, the absorption and emission of radiation by
dust, and also radio emission (thermal and synchrotron) powered
by massive stars (Bressan et al. 2002).
2.2.1 SED model
The main features of the GRASIL model are as follows:
(i) The stars are assumed to have an axisymmetric distribution in a
disk and a bulge. Given the distribution of stars in age and metal-
licity (obtained from the star formation and chemical enrichment
history), the SED of the stellar population is calculated using a
population synthesis model based on the Padova stellar evolution
tracks and Kurucz model atmospheres (Bressan et al. 1998). This
is done separately for the disk and bulge.
(ii) The cold gas and dust in a galaxy are assumed to be in a 2-phase
medium, consisting of dense gas in giant molecular clouds embed-
ded in a lower-density diffuse component. In a quiescent galaxy,
the dust and gas are assumed to be confined to the disk, while for
a galaxy undergoing a burst, the dust and gas are confined to the
spheroidal burst component.
(iii) Stars are assumed to be born inside molecular clouds, and then
to leak out into the diffuse medium on a timescale tesc. As a result,
the youngest and most massive stars are concentrated in the dustiest
regions, so they experience larger dust extinctions than older, typ-
ically lower-mass stars, and dust in the clouds is also much more
strongly heated than dust in the diffuse medium.
(iv) The extinction of the starlight by dust is computed using a ra-
diative transfer code; this is used also to compute the intensity of
the stellar radiation field heating the dust at each point in a galaxy.
(v) The dust is modelled as a mixture of graphite and silicate grains
with a continuous distribution of grain sizes (varying between 8Å
and 0.25 µm), and also Polycyclic Aromatic Hydrocarbon (PAH)
molecules with a distribution of sizes. The equilibrium temperature
in the local interstellar radiation field is calculated for each type
and size of grain, at each point in the galaxy, and this information
is then used to calculate the emission from each grain. In the case of
very small grains and PAH molecules, temperature fluctuations are
important, and the probability distribution of the temperature is cal-
culated. The detailed spectrum of the PAH emission is obtained us-
ing the PAH cross-sections from Li & Draine (2001), as described
in Vega et al. (2005). The grain size distribution is chosen to match
the mean dust extinction curve and emissivity in the local ISM, and
is not varied, except that the PAH abundance in molecular clouds
is assumed to be 10−3 of that in the diffuse medium (Vega et al.
2005).
(vi) Radio emission from ionized gas in HII regions and from syn-
chrotron radiation from relativistic electrons accelerated in super-
nova remnant shocks are calculated as described in Bressan et al.
(2002).
The output from GRASIL is then the complete SED of a
galaxy from the far-UV to the radio (wavelengths 100Å . λ .
1m). The SED of the dust emission is computed as a sum over the
different types of grains, having different temperatures depending
on their size and their position in the galaxy. The dust SED is thus
intrinsically multi-temperature. GRASIL has been shown to give an
excellent match to the measured SEDs of both quiescent (e.g. M51)
and starburst (e.g. M82) galaxies (Silva et al. 1998; Bressan et al.
2002).
The assumption of axisymmetry in GRASIL is a limitation
when considering starbursts triggered by galaxy mergers. However,
observations of local ULIRGs imply that most of the star formation
happens in a single burst component after the galaxy merger is sub-
stantially complete, so the assumption of axisymmetry for the burst
component may not be so bad.
2.2.2 GRASIL parameters
The main parameters in the GRASIL dust model are the fraction
fmc of the cold gas which is in molecular clouds, the timescale tesc
for newly-formed stars to escape from their parent molecular cloud,
and the cloud masses Mc and radii rc in the combination Mc/r
which determines the dust optical depth of the clouds. We assume
fmc = 0.25, Mc = 10
6M⊙ and rc = 16pc as in Granato et al.
(2000), and also adopt the same geometrical parameters as in that
paper. We make the following two changes in GRASIL parame-
ters relative to Granato et al. , as discussed in Baugh et al. (2005):
(a) We assume tesc = 1Myr in both disks and bursts (instead
of the Granato et al. values tesc = 2 and 10Myr respectively).
This value was chosen in order to obtain a better match of the pre-
dicted rest-frame far-UV luminosity function of galaxies at z ∼ 3
to that measured for Lyman-break galaxies. (b) The dust emissivity
law in bursts at long wavelengths is modified from ǫν ∝ ν
−2 to
ǫν ∝ ν
−1.5 for λ > 100µm. This was done in order to improve
slightly the fit of the model to the observed sub-mm number counts.
In applying GRASIL to model the SEDs of a sample of nearby
galaxies, Silva et al. (1998) found that a similar modification (to
ǫν ∝ ν
−1.6) seemed to be required in the case of Arp220 (the only
ultra-luminous starburst in their sample), in order to reproduce the
observed sub-mm data for that galaxy. This modification in fact has
little effect on the IR predictions presented in the present paper, but
we retain it for consistency with Baugh et al. (2005).
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 7
2.2.3 Interface with GALFORM
For calculating the statistical properties of the galaxy population
from the combined GALFORM+GRASIL model, we follow the
same strategy as described in Granato et al. (2000). We first run
the GALFORM code to generate a large catalogue of model galaxies
at any redshift, and then run the GRASIL code on subsamples of
these. For the quiescent galaxies, we select a subsample which has
equal numbers of galaxies in equal logarithmic bins of stellar mass,
while for the bursting galaxies, we select a subsample with equal
numbers of galaxies in equal logarithmic bins of burst mass. For the
burst sample, we compute SEDs at several different representative
stages in the burst evolution, while for the quiescent sample, we
only compute SEDs at a single epoch. Using this sampling strat-
egy, we obtain a good coverage of all the different masses, types
and evolutionary stages of galaxies, while minimizing the compu-
tational cost of running the GRASIL code. The statistical properties
of the galaxy population are then obtained by assigning the model
galaxies appropriate weights depending on their predicted number
density in a representative cosmological volume.
The outputs from the GALFORM galaxy formation model re-
quired by GRASIL to calculate the galaxy SEDs are: the combined
star formation history and metallicity distribution for the disk and
bulge, the radii of both components, and the total mass of dust. The
dust mass is calculated from the mass and metallicity of the cold
gas in the galaxy, assuming that the dust-to-gas ratio is proportional
to the metallicity. Since the gas mass and metallicity both evolve,
so does the dust mass, and this evolution is fully taken into account
in GRASIL. For simplicity, we assume that the size distribution of
the dust grains and PAH molecules does not evolve, apart from the
normalization.
Once we have calculated the SEDs for the model galaxies, we
compute luminosities in different observed bands (e.g. the optical
B-band or the Spitzer 24µm band) by convolving the SED with the
filter+detector response function for that band. For computing the
predicted fluxes from galaxies in a fixed observer-frame band, we
redshift the SED before doing the convolution.
The GRASIL code is quite CPU-intensive, requiring several
minutes of CPU time per galaxy. Consequently, we are limited to
running samples of a few thousand galaxies at each redshift. As a
result, quantities such as luminosity functions and redshift distri-
butions still show some small amount of noise, rather than being
completely smooth curves, as can be seen in many of the figures in
this paper.
2.3 Choice of parameters in the GALFORM+GRASIL model
The combined GALFORM+GRASIL model has a significant num-
ber of parameters, but this is inevitable given the very wide range
of physical processes which are included. The parameters are con-
strained by requiring the model predictions to reproduce a limited
set of observational data - once this is done, there is rather little
freedom in the choice of parameters. We have described above how
the main parameters are fixed, and more details can be found in
Cole et al. (2000) and Baugh et al. (2005). For both of these pa-
pers, large grids of GALFORM models were run with different pa-
rameters, in order to decide which set of parameters gave the best
overall fit to the set of calibrating observational data. These papers
also show the effects of varying some of the main model parameters
around their best-fit values. The parameters in the standard model
for which we present results in this paper were chosen to reproduce
the following properties for present-day galaxies: the luminosity
functions in the B- and K-bands and at 60µm, the relations between
gas mass and luminosity and metallicity and luminosity, the size-
luminosity relation for galaxy disks, and the fraction of spheroidal
galaxies. In addition, the model was required to reproduce the ob-
served rest-frame far-UV (1500Å) luminosity function at z = 3,
and the observed sub-mm number counts and redshift distribution
at 850µm (Baugh et al. 2005). The sub-mm number counts are the
main factor driving the need to include a top-heavy IMF in bursts.
The parameters for our standard model are exactly the same as
in Baugh et al. (2005), which were chosen before Spitzer data be-
came available. Since these parameters were not adjusted to match
any data obtained with Spitzer, the predictions of our model in the
Spitzer bands are genuine predictions. We could obviously have
fine-tuned our parameters in order to match better the observational
data we considered in this paper, but this would have conflicted
with our main goal, which is to present predictions for a wide set of
observable properties based on a single physical model in a series
of papers.
Since our assumption of a top-heavy IMF in bursts is a con-
troversial one, we will also show some predictions from a variant
model, which is identical to the standard model, except that we
assume the same solar neighbourhood (Kennicutt) IMF in bursts
and in disks. Comparing the predictions for the standard and vari-
ant models then shows directly the effects of changing the IMF
in bursts. We note that the variant model matches the present-day
optical and near-IR luminosity functions almost as well as the stan-
dard model, though it is a poorer fit to the local 60µm luminosity
function for the brightest galaxies (see Fig. 9). The variant model
underpredicts the 850µm counts by a factor of 10–30.
3 NUMBER COUNTS
We begin our comparison of the predictions of our galaxy forma-
tion model against Spitzer data with the galaxy number counts.
Fig. 1 shows number counts in the four IRAC bands (3.6, 4.5, 5.8
and 8.0 µm), and Fig. 2 does the same for the three MIPS bands
(24, 70 and 160 µm). Each panel is split in two: the upper sub-
panel plots the counts per logarithmic flux interval, dN/d lnSν ,
while the lower sub-panel instead plots SνdN/d lnSν . The latter
is designed to take out much of the trend with flux, in order to
show more clearly the differences between the model and the on-
servational data. In each case we plot three curves for our standard
model: the solid blue line shows the total number counts includ-
ing both extinction and emission by dust, the solid red line shows
the contribution to this from galaxies currently forming stars in a
burst, and the solid green line shows the contribution from all other
galaxies (star-forming or not), which we denote as “quiescent”. In
Fig. 1, we also plot a dashed blue line which shows the predicted
total counts if we ignore absorption and emission from interstellar
dust (emission from dust in the envelopes of AGB stars is still in-
cluded in the stellar contribution, however). In the MIPS bands, the
predicted counts are negligible in the absence of interstellar dust,
so we do not plot them in Fig. 2. In the lower sub-panels, we also
show by a dashed magenta line the prediction from a variant model
which assumes a normal (Kennicutt) IMF for all star formation, but
is otherwise identical to our standard model (which has a top-heavy
IMF in bursts). This variant model fits the local B- and K-band and
60 µm luminosity functions about as well as our standard model,
but dramatically underpredicts the 850 µm number counts. The ob-
served number counts are shown by black symbols with error bars.
Overall, the agreement between the predictions of our stan-
c© 0000 RAS, MNRAS 000, 000–000
8 Lacey et al.
Figure 1. Galaxy differential number counts in the four IRAC bands. The curves show model predictions, while the symbols with error bars show observational
data from Fazio et al. (2004) (with different symbols for data from different survey fields). Each panel is split in two: the upper sub-panel plots the counts
as dN/d lnSν vs Sν , while the lower sub-panel plots SνdN/d lnSν (in units mJy deg
−2) on the same horizontal scale. The upper sub-panels show four
different curves for our standard model - solid blue: total counts including dust extinction and emission; dashed blue: total counts excluding interstellar dust;
solid red: ongoing bursts (including dust); solid green: quiescent galaxies (including dust). The lower sub-panels compare the total counts including dust for
the standard model (solid blue line) with those for a variant model with a normal IMF for all stars (dashed magenta line). The vertical dashed line shows the
estimated confusion limit for the model. (a) 3.6 µm. (b) 4.5 µm. (c) 5.8 µm. (d) 8.0 µm.
dard model and the observed counts is remarkably good, when one
takes account of the fact that no parameters of the model were ad-
justed to improve the fit to any data from Spitzer. Consider first the
results for the IRAC bands, shown in Fig. 1. Here, the agreement
of the model with observations seems best at 3.6 and 8.0 µm, and
somewhat poorer at 5.8 µm. The model predicts somewhat too few
objects at fainter fluxes in all of the IRAC bands. Comparing the red
and green curves, we see that quiescent galaxies rather than bursts
dominate the counts at all observed fluxes in all of the IRAC bands,
but especially at the shorter wavelengths, consistent with the expec-
tation that at 3.6 and 4.5 µm, we are seeing mostly light from old
stellar populations. Comparing the solid and dashed blue lines, we
see that the effects of dust are small at 3.6 and 4.5 µm, with a small
amount of extinction at faint fluxes (and thus higher average red-
shifts), but negligible extinction for brighter fluxes (and thus lower
redshifts). On the other hand, dust has large effects at 8.0 µm, with
dust emission (due to strong PAH features at λ ∼ 6 − 9µm) be-
coming very important at bright fluxes (which correspond to low
average redshifts - see Fig. A1(b) in the Appendix). The 8.0 µm
counts thus are predicted to be dominated by dust emission from
quiescently star-forming galaxies, except at the faintest fluxes. The
counts at 5.8 µm show behaviour which is intermediate, with mild
emission effects at bright fluxes and mild extinction at faint fluxes.
Comparing the solid blue and dashed magenta lines, we see that the
predicted number counts in the IRAC bands are almost the same
whether or not we assume a top-heavy IMF in bursts, consistent
with the counts being dominated by quiescent galaxies.
Consider next the results for the MIPS bands, shown in Fig. 2.
We again see remarkably good agreement of the standard model
with the observational data. The agreement is especially good at
faint fluxes (corresponding to higher redshifts). In particular, the
model matches well the observed 24 µm counts at the “bump”
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 9
around fluxes Sν ∼ 0.1 − 1mJy. Accurate modelling of the PAH
emission features is obviously crucial for modelling the 24 µm
number counts, since the PAH features dominate the flux in the
24 µm band as they are redshifted into the band at z & 0.5.
On the other hand, the standard model overpredicts the number
counts at bright fluxes (corresponding to low redshifts) in all three
MIPS bands. The evolution at these wavelengths predicted by our
ΛCDM-based model thus seems to be not quite as strong as indi-
cated by observations.
In the MIPS bands, emission from galaxies is completely
dominated by dust, which is why no dashed blue lines are shown in
Fig. 2. Comparing the red and green curves, we see that quiescent
(but star-forming) galaxies tend to dominate the number counts in
these bands at brighter fluxes, and bursts at fainter fluxes. This re-
flects the increasing dominance of bursts in the mid- and far-IR
luminosity function at higher redshifts. Comparing the solid blue
and dashed magenta curves, we see that our standard model with a
top-heavy IMF in bursts provides a significantly better overall fit to
the observed 24 µm counts than the variant model with a normal
IMF in bursts (although at the brightest fluxes, the variant model
fits better). The faint number counts at 70 µm also favour the top-
heavy IMF model, while the number counts at 160 µm cover a
smaller flux range, and do not usefully distinguish between the two
variants of our model with different burst IMFs.
We can use our model to predict the flux levels at which
sources should become confused in the different Spitzer bands.
We estimate the confusion limit using the source density cri-
terion (e.g. Vaisanen et al. 2001; Dole et al. 2003): if the tele-
scope has an FWHM beamwidth of θFWHM , we define the ef-
fective beam solid angle as ωbeam = (π/(4 ln 2)) θ
FWHM =
1.13θ2FWHM , and then define the confusion limited flux Sconf
to be such that N(> Sconf ) = 1/(Nbeamωbeam), where
N(> S) is the number per solid angle of sources brighter than
flux S. We choose Nbeam = 20 for the number of beams
per source, which gives similar results to more detailed analy-
ses (e.g. Vaisanen et al. 2001; Dole et al. 2004b). We use values
of the beamsize θFWHM = (1.66, 1.72, 1.88, 1.98) arcsec for
the four IRAC bands (Fazio et al. 2004b) and (5.6, 16.7, 35.2)
arcsec for the three MIPS bands (Dole et al. 2003). Our stan-
dard model then predicts confusion-limited fluxes of Sconf =
(0.62, 0.62, 0.69, 0.70)µJy in the (3.6, 4.5, 5.8, 8.0)µm IRAC
bands, and Sconf = (0.072, 2.6, 43)mJy in the (24, 70, 160)µm
MIPS bands. These confusion estimates for the MIPS bands are
similar to those of Dole et al. (2004b), which were based on ex-
trapolating from the observed counts. These values for the confu-
sion limits are indicated in Figs. 1 and 2 by vertical dashed lines.
Our galaxy evolution model does not compute the contribution
of AGN to the IR luminosities of galaxies. On the other hand, the
observed number counts to which we compare include both normal
galaxies, in which the IR emission is powered by stellar popula-
tions, and AGN, in which there is also IR emission from a dust
torus, which is expected to be most prominent in the mid-IR. How-
ever, multi-wavelength studies using optical, IR and X-ray data in-
dicate that even at 24 µm, the fraction of sources dominated at
that wavelength by AGN is only 10-20% (e.g. Franceschini et al.
2005), and the contribution of AGN-dominated sources in the other
Spitzer bands is likely to be smaller. Therefore we should not make
any serious error by comparing our model predictions directly with
the total number counts, as we have done here.
Figure 2. Galaxy differential number counts in the three MIPS bands. The
curves show model predictions while the symbols with error bars show ob-
servational data. The meaning of the different model lines is the same as
in Fig. 1. (a) 24 µm, with observational data from Papovich et al. (2004).
(b) 70 µm, with observational data from Dole et al. (2004a) (filled sym-
bols), Frayer et al. (2006a) (crosses), and Frayer et al. (2006b) (open sym-
bols). (c) 160 µm (bottom panel), with observational data from Dole et al.
(2004a) (filled symbols) and Frayer et al. (2006a) (crosses).
c© 0000 RAS, MNRAS 000, 000–000
10 Lacey et al.
Figure 3. Predicted evolution of the galaxy luminosity function in our standard model (including dust) at rest-frame wavelengths of (a) 3.6 and (b) 8.0 µm for
redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key.
4 EVOLUTION OF THE GALAXY LUMINOSITY
FUNCTION
While galaxy number counts provide interesting constraints on the-
oretical models, it is more physically revealing to compare with
galaxy luminosity functions, since these isolate behaviour at partic-
ular redshifts, luminosities and rest-frame wavelengths. In the fol-
lowing subsections, we compare our model predictions with recent
estimates of luminosity function (LF) evolution based on Spitzer
data.
4.1 Evolution of the galaxy luminosity function at 3-8 µm
We consider first the evolution of the luminosity function in the
wavelength range covered by the IRAC bands, i.e. 3.6-8.0 µm.
Fig. 3 shows what our standard model with a top-heavy IMF in
bursts predicts for LF evolution at rest-frame wavelengths of 3.6
and 8.0 µm for redshifts z = 0 − 3 3. We see that at a rest-frame
wavelength of 3.6 µm, the model LF hardly evolves at all over the
whole redshift range z = 0 − 3. This lack of evolution appears to
be somewhat fortuitous. Galaxy luminosities at a rest-frame wave-
length of 3.6 µm are dominated by the emission from moderately
old stars, but the stellar mass function in the model evolves quite
strongly over the range z = 0 − 3 (as we show in §5). The weak
evolution in the 3.6 µm LF results from a cancellation between a
declining luminosity-to-stellar-mass ratio with increasing time and
increasing stellar masses (see Figs. 13(a) and (e)). On the other
hand, at a rest-frame wavelength of 8.0 µm, the model LF becomes
significantly brighter in going from z = 0 to z = 3. Galaxy lu-
minosities at a rest-frame wavelength of 8.0 µm are dominated by
emission from dust heated by young stars, so this evolution reflects
the increase in star formation activity with increasing redshift (see
Fig. 13(b) in §5).
In Fig. 4, we compare the model predictions for evolu-
tion of the LF at 3.6 µm with observational estimates from
3 In this figure, and in Figs. 4, 5, 8, and 10, the luminosities Lν are calcu-
lated through the corresponding Spitzer passbands.
Babbedge et al. (2006) and Franceschini et al. (2006)4. The
model predictions are given for redshifts z = 0, 0.5 and 1. For
the observational data, the mean redshifts for the different redshift
bins used do not exactly coincide with the model redshifts, so we
plot them with the model output closest in redshift 5. The observa-
tional estimates of the 3.6 µm LF rely on the measured redshifts.
In the case of Babbedge et al. (2006), these are mostly photomet-
ric, using optical and NIR (including 3.6 and 4.5 µm) fluxes, while
for the Franceschini et al. sample, about 50% of the redshifts are
spectroscopic and the remainder photometric. In both samples, the
measured 3.6 µm fluxes were k-corrected to estimate the rest-frame
3.6 µm luminosities.
We see from comparing the blue curve with the observational
data in Fig. 4 that the 3.6 µm LF predicted by our standard model
is in very good agreement with the observations. In particular, the
observational data show very little evolution in the 3.6 µm LF over
the redshift range z = 0 − 1. The largest difference seen is at
z = 1, where the Babbedge et al. data show a tail of objects to
very high luminosities, which is not seen in the model predictions.
However, this tail is not seen in the Franceschini et al. data at the
same redshift, and is also not present in the observational data at
the lower redshifts. More spectroscopic redshifts are needed for the
Babbedge et al. sample to clarify whether this high-luminosity tail
is real. Comparing the red, green and blue lines for the standard
model shows that the model luminosity function is dominated by
quiescent galaxies at low luminosity, but the contribution of bursts
becomes comparable to that of quiescent galaxies at high luminosi-
ties. We have not shown model LFs excluding dust extinction in
this figure, since they are almost identical to the predictions includ-
ing dust. The dashed magenta lines show the predicted LFs for the
4 Babbedge et al. (2006) also compared their measured LFs at 3.6, 8.0 and
24 µm with predictions from a preliminary version of the model described
in this paper
5 Specifically, for z = 0, we compare with the z = 0.1 data from
Babbedge et al. , for z = 0.5 we compare with the z = 0.5 data
from Babbedge et al. and z = 0.3 data from Franceschini et al. , and for
z = 1, we compare with the z = 0.75 (open symbols) and z = 1.25
(filled symbols) data from Babbedge et al. and z = 1.15 data from
Franceschini et al.
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 11
Figure 4. Predicted evolution of the galaxy luminosity function at rest-
frame 3.6 µm compared to observational data. The different panels show
redshifts (a) z = 0, (b) z = 0.5 and (c) z = 1. The predictions for our stan-
dard model are shown by the blue line, with the red and green lines showing
the separate contributions from ongoing bursts and quiescent galaxies. The
dashed magenta line shows the prediction for a variant model with a nor-
mal IMF for all stars. The error bars on the model lines indicate the Poisson
uncertainties due to the finite number of galaxies simulated. The black sym-
bols with error bars show observational data from Babbedge et al. (2006)
(open circles and triangles, for z = 0, 0.5 and 1) and Franceschini et al.
(2006) (filled squares, for z = 0.5 and 1).
Figure 5. Predicted evolution of the galaxy luminosity function at rest-
frame 8.0 µm compared to observational data. The different panels show
redshifts (a) z = 0, (b) z = 1 and (c) z = 2. The coloured lines showing
the model predictions have the same meaning as in Fig. 4. The black sym-
bols with error bars show observational data from Babbedge et al. (2006)
(open circles for z = 0 and 0.7, triangles for z = 1.2), Huang et al. (2007)
(filled circles for z = 0) and Caputi et al. (2007) (filled circles for z = 1
and 2). The observed LFs are for normal galaxies and exclude AGN.
c© 0000 RAS, MNRAS 000, 000–000
12 Lacey et al.
variant model with a normal IMF in bursts. We see that these differ
only slightly from our standard model, but are a somewhat poorer
fit to the observational data at higher luminosities.
In Fig. 5 we show a similar comparison for the LF evolution at
a rest-frame wavelength of 8 µm. The model predictions are given
for redshifts z = 0, 1 and 2, and are compared with observational
estimates by Huang et al. (2007) (for z ∼ 0), Babbedge et al.
(2006) (for z ∼ 0 and z ∼ 1) and Caputi et al. (2007) (for z ∼ 1
and z ∼ 2). These papers all classified objects in their samples
as either galaxies or AGN, and then computed separate LFs for
the two types of objects 6. Our model does not make any predic-
tions for AGN, so we compare our model predictions with the ob-
served LFs for objects classified as galaxies only. We see that for
redshifts around z = 1, the observed LFs from Babbedge et al.
and Caputi et al. are in very poor agreement with each other, with
the Caputi et al. LF being around 10 times higher in number den-
sity at the same luminosity. This differerence presumably results
from some combination of: (a) different methods of classifying ob-
jects as galaxies or AGN (Babbedge et al. used only optical and
IR fluxes to do this, while Caputi et al. also used X-ray data); (b)
different photometric redshift estimators; and (c) different meth-
ods for k-correcting luminosities to a rest-frame wavelength of 8
µm. There are smaller differences between the Huang et al. and
Babbedge et al. LFs at z ∼ 0. Futher observational investigation
appears to be necessary to resolve these issues. Our standard model
is in reasonable agreement with the Babbedge et al. observed LF
at z ∼ 0, and with the Caputi et al. observed LFs at z ∼ 1 and
z ∼ 2, but not with the Babbedge et al. observed LF at z ∼ 1. The
comparison with Caputi et al. favours our standard model with a
top-heavy IMF in starbursts over the variant model with a normal
4.2 Evolution of the galaxy luminosity function at 12-24 µm
In this subsection, we consider the evolution of the galaxy lumi-
nosity function at mid-IR wavelengths, and compare with data ob-
tained using mainly the MIPS 24 µm band.
Fig. 6 shows what our standard model with a top-heavy IMF
in bursts predicts for the evolution of the galaxy LF at rest-frame
wavelengths of 15 and 24 µm for redshifts z = 0 − 3 7. At rest-
frame wavelengths of 15 and 24 µm, galaxy luminosities are typi-
cally dominated by the continuum emission from warm dust grains
heated by young stars (although PAH emission is also significant
at some nearby wavelengths). Fig. 6 shows strong evolution in the
model LFs over the redshift range z = 0− 3 at both wavelengths,
reflecting both the increase in star formation activity with increas-
ing redshift (see Fig. 13(b)) and the increasing dominance of the
burst mode of star formation, for which the top-heavy IMF fur-
ther boosts the mid- and far-IR luminosities compared to a normal
IMF. Comparing Fig. 6 with Fig. 3(a), we also see a difference in
the shape of the bright end of the LF: at 3.6 µm, where the LF is
dominated by emission from stars, the bright end cuts off roughly
6 Note that a variety of criteria have been used for classifying observed IR
sources as AGN or normal galaxies, and these do not all give equivalent
results. Even if an object is classified as an AGN, it is also not clear that in
all cases the AGN luminosity dominates over that of the host galaxy in all
Spitzer bands
7 In this figure, and in Figs. 7 and 8, the 24 µm luminosities are calculated
through the corresponding MIPS passband, while the 15 µm luminosities
are calculated through a top-hat filter with a fractional width of 10% in
wavelength.
Figure 8. Predicted evolution of the galaxy luminosity function at rest-
frame wavelength 24 µm compared to observational data from Shupe et al.
(1998) (at z = 0, open symbols) and from Babbedge et al. (2006) (for the
same redshifts as in Fig. 4). The meaning of the curves showing the model
predictions is the same as in Fig. 4. (a) z = 0, (b) z = 0.5 and (c) z = 1.
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 13
Figure 6. Predicted evolution of the galaxy luminosity function in our standard model at rest-frame wavelengths(a) 15 µm (left) and (b) 24 µm (right) for
redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key.
exponentially, while at 15 and 24 µm, where the LF is dominated
by emission from warm dust, the bright end declines more gradu-
ally, roughly as a power-law. This difference reflects the difference
in shape of the galaxy stellar mass function (GSMF) and the galaxy
star formation rate distribution (GSFRD). The GSMF shows an
exponential-like cutoff at high masses, while the GSFRD shows a
more gradual cutoff at high SFRs because of starbursts triggered by
galaxy mergers (see Figs. 13(a) and (b) in §5). This difference was
noticed earlier by observers comparing optical and far-IR LFs of
galaxies, but its origin was not understood (Lawrence et al. 1986;
Soifer et al. 1987b).
In Fig. 7, we compare the model LFs at rest-frame wave-
lengths 12 and 15 µm with observational estimates. For z = 0,
we plot the observational estimates from Soifer & Neugebauer
(1991) and Rush et al. (1993), based on IRAS 12 µm data (with
AGN removed). For z = 0.5 − 1 and z = 1.5 − 2.5, we
plot the data of Le Floc’h et al. (2005) and Perez-Gonzalez et al.
(2005) respectively, which were obtained from galaxy samples
selected on Spitzer 24 µm flux. Le Floc’h et al. k-corrected
their measured 24 µm fluxes to 15 µm rest-frame luminosities,
while Perez-Gonzalez et al. k-corrected to 12 µm rest-frame8.
Le Floc’h et al. obtained most of their redshifts from photometric
redshifts based on optical data, while Perez-Gonzalez et al. used a
new photometric redshift technique based on fitting empirical SEDs
to all of the available broad-band data from the far-UV to 24 µm,
and also removed “extreme” AGN from their observed LF. Note
that the redshifts for the observed LFs do not exactly coincide with
model redshifts in all cases, but are close.
We see from comparing the blue line to the observational dat-
apoints in Fig. 7 that our standard model with a top-heavy IMF in
bursts fits the observations remarkably well up to z = 2. In partic-
ular, the model matches the strong evolution in the mid-IR LF seen
8 The exact passband used for the model LF in each panel depends on
which observational data we are comparing with. For z = 0, we use the
IRAS 12 µm passband; at z = 0.5 and z = 1 we use a top-hat passband
centred at 15 µm; and at z = 1.5, 2 and 2.5, we use a top-hat passband
centred at 12 µm (both top-hat passbands having fractional width 10% in
wavelength).
in the observational data. The model falls below the observational
data at z = 2.5, but here both the photometric redshifts and the
k-corrections are probably the most uncertain. The standard model
also does not provide a perfect fit to the z = 0 data, predicting
somewhat too many very bright galaxies and somewhat too few
very faint galaxies (though the latter discrepancy might be affected
by local galaxy clustering in the IRAS data). Comparing the red,
green and blue lines for the standard model in the figure, we see
that the bright end of the 12 or 15 µm LF is dominated by bursts
at all redshifts. The figure also shows by a dashed magenta line the
predictions for the variant model with a normal IMF in bursts. This
latter model predicts much less evolution in the bright end of the LF
than is observed. This comparison thus strongly favours the model
with the top-heavy IMF in bursts.
Finally, in Fig. 8, we carry out a similar comparison of the evo-
lution of predicted and observed LFs at a rest-frame wavelength
of 24 µm over the redshift range z = 0 − 1, in this case com-
paring with observational estimates from Shupe et al. (1998) (for
z = 0), based on IRAS data, and from Babbedge et al. (2006) (for
z = 0 − 1), based on Spitzer data9. The galaxy redshifts for the
Babbedge et al. data were obtained in the same way as for the 3.6
µm LFs shown in Fig. 4, and the luminosities were k-corrected
from observer-frame 24 µm to rest-frame 24 µm. The LF plotted
from Babbedge et al. is that for normal galaxies, with AGN ex-
cluded.
The conclusions from comparing the model with the 24 µm
LFs are similar to those from the comparison with the 12 and 15
µm LFs. The data favour our standard model over the variant with
a normal IMF in bursts (except possibly for z = 0.5), as the latter
predicts too little evolution at the bright end. At z = 0, the model
fits the 24 µm data rather better than for the corresponding com-
parison at 12 µm. On the other hand, at z = 0.5 and z = 1, the
model LF is a somewhat worse fit to the observational data at 24
µm than at 15 µm. These differences between the 12/15 and 24
µm comparisons might result from the different photometric red-
shifts and k-corrections used in the observational samples in the
9 The model luminosities are all computed through the Spitzer 24 µm pass-
band.
c© 0000 RAS, MNRAS 000, 000–000
14 Lacey et al.
Figure 7. Predicted evolution of the galaxy luminosity function at rest-frame wavelength 12 or 15 µm compared to observational data. The different panels
show redshifts: (a) z = 0, (b) z = 0.5, (c) z = 1, (d) z = 1.5, (e) z = 2 and (f) z = 2.5. The meaning of the curves showing the model predictions is the
same as in Fig. 4. In panel (a), the predictions at 12µm are compared to observational determinations from Soifer & Neugebauer (1991) (open symbols) and
Rush et al. (1993) (filled symbols) based on IRAS data. In panels (b) and (c), the predictions at 15µm are compared to observational data from Le Floc’h et al.
(2005). In panels (d), (e) and (f), the predictions at 12µm are compared to observational data from Perez-Gonzalez et al. (2005).
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 15
Figure 9. The predicted galaxy luminosity function at 60 µm compared
to observational data from IRAS. The meaning of the different lines is
the same as in Fig. 4. The black symbols show observational data from
Saunders et al. (1990) (crosses), Soifer & Neugebauer (1991) (open cir-
cles), and Takeuchi et al. (2003) (filled circles).
two cases. Alternatively, they might result from problems in mod-
elling the dust SEDs in the complex mid-IR range.
4.3 Evolution of the galaxy luminosity function at 70-160 µm
We now briefly consider the evolution of the luminosity function
in the far-IR. The far-IR is the wavelength range where most of
the luminosity from dust in normal galaxies is emitted. The lo-
cal 60 µm luminosity function was very well measured by sur-
veys with IRAS, and so is commonly used as a starting point or
benchmark for modelling the evolution of the galaxy population in
the far-IR. We therefore present in Fig. 9 the model prediction for
the 60 µm luminosity function at z = 0, compared with obser-
vational data from Saunders et al. (1990), Soifer & Neugebauer
(1991) and Takeuchi et al. (2003). As discussed in Baugh et al.
(2005), the local 60 µm luminosity function was used as one of the
primary constraints in fixing the parameters of our galaxy forma-
tion model, and the figure shows that our standard model provides
a good match to the data. The variant model with a normal IMF in
bursts underpredicts the abundance of the brightest 60 µm galaxies.
In Fig. 10, we show the model predictions for the evolution of
the luminosity function in the two longer wavelength MIPS bands,
at rest-frame wavelengths of 70 and 160 µm, from z = 0 to z = 3.
At 70 µm, the luminosity function at high luminosities is predicted
to brighten by about a factor 10 going from z = 0 to z = 2. This
is about a factor 2 less than the brightening predicted in the mid-IR
at 15 µm (compare to Fig. 6), but nearly a factor 2 more evolution
than is predicted at 160 µm. These differences between the amount
of evolution seen at different IR wavelengths reflect evolution in
the shapes of the SEDs of the galaxies responsible for the bulk of
the IR emission. No observational estimates of the evolution of the
luminosity function at 70 and 160 µm have yet been published,
but they are expected to be forthcoming from ongoing surveys with
Spitzer.
Figure 11. Predicted evolution of the total mid+far-IR (8-1000 µm) galaxy
luminosity function for our standard model, for redshifts z = 0, 1, 2, 3, 4
and 6, as shown in the key.
4.4 Evolution of the total mid+far-IR luminosity function
The total mid+far IR luminosity of a galaxy, LIR, integrated over
the whole wavelength range 8-1000 µm, is a very good approxi-
mation to the total luminosity emitted by interstellar dust grains in
all galaxies except those with very small dust contents. In galax-
ies with significant star formation, LIR is mostly powered by dust
heated by young stars, and so provides a quantitative indicator of
the amount of dust-obscured star formation which is independent
of the shape of the IR SED (though still subject to uncertainties
about the IMF). The evolution of the luminosity function in LIR
is therefore a very interesting quantity to compare between models
and observations. We show in Fig. 11 what our standard model pre-
dicts for the evolution of the IR LF over the range z = 0 − 6. We
see that the model predicts substantial evolution in this LF, with the
high luminosity end brightening by a factor ∼ 10 from z = 0 to
z = 2, followed by a “plateau” from z = 2 to z = 4, and a decline
from z = 4 to z = 6.
In Fig. 12, we compare our model predictions with existing
observational estimates of the total IR LF for z = 0 − 2. These
observational estimates are only robust for z = 0, where they are
based on IRAS measurements covering the wavelength range 12-
100 µm. At all of the higher redshifts plotted, the observational
estimates are based on measurements of the mid-IR luminosity de-
rived from Spitzer 24 µm fluxes, converted to total IR luminosi-
ties by assuming SED shapes for the mid- to far-IR emission. The
bolometric correction from the observed mid-IR luminosity to the
inferred total IR luminosity is typically a factor ∼ 10, and is sig-
nificantly uncertain. Therefore, the most robust way to compare
the models with the observations is to compare them at the mid-
IR wavelengths where the measurements are actually made, as we
have done in §4.1 and §4.2. Nonetheless, if we take the observa-
tional determinations at face value, then we see that observed evo-
lution of the total IR LF agrees remarkably well with the predic-
tions of our standard model with a top-heavy IMF. On the other
hand, the variant model with a normal IMF predicts far too few
c© 0000 RAS, MNRAS 000, 000–000
16 Lacey et al.
Figure 10. Predicted evolution of the galaxy luminosity function in our standard model (including dust) at rest-frame wavelengths (a) 70 µm and (b) 160 µm,
for redshifts z = 0, 0.5, 1, 1.5, 2 and 3, as shown in the key.
high LIR galaxies at higher z, and is strongly disfavoured by the
existing data.
5 INFERRING STELLAR MASSES AND STAR
FORMATION RATES FROM Spitzer DATA
In this section, we consider what the models imply about how well
we can infer the stellar masses and star formation rates (SFRs) in
galaxies from measurements of rest-frame IR luminosities. The top
two panels of Fig. 13 show the predicted galaxy stellar mass func-
tion (GSMF, left panel) and galaxy star formation rate distribution
(GSFRD, right panel), for redshifts z = 0− 6. We see that the pre-
dicted stellar mass function shows dramatic evolution over this red-
shift range, with a monotonic decline in the number of high-mass
galaxies with increasing redshift. On the other hand, the SFR distri-
bution shows much less dramatic evolution over this redshift range,
with a mild increase in the number of high-SFR objects up to z ∼ 3,
followed by a decline above that. The lower four panels in Fig. 13
show the relation in the models between stellar masses and SFRs
and rest-frame luminosities at different IR wavelengths. (Note that
in all cases, luminosities are measured in units of the bolometric
solar luminosity.) The middle and bottom left panels respectively
show the mean ratio of luminosity in the rest-frame K (2.2µm) or
3.6 µm bands to stellar mass as a function of stellar mass. The
middle and bottom right panels respectively show the mean ratio
of total mid+far-IR (8− 1000µm) or rest-frame 15 µm luminosity
to SFR as a function of SFR. (The mean L/M∗ or L/SFR ratios
plotted are computed by dividing the total luminosity by the total
mass or SFR, in each bin of mass or SFR.)
The near-IR luminosity is often used as a tracer of stellar mass.
The left panels of Fig. 13 show that the L/M∗ ratio varies strongly
with redshift, reflecting the difference in the ages of the stellar pop-
ulations. At higher redshifts it also shows a significant dependence
on stellar mass, presumably reflecting a trend of age with mass.
However, the variation of mean L/M∗ with redshift is seen to be
much smaller at 2.2 µm than at 3.6 µm, implying that the rest-
frame K-band light should provide a more robust estimator of stel-
lar mass than the light at longer wavelengths. The differences be-
tween L/M∗ values at 2.2 µm and 3.6 µm reflect the larger contri-
bution from AGB compared to RGB stars at the longer wavelength.
AGB stars have higher masses and younger ages than RGB stars,
and so are more sensitive to star formation at recent epochs. The
scatter in L/M∗ at a given mass is also found in the models to in-
crease with redshift. In the K-band, it increases from ∼ 40% at
z ∼ 0 to a factor ∼ 3 at z ∼ 6. The large scatter at high redshifts
results in part from having two different IMFs.
The luminosity in the mid- and far-IR is widely used as a
tracer of dust-obscured star formation (although in galaxies with
very low star formation rates, the dust heating can be dominated
by older stars). The total mid+far-IR (rest-frame 8-1000 µm) lumi-
nosity is expected to provide a more robust tracer of star formation
than the luminosity at any single IR wavelength, since the shape of
the SED of dust emission depends on the dust temperature distri-
bution (as well as on the dust grain properties). This is borne out
by our model predictions. The middle right panel of Fig. 13 shows
that the LIR/SFR ratio depends weakly on both SFR and red-
shift. This behaviour results mostly from having different IMFs in
the model in quiescent and bursting galaxies, with the fractional
contribution of the bursts increasing both with SFR and with red-
shift. If we look at quiescent and bursting galaxies separately, we
find roughly constant ratios LIR/SFR ≈ 6 × 10
9h−1L⊙/M⊙
and LIR/SFR = 2× 10
10h−1L⊙/M⊙ respectively, for galaxies
where LIR is powered mostly by young stars. However, there is
also a trend at lower redshift for LIR/SFR to be larger at lower
SFR - this reflects the larger fraction of dust heating from older
stars in galaxies with lower SFRs, which more than compensates
for the lower average dust obscuration in these galaxies. The lower
right panel of Fig. 13 shows that the L/SFR ratio in the mid-IR
(in this case at 15 µm in the rest-frame) shows more variation with
SFR and redshift than the ratio for the total IR luminosity. This re-
flects the variation in the mid- to far-IR SED shapes in the model.
The scatter in the L/SFR ratio is roughly a factor 2 around the
average relation for the total IR luminosity, but is larger for the 15
µm luminosity.
The results of this section illustrate why it is not straightfor-
ward to compare theoretical predictions for the evolution of the
galaxy stellar mass function and star formation rate distribution (or
even the stellar mass and star formation rate densities) with obser-
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 17
Figure 12. Predicted evolution of the total mid+far IR (8-1000µm) galaxy LF compared to observational data. The different panels show redshifts (a) z = 0,
(b) z = 0.5, (c) z = 1 and (d) z = 2. For z = 0, we compare with observational data from Sanders et al. (2003) (filled symbols) and Takeuchi et al. (2003)
(open symbols, converting his 60 µm LF to a total IR LF assuming a constant conversion factor, LIR/νLν(60µm) = 2.5). We compare with data from
Le Floc’h et al. (2005) for z = 0.5 and z = 1 (filled and open symbols), and with Caputi et al. (2007) for z = 1 and z = 2 (crosses).
vational estimates. In addition to assumptions about galaxy star for-
mation histories and metallicities (for stellar mass estimates), and
about the SED shapes for dust emission (for SFR estimates from
IR and sub-mm data), observational estimates all rest on some as-
sumed form for the IMF. If the IMF assumed in the observational
analysis is different from the true IMF, the observational estimates
for stellar masses and SFRs can be wrong by large factors. If the
IMFs differ only below 1M⊙, then one can apply a simple rescal-
ing to relate stellar mass and SFR estimates for different IMFs.
However, if our current galaxy formation model is correct, stars
form with different IMFs in quiescent disks and in merger-driven
bursts, and so no observational estimate based on assuming a sin-
gle IMF can give the correct GSMFs and GSFRDs, nor the correct
stellar mass and SFR densities. A direct comparison of the GSMF
and GSFRD evolution predicted by our model with observational
estimates is therefore not meaningful. Instead, the comparison be-
tween models and observations must be made via directly observ-
able (rather than inferred) quantities, such as the K-band luminosi-
ties to constrain stellar masses, and the total IR luminosities to con-
strain SFRs.
6 CONCLUSIONS
We have computed predictions for the evolution of the galaxy pop-
ulation at infrared wavelengths using a detailed model of hierar-
chical galaxy formation and of the reprocessing of starlight by
dust, and compared these predictions with observational data from
the Spitzer Space Telescope. We calculated galaxy formation in
the framework of the ΛCDM model using the GALFORM semi-
analytical model, which includes physical treatments of the hier-
archical assembly of dark matter halos, shock-heating and cool-
ing of gas, star formation, feedback from supernova explosions
and photo-ionization of the IGM, galaxy mergers and chemical en-
richment. We computed the IR luminosities and SEDs of galaxies
using the GRASIL multi-wavelength spectrophotometric model,
which computes the luminosities of the stellar populations in galax-
c© 0000 RAS, MNRAS 000, 000–000
18 Lacey et al.
Figure 13. Model predictions for properties related to stellar masses (left column) and star formation rates (right column), for redshifts z = 0, 1, 2, 3, 4, and
6: (a) galaxy stellar mass function (GSMF); (b) galaxy star formation rate distribution (GSFRD); (c) mean ratio of rest-frame K-band luminosity to stellar
mass, as a function of stellar mass; (d) mean ratio of total mid+far IR luminosity to SFR, as a function of SFR; (e) mean ratio of rest-frame 3.6 µm luminosity
to stellar mass, as a function of stellar mass; (f) mean ratio of rest-frame 15 µm luminosity to SFR, as a function of SFR. (The 15 µm luminosity is here
calculated through top-hat filter with a fractional wavelength width of 10%.)
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 19
ies, and then the reprocessing of this radiation by dust, including
radiative transfer through a two-phase dust medium, and a self-
consistent calculation of the distribution of grain temperatures in
each galaxy based on a local balance between heating and cool-
ing. The GRASIL model includes a treatment of the emission from
PAH molecules, which is essential for understanding the mid-IR
emission from galaxies.
Our galaxy formation model incorporates two different IMFs:
quiescent star formation in galaxy disks occurs with a normal so-
lar neighbourhood IMF, but star formation in bursts triggered by
galaxy mergers happens with a top-heavy x = 0 IMF. In a previ-
ous paper (Baugh et al. 2005), we found that the top-heavy IMF in
bursts was required in order that the model reproduces the observed
number counts of the faint sub-mm galaxies detected at 850 µm,
which are typically ultra-luminous starbursts at z ∼ 2, with total IR
luminosities LIR ∼ 10
− 1013L⊙. This conclusion was arrived
at following a search of a large grid of model parameters, with the
imposition of a variety of detailed observational constraints. The
parameters in the Baugh et al. (2005) model were chosen before
the publication of any results from Spitzer, without reference to any
IR data apart from the local 60 µm luminosity function and the 850
µm galaxy counts. We have kept the same parameter values in the
present paper, in order to test what the same model predicts at other
wavelengths and other redshifts. By doing this, we hope to address
the criticism made of many semi-analytical models that they have
no predictive power, because their parameters are always adjusted
to match the observational data being analysed at that instant.
We first compared the predictions from our model with the
galaxy number counts measured in all 7 Spitzer bands, from 3.6 to
160 µm. We found broad agreement between the model and the
observations. In the 4 IRAC bands (3.6-8.0 µm), where the counts
are mostly dominated by emission from older stellar populations,
we found that the predicted counts were insensitive to whether we
had a top-heavy or normal IMF in bursts. On the other hand, in
the MIPS bands (24-160 µm), where the counts are dominated by
emission from dust in star-forming galaxies, the predicted counts
are more sensitive to the choice of IMF, and the counts are fit better
by the model with a top-heavy IMF. We next investigated the evo-
lution of the galaxy luminosity function at IR wavelengths, where
several groups have now used Spitzer data to try to measure the
evolution of the galaxy luminosity function over the redshift range
z ∼ 0− 2, at rest-frame wavelengths from 3.6 to 24 µm.
Our model predicts that at mid- and far-IR rest-frame wave-
lengths, the luminosity function evolution is very sensitive to the
choice of IMF in bursts. We found that our standard model with a
top-heavy IMF in bursts fits the measured evolution of the mid-IR
luminosity function remarkably well (when allowance is made for
complexity of predicting dust emission in the mid-IR), without any
adjustment of the parameters. On the other hand, a model with a
normal IMF in bursts predicts far too little evolution in the mid-IR
luminosity function compared to what is observed. We made a sim-
ilar comparison with the evolution of the total IR luminosity func-
tion, where in the case of the observations, the total IR luminosities
at high redshifts have been inferred from the 24 µm fluxes by fit-
ting SEDs, and reached the same conclusion. The evolution of the
galaxy luminosity function in the mid-IR found by Spitzer thus sup-
ports our original conclusion about the need for a top-heavy IMF
in bursts, which was based only on the sub-mm counts. This con-
clusion will be further tested by ongoing Spitzer surveys at longer
wavelengths. To assist this, we have also presented predictions for
the evolution of the luminosity function in the Spitzer 70µm and
160µm bands.
We have also presented predictions for the evolution of the
stellar mass function and star formation rate distribution of galax-
ies. We investigated how the L/M∗ and L/SFR ratios varied with
galaxy mass, SFR and redshift in different IR wavelength ranges,
and considered the implications for observational estimates of stel-
lar masses and SFRs from IR observations. Even in the near-IR, the
predicted variations inL/M∗ with mass and redshift can be surpris-
ingly large. The variations in L/M∗ are much larger at a rest-frame
wavelength of 3.6 µm than at 2.2 µm, implying that the 2.2 µm
luminosity is a more robust tracer of stellar mass.
Finally, we have presented in an Appendix the predictions of
our model for the redshift distributions of galaxies selected at dif-
ferent IR fluxes in the Spitzer bands.
One significant limitation of our model is that it does not in-
clude the effects of AGN. Two effects are relevant here. The first is
feedback from AGN on galaxy formation. In several recent galaxy
formation models, AGN feedback is invoked to prevent the forma-
tion of too many massive galaxies at the present day. In the model
presented here, we instead posit feedback from supernova-driven
galactic superwinds, which perform a similar role to AGN feed-
back in suppressing the formation of very massive galaxies. Both
the superwind and AGN feedback models include free parameters
which are tuned to give a match to the present-day optical galaxy
luminosity function. However, the redshift dependence of the feed-
back will be different between our superwind model and the various
AGN feedback models, so in general they will all predict different
evolution of the galaxy population with redshift. We will investi-
gate galaxy evolution in the IR in a model with AGN feedback in
a future paper. The second effect of AGN which we have not in-
cluded is the emission from AGN and their associated dust tori. In
order to compensate for this, we have wherever possible compared
our model predictions with observations from which the AGN con-
tribution has been subtracted out. This was possible for most of our
comparisons of luminosity function evolution. This was not possi-
ble for the number counts comparisons, but in this case the contri-
bution from AGN is thought (based on observations) to be a small
fraction of the total over the flux range explored by Spitzer, even in
the mid-IR where the dust tori are the most prominent. We therefore
believe that emission from AGN does not seriously affect our con-
clusions about the IR evolution of star-forming galaxies. We hope
to include AGN emission directly into our models in the future.
We have thus shown that Spitzer data provide a stringent test of
galaxy formation theory, by probing galaxy evolution, constraining
star formation rates and the role of dust to z ∼ 2. We find that
an ab initio ΛCDM model gives an acceptable fit to the Spitzer
data provided that ∼ 10% of the stars in galaxies today formed
in bursts of star formation with a top-heavy IMF. Future facilities
like Herschel, SPICA, JWST and ALMA will continue to exploit the
valuable information on galaxy formation contained in the IR part
of the electromagnetic spectrum.
ACKNOWLEDGEMENTS
We thank T. Babbedge, K. Caputi, A. Franceschini, E. Le Floch,
and P. Perez-Gonzalez, for providing us with their observational
data in a convenient form. CMB acknowledges the receipt of a
Royal Society University Research Fellowship. CSF is the recip-
ient of a Royal Society Wolfson Research Merit Award. This work
was also supported by the PPARC rolling grant for extragalactic
astronomy and cosmology at Durham.
c© 0000 RAS, MNRAS 000, 000–000
20 Lacey et al.
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APPENDIX A: REDSHIFT DISTRIBUTIONS
In this Appendix, we present some predictions from our standard
model for the redshift distributions of galaxies selected at different
fluxes in the Spitzer bands. This is principally for completeness,
to assist in interpreting data from current surveys, and to assist in
planning future surveys based on Spitzer data. The set of redshift
distributions at all observed fluxes in principle contains equivalent
information to that in the luminosity functions at different wave-
lengths and redshifts. However, comparing models with observa-
tions via luminosity functions is more physically transparent than
making the comparison via redshift distributions, which is why we
have presented our results on luminosity functions in the main part
of the paper, and why we make only a limited direct comparison
with observed redshift distributions in this Appendix. In addition,
if one only compares the predicted and observed redshift distribu-
tions for galaxies above a single flux limit (e.g. the flux limit of
a survey), this has less information than comparing the luminosity
functions at different redshifts.
We first show in Fig. A1 how the median redshift, and the 10-
90 percentile range, are predicted to change with flux for galax-
ies selected in one of the four Spitzer bands 3.6, 8.0, 24 or 70
µm. While at most wavelengths the median redshift is predicted
to increase smoothly and monotonically with decreasing flux, this
is not true at 24 µm, where there is a bump around Sν ∼ 100µJy.
The structure seen for the 24 µm band as compared to the other
wavelengths results from different PAH emission features moving
through the band with increasing redshift.
In Fig. A2, we show the predictions from our standard model
for the redshift distributions of galaxies in the four IRAC bands.
For each band, we show the redshift distribution for galaxies se-
lected to be brighter than Sν > 10µJy in that band. The flux limit
Sν > 10µJy has been chosen to match that in the observed deep
sample selected at 3.6µm by Franceschini et al. (2006). In each
panel, the blue curve shows the predicted dN/dz for all galax-
ies, normalized to unit area under the curve, and the red and green
curves show the separate contributions of bursting and quiescent
galaxies to the total. For 3.6µm, the black line shows the ob-
served redshift distribution from Franceschini et al. (2006), which
has also been normalized to unit area under the curve. We see that
the observed redshift distribution peaks at a slightly higher redshift
than in the model. However, the luminosity function evolution de-
rived from this same sample is in reasonable agreement with the
model, as was already shown in Fig. 4. Franceschini et al. (2006)
note that the peak seen in their data at z ∼ 0.8 is partly contributed
by large-scale structures in the CDFS field.
In Fig. A3, we show predicted redshift distributions for galax-
ies selected to be at a set of different fluxes in the four IRAC bands.
The curves for the different fluxes are all normalized to have unit
area as before, but in this figure the galaxies are selected to be at a
particular flux, rather than being brighter than a certain flux. As one
would expect, the typical redshift increases as the flux decreases.
Figs. A4 and A5 show for the three MIPS bands the equiv-
alent of Figs. A2 and A3 for the IRAC bands. In Fig. A4, we
show the predicted redshift distributions for galaxies brighter than
a particular flux, where this flux limit is taken to be 83 µJy at 24
µm, 10 mJy at 70 µm and 100 mJy at 160 µm. The flux limit
at 24 µm has been chosen to match that used in the deep obser-
vational samples of Le Floc’h et al. (2005), Perez-Gonzalez et al.
(2005) and Caputi et al. (2006a), while the flux limits at 70 µm
and 160 µm have been chosen to be roughly 3 times brighter than
the source confusion limits in these bands. We see in Fig. A5 that
c© 0000 RAS, MNRAS 000, 000–000
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22 Lacey et al.
Figure A1. Model predictions for the median redshift as a function of flux in four Spitzer bands. (a) 3.6 µm, 8.0 µm, (c) 24 µm, (d) 70 µm. In each panel,
the median redshift for galaxies at each flux is shown by a solid line, and the 10- and 90-percentiles are shown by dashed lines.
the redshift distributions at 24 µm show much more structure than
at other wavelengths. This results from different PAH emission fea-
tures moving through the 24 µm band with changing redshift.
In Fig. A4(a), we compare the predicted redshift dis-
tribution at 24 µm with observational determinations from
Perez-Gonzalez et al. (2005) (dashed black line) and Caputi et al.
(2006a) (solid black line). The observed distributions have been
separately normalized to unit area under the curve, as for the
model distribution. Both observed distributions are based pri-
marily on photometric redshifts, but the photometric redshifts of
Caputi et al. (2006a) are likely to be more accurate than those of
Perez-Gonzalez et al. (2005), since the former are based on deeper
optical and K-band data than the latter. (Perez-Gonzalez et al.
found optical counterparts with BAB . 24.7 or RAB . 23.7 for
∼ 70% of their Sν(24µm) > 83µJy sources, but relied on IRAC
fluxes in deriving photo-z’s for the remaining ∼ 30% of their sam-
ple. On the other hand, Caputi et al. found K-band counterparts
with K(V ega) < 21.5 for 95% of their Sν(24µm) > 80µJy
sample, and derived photo-z’s for essentially all of these sources
using optical and K-band data alone). Both observed distributions
are similar, but the Caputi et al. distribution shows more structure.
This is a combination of the effects of more accurate photometric
redshifts but also a 9 times smaller survey area, which means that
fluctuations due to galaxy clustering are larger. Caputi et al. argue
that the separate peaks at z ∼ 0.7 and 1.1 result from large-scale
structure, but that the bump at z ∼ 1.9 results from PAH emis-
sion features entering the observed 24 µm band. We see that the
model also predicts peaks in the redshift distribution at z ∼ 0.3,
z ∼ 1 and z ∼ 2, which can be explained by different PAH fea-
tures moving through the 24 µm band, although the z ∼ 2 peak is
more prominent than is seen in the observational data. Overall, the
model redshift distribution at this flux limit is too skewed to high
redshift compared to the observations, predicting too few galaxies
at z ∼ 0.5− 1, and too many in the peak at z ∼ 2.
We investigate further this apparent discrepancy in the 24 µm
redshift distribution in Fig. A6, where we show the effects of appar-
ent magnitude limits in the R and K-bands on the predicted redshift
distributions for Sν(24µm) > 83µJy. In this plot, the redshift
distributions are plotted as number per solid angle, without nor-
malizing to unit area under the curve. The left and right panels re-
spectively have the redshift distributions of Perez-Gonzalez et al.
and Caputi et al. overplotted. We concentrate on the comparison
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 23
Figure A2. Predicted galaxy redshift distributions in the four IRAC bands, for galaxies brighter than Sν = 10µJy. (a) 3.6 µm, (b) 4.5 µm, (c) 5.8 µm, and
(d) 8.0 µm. The model curves (which all include the effects of dust) are as follows - blue: total; red: ongoing bursts; green: quiescent galaxies. The curves are
normalized to unit area under the curve for the total counts. The median (z50) and 10- and 90-percentile (z10, z90) redshifts for the total counts in each band
are also given in each panel. For 3.6 µm, the model predictions are compared with observational data from Franceschini et al. (2006) (black dashed line),
normalized to unit area as for the models. The error bars plotted on the observational data include Poisson errors only.
with Caputi et al. , since this has the simpler sample selection and
more accurate redshifts. The model prediction for K < 21.5
(which is the magnitude limit used by Caputi et al. ) is shown
by the short-dashed blue line, while the prediction with no limit
on the K-magnitude is shown by the solid blue line. The model
dN/dz with no limit on the K magnitude is most discrepant with
the Caputi et al. data at z ∼ 2, where it predicts ∼ 2 times too
many galaxies. This is directly related to the fact that the pre-
dicted luminosity function at z = 2 at rest-frame wavelength
8 µm (corresponding to observed wavelength 24 µm) and lumi-
nosity ∼ 1011L⊙ is also ∼ 2 times too high compared to what
Caputi et al. estimate from their data, as shown in Fig. 5(c). When
the effect of the K < 21.5 limit is included, the predicted redshift
distribution is closer to the observational data, but only 58% of the
model galaxies are brighter than this K-band magnitude limit, as
against 95% in the observed sample of Caputi et al. . We conclude
that the main reason for the discrepancy between the predicted and
observed redshift distributions at 24 µm is that the model predicts
a rest-frame 8 µm luminosity function at z ∼ 2 which is somewhat
too high at luminosities ∼ 1011L⊙, even though it reproduces quite
well the general features of the evolution of the mid-IR luminosity
function.
c© 0000 RAS, MNRAS 000, 000–000
24 Lacey et al.
Figure A3. Predicted galaxy redshift distributions in the four IRAC bands, for different fluxes. (a) 3.6 µm, (b) 4.5 µm, (c) 5.8 µm, and (d) 8.0 µm. In this
figure, the redshift distributions are for galaxies at a particular flux. Predictions are shown for fluxes Sν = 0.1, 1, 10, 100 and 1000 µJy, as shown in the key.
In all cases, the model curves are normalized to unit area, and include the effects of dust.
c© 0000 RAS, MNRAS 000, 000–000
Galaxy evolution in the IR 25
Figure A4. Predicted galaxy redshift distributions in the three MIPS bands,
for galaxies brighter than a specified flux. (a) 24 µm, Sν > 83µJy, (b) 70
µm, Sν > 10mJy, and (c) 160 µm, Sν > 100mJy. The model curves
are as follows - blue: total; red: ongoing bursts; green: quiescent galax-
ies. The curves are normalized to unit area under the curve for the total
counts. The median (z50) and 10- and 90-percentile (z10, z90) redshifts for
the total counts in each band are also given in each panel. For 24 µm, the
model predictions are compared with observational data from Caputi et al.
(2006a) (solid black line) and Perez-Gonzalez et al. (2005) (dashed black
line), normalized to unit area as for the models. The error bars plotted on
the observational data include Poisson errors only for Caputi et al. , but also
include errors in photometric redshifts for Perez-Gonzalez et al.
Figure A5. Predicted galaxy redshift distributions in the three MIPS bands,
for different fluxes. (a) 24 µm, (b) 70 µm, and (c) 160 µm. In this figure,
the redshift distributions are for galaxies at a particular flux, as shown in the
key in each panel. In all cases, the model curves are normalized to unit area,
and include the effects of dust.
c© 0000 RAS, MNRAS 000, 000–000
26 Lacey et al.
Figure A6. Predicted redshift distributions at 24µm, showing the effects of optical or near-IR magnitude limits. Model galaxies are selected with Sν > 83µJy
together with the optical/NIR magnitude limits as shown in the key. The fraction of 24 µm sources brighter than each magnitude limit is also given. (a) R-band
magnitude limit. The observed redshift distribution from Perez-Gonzalez et al. (2005) is overplotted in black. Note Le Floc’h et al. (2005) used R < 24
and obtained 54% completeness. (b) K-band magnitude limit. The observed redshift distribution from Caputi et al. (2006a) (with K < 21.5) is overplotted.
Magnitudes are on the Vega system.
c© 0000 RAS, MNRAS 000, 000–000
Introduction
Model
GALFORM galaxy formation model
GRASIL model for stellar and dust emission
Choice of parameters in the GALFORM+GRASIL model
Number counts
Evolution of the galaxy luminosity function
Evolution of the galaxy luminosity function at 3-8 m
Evolution of the galaxy luminosity function at 12-24 m
Evolution of the galaxy luminosity function at 70-160 m
Evolution of the total mid+far-IR luminosity function
Inferring stellar masses and star formation rates from Spitzer data
Conclusions
Redshift distributions
|
0704.1563 | Use of Triangular Elements for Nearly Exact BEM Solutions | Use of Triangular Elements for Nearly Exact BEM
Solutions
Supratik Mukhopadhyay, Nayana Majumdar
INO Section, Saha Institute of Nuclear Physics
1/AF, Sector 1, Bidhannagar, Kolkata 700064, WB, India
[email protected], [email protected]
Abstract
A library of C functions yielding exact solutions of potential and flux influences due to
uniform surface distribution of singularities on flat triangular and rectangular elements has
been developed. This library, ISLES, has been used to develop the neBEM solver that is both
precise and fast in solving a wide range of problems of scientific and technological interest.
Here we present the exact expressions proposed for computing the influence of uniform
singularity distributions on triangular elements and illustrate their accuracy. We also present
a study concerning the time taken to evaluate these long and complicated expressions vis
a vis that spent in carrying out simple quadratures. Finally, we solve a classic benchmark
problem in electrostatics, namely, estimation of the capacitance of a unit square plate raised
to unit volt. For this problem, we present the estimated values of capacitance and compare
them successfully with some of the most accurate results available in the literature. In
addition, we present the variation of the charge density close to the corner of the plate for
various degrees of discretization. The variations are found to be smooth and converging.
This is in clear contrast to the criticism commonly leveled against usual BEM solvers.
Keywords: Boundary element method, triangular element, potential, flux, unit square
plate, charge density, capacitance.
1 Introduction
One of the elegant methods for solving the Laplace / Poisson equations (normally an integral
expression of the inverse square law) is to set up the Boundary Integral Equations (BIE) which
lead to the moderately popular Boundary Element Method (BEM). In the forward version of
the BEM, surfaces of a given geometry are replaced by a distribution of point singularities
such as source / dipole of unknown strengths. The strengths of these singularities are obtained
through the satisfaction of a given set of boundary conditions that can be Dirichlet, Neumann
or of the Robin type. The numerical implementation requires considerable care [1] because it
involves evaluation of singular (weak, strong and hyper) integrals. Some of the notable two-
dimensional and three-dimensional approaches are [1] and [2, 3, 4, 5, 6] and the references in
these papers. Despite a large body of literature, closed form analytic expressions for computing
the effects of distributed singularities are rare [7, 8] and complicated to implement. Thus, for
solving difficult but realistic problems involving, for example, sharp edges and corners or thin
http://arxiv.org/abs/0704.1563v1
elements, introduction of complicated mathematics and special formulations becomes a necessity
[9, 10]. These drawbacks are some of the major reasons behind the relative unpopularity of the
BEM despite its significant advantages over domain approaches such as the finite-difference and
finite-element methods (FDM and FEM) while solving non-dissipative problems [11]. It is well-
understood that most of the difficulties in the available BEM solvers stem from the assumption
of nodal concentration of singularities which leads to various mathematical difficulties and to the
infamous numerical boundary layers [9, 12]. The Inverse Square Law Exact Solutions (ISLES)
library, in contrast, is capable of truly modeling the effect of distributed singularities precisely
and, thus, its application is not limited by the proximity of other singular surfaces or their
curvature or their size and aspect ratio. The library consists of exact solutions for both potential
and flux due to uniform distribution of singularity on flat rectangular and triangular elements.
While the rectangular element can be of any arbitrary size [13, 14], the triangular element can
be a right angled triangle of arbitrary size [15]. Since any real geometry can be represented
through elements of the above two types (or by the triangular type alone), this library can help
in developing solvers capable of solving three-dimensional potential problems for any geometry.
It may be noted here that any non-right-angled triangle can be easily decomposed in to two
right-angled triangles. Thus, the right-angled triangles considered here, in fact, can take care
of any three-dimensional geometry. Several difficulties were faced in developing the library
which arose due to the various terms of the integrals and also from the approximate nature of
computation in digital computers. In this paper, we have discussed these difficulties, solutions
adopted at present and possible ways of future improvement.
The classic benchmark problem of estimating the capacitance of a unit square plate raised
to unit volt has been addressed using a solver based on ISLES, namely, the nearly exact BEM
(neBEM) solver. Results obtained using neBEM have been compared with other precise results
available in the literature. The comparison clearly indicates the excellent precision and efficiency
achievable using ISLES and neBEM. In addition, we have also presented the variation of charge
density close to the corner of the square plate. Usually, using BEM, it is difficult to obtain
physically consistent results close to these geometric singularities. Wild variations in the magni-
tude of the charge density has been observed with the change in the degree of discretization, the
reason once again being associated with the nodal model of singularities [16]. In contrast, using
neBEM, we have obtained very smooth variation close to the corner. Moreover, the magnitudes
of the charge density have been found to be consistently converging to physically realistic values.
These results clearly indicate that since the foundation expressions of the solver are exact, it
is possible to find the potential and flux accurately in the complete physical domain, includ-
ing the critical near-field domain using neBEM. In addition, since singularities are no longer
assumed to be nodal and we have the exact expressions for potential and flux throughout the
physical domain, the boundary conditions no longer need to be satisfied at special points such
as the centroid of an element. Although consequences of this considerable advantage is still
under study, it is expected that this feature will allow neBEM to yield accurate estimates for
problems involving corners and edges that are very important in a large number of scientific and
technological studies.
It should be noted here that the exact expressions for triangular elements consist of a signifi-
cantly larger number of mathematical operations than those for rectangular elements. Thus, for
the solver, it is more economical if we use a mixed mesh of rectangular and triangular elements
using rectangular elements as much as possible. However, in the present work, we have inten-
tionally concentrated on the performance of the triangular elements and results shown here are
those obtained using only triangular elements.
2 Exact Solutions
The expressions for potential and flux at a point (X,Y,Z) in free space due to uniform source
distributed on a rectangular flat surface having corners situated at (x1, z1) and (x2, z2) has been
presented, validated and used in [13, 14] and, thus, is not being repeated here.
Here, we present the exact expressions necessary to compute the potential and flux due to a
right-angled triangular element of arbitrary size, as shown in Fig.1. It may be noted here that
the length in the X direction has been normalized, while that in the Z direction has been allowed
to be of any arbitrary magnitude, zM . From the figure, it is easy to see that in order to find
out the influence due to triangular element, we have imposed another restriction, namely, the
necessity that the X and Z axes coincide with the perpendicular sides of the right-angled triangle.
Both these restrictions are trivial and can be taken care of by carrying out suitable scaling and
appropriate vector transformations. It may be noted here that closed-form expressions for the
influence of rectangular and triangular elements having uniform singularity distributions have
been previously presented in [7, 8]. However, in these works, the expressions presented are quite
complicated and difficult to implement. In [13] and in the present work, the expressions we have
presented are lengthy, but completely straight-forward. As a result, the implementation issues
of the present expressions, in terms of the development of the ISLES library and the neBEM
solver are managed quite easily.
Figure 1: Right-angled triangular element with x-length 1 and an arbitrary z-length, zM ; P is
the point where the influence (potential and flux) is being computed
It is easy to show that the influence (potential) at a point P (X,Y,Z) due to uniform source
distributed on a right-angled triangular element as depicted in Fig.1 can be represented as a
multiple of
φ(X,Y,Z) =
∫ z(x)
dx dz
(X − x)2 + Y 2 + (Z − z)2
in which we have assumed that x1 = 0, z1 = 0, x2 = 1 and z2 = zM , as shown in the geometry
of the triangular element. The closed-form expression for the potential has been obtained using
symbolic integration [17] which was subsequently simplified through substantial effort. It is found
to be significantly more complicated in comparison to the expression for rectangular elements
presented in [13] and can be written as
( (zMY
2 −XG)(LP1 + LM1 − LP2 − LM2) + i |Y | (zMX +G)(LP1 − LM1 − LP2 + LM2)
−S1X(tanh−1(
R1 + iI1
D11|Z|
) + tanh−1(
R1 − iI1
D11|Z|
)− tanh−1(
R1 + iI2
D21|Z|
)− tanh−1(
R1 − iI2
D21|Z|
+iS1|Y |(tanh−1(
R1 + iI1
D11|Z|
)− tanh−1(R1 − iI1
D11|Z|
)− tanh−1(R1 + iI2
D21|Z|
) + tanh−1(
R1 − iI2
D21|Z|
1 + zM 2
log (
1 + zM 2D12 − E1√
1 + zM 2D21 − E2
) + 2Z log
D21 −X + 1
D11 −X
) + C (2)
where,
D11 =
(X − x1)2 + Y 2 + (Z − z1)2; D12 =
(X − x1)2 + Y 2 + (Z − z2)2
D21 =
(X − x2)2 + Y 2 + (Z − z1)2; I1 = (X − x1) |Y | ; I2 = (X − x2) |Y |
S1 = sign(z1 − Z); R1 = Y 2 + (Z − z1)2
E1 = (X + zM
2 − zMZ); E2 = (X − 1− zMZ),
G = zM (X − 1) + Z; H1 = Y 2 +G(Z − zM ); H2 = Y 2 +GZ
LP1 =
G− izM |Y |
(H1 +GD12) + i|Y |(E1 − izMD12)
−X + i|Y |
LM1 =
G+ izM |Y |
(H1 +GD12)− i|Y |(E1 − izMD12)
−X − i|Y |
LP2 =
G− izM |Y |
(H2 +GD21) + i|Y |(E2 − izMD21)
1−X + i|Y |
LM2 =
G+ izM |Y |
(H2 +GD21)− i|Y |(E2 − izMD21)
1−X − i|Y |
and C denotes a constant of integration.
Similarly, the flux components due to the above singularity distribution can also be repre-
sented through closed-form expressions as shown below:
Fx = −
( (G)(LP1 + LM1 − LP2 − LM2)− i |Y | (zM )(LP1 − LM1 − LP2 + LM2)
+S1(tanh
R1 + iI1
D11|Z|
) + tanh−1(
R1 − iI1
D11|Z|
)− tanh−1(
R1 + iI2
D21|Z|
)− tanh−1(
R1 − iI2
D21|Z|
1 + zM 2
log (
1 + zM 2D12 − E1√
1 + zM 2D21 − E2
) ) + C (3)
Fy = −
( (2zMY )(LP1 + LM1 − LP2 − LM2) + i |Y | (Sn(Y )G)(LP1 − LM1 − LP2 + LM2)
+iS1Sn(Y )(tanh
R1 + iI1
D11|Z|
)− tanh−1(
R1 − iI1
D11|Z|
)− tanh−1(
R1 + iI2
D21|Z|
) + tanh−1(
R1 − iI2
D21|Z|
)) ) + C
Fz = −
1 + zM
log (
1 + zM
2D21 − E2√
1 + zM
2D12 − E1
) + log
D11 −X
D21 −X + 1
) + C (5)
where Sn(Y ) implies the sign of the Y-coordinate and C indicates constants of integrations. It
is to be noted that the constants of different integrations are not the same. These expression
are expected to be useful in the mathematical modeling of physical processes governed by the
inverse square laws. Being exact and valid throughout the physical domain, they can be used to
formulate versatile solvers to solve multi-scale multi-physics problems governed by the Laplace
/ Poisson equations involving Dirichlet, Neumann or Robin boundary conditions.
3 Development of the ISLES library
Due to the tremendous popularity of the C language we have written the codes in the C pro-
gramming language. However, it should be quite simple to translate the library to other popular
languages such as FORTRAN or C++, since no special feature of the C language has been used
to develop the codes.
3.1 Validation of the exact expressions
The expressions for the rectangular element have been validated in detail in [13]. Here, we present
the results for triangular elements in fair detail. In Fig.2, we have presented a comparison of
potentials evaluated for a unit triangular element by using the exact expressions, as well as
by using numerical quadrature of high accuracy. The two results are found to compare very
well throughout. Please note that contours have been obtained on the plane of the element,
and thus, represents a rather critical situation. Similarly, Fig.3 shows a comparison between
the results obtained using closed-form expressions for flux and those obtained using numerical
quadrature. The flux considered here is in the Y direction and is along a line beginning from
(−2,−2,−2) and ending at (2, 2, 2). The comparison shows the commendable accuracy expected
from closed form expressions. In Fig.4(a) and 4(b), the surface plots of potential on the element
plane (XZ plane) and Y -flux on the XY plane have been presented from which the expected
significant increase in potential and sharp change in the flux value on the element is observed.
Thus, by using a small fraction of computational resources in comparison to those consumed in
numerical quadratures, ISLES can compute the exact value of potential and flux for singularities
distributed on triangular elements.
Figure 2: Potential contours on a triangular element computed using exact expressions and by
numerical quadrature
3.2 Near-field performance
In order to emphasize the accuracy of ISLES, we have considered the following severe situations
in the near-field region in which it is observed that the quadratures can match the accuracy of
ISLES only when a high degree of discretization is used. Please note that in these cases, the
value of zM has been considered to be 10. In Fig.5 we have presented the variation of potential
along a line on the element surface running parallel to the Z-axis of the triangular element (see
Fig.1) and going through the centroid of the element. It is observed that results obtained using
even a 100× 100 quadrature is quite unacceptable. In fact, by zooming on to the image, it can
-4 -3 -2 -1 0 1 2 3 4
Distance (a.u.)
100X100
Exact
Figure 3: Comparison of flux (in the Y direction) as computed by ISLES and numerical quadra-
ture along a diagonal line
be found that only the maximum discretization yields results that match closely to the exact
solution. It may be noted here that the potential is a relatively easier property to compute.
The difficulty of achieving accurate flux estimates is illustrated in the two following figures. The
variation of flux in the X-direction along the same line as used in Fig.5 has been presented in
Fig.6. Similarly, variation of Y -flux along a diagonal line (beginning at (-10,-10,-10) and ending
at (10,10,10) and piercing the element at the centroid) has been presented in Fig.7. From these
figures we see that the flux values obtained using the quadrature are always inaccurate even
if the discretization is as high as 100 × 100. We also observe that the estimates are locally
inaccurate despite the use of very high amount of discretization (200× 200 or 500). Specifically,
in the latter figure, even the highest discretization can not match the exact values at the peak,
while in the former only the highest one can correctly emulate the sharp change in the flux
value. It is also heartening to note that the values from the quadrature using higher amount of
discretization consistently converge towards the ISLES values.
3.3 Far field performance
It is expected that beyond a certain distance, the effect of the singularity distribution can
be considered to be the same as that of a centroidally concentrated singularity or a simple
quadrature. The optimized amount of discretization to be used for the quadrature can be
determined from a study of the speed of execution of each of the functions in the library and has
been presented separately in a following sub-section. If we plan to replace the exact expressions
by quadratures (in order to reduce the computational expenses, presumably) beyond a certain
given distance, the quadrature should necessarily be efficient enough to justify the replacement.
While standard but more elaborate algorithms similar to the fast multipole method (FMM) [18]
along with the GMRES [19] matrix solver can lead to further of computational efficiency, the
(a) Potential surface (b) Flux surface
Figure 4: (a) Potential surface due to a triangular source distribution on the element plane, (b)
Flux (in the Y direction) surface due to a triangular source distribution on the XY plane at Z=0
simple approach as outlined above can help in reducing a fair amount of computational effort.
In the following, we present the results of numerical experiments that help us in determining
the far-field performance of the exact expressions and quadratures of various degrees that, in
turn, help us in choosing the more efficient approach for a desired level of accuracy.
In Fig.8 we have presented potential values obtained using the exact approach, 100 × 100,
10 × 10 and no discretization, i.e., the usual BEM approximation while using the zeroth order
piecewise uniform charge density assumption. The potentials are computed along a diagonal line
running from (-1000, -1000, -1000) to (1000, 1000, 1000) which pierces a triangular element of
zM = 10. It can be seen that results obtained using the usual BEM approach yields inaccurate
results as we move closer than distances of 10 units, while the 10 × 10 discretization yields
acceptable results up to a distance of 1.0 unit. In order to visualize the errors incurred due
to the use of quadratures, we have plotted Fig.9 where the errors incurred (normalized with
respect to the exact value) have been plotted. From this figure we can conclude that for the
given diagonal line, the error due to the usual BEM approximation falls below 1% if the distance
is larger than 20 units while for the simple 10×10 discretization, it is 2 units. It may be mentioned
here that along the axes the error turns out to be significantly more [13] and the limits need
to be effectively doubled to achieve the accuracy for all cases possible. Thus, for achieving 1%
accuracy, the usual BEM is satisfactory only if the distance of the influenced point is five times
the longer side of an element. Please note here that the error drops to 1 out of 106 as the
distance becomes fifty times the longer side. Besides proving that the exact expressions work
equally well in the near-field as well as the far-field, this fact justifies the usual BEM approach
for much of the computational domain leading to substantial savings in computational expenses.
The accuracy of the exact expressions used in the ISLES library is confirmed from the above
comparisons. However, there are several other important issues related to the development of
the library that are discussed below briefly.
3.4 Evaluation of the component functions
Many of the irrational and transcendental functions have domains and ranges in which they are
defined. Moreover, they are often multiply defined in the complex domain; for example, there
0 2 4 6 8 10
Distance along Z
10by10
50by50
100by100
200by200
500by500
IslesLib
AppFlag
Figure 5: Variation of potential along a centroidal line on the XZ plane parallel to the Z axis
for a triangular element: comparison among values obtained using the exact expressions and
numerical quadratures
are an infinite number of complex values for the logarithm function. In such cases, only one
principal value must be returned by the function. In general, such values cannot be chosen so
as to make the range continuous and thus, lines in the domain called branch cuts need to be
defined, which in turn define the discontinuities in the range. While evaluating expressions such
as the ones displayed in eqns.(2 - 5) a number of such problems are expected to occur. However,
when the expressions are analyzed at critical locations such as the corners and edges of the
element, it is observed that the terms likely to create difficulties while evaluating potentials are
either cancelled out or are themselves multiplied by zero. As a result, at these locations of likely
geometric and mathematical singularities, the solution behaves nicely. However, the same is
not true for the expressions related to the flux components. For these, we have to deal with
branch-cut problems in relation to tanh−1 and problems related to the evaluation of log(0). It
should be noted here that these singularities associated to the edges and corners of the elements
are of the weak type and it is expected that exact evaluation of these terms as well will be
possible through further work.
However, difficulties of a different nature crop up in these calculations which can be linked
directly to the limitation of the computer itself, namely, round-off errors [15]. These errors can
lead to severe problems while handling multi-scale problems such as those described in [13]. A
completely different approach is necessary to cope up with these difficulties, for example, the
use of extended range arithmetic [20], interval arithmetic [21] or the use of specialized libraries
such as the CORE library of the Exact Geometric Computation (EGC) initiative [22]. In the
present version of ISLES, a simple approach has been implemented which sets a lower limit to
various distance values. Below this value, the distance is considered to be zero. Plan of future
improvements in this regard has been kept at a high priority.
5 5.5 6 6.5 7 7.5 8
Distance along Z
10by10
50by50
100by100
200by200
500by500
IslesLib
AppFlag
Figure 6: Variation of flux in the X direction along a line on the XZ plane parallel to the Z axis
for a triangular element: comparison among values obtained using the exact expressions and
numerical quadratures
3.4.1 Algorithm
As discussed above, there are possibilities of facing problems while using the exact expressions
which may be due to the functions being evaluated or due to round-off errors leading to erro-
neous results. Moreover, despite providing many checks during the computation there is finite
possibility of ending up with a wrong value of a property indicated by its being Nan or inf
or potential due to unit positive singularity strength turning out to be negative. In order to
maintain the robustness of the library, we have tried to keep checks on the intermediate and final
values during the course of the computation. When the results are found to be unsatisfactory,
unphysical, we have re-estimated the results by using numerical quadrature and kept a track of
the cause by raising a unique approximation flag which is specific for a problem. As a result,
the steps for the calculation for a property can be written as follows:
• Get the required inputs - geometry of the element and the position where the effect needs
to be evaluated; Check whether the element size and distances are large enough so that
the results do not suffer from round-off errors.
• Check whether the location coincides with one of the special ones, such as corners or edges.
• Evaluate the necessary expressions in accordance with the foregoing results. If necessary,
consider each term in the expressions separately to sort out difficulties related to singular-
ities, branch-cuts or round-off errors. Note that if the multiplier is zero, rest of the term
does not need evaluation.
• If direct evaluation of the expressions fail, raise a unique approximation flag specific to this
problem and term and return the value of the property by using numerical quadrature.
-1 -0.5 0 0.5 1 1.5
Distance along diagonal
10by10
50by50
100by100
200by200
500by500
IslesLib
AppFlag
Figure 7: Comparison of flux (in the Y direction) along a diagonal line piercing the triangular
element at the centroid: comparison among values obtained using the exact expressions and
numerical quadratures
• Compute all the terms and find the final value, Check whether the final value is a number
and physically meaningful. If not satisfied, recompute the result using numerical quadra-
ture and raise the relevant approximation flag.
3.5 Speed of execution
The time taken to compute the potential and flux is an important parameter related to the
overall computational efficiency of the codes. This is true despite the fact that, in a typical
simulation, the time taken to solve the system of algebraic equation is far greater than the time
taken to build the influence coefficient matrix and post-processing. Moreover, the amount of
time taken to solve the system of equations tend to increase at a greater rate than the time
taken to complete the other two. It should be mentioned here that the time taken in each of
these steps can vary to a significant amount depending on the algorithm of the solver. In the
present case, the system of equations has been solved using lower upper decomposition using
the well known Crout’s partial pivoting. Although this method is known to be very rugged
and accurate, it is not efficient as far as number of arithmetic operations, and thus, time is
concerned. It is also possible to reduce the time taken to pre-process (generation of mesh and
creation of influence matrices), solve the system of algebraic equations and that for post-process
(computation of potential and flux at the required locations) can be significantly reduced by
adopting faster algorithms, including those involving parallelization.
In order to optimize the time taken to generate the influence coefficient matrix and that to
carry out the post-processing, we carried out a small numerical study to determine the amount
of time taken to complete the various functions being used in ISLES, especially those being used
to evaluate the exact expressions and those being used to carry out the quadratures. The results
1 10 100
Exact
Usual BEM
10 X 10
100 X 100
Figure 8: Potential along a diagonal through the triangular element computed using exact,
100 × 100, 10 × 10 and usual BEM approach
of the study (which was carried out using the linux system command gprof ) has been presented
in the following Table1.
Table 1: Time taken to evaluate exact expression and various quadratures
Method Exact Usual BEM 10× 10 100 × 100 500× 500
Time 0.8 µs 25 ns 1 µs 200 µs 5 ms
Please note that the numbers presented in this table are representative and are likely to
have statistical fluctuations. However, despite the fluctuations, it may be safely concluded that
a quadrature having only 10 × 10 discretization is already consuming time that is comparable
to that needed exact evaluation. Thus, the exact expressions, despite their complexity, are
extremely efficient in the near-field which can be considered at least as large as 0.5 times the
larger side of a triangular element (please refer to Fig.9). In making this statement, we have as-
sumed that the required accuracy for generating the influence coefficient matrix and subsequent
potential and flux calculations is 1%. This may not be acceptable at all under many practical
circumstances, in which case the near-field would imply a larger volume.
3.6 Salient features of ISLES
Development of usual BEM solvers are dependent on the two following assumptions:
• While computing the influences of the singularities, the singularities are modeled by a sum
of known basis functions with constant unknown coefficients. For example, in the constant
element approach, the singularities are assumed to be concentrated at the centroid of the
element, except for special cases such as self influence. This becomes necessary because
1e-07
1e-06
1e-05
1e-04
0.001
0.01
1 10 100 1000
100 X 100
10 X 10
Usual BEM
Figure 9: Error along a diagonal through the triangular element computed using 100 × 100,
10× 10 and usual BEM approach
closed form expressions for the influences are not, in general, available for surface elements.
An approximate and computationally rather expensive way of circumventing this limitation
is to use numerical integration over each element or to use linear or higher order basis
functions.
• The strengths of the singularities are solved depending upon the boundary conditions,
which, in turn, are modeled by the shape functions. For example, in the constant element
approach, it is assumed that it is sufficient to satisfy the boundary conditions at the
centroids of the elements. In this approach, the position of the singularity and the point
where the boundary condition is satisfied for a given element usually matches and is called
the collocation point.
The first (and possibly, the more damaging) approximation for BEM solvers can be relaxed by
using ISLES and can be restated as,
• The singularities distributed on the boundary elements are assumed to be uniform on a
particular element. The strength of the singularity may change from element to element.
This improvement turns out to be very significant as demonstrated in the following section and
some of our other studies involving microelectromechanical systems (MEMS) and gas detectors
for nuclear applications [13, 14]. Some of the advantages of using ISLES are itemized below:
• For a given level of discretization, the estimates are more accurate,
• Effective efficiency of the solver improves, as a result,
• Large variation of length-scales, aspect ratios can be tackled,
• Thinness of members or nearness of surfaces does not pose any problem,
• Curvature has no detrimental effect on the solution,
• The boundary condition can be satisfied anywhere on the elements, i.e., points other than
the centroidal points can be easily used, if necessary (for a corner problem, may be),
• The same formulation, library and solver is expected to work in majority of physical
situations. As a result, the necessity for specialized formulations of BEM can be greatly
minimized.
4 Capacitance of a unit square plate - a classic benchmark prob-
Using the neBEM solver, we have computed the capacitance of a unit square conducting plate
raised to a unit volt. This problem is still considered to be one of the major unsolved problems of
electrostatic theory [23, 8, 26, 16] and no analytical solution for this problem has been obtained
so far. The capacitance value estimated by the present method has been compared with very
accurate results available in the literature (using BEM and other methods). The results obtained
using the neBEM solver is found to be among the most accurate ones available till date as shown
in Table.2. Please note that we have not invoked symmetry or used extrapolation techniques to
arrive at our result presented in the table.
Table 2: Comparison of capacitance values
Reference Method Capacitance (pF) / 4 πǫ0
[23] Surface Charge 0.3607
[24] Surface Charge 0.362
[25] Surface Charge 0.367
[8] Refined Surface Charge 0.3667892 ± 1.1× 10−6
and Extrapolation
[26] Refined Boundary Element 0.3667874 ± 1× 10−7
and Extrapolation
[27] Numerical Path Integration 0.36684
[16] Random Walk 0.36 ± 0.01
This work neBEM 0.3660587
Finally, we consider the corner problem related to the electrostatics of the above conducting
plate. Problems of this nature are considered to be challenging for any numerical tool and
especially so for the BEM approach. The inadequacy of the BEM approach, especially in
solving the present problem, has been mentioned even quite recently [16] in which it has been
correctly mentioned that since the method can not extend its mathematical model to include
the edges and corners in reality, it is unlikely that it will ever succeed in modeling the edge
/ corner singularities correctly. As a result, with change in discretization, the properties near
these geometric singularities are expected to oscillate significantly leading to erroneous results.
However, as discussed above, the neBEM does extend its singularities distributed on the surface
elements right till an edge or a corner. Moreover, using neBEM, it is also possible to satisfy the
boundary conditions (both potential and flux) as close to the edge / corner as is required. In
fact, it should be possible to specify the potentials right at the edge / corner.
In the following study, we have presented estimates of charge density very close the flat plate
corner as obtained using neBEM. Please note that the boundary conditions have been satisfied
at the centroids of each element although we plan to carry out detailed studies of changing the
position of these points, especially in relation to problems involving edges / corners. In Fig.10,
charge densities very close to the corner of the flat plate estimated by neBEM using various
amounts of discretization have been presented. It can be seen that each curve follows the same
general trend, does not suffer from any oscillation and seems to be converging to a single curve.
This is true despite the fact that there has been almost an order of magnitude variation in the
element lengths.
Finally, in Fig.11, we present a least-square fitted straight line matching the charge density
as obtained the highest discretization in this study. It is found that the slope of the straight
line is 0.713567, which compares very well with both old and recent estimates of 0.7034 [28, 29].
This is despite the fact that here we have used a relatively coarse discretization near the corner.
It should be mentioned that none of the earlier references cited here used the BEM approach.
While the former used a singular perturbation technique, the latter used a diffusion based Monte-
Carlo method. Thus, it is extremely encouraging to note that using the neBEM approach, we
have been able to match the accuracy of these sophisticated techniques.
0 0.02 0.04 0.06 0.08 0.1
Distance from corner
0.0182
0.0091
0.0067
0.0047
0.0047
Figure 10: Corner charge density estimated by neBEM using various sizes of triangular elements
5 Conclusion
An efficient and robust library for solving potential problems in a large variety of science and
engineering problems has been developed. Exact closed-form expressions used to develop ISLES
have been validated throughout the physical domain (including the critical near-field region)
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5
log(r)
neBEM
Fitted line
Figure 11: Variation of charge density with increasing distance from the corner of the unit square
plate and a least-square fitted straight line: slope of the fitted line is 0.713567
by comparing these results with results obtained using numerical quadrature of high accuracy.
Algorithmic aspects of this development have also been touched upon. A classic benchmark
problem of electrostatics has been successfully simulated to very high precision. Charge density
values at critical geometric locations like corners have been found to be numerically stable and
physically acceptable. Several advantages over usual BEM solvers and other specialized BEM
solvers have been briefly mentioned. Work is under way to make the code more robust and
efficient through the implementation of more efficient algorithms and parallelization.
Acknowledgements
We would like to thank Professor Bikas Sinha, Director, SINP and Professor Sudeb Bhat-
tacharya, Head, INO Section, SINP for their support and encouragement during the course
of this work.
References
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a nearly exact BEM solver”, Eng Anal Boundary Elem, 30, pp.687-696.
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A, 566, pp.489-494.
[15] Mukhopadhyay, S., Majumdar, N., 2007, “Use of rectangular and triangular elements for
nearly exact BEM solutions”, Emerging Mechanical Technology - Macro to Nano, Research
Publishing Services, Chennai, India, pp.107-114 (ISBN: 81-904262-8-1).
[16] Wintle, H.J., 2004, “The capacitance of the cube and square plate by random walk meth-
ods”, J Electrostatics, 62 pp.51-62.
[17] Etter, D.M., 1997, Engineering Problem Solving with MatLab, Prentice Hall, International,
Inc., New Jersey 07458, USA.
[18] Greengard, L., Rokhlin, V., 1987, “A fast algorithm for particle simulation”, Journal of
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[19] Saad, Y., Schultz, M., 1986, “GMRES: A generalized minimal residual algorithm for solving
nonsymmetric linear systems”, SIAM J. Sci. Statist. Comput., 7 pp.856-869.
[20] Smith, J.M., Olver, F.W., Lozier, D.W., 1981, “Extended-Range Arithmetic and Normal-
ized Legendre Polynomial”, ACM Transactions on Mathematical Software, 7, pp.93-105.
[21] Alefeld, G., Herzberger, J., 1983, Introduction to interval analysis, Academic Press.
[22] http://cs.nyu.edu/exact/
[23] Maxwell, J.C., 1878, Electrical Research of the Honorable Henry Cavendish, p.426, Cam-
bridge University Press, Cambridge, UK.
[24] Reitan, D.K., Higgins, R.J., 1957, “Accurate determination of capacitance of a thin rect-
angular plate”, Trans AIEE, Part 1, 75, pp.761-766.
[25] Solomon, L., 1964, C.R.Acad.Sci III, 258, pp.64.
[26] Read, F.H., 1997, “Improved extrapolation technique in the boundary element method to
find the capacitance of the unit square and cube”, J Comput Phys, 133, pp.1-5.
[27] Mansfield, M.L., Douglas, J.F., Garboczi, E.J., 2001, “Intrinsic viscosity and the electrical
polarizability of arbitrarily shaped objects”, Phys Rev E, 64, 6, pp.061401-16.
[28] Morrison, J.A., Lewis, J.A., 1975, “Charge singularity at the corner of a flat plate”, SIAM
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http://cs.nyu.edu/exact/
Introduction
Exact Solutions
Development of the ISLES library
Validation of the exact expressions
Near-field performance
Far field performance
Evaluation of the component functions
Algorithm
Speed of execution
Salient features of ISLES
Capacitance of a unit square plate - a classic benchmark problem
Conclusion
|
0704.1564 | Entropy of eigenfunctions | ENTROPY OF EIGENFUNCTIONS
NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER
Abstra
t. We study the high�energy limit for eigenfun
tions of the lapla
ian, on a
ompa
t negatively
urved manifold. We review the re
ent result of Anantharaman�
Nonnenma
her [4℄ giving a lower bound on the Kolmogorov�Sinai entropy of semi
lassi
al
measures. The bound proved here improves the result of [4℄ in the
ase of variable negative
urvature.
1. Motivations
The theory of quantum
haos tries to understand how the
haoti
behaviour of a
lassi-
al Hamiltonian system is re�e
ted in its quantum
ounterpart. For instan
e, let M be a
ompa
t Riemannian C∞ manifold, with negative se
tional
urvatures. The geodesi
�ow
has the Anosov property, whi
h is
onsidered as the ideal
haoti
behaviour in the theory
of dynami
al systems. The
orresponding quantum dynami
s is the unitary �ow gener-
ated by the Lapla
e-Beltrami operator on L2(M). One expe
ts that the
haoti
properties
of the geodesi
�ow in�uen
e the spe
tral theory of the Lapla
ian. The Random Matrix
onje
ture [7℄ asserts that the large eigenvalues should, after proper unfolding, statisti-
ally resemble those of a large random matrix, at least for a generi
Anosov metri
. The
Quantum Unique Ergodi
ity
onje
ture [26℄ (see also [6, 30℄) des
ribes the
orresponding
eigenfun
tions ψk: it
laims that the probability measure |ψk(x)|
2dx should approa
h (in
the weak topology) the Riemannian volume, when the eigenvalue tends to in�nity. In fa
t
a stronger property should hold for the Wigner transform Wψ, a fun
tion on the
otangent
bundle T ∗M , (the
lassi
al phase spa
e) whi
h simultaneously des
ribes the lo
alization of
the wave fun
tion ψ in position and momentum.
We will adopt a semi
lassi
al point of view, that is
onsider the eigenstates of eigenvalue
unity of the semi
lassi
al Lapla
ian −~2△, thereby repla
ing the high-energy limit by the
semi
lassi
al limit ~ → 0. We denote by (ψk)k∈N an orthonormal basis of L
2(M) made of
eigenfun
tions of the Lapla
ian, and by (− 1
)k∈N the
orresponding eigenvalues:
(1.1) − ~2k△ψk = ψk, with ~k+1 ≤ ~k .
We are interested in the high-energy eigenfun
tions of −△, in other words the semi
lassi
al
limit ~k → 0.
The Wigner distribution asso
iated to an eigenfun
tion ψk is de�ned by
Wk(a) = 〈Op~k(a)ψk, ψk〉L2(M), a ∈ C
∗M) .
Here Op
is a quantization pro
edure, set at the s
ale (wavelength) ~k, whi
h asso
iates to
any smooth phase spa
e fun
tion a (with ni
e behaviour at in�nity) a bounded operator on
http://arxiv.org/abs/0704.1564v1
2 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
L2(M). See for instan
e [13℄ or [14℄ for various quantizations Op
. On a manifold,
one
an use lo
al
oordinates to de�ne Op in a �nite system of
harts, then glue the obje
ts
de�ned lo
ally thanks to a smooth partition of unity [11℄. For standard quantizations Op
the Wigner distribution is of the form Wk(x, ξ) dx dξ, where Wk(x, ξ) is a smooth fun
tion
on T ∗M ,
alled the Wigner transform of ψ. If a is a fun
tion on the manifold M , Op
an be taken as the multipli
ation by a, and thus we have Wk(a) =
a(x)|ψk(x)|
2dx: the
Wigner transform is thus a mi
rolo
al lift of the density |ψk(x)|
. Although the de�nition
ofWk depends on a
ertain number of
hoi
es, like the
hoi
e of lo
al
oordinates, or of the
quantization pro
edure (Weyl, anti-Wi
k, �right� or �left� quantization...), its asymptoti
behaviour when ~k −→ 0 does not. A
ordingly, we
all semi
lassi
al measures the limit
points of the sequen
e (Wk)k∈N, in the distribution topology.
In the semi
lassi
al limit, �quantum me
hani
s
onverges to
lassi
al me
hani
s�. We
will denote |·|x the norm on T
xM given by the metri
. The geodesi
�ow (g
t)t∈R is the
Hamiltonian �ow on T ∗M generated by the Hamiltonian H(x, ξ) =
|ξ|2x
. A quantization
of this Hamiltonian is given by the res
aled Lapla
ian −~
, whi
h generates the unitary
�ow (U t
) = (exp(it~△
)) a
ting on L2(M). The semi
lassi
al
orresponden
e of the �ows
) and (gt) is expressed through the Egorov Theorem :
Theorem 1.1. Let a ∈ C∞c (T
∗M). Then, for any given t in R,
(1.2) ‖U−t
(a)U t
(a ◦ gt)‖L2(M) = O(~) , ~ → 0 .
The
onstant implied in the remainder grows (often exponentially) with t, whi
h rep-
resents a notorious problem when one wants to study the large time behaviour of (U t
Typi
ally, the quantum-
lassi
al
orresponden
e will break down for times t of the order
of the Ehrenfest time (3.25).
Using (1.2) and other standard semi
lassi
al arguments, one shows the following :
Proposition 1.2. Any semi
lassi
al measure is a probability measure
arried on the energy
layer E = H−1(1
) (whi
h
oin
ides with the unit
otangent bundle S∗M). This measure is
invariant under the geodesi
�ow.
Let us
all M the set of gt-invariant probability measures on E . This set is
onvex
and
ompa
t for the weak topology. If the geodesi
�ow has the Anosov property � for
instan
e if M has negative se
tional
urvature � that set is very large. The geodesi
�ow has
ountably many periodi
orbits, ea
h of them
arrying an invariant probability
measure. There are many other invariant measures, like the equilibrium states obtained
by variational prin
iples [19℄, among them the Liouville measure µLiouv, and the measure
of maximal entropy. Note that, for all these examples of measures, the geodesi
�ow
a
ts ergodi
ally, meaning that these examples are extremal points in M. Our aim is to
determine, at least partially, the set Msc formed by all possible semi
lassi
al measures. By
its de�nition, Msc is a
losed subset of M, in the weak topology.
For manifolds su
h that the geodesi
�ow is ergodi
with respe
t to the Liouville measure,
it has been known for some time that almost all eigenfun
tions be
ome equidistributed over
E , in the semi
lassi
al limit. This property is dubbed as Quantum Ergodi
ity :
ENTROPY OF EIGENFUNCTIONS 3
Theorem 1.3. [27, 32, 11℄ Let M be a
ompa
t Riemannian manifold, assume that the
a
tion of the geodesi
�ow on E = S∗M is ergodi
with respe
t to the Liouville measure.
Let (ψk)k∈N be an orthonormal basis of L
2(M)
onsisting of eigenfun
tions of the Lapla
ian
(1.1), and let (Wk) be the asso
iated Wigner distributions on T
Then, there exists a subset S ⊂ N of density 1, su
h that
(1.3) Wk −→µLiouv, k → ∞, k ∈ S.
The question of existen
e of �ex
eptional� subsequen
es of eigenstates with a di�erent
behaviour is still open. On a negatively
urved manifold, the geodesi
�ow satis�es the
ergodi
ity assumption, and in fa
t mu
h stronger properties : mixing, K�property, et
.
For su
h manifolds, it has been postulated in the Quantum Unique Ergodi
ity
onje
ture
[26℄ that the full sequen
e of eigenstates be
omes semi
lassi
ally equidistributed over E :
one
an take S = N in the limit (1.3). In other words, this
onje
ture states that there
exists a unique semi
lassi
al measure, and Msc = {µLiouv}.
So far the most pre
ise results on this question were obtained for manifolds M with
onstant negative
urvature and arithmeti
properties: see Rudni
k�Sarnak [26℄, Wolpert
[31℄. In that very parti
ular situation, there exists a
ountable
ommutative family of
self�adjoint operators
ommuting with the Lapla
ian : the He
ke operators. One may
thus de
ide to restri
t the attention to
ommon bases of eigenfun
tions, often
alled �arith-
meti
� eigenstates, or He
ke eigenstates. A few years ago, Lindenstrauss [24℄ proved that
any sequen
e of arithmeti
eigenstates be
ome asymptoti
ally equidistributed. If there is
some degenera
y in the spe
trum of the Lapla
ian, note that it
ould be possible that the
Quantum Unique Ergodi
ity
onje
tured by Rudni
k and Sarnak holds for one orthonormal
basis but not for another. On su
h arithmeti
manifolds, it is believed that the spe
trum
of the Lapla
ian has bounded multipli
ity: if this is really the
ase, then the semi
lassi
al
equidistribution easily extends to any sequen
e of eigenstates.
Nevertheless, one may be less optimisti
when extending the Quantum Unique Ergod-
i
ity
onje
ture to more general systems. One of the simplest example of a symple
ti
Anosov dynami
al system is given by linear hyperboli
automorphisms of the 2-torus, e.g.
Arnold's �
at map�
. This system
an be quantized into a sequen
e ofN×N unitary
matri
es � the propagators, where N ∼ ~−1 [18℄. The eigenstates of these matri
es satisfy
a Quantum Ergodi
ity theorem similar with Theorem 1.3, meaning that almost all eigen-
states be
ome equidistributed on the torus in the semi
lassi
al limit [9℄. Besides, one
an
hoose orthonormal eigenbases of the propagators, su
h that the whole sequen
e of eigen-
states is semi
lassi
ally equidistributed [22℄. Still, be
ause the spe
tra of the propagators
are highly degenerate, one
an also
onstru
t sequen
es of eigenstates with a di�erent limit
measure [15℄, for instan
e, a semi
lassi
al measure
onsisting in two ergodi
omponents:
half of it is the Liouville measure, while the other half is a Dira
peak on a single (unsta-
ble) periodi
orbit. It was also shown that this half-lo
alization is maximal for this model
[16℄ : a semi
lassi
al measure
annot have more than half its mass
arried by a
ountable
union of periodi
orbits. The same type of half-lo
alized eigenstates were
onstru
ted by
two of the authors for another solvable model, namely the �Walsh quantization� of the
4 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
baker's map on the torus [3℄; for that model, there exist ergodi
semi
lassi
al measures of
purely fra
tal type (that is, without any Liouville
omponent). Another type of semi
las-
si
al measure was re
ently obtained by Kelmer for quantized hyperboli
automorphisms
on higher-dimensional tori [20℄: it
onsists in the Lebesgue measure on some invariant
o-isotropi
subspa
e of the torus.
For these Anosov models on tori, the
onstru
tion of ex
eptional eigenstates strongly
uses nongeneri
algebrai
properties of the
lassi
al and quantized systems, and
annot be
generalized to nonlinear systems.
2. Main result.
In order to understand the set Msc, we will attempt to
ompute the Kolmogorov�Sinai
entropies of semi
lassi
al measures. We work on a
ompa
t Riemannian manifold M of
arbitrary dimension, and assume that the geodesi
�ow has the Anosov property. A
tually,
our method
an without doubt be adapted to more general Anosov Hamiltonian systems.
The Kolmogorov�Sinai entropy, also
alled metri
entropy, of a (gt)-invariant probability
measure µ is a nonnegative number hKS(µ) that des
ribes, in some sense, the
omplexity of
a µ-typi
al orbit of the �ow. The pre
ise de�nition will be given later, but for the moment
let us just give a few fa
ts. A measure
arried on a
losed geodesi
has vanishing entropy.
In
onstant
urvature, the entropy is maximal for the Liouville measure. More generally,
for any Anosov �ow, the energy layer E is foliated into unstable manifolds of the �ow. An
upper bound on the entropy of an invariant probability measure is then provided by the
Ruelle inequality:
(2.1) hKS(µ) ≤
log Ju(ρ)dµ(ρ)
In this inequality, Ju(ρ) is the unstable Ja
obian of the �ow at the point ρ ∈ E , de�ned
as the Ja
obian of the map g−1 restri
ted to the unstable manifold at the point g1ρ (note
that the average of log Ju over any invariant measure is negative). The equality holds in
(2.1) if and only if µ is the Liouville measure on E [23℄. If M has dimension d and has
onstant se
tional
urvature −1, the above inequality just reads hKS(µ) ≤ d− 1.
Finally, an important property of the metri
entropy is that it is an a�ne fun
tional on
M. A
ording to the Birkho� ergodi
theorem, for any µ ∈ M and for µ�almost every
ρ ∈ E , the weak limit
µρ = lim
|t|−→∞
δgsρds
exists, and is an ergodi
probability measure. We
an then write
µρdµ(ρ),
whi
h realizes the ergodi
de
omposition of µ. The a�neness of the KS entropy means
hKS(µ) =
hKS(µ
ρ)dµ(ρ).
ENTROPY OF EIGENFUNCTIONS 5
An obvious
onsequen
e is the fa
t that the range of hKS on M is an interval [0, hmax].
In the whole arti
le, we
onsider a
ertain subsequen
e of eigenstates (ψkj )j∈N of the
Lapla
ian, su
h that the
orresponding sequen
e of Wigner distributions (Wkj)
onverges
to a semi
lassi
al measure µ. In the following, the subsequen
e (ψkj )j∈N will simply be
denoted by (ψ~)~→0, using the slightly abusive notation ψ~ = ψ~kj for the eigenstate ψkj .
Ea
h eigenstate ψ~ thus satis�es
(2.2) (−~2 △−1)ψ~ = 0 .
In [2℄ the �rst author proved that the entropy of any µ ∈ Msc is stri
tly positive. In [4℄,
more expli
it lower bounds were obtained. The aim of this paper is to improve the lower
bounds of [4℄ into the following
Theorem 2.1. Let µ be a semi
lassi
al measure asso
iated to the eigenfun
tions of the
Lapla
ian on M . Then its metri
entropy satis�es
(2.3) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
(d− 1)
λmax ,
where d = dimM and λmax = limt→±∞
log supρ∈E |dg
ρ| is the maximal expansion rate of
the geodesi
�ow on E .
In parti
ular, if M has
onstant se
tional
urvature −1, we have
(2.4) hKS(µ) ≥
In dimension d, we always have
log Ju(ρ)dµ(ρ)
≤ (d− 1)λmax ,
so the above bound is an improvement over the one obtained in [4℄,
(2.5) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
− (d− 1)λmax .
In the
ase of
onstant or little-varying
urvature, the bound (2.4) is mu
h sharper than
the one proved in [2℄. On the other hand, if the
urvature varies a lot (still being negative
everywhere), the right hand side of (2.3) may a
tually be negative, in whi
h
ase the
bound is trivial. We believe this �problem� to be a te
hni
al short
oming of our method,
and a
tually
onje
ture the following bound:
(2.6) hKS(µ) ≥
log Ju(ρ)dµ(ρ)
Extended to the
ase of the quantized torus automorphisms or the Walsh-quantized baker's
map, this bound is saturated for the half-lo
alized semi
lassi
al measures
onstru
ted in
[15℄, as well as those obtained in [20, 3℄. This bound allows
ertain ergodi
omponents
to be
arried by
losed geodesi
s, as long as other
omponents have positive entropy. This
may be
ompared with the following result obtained by Bourgain and Lindenstrauss in the
ase of arithmeti
surfa
es :
6 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
Theorem 2.2. [8℄ Let M be a
ongruen
e arithmeti
surfa
e, and (ψj) an orthonormal
basis of eigenfun
tions for the Lapla
ian and the He
ke operators.
Let µ be a
orresponding semi
lassi
al measure, with ergodi
de
omposition µ =
µρdµ(ρ).
Then, for µ-almost all ergodi
omponents we have hKS(µ
ρ) ≥ 1
As dis
ussed above, the Liouville measure is the only one satisfying hKS(µ) =
log Ju(ρ) dµ(ρ)
[23℄, so the Quantum Unique Ergodi
ity would be proven in one
ould repla
e 1/2 by 1 on
the right hand side of (2.6). However, we believe that (2.6) is the optimal result that
an
be obtained without using mu
h more pre
ise information, like for instan
e a sharp
ontrol
on the spe
tral degenera
ies, or �ne information on the lengths of
losed geodesi
s.
Indeed, in the above mentioned examples of Anosov systems where the Quantum Unique
Ergodi
ity
onje
ture is wrong and the bound (2.6) sharp, the quantum spe
trum has very
high degenera
ies, whi
h
ould be responsible for the possibility to
onstru
t ex
eptional
eigenstates. Su
h high degenera
ies are not expe
ted in the
ase of the Lapla
ian on a neg-
atively
urved manifold. For the moment, however, there is no
lear understanding of the
pre
ise relation between spe
tral degenera
ies and failure of Quantum Unique Ergodi
ity.
A
knowledgements. N.A and S.N. were partially supported by the Agen
e Nationale
de la Re
her
he, under the grant ANR-05-JCJC-0107-01. They bene�ted from numerous
dis
ussions with Y. Colin de Verdière and M. Zworski. S.N. is grateful to the Mathemati
al
Department in Bonn for its hospitality in De
ember 2006.
3. Outline of the proof
We start by re
alling the de�nition and some properties of the metri
entropy asso
iated
with a probability measure on T ∗M , invariant through the geodesi
�ow. In �3.2 we extend
the notion of entropy to the quantum framework. Our approa
h is semi
lassi
al, so we want
the
lassi
al and quantum entropies to be
onne
ted in some way when ~ → 0. The weights
appearing in our quantum entropy are estimated in Thm. 3.1, whi
h was proven and used
in [2℄. In �3.2.1 we also
ompare our quantum entropy with several �quantum dynami
al
entropies� previously de�ned in the literature. The proof of Thm. 2.1 a
tually starts in
�3.3, where we present the algebrai
tool allowing us to take advantage of our estimates
(3.9) (or their optimized version given in Thm. 3.5), namely an �entropi
un
ertainty
prin
iple� spe
i�
of the quantum framework. From �3.4 on, we apply this �prin
iple� to
the quantum entropies appearing in our problem, and pro
eed to prove Thm. 2.1. Although
the method is basi
ally the same as in [4℄, several small modi�
ations allow to �nally obtain
the improved lower bound (2.3), and also simplify some intermediate proofs, as explained
in Remark 3.6.
3.1. De�nition of the metri
entropy. In this paper we will meet several types of
entropies, all of whi
h are de�ned using the fun
tion η(s) = −s log s, for s ∈ [0, 1]. We
start with the Kolmogorov-Sinai entropy of the geodesi
�ow with respe
t to an invariant
probability measure.
Let µ be a probability measure on the
otangent bundle T ∗M . Let P = (E1, . . . , EK) be
a �nite measurable partition of T ∗M : T ∗M =
i=1Ei. We will denote the set of indi
es
ENTROPY OF EIGENFUNCTIONS 7
{1, . . . , K} = [[1, K]]. The Shannon entropy of µ with respe
t to the partition P is de�ned
hP(µ) = −
µ(Ek) logµ(Ek) =
µ(Ek)
For any integer n ≥ 1, we denote by P∨n the partition formed by the sets
(3.1) E
= Eα0 ∩ g
−1Eα1 . . . ∩ g
−n+1Eαn−1 ,
where α = (α0, . . . , αn−1)
an be any sequen
e in [[1, K]]
(su
h a sequen
e is said to be
of length |α| = n). The partition P∨n is
alled the n-th re�nement of the initial partition
P = P∨1. The entropy of µ with respe
t to P∨n is denoted by
(3.2) hn(µ,P) = hP∨n(µ) =
α∈[[1,K]]n
If µ is (gt)�invariant, it follows from the
onvexity of the logarithm that
(3.3) ∀n,m ≥ 1, hn+m(µ,P) ≤ hn(µ,P) + hm(µ,P),
in other words the sequen
e (hn(µ,P))n∈N is subadditive. The entropy of µ with respe
t
to the a
tion of the geodesi
�ow and to the partition P is de�ned by
(3.4) hKS(µ,P) = lim
hn(µ,P)
= inf
hn(µ,P)
Ea
h weight µ(E
) measures the µ�probability to visit su
essively Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 through the geodesi
�ow. Roughly speaking, the entropy measures
the exponential de
ay of these probabilities when n gets large. It is easy to see that
hKS(µ,P) ≥ β if there exists C su
h that µ(Eα) ≤ C e
, for all n and all α ∈ [[1, K]]
Finally, the Kolmogorov-Sinai entropy of µ with respe
t to the a
tion of the geodesi
�ow is de�ned as
(3.5) hKS(µ) = sup
hKS(µ,P),
the supremum running over all �nite measurable partitions P. The
hoi
e to
onsider the
time 1 of the geodesi
�ow in the de�nition (3.1) may seem arbitrary, but the entropy has
a natural s
aling property : the entropy of µ with respe
t to the �ow (gat) is |a|�times its
entropy with respe
t to (gt).
Assume µ is
arried on the energy layer E . Due to the Anosov property of the geodesi
�ow on E , it is known that the supremum (3.5) is rea
hed as soon as the diameter of the
partition P ∩ E (that is, the maximum diameter of its elements Ek ∩ E) is small enough.
Furthermore, let us assume (without loss of generality) that the inje
tivity radius of M is
larger than 1. Then, we may restri
t our attention to partitions P obtained by lifting on
E a partition of the manifoldM , that is take M =
k=1Mk and then Ek = T
∗Mk. In fa
t,
if the diameter of Mk in M is of order ε, then the diameter of the partition P
∨2 ∩E in E is
also of order ε. This spe
ial
hoi
e of our partition is not
ru
ial, but it simpli�es
ertain
aspe
ts of the analysis.
8 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The existen
e of the limit in (3.4), and the fa
t that it
oin
ides with the in�mum, follow
from a standard subadditivity argument. It has a
ru
ial
onsequen
e : if (µi) is a sequen
e
of (gt)�invariant probability measures on T ∗M , weakly
onverging to a probability µ, and
if µ does not
harge the boundary of the partition P, we have
hKS(µ,P) ≥ lim sup
hKS(µi,P) .
In parti
ular, assume that for i large enough, the following estimates hold :
(3.6) ∀n ≥ 1, ∀α ∈ [[1, K]]
, µi(Eα) ≤ Ci e
−βn ,
with β independent of i. This implies for i large enough hKS(µi,P) ≥ β, and this estimate
goes to the limit to yield hKS(µ) ≥ β.
3.2. From
lassi
al to quantum dynami
al entropy. Sin
e our semi
lassi
al measure
µ is de�ned as a limit of Wigner distributions W~, a naive idea would be to estimate
from below the KS entropy of W~ and then take the limit ~ → 0. This idea
annot work
dire
tly, be
ause the Wigner transformsW~ are neither positive, nor are they (g
t)�invariant.
Therefore, one
annot dire
tly use the (formal) integrals W~(Eα) =
W~(x, ξ) dx dξ to
ompute the entropy of the semi
lassi
al measure.
Instead, the method initiated by the �rst author in [2℄ is based on the following remarks.
Ea
h integral W~(Eα)
an also be written as W~(1lEα) =
W~ 1lEα , where 1lEα is the
hara
teristi
fun
tion on the set E
, that is
(3.7) 1lEα = (1lEαn−1 ◦ g
n−1)× . . .× (1lEα1 ◦ g)× 1lEα0 .
Remember we took Ek = T
∗Mk, where the Mk form a partition of M .
From the de�nition of the Wigner distribution, this integral
orresponds formally to the
overlap 〈ψ~,Op~(1lEα )ψ~〉. Yet, the
hara
teristi
fun
tions 1lEα have sharp dis
ontinuities,
so their quantizations
annot be in
orporated in a ni
e pseudodi�erential
al
ulus. Besides,
the set E
is not
ompa
tly supported, and shrinks in the unstable dire
tion when n =
|α| −→ +∞, so that the operator Op
(1lEα ) is very problemati
.
We also note that an overlap of the form 〈ψ~,Op~(1lEα)ψ~〉 is a hybrid expression: this is
a quantum matrix element of an operator de�ned in terms of the
lassi
al evolution (3.7).
From the point of view of quantum me
hani
s, it is more natural to
onsider, instead, the
operator obtained as the produ
t of Heisenberg-evolved quantized fun
tions, namely
(3.8) (U−n+1
Pαn−1U
) (U−n+2
Pαn−2U
) · · · (U−1
Pα1U~)Pα0 .
Here we used the shorthand notation Pk = 1lMk , k ∈ [[1, K]] (multipli
ation operators). To
remedy the fa
t that the fun
tions 1lMk are not smooth, whi
h would prevent us from using
a semi
lassi
al
al
ulus, we apply a
onvolution kernel to smooth them, obtain fun
tions
1lsmMk ∈ C
∞(M), and
onsider Pk
= 1lsmMk (we
an do this keeping the property
k=1 1l
In the following, we will use the notation A(t)
= U−t
for the Heisenberg evolution
of the operator A though the S
hrödinger �ow U t
= exp(−it~△
). The norm ‖•‖ will denote
ENTROPY OF EIGENFUNCTIONS 9
either the Hilbert norm on L2(M), or the
orresponding operator norm. The subsequent
�purely quantum� norms were estimated in [2, Thm. 1.3.3℄:
Theorem 3.1. (The main estimate [2℄) Set as above Pk
= 1lsmMk . For every K > 0,
there exists ~K > 0 su
h that, uniformly for all ~ < ~K, for all n ≤ K| log ~|, for all
(α0, . . . , αn−1) ∈ [[1, K]]
(3.9) ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 ψ~‖ ≤ 2(2π~)
−d/2 e−
n(1 +O(ε))n.
The exponent Λ is given by the �smallest expansion rate�:
Λ = − sup
log Ju(ρ)dν(ρ) = inf
λ+i (γ).
The in�mum on the right hand side runs over the set of
losed orbits on E , and the λ+i denote
the positive Lyapunov exponents along the orbit, that is the logarithms of the expanding
eigenvalues of the Poin
aré map, divided by the period of the orbit. The parameter ε > 0
is an upper bound on the diameters of the supports of the fun
tions 1lsmMk in M .
From now on we will
all the produ
t operator
(3.10) P
= Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0 , α ∈ [[1, K]]
To prove the above estimate, one a
tually
ontrols the operator norm
(3.11) ‖P
(χ)‖ ≤ 2(2π~)−d/2 e−
n(1 +O(ε))n ,
where χ ∈ C∞c (E
ε) is an energy
uto� su
h that χ = 1 near E , supported inside a neigh-
bourhood Eε = H−1([1
− ε, 1
+ ε]) of E .
In quantum me
hani
s, the matrix element 〈ψ~, Pαψ~〉 looks like the �probability�, for a
parti
le in the state ψ~, to visit su
essively the phase spa
e regions Eα0 , Eα1 , . . . , Eαn−1 at
times 0, 1, . . . , n − 1 of the S
hrödinger �ow. Theorem 3.1 implies that this �probability�
de
ays exponentially fast with n, with rate Λ
, but this de
ay only starts around the time
(3.12) n1
d| log ~|
whi
h is a kind of �Ehrenfest time� (see (3.25) for another de�nition of Ehrenfest time).
Yet, be
ause the matrix elements 〈ψ~, Pαψ~〉 are not real in general, they
an hardly be
used to de�ne a �quantum measure�. Another possibility to de�ne the probability for the
parti
le to visit the sets Eαk at times k, is to take the squares of the norms appearing in
(3.9):
(3.13) ‖P
2 = ‖Pαn−1(n− 1)Pαn−2(n− 2) · · ·Pα0ψ~‖
Now we require the smoothed
hara
teristi
fun
tions 1lsmMi to satisfy the identity
(3.14)
1lsmMk(x)
= 1 for any point x ∈M .
10 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
We denote by Psm the smooth partition of M made by the fun
tions
(1lsmMk)
. The
orresponding set of multipli
ation operators (Pk)
= Pq forms a �quantum partition of
unity� :
(3.15)
P 2k = IdL2 .
For any n ≥ 1, we re�ne the quantum partition Pq into (Pα)|α|, as in (3.10). The weights
(3.13) exa
tly add up to unity, so it makes sense to
onsider the entropy
(3.16) hn(ψ~,Pq)
α∈[[1,K]]n
3.2.1. Conne
tion with other quantum entropies. This entropy appears to be a parti
ular
ase of the �general quantum entropies� des
ribed by Sªom
zy«ski and �y
zkowski [28℄,
who already had in mind appli
ations to quantum
haos. In their terminology, a family of
bounded operators π = (πk)
k=1 on a Hilbert spa
e H satisfying
(3.17)
π∗k πk = IdH
provides an �instrument� whi
h, to ea
h index k ∈ [[1,N ]], asso
iates the following map on
density matri
es:
ρ 7→ I(k)ρ = πk ρ π
k , a nonnegative operator with tr(I(k)ρ) ≤ 1 .
From a unitary propagator U and its adjoint a
tion Uρ = UρU−1, they propose to
onstru
t
the re�ned instrument
I(α)ρ
= I(αn−1) ◦ · · · U ◦ I(α1) ◦ U ◦ I(α0)ρ = U
−n+1 π
Un−1 , α ∈ [[1,N ]]
where we used (3.10) to re�ne the operators πk into πα. We obtain the probability weights
(3.18) tr(I(α)ρ) = tr(π
) , α ∈ [[1,N ]]
For any U-invariant density ρ, these weights provide an entropy
(3.19) hn(ρ, I) =
α∈[[1,N ]]n
tr(I(α)ρ)
One easily
he
ks that our quantum partition Pq = (Pk)
k=1 satis�es (3.17), and that if
one takes ρ = |ψ~〉〈ψ~| the weights tr(I(α)ρ) exa
tly
orrespond to our weights ‖Pαψ‖
Hen
e, the entropy (3.19)
oin
ides with (3.16).
Around the same time, Ali
ki and Fannes [1℄ used the same quantum partition (3.17)
(whi
h they
alled ��nite operational partitions of unity�) to de�ne a di�erent type of
entropy, now
alled the �Ali
ki-Fannes entropy� (the de�nition extends to general C∗-
dynami
al systems). For ea
h n ≥ 1 they extend the weights (3.18) to �o�-diagonal entries�
to form a N n ×N n density matrix ρn:
(3.20) [ρn]α′,α = tr(πα′ ρ π
), α,α′ ∈ [[1,N ]]
ENTROPY OF EIGENFUNCTIONS 11
The AF entropy of the system (U , ρ) is then de�ned as follows: take the Von Neumann
entropy of these density matri
es, hAFn (ρ, π) = tr η(ρn), then take lim supn→∞
hAFn (ρ, π)
and �nally take the supremum over all possible �nite operational partitions of unity π.
We mention that tra
es of the form (3.20) also appear in the �quantum histories� ap-
proa
h to quantum me
hani
s (see e.g. [17℄, and [28, Appendix D℄ for referen
es).
3.2.2. Naive treatment of the entropy hn(ψ~,Pq). For �xed |α| > 0, the Egorov theorem
shows that ‖P
onverges to the
lassi
al weight µ
(1lsmMα )
when ~ → 0, so for �xed
n > 0 the entropy hn(ψ~,Pq)
onverges to hn(µ,Psm), de�ned as in (3.2), the
hara
teristi
fun
tions 1lMk being repla
ed by their smoothed versions (1l
)2. On the other hand, from
the estimate (3.11) the entropies hn(ψ~,Pq) satisfy, for ~ small enough,
(3.21) hn(ψ~,Pq) ≥ n
Λ+O(ε)
− d| log ~|+O(1) ,
for any time n ≤ K| log ~|. For large times n ≈ K| log ~|, this provides a lower bound
hn(ψ~,Pq) ≥
Λ +O(ε)
+O(1/| log ~|) ,
whi
h looks very promising sin
e K
an be taken arbitrary large: we
ould be tempted to
take the semi
lassi
al limit, and dedu
e a lower bound hKS(µ) ≥ Λ.
Unfortunately, this does not work, be
ause in the range {n > n1} where the estimate
(3.21) is useful, the Egorov theorem breaks down, the weights (3.13) do not approximate
the
lassi
al weights µ
(1lsmMα )
, and there is no relationship between hn(ψ,Pq) and the
lassi
al entropies hn(µ,Psm).
This breakdown of the quantum-
lassi
al
orresponden
e around the Ehrenfest time is
ubiquitous for
haoti
dynami
s. It has been observed before when studying the
onne
tion
between the Ali
ki-Fannes entropy for the quantized torus automorphisms and the KS
entropy of the
lassi
al dynami
s [5℄: the quantum entropies hAFn (ψ~,Pq) follow the
lassi
al
hn(µ,Psm) until the Ehrenfest time (and therefore grow linearly with n), after whi
h they
�saturate�, to produ
e a vanishing entropy lim supn→∞
hAFn (ψ~,Pq).
To prove the Theorem 2.1, we will still use the estimates (3.11), but in a more subtle
way, namely by referring to an entropi
un
ertainty prin
iple.
3.3. Entropi
un
ertainty prin
iple. The theorem below is an adaptation of the en-
tropi
un
ertainty prin
iple
onje
tured by Deuts
h and Kraus [12, 21℄ and proved by
Massen and U�nk [25℄. These authors were investigating the theory of measurement in
quantum me
hani
s. Roughly speaking, this result states that if a unitary matrix has
�small� entries, then any of its eigenve
tors must have a �large� Shannon entropy.
Let (H, 〈., .〉) be a
omplex Hilbert spa
e, and denote ‖ψ‖ =
〈ψ, ψ〉 the asso
iated
norm. Consider a quantum partition of unity (πk)
k=1 on H as in (3.17). If ‖ψ‖ = 1,
we de�ne the entropy of ψ with respe
t to the partition π as in (3.16), namely hπ(ψ) =
12 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
‖πk ψ‖
. We extend this de�nition by introdu
ing the notion of pressure, asso
i-
ated to a family v = (vk)k=1,...,N of positive real numbers: the pressure is de�ned by
pπ,v(ψ)
‖πk ψ‖
‖πk ψ‖
2 log v2k.
In Theorem 3.2, we a
tually need two partitions of unity (πk)
k=1 and (τj)
j=1, and two
families of weights v = (vk)
k=1, w = (wj)
j=1, and
onsider the
orresponding pressures
pπ,v(ψ), pτ,w(ψ). Besides the appearan
e of the weights v, w, we bring another modi�
ation
to the statement in [25℄ by introdu
ing an auxiliary operator O.
Theorem 3.2. [4, Thm. 6.5℄ Let O be a bounded operator and U be an isometry on H.
De�ne c
(v,w)
O (U)
= supj,k wj vk ‖τj U π
kO‖, and V = maxk vk, W = maxj wj.
Then, for any ǫ ≥ 0, for any normalized ψ ∈ H satisfying
(3.22) ∀k = 1, . . . ,N , ‖(Id−O) πk ψ‖ ≤ ǫ ,
the pressures pτ,w
, pπ,v
satisfy
+ pπ,v
≥ −2 log
(v,w)
O (U) +N V W ǫ
Example 1. The original result of [25℄
orresponds to the
ase where H = CN , O = Id,
ǫ = 0, N = M, vk = wj = 1, and the operators πk = τk are the orthogonal proje
tors on
some orthonormal basis (ek)
k=1 of H. In this
ase, the theorem asserts that
hπ(U ψ) + hπ(ψ) ≥ −2 log c(U)
where c(U) = supj,k |〈ek,Uej〉| is the supremum of all matrix elements of U in the orthonor-
mal basis (ek). As a spe
ial
ase, one gets hπ(ψ) ≥ − log c(U) if ψ is an eigenfun
tion of
3.4. Applying the entropi
un
ertainty prin
iple to the Lapla
ian eigenstates.
In this se
tion we explain how to use Theorem 3.2 in order to obtain nontrivial information
on the quantum entropies (3.16) and then hKS(µ). For this we need to de�ne the data
to input in the theorem. Ex
ept the Hilbert spa
e H = L2(M), all other data depend on
the semi
lassi
al parameter ~: the quantum partition π, the operator O, the positive real
number ǫ, the weights (vj), (wk) and the unitary operator U .
As explained in se
tion 3.2, we partition M into M = ⊔Kk=1Mk,
onsider open sets
Ωk ⊃⊃Mk (whi
h we assume to have diameters ≤ ε), and
onsider smoothed
hara
teristi
fun
tions 1lsmMk supported respe
tively inside Ωk, and satisfying the identity (3.14). The
asso
iated multipli
ation operators on H are form a quantum partition (Pk)
k=1, whi
h we
had
alled Pq. To alleviate notations, we will drop the subs
ript q.
From (3.15), and using the unitarity of U~, one realizes that for any n ≥ 1, the families
of operators P∨n = (P ∗
)|α|=n and T
∨n = (P
)|α|=n (see (3.10)) make up two quantum
partitions of unity as in (3.17), of
ardinal Kn.
ENTROPY OF EIGENFUNCTIONS 13
3.4.1. Sharp energy lo
alization. In the estimate (3.11), we introdu
ed an energy
uto� χ
on a �nite energy strip Eε, with χ ≡ 1 near E . This
uto� does not appear in the estimate
(3.9), be
ause, when applied to the eigenstate ψ~, the operator Op~(χ) essentially a
ts like
the identity.
The estimate (3.11) will a
tually not su�
e to prove Theorem 2.1. We will need to
optimize it by repla
ing χ in (3.11) with a �sharp� energy
uto�. For some �xed (small)
δ ∈ (0, 1), we
onsider a smooth fun
tion χδ ∈ C
∞(R; [0, 1]), with χδ(t) = 1 for |t| ≤ e
and χδ(t) = 0 for |t| ≥ 1. Then, we res
ale that fun
tion to obtain the following family of
~-dependent
uto�s near E :
(3.23) ∀~ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗M, χ(n)(ρ; ~)
e−nδ ~−1+δ(H(ρ)− 1/2)
The
uto� χ(n) is supported in a tubular neighbourhood of E of width 2~1−δ enδ. We will
always assume that this width is << ~1/2 in the semi
lassi
al limit, whi
h is the
ase if we
ensure that n ≤ Cδ| log ~| for some 0 < Cδ < (2δ)
−1−1. In spite of their singular behaviour,
these
uto�s
an be quantized into pseudodi�erential operators Op(χ(n)) des
ribed in [4℄
(the quantization uses a pseudodi�erential
al
ulus adapted to the energy layer E , drawn
from [29℄). The eigenstate ψ~ is indeed very lo
alized near E , sin
e it satis�es
(3.24) ‖
Op(χ(0))− 1
ψ~‖ = O(~
∞) ‖ψ~‖ .
In the rest of the paper, we also �x a small δ′ > 0, and
all �Ehrenfest time� the ~-dependent
integer
(3.25) nE(~)
⌊(1− δ′)| log ~|
Noti
e the resemblan
e with the time n1 de�ned in (3.12). The signi�
an
e of this time
s
ale will be dis
ussed in �3.4.5.
The following proposition states that the operators (P ∗
)|α|=nE , almost preserve the en-
ergy lo
alization of ψ~ :
Proposition 3.3. For any L > 0, there exists ~L su
h that, for any ~ ≤ ~L, the Lapla
ian
eigenstate satis�es
(3.26) ∀α, |α| = nE , ‖
Op(χ(nE))− Id
ψ~‖ ≤ ~
L‖ψ~‖ .
We re
ognize here a
ondition of the form (3.22).
3.4.2. Applying Theorem 3.2: Step 1. We now pre
ise some of the data we will use in the
entropi
un
ertainty prin
iple, Theorem 3.2. As opposed to the
hoi
e made in [4℄, we will
use two di�erent partitions π, τ .
• the quantum partitions π and τ are given respe
tively by the families of operators
π = P∨nE = (P ∗
)|α|=nE , τ = T
∨nE = (P
)|α|=nE . Noti
e that these partitions
only di�er by the ordering of the operators Pαi(i) inside the produ
ts. In the
semi
lassi
al limit, these partitions have
ardinality N = KnE ≍ ~−K0 for some
�xed K0 > 0.
• the isometry will be the propagator at the Ehrenfest time, U = UnE
14 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
• the auxiliarly operator is given as O = Op(χ(nE)), and the error ǫ = ~L, where L
will be
hosen very large (see �3.4.4).
• the weights v
will be sele
ted in �3.4.4. They will be semi
lassi
ally tempered,
meaning that there exists K1 > 0 su
h that, for ~ small enough, all vα, wα are
ontained in the interval [1, ~−K1].
The entropy and pressures asso
iated with a state ψ ∈ H are given by
hπ(ψ) =
|α|=nE
,(3.27)
pπ,v(ψ) = hπ(ψ)− 2
|α|=nE
ψ‖2 log v
.(3.28)
With respe
t to the se
ond partition, we have
hτ (ψ) =
|α|=nE
,(3.29)
pτ,w(ψ) = hτ (ψ)− 2
|α|=nE
ψ‖2 logw
.(3.30)
We noti
e that the entropy hτ (ψ) exa
tly
orresponds to the formula (3.16), while hπ(ψ)
is built from the norms
ψ‖2 = ‖Pα0Pα1(1) · · ·Pαn−1(n− 1)ψ‖
If ψ is an eigenfun
tion of U~, the above norm
an be obtained from (3.13) by ex
hanging
U~ with U
, and repla
ing the sequen
e α = (α0, . . . , αn−1) by ᾱ
= (αn−1, . . . , α0). So the
entropies hπ(ψ) and hτ (ψ) are mapped to one another through the time reversal U~ → U
With these data, we draw from Theorem 3.2 the following
Corollary 3.4. For ~ > 0 small enough
onsider the data π, τ , U , O as de�ned above.
(3.31) c
O (U)
= max
|α|=|α′|=nE
Op(χ(nE))‖
Then for any normalized state φ satisfying (3.26),
pτ,w(U
φ) + pπ,v(φ) ≥ −2 log
O (U) + h
L−K0−2K1
From (3.26), we see that the above
orollary applies to the eigenstate ψ~ if ~ is small
enough.
The reason to take the same value nE for the re�ned partitions P
, T ∨nE and the
propagator U
is the following : the produ
ts appearing in c
O (U)
an be rewritten
(with U ≡ U~):
′ UnE P
= U−nE+1Pα′
U · · ·UPα′0UPαnE−1U · · ·UPα0 = U
ENTROPY OF EIGENFUNCTIONS 15
Thus, the estimate (3.11) with n = 2nE already provides an upper bound for the norms
appearing in (3.31) � the repla
ement of χ by the sharp
uto� χ(nE) does not harm the
estimate.
To prove Theorem 2.1, we a
tually need to improve the estimate (3.11), as was done in
[4℄, see Theorem 3.5 below. This improvement is done at two levels: we will use the fa
t
that the
uto�s χ(nE) are sharper than χ, and also the fa
t that the expansion rate of the
geodesi
�ow (whi
h governs the upper bound in (3.11)) is not uniform, but depends on
the sequen
e α.
Our
hoi
e for the weights v
will then be guided by the α-dependent upper bounds
given in Theorem 3.5. To state that theorem, we introdu
e some notations.
3.4.3. Coarse-grained unstable Ja
obian. We re
all that, for any energy λ > 0, the geodesi
�ow gt on the energy layer E(λ) = H−1(λ) ⊂ T ∗M is Anosov, so that the tangent spa
e
TρE(λ) at ea
h ρ ∈ T
∗M , H(ρ) > 0 splits into
TρE(λ) = E
u(ρ)⊕Es(ρ)⊕ RXH(ρ)
where Eu (resp. Es) is the unstable (resp. stable) subspa
e. The unstable Ja
obian Ju(ρ)
is de�ned by Ju(ρ) = det
|Eu(g1ρ)
(the unstable spa
es at ρ and g1ρ are equipped with
the indu
ed Riemannian metri
).
This Ja
obian
an be �dis
retized� as follows in the energy strip Eε ⊃ E . For any pair of
indi
es (α0, α1) ∈ [[1, K]]
, we de�ne
(3.32) Ju1 (α0, α1)
= sup
Ju(ρ) : ρ ∈ T ∗Ωα0 ∩ E
ε, g1ρ ∈ T ∗Ωα1
if the set on the right hand side is not empty, and Ju1 (α0, α1) = e
otherwise, where R > 0
is a �xed large number. For any sequen
e of symbols α of length n, we de�ne
(3.33) Jun(α)
= Ju1 (α0, α1) · · ·J
1 (αn−2, αn−1) .
Although Ju and Ju1 (α0, α1) are not ne
essarily everywhere smaller than unity, there exists
C, λ+, λ− > 0 su
h that, for any n > 0, for any α with |α| = n,
(3.34) C−1 e−n(d−1) λ+ ≤ Jun(α) ≤ C e
−n(d−1) λ− .
One
an take λ+ = λmax(1+ε), where λmax is the maximal expanding rate in Theorem. 2.1.
We now give our
entral estimate, easy to draw from [4, Corollary 3.4℄.
Theorem 3.5. Fix small positive
onstants ε, δ, δ′ and a
onstant 0 < Cδ < (2δ)
−1 − 1.
Take an open
over M =
k Ωk of diameter ≤ ε and an asso
iated quantum partition P =
k=1. There exists ~0 su
h that, for any ~ ≤ ~0, for any positive integer n ≤ Cδ| log ~|,
and any pair of sequen
es α, α
of length n,
(3.35) ‖P
′ Op(χ(n))‖ = ‖P
Op(χ(n))‖ ≤ C ~−
−δ enδ
Jun(α) J
The
onstant C only depends on the Riemannian manifold (M, g). If we take n = nE, this
takes the form
(3.36) ‖P
Op(χ(nE))‖ ≤ C ~−
d−1+cδ
JunE(α) J
(α′) ,
16 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
where c = 2 + 2λ−1max.
The idea of proof in Theorem 3.5 is rather simple, although the te
hni
al implementation
is
umbersome. We �rst show that for any normalized state ψ, the state Op(χ(n))ψ
an
be essentially de
omposed into a superposition of ~
−d| suppχ(n)| normalized Lagrangian
states, supported on Lagrangian manifolds transverse to the stable foliation. In fa
t the
Lagrangian states we work with are trun
ated δ�fun
tions, supported on lagrangians of the
form ∪tg
tS∗zM . The a
tion of the operator U
′ = Pα′
U · · ·UPα0 on su
h Lagrangian
states
an be analyzed through WKB methods, and is simple to understand at the
lassi
al
level : ea
h appli
ation of the propagator U stret
hes the Lagrangian along the unstable
dire
tion (the rate of stret
hing being des
ribed by the lo
al unstable Ja
obian), whereas
ea
h operator Pk �proje
ts� on a pie
e of Lagrangian of diameter ε. This iteration of
stret
hing and
utting a
ounts for the exponential de
ay. The αα
-independent fa
tor
on the right of (3.36) results from adding together the
ontributions of all the initial
Lagrangian states. Noti
e that this prefa
tor is smaller than in Theorem. 3.1 due to the
ondition Cδ < (2δ)
−1 − 1.
Remark 3.6. In [4℄ we used the same quantum partition P∨nE for π and τ in Theorem. 3.2.
As a result, we needed to estimate from above the norms ‖P ∗
′ UnE PαOp(χ
(nE))‖ (see [4,
Theorem. 2.6℄). The proof of this estimate was mu
h more involved than the one for
(3.36), sin
e it required to
ontrol long pie
es of unstable manifolds. By using instead the
two partitions P(n), T (n), we not only prove a more pre
ise lower bound (2.3) on the KS
entropy, but also short-
ir
uit some �ne dynami
al analysis.
3.4.4. Applying Theorem 3.2: Step 2. There remains to
hoose the weights (v
) to use
in Theorem 3.2. Our
hoi
e is guided by the following idea: in (3.31), the weights should
balan
e the variations (with respe
t to α,α′) in the norms, su
h as to make all terms in
(3.31) of the same order. Using the upper bounds (3.36), we end up with the following
hoi
e for all α of length nE :
= JunE(α)
−1/2 .
From (3.34), there exists K1 > 0 su
h that, for ~ small enough, all the weights are
ontained in the interval [1, ~−K1], as announ
ed in �3.4.2. Using these weights, the estimate
(3.36) implies the following bound on the
oe�
ient (3.31):
∀~ < ~0, c
O (U) ≤ C ~
− d−1+cδ
We
an now apply Corollary 3.4 to the parti
ular
ase of the eigenstates ψ~. We
hoose L
su
h that L−K0 − 2K1 > −
d−1+cδ
, so from Corollary 3.4 we draw the following
Proposition 3.7. Let (ψ~)~→0 be our sequen
e of eigenstates (2.2). In the semi
lassi
al
limit, the pressures of ψ~ satisfy
(3.37) pP∨nE ,v(ψ~) + pT ∨nE ,w(ψ~) ≥ −
(d − 1 + cδ)λmax
(1− δ′)
nE +O(1) .
ENTROPY OF EIGENFUNCTIONS 17
If M has
onstant
urvature we have log Jn
≤ −n(d − 1)λmax(1 − O(ε)) for all α of
length n, and the above lower bound
an be written
hP∨nE (ψ~) + hT ∨nE (ψ~) ≥ (d− 1)λmax
1 +O(ε, δ, δ′)
As opposed to (3.21), the above inequality provides a nontrivial lower bound for the quan-
tum entropies at the time nE , whi
h is smaller than the time n1 of (3.12), and will allow
to
onne
t those entropies to the KS entropy of the semi
lassi
al measure (see below).
3.4.5. Subadditivity until the Ehrenfest time. Even at the relatively small time nE , the
onne
tion between the quantum entropy h(ψ~,P
∨nE) and the
lassi
al h(µ,P∨nEsm ) is not
ompletely obvious: both are sums of a large number of terms (≍ ~−K0). Before taking
the limit ~ → 0, we will prove that a lower bound of the form (3.37) still holds if we
repla
e nE ≍ | log ~| by some �xed no ∈ N, and P
by the
orresponding quantum
partition P∨no . The link between quantum pressures at times nE and no is provided by
the following subadditivity property, whi
h is the semi
lassi
al analogue of the
lassi
al
subadditivity of pressures for invariant measures (see (3.3)).
Proposition 3.8 (Subadditivity). Let δ′ > 0. There is a fun
tion R(no, ~), and a real
number R > 0 independent of δ′, su
h that, for any integer no ≥ 1,
lim sup
|R(no, ~)| ≤ R
and with the following properties. For any small enough ~ > 0, any integers no, n ∈ N with
no + n ≤ nE(~), for any ψ~ normalized eigenstate satisfying (2.2), the following inequality
holds:
pP∨(no+n),v(ψ~) ≤ pP∨no ,v(ψ~) + pP∨n,v(ψ~) +R(no, ~) .
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~).
To prove this proposition, one uses a re�ned version of Egorov's theorem [10℄ to show that
the non�
ommutative dynami
al system formed by (U t
) a
ting (through Heisenberg) on
observables supported near E is (approximately)
ommutative on time intervals of length
nE(~). Pre
isely, we showed in [4℄ that, provided ε is small enough, for any a, b ∈ C
∀t ∈ [−nE(~), nE(~)], ‖[Op~(a)(t),Op~(b)]‖ = O(~
cδ′), ~ → 0 ,
and the implied
onstant is uniform with respe
t to t. Within that time interval, the oper-
ators Pαj (j) appearing in the de�nition of the pressures
ommute up to small semi
lassi
al
errors. This almost
ommutativity explains why the quantum pressures pP∨n,v(ψ~) satisfy
the same subadditivity property as the
lassi
al entropy (3.3), for times smaller than nE .
Thanks to this subadditivity, we may �nish the proof of Theorem. 2.1. Fixing no, using
for ea
h ~ the Eu
lidean division nE(~) = q(~)no + r(~) (with r(~) < no), Proposition 3.8
implies that for ~ small enough,
pP∨nE ,v(ψ~)
pP∨no ,v(ψ~)
pP∨r,v(ψ~)
R(no, ~)
18 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
The same inequality is satis�ed by the pressures pT ∨n,w(ψ~). Using (3.37) and the fa
t
that pP∨r,v(ψ~) stays uniformly bounded when ~ → 0, we �nd
(3.38)
pP∨no ,v(ψ~) + pT ∨no ,w(ψ~)
2(d− 1 + cδ)λmax
2(1− δ′)
2R(no, ~)
+Ono(1/nE) .
We are now dealing with quantum partitions P∨no , T ∨no , for n0 ∈ N independent of ~. At
this level the quantum and
lassi
al entropies are related through the (�nite time) Egorov
theorem, as we had noti
ed in �3.2.2. For any α of length no, the weights ‖Pαψ~‖
both
onverge to µ
(1lsmMα)
, where we re
all that
1lsmMα = (1l
Mαno−1
◦ gno−1)× . . .× (1lsmMα1
◦ g)× 1lsmMα0
Thus, both entropies hP∨no (ψ~), hT ∨no (ψ~) semi
lassi
ally
onverge to the
lassi
al entropy
hno(µ,Psm). As a result, the left hand side of (3.38)
onverges to
(3.39) 2
hno(µ,Psm)
|α|=no
(1lsmMα )
log Juno(α) .
Sin
e µ is gt-invariant and Juno has the multipli
ative stru
ture (3.33), the se
ond term in
(3.39)
an be simpli�ed:
|α|=no
(1lsmMα )
log Juno(α) = (no − 1)
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1) .
We have thus obtained the lower bound
(3.40)
hno(µ,Psm)
no − 1
α0,α1
(1lsmM(α0,α1)
log Ju1 (α0, α1)−
(d− 1 + cδ)λmax
2(1− δ′)
At this stage we may forget about δ and δ′. The above lower bound does not depend on
the derivatives of the fun
tions 1lsmMα , so the same bound
arries over if we repla
e 1l
the
hara
teristi
fun
tions 1lMα . We
an �nally let no tend to +∞, then let the diameter
ε tend to 0. The left hand side
onverges to hKS(µ) while, from the de�nition (3.32), the
sum in the right hand side of (3.40)
onverges to the integral
log Ju(ρ)dµ(ρ) as ε → 0,
whi
h proves (2.3).
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Math. 183, 191�253 (1999)
[30℄ A. Voros, Semi
lassi
al ergodi
ity of quantum eigenstates in the Wigner representation, Le
t.
Notes Phys. 93, 326-333 (1979) in: Sto
hasti
Behavior in Classi
al and Quantum Hamiltonian
Systems, G. Casati, J. Ford, eds., Pro
eedings of the Volta Memorial Conferen
e, Como, Italy,
1977, Springer, Berlin
http://math.berkeley.edu/~zworski
20 N. ANANTHARAMAN, H. KOCH, AND S. NONNENMACHER
[31℄ S.A. Wolpert, The modulus of
ontinuity for Γ0(m)/H semi-
lassi
al limits, Commun. Math.
Phys. 216, 313�323 (2001)
[32℄ S. Zeldit
h, Uniform distribution of the eigenfun
tions on
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CMLS, É
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hnique, 91128 Palaiseau, Fran
e
E-mail address : nalini�math.polyte
hnique.fr
Mathemati
al Institute, University of Bonn, Beringstraÿe 1,D-53115 Bonn, Germany
E-mail address : ko
h�math.uni-bonn.de
Servi
e de Physique Théorique, CEA/DSM/PhT, Unité de re
her
he asso
iée au CNRS,
CEA/Sa
lay, 91191 Gif-sur-Yvette, Fran
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E-mail address : snonnenma
her�
ea.fr
1. Motivations
2. Main result.
Acknowledgements
3. Outline of the proof
3.1. Definition of the metric entropy
3.2. From classical to quantum dynamical entropy
3.3. Entropic uncertainty principle
3.4. Applying the entropic uncertainty principle to the Laplacian eigenstates
References
|
0704.1565 | Vector meson production from a polarized nucleon | arXiv:0704.1565v2 [hep-ph] 29 Aug 2007
Preprint typeset in JHEP style - PAPER VERSION DESY-07-049
arXiv:0704.1565 [hep-ph]
Vector meson production from a polarized nucleon
M. Diehl
Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany
Abstract: We provide a framework to analyze the electroproduction process ep → epρ
with a polarized target, writing the angular distribution of the ρ decay products in terms
of spin density matrix elements that parameterize the hadronic subprocess γ∗p → ρp.
Using the helicity basis for both photon and meson, we find a representation in which the
expressions for a polarized and unpolarized target are related by simple substitution rules.
Keywords: Lepton-Nucleon Scattering, Spin and Polarization Effects.
http://arxiv.org/abs/0704.1565v2
Contents
1. Introduction 1
2. Kinematics and target polarization 2
3. Helicity amplitudes and spin density matrix 4
4. The angular distribution 9
5. Natural and unnatural parity 17
6. Positivity constraints 21
7. Mixing between transverse and longitudinal polarization 23
8. A note on non-resonant contributions 27
9. Summary 28
1. Introduction
Exclusive vector meson production has long played an important role in studying the
strong interaction. The seminal work [1, 2] has renewed interest in this process, showing
that in Bjorken kinematics it provides access to generalized parton distributions and thus
to a wealth of information on the structure of the proton. While most theoretical and
experimental studies so far are for an unpolarized proton, the particular interest of target
polarization became clear when it was pointed out that meson production on a transversely
polarized target is sensitive to the nucleon helicity-flip distribution E [3, 4]. This distri-
bution offers unique views on the orbital angular momentum carried by partons in the
proton [5, 6] and on the correlation between polarization and the spatial distribution of
partons [7]. Whereas the corresponding polarization asymmetry in deeply virtual Compton
scattering is under better theoretical control, vector meson production has the advantage
of a greater sensitivity to the distribution of gluons (which in Compton scattering only
enters at next-to-leading order in αs). This holds not only in the high-energy regime but
even in a wide range of fixed-target kinematics [8, 9, 10], where polarization measurements
are feasible at existing or planned experimental facilities.
A different motivation to study polarized exclusive ρ production is that this channel
plays a rather prominent role in semi-inclusive pion production [11, 12, 9], which has become
a privileged tool to study a variety of spin effects, see e.g. [13]. It is important to identify
– 1 –
kinematical regions where the exclusive channel ep → epρ → epπ+π− dominates semi-
inclusive observables, because in these regions great care must be taken when interpreting
the data in terms of semi-inclusive factorization.
Even with an unpolarized target, the spin structure of the process ep → epρ→ epπ+π−
is very rich, because the angular distribution of the final state contains information on the
helicities of the exchanged virtual photon and of the ρ meson, as was worked out in the
classical analysis of Schilling and Wolf [14]. Yet more detailed information is available with
target polarization [15]. Experiments on unpolarized targets have found that s-channel
helicity is approximately conserved in the transition from the γ∗ to the ρ, with helicity
changing amplitudes occurring at most at the 10% level [16, 17, 18, 19, 20]. This greatly
simplifies the spin structure of the process. The aim of the present paper is to provide
an analysis framework for exclusive ρ production on a polarized nucleon target, making as
explicit as possible the relation between the angular dependence of the cross section and
the helicity amplitudes describing the hadronic subprocess γ∗p→ ρp. We will present our
results in a form that emphasizes the close similarity in structure between an unpolarized
and a polarized target. Using the helicity basis for both virtual photon and meson, we also
provide an alternative to the representation of the unpolarized cross section in [14].
The following section gives the definitions of the kinematics and polarization variables
for the reaction under study. In Section 3 we define the helicity amplitudes and the spin
density matrix elements describing the process and discuss some of their general properties.
In Section 4 we express the angular distribution of the polarized cross section in terms of
these spin density matrix elements and point out some salient features of this representa-
tion. The simplifications arising from distinguishing natural and unnatural parity exchange
in the reaction are discussed in Section 5. A number of positivity bounds relating different
spin density matrix elements are given in Section 6. In Section 7 we explain the complica-
tions arising from the distinction between target polarization relative to the momentum of
either the incident lepton or the virtual photon. The role of non-resonant contributions in
π+π− production is briefly discussed in Section 8. Our results are summarized in Section 9.
2. Kinematics and target polarization
Let us consider the electroproduction process
e(l) + p(p) → e(l′) + p(p′) + ρ(q′) (2.1)
followed by the decay
ρ(q′) → π+(k) + π−(k′), (2.2)
where four-momenta are given in parentheses. Throughout this work we use the one-photon
exchange approximation. All or results are equally valid for the production of a φ followed
by the decay φ→ K+K−. They also hold if the scattered proton is replaced by an inclusive
system X with four-momentum p′, as explained at the end of Section 3.
To describe the kinematics we use the conventional variables for deep inelastic pro-
cesses, Q2 = −q2, xB = Q2/(2p · q) and y = (p · q)/(p · l). We neglect the lepton mass
– 2 –
lepton plane
hadron plane
θγ ST
Figure 1: Kinematics of ep → epρ in the target rest frame. ST is the transverse component of
the target spin vector w.r.t. the virtual photon direction.
throughout and denote the longitudinal lepton beam polarization by Pℓ, with Pℓ = +1
corresponding to a purely right-handed and Pℓ = −1 to a purely left-handed beam. Let us
now go to the target rest frame and introduce the right-handed coordinate system (x, y, z)
of Fig. 1 such that q points in the positive z direction and l has a positive x component. In
this system we have l = |l|(sin θγ , 0, cos θγ) and q = |q|(0, 0, 1), where the angle θγ between
l and q is defined to be between 0 and π. In accordance with the Trento convention [21]
we define the angle φ between the lepton and the hadron plane as the azimuthal angle of
q′ in this coordinate system, and φS as the azimuthal angle of the target spin vector S.
Following [22] we write S = (ST cosφS , ST sinφS ,−SL) with 0 ≤ ST ≤ 1 and −1 ≤ SL ≤ 1,
so that ST and SL describe transverse and longitudinal polarization with respect to the
virtual photon momentum, with SL = 1 corresponding to a right-handed proton in the
γ∗p c.m.
*γ c.m.p
boost
π π+ c.m.−
Figure 2: Kinematics of the hadronic subprocess γ∗p → ρp followed by the decay ρ → π+π−.
The coordinate systems (x, y, z) and (x′, y′, z′) differ from those in Fig. 1.
– 3 –
To describe the target polarization of a given experimental setup, we introduce an-
other right-handed coordinate system (x′, y′, z′) in the target rest frame such that l =
|l|(0, 0, 1) and q = |q|(− sin θγ , 0, cos θγ) as shown in Fig. 1. In this system we write
S = (PT cosψ,PT sinψ,−PL) with 0 ≤ PT ≤ 1 and −1 ≤ PL ≤ 1, following again [22]. PT
and PL describe transverse and longitudinal polarization with respect to the lepton beam
direction, with PL = 1 corresponding to a right-handed proton in the ep c.m. The two sets
of variables describing the target polarization are related by
ST cosφS = cos θγPT cosψ − sin θγPL ,
ST sinφS = PT sinψ ,
SL = sin θγPT cosψ + cos θγPL , (2.3)
which we will use in Sect. 7. In terms of invariants the mixing angle θγ is given by
sin θγ = γ
1− y − 1
1 + γ2
, γ =
2xBMN
, (2.4)
whereMN is the nucleon mass. In Bjorken kinematics γ is small, and so is sin θγ ≈ γ
1− y.
We finally specify the variables describing the vector meson decay (2.2). This is conve-
niently done in the π+π− c.m., which can be obtained from the γ∗p c.m. by a boost in the
direction of the scattered nucleon as shown in Fig. 2. In the π+π− c.m. we introduce the
right-handed coordinate system (x, y, z) shown in Fig. 2, where p′ = |p′|(0, 0,−1) and where
the target momentum p has a positive x component. In this system we define ϑ and ϕ as the
polar and azimuthal angle of the π+ momentum, i.e. k = |k|(sinϑ cosϕ, sinϑ sinϕ, cos ϑ).
The relation between our notation here and the one of Schilling and Wolf is1
φhere = −Φ[14] , ϕhere = φ [14] , ϑhere = θ [14] . (2.5)
3. Helicity amplitudes and spin density matrix
The strong-interaction dynamics of the electroproduction process (2.1) is fully contained
in the helicity amplitudes for the subprocess γ∗p→ ρp. From these we will construct spin
density matrix elements which describe the angular distribution of the overall reaction
ep → ep π+π− and its dependence on the target polarization.
Since we will deal with interference terms we must specify our phase conventions. We
do this in the γ∗p c.m. and use the right-handed coordinate system (x′, y′, z′) shown in
Fig. 2. In this system we have q = |q|(0, 0,−1) and q′ = |q′|(sinΘ, 0,− cos Θ), with the
scattering angle Θ of the vector meson defined to be between 0 and π. Note that the positive
z′ axis points along p rather than q, as is often preferred for theoretical calculations. We
specify polarization states of the target proton by two-component spinors χ+1/2 = (1, 0)
1We remark that the expression for sinΦ given in eq. (13) of [14] is incorrect since it is always positive.
A correct definition is given in [23].
– 4 –
for positive and χ−1/2 = (0, 1) for negative helicity. For the polarization vectors of the
virtual photon we choose
ε+1 = −
0, 1,−i, 0
, ε−1 =
0, 1, i, 0
εα0 = Nε
qα − q
p · q p
, (3.1)
and for the polarization vectors of the ρ
e+1 = −
0, cos Θ,−i, sinΘ
, e−1 =
0, cos Θ, i, sinΘ
eα0 = Ne
q′α − q
p′ · q′ p
, (3.2)
where the subscripts indicate helicities. Nε and Ne are positive constants ensuring the
proper normalization ε20 = 1 and e
0 = −1 of the longitudinal polarization vectors. In the
ρ rest frame and the coordinate system (x, y, z) of Fig. 2, our meson polarization vectors
have the standard form e+1 = −(0, 1, i, 0)/
2, e−1 = (0, 1,−i, 0)/
2 and e0 = (0, 0, 0, 1).
Our phase conventions for the proton and the virtual photon are as in [22].
We now introduce amplitudes T νσµλ for the subprocess γ
∗(µ)+ p(λ) → ρ(ν)+ p(σ) with
definite helicities µ, ν, λ, σ. Since the above phase conventions are defined with reference
only to momentum vectors of this subprocess, the helicity amplitudes only depend on the
photon virtuality, the γ∗p scattering energy and the scattering angle Θ, or equivalently on
Q2, xB and t = (p− p′)2. With our phase conventions they obey the usual parity relations
T−ν−σ−µ−λ = (−1)
ν−µ−σ+λ T νσµλ (3.3)
for equal Q2, xB and t on both sides. With these helicity amplitudes we define
µµ′,λλ′ = (NT + ǫNL)
T νσµλ
. (3.4)
Regarding the upper indices this is the spin density matrix of the vector meson, whereas the
lower indices specify the polarizations in the γ∗p state from which the meson is produced.2
The normalization factors
λ,ν,σ
∣T νσ+λ
, NL =
λ,ν,σ
∣T νσ0λ
(3.5)
are proportional to the differential cross sections dσT /dt and dσL/dt for transverse and
longitudinal photon polarization, respectively, and
1− y − 1
1− y + 1
y2 + 1
(3.6)
2Taking the trace in the meson polarization indices we obtain the relation
ρννµµ′,λλ′ ∝ dσ
µ′µ/dt
between the spin density matrix ρ introduced here and the cross sections and interference terms used in
[22]. Compared with [22] we take the opposite order of indices in ρ, so that ν and ν′ appear in the standard
order for a spin density matrix.
– 5 –
is the usual ratio of longitudinal and transverse photon flux. In addition to Q2, xB and
t, the spin density matrix elements ρνν
µµ′,λλ′ depend on ǫ through the normalization factor
(NT + ǫNL). If one can perform a Rosenbluth separation by measuring at different ǫ but
equal Q2 and xB, it is advantageous to normalize them instead to NT , NL or
NTNL as
was done in [14]. It is straightforward to implement such a change in the formulae we give
in the following.
We find it useful to introduce the combinations
µµ′ =
µµ′,++ + ρ
µµ′,−−
, lνν
µµ′ =
µµ′,++ − ρνν
µµ′,−−
(3.7)
for an unpolarized and a longitudinally polarized target, where for the sake legibility we
have labeled the target polarization by ± instead of ±1
. The combinations
µµ′ =
µµ′,+− + ρ
µµ′,−+
, nνν
µµ′ =
µµ′,+− − ρνν
µµ′,−+
(3.8)
respectively describe transverse target polarization in the hadron plane (“sideways”) and
perpendicular to it (“normal”). One readily finds that the matrices u , l and s are hermi-
tian, whereas n is antihermitian,
µ′µ =
µ′µ =
µ′µ =
µ′µ = −
. (3.9)
The diagonal elements uννµµ, l
µµ and s
µµ are therefore purely real, whereas n
µµ is purely
imaginary. Furthermore, the parity relations (3.3) translate into
u−ν−ν
−µ−µ′
= (−1)ν−µ−ν′+µ′ uνν′µµ′ , l−ν−ν
−µ−µ′
= −(−1)ν−µ−ν′+µ′ lνν′µµ′ ,
n−ν−ν
−µ−µ′ = (−1)
ν−µ−ν′+µ′ nνν
µµ′ , s
−ν−ν′
−µ−µ′ = −(−1)
ν−µ−ν′+µ′ sνν
µµ′ . (3.10)
As a consequence the matrix elements
u−+−+ , u
−+ , u
0 0 , u
−+ (3.11)
are purely real, whereas the corresponding elements of l , s and n are purely imaginary.
Both experiment and theory indicate that s-channel helicity is approximately conserved
in the γ∗ → ρ transition for small invariant momentum transfer t. Correspondingly, one
expects that spin density matrix elements involving the product of two helicity conserving
amplitudes are greater than interference terms between a helicity conserving and a helicity
changing amplitude, and that those are greater than matrix elements involving the product
of two helicity changing amplitudes (where we refer to the helicities of the photon and the
ρ but not of the nucleon). Exceptions to this rule are however possible, since two large
amplitudes can have a small interference term because of their relative phase, and since
there can be cancellation of individually large terms in the linear combinations (3.7) and
(3.8) associated with different target polarizations. With this caveat in mind one can
readily assess the expected size of the spin density matrix elements (3.7) and (3.8) by
comparing the upper with the lower indices.
– 6 –
Let us now investigate the behavior of our matrix elements for Θ → 0, i.e. in the limit
of forward scattering γ∗p→ ρp. To this end we perform a partial wave decomposition
T νσµλ (Θ) =
tνσµλ(J) d
λ−µ,σ−ν(Θ) (3.12)
where we have suppressed the dependence of T and the partial wave amplitudes t(J) on
Q2 and xB . Using the behavior d
m,n(Θ) ∼ Θ|m−n| of the rotation functions for Θ → 0 we
readily find
µµ′ , l
µµ′ ∼ Θp , nνν
µµ′ , s
µµ′ ∼ Θq (3.13)
p ≥ pmin = min
σ,λ=±1/2
∣ν − µ− σ + λ
∣ν ′ − µ′ − σ + λ
q ≥ qmin = min
σ,λ=±1/2
∣ν − µ− σ + λ
∣ν ′ − µ′ − σ − λ
. (3.14)
With Θ ∝ (t0 − t)1/2 for small Θ, we can rewrite (3.13) as
µµ′ , l
µµ′ ∼
(t0 − t)p/2 , nνν
µµ′ , s
µµ′ ∼
(t0 − t)q/2 , (3.15)
where t0 is the value of t for Θ = 0 at given Q
2 and xB. In Tables 1 and 2 we give
the corresponding powers for the linear combinations of spin density matrix elements that
will appear in our results for the cross section in Section 4. We have ordered the entries
according to the hierarchy discussed after (3.11), listing first terms containing the product
of two helicity conserving amplitudes, then terms containing the interference between a
helicity conserving and a helicity changing amplitude, and finally terms which only involve
helicity changing amplitudes (with helicities always referring to the photon and the ρ but
not to the nucleon).
We emphasize that certain partial wave amplitudes tνσµλ(J) in (3.12) may be zero or
negligibly small for dynamical reasons. The actual powers of (t0 − t)1/2 in (3.15) can
thus be larger than the minimum values pmin and qmin required by angular momentum
conservation. If there is for instance no s-channel helicity transferred between the proton-
proton and the photon-meson transitions, then the relevant powers for n and s are given
by q = pmin+1, which is equal to qmin+2 for all but the first four entries in Tables 1 and 2.
A concrete realization of this scenario is the calculation in [24], where the proton-proton
transition is described by the generalized parton distributions H, E and H̃, Ẽ, which do
not allow for helicity transfer to the photon-meson transition.
In the limit of large Q2 at fixed xB and t, the proof of the factorization theorem
in [2] implies that the transition from a longitudinal photon to a longitudinal ρ becomes
dominant, with all other transitions suppressed by powers of 1/Q. In this limit only the
spin density matrix elements u 0 00 0 and n
0 0 survive and can be expressed as convolutions of
hard-scattering kernels with generalized parton distributions and the light-cone distribution
amplitude of the ρ. To leading order in 1/Q one has in particular
Imn 0 00 0
u 0 00 0
t0 − t
1− ξ2 Im
(1− ξ2) |H|2 −
ξ2 + t/(4M2N )
|E|2 − 2ξ2 Re
, (3.16)
– 7 –
matrix elements pmin
u 0 0++ + ǫu
0 0 0
u 0+0+ − u
0+ − l
u++++ + u
++ + 2ǫu
0 0 l
++ + l
u−+−+ l
u 0 00+ l
u 0+++ − u−0++ + 2Re ǫu 0+0 0 l
++ − l−0++ + 2i Im ǫl 0+0 0 1
u 0+−+ l
u 0−0+ − u
0+ − l
u−+++ + ǫu
0 0 l
++ + ǫl
0 0 2
u++−+ l
u++0+ + u
0+ + l
u−+0+ l
l 0 0++ 2
u 0 0−+ l
u+0−+ l
u+−0+ l
u+−−+ l
Table 1: Minimum values of the powers which control the t→ t0 behavior of combinations of spin
density matrix elements u and l as in (3.15). Some of the combinations are purely real or purely
imaginary because of the symmetry relations (3.9) and (3.10), whereas others are complex valued.
where ξ = xB/(2−xB) and the convolution integrals H and E are for instance given in [22].
Experimental results and phenomenological analysis show however that 1/Q2 suppressed
effects can be numerically significant for Q2 of several GeV2, see e.g. [25, 24, 9, 10]. This
concerns both power corrections within u 0 00 0 or n
0 0 and formally power suppressed spin
density matrix elements such as u++++ or u
0+ . The detailed analysis in [2] reveals that
beyond leading-power accuracy in 1/Q, factorization of meson production into a hard-
scattering subprocess and nonperturbative quantities pertaining either to the target or
to the meson may be broken. On the other hand, factorization based approaches which
go beyond leading power in 1/Q and in particular also evaluate transition amplitudes for
transverse polarization of the γ∗ or ρ have been phenomenologically rather successful, see
e.g. [26, 24]
Let us finally generalize our considerations to the process
e(l) + p(p) → e(l′) +X(p′) + ρ(q′) , (3.17)
where the target proton dissociates into a hadronic system X. In analogy to the elastic case
one can introduce helicity amplitudes T
and combine them into spin density matrix
– 8 –
matrix elements qmin
n 0 0++ + ǫn
0 0 1
n 0+0+ − n
0+ − s
n++++ + n
++ + 2ǫn
0 0 s
++ + s
n−+−+ s
n 0 00+ s
n 0+++ − n−0++ + 2i Im ǫn 0+0 0 s
++ − s−0++ + 2i Im ǫs 0+0 0 0
n 0+−+ s
n 0−0+ − n
0+ − s
n−+++ + ǫn
0 0 s
++ + ǫs
0 0 1
n++−+ s
n++0+ + n
0+ + s
n−+0+ s
s 0 0++ 1
n 0 0−+ s
n+0−+ s
n+−0+ s
n+−−+ s
Table 2: As Table 1 but for combinations of spin density matrix elements n and s .
elements
µµ′,λλ′ = (NT + ǫNL)
ν′σ,X
. (3.18)
The normalization factors NT and NL are defined as in (3.5) but with an additional sum
over all hadronic states X of given invariant mass MX , on which ρ
µµ′,λλ′ now depends
in addition to Q2, xB , t and ǫ. The combinations (3.7) and (3.8) for different target
polarization have the same symmetry properties (3.9) and (3.10) as in the elastic case.
Their behavior for t→ t0 can be different, since in (3.14) one must now take the minimum
over all possible helicities σ = ±1
, . . . of the hadronic system X. One finds however
that the powers pmin and qmin for the combinations of spin density matrix elements in
Tables 1 and 2 are the same as in the elastic case. The results in the remainder of this
work only depend on the properties (3.9) and (3.10) and thus immediately generalize to
the case of target dissociation.
4. The angular distribution
The calculation of the cross section for ep → ep π+π− proceeds by using standard methods
and we shall only sketch the essential steps. More details are for instance given in [14, 27,
– 9 –
22]. With our phase conventions the polarization state of the proton target is described by
the spin density matrix
τλλ′ =
1 + SL ST e
−i(φ−φS)
i(φ−φS) 1− SL
, (4.1)
which is to be contracted with the matrix in (3.4). The result is conveniently expressed in
terms of the combinations (3.7) and (3.8) as
τλλ′ ρ
µµ′,λλ′ = u
µµ′ + SL l
µµ′ + ST cos(φ− φS) sνν
µµ′ − ST sin(φ− φS) inνν
µµ′ (4.2)
and describes the subprocess γ∗p → ρp. The decay ρ → π+π− is taken into account by
multiplication with the spherical harmonics,
ρµµ′ =
τλλ′ ρ
µµ′,λλ′ Y1ν(ϕ, ϑ)Y
1ν′(ϕ, ϑ) , (4.3)
where
Y1+1 = −
sinϑ eiϕ , Y10 =
cos ϑ , Y1−1 =
sinϑ e−iϕ . (4.4)
To obtain the cross section for the overall process ep→ epπ+π− one must finally contract
the matrix ρµµ′ in (4.3) with the spin density matrix of the virtual photon.
3 The cross
section can be written as
dψ dφdϕd(cos ϑ) dxB dQ2 dt
(2π)2
dxB dQ2 dt
WUU + PℓWLU + SLWUL + PℓSLWLL + STWUT + PℓSTWLT
(4.5)
dxB dQ2 dt
1− xB
, (4.6)
where dσT /dt and dσL/dt are the usual γ
∗p cross sections for a transverse and longitudinal
photon and an unpolarized proton, with Hand’s convention for virtual photon flux. The
angular distribution is described by the quantities WXY , where X specifies the beam and
Y the target polarization. The normalization of the unpolarized term WUU is
dϕ d(cos ϑ)WUU(φ,ϕ, ϑ) = 1 . (4.7)
To limit the length of subsequent expressions, we further decompose the coefficients ac-
cording to the ρ polarization and write
WXY (φ,ϕ, ϑ)
cos2ϑ WLLXY (φ) +
2 cos ϑ sinϑ WLTXY (φ,ϕ) + sin
2ϑ W TTXY (φ,ϕ)
(4.8)
for X,Y = U,L. The production of a longitudinal ρ is described by WLLXY , the production
3Up to a global factor, the result of this contraction can e.g. be obtained from eq. (3.20) of [27], with
ρµµ′ in the present work corresponding to σ
µ′µ in [27] and φhere = −ϕ[27].
– 10 –
of a transverse ρ (including the interference between positive and negative ρ helicity) by
W TTXY , and the interference between longitudinal and transverse ρ polarization by W
For a transversely polarized target we have in addition a dependence on φS ,
WXT (φS , φ, ϕ, ϑ)
cos2ϑ WLLXT (φS , φ) +
2 cos ϑ sinϑ WLTXT (φS , φ, ϕ) + sin
2ϑ W TTXT (φS , φ, ϕ)
(4.9)
with X = U,L. In addition to the angles, all coefficients WXY depend on Q
2, xB and t,
which we have not displayed for the sake of legibility.
For unpolarized target and beam we have
WLLUU(φ) =
u 0 0++ + ǫu
− 2 cos φ
ǫ(1 + ǫ) Reu 0 00+ − cos(2φ) ǫu 0 0−+ ,
WLTUU (φ,ϕ) = cos(φ+ ϕ)
ǫ(1 + ǫ) Re
u 0+0+ − u
− cosϕ Re
u 0+++ − u−0++ + 2ǫu 0+0 0
+ cos(2φ+ ϕ) ǫRe u 0+−+
− cos(φ− ϕ)
ǫ(1 + ǫ) Re
u 0−0+ − u
+ cos(2φ − ϕ) ǫRe u+0−+ ,
W TTUU (φ,ϕ) =
u++++ + u
++ + 2ǫu
cos(2φ+ 2ϕ) ǫu−+−+
− cosφ
ǫ(1 + ǫ) Re
u++0+ + u
+ cos(φ+ 2ϕ)
ǫ(1 + ǫ) Re u−+0+
− cos(2ϕ) Re
u−+++ + ǫu
− cos(2φ) ǫRe u++−+
+ cos(φ− 2ϕ)
ǫ(1 + ǫ) Reu+−0+ +
cos(2φ− 2ϕ) ǫu+−−+ . (4.10)
Here and in the following we order terms according to the hierarchy discussed after (3.11),
as already done in Table 1. The terms independent of φ and ϕ in WLLUU and W
UU are
related by
u++++ + u
++ + 2ǫu
0 0 = 1−
u 0 0++ + ǫu
, (4.11)
which ensures the normalization condition (4.7). The terms for beam polarization with an
unpolarized target read
WLLLU (φ) = −2 sinφ
ǫ(1− ǫ) Imu 0 00+ ,
WLTLU (φ,ϕ) = sin(φ+ ϕ)
ǫ(1− ǫ) Im
u 0+0+ − u
− sinϕ
1− ǫ2 Im
u 0+++ − u−0++
− sin(φ− ϕ)
ǫ(1− ǫ) Im
u 0−0+ − u
W TTLU (φ,ϕ) = − sinφ
ǫ(1− ǫ) Im
u++0+ + u
+ sin(φ+ 2ϕ)
ǫ(1− ǫ) Imu−+0+
− sin(2ϕ)
1− ǫ2 Imu−+++
+ sin(φ− 2ϕ)
ǫ(1− ǫ) Imu+−0+ . (4.12)
– 11 –
The results for longitudinal target polarization are very similar, with
WLLUL(φ) = −2 sinφ
ǫ(1 + ǫ) Im l 0 00+ − sin(2φ) ǫ Im l 0 0−+ ,
WLTUL (φ,ϕ) = sin(φ+ ϕ)
ǫ(1 + ǫ) Im
l 0+0+ − l
− sinϕ Im
l 0+++ − l−0++ + 2ǫl 0+0 0
+ sin(2φ+ ϕ) ǫ Im l 0+−+
− sin(φ− ϕ)
ǫ(1 + ǫ) Im
l 0−0+ − l
+ sin(2φ− ϕ) ǫ Im l+0−+ ,
W TTUL (φ,ϕ) =
sin(2φ+ 2ϕ) ǫ Im l−+−+
− sinφ
ǫ(1 + ǫ) Im
l++0+ + l
+ sin(φ+ 2ϕ)
ǫ(1 + ǫ) Im l−+0+
− sin(2ϕ) Im
l−+++ + ǫl
− sin(2φ) ǫ Im l++−+
+ sin(φ− 2ϕ)
ǫ(1 + ǫ) Im l+−0+ +
sin(2φ− 2ϕ) ǫ Im l+−−+ (4.13)
for an unpolarized beam, and
WLLLL (φ) = −2 cosφ
ǫ(1− ǫ) Re l 0 00+ +
1− ǫ2 l 0 0++ ,
WLTLL (φ,ϕ) = cos(φ+ ϕ)
ǫ(1− ǫ) Re
l 0+0+ − l
− cosϕ
1− ǫ2 Re
l 0+++ − l−0++
− cos(φ− ϕ)
ǫ(1− ǫ) Re
l 0−0+ − l
W TTLL (φ,ϕ) =
1− ǫ2 1
l++++ + l
− cosφ
ǫ(1− ǫ) Re
l++0+ + l
+ cos(φ+ 2ϕ)
ǫ(1− ǫ) Re l−+0+
− cos(2ϕ)
1− ǫ2 Re l−+++
+ cos(φ− 2ϕ)
ǫ(1− ǫ) Re l+−0+ (4.14)
for beam polarization. In (4.10) to (4.14) we have used the symmetry relations (3.9) and
(3.10) to write our results with a minimal set of matrix elements uνν
µµ′ or l
µµ′ . Although
they are a little lengthy, their structure is quite simple:
1. The combinations u++
+ u−−
, u 0+
− u−0
and u 0−
− u+0
and their analogs for l
always appear together because the corresponding products of spherical harmonics
are identical, Y1+1Y
1+1 = Y1−1Y
1−1 and Y10Y
1+1 = −Y1−1Y ∗10. In some cases the
corresponding sum can be simplified using symmetry relations like u++0 0 + u
0 0 =
2u++0 0 , but in others one remains with a linear combination of matrix elements that
cannot be separated. With the caveats discussed after (3.11) one finds however that
these combinations are dominated by a single term. Exceptions are Re
u 0+++ −u−0++ +
2ǫu 0+0 0
and Im
l 0+++ − l−0++ + 2ǫl 0+0 0
, each of which contains two interference terms
between a helicity conserving and a helicity changing amplitude.
– 12 –
2. An angular dependence through (kφ + mϕ) is associated with the interference be-
tween transverse and longitudinal ρ polarization for |m| = 1, the interference between
positive and negative ρ helicity for |m| = 2, and equal ρ polarization in the amplitude
and its conjugate for m = 0. In the same way |k| = 1, |k| = 2 and k = 0 are related to
the virtual photon polarization. Notice that for m = 0 one can distinguish transverse
and longitudinal ρ production by the ϑ dependence in (4.8), whereas for k = 0 the
separation of terms for transverse and longitudinal photons requires variation of ǫ.
The beam spin asymmetries WLU and WLL contain no terms with |k| = 2, because
there is no term with Pℓ cos 2φ or Pℓ sin 2φ in the spin density matrix of the virtual
photon.
3. The unpolarized or doubly polarized terms WUU and WLL depend on Reu or Re l
and are even under the reflection (φ,ϕ) → (−φ,−ϕ) of the azimuthal angles, whereas
the single spin asymmetriesWLU andWUL depend on Imu or Im l and are odd under
(φ,ϕ) → (−φ,−ϕ). This is a consequence of parity and time reversal invariance.
4. As we have written our results, the angular distribution for longitudinal target po-
larization can be obtained from the one for an unpolarized target by replacing
cos(kφ+mϕ) Re u → sin(kφ+mϕ) Im l ,
sin(kφ+mϕ) Imu → cos(kφ+mϕ) Re l . (4.15)
Terms with k = m = 0 in WUU and WLL are independent of φ and ϕ, and have of
course no counterparts inWUL orWLU . This corresponds to 16 terms with a different
angular dependence in WUU and 14 terms in WUL, and to 10 terms in WLL and 8
terms in WLU .
The symmetry properties (3.9) and (3.10), which we used to obtain (4.10) to (4.14), are
identical for uνν
µµ′ and in
µµ′ , as well as for l
µµ′ and s
µµ′ . According to (4.2) the cross section
for a transversely polarized target can therefore be obtained from the one for longitudinal
and no target polarization by the replacements
Reu → ST sin(φ− φS) Imn , SL Im l → ST cos(φ− φS) Im s ,
Imu → −ST sin(φ− φS) Ren , SLRe l → ST cos(φ− φS) Re s . (4.16)
We thus simply have
WLLUT (φS , φ) = sin(φ− φS)
n 0 0++ + ǫn
− 2 cos φ
ǫ(1 + ǫ) Imn 0 00+ − cos(2φ) ǫ Imn 0 0−+
+ cos(φ− φS)
−2 sinφ
ǫ(1 + ǫ) Im s 0 00+ − sin(2φ) ǫ Im s 0 0−+
WLTUT (φS , φ, ϕ) = sin(φ− φS)
cos(φ+ ϕ)
ǫ(1 + ǫ) Im
n 0+0+ − n
− cosϕ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
+ cos(2φ+ ϕ) ǫ Imn 0+−+
− cos(φ− ϕ)
ǫ(1 + ǫ) Im
n 0−0+ − n
+ cos(2φ− ϕ) ǫ Imn+0−+
– 13 –
+ cos(φ− φS)
sin(φ+ ϕ)
ǫ(1 + ǫ) Im
s 0+0+ − s
− sinϕ Im
s 0+++ − s−0++ + 2ǫs 0+0 0
+ sin(2φ + ϕ) ǫ Im s 0+−+
− sin(φ− ϕ)
ǫ(1 + ǫ) Im
s 0−0+ − s
+ sin(2φ− ϕ) ǫ Im s+0−+
W TTUT (φS , φ, ϕ) = sin(φ− φS)
n++++ + n
++ + 2ǫn
cos(2φ+ 2ϕ) ǫ Imn−+−+
− cosφ
ǫ(1 + ǫ) Im
n++0+ + n
+ cos(φ+ 2ϕ)
ǫ(1 + ǫ) Imn−+0+
− cos(2ϕ) Im
n−+++ + ǫn
− cos(2φ) ǫ Imn++−+
+ cos(φ− 2ϕ)
ǫ(1 + ǫ) Imn+−0+ +
cos(2φ− 2ϕ) ǫ Im n+−−+
+ cos(φ− φS)
sin(2φ+ 2ϕ) ǫ Im s−+−+
− sinφ
ǫ(1 + ǫ) Im
s++0+ + s
+ sin(φ+ 2ϕ)
ǫ(1 + ǫ) Im s−+0+
− sin(2ϕ) Im
s−+++ + ǫs
− sin(2φ) ǫ Im s++−+
+ sin(φ− 2ϕ)
ǫ(1 + ǫ) Im s+−0+ +
sin(2φ− 2ϕ) ǫ Im s+−−+
(4.17)
for an unpolarized beam, and
WLLLT (φS , φ) = sin(φ− φS)
2 sinφ
ǫ(1− ǫ) Ren 0 00+
+ cos(φ− φS)
−2 cos φ
ǫ(1− ǫ) Re s 0 00+ +
1− ǫ2 s 0 0++
WLTLT (φS , φ, ϕ) = sin(φ− φS)
− sin(φ+ ϕ)
ǫ(1− ǫ) Re
n 0+0+ − n
+ sinϕ
1− ǫ2 Re
n 0+++ − n−0++
+ sin(φ− ϕ)
ǫ(1− ǫ) Re
n 0−0+ − n
+ cos(φ− φS)
cos(φ+ ϕ)
ǫ(1− ǫ) Re
s 0+0+ − s
− cosϕ
1− ǫ2 Re
s 0+++ − s−0++
− cos(φ− ϕ)
ǫ(1− ǫ) Re
s 0−0+ − s
W TTLT (φS , φ, ϕ) = sin(φ− φS)
ǫ(1− ǫ) Re
n++0+ + n
− sin(φ+ 2ϕ)
ǫ(1− ǫ) Ren−+0+
+ sin(2ϕ)
1− ǫ2 Ren−+++
− sin(φ− 2ϕ)
ǫ(1− ǫ) Ren+−0+
+ cos(φ− φS)
1− ǫ2 1
s++++ + s
– 14 –
unpolarized beam polarized beam
WUU WUL WUT WLU WLL WLT
Re u Im l Imn Im s Imu Re l Ren Re s
15 14 16 14 8 10 8 10
Table 3: Number of linear combinations of spin density matrix elements describing the angular
distribution of the cross section (4.5). The number of independent combinations for Reu is one
less than for Imn because of the relation (4.11).
− cosφ
ǫ(1− ǫ) Re
s++0+ + s
+ cos(φ+ 2ϕ)
ǫ(1− ǫ) Re s−+0+
− cos(2ϕ)
1− ǫ2 Re s−+++
+ cos(φ− 2ϕ)
ǫ(1− ǫ) Re s+−0+
(4.18)
for beam polarization. With obvious adjustments, the general structure discussed in points
1 to 3 above is found again for a transverse target. Note that the terms u 0 0++ + ǫu
0 0 and
u++++ + u
++ + 2ǫu
0 0 in the unpolarized coefficients W
UU and W
UU add up to 1 according
to (4.11), whereas their counterparts Im
n 0 0++ + ǫn
and Im
n++++ + n
++ + 2ǫn
WLLUT and W
UT are independent quantities. To keep the close similarity between the two
cases we have not used (4.11) to simplify (4.10).
Since there are two independent transverse polarizations relative to the hadron plane
(normal and sideways) we have a rather large number of terms with different angular
dependence in (4.17) and (4.18). The single spin asymmetry WUT contains 16 terms
with Imn and 14 terms with Im s , whereas the double spin asymmetry WLT contains 8
terms with Ren and 10 terms with Re s . Table 3 lists the number of independent linear
combinations of spin density matrix elements describing the angular distribution for the
different combinations of beam and target spin. For reasons discussed in Section 5 it is
useful to consider the spin density matrices n and s separately. It is then natural to work
in the basis of angular functions given by the product of sin(φ − φS) or cos(φ − φS) with
sin(kφ +mϕ) or cos(kφ +mϕ). With the replacement rules (4.15) and (4.16) we obtain
the combinations
sin(φ− φS) cos(kφ+mϕ) Imn + cos(φ− φS) sin(kφ+mϕ) Im s ,
− sin(φ− φS) sin(kφ+mϕ) Ren + cos(φ− φS) cos(kφ+mϕ) Re s (4.19)
in WUT and WLT , respectively.
We conclude this section by giving the relation between our spin density matrix ele-
ments for an unpolarized target and those in the classical work [14] of Schilling and Wolf.
We have
u 0 0++ + ǫu
0 0 = r
u 0+0+ − u
Im r610 − Re r510
u++++ + u
++ + 2ǫu
0 0 = 1− r0400 ,
u−+−+ = r
1−1 − Im r21−1 ,
– 15 –
Re u 0 00+ = −r500/
u 0+++ − u−0++ + 2ǫu 0+0 0
= 2Re r0410 ,
Re u 0+−+ = Re r
10 − Im r210 ,
u 0−0+ − u
Im r610 +Re r
u−+++ + ǫu
= r041−1 ,
Re u++−+ = r
u++0+ + u
2 r511 ,
Re u−+0+ =
Im r61−1 − r51−1
u 0 0−+ = r
Re u+0−+ = Re r
10 + Im r
Re u+−0+ = −
Im r61−1 + r
u+−−+ = r
1−1 + Im r
1−1 (4.20)
u 0+0+ − u
Im r710 +Re r
Imu 0 00+ = r
u 0+++ − u−0++
= −2 Im r310 ,
u 0−0+ − u
Im r710 − Re r810
Imu−+++ = − Im r31−1 ,
u++0+ + u
2 r811 ,
Imu−+0+ =
Im r71−1 + r
Imu+−0+ = −
Im r71−1 − r81−1
2 . (4.21)
The lower indices in the matrix elements of Schilling and Wolf refer to the ρ helicity and
correspond to the upper indices of u in our notation. Their upper indices correspond to a
representation of the virtual photon spin density matrix which refers partly to circular and
partly to linear polarization, whereas we use the helicity basis for the photon throughout.
The consequences of approximate s-channel helicity conservation are more explicit in our
notation: the relation Im r610 ≈ −Re r510 for instance corresponds to
u 0+0+ − u
u 0−0+ − u
∣. Notice also that the simple relation between single-spin asymmetries
and imaginary parts of spin density matrix elements discussed in point 3 above holds in
the helicity basis but not for linear polarization.
We note that our phase convention (3.1) for the helicity states of the virtual photon
differs from the one in [14] by a relative minus sign between transverse and longitudinal
polarization, and that our normalization factors NT and NL in (3.5) differ from those in
[14] by a factor of two. The combinations of helicity amplitudes corresponding to the spin
density matrix elements in (4.20) and (4.21) should be compared according to
NT + ǫNL
T νσµλ
= ηµµ′
NT + ǫNL
Tνσ,µλ T
ν′σ,µ′λ
, (4.22)
– 16 –
where η0± = η±0 = −1 for the interference of transverse and longitudinal photon polar-
ization, and ηµµ′ = +1 in all other cases.
5. Natural and unnatural parity
The exclusive process γ∗p → ρp is described by eighteen independent helicity amplitudes,
and we have already used approximate s-channel helicity conservation to establish a hierar-
chy among these amplitudes and the spin density matrix elements constructed from them.
A further dynamical criterion to order these quantities is given by natural and unnatural
parity exchange, which we shall now discuss.
Following [14] we define amplitudes N for natural and U unnatural parity exchange as
linear combinations
Nνσµλ =
T νσµλ + (−1)ν−µ T−νσ−µλ
T νσµλ + (−1)λ−σ T ν−σµ−λ
Uνσµλ =
T νσµλ − (−1)ν−µ T−νσ−µλ
T νσµλ − (−1)λ−σ T ν−σµ−λ
. (5.1)
With respect to the photon and meson helicity, the amplitudes N have the same symmetry
behavior as the amplitudes for γ∗t→ ρt on a spin-zero target t, whereas the corresponding
relation for the amplitudes U has an additional minus sign,
= (−1)ν−µNνσµλ , U−νσ−µλ = −(−1)
ν−µ Uνσµλ . (5.2)
For the proton helicity we have relations Nν+µ+ = N
µ− and N
µ− = −Nν−µ+ for natural parity
exchange, compared to Uν+µ+ = −Uν−µ− and Uν+µ− = Uν−µ+ for unnatural parity exchange. This
symmetry behavior immediately implies that in a dynamical description using generalized
parton distributions, amplitudes N go with distributions H and E, whereas amplitudes U
go with distributions H̃ and Ẽ. This is explicitly borne out in the calculation of [24]. Since
U 0σ0λ = 0 according to (5.2), unnatural parity exchange amplitudes are power suppressed
at large Q2 and the leading-twist factorization theorem [2] only applies to the natural
parity exchange amplitudes N 0σ0λ . We remark that in the context of low-energy dynamics
t-channel exchange of a pion plays a prominent role for unnatural parity exchange ampli-
tudes, see e.g. [15]. This has a natural counterpart in the framework of generalized parton
distributions, where pion exchange gives an essential contribution to the distribution Ẽ in
the isovector channel [28, 3, 29].
4The correspondence in (4.20) to (4.22) is obtained from comparing our results (4.10) and (4.12) for the
angular distribution with the ones in eqs. (92) and (92a) of [14], together with the relation between spin
density matrix elements and helicity amplitudes specified in eq. (91) and Appendix A of [14]. We have not
found an explicit specification of the phase convention for the virtual photon polarizations used in [14].
– 17 –
For the spin density matrix elements one readily finds
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
µµ′ = (NT + ǫNL)
Nνσµ+
+ Uνσµ+
. (5.3)
The matrix elements u and n hence involve a product of two natural parity exchange
amplitudes plus a product of two amplitudes for unnatural parity exchange, whereas l
and s involve the interference between natural and unnatural parity exchange [15]. To
the extent that amplitudes U are smaller than their counterparts N , one can thus expect
that matrix elements l and s are small compared with u and n for equal helicity indices.
Exceptions to this guideline are possible since products Nνσµ+
or Nνσµ−
have a small real or imaginary part due to the relative phase between the two amplitudes.
If amplitudes U are smaller than N , one can furthermore neglect the terms
µµ′ = (NT + ǫNL)
Uνσµ+
µµ′ = (NT + ǫNL)
Uνσµ+
(5.4)
involving unnatural parity exchange in the matrix elements u and n . Using the relations
(−1)ν−µ u−νν′
= uνν
µµ′ − 2ũνν
µµ′ (5.5)
following from (5.2) and (5.3), we have in particular
−u 0+−+ = u 0+++ − 2ũ 0+++ , u−+−+ = u++++ − 2ũ++++ ,
−u−+0+ = u
0+ − 2ũ
0+ , u
−+ = u
++ − 2ũ−+++ . (5.6)
This allows us to rewrite
WLTUU = − cosϕ Re
u 0+++ − u−0++ + 2ǫu 0+0 0
− cos(2φ+ ϕ) ǫRe
u 0+++ − 2ũ 0+++
+ . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ− ϕ) ,
W TTUU =
u++++ + u
++ + 2ǫu
cos(2φ+ 2ϕ) ǫ
u++++ − 2ũ++++
− cosφ
ǫ(1 + ǫ) Re
u++0+ + u
− cos(φ+ 2ϕ)
ǫ(1 + ǫ) Re
u++0+ − 2ũ
− cos(2ϕ) Re
u−+++ + ǫu
− cos(2φ) ǫRe
u−+++ − 2ũ−+++
+ . . . cos(φ− 2ϕ) + . . . cos(2φ− 2ϕ) ,
– 18 –
W TTLU = − sinφ
ǫ(1− ǫ) Im
u++0+ + u
− sin(φ+ 2ϕ)
ǫ(1− ǫ) Im
u++0+ − 2ũ
+ . . . sin(2ϕ) + . . . sin(φ− 2ϕ) , (5.7)
where terms indicated by . . . are the same as in the original expressions (4.10) and (4.12)
and have not been repeated for brevity. We see that the coefficients of adjacent terms in
(5.7) will be approximately equal to the extent that unnatural parity exchange is suppressed
and s-channel helicity approximately conserved. This can be tested experimentally by
measuring the angular distribution of the final-state particles.
The relations (5.6) and their counterparts for other index combinations can also be
used to approximately isolate spin density matrix elements of particular interest. Consider
as an example the leading-twist matrix element u 0 00 0 , which in the angular distribution
appears only in the combination u 0 0++ + ǫu
0 0 , i.e. together with a matrix element that
should be suppressed since it does not conserve s-channel helicity. If unnatural parity
exchange is strongly suppressed, an even better approximation for u 0 00 0 can be obtained
from the linear combination
ǫu 0 00 0 + 2ũ
u 0 0++ + ǫu
+ u 0 0−+ , (5.8)
whose r.h.s. can be extracted from the angular distribution. Similarly, one can approxi-
mately isolate the matrix element Re u 0+0 0 in the combination
ǫRe u 0+0 0 +Re
ũ 0+++ − ũ−0++
u 0+++ − u−0++ + 2ǫu 0+0 0
+Re u 0+−+ +Reu
. (5.9)
Conversely, one can extract from the angular distribution the linear combinations
ũ++++ + ũ
−− + 2ǫũ
0 0 − 2Re ũ
u++++ + u
++ + 2ǫu
u−+−+ − 12 u
u−+++ + ǫu
−Re u++−+ ,
ũ++0+ + ũ
u++0+ + u
u−+0+ +
u+−0+ , (5.10)
which only involve unnatural parity exchange. In a dynamical approach based on gen-
eralized parton distributions, these combinations are interesting because they isolate the
polarized distributions H̃ and Ẽ and in particular involve these distributions for gluons,
which are very hard to access in any other process.5 The price to pay for this is that
the corresponding amplitudes are power suppressed and cannot be calculated with the
theoretical rigor provided by the leading-twist factorization theorem. On the other hand,
phenomenological analysis indicates that a quantitative description of meson production
at Q2 of a few GeV2 requires the inclusion of power-suppressed effects also for the leading
matrix element u 0 00 0 .
The discussion of the matrix elements for transverse target polarization normal to the
hadron plane proceeds in full analogy to the unpolarized case. With
(−1)ν−µ n−νν′
= nνν
µµ′ − 2ñνν
µµ′ (5.11)
5In contrast to their quark counterparts, H̃g and Ẽg do not appear in pseudoscalar meson production
at leading twist and leading order in αs, see e.g. Section 5.1.1 of [30].
– 19 –
we have
−n 0+−+ = n 0+++ − 2ñ 0+++ , n−+−+ = n++++ − 2ñ++++ ,
−n−+0+ = n
0+ − 2ñ
0+ , n
−+ = n
++ − 2ñ−+++ (5.12)
and can write
WLTUT = cos(φ− φS)
. . .
+ sin(φ− φS)
− cosϕ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
− cos(2φ+ ϕ) ǫ Im
n 0+++ − 2ñ 0+++
+ . . . cos(φ+ ϕ) + . . . cos(φ− ϕ) + . . . cos(2φ − ϕ)
W TTUT = cos(φ− φS)
. . .
+ sin(φ− φS)
n++++ + n
++ + 2ǫn
cos(2φ+ 2ϕ) ǫ Im
n++++ − 2ñ++++
− cosφ
ǫ(1 + ǫ) Im
n++0+ + n
− cos(φ+ 2ϕ)
ǫ(1 + ǫ) Im
n++0+ − 2ñ
− cos(2ϕ) Im
n−+++ + ǫn
− cos(2φ) ǫ Im
n−+++ − 2ñ−+++
+ . . . cos(φ− 2ϕ) + . . . cos(2φ − 2ϕ)
W TTLT = cos(φ− φS)
. . .
+ sin(φ− φS)
ǫ(1− ǫ) Re
n++0+ + n
+ sin(φ+ 2ϕ)
ǫ(1− ǫ) Re
n++0+ − 2ñ
+ . . . sin(2ϕ) + . . . sin(φ− 2ϕ)
, (5.13)
where terms denoted by . . . are as in the original expressions (4.17) and (4.18). Again, the
coefficients of adjacent terms should be approximately equal to the extent that unnatural
parity exchange is suppressed and s-channel helicity approximately conserved. The matrix
elements Imn 0 00 0 and Imn
0 0 can be approximately isolated in
ǫ Imn 0 00 0 + 2 Im ñ
++ = Im
n 0 0++ + ǫn
+ Imn 0 0−+ (5.14)
ǫ Imn 0+0 0 + Im
ñ 0+++ − ñ−0++
n 0+++ − n−0++ + 2ǫn 0+0 0
+ Imn 0+−+ + Imn
. (5.15)
In turn, the linear combinations
ñ++++ + ñ
−− + 2ǫñ
0 0 − 2ñ
n++++ + n
++ + 2ǫn
Imn−+−+ − 12 Imn
n−+++ + ǫn
− Imn++−+ ,
ñ++0+ + ñ
n++0+ + n
n−+0+ +
n+−0+ (5.16)
involve only unnatural parity exchange.
– 20 –
6. Positivity constraints
From the definition (3.4) of the spin-density matrix elements one readily finds
ν′µ′λ′
cνµλ ρ
µµ′,λλ′
= (NT + ǫNL)
cνµλ T
≥ 0 (6.1)
for arbitrary complex numbers cνµλ. Hence ρ
µµ′,λλ′ is a positive semidefinite matrix, with
row indices specified by {νµλ} and column indices by {ν ′µ′λ′}. This implies inequalities
among the spin density matrix elements, which extend those given e.g. in [22, 27]. We
do not attempt here to study the bounds following from positivity of the full 18 × 18
matrix ρνν
µµ′,λλ′ , which is quite unwieldy. Instead, we consider the subset of matrix elements
conserving s-channel helicity for the photon-meson transition and derive a number of simple
inequalities, which may be useful in practice. Ordering the row and column indices as
{+++}, {0 0+}, {−−+}, {++−}, {0 0−}, {−−−}, we have a positive semidefinite matrix
C, which can be written in block form as
A+ B+
B− A−
(6.2)
u++++ + η l
u 0+0+ + η l
u−+−+ − η l−+−+
u 0+0+ + η l
0 0 u
0+ − η l
u−+−+ + η l
u 0+0+ − η l
u++++ − η l++++
(6.3)
s++++ + η n
s 0+0+ − η n
)∗ −s−+−+ + η n−+−+
s 0+0+ + η n
0+ η n
0 0 −s 0+0+ + η n
s−+−+ + η n
s 0+0+ + η n
)∗ −s++++ + η n++++
, (6.4)
where η = ±1. Concentrating first on the matrix elements for an unpolarized or longitu-
dinally polarized target, we find that the matrix Aη has eigenvalues whose expressions are
very lengthy and therefore restrict our attention to 2×2 submatrices. The matrix obtained
from the first and third rows and columns of A+ has eigenvalues
u++++ ±
u−+−+
l++++
Im l−+−+
, (6.5)
whose positivity implies a bound
l++++
Im l−+−+
u++++
u−+−+
. (6.6)
Similarly, the matrix obtained from the first and second and the matrix obtained from the
second and third rows and columns of A+ have respective eigenvalues
u++++ + l
++ + u
u++++ + l
++ − u 0 00 0
∣u 0+0+ + l
u++++ − l++++ + u 0 00 0
u++++ − l++++ − u 0 00 0
∣u 0+0+ − l
, (6.7)
– 21 –
whose positivity gives bounds
Reu 0+0+ +Re l
Imu 0+0+ + Im l
)2 ≤ u 0 00 0
u++++ + l
Reu 0+0+ − Re l
Imu 0+0+ − Im l
)2 ≤ u 0 00 0
u++++ − l++++
. (6.8)
A weaker condition is obtained by taking the sum of these two bounds,
Re l 0+0+
Im l 0+0+
)2 ≤ u 0 00 0 u++++ −
Re u 0+0+
Imu 0+0+
. (6.9)
The bounds (6.6) and (6.9) have right-hand sides involving only matrix elements accessi-
ble with an unpolarized target and constrain the matrix elements for longitudinal target
polarization on their left-hand sides.
As a second example let us derive conditions which involve only matrix elements u
and n . To this end we consider the matrix
C′ = 1
C+D†CD
(6.10)
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
, (6.11)
which is half the sum of the positive semidefinite matrices C and D†CD and hence positive
semidefinite itself. One readily finds that matrix elements l and s drop out in C′, which
reads
u++++
u 0+0+
u−+−+ n
n 0+0+
n−+−+
u 0+0+ u
0 0 u
0 0 n
u−+−+
u 0+0+
u++++ n
n 0+0+
n++++
−n++++
n 0+0+
)∗ −n−+−+ u++++
u 0+0+
u−+−+
−n 0+0+ −n 0 00 0 −n
0 0 u
−n−+−+
n 0+0+
)∗ −n++++ u−+−+
u 0+0+
u++++
. (6.12)
This matrix has three eigenvalues
u++++ − u−+−+ + Imn++++ − Imn−+−+ ,
u++++ + u
−+ + Imn
++ + Imn
−+ + u
0 0 + Imn
u++++ + u
−+ + Imn
++ + Imn
−+ − u 0 00 0 − Imn 0 00 0
∣u 0+0+ − in
(6.13)
and three further eigenvalues obtained by reversing the sign of all matrix elements n . Their
positivity results in the bounds
Imn++++ − Imn−+−+
u++++ − u−+−+
(6.14)
– 22 –
Reu 0+0+ + Imn
Imu 0+0+ − Ren
u 0 00 0 + Imn
u++++ + u
−+ + Imn
++ + Imn
Reu 0+0+ − Imn
Imu 0+0+ +Ren
u 0 00 0 − Imn 0 00 0
u++++ + u
−+ − Imn++++ − Imn−+−+
. (6.15)
Omitting the terms with Imu 0+0+ and Ren
0+ , one obtains bounds involving only matrix
elements that are accessible with an unpolarized lepton beam.
As we have seen in Section 4, s-channel helicity conserving matrix elements can be
extracted from the angular distribution under the approximation that s-channel helicity
changing transitions are suppressed. The bounds derived in this section may be used to
check the consistency of this approximation.
7. Mixing between transverse and longitudinal polarization
So far we have discussed target polarization longitudinal or transverse to the virtual photon
direction in the target rest frame, which is natural from the point of view of the strong-
interaction dynamics. In an experimental setup one has however definite target polarization
with respect to the lepton beam direction. The transformation from one polarization basis
to the other is readily performed using the relations (2.3). For a target having longitudinal
polarization PL with respect to the lepton beam one finds
dφ dϕd(cos ϑ) dxB dQ2 dt
dxB dQ2 dt
WUU + PL
cos θγWUL − sin θγWUT (φS = 0)
+ PℓWLU + PℓPL
cos θγ WLL − sin θγ WLT (φS = 0)
. (7.1)
Note that in this case the azimuthal angle ψ in (4.5) needs to be defined with respect
to some fixed spatial direction in the target rest frame, rather than with respect to the
(vanishing) transverse component of the target polarization relative to the lepton beam.
We have integrated over this angle in (7.1) because the cross section does not depend on it.
For a target having transverse polarization PT with respect to the lepton beam one
dφS dφ dϕd(cos ϑ) dxB dQ2 dt
(2π)2
dxB dQ2 dt
cos θγ
1− sin2θγ sin2φS
WUU + PT
cos θγ WUT + sin θγ cosφS WUL
1− sin2θγ sin2φS
+ PℓWLU + PℓPT
cos θγ WLT + sin θγ cosφSWLL
1− sin2θγ sin2φS
. (7.2)
The factor cos θγ /(1 − sin2θγ sin2φS) comes from the change of variables from dψ to dφS
– 23 –
in the cross section. The relation between these two angles is readily obtained by setting
PL = 0 in (2.3) and given in [22].
It is a straightforward (if somewhat lengthy) exercise to insert our results (4.13), (4.14)
and (4.17), (4.18) into (7.1) and (7.2) and to rewrite the expressions in terms of a suitable
basis of functions depending on the azimuthal angles. Here we only give the combinations
needed in (7.2) for a transversely polarized target and an unpolarized beam,
cos θγ W
UT (φS , φ) + sin θγ cosφS W
UL(φ)
= sin(φ− φS)
cos θγ Im
n 0 0++ + ǫn
− sin θγ
ǫ(1 + ǫ) Im l 0 00+
− cos(2φ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l 0 00+
− 2 cosφ
cos θγ
ǫ(1 + ǫ) Imn 0 00+ +
sin θγ ǫ Im l
+ cos(φ− φS)
− sin(2φ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l 0 00+
− 2 sinφ
cos θγ
ǫ(1 + ǫ) Im s 0 00+ +
sin θγ ǫ Im l
sin θγ sin(φS + 2φ) ǫ Im l
−+ , (7.3)
cos θγW
UT (φS , φ, ϕ) + sin θγ cosφSW
UL (φ,ϕ)
= sin(φ− φS)
cos(φ+ ϕ)
cos θγ
ǫ(1 + ǫ) Im
n 0+0+ − n
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
+ ǫ Im l 0+−+
− cos(φ− ϕ)
cos θγ
ǫ(1 + ǫ) Im
n 0−0+ − n
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
− ǫ Im l+0−+
+cos(2φ+ ϕ)
cos θγ ǫ Imn
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l 0+0+ − l
+cos(2φ− ϕ)
cos θγ ǫ Imn
sin θγ
ǫ(1 + ǫ) Im
l 0−0+ − l
− cosϕ
cos θγ Im
n 0+++ − n−0++ + 2ǫn 0+0 0
sin θγ
ǫ(1 + ǫ)
l 0+0+ − l
l 0−0+ − l
+ cos(φ− φS)
sin(φ+ ϕ)
cos θγ
ǫ(1 + ǫ) Im
s 0+0+ − s
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
− ǫ Im l 0+−+
– 24 –
− sin(φ− ϕ)
cos θγ
ǫ(1 + ǫ) Im
s 0−0+ − s
sin θγ
l 0+++ − l−0++ + 2ǫl 0+0 0
+ ǫ Im l+0−+
+sin(2φ+ ϕ)
cos θγ ǫ Im s
sin θγ
ǫ(1 + ǫ) Im
l 0+0+ − l
+sin(2φ− ϕ)
cos θγ ǫ Im s
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l 0−0+ − l
− sinϕ
cos θγ Im
s 0+++ − s−0++ + 2ǫs 0+0 0
sin θγ
ǫ(1 + ǫ)
l 0+0+ − l
l 0−0+ − l
sin θγ
sin(φS + 2φ+ ϕ) ǫ Im l
−+ + sin(φS + 2φ− ϕ) ǫ Im l+0−+
, (7.4)
cos θγW
UT (φS , φ, ϕ) + sin θγ cosφSW
UL (φ,ϕ)
= sin(φ− φS)
cos θγ Im
n++++ + n
++ + 2ǫn
sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− cos(2φ)
cos θγ ǫ Imn
−+ − 12 sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− cosφ
cos θγ
ǫ(1 + ǫ) Im
n++0+ + n
sin θγ ǫ Im l
cos(2φ+ 2ϕ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l−+0+
cos(2φ− 2ϕ)
cos θγ ǫ Imn
−+ − sin θγ
ǫ(1 + ǫ) Im l+−0+
− cos(2ϕ)
cos θγ Im
n−+++ + ǫn
sin θγ
ǫ(1 + ǫ)
Im l−+0+ + Im l
+ cos(φ+ 2ϕ)
cos θγ
ǫ(1 + ǫ) Imn−+0+ +
sin θγ
ǫ Im l−+−+ + 2 Im
l−+++ + ǫl
+ cos(φ− 2ϕ)
cos θγ
ǫ(1 + ǫ) Imn+−0+ +
sin θγ
ǫ Im l+−−+ − 2 Im
l−+++ + ǫl
+ cos(φ− φS)
− sin(2φ)
cos θγ ǫ Im s
sin θγ
ǫ(1 + ǫ) Im
l++0+ + l
− sinφ
cos θγ
ǫ(1 + ǫ) Im
s++0+ + s
sin θγ ǫ Im l
sin(2φ+ 2ϕ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l−+0+
sin(2φ− 2ϕ)
cos θγ ǫ Im s
−+ + sin θγ
ǫ(1 + ǫ) Im l+−0+
− sin(2ϕ)
cos θγ Im
s−+++ + ǫs
sin θγ
ǫ(1 + ǫ)
Im l−+0+ − Im l
– 25 –
+ sin(φ+ 2ϕ)
cos θγ
ǫ(1 + ǫ) Im s−+0+ +
sin θγ
ǫ Im l−+−+ − 2 Im
l−+++ + ǫl
+ sin(φ− 2ϕ)
cos θγ
ǫ(1 + ǫ) Im s+−0+ +
sin θγ
ǫ Im l+−−+ + 2 Im
l−+++ + ǫl
sin θγ
sin(φS + 2φ+ 2ϕ) ǫ Im l
−+ + sin(φS + 2φ− 2ϕ) ǫ Im l+−−+
sin θγ sin(φS + 2φ) ǫ Im l
−+ . (7.5)
Compared with (4.17) and (4.18) we have changed the order of terms such that one readily
sees which coefficients cos θγ Imn or cos θγ Im s receive an admixture from the same coef-
ficients sin θγ Im l . The terms in the last lines of (7.3) and (7.4) and in the last two lines
of (7.5) involve only coefficients sin θγ Im l . They come with an angular dependence which
is absent for sin θγ = 0, as is readily seen by rewriting
sin(φS + 2φ+mϕ) = − sin(φ− φS) cos(3φ+mϕ) + cos(φ− φS) sin(3φ +mϕ) . (7.6)
We see in (7.3) to (7.5) that from the angular dependence of the cross section for
transverse target polarization one can extract linear combinations of terms cos θγ Imn and
sin θγ Im l or of cos θγ Im s and sin θγ Im l . To separate these terms requires an additional
measurement with longitudinal target polarization.6 The expressions (7.3) to (7.5) allow
us to see for which terms the admixture of sin θγ Im l terms can be expected to be small, so
that Imn and Im s may be determined with reasonable accuracy without such an additional
measurement. Let us discuss a few examples.
1. The leading-twist matrix element n 0 00 0 appears in the linear combination
c0 = cos θγ Im
n 0 0++ + ǫn
− sin θγ
ǫ(1 + ǫ) Im l 0 00+ (7.7)
in (7.3) and thus has an admixture from l 0 00+ , which involves one s-channel helicity
changing amplitude. According to Section 5 this admixture is additionally suppressed
if unnatural parity exchange is small compared with natural parity exchange. One
may also add to c0 the angular coefficient
c1 = − cos θγ ǫ Imn 0 0−+ + sin θγ
ǫ(1 + ǫ) Im l 0 00+ (7.8)
from (7.3), thus trading the admixture of sin θγ l
0+ for an admixture of cos θγ n
which involves two s-channel helicity changing amplitudes (but lacks the relative
factor tan θγ and is not suppressed by unnatural parity exchange). We remark that
the linear combination of matrix elements in (5.14) corresponds to c0 − c1/ǫ, where
l 0 00+ does not drop out. Whether c0, c0+ c1 or c0− c1/ǫ gives the best approximation
to cos θγ ǫ Imn
0 0 will thus depend on the detailed magnitude of the relevant terms.
In practice one might for instance use the difference between these terms as a measure
for the uncertainty of this approximation.
6A corresponding separation for semi-inclusive pion production ep → eπX has recently been performed
in [31].
– 26 –
2. The s-channel helicity conserving matrix elements n 0+0+ in (7.4) and n
++ , n
−+ in
(7.5) come together with terms involving at least one s-channel helicity changing
amplitude. These admixtures should hence be negligible unless the corresponding
s-channel helicity conserving matrix element is small itself. For Imn 0+0+ this may for
instance happen because of the relative phase between the interfering amplitudes.
3. The matrix element n 0 00+ in (7.3) comes with an admixture from l
−+ , which involves
two s-channel helicity changing amplitudes and should hence again be suppressed. In
addition, one can extract Im l 0 0−+ from the angular dependence itself, given the last
term in (7.3). We remark that the unpolarized analog u 0 00+ of n
0+ has a real part
which is experimentally seen to be nonzero [17, 19], providing evidence that s-channel
helicity is not strictly conserved in electroproduction. (In the notation of Schilling
and Wolf one has r500 = −
2Reu 0 00+ .)
4. The only s-channel helicity conserving matrix elements for sideways transverse target
polarization in (7.3) to (7.5) are s 0+0+ and s
−+ . They come together with terms
involving at least one s-channel helicity changing amplitude, so that the situation
is similar to the one in point 2. Note however that in the present case there is no
additional suppression of the admixture terms due to unnatural parity exchange,
since both s and l contain one unnatural parity exchange amplitude.
In these examples one thus has the favorable situation that the admixture from longitudinal
polarization terms is probably small and in some cases may even be removed or traded for
yet smaller terms. This does not always happen: the matrix elements n 0+−+ and s
−+ in
(7.4) receive for instance an admixture from the s-channel helicity conserving term l 0+0+ ,
which may not be small itself, so that from the coefficients of sin(φ − φS) cos(2φ + ϕ) or
cos(φ−φS) sin(2φ+ϕ) one cannot directly infer on the matrix elements Imn 0+−+ or Im s 0+−+ .
To make a more precise statement about their size one needs independent information on
Im l 0+0+ , for instance from the positivity bound (6.9).
8. A note on non-resonant contributions
So far we have treated the production of two pions in a two-step picture, where a ρ is first
produced in ep → epρ and then decays as ρ→ π+π−. For deriving the angular distribution
and polarization dependence we have used that the pion pair is in the L = 1 partial wave,
as can be seen in (4.3). We did however not use the narrow-width approximation for
the ρ or make any assumption about its line shape. In fact, our results for the angular
distribution can readily be used at any given invariant mass mππ of the pion pair, with the
ep cross sections on the left- and right-hand sides of (4.5) made differential in mππ. The
spin-density matrix ρνν
µµ′,λλ′ and its linear combinations u , l , s , n then depend on mππ and
refer not to γ∗p → ρp but to γ∗p → π+π− p with π+π− in the L = 1 partial wave. No
explicit reference to the ρ resonance needs to be made in this case.
The situation is more complicated if one considers other partial waves of the pion pair,
which can arise from non-resonant production mechanisms. To describe a general π+π−
– 27 –
state, one should replace ρνν
µµ′,λλ′ with the spin-density matrix ρ
νν′,LL′
µµ′,λλ′
for a pion pair with
angular momentum L in the amplitude and L′ in the conjugate amplitude. One then has to
take YLν(ϕ, ϑ)Y
L′ν′(ϕ, ϑ) instead of Y1ν(ϕ, ϑ)Y
1ν′(ϕ, ϑ) in (4.3) and will obviously obtain
a different angular dependence of the ep cross section. The distribution in ϕ and ϑ for a
pion pair with L = 0, 1, 2 has been discussed in [32].
It is quite simple to test for the presence of L = 0 or L = 2 partial waves in data
by using discrete symmetry properties, and for mππ around the ρ mass one can expect
that partial waves with L = 3 or higher are strongly phase space suppressed. Since even
partial waves of the π+π− system have charge conjugation parity C = +1 and odd partial
waves have C = −1, the interference of L = 1 with L = 0 or L = 2 gives rise to terms in
the angular distribution which are odd under interchange of the π+ and π− momenta, i.e.
under the replacement
ϑ→ π − ϑ , ϕ→ ϕ+ π . (8.1)
Simple examples are an angular dependence like cos ϑ or like an odd polynomial in cos ϑ.
Corresponding observables provide a way to study the L = 0 and L = 2 partial waves as a
“signal” interfering with the ρ resonance “background” [33, 34]. This has been used in the
experimental analysis [35], which did see such interference away from the ρ resonance peak,
whereas close to the peak the predominance of the ρ was too strong to observe a significant
contribution from any partial wave with L 6= 1. If on the other hand one is interested in a
precise study of the L = 1 component, one can eliminate its interference with even partial
waves by symmetrizing the angular distribution according to (8.1). One is then left with
contributions from L = 0 and L = 2 in both the amplitude and its conjugate, which should
be very small around the ρ peak.
9. Summary
We have expressed the fully differential cross section for exclusive ρ production on a po-
larized nucleon in terms of spin density matrix element for the subprocess γ∗p → ρp. We
work in the helicity basis for both γ∗ and ρ and obtain very similar forms for the unpolar-
ized and polarized parts of the cross sections, with the substitution rules (4.15) and (4.16).
The terms for transverse target polarization normal to the hadron plane closely resemble
those for an unpolarized target, and in both cases the number of independent spin density
matrix elements is reduced if one neglects unnatural parity exchange compared with nat-
ural parity exchange. The spin density matrix elements for transverse target polarization
in the hadron plane closely resemble those for a longitudinally polarized target, with both
types of matrix elements involving the interference between natural and unnatural parity
exchange. We have given simple positivity bounds which involve only matrix elements for
an unpolarized target and either those for longitudinal target polarization or for transverse
target polarization normal to the hadron plane. Furthermore, we have investigated the
admixture of longitudinal target polarization relative to the virtual photon momentum for
a target polarized transversely to the lepton beam. This admixture should be small for
the spin density matrix elements which conserve s-channel helicity in the transition from
– 28 –
γ∗ to ρ, but it may be important for s-channel helicity changing matrix elements. Finally,
we have briefly discussed how the results obtained in this paper can be used and extended
for analyzing the production of pion pairs not associated with the ρ resonance.
Acknowledgments
It is a pleasure to thank my colleagues from HERMES for their interest in this work and
for many discussions, especially A. Borissov, J. Dreschler, D. Hasch and A. Rostomyan.
I also gratefully acknowledge helpful discussions with P. Kroll and A. Schäfer. This work
is supported by the Helmholtz Association, contract number VH-NG-004.
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|
0704.1566 | The effect of the solar corona on the attenuation of small-amplitude
prominence oscillations. I. Longitudinal magnetic field | Astronomy & Astrophysics manuscript no. attIarxiv c© ESO 2018
October 31, 2018
The effect of the solar corona on the attenuation of
small-amplitude prominence oscillations
I. Longitudinal magnetic field
R. Soler, R. Oliver, and J. L. Ballester
Departament de Fı́sica, Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain
e-mail: [roberto.soler;ramon.oliver;dfsjlb0]@uib.es
Received xxxx; accepted xxxx
ABSTRACT
Context. One of the typical features shown by observations of solar prominence oscillations is that they are damped in time and
that the values of the damping times are usually between one and three times the corresponding oscillatory period. However, the
mechanism responsible for the attenuation is still not well-known.
Aims. Thermal conduction, optically thin or thick radiation and heating are taken into account in the energy equation, and their role
on the attenuation of prominence oscillations is evaluated.
Methods. The dispersion relation for linear non-adiabatic magnetoacoustic waves is derived considering an equilibrium made of
a prominence plasma slab embedded in an unbounded corona. The magnetic field is orientated along the direction parallel to the
slab axis and has the same strength in all regions. By solving the dispersion relation for a fixed wavenumber, a complex oscillatory
frequency is obtained, and the period and the damping time are computed.
Results. The effect of conduction and radiation losses is different for each magnetoacoustic mode and depends on the wavenumber. In
the observed range of wavelengths the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the
combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. The consideration of
the external corona is of paramount importance in the case of the fast and external slow modes, whereas it does not affect the internal
slow modes at all. When a thinner slab representing a filament thread is considered the fast mode is less attenuatted whereas both
internal and external slow modes are not affected.
Conclusions. Non-adiabatic effects are efficient damping mechanisms for magnetoacoustic modes, and the values of the obtained
damping times are compatible with those observed.
Key words. Sun: oscillations – Sun: magnetic fields – Sun: corona – Sun: prominences
1. Introduction
Prominences are dense coronal structures which appear as thin,
dark filaments on the solar disc when observed in Hα. On the
contrary, they show up as bright objects above the solar limb.
The coronal magnetic field is responsible for the support of
prominences against gravity, and it also plays a fundamental role
in the thermal confinement of the cool prominence plasma em-
bedded in the much hotter coronal environment. Nevertheless,
the structure, orientation and strength of the magnetic field in
prominences and the surrounding corona is still enigmatic and
not well-known. High resolution observations reveal that promi-
nences are composed by numerous very thin, thread-like struc-
tures, called fibrils, piled up to form the body of the prominence
(Lin et al. 2003; Lin et al. 2005, Lin et al. 2007) and measures
also indicate that magnetic field lines are orientated along these
thin threads.
The observational evidence of small-amplitude oscillations
in quiescent solar prominences goes back to 40 years ago
(Harvey 1969). The amplitude of these oscillations typically
goes from less than 0.1 km s−1 to 2–3 km s−1, and have been
historically classified, according to their periods, in short- (P <
10 min), intermediate- (10 min < P < 40 min) and long-period
Send offprint requests to: R. Soler
oscillations (P > 40 min), although very short-periods of less
than 1 min (Balthasar et al. 1993) and extreme ultra-long-periods
of more than 8 hours (Foullon et al. 2004) have been reported.
Nevertheless, the value of the period seems not to be related with
the nature or the source of the trigger and probably is linked to
the prominence eigenmode that is excited. There are also a few
determinations of the wavelength and phase speed of standing
oscillations and propagating waves in large regions of promi-
nences (Molowny-Horas et al. 1997; Terradas et al. 2002) and
in single filament threads (Lin et al. 2007). On the other hand,
several observations (Molowny-Horas et al. 1999; Terradas et al.
2002) have informed about the evidence of the attenuation of the
oscillations in Doppler velocity time series, which is a common
feature observed in large areas. By fitting a sinusoidal function
multiplied by a factor exp(−t/τD) to the Doppler series, these
authors have obtained values of the damping time, τD, which
are usually between 1 and 3 times the corresponding oscilla-
tory period. The reader is referred to some recent reviews for
more information about the observational background (Oliver &
Ballester 2002, Wiehr 2004, Engvold 2004, Ballester 2006).
From the theoretical point of view, small-amplitude promi-
nence oscillations can be interpreted in terms of linear magne-
tohydrodynamic (MHD) waves. Although, there is a wide bib-
liography of works that investigate the ideal MHD wave modes
http://arxiv.org/abs/0704.1566v2
2 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
supported by prominence models (see Oliver & Ballester 2002
for an extensive review of theoretical studies), the investigation
of the wave damping has been broached in few papers. By re-
moving the ideal assumption and including dissipative terms in
the basic MHD equations, several works have studied the atten-
uation of prominence oscillations considering radiative losses
based on the Newtonian law of cooling with a constant relax-
ation time (Terradas et al 2001), or performing a more complete
treatment of non-adiabatic effects, assuming optically thin radi-
ation, heating and thermal conduction (Carbonell et al. 2004;
Terradas et al. 2005). The main conclusion of these previous
studies is that only the slow wave is attenuated by thermal
effects, radiation being the dominant damping mechanism in
the range of typically observed wavelengths in prominences,
but the fast wave remains practically undamped. On the other
hand, Forteza et al. (2007) proposed ion-neutral collisions as a
damping mechanism on the basis that prominences are partially
ionised plasmas, but they found that this mechanism is only ef-
ficient in attenuating the fast mode in quasi-neutral plasmas, the
slow mode being almost unaffected.
In the light of these referred studies, it is likely that non-
adiabatic effects are the best candidates for the damping of
small-amplitude oscillations, at least for slow modes. However,
previous results do not asses the influence of the corona. The
main aim of the present work is to perform a step forward in the
investigation of the effect of non-adiabatic mechanisms (radia-
tion losses, thermal conduction and heating) on the time damp-
ing of prominence oscillations. We consider a slab model with
a longitudinal magnetic field and take into account the external
coronal medium. So, we explore for the first time the joint ef-
fect of prominence and coronal mechanisms on the attenuation
of oscillations. The magnetoacoustic normal modes of this equi-
librium have been previously investigated by Edwin & Roberts
(1982) and Joarder & Roberts (1992) in the adiabatic case. Later,
a revision of these works has been done in Soler et al. (2007),
hereafter Paper I, and the normal modes have been studied and
reclassified according to their magnetoacoustic properties.
This paper is organised as follows. The description of the
equilibrium model and the linear non-adiabatic wave equations
are given in Sect. 2, whereas the dispersion relation for the mag-
netoacoustic modes is derived in Sect. 3. Then, the results are
plotted and investigated in Sect. 4. Finally, Sect. 5 contains the
conclusions of this work.
2. Equilibrium and basic equations
Our equilibrium configuration (Fig. 1) is made of a homoge-
neous plasma layer with prominence conditions (density ρp and
temperature Tp) embedded in an unbounded corona (density ρc
and temperature Tc). The coronal density is computed by fix-
ing the coronal temperature and imposing pressure continuity
across the interfaces. The magnetic field is B0 = B0êx, with B0
a constant everywhere. Both media are unlimited in the x- and
y-directions. The half-width of the prominence slab is zp.
The basic magnetohydrodynamic equations for the discus-
sion of non-adiabatic processes are:
+ ρ∇ · v = 0, (1)
= −∇p +
(∇ × B) × B, (2)
+ (γ − 1)[ρL(ρ, T ) − ∇ · (κ · ∇T )] = 0, (3)
Fig. 1. Sketch of the equilibrium.
Table 1. Parameter values of the radiative loss function
corresponding to the considered regimes. The three promi-
nence regimes represent different plasma optical thicknesses.
Prominence (1) regime corresponds to an optically thin plasma
whereas Prominence (2) and Prominence (3) regimes represent
greater optical thicknesses. All quantities are expressed in MKS
units.
Regime χ∗ α Reference
Prominence (1) 1.76 × 10−13 7.4 Hildner (1974)
Prominence (2) 1.76 × 10−53 17.4 Milne et al. (1979)
Prominence (3) 7.01 × 10−104 30 Rosner et al. (1978)
Corona 1.97 × 1024 −1 Hildner (1974)
= ∇ × (v × B), (4)
∇ · B = 0, (5)
, (6)
where D
+ v · ∇ is the material derivative for time vari-
ations following the motion and all quantities have their usual
meaning. Equation (3) is the energy equation, which in the
present form takes into account non-adiabatic effects (radiation
losses, thermal conduction and heating) and whose terms are ex-
plained in detail in Carbonell et al. (2004) and Terradas et al.
(2005). Following these works, only thermal conduction par-
allel to the magnetic field is assumed and we use the typical
value for the parallel conductivity in prominence and coronal
applications, κ‖ = 10
−11T 5/2 W m−1 K−1. Radiative losses and
heating are evaluated together through the heat-loss function,
L(ρ, T ) = χ∗ρTα − hρaT b, where radiation is parametrised with
χ∗ and α (see Table 1) and the heating scenario is given by ex-
ponents a and b. The heating mechanisms taken into account in
this work are (Rosner et al. 1978; Dahlburg & Mariska 1988):
– constant heating per unit volume (a = b = 0);
– constant heating per unit mass (a = 1, b = 0);
– heating by coronal current dissipation (a = b = 1);
– heating by Alfvén mode/mode conversion (a = b = 7/6);
– heating by Alfvén mode/anomalous conduction damping
(a = 1/2, b = −1/2).
Following the same process as in Carbonell et al. (2004),
we consider small perturbations from the equilibrium state, lin-
earise the basic Eqs. (1)–(6) and obtain their Eqs. (9)–(14). Since
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 3
our model is unlimited in the x- and y-directions, we assume all
perturbations are in the form f1(z) exp i(ωt+ kxx+ kyy), and con-
sidering only motions and propagation in the xz-plane (vy = 0,
ky = 0), which excludes Alfvén waves, the linearised equations
become
iωρ1 + ρ0
ikxvx +
= 0, (7)
iωρ0vx = −ikx p1, (8)
iωρ0vz = −
ikxB1z −
, (9)
p1 − c
= −(γ − 1)
k2xκ‖T1 +
ωρρ1 +
ωT T1
, (10)
iωB1x = −B0
, (11)
iωB1z = B0ikxvz, (12)
where c2s =
is the adiabatic sound speed squared and
L + ρ0Lρ
, ωT ≡
T0LT ,
Lρ, LT being the partial derivatives of the heat-loss function with
respect to density and temperature, respectively,
, LT ≡
Now, it is possible to eliminate all perturbations in favour of
vz to obtain a single differential equation
+ k2z vz = 0, (13)
in which
k2z =
ω2 − k2xv
ω2 − k2xΛ
v2A + Λ
ω2 − k2xc̃
) , (14)
where v2A =
is the Alfvén speed squared. Λ2 and c̃2T are the
modified sound and cusp (or tube) speed squared, respectively,
(γ − 1)
x + ωT − ωρ
+ iγω
(γ − 1)
κ‖k2x + ωT
, (15)
c̃2T ≡
v2A + Λ
. (16)
Expressions for the perturbations in terms of vz are given in
App. A. In all the following formulae, subscripts p or c denote
quantities computed using prominence or coronal values, respec-
tively.
3. Dispersion relation
We impose some restrictions on the solutions of Eq. (13) in or-
der to obtain the dispersion relation for the linear non-adiabatic
magnetoacoustic waves. We restrict this analysis to body waves
which are evanescent in the corona, since we are looking for
solutions which are essentially confined to the slab. For such so-
lutions, vz(z) is of the form
vz(z) =
A1 exp
z + zp
, if z ≤ −zp,
A2 cos
+ A3 sin
, if −zp ≤ z ≤ zp,
A4 exp
z − zp
, if z ≥ zp.
withℜ(kzp) > 0 andℜ(kzc) > 0.
Imposing continuity of vz and the total (gas plus magnetic)
pressure perturbation across the interfaces, we find four alge-
braic relations between the constants A1, A2, A3 and A4. The
non-trivial solution of this system gives us the dispersion rela-
Ac − ω
kzpzp
Ap − ω
kzc = 0, (18)
where cot/tan terms and ± signs are related with the symmetry of
the perturbations. The cot term and the + sign correspond to kink
modes (A3 = 0), whereas the tan term and the − sign correspond
to sausage modes (A2 = 0).
The dispersion relation for the magnetoacoustic waves pre-
sented in Eq. (18) is equivalent to the relation investigated in
Edwin & Roberts (1982) and Joarder & Roberts (1992), and re-
vised Paper I, in the case of adiabatic perturbations, since all
non-adiabatic terms are now enclosed in kzp and kzc through
Eq. (14).
4. Results
Now, we assume Prominence (1) conditions inside the slab (i.e.
an optically thin prominence) and a heating mechanism given
by a = b = 0. Unless otherwise stated, the following equilibrium
parameters are considered in all computations: Tp = 8000 K,
ρp = 5 × 10
−11 kg m−3, Tc = 10
6 K, ρc = 2.5 × 10
−13 kg m−3,
B0 = 5 G and zp = 3000 km. The solution of the dispersion
relation (Eq. [18]) for a fixed real kx gives us a complex fre-
quency ω = ωR + iωI. We then compute the oscillatory period,
the damping time and the ratio of the damping time to the pe-
riod because this is an important quantity from the observational
point of view,
, τD =
Since we are interested in studying the behaviour of the most
relevant solutions of the dispersion relation, we only compute the
results for the fundamental modes, which are labelled, accord-
ing to the classification of Paper I, as internal and external slow
modes and fast modes. The band structure described in Paper I is
slightly modified when non-adiabatic terms are considered (see
Fig. 2). The phase speed of the internal slow modes is now en-
closed in the bandℜ(c̃Tp) < ωR/kx < ℜ(Λp). The adiabatic fast
modes exist in two separated bands in the phase speed diagram
due to the presence of a forbidden region (cTc < ωR/kx < csc),
but now the forbidden band is avoided and a continuous fast
mode is found with vAp < ωR/kx < vAc. Finally, and like in
the adiabatic case, among the external slow modes only the fun-
damental kink one exists as a non-leaky solution in a restricted
4 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
wavenumber range and couples with the fundamental fast kink
mode. Its phase speed is ωR/kx ≈ ℜ(Λc). Therefore, we see that
in the non-adiabatic case Λ plays the role of cs in the adiabatic
case.
Fig. 2. Phase speed versus the dimensionless wavenumber for
the three fundamental oscillatory modes. Solid lines denote kink
modes whereas dotted lines correspond to sausage modes. The
shaded zones are projections of the forbidden (or leaky) regions
on the plane of this diagram. Note that the vertical axis is not
drawn to scale.
In Fig. 3 P, τD and τD/P are represented for the fundamental
modes and for a range of the longitudinal wavenumber between
10−10 m−1 and 103 m−1. The shaded zones correspond to wave-
lengths between 5 × 103 km and 105 km, the typically observed
values. It turns out that the values of the period are very sim-
ilar to those obtained in the adiabatic case (Joarder & Roberts
1992; Paper I). The damping time presents a strong dependence
with the wavenumber and its behaviour is very different from
one mode to another. This fact suggests that the non-adiabatic
mechanisms can affect each mode in a different way (Carbonell
et al. 2004). This is studied in detail in Sect 4.1. Observations
show that prominence oscillations are typically attenuated in a
few periods (Terradas et al. 2002), so a damping time of the order
of the period is expected. In our results, the fundamental modes
present values of τD/P in the range 1 to 10 in the observed wave-
length region, which is in agreement with observations.
4.1. Regions of dominance of the damping mechanisms
The importance of the different non-adiabatic terms included in
the energy equation (Eq. [3]) depends on the wavenumber. In
order to know which is the range of dominance of each mech-
anism, we compare the damping time obtained when consider-
ing all non-adiabatic terms (displayed in the middle column of
Fig. 3) with the results obtained when a specific mechanism is
removed from the energy equation. With this analysis, we are
able to know where the omitted mechanism has an appreciable
effect on the damping. The results of these computations for the
fundamental kink modes (Fig. 4) are summarised as follows:
– The fundamental internal slow kink mode is not affected by
the mechanisms related with the corona. This is a conse-
quence of the nature of this mode, which propagates strictly
along the prominence without disturbing the corona (see Fig.
4, top row, of Paper I). For this reason, in the adiabatic case
it is also independent of the coronal conditions. On the other
hand, the prominence-related mechanisms show different ef-
fects in two different ranges of kx. For kx . 10
−3 m−1 promi-
nence radiation dominates, while for kx & 10
−3 m−1 promi-
nence conduction is the dominant mechanism. Beginning
from small values of the wavenumber, prominence radiation
becomes more efficient as kx grows and the damping time
falls following a power law until kx ≈ 10
−5 m−1, where τD
saturates in a plateau between kx ≈ 10
−5 and kx ≈ 10
−3 m−1.
Then, prominence conduction becomes the dominant mech-
anism and the damping time falls again until kx ≈ 10
−1 m−1
where a new plateau begins. This last part of the curve cor-
responds to the isothermal or superconductive regime, in
which the amplitude of the temperature perturbation drops
dramatically (Carbonell et al. 2006). Prominence radiation
is responsible for the attenuation of the slow mode in the
observed wavelength range. An approximate dispersion re-
lation for the internal slow modes is included in App. B.
– The fundamental fast kink mode is affected by the four
mechanisms. For kx . 3 × 10
−9 m−1 coronal radiation dom-
inates but for 3 × 10−9 m−1 . kx . 5 × 10
−7 m−1 the effect
of coronal conduction grows and becomes the main damping
mechanism. Then, for kx & 5×10
−7 m−1 the corona loses dra-
matically its influence and prominence mechanisms become
responsible for the attenuation of this mode. First, promi-
nence radiation is dominant in the range 5 × 10−7 m−1 .
kx . 10
−3 m−1, then prominence conduction governs the
wave damping for kx & 10
−3 m−1 and finally the isother-
mal regime begins for kx ≈ 10
0 m−1. The minimum of τD
occurs into the coronal conduction regime, for the value of
kx which corresponds to the coupling with the external slow
mode. The transition between the coronal conduction regime
and the prominence radiation regime occurs in the observed
wavelength range. The reason for the sensitivity of the fast
mode damping time on prominence and coronal conditions
is that this wave has a considerable amplitude both inside the
prominence and in the corona, the later becoming more im-
portant for long wavelengths (see the second and third rows
of Fig. 4 of Paper I).
– The behaviour of the damping time of the fundamental exter-
nal slow kink mode is entirely dominated by coronal mech-
anisms whereas the prominence mechanisms do not affect it
at all. This behaviour is a result of the negligible amplitude
of this wave in the prominence (see the fourth and fifth rows
of Fig. 4 of Paper I). For kx . 3× 10
−9 m−1 coronal radiation
dominates, but for shorter wavelengths coronal conduction
becomes more relevant and is responsible for the damping
in the observed wavelength range until the frequency cut-off
is reached. At the cut-off, τD has a value of the order of the
period.
Regarding the fundamental sausage modes, the behaviour of
the internal slow sausage mode is exactly that of the slow kink
mode, so no additional comments are needed. The fundamen-
tal fast sausage mode (Fig. 5) presents the same scheme as the
fundamental fast kink mode for kx & 10
−8 m−1. The main dif-
ference between the fast kink and sausage modes happens in the
observed wavelength range, where the effect of coronal conduc-
tion on the sausage mode is less relevant. If coronal conduction
is omitted, the fundamental fast sausage mode is not able to tra-
verse the forbidden region in the dispersion diagram and then
shows frequency cut-offs as in the adiabatic case. This means
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 5
Fig. 3. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus the longitudinal wavenumber
for the fundamental oscillatory modes. Upper panels: internal slow kink (solid line), fast kink (dotted line) and external slow kink
(dashed line). Lower panels: internal slow sausage (solid line) and fast sausage (dotted line). Shaded zones correspond to those
wavelengths typically observed. Note the cut-offs of the external slow kink mode and the fast sausage mode. Prominence (1)
radiation conditions have been taken for the prominence plasma and the heating scenario is given by a = b = 0.
Fig. 4. Damping time versus the longitudinal wavenumber for the three fundamental kink oscillatory modes: internal slow (left),
fast (centre) and external slow (right). Different linestyles represent the omitted mechanism: all mechanisms considered (solid line),
prominence conduction eliminated (dotted line), prominence radiation eliminated (dashed line), coronal conduction eliminated (dot-
dashed line) and coronal radiation eliminated (three dot-dashed line). Prominence (1) radiation conditions have been taken for the
prominence plasma and the heating scenario is given by a = b = 0.
that coronal conduction causes the fast mode to cross the forbid-
den region in the dispersion diagram in the non-adiabatic case.
Approximate values of kx for which the transitions between
regimes take place can be computed by following a process simi-
lar to that in Carbonell et al. (2006). The thermal ratio, d, and the
radiation ratio, r, quantify the importance of thermal conduction
and radiation, respectively (De Moortel & Hood 2004),
(γ − 1)κ‖T0ρ0
γ2 p20τs
τcond
, (19)
(γ − 1)τsρ
, (20)
where τs is the sound travel time and τcond and τrad are character-
istic conductive and radiative time scales. Taking τs = 2π/k
the value of k∗ for which the condition d = r is satisfied is
k∗ = 2πρ0
χ∗Tα−10
. (21)
Now, we use prominence values to compute k∗ for the promi-
nence radiation–prominence conduction transition (k∗p), and
6 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 6. Damping time versus the longitudinal wavenumber for the fundamental internal slow kink mode (left), the fundamental fast
kink mode (centre) and the fundamental external slow kink mode (right). The different linestyles represent different values of the
prominence temperature: Tp = 8000 K (solid line), Tp = 5000 K (dotted line) and Tp = 13000 K (dashed line). The heating scenario
is given by a = b = 0 and the optical thickness for the prominence plasma is Prominence (1).
Fig. 7. Same as Fig. 6 with ρp = 5 × 10−11 kg m−3 (solid line), ρp = 2 × 10−11 kg m−3 (dotted line) and ρp = 10−10 kg m−3 (dashed
line).
Fig. 8. Same as Fig. 6 with B0 = 5 G (solid line), B0 = 2 G (dotted line) and B0 = 10 G (dashed line).
coronal values for the coronal radiation–coronal conduction
transition (k∗c). This gives the values k
p ≈ 1.7 × 10
−3 m−1, and
k∗c ≈ 2.2 × 10
−8 m−1. For the transition of the fast kink mode
between the coronal conduction and the prominence radiation
regimes, the boundary wavenumber k∗p↔c can be roughly calcu-
lated by imposing dc = rp, that gives
k∗p↔c = 2πρp
cscχ∗pT
cspκ‖cTc
, (22)
and whose numerical value is k∗p↔c ≈ 1.4 × 10
−6 m−1. All these
wavenumbers for the transitions between different regimes are
independent of the wave type, be it fast or slow, internal or ex-
ternal (this agrees with Figs. 4 and 5). On the other hand, the be-
ginning of the isothermal regime can be estimated by following
Porter et al. (1994). Considering c2sp/v
Ap ≪ 1 and the approxi-
mations ωR ≈ kxcsp for the slow wave and ωR ≈ kxvAp for the
fast wave, the critical wavenumber is
kcrit−slow =
2ρpkBcsp
κ‖pmp cos θ
, (23)
for the internal slow mode, and
kcrit−fast =
2ρpkBvAp
κ‖pmp cos2 θ
, (24)
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 7
Fig. 5. Same as Fig. 4 for the fundamental fast sausage mode.
for the fast mode, where mp is the proton mass, kB is the
Boltzmann constant and θ is the angle between B and k. Taking
cos θ = 1 for simplicity, the approximate critical values are
kcrit−slow ≈ 1.7 × 10
−1 m−1 and kcrit−fast ≈ 9.1 × 10
−1 m−1. We
note that all these approximate values describe correctly the tran-
sitions between the diverse regimes shown in Figs. 4 and 5, but
their numerical values overestimate by almost an order of mag-
nitude the actual critical wavenumbers.
4.2. Exploring the parameter space
4.2.1. Dependence on the equilibrium physical conditions
In this section, we compute the solutions for different values
of the equilibrium physical conditions. We only present the re-
sults for the fundamental kink modes since they are equivalent to
those of sausage modes. Figures 6, 7 and 8 display the damping
time as function of kx for some selected values of the prominence
temperature, the prominence density and the magnetic field, re-
spectively.
For the internal slow mode, a decrease of the prominence
temperature or the prominence density raises the position of
the radiative plateau and increases its length. The opposite be-
haviour is seen when the density or the temperature are in-
creased. However, the value of the magnetic field does not in-
fluence the attenuation of this mode, such as expected for a slow
wave.
Increasing the value of the prominence temperature causes
a vertical displacement of τD of the fast mode in those regions
in which prominence mechanisms dominate. The value of the
prominence density has a smaller effect and its main influence
is in changing the coupling point with the external slow mode,
which moves to higher kx for greater values of the density. The
magnetic field strength has a more complex effect on τD and also
modifies the coupling point.
Finally, the external slow mode is only slightly affected by
a modification of the prominence physical parameters since it
is mainly dominated by coronal conditions, and the influence of
the magnetic field is very small due to the slow-like magnetoa-
coustic character of this solution.
4.2.2. Dependence on the prominence optical thickness
The optically thin radiation assumption is a reasonable approxi-
mation in a plasma with coronal conditions but prominence plas-
mas often are optically thick. In this section we compare the
results obtained considering different optical thicknesses for the
prominence plasma (see Fig. 9 for the fundamental kink modes).
The results corresponding to the slow sausage mode have not
been plotted since they are equivalent to those obtained for slow
kink mode; those for the fundamental fast sausage mode, how-
ever, are displayed in Fig. 10.
The variation of the prominence optical thickness modi-
fies the prominence conduction–prominence radiation critical
wavenumber, k∗p (see analytical approximation of Eq. [21]). For
the internal slow mode, an increase in the optical thickness
raises the position of the radiative plateau and shifts it to smaller
wavenumbers. This fact causes an a priori surprising result in the
observed wavelength range, since τD has a smaller value for opti-
cally thick radiation, Prominence (3), than for optically thin radi-
ation, Prominence (1). Regarding fast modes, the damping time
increases when the optical thickness is increased, but only in
the region in which prominence radiation dominates. The value
of τD inside the observed wavelength range is partially affected
and raises an order of magnitude for Prominence (3) conditions
in comparison with the results for Prominence (1) conditions.
Finally, the damping time of the external slow mode is not af-
fected by the prominence optical thickness since it is entirely
dominated by the corona, as it has been noticed in Sect. 4.1.
Fig. 10. Same as Fig. 9 for the fundamental fast sausage mode.
4.2.3. Dependence on the heating scenario
Now, we compute the damping time for the five possible heating
scenarios. For simplicity, we only consider the fundamental kink
modes (Fig. 11). Carbonell et al. (2004) showed that in a plasma
with prominence conditions the different heating scenarios have
no significant influence on the damping time. Nevertheless, in
coronal conditions wave instabilities can appear depending on
the heating mechanism. In our results, we see that the heating
scenario affects the value of τD only in the ranges of kx in which
radiation is the dominant damping mechanism. The heating sce-
nario has a negligible effect when prominence radiation domi-
nates, since τD is only slightly modified. On the contrary, wave
instabilities appear in those regions in which coronal radiation
dominates. Thermal destabilisation occurs when the imaginary
part of the frequency becomes negative, so oscillations are not
attenuated but amplified in time. Instabilities only occur in the
fundamental fast kink and the external slow modes for very small
values of kx, outside the observed wavelength range.
8 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 9. Same as Fig. 6 with the prominence optical thickness given by Prominence (1) (solid line), Prominence (2) (dotted line) and
Prominence (3) (dashed line) conditions.
Fig. 11. Same as Fig. 6 with the heating scenario given by a = b = 0 (solid line); a = 1, b = 0 (dotted line); a = b = 1 (dashed line);
a = b = 7/6 (dot-dashed line); a = 1/2, b = −1/2 (three dot-dashed line).
4.3. Comparison with the solution for an isolated slab
In order to assess the effects arising from the presence of two
different media in the equilibrium, a comparison between the
previous results and those corresponding to a single medium is
suitable. So, we consider a simpler equilibrium made of an iso-
lated prominence slab with the magnetic field parallel to its axis.
The external medium is not taken into account. Magnetoacoustic
non-adiabatic perturbations are governed by Eq. (13), and rigid
boundary conditions for vz are imposed at the edges of the promi-
nence slab,
vz(−zp) = vz(zp) = 0. (25)
Then, the solution is of the form
vz(z) = C1 cos
+C2 sin
, (26)
and after imposing boundary conditions (Eq. [25]), we deduce
the dispersion relation for the magnetoacoustic slow and fast
non-adiabatic waves,
kzpzp =
π, (n = 0, 1, 2, . . .), (27)
for the kink modes, and
kzpzp = nπ, (n = 1, 2, 3, . . .), (28)
for the sausage modes. Inserting expressions (14) and (15) for
kzp and Λp respectively, one can rewrite the dispersion relations
(27) and (28) as polynomial equations in ω. See App. C for the
details.
Next, considering only the fundamental kink modes for sim-
plicity, we compute the period and the damping time and com-
pare with those obtained when the surrounding corona is taken
into account (Fig. 12). We see that there is a perfect agree-
ment between both results in the case of the internal slow mode,
whereas the solutions for the fast mode only coincide for inter-
mediate and large wavenumbers, and show an absolutely differ-
ent behaviour in the observed wavelength range and for smaller
wavenumbers. Additionally, one must bear in mind that the ex-
ternal slow mode exists because of the presence of the coronal
medium, hence it is not supported by an isolated slab.
In Paper I we proved that the internal slow mode is essen-
tially confined within the prominence slab and that the effect of
the corona on its oscillatory period can be neglected. Now, we
see that the corona has no influence on the damping time either.
On the other hand, the confinement of the fast mode is poor for
small wavenumbers, the isolated slab approximation not being
valid. As it has been noted in Section 4.1, the corona has an es-
sential effect on the attenuation of the fast mode in the observed
wavelength range.
4.4. Application to a prominence fibril
Since magnetic field lines are orientated along fibrils, our model
can also be applied to study the oscillatory modes supported by
a single prominence fibril. In order to perform this investigation,
we reduce the slab half-width, zp, to a value according to the
typical observed size of filament threads, which is between 0.2
to 0.6 arcsec (Lin et al. 2005). Since these values are close to the
resolution limit of present-day telescopes, it is likely that thinner
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 9
Fig. 12. Comparison between the solutions for a prominence
plus corona system and for an isolated slab with prominence
conditions. The upper panels correspond to the fundamental in-
ternal slow kink mode and the lower panels to the fundamental
fast kink mode. The solid lines are the solutions for a prominence
plus corona equilibrium whereas the dotted lines with diamonds
represent the solutions for an isolated slab. Prominence (1) pa-
rameters and a = b = 0 have been used in the computations.
threads could exist. So, assuming now zp = 30 km, we compute
P, τD and τD/P for the fundamental kink modes and compare
these results with those obtained for zp = 3000 km.
Such as displayed in Fig. 13, both internal and external slow
modes are not affected by the width of the prominence slab since
they are essentially polarised along the x-direction and so they
are not influenced by the equilibrium structure in the z-direction.
Nevertheless, the location of the cut-off of the external slow
mode and the coupling point with the fast mode are shifted to
larger values of kx when the slab width is reduced. On the other
hand, the fast mode, which is responsible for transverse motions,
is highly influenced by the value of zp. The τD curve for the
fast mode is displaced to larger values of kx when smaller zp is
considered. This causes that higher values of τD/P are obtained
in the observed wavelength range. Hence, these results suggest
that local prominence oscillations related with transverse fast
modes supported by a single fibril could be less affected by non-
adiabatic mechanisms than global fast modes supported by the
whole or large regions of the prominence. However, according to
the results pointed out by Dı́az et al. (2005) and Dı́az & Roberts
(2006), large groups of fibrils tend to oscillate together since the
separation between individual fibrils is of the order of their thick-
ness. In a very rough approximation one can consider that a thick
prominence slab could represent many near threads which oscil-
late together and that the larger the slab width, the more threads
fit inside it. So, our results show that the slab size (i.e. the number
of threads which oscillate together in this rough approximation)
has important repercussions on the damping time of collective
transverse oscillations, hence the oscillations could be more at-
tenuated when the number of oscillating threads is larger. This
affirmation should be verified by investigating the damping in
multifibril models.
5. Conclusions
In this paper, we have studied the time damping of magne-
toacoustic waves in a prominence-corona system considering
non-adiabatic terms (thermal conduction, radiation losses and
heating) in the energy equation. Small amplitude perturbations
have been assumed, so the linearised non-adiabatic MHD equa-
tions have been considered and the dispersion relation for the
slow and fast magnetoacoustic modes has been found assuming
evanescent-like perturbations in the coronal medium. Finally, the
damping time of the fundamental oscillatory modes has been
computed and the relevance of each non-adiabatic mechanism
on the attenuation has been assessed. Next, we summarise the
main conclusions of this work:
1. Non-adiabatic effects are an efficient mechanism to obtain
small ratios of the damping time to the period in the range of
typically observed wavelengths of small-amplitude promi-
nence oscillations.
2. The mechanism responsible for the attenuation of oscilla-
tions is different for each magnetoacoustic mode and de-
pends on the wavenumber.
3. The damping of the internal slow mode is dominated by
prominence-related mechanisms, prominence radiation be-
ing responsible for the attenuation in the observed wave-
length range. Such as happens in the adiabatic case (see
Paper I) the corona does not affect the slow mode at all, and
these results are in perfect agreement with those for an iso-
lated prominence slab.
4. The attenuation of the fast mode in the observed wavelength
range is governed by a combined effect of prominence radia-
tion and coronal conduction. The presence of the corona is of
paramount importance to explain the behaviour of the damp-
ing time for small wavenumbers within the observed range of
wavelengths. Non-adiabatic mechanisms in both the promi-
nence and the corona are significant because the fast mode
achieves large amplitudes in both regions.
5. Since the external slow mode is principally supported by
the corona, its damping time is entirely governed by coro-
nal mechanisms, coronal conduction being the dominant one
in the observed wavelength range.
6. The consideration of different optical thicknesses for the
prominence plasma causes an important variation of the
damping time of the internal slow and fast modes in the ob-
served wavelength range. Hence a precise knowledge of the
radiative processes of prominence plasmas is needed to ob-
tain more realistic theoretical results.
7. The heating scenario has a negligible effect on the damp-
ing time of all solutions in the observed wavelength range.
Depending on the scenario considered, thermal instabilities
can appear for small values of the wavenumber, in which
coronal radiation dominates.
8. The width of the prominence slab does not affect the results
for both internal and external slow modes. However, fast
modes are less attenuated in the range of observed wave-
lengths when thinner slabs or filaments threads are consid-
ered.
Taking into account the results in the observed range of
wavelengths, one can conclude that radiative effects of the
prominence plasma are responsible for the attenuation of the in-
ternal slow modes, which can be connected with intermediate-
and long-period prominence oscillations, whereas a combined
effect of prominence radiation and coronal thermal conduction
10 R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I
Fig. 13. Period (left), damping time (centre) and ratio of the damping time to the period (right) versus kx for the fundamental
kink oscillatory modes: internal slow (top panels), fast (mid panels) and external slow (bottom panels). Solid lines correspond
to zp = 3000 km whereas dotted lines correspond to zp = 30 km. Prominence (1) radiation conditions have been taken for the
prominence plasma and the heating scenario is given by a = b = 0.
governs the damping of fast modes, whose periods are compati-
ble with those of short-period oscillations.
Acknowledgements. The authors acknowledge the financial support received
from the Spanish Ministerio de Ciencia y Tecnologı́a under grant AYA2006-
07637. R. Soler thanks the Conselleria d’Economia, Hisenda i Innovació for a
fellowship.
Appendix A: Expressions for the perturbations
Combining Eqs. (7)–(12), one can obtain the expressions for the
perturbed quantities as functions of vz and its derivative
−ikxΛ
ω2 − k2xΛ
, (A.1)
ω2 − k2xΛ
, (A.2)
iωρ0Λ
ω2 − k2xΛ
, (A.3)
ω2 − k2xΛ
, (A.4)
B1x =
, (A.5)
B1z =
vz. (A.6)
Now, we write the expressions for the perturbations to the mag-
netic pressure, p1m, and the total pressure, p1T,
p1m =
B1x =
, (A.7)
p1T = p1 + p1m =
ω2 − k2xv
. (A.8)
In the limit Λ → cs (i.e. in the absence of conduction, radiation
losses and heating), all the expressions reduce to those corre-
sponding to the adiabatic case.
R. Soler et al.: The effect of the solar corona on the attenuation of prominence oscillations I 11
Appendix B: Approximate dispersion relation for
the internal slow modes
Internal slow modes are almost non-dispersive and for adia-
batic perturbations a good approximation for the frequency is
ω ≈ cspkx, csp being the prominence sound speed. In the non-
adiabatic case, we can consider the equivalence between cs and
Λ to propose ω ≈ Λpkx as an approximate dispersion relation.
Taking into account Eq. (15) for Λ, the approximate dispersion
relation for the internal slow modes is a third order polynomial
in ω,
− iBω2 − k2xc
spω + i
Ak2x = 0, (B.1)
A = (γ − 1)
κ̂‖pk
x + ωTp − ωρp
, (B.2)
B = (γ − 1)
κ̂‖pk
x + ωTp
, (B.3)
κ̂‖p = κ‖p
In Fig. B.1 a comparison between the exact and approximate
solutions is displayed and a perfect agreement is seen.
Fig. B.1. Period (left) and damping time (right) versus the lon-
gitudinal wavenumber for the fundamental internal slow kink
mode. The solid line corresponds to the exact solution and the di-
amonds correspond to the approximate solution. Prominence (1)
parameters and a = b = 0 have been used in the computations.
Appendix C: Dispersion relation for an isolated slab
We here deduce a polynomial dispersion relation for the magne-
toacoustic normal modes of a slab with a longitudinal magnetic
field. Taking Eqs. (27) and (28) as the dispersion relations for
the kink and sausage modes, respectively, one can replace kz and
Λ with their correspondent expressions (Eqs. [14]–[15]), and the
following fifth order polynomial equation is found,
− iBω4 −
v2A + c
v2AB +A
+ v2Ac
iAv2Ac
= 0, (C.1)
= k2x +
(n + 1/2)2 π2
, (n = 0, 1, 2, . . .),
for the kink modes, and
= k2x +
, (n = 1, 2, 3, . . .),
for the sausage modes. Quantities A and B are given by
Eqs. (B.2) and (B.3), respectively.
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Introduction
Equilibrium and basic equations
Dispersion relation
Results
Regions of dominance of the damping mechanisms
Exploring the parameter space
Dependence on the equilibrium physical conditions
Dependence on the prominence optical thickness
Dependence on the heating scenario
Comparison with the solution for an isolated slab
Application to a prominence fibril
Conclusions
Expressions for the perturbations
Approximate dispersion relation for the internal slow modes
Dispersion relation for an isolated slab
|
0704.1567 | Energy and Momentum Distributions of Kantowski and Sachs Space-time | arXiv:0704.1567v2 [gr-qc] 15 Apr 2007
Energy and Momentum Distributions of
Kantowski and Sachs Space-time
Ragab M. Gad1 and A. Fouad
Mathematics Department, Faculty of Science,
Minia University, 61915 El-Minia, EGYPT.
Abstract
We use the Einstein, Bergmann-Thomson, Landau-Lifshitz and Pa-
papetrou energy-momentum complexes to calculate the energy and mo-
mentum distributions of Kantowski and Sachs space-time. We show
that the Einstein and Bergmann-Thomson definitions furnish a consis-
tent result for the energy distribution, but the definition of Landau-
Lifshitz do not agree with them. We show that a signature switch
should affect about everything including energy distribution in the
case of Einstein and Papapetrou prescriptions but not in Bergmann-
Thomson and Landau-Lifshitz prescriptions.
1 Introduction
One of the most interesting and intricate problems still unsolved since the
outset of general relativity is the energy-momentum localization. Einstein
himself proposed the first energy-momentum complex in an attempt to de-
fine the local distribution of energy and momentum [1]. After this attempt, a
plethora of different energy-momentum complexes were proposed, including
formulations by Tolman [2], Papapetrou [3], Møller [4], Landau and Lifshitz
[5], Weinberg [6], Bergmann-Thomson [7] and others. This approach was
abandoned for a long time due to severe criticism for a number of reasons.
Recently, Virbhadra re-opened the subject of energy-momentum com-
plexes [8]. He pointed out that though these complexes are non-tensors,
they yield reasonable and consistent results for a given space-time. Aguir-
regabiria et al [9] found that for any metric of the Kerr-Schild class, sev-
eral different definitions of the energy-momentum complex yield precisely
the same results.Virbhadra [10] investigated whether or not these energy
momentum complexes lead to the same results for the most general non-
static spherically symmetric metric and found that they disagree. He noted
that the energy-momentum complexes of Landau and Lifshitz, Papapetrou
1Email Address: [email protected]
http://arxiv.org/abs/0704.1567v2
and Weinberg give the same results as in the Einstein definition if the cal-
culations are performed in Kerr-Schild Cartesian coordinates. However,
these energy-momentum complexes disagree if computations are done in
”Schwarzschild Cartesian” coordinates2. In a detail study of the question,
Xulu [11] has confirmed this suggestion. He obtained the energy distribu-
tion for the most general non-static spherically symmetric using Møller’s
definition and found different results in general from those obtained us-
ing Einstein’s definition. These results agree for the Schwarzschild, Vaidya
and Janis-Newmann-Winicour space-times, but disagree for the Reissner-
Nordström space-time. Many authors had similarly successfully applied the
aforementioned energy-momentum complexes to various black hole configu-
rations [12].
It has been remained a controversial problem whether or not energy and
momentum are localizable. There are different opinions on this subject.
Contradicting the viewpoint of Misner et al. [13] that the energy is local-
izable only for spherical systems, Cooperstock and Sarracino [14] argued
that if the energy localization is meaningful for spherical systems then it
is meaningful for all systems. Bondi [15] expressed that a non-localizable
form of energy is inadmissible in relativity and its location can in prin-
ciple be found. These contradictory viewpoints bear significantly on the
study of gravitational waves. It is an interesting question whether or not
gravitational waves have energy and momentum content. In a series of pa-
pers, Cooperstock [16] hypothesized that in a curved space-time energy and
momentum are confined to the region of non-vanishing energy-momentum
tensor T ab and consequently the gravitational waves are not carriers of energy
and momentum in vacuum space-times. This hypothesis has neither been
proved nor disproved. There are many results supporting this hypothesis
(see for example, [17, 18]).
In this paper we evaluate the energy and momentum distributions of
the Kantowski and Sachs space-time, using Einstein, Bergmann-Thomson,
Landau-Lifshitz and Papapetrou energy-momentum complexes.
Through this paper we use G = 1 and c = 1 units and follow the conven-
tion that Latin indices take value from 0 to 3 and Greek indices take value
from 1 to 3.
2Schwarzschild metric in “Schwarzschild Cartesian coordinates” is obtained by trans-
forming this metric (in usual Schwarzschild coordinates {t, r, θ, φ}) to {t, x, y, z} using
x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ.
2 Kantowski and Sachs Space-time
The standard representation of Kantowski and Sachs space-times are given
by [19]
dS2 = dt2 −A2(t)dr2 −B2(t)(d2θ + sin2 θd2φ), (2.1)
where the functions A(t) and B(t) are to be determined from the field equa-
tions.
The solutions of the Einstein field equations for the above metric were con-
sidered with dust source [19], but they were generalized (in fact earlier) to
the general perfect fluid source [21].
From the geometrically point of view, this line element admits a four
parameter continuous group of isometries which acts on space-like hypersur-
face, and has no three parameter subgroup that would be simply transitive
on the orbits (for more detailed description see Kantowski and Sachs [19]
and Collins [20]).
From the physical point of view, the metric (2.1) automatically defines an
energy-momentum tensor of a fluid with anisotropic pressure, and the coor-
dinates of (2.1) are comoving. The rotation and acceleration are zero, but
if the source is to be a perfect fluid, then the shear is necessarily non-zero.
It is well known that if the calculations are performed in quasi-Cartesian
coordinates, all the energy-momentum complexes give meaningful results.
According to the following transformations
x2 + y2 + z2, φ = arctan(
the line element (2.1) written in terms of quasi-Cartesian coordinates reads:
dS2 = dt2+
(dx2+ dy2+ dz2)−
(xdx+ ydy+ zdz)
. (2.2)
For the above metric the determinant of the metric tensor and the con-
travariant components of the tensor are given, respectively, as follows
det(g) = −A
g00 = 1,
g11 = x2
g12 = xy
g13 = xz
g22 = y2
g23 = yz
g33 = z2
(2.3)
3 Einstein’s Energy-momentum Complex
The energy-momentum complex as defined by Einstein [1] is given by
θki =
Hkli,l, (3.4)
where the Einstein’s superpotential Hkli is of the form
Hkli = −H
[− g(gknglm − glngkm)],m. (3.5)
and θ0α are the energy and momentum density components, respectively.
The Einstein energy-momentum satisfies the local conservation law
The energy and momentum in the Einstein’s prescription are given by
∫ ∫ ∫
θ0i dx
1dx2dx3. (3.6)
Using the Gauss theorem we obtain
H0αi nαds, (3.7)
where nα = (
) are the components of a normal vector over an in-
finitesimal surface element ds = r2 sin θdθdφ.
The required non zero components of Hkli for the line element (2.1) are
given by
A2r2 +B2
A2r2 +B2
A2r2 +B2
(ȦB −AḂ)− (ȦB +AḂ)
= H02
= 2xyB
(ȦB −AḂ),
= H03
= 2xzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
= H03
= 2yzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
(3.8)
Using the components (3.8) we obtain the components of energy and mo-
mentum densities in the form
8πAr4
(A2r2 −B2),
(ȦB +AḂ),
(ȦB +AḂ),
(ȦB +AḂ).
(3.9)
Using equations (3.8) in equation (3.7), the energy and momentum distri-
butions are the following
EEin = P0 =
(A2r2 +B2),
P1 = P2 = P3 = 0.
We notice that if the signature of the space-time under study is changed
from +2 to -2, we find that the values of energy and momentum densities
as well as the energy distribution are changed from positive to negative.
4 The Energy-Momentum Complex of Bergmann-
Thomson
The Bergmann-Thomson energy-momentum complex [7] is given by
[gilBkml ],m, (4.1)
where
Bkml =
gkngmp − gmngkp
B00 and B0α are the energy and momentum density components.
The energy and momentum are given by
P i =
∫ ∫ ∫
Bi0dx1dx2dx3. (4.2)
Using the Gauss theorem we have
P i =
Bi0αnαdS. (4.3)
In order to calculate the energy and momentum distributions for the
space-time under consideration, using Bergmann-Thomson energy-momentum
complex, we require the following non-vanishing components of Bkml
A2r2 +B2
A2r2 +B2
A2r2 +B2
(ȦB −AḂ)− (ȦB +AḂ)
= B02
(ȦB −AḂ),
= B03
= 2xzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
= B03
= 2yzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
(4.4)
Using the components (4.4) in (4.1), the components of energy and mo-
mentum densities are as follows
00 = 1
8πAr4
(A2r2 −B2),
01 = − x
Ȧr2 + B
(2AḂ − ȦB
02 = − y
Ȧr2 + B
(2AḂ − ȦB
03 = − z
Ȧr2 + B
(2AḂ − ȦB
(4.5)
Using equations (4.4) in equation (4.3), we obtain the energy and momentum
distributions in the following form
EBerg = P
(A2r2 +B2),
P1 = P2 = P3 = 0.
The above energy density and energy distribution are agreement with that
obtained before, using Einstein’s energy-momentum complex. In the case of
Bergmann’s energy-momentum complex, we notice that a signature switch
do not affect about everything including energy distribution. Consequently,
the Einstein and Bergmann-Thomson prescription do not give the same
results when the signature of the space-time under study is -2.
5 Landau-Lifshitz’s Energy-momentum Complex
Landau-Lifshitz’s energy-momentum complex [5] is given by
Lij =
, (5.1)
where
Sikjl = −g(gijgkl − gilgkj). (5.2)
Lij is symmetric in its indices, L00 is the energy density and L0α are the
momentum (energy current) density components. Sikjl has the symmetries
of the Riemann curvature tensor.
The energy and momentum are given by
P i =
∫ ∫ ∫
Li0dx1dx2dx3. (5.3)
Using the Gauss theorem we have
P i =
Si0αnαdS. (5.4)
The required non-vanishing components of Sikjl are
S0101 = B2
(A2r2 −B2)− A
S0102 =
B2(A2r2 −B2),
S0103 = xz
B2(A2r2 −B2),
S0202 = B2
(A2r2 −B2)− A
S0203 =
B2(A2r2 −B2),
S0303 = B2
(A2r2 −B2)− A
(5.5)
Using these components in equation (5.1), we obtained the energy and mo-
mentum densities are
L00 = − B
[A2r2 + 3B2],
L10 = − xB
[A(AḂ + ȦB)r2 + 2B2Ḃ],
L20 = − yB
[A(AḂ + ȦB)r2 + 2B2Ḃ],
L30 = − zB
[A(AḂ + ȦB)r2 + 2B2Ḃ].
(5.6)
Using equations (5.1) in equation (5.4), we obtain the energy and momentum
distributions
ELL =
(A2r2 +B2). (5.7)
P 1LL = P
LL = P
LL = 0. (5.8)
The above results do not agree with the results obtained before, using Ein-
stein and Bergmann-Thomson energy-momentum complexes. A signature
switch does not affect about everything including energy distribution.
6 Papapetrou’s Energy-momentum Complex
The symmetric energy-momentum complex of Papapetrou [3] is given by
Ωij =
, (6.1)
where
Υijkl =
−g(gijηkl − gikηjl + gklηij − gjlηik), (6.2)
and ηik is the Minkowski metric with signature +2.
Ω00 and Ωα0 are the energy and momentum density components.
The energy and momentum, using the Papapetrou prescription are given by
P i =
∫ ∫ ∫
Ωi0dx1dx2dx3. (6.3)
Using the Gauss theorem we obtain
P i =
Υi0αl,l nαdS. (6.4)
The non-vanishing components of Υijkl are as follows
Υ0011 = 1
(A2r2 −B2)−A2
Υ0012 = xy
(A2r2 −B2),
Υ0013 = xz
(A2r2 −B2),
Υ0022 = 1
(A2r2 −B2)−A2
Υ0023 =
(A2r2 −B2),
Υ0033 = 1
(A2r2 −B2)−A2
(6.5)
Using these components in (6.1), we get the following energy and momentum
density components
Ω00 = A
[r2 −B2],
Ω10 = − x
[r2Ȧ+B(ȦB + 2AḂ)],
Ω20 = − y
[r2Ȧ+B(ȦB + 2AḂ)],
Ω30 = − z
[r2Ȧ+B(ȦB + 2AḂ)].
(6.6)
Using equations (6.5) in equation (6.4), we obtain the energy and momentum
distributions
(r2 +B2) (6.7)
P 1 = P 2 = P 3 = 0 (6.8)
The above results do not agree with the results obtained before, using Ein-
stein, Bergmann-Thomson and Landau and Lifshitz energy-momentum com-
plexes. A signature switch should affect about everything including energy
distribution.
Discussion
We investigated the energy and momentum (due to matter plus fields in-
cluding gravity) distribution of the Kantowski and Sachs space-time using
the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-
momentum complexes. We found that the quantities of energy and mo-
mentum densities as well as energy distribution are well-defined and well-
behaved. For the space-time under consideration, we found that the energy-
momentum complexes of Einstein and Bergmann-Thomson give the same
results, while Landau-Lifshitz and Papapetrou do not give the same results
and not agree with the aforementioned complexes. We have shown that a
signature switch affects about every thing (by changing the sign of the values
of energy and momentum densities as well as energy distribution) including
energy distribution. These changes occur in the case of Einstein and Pa-
papetrou prescriptions but not in Bergmann-Thomson and Landau-Lifshitz
prescriptions.
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|
0704.1568 | Green functions and nonlinear systems: Short time expansion | Green functions and nonlinear systems: Short time expansion
Marco Frasca∗
Via Erasmo Gattamelata, 3
00176 Roma (Italy)
(Dated: February 1, 2008)
We show that Green function methods can be straightforwardly applied to nonlinear equations
appearing as the leading order of a short time expansion. Higher order corrections can be then
computed giving a satisfactory agreement with numerical results. The relevance of these results
relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field
theory granting the existence of a strong coupling expansion. Having a Green function in this regime
in quantum field theory amounts to obtain the corresponding spectrum of the theory.
I. INTRODUCTION
Nonlinear equations represent a class of very difficult mathematical problems to manage by analytical methods. A
lot of fundamental aspects of physics are described by these equations making not easy their understanding due to
the lack of useful techniques.
In this paper we present an approach that is based on an unexpected result for Green functions. First hints
in this direction were obtained in [1, 2] where we showed that for a part of the integration interval, a nonlinear
differential equation like φ̈ + φ3 = j, being j a source term, can be solved by its Green function G̈ + G3 = δ(t) as
φ(t) ≈
G(t − t′)j(t′)dt′.
Although this solution was put forward, the knowledge of this result is hardly useful unless we are not able to
understand how to get higher order corrections. The aim of this paper is to give a proper understanding of this
solution and to give a technique to get higher order corrections in order to improve it. We will show that it represents
a short time solution. Then, a form factor described by polynomial terms in time can correct properly the propagator
to improve in some cases this approximation.
The reason to give such a solution relies on the possibility to treat strong coupled quantum field theories that at the
leading order produce nonlinear equations driven by a source. Strongly coupled theory can be managed by a gradient
expansion that is the dual perturbation series to a weak coupling expansion as we proved in [1, 3, 4]. By “dual” we
mean that two series can be obtained by simply interchanging the terms of the expansion producing in a case a series
with an expansion parameter being the inverse of the expansion parameter of the other series, the asymptotic series
so obtained holding in the proper limit where this parameter gives a converging expansion (coupling going to infinity
in a case while going to zero in the other). This approach is true for any differential equation set and we applied it
also in general relativity [3] obtaining a sound proof of the Belinski-Khalatnikov-Lifshitz conjecture [5, 6, 7] as this is
a result of a gradient expansion.
A strongly coupled system in quantum mechanics is known to be a classical system as was firstly shown by Simon
[8]. We revised this approach in [9] where we have seen that the gradient expansion for the Schrödinger equation,
also known as Wigner-Kirkwood expansion, gives rise to a Thomas-Fermi approximation to the leading order for
a many-body system [10] and has the same eigenvalue expansion as for a WKB approximation. Wigner-Kirkwood
expansion is indeed the gradient expansion of the Schrödinger equation.
Gradient expansions in quantum field theories were not widely used before while their proper understanding is not
that easy. Our aim in this paper is to fully exploit this perturbation approach and its application in quantum field
theory wherever possible. This method may pave the way to manage analytically some problems that now appear
difficult to manage also in a wide variety of fields where nonlinear equations are at the foundations.
The paper is structured in the following way. In section II we show how to derive a gradient expansion out of a
duality principle in perturbation theory with the proper understanding of the expansion parameter. In section III
we present the main motivation for this paper, that is the continuum limit of a scalar quantum field theory giving
rise to a model nonlinear equation we will use throughout the paper. In section IV we present the method firstly
applied to a simple Riccati equation having a known analytical solution and then we generalize our method to the
case of the leading order equation of a scalar field theory. In section V we give the numerical results showing how the
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http://arXiv.org/abs/0704.1568v3
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approximation improves with higher order corrections varying also the forcing term into the equation. In section VI we
compare our approach with functional iteration method, a well known method used to analyze non-linear differential
equations. This will give a proper understanding of the speed of convergence of our method. Finally, in section VII
conclusions are given.
II. DUAL EXPANSION FOR NONLINEAR PDES
In order to make the paper self-contained we present here some material already given in Ref.[4]. We specialize the
presentation to a λφ4 model that is our reference model.
The Hamiltonian of the model is given by
dD−1x
(∇φ)2 + V (φ)
being D the spacetime dimensionality and V (φ) = 1
φ2 + λ
φ4 and we take the case of a single component for the sake
of simplicity. Hamilton equations are
∂tφ = π (2)
∂tπ = ∇2φ − φ − λφ3.
We can see at glance that we can chose to do perturbation theory by two different choices. One can take either λφ3
or ∇2φ − φ as a small term. What we want to understand is the link between the two series with respect to the
parameter λ.
By choosing λφ3 as a small term one gets the small perturbation series
∂tφ0 = π0 (3)
∂tφ1 = π1
∂tφ2 = π2
∂tπ0 = ∇2φ0 − φ0
∂tπ1 = ∇2φ1 − φ1 − φ30
∂tπ2 = ∇2φ2 − φ2 − 3φ20φ1
where it easily seen that the free theory, �φ0 + φ0 = 0, is the leading order solution. Our aim is to derive a dual
perturbation series to this one meaning by this that we want a series with a development parameter going as 1
In order to reach our aim, following the principle of duality in perturbation theory [11] we put
λt (4)
π2 + . . .
φ = φ0 +
φ2 + . . . .
The following non trivial set of equations is obtained
∂τφ0 = π0 (5)
∂τφ1 = π1
∂τφ2 = π2
∂τπ0 = −φ30
∂τπ1 = ∇2φ0 − φ0 − 3φ20φ1
∂τπ2 = ∇2φ1 − φ1 − 3φ0φ21 − 3φ20φ2
whose solution proves the existence of a dual perturbation series for the classical λφ4 theory. We easily realize that
this set of equations would have been obtained if one takes as a small term ∇2φ − φ giving rise in this case to a
gradient expansion, that is a series having derivatives in space as small terms. So, strong coupling expansion and
weak coupling expansion are related by the duality principle in perturbation theory [11] producing in the former case
a gradient expansion. This result can be easily generalized to any kind of PDE [3, 4].
The point to be noted is that to have an analytical result for a strong coupling expansion we have to solve a
nonlinear differential equation that in this case is given by
∂2τφ0 + φ
0 = 0. (6)
Things can be more involved when a source term is present as is generally the case in quantum field theory and a
meaning should be attached to the leading order equation
∂2τφ0 + φ
0 = j. (7)
being j a source term. The aim of this paper is to show that indeed an approach through Green functions is applicable
in these cases, that is, as already shown by numerical methods in [1, 2], a first approximated solution is given by
dτ ′G(τ − τ ′)j(τ ′) (8)
being G(τ) the Green function solving the equation
∂2τG(τ) + G(τ)
3 = δ(τ). (9)
We will give in this paper a general approach to compute higher order corrections to this result.
We will note that the method can be applicable when a solution is known to an equation like
τG(τ) + F (G(τ)) = aδ(τ) (10)
being a a constant and F (G(τ)) a generic term. Otherwise we are not able to get analytical results and we have to
resort to numerical methods. Anyhow, the situation is favorable for the most common models.
III. GRADIENT EXPANSION AND QUANTUM FIELD THEORY
Quantum field theory of the model we are considering is given by the generating functional
Z[j] =
[dφ]e{i
(∂tφ)
4+jφ]}e{−i
(∇φ)2+ 1
2]} (11)
that we have written separating the spatial part from the rest. We did this in order to derive the strong coupling
expansion to this case as already done in sec.II for the classical model. By doing the expansion, considering the
gradient term 1
(∇φ)2 + 1
φ2 as small, the leading order term to be computed is
Z0[j] =
[dφ]e{i
(∂tφ)
4+jφ]} (12)
and in the end we are left with the equation to solve
∂2t φ + λφ
3 = j (13)
that is the leading order of our gradient expansion as already seen in sec.II.
The applicability of the Green function method implies that also in a strong coupling regime one can obtain
information on the spectrum of the theory in this limit. We can exploit this point easily for a our case. Firstly, we
use the mass µ0 of the theory to make all adimensional putting x → µ0x, φ2 → µ2−D0 φ2 and introducing the coupling
constant g = λ
. Then, let us consider the equation
∂2t G + gG
3 = δ(t) (14)
that has solution [1]
G(t) = θ(t)
being θ(t) the Heaviside function and sn a Jacobi elliptical function. Being the equation second order we have that
also the time reversed solution holds. It is known [12] that the following series holds for this Jacobi function
sn(u, i) =
(−1)ne−(n+ 12 )π
1 + e−(2n+1)π
(2n + 1)
2K(i)
being K(i) =
1+sin2 θ
≈ 1.3111028777 a constant. Then the mass spectrum of the theory in the limit of a very
large g is given by En = (2n + 1)
2K(i)
4 µ0 that we can recognize as the one of a harmonic oscillator.
So, the main physical motivation to study our approach through Green functions for nonlinear equations is to have
a deeper understanding in quantum field theories but the method is rather general and could find applications in a
lot of other fields.
IV. GREEN FUNCTION METHOD FOR NONLINEAR DIFFERENTIAL EQUATIONS
In order to make our approach as clearer as possible, we consider the trivial problem of a Riccati equation
ẏ + y2 = 1 (17)
with the initial condition y(0) = 0. The solution is given by y(t) = tanh(t). A Green function is easy to compute for
this equation being given by
G(t) = θ(t)
1 + t
. (18)
So, let us consider the following small time expansion as a solution of the above Riccati equation
y(t) ≈
1 + t − t′
1 + t − t′
(t − t′) (19)
1 + t − t′
(t − t′)2 + c
1 + t − t′
(t − t′)3 + . . .
being a, b and c constants to be computed. In order to compute these constants we consider the equation we started
with and compute all the derivatives till the order we are interested in. Then, we compare these derivatives with
the one obtained through equation (19) fixing in this way the values of the constants to make them equal. So, from
eq.(19) one gets
y(t) ≈ ln(t + 1) + a[− ln(t + 1) + t] + b[ln(t + 1) + t
− t] + c[− ln(t + 1) + t
+ t − t
] + . . . (20)
and from this we can compute y(0), ẏ(0), ÿ(0) and so on. From the Riccati equation we have y(0) = 0, ẏ(0) = 1,
ÿ(0) = 0 and so on giving finally a = 1, b = −1 and c = −1 for our case yielding y(t) = t − t
that are the first
two terms of the Taylor series of the tanh(t), the exact solution of the equation. From this exercise we learn that the
series (19) is a small time series solution of the original equation and that the convergence may be really slow. It is a
rather interesting aspect of this approach that the Green function method has such a way to be applied to nonlinear
equations. The case we considered here is a rather trivial one but things are made more interesting for the case of a
λφ4 when we go to a numerical comparison.
We want to apply the above approach to the case of equation (13). So, let us seek a solution in the form (properly
normalized by µ0)
φ(t) ≈
(t − t′), i
j(t′)dt′ (21)
(t − t′), i
(t − t′)4j(t′)dt′
(t − t′), i
(t − t′)6j(t′)dt′ + . . .
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Numerical
2 terms
1 term
FIG. 1: Comparison for a driving source j(t) = sin(2πt).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Numerical
2 terms
3 terms
1 term
FIG. 2: Comparison for a driving source j(t) = exp(−t).
and after computing derivatives of this equation and eq.(13) we get easily a = g
and b = − g
[j(0)]2. This gives
the result we aimed for. We have got the proper expansion by Green function method of a solution to a nonlinear
differential equation. What we want to see is how good is this approximation when compared to numerical results.
This will be shown in the following section.
V. NUMERICAL RESULTS
In order to verify the quality of our approximation we solve the equation (13) for two different driving sources and
take the coupling constant g = 1. Firstly we considered j(t) = sin(2πt) and the results are given in fig.1. In this case
we can only have a first order correction as j(0) = 0. The agreement is very satisfactory till the end of the integration
interval.
The second case we considered j(t) = e−t permits to introduce another correction term but we notice no significant
improvement due to the slow convergence of the approximation as can be seen from fig.2.
The quality of the approximation depends on j(t) that can make very demanding the need for higher order correc-
tions.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Numerical
Functional Iteration
Green Function
FIG. 3: Comparison between our approach and functional iteration method.
VI. COMPARISON WITH OTHER METHODS
There exist different techniques to manage non-linear differential equations. In order to have a proper comparison
we have to limit our analysis to small time methods. In this sense, the most similar approach to ours is the functional
iteration method [13]. This method proves to have a rapid convergence to a good approximant of the true solution
when the equation is not stiff. This is exactly our case. So, let us consider the equation
φ̈(t) + φ3(t) = sin(2πt) (22)
where dot means a time derivative. Functional iteration method implies that we solve the above equation iteratively.
We assume φ(0) = 0 and φ̇(0) = 0. We take as zero order iterate φ0(t) = φ(0) = 0 and then for the successive iterates
we have
φ̈ν+1(t) = −φ3ν(t) + sin(2πt) (23)
starting with ν = 0. Already at the second iterate we get a very good approximation to the true solution in the range
we are considering. Then, we can compare this approximation with our method considering two terms. The results
are presented in fig.3.
This result shows that functional iteration method has a faster convergence and at least another term should be
computed with our approach to reach an identical precision in the required range. This result should also be expected
on the ground of efficiency of iterative methods with respect to series solutions. So, in order to decide the proper
method to use one should properly analyze the problem at hand.
VII. CONCLUSIONS
We have shown an approximation method to solve nonlinear differential equations using Green function methods.
This method proves to be a small time expansion and the convergence in some cases may turn out really slow. The
main point to be emphasized is the unexpected utility of this approach generally assumed to hold only for linear
differential equations. This implies that a gradient expansion for nonlinear PDE can also be applied successfully
and a quantum field theory obtained. In this latter case one should consider that a gradient expansion is a strong
coupling expansion and then, information in this regime of the corresponding quantum field theory is given. This
yields another method to approach these problems generally very difficulty to manage with analytical methods.
[1] M. Frasca, Phys. Rev. D 73, 027701 (2006); Erratum-ibid., 049902 (2006).
[2] M. Frasca, Mod. Phys. Lett. A 22, 1293 (2007).
[3] M. Frasca, Int. J. Mod. Phys. D 15, 1373 (2006).
[4] M. Frasca, Int. J. Mod. Phys. A 22, 1441 (2007).
[5] I. M. Kalathnikov, and E. M. Lifshitz, Phys. Rev. Lett. 24, 76 (1970).
[6] V. A. Belinski, I. M. Kalathnikov, and E. M. Lifshitz, Adv. Phys. 19, 525 (1970).
[7] V. A. Belinski, I. M. Kalathnikov, and E. M. Lifshitz, Adv. Phys. 31, 639 (1982).
[8] B. Simon, Functional Integration and Quantum Physics, (AMS, Providence, 2005).
[9] M. Frasca, Proc. R. Soc. A 463, 2195 (2007).
[10] P. Ring and P. Schuck, The Nuclear Many-Body Problem, (Springer, Berlin, 1980).
[11] M. Frasca, Phys. Rev. A 58, 3439 (1998).
[12] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, 2000).
[13] D. Zwillinger, Handbook of Differential Equations, (Academic Press, San Diego, 1989).
Introduction
Dual expansion for nonlinear PDEs
Gradient expansion and quantum field theory
Green function method for nonlinear differential equations
Numerical results
Comparison with other methods
Conclusions
References
|
0704.1569 | One-way permutations, computational asymmetry and distortion | One-way permutations, computational asymmetry and distortion
Jean-Camille Birget ∗
October 30, 2018
Abstract
Computational asymmetry, i.e., the discrepancy between the complexity of transformations
and the complexity of their inverses, is at the core of one-way transformations. We introduce a
computational asymmetry function that measures the amount of one-wayness of permutations. We
also introduce the word-length asymmetry function for groups, which is an algebraic analogue of
computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a
fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean
circuits have a representation in terms of elements of a Thompson group, in such a way that circuit
size is polynomially equivalent to word-length. We show that circuits built with gates that are
not constrained to have fixed-length inputs and outputs, are at most quadratically more compact
than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show
that the computational asymmetry function is closely related to certain distortion functions: The
computational asymmetry function is polynomially equivalent to the distortion of the path length
in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of
certain Thompson monoids. We also show that the results of Razborov and others on monotone
circuit complexity lead to exponential lower bounds on certain distortions.
1 Introduction
The existence of one-way functions, i.e., functions that are “easy to evaluate” but “hard to invert”,
is a major open problem. Much of cryptography depends on one-way functions; moreover, indirectly,
their existence is connected to the question whether P is different from NP. In this paper we give some
connections between these questions and some group-theoretic concepts:
(1) We continue the work of [7], [8], and [9], on the relation between combinational circuits, on the one
hand, and Thompson groups and monoids on the other hand. We give a representation of any circuit
by a word over the Thompson group, such that circuit size is polynomially equivalent to word-length.
(2) We establish connections between the existence of one-way permutations and the distortion func-
tion in a certain Thompson group. Distortion is an important concept in metric spaces (e.g., Bourgain
[10]) and in combinatorial group theory (e.g., Gromov [17], Farb [14]).
Overview:
Subsections 1.1 - 1.6 of the present Section define and motivate the concepts used: One-way functions
and one-way permutations; computational asymmetry; word-length asymmetry; reversible computing;
distortion; Thompson groups and monoids. In Section 2 we show that circuits can be represented by
elements of Thompson monoids: A boolean circuit is equivalent to a word over a fixed generating set
∗Supported by NSF grant CCR-0310793. Some of the results of this paper were presented at the AMS Section
Meeting, Oct. 21-23, 2005, Lincoln, Nebraska (http://www.ams.org/amsmtgs/2117 program.html), and at the conference
“Various Faces of Cryptography”, 10 Nov. 2006 at City College of CUNY, New York.
http://arxiv.org/abs/0704.1569v1
http://www.ams.org/amsmtgs/2117_program.html
of a Thompson monoid, with circuit size being equal (or linearly equivalent) to word-length over the
generating set. The Thompson monoids that appear here are monoid generalizations of the Thompson
group G2,1, obtained when bijections are generalized to partial functions [9]. Section 3 shows that
computational asymmetry and word-length asymmetry (for the Thompson groups and monoids) are
linearly related. In Section 4 we give a representation of arbitrary (not necessarily bijective) circuits
by elements of the Thompson group G2,1; circuit size is polynomially equivalent to word-length over
a certain generating set in the Thompson group. In Section 5 we show that the computational
asymmetry function of permutations is polynomially related to a certain distortion in a Thompson
group. Section 6 contains miscellaneous results, in particular that the work of Razborov and others
on monotone circuit complexity leads to exponential lower bounds on certain distortion functions.
1.1 One-way functions and one-way permutations
Intuitively, a one-way function is a function f (mapping words to words, over a finite alphabet), such
that f is “easy to evaluate” (i.e., given x0 in the domain, it is “easy” to compute f(x0)), but “hard
to invert” (i.e., given y0 in the range, it is “hard” to find any x0 such that f(x0) = y0). The concept
was introduced by Diffie and Hellman [13].
There are many ways of defining the words “easy” and “hard”, and accordingly there exist many
different rigorous notions of a one-way function, all corresponding to a similar intuition. It remains
an open problem whether one-way functions exist, for any “reasonable” definition. Moreover, for
certain definitional choices, this problem is a generalization of the famous question whether P 6= NP
[16, 34, 11].
We will base our one-way functions on combinational circuits and their size. The size of a circuit
will also be called its complexity. Below, {0, 1}n (for any integer n ≥ 0) denotes the set of all bitstrings
of length n. A combinational circuit with input-output function f : {0, 1}m → {0, 1}n is an acyclic
boolean circuit with m input wires (or “input ports”) and n output wires (or “output ports”). The
circuit is made from gates of type not, and, or, fork, as well as wire-crossings or wire-swappings. These
gates are very traditional and are defined as follows.
and: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 = x2 = 1, and y = 0 otherwise.
or: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 0 if x1 = x2 = 0, and y = 1 otherwise.
not: x ∈ {0, 1} 7−→ y ∈ {0, 1}, where y = 0 if x = 1, y = 1 otherwise.
fork: x ∈ {0, 1} 7−→ (x, x) ∈ {0, 1}2.
Another gate that is often used is the exclusive-or gate,
xor: (x1, x2) ∈ {0, 1}2 7−→ y ∈ {0, 1}, where y = 1 if x1 6= x2, and y = 0 otherwise.
The wire-swapping of the ith and jth wire (i < j) is described by the bit transposition (or bit position
transposition)
τi,j : uxivxjw ∈ {0, 1}ℓ 7−→ uxjvxiw ∈ {0, 1}ℓ, where |u| = i− 1, |v| = j − i− 1, |w| = ℓ− j − 1.
The fork and wire-swapping operations, although heavily used, are usually not explicitly called “gates”;
but because of their important role we will need to consider them explicitly. Other notations for the
gates: and(x1, x2) = x1 ∧ x2, or(x1, x2) = x1 ∨ x2, not(x) = x, xor(x1, x2) = x1 ⊕ x2.
A combinational circuit for a function f : {0, 1}m → {0, 1}n is defined by an acyclic directed graph
drawn in the plane (with crossing of edges allowed). In the circuit drawing, the m input ports are
vertices lined up in a vertical column on the left end of the circuit, and the n output ports are vertices
lined up in a vertical column on the right end of the circuit. The input and output ports and the gates
of the circuit (including the fork gates, but not the wire transpositions) form the vertices of the circuit
graph. We often view the circuit as cut into vertical slices. A slice can be any collection of gates and
wires in the circuit such that no gate in a slice is an ancestor of another gate in the same slice, and
no wire in a slice is an ancestor of another wire in the same slice (unless these two wires are an input
wire and an output wire of a same gate). Two slices do not overlap, and every wire and every gate
belongs to some slice. For more details on combinational circuits, see [32, 43, 11].
The size of a combinational circuit is defined to be the number of gates in the circuit, including
forks and wire-swappings, as well as the input ports and the output ports. For a function f : {0, 1}m →
{0, 1}n, the circuit complexity (denoted C(f)) is the smallest size of any combinational circuit with
input-output function f .
A cause of confusion about gates in a circuit is that gates of a certain type (e.g., and) are tradi-
tionally considered the same, no matter where they occur in the circuit. However, gates applied to
different wires in a circuit are different functions; e.g., for the and gate, (x1, x2, x3) 7→ (x1 ∧ x2, x3) is
a different function than (x1, x2, x3) 7→ (x1, x2 ∧ x3).
1.2 Computational Asymmetry
Computational asymmetry is the core property of one-way functions. Below we will define computa-
tional asymmetry in a quantitative way, and in a later Section we will relate it to the group-theoretic
notion of distortion.
For the existence of one-way functions, it is mainly the relation between the circuit complexity
C(f) of f and the circuit complexity C(f−1) of f−1 that matters, not the complexities of f and of
f−1 themselves. Indeed, a classical padding argument can be used: If we add C(f) “identity wires” to
a circuit for f , then the resulting circuit has linear size as a function of its number of input wires; see
Proposition 1.2 below. (An identity wire is a wire that goes directly from an input port to an output
port, without being connected to any gate.)
In [11] (page 230) Boppana and Lagarias considered logC(f ′)/logC(f) as a measure of one-
wayness; here, f ′ denotes an inverse of f , i.e., any function such that f ◦ f ′ ◦ f = f . Massey and
Hiltgen [25, 19] introduced the phrases complexity asymmetry and computational asymmetry for injec-
tive functions, in reference to the situation where the circuit complexities C(f) and C(f−1) are very
different. The concept of computational asymmetry can be generalized to arbitrary (non-injective)
functions, with the meaning that for every inverse f ′ of f , C(f) and C(f ′) are very different.
In [25] Massey made the following observation. For any large-enough fixed m and for almost all
permutations f of {0, 1}m, the circuit complexities C(f) and C(f−1) are very similar:
C(f) ≤ C(f−1) ≤ 10 C(f)
Massey’s proof is adapted from the Shannon lower bound [35] and the Lupanov upper bound [23] (see
also [19], [32]), from which it follows that almost all functions and almost all permutations (and their
inverses) have circuit complexity close to the Shannon bounds. Massey’s observation can be extended
to the set of all functions f : {0, 1}m → {0, 1}n, i.e., for almost all f and for every inverse f ′ of f , the
complexities C(f) and C(f ′) are within constant factors of each other.
Hence, computationally asymmetric permutations are rare among the boolean permutations overall
(and similarly for functions). This is an interesting fact about computational asymmetry, but by
itself it does not imply anything about the existence or non-existence of one-way functions, not even
heuristically. Indeed, Massey proved his linear relation C(f) = Θ(C(f ′)) in the situation where
C(f) = Θ(2m), and then uses the fact that the condition C(f) = Θ(2m) holds for almost all boolean
permutations and for almost all boolean functions. But there also exist functions with C(f) = O(mk),
with k a small constant. In particular, one-way functions (if they exist) have small circuits; by
definition, one-way functions violate the condition C(f) = Θ(2m).
A well-known candidate for a one-way permutation is the following. For a large prime number p
and a primitive root r modulo p, consider the map x ∈ {0, 1, . . . , p−2} 7−→ rx−1 ∈ {0, 1, . . . , p−2}.
This is a permutation whose inverse, known as the discrete logarithm, is believed to be difficult to
compute.
Measuring computational asymmetry:
Let S{0,1}m denote the set of all permutations of {0, 1}m, i.e., S{0,1}m is the symmetric group. We
will measure the computational asymmetry of all permutations of {0, 1}m (for all m > 0) by defining
a computational asymmetry function, as follows. A function a : N → N is an upper bound on the
computational asymmetry function iff for all all m > 0 and all permutations f of {0, 1}m we have:
C(f−1) ≤ a
. The computational asymmetry function α of the boolean permutations is the
least such function a(.). Hence:
Definition 1.1 The computational asymmetry function α of the boolean permutations is defined as
follows for all s ∈ N : α(s) = max
C(f−1) : C(f) ≤ s, f ∈ S{0,1}m , m > 0
Note that in this definition we look at all combinational circuits, for all permutations in
m>0 S{0,1}m ;
we don’t need to work with non-uniform or uniform families of circuits.
Computational asymmetry is closely related to one-wayness, as the next proposition shows.
Proposition 1.2
(1) For infinitely many n we have: There exists a permutation fn of {0, 1}n such that fn is computed
by a circuit of size ≤ 3n, but f−1n has no circuit of size < α(n).
(2) Suppose that α is exponential, i.e., there is k > 1 such that for all n, α(n) ≥ kn. Then k ≤ 2,
and there is a constant c > 1 such that we have: For every integer n ≥ 1 there exists a permutation
Fn of {0, 1}n which is computed by a circuit of size ≤ c n, but F−1n has no circuit of size < kn.
Proof. (1) By the definition of α, for every m > 0 there exists a permutation F of {0, 1}m such that
F is computed by a circuit of some size CF , but F
−1 has no circuit of size < α(CF ). Let n = CF ,
and let us consider the function fn : {0, 1}CF → {0, 1}CF defined by fn : (x,w) 7−→ (F (x), w), for all
x ∈ {0, 1}m and w ∈ {0, 1}CF−m.
Then fn(x,w) is computed by a circuit of size CF + 2 (CF − m); the term “2 (CF −m)” comes
from counting the input-output wires of w. Hence fn has a circuit of size ≤ 3n. On the other hand,
(y,w) 7−→ f−1n (y,w) = (F−1(y), w) is not computed by any circuit of size < α(CF ), so f−1n has no
circuit of size < α(n).
(2) For every n ≥ 1 there exists a permutation F of {0, 1}n such that F is computed by a circuit of
some size CF , and F
−1 has a circuit of size CF−1 = α(CF ) ≥ kCF ; moreover, F−1 has no circuit of
size < α(CF ). Thus, k
CF ≤ CF−1 ≤ 2n (1 + co lognn ), for some constant co > 1; the latter inequality
comes from the Lupanov upper bound [23] (or see Theorem 2.13.2 in [32]). Hence, k ≤ 2 and n ≤
CF ≤ 1log2 k n + c1
, for some constant c1 > 0. Hence, for all n ≥ 1 there exists a permutation F
of {0, 1}n with circuit size CF ∈ [n, 1log2 k · n+ c1 ·
], such that CF−1 = α(CF ) ≥ kCF ≥ kn. ✷
We will show later that the computational asymmetry function is closely related to the distortion
of certain groups within certain monoids.
Remarks:
Although in this paper we only use the computational asymmetry function of the boolean permuta-
tions, the concept can be generalized. Let Inj({0, 1}m, {0, 1}n) denote the set of all injective functions
{0, 1}m → {0, 1}n. The computational asymmetry function αinj of the injective boolean functions is
defined by
αinj(s) = max
C(f−1) : C(f) ≤ s, f ∈ Inj({0, 1}m, {0, 1}n), m > 0, n > 0
More generally, let ({0, 1}n){0,1}m denote the set of all functions {0, 1}m → {0, 1}n. The computa-
tional asymmetry of all finite boolean functions is defined by
αfunc(s) = max
C(f ′) : C(f) ≤ s, ff ′f = f, f, f ′ ∈ ({0, 1}n){0,1}m , n > 0,m > 0
When we compare functions we will be mostly interested in their asymptotic growth pattern.
Hence we will often use the big-O notation, and the following definitions.
By definition, two functions f1 : N → N and f2 : N → N are linearly equivalent iff there are
constants c0, c1, c2 > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1n) and f2(n) ≤ c2 f1(c2n).
Notation: f1 ≃lin f2.
Two functions f1 and f2 (from N to N) are called polynomially equivalent iff there are constants
c0, c1, c2, d, e > 0 such that for all n ≥ c0 : f1(n) ≤ c1 f2(c1nd)d and f2(n) ≤ c2 f1(c2ne)e. Notation:
f1 ≃poly f2.
1.3 Wordlength asymmetry
We introduce an algebraic notion that looks very similar to computational asymmetry:
Definition 1.3 Let G be a group, let M be a monoid with generating set Γ (finite or infinite), and
suppose G ⊆M . The word-length asymmetry function of G within M (over Γ) is
λ(n) = max{ |g−1|Γ : |g|Γ ≤ n, g ∈ G}.
The word-length asymmetry function λ depends on G, M , Γ, and the embedding of G in M .
Consider the right Cayley graph of the monoid M with generating set Γ; its vertex set is M and
the edges have the form x
γ−→ γx (for x ∈ M , γ ∈ Γ). For x, y ∈ M , the directed distance d(x, y)
in the Cayley graph is the shortest length over all paths from x to y in the Cayley graph; if no path
from x to y exists, the directed distance is infinite. By “path” we always mean directed path.
Lemma 1.4 Under the above conditions on G, M , Γ, we have for every g ∈ G : d(1, g−1) = d(g,1)
and d(1, g) = d(g−1,1).
Proof. Let η : Γ∗ →M be the map that evaluates generator sequences in M . If v ∈ Γ∗ is the label of
a shortest path from 1 to g−1 in the Cayley graph then g ·η(v) = 1 inM , hence η(v) = g−1. Therefore,
the path starting at g and labeled by v ends at 1; hence d(g,1) ≤ |v| = d(1, g−1). In a similar way
one proves that d(1, g−1) ≤ d(g,1). The equality d(1, g) = d(g−1,1) is also proved in a similar way.
Since |g|Γ is the distance d(1, g) in the graph of M , and since |g−1|Γ = d(1, g−1) = d(g,1), the
word-length asymmetry also measures the asymmetry of the directed distance, to or from the identity
element 1 in the Cayley graph of M , restricted to vertices in the subgroup G.
For distances to or from the identity element of M it does not matter whether we consider the left
Caley graph or the right Caley graph.
1.4 Computational asymmetry and reversible computing
Reversible computing deals with the following questions: If a function f is injective (or bijective) and
computable, can f be computed in such a way that each elementary computation step is injective
(respectively bijective)? And if such injective (or bijective) computations are possible, what is their
complexity, compared to the usual (non-injective) complexity?
One of the main results is the following (Bennett’s theorem [4, 5], and earlier work of Lecerf
[22]): Let f be an injective function, and assume f and f−1 are computable by deterministic Turing
machines with time complexity Tf (.), respectively Tf−1(.). Then f (and also f
−1) is computable
by a reversible Turing machine (in which every transition is deterministic and injective) with time
complexity O(Tf + Tf−1). Note that only injectiveness (not bijectiveness) is used here.
Bennett’s theorem has the following important consequence, which relates reversible computing
to one-way functions: Injective one-way functions exist iff there exist injective functions that have
efficient traditional algorithms but that do not have efficient reversible algorithms.
Toffoli representation
Remarkably, it is possible to “simulate” any function f : {0, 1}m → {0, 1}n (injective or not, one-
way or not) by a bijective circuit; a circuit is called bijective iff the circuit is made from bijective
gates. Here, bijective circuits will be built from the wire swapping operations and the following
bijective gates: not (negation), c-not (the Controlled Not, also called “Feynman gate”) defined by
(x1, x2) ∈ {0, 1}2 7−→ (x1, x1 ⊕ x2) ∈ {0, 1}2, and cc-not (the Doubly Controlled Not, also called
“Toffoli gate”) defined by (x1, x2, x3) ∈ {0, 1}3 7−→ (x1, x2, (x1 ∧ x2)⊕ x3) ∈ {0, 1}3.
Theorem 1.5 (Toffoli [40]). For every boolean function f : {0, 1}m → {0, 1}n there exists a bijective
boolean circuit βf (over the bijective gates not, c-not, cc-not, and wire transpositions), with input-
output function βf : x 0
n ∈ {0, 1}m+n 7−→ f(x) x ∈ {0, 1}n+m.
In other words, f(x) consists of the projection onto the first n bits of βf (x 0
n); equivalently, f(.) =
projn ◦ βf ◦ concat0n(.), where projn projects a string of length n+m to the first n bits, and concat0n
concatenates 0n to the right of a string. See Theorems 4.1, 5.3 and 5.4 of [40], and see Fig. 1 below.
✲ f(x)
Fig. 1: Toffoli representation of the function f .
The Toffoli representation contains two non-bijective actions: The projection at the output, and
the forced setting of the value of some of the input wires.
Toffoli’s proofs and constructions are based on truth tables, and he does not prove anything about
the circuit size of βf (counting the bijective gates), compared to the circuit size of f . The following
gives a polynomial bound on the size of the bijective circuit, at the expense of a large number of input-
and output-wires.
Theorem 1.6 (E. Fredkin, T. Toffoli [15]). For every boolean function f : {0, 1}m → {0, 1}n with
circuit size C(f) there exists a bijective boolean circuit Bf (over a bounded collection of bijective gates,
e.g., not, c-not, cc-not, and wire transpositions), with input-output function
Bf : x 0
n+C(f) ∈ {0, 1}m+n+C(f) 7−→ f(x) z(x) ∈ {0, 1}m+n+C(f)
for some z(x) ∈ {0, 1}m+C(f).
If g : {0, 1}m → {0, 1}m is a permutation then there exists a bijective boolean circuit Ug (over
bijective gates), with input-output function
Ug : x 1
m 0m+C ∈ {0, 1}3m+C 7−→ g(x) g(x) x 0C ∈ {0, 1}3m+C
where C = max{C(g), C(g−1)}, and g(x) is the bitwise complement of g(x).
Later we will introduce another reversible representation of boolean functions by bijective gates;
we will need only one 0-wire, but the gates will be taken from the Thompson group G2,1, i.e., we will
also use non-length-preserving transformations of bitstrings (Theorems 4.1 and 4.2 below).
1.5 Distortion
We will prove later (Theorem 5.10) that computational asymmetry has a lot to do with distortion,
a concept introduced into group theory by Gromov [17] and Farb [14]. Distortion is already known
to have connections with isoperimetric functions (see [28], [29], [24]). A somewhat different problem
about distortion (for finite metric spaces) was tackled by Bourgain [10].
We will use a slightly more general notion of distortion, based on (possibly directed) countably
infinite rooted graphs, and their (directed) path metric.
A weighted directed graph is a structure (V,E, ω) where V is a set (called the vertex set), E ⊆ V ×V
(called the edge set), and ω : E 7−→ R>0 is a function (called the weight function); note that every edge
has a strictly positive weight. It is sometimes convenient to define ω(u, v) = ∞ when (u, v) ∈ V ×V −E.
A path in (V,E) is a sequence of edges (ui, vi) (1 ≤ i ≤ n) such that ui+1 = vi for all i < n, and
such that all elements in {ui : 1 ≤ i ≤ n} ∪ {vn} are distinct; u1 is called the start vertex of this
path, and vn is called the end vertex of this path; the sum of weights
i=1 ω(ui, vi) over the edges
in the path is called the length of the path. Here we do not consider any paths with infinitely many
edges; but we allow V and E to be countably infinite. A vertex w2 is said to be reachable from a
vertex w1 in (V,E) iff there exists a path with start vertex w1 and end vertex w2. If w2 is reachable
from w1 then the minimum length over all paths from w1 to w2 is called the directed distance from
w1 to w2, denoted d(w1, w2); since we only consider finite paths here, this minimum exists. If w2 is
not reachable from w1 then we define d(w1, w2) to be ∞. Clearly we have w1 = w2 iff d(w1, w2) = 0,
and for all u, v, w ∈ V , d(u,w) ≤ d(u, v) + d(v,w). In a directed graph, the function d(., .) need not
be symmetric. The function d : V ×V → R≥0 ∪{∞} is called the directed path metric of (V,E, ω). A
rooted directed weighted graph is a structure (V,E, ω, r) where (V,E, ω) is a directed weighted graph,
r ∈ V , and all vertices in V are reachable from r.
A set M with a function d : M × M → R≥0 ∪ {∞}, satisfying the two axioms w1 = w2 iff
d(w1, w2) = 0, and d(u,w) ≤ d(u, v) + d(v,w), will be called directed metric space (a.k.a. quasi-metric
space).
Any subset G embedded in a directed metric space M becomes a directed metric space by using
the directed distance of M . We call this the directed distance on G inherited from M .
If G ⊆ V for a rooted directed weighted graph (V,E, ω, r), we consider the function ℓ : g ∈ G 7−→
d(r, g) ∈ R≥0, which we call the directed length function on G inherited from (V,E, ω, r). (The value
∞ will not appear here since all of G is reachable from r.)
We now define distortion in a very general way. Intuitively, distortion in a set is a quantitative
comparison between two (directed) length functions that are defined on the same set.
Definition 1.7 Let G be a set, and let ℓ1 and ℓ2 be two functions G → R≥0. The distortion of ℓ1
with respect to ℓ2 is the function δℓ1,ℓ2 : R≥0 → R≥0 defined by
δℓ1,ℓ2(n) = max{ℓ1(g) : g ∈ G, ℓ2(g) ≤ n}.
We will also use the notation δ[ℓ1, ℓ2](.) for δℓ1,ℓ2(.). When we consider a distortion δℓ1,ℓ2(.) we often
assume that ℓ2 ≤ ℓ1 or ℓ2 ≤ O(ℓ1); this insures that the distortion is at least linear, i.e., δℓ1,ℓ2(n) ≥ c n,
for some constant c > 0. We will only deal with functions obtained from the lengths of finite paths
in countable directed graphs, so in that case the functions ℓi are discrete, and the distortion function
exists. The next Lemma generalizes the distortion result of Prop. 4.2 of [14].
Lemma 1.8 Let G be a set and consider three functions ℓ3, ℓ2, ℓ1 : G→ R≥0 such that ℓ1(.) ≥ ℓ2(.) ≥
ℓ3(.). Then the corresponding distortions satisfy: δℓ1,ℓ3(.) ≤ δℓ1,ℓ2 ◦ δℓ2,ℓ3(.).
Proof. The inequalities ℓ1(.) ≥ ℓ2(.) ≥ ℓ3(.) guarantee that the three distortions δℓ1,ℓ3 , δℓ1,ℓ2 , and
δℓ2,ℓ3 are at least as large as the identity map. By definition,
δℓ1,ℓ2
δℓ2,ℓ3(n)
= max{ℓ1(x) : x ∈ G, ℓ2(x) ≤ δℓ2,ℓ3(n)}
= max
ℓ1(x) : x ∈ G, ℓ2(x) ≤ max{ℓ2(z) : z ∈ G, ℓ3(z) ≤ n}
= max
ℓ1(x) : x ∈ G, (∃z ∈ G)
ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n
≥ max{ℓ1(x) : x ∈ G, ℓ3(x) ≤ n} = δℓ1,ℓ3(n).
The last inequality follows from the fact that if ℓ3(x) ≤ n then for some z (e.g., for z = x):
ℓ2(x) ≤ ℓ2(z) and ℓ3(z) ≤ n. ✷
Examples of distortion:
Distortion and asymmetry are unifying concepts that apply to many fields.
1. Gromov distortion: Let G be a subgroup of a group H, with generating sets ΓG, respectively
ΓH , such that ΓG ⊆ ΓH , and such that ΓG = Γ−1G and ΓH = Γ
H . This determines a Cayley graph
for G and a Cayley graph for H. Now we have two distance functions on G, one obtained from the
Cayley graph of G itself (based on ΓG), and the other inherited from the embedding of G in H. See
[17], [10], and [14].
The Gromov distortion function is a natural measure of the difficulty of the generalized word
problem. A very important case is when both ΓG and ΓH are finite. Here are some results for that
case:
Theorem of Ol′shanskii and Sapir [29] (making precise and proving the outline on pp. 66-67 in [17]):
All Dehn functions of finitely presented groups (and “approximately all” time complexity functions of
nondeterministic Turing machines) are Gromov distortion functions of finitely generated subgroups
of FG2×FG2; here, FG2 denotes the 2-generated free group. Moreover, in [6] it was proved that
FG2×FG2 is embeddable with linear distortion in the Thompson group G2,1. So the theorem of
Ol′shanskii and Sapir also holds for the finitely generated subgroups of G2,1.
Actually, Gromov [17] and Bourgain [10] defined the distortion to be 1
·max{|g|ΓG : |g|ΓH ≤ n, g ∈
G}, i.e., they use an extra factor 1
. However, the connections between distortion, the generalized word
problem, and complexity (as we just saw, and will further see in the present paper) are more direct
without the factor 1
2. Bourgain’s distortion theorem: Given a finite metric space G with n elements, the aim is to
find embeddings of G into a finite-dimensional euclidean space. The two distances of G are its given
distance and the inherited euclidean distance. In this problem the goal is to have small distortion,
as a function of the cardinality of G, while also keeping the dimension of the euclidean space small.
Bourgain [10] found a bound O(n log n) for the distortion (or “O(log n)” in Bourgain’s and Gromov’s
terminology). This is an important result. See also [21], [2], [3].
3. Generator distortion: A variant of Gromov’s distortion is obtained when G = H, but ΓG $ ΓH .
So here we look at the distorting effect of a change of generators in a given group. When ΓG and
ΓH are both finite the generator distortion is linear; however, when ΓG is finite and ΓH is infinite
the distortion becomes interesting. E.g., for the Thompson group G2,1 let us take ΓG to be any finite
generating set, and for ΓH let us take ΓG ∪ {τi,j : 1 ≤ i < j}; here τi,j is the position transposition
defined earlier. Then the generator distortion is exponential (see [7]). Also, the word problem of G2,1
over any finite generating set ΓG is in P, but the word problem of G2,1 over ΓG ∪ {τi,j : 1 ≤ i < j} is
coNP-complete (see [7] and [8]).
4. Monoids and directed distance: Gromov’s distortion and the generator distortion can be
generalized to monoids. We repeat what we said about Gromov distortion, but G and H are now
monoids, and ΓG, respectively ΓH , are monoid generating sets which are used to define monoid Cayley
graphs. We will use the left Cayley graphs. We assume ΓG ⊆ ΓH . In each Cayley graph there is
a directed distance, defined by the lengths of directed paths. The monoid G now has two directed
distance functions, the distance in the Cayley graph ofG itself, and the directed distance thatG inherits
from its embedding into the Cayley graph of H. We denote the word-length of g ∈ G over ΓG by |g|G;
this is the minimum length of all words over ΓG that represent g; it is also the length of a shortest
path from the identity to g in the Cayley graph of G. Similarly, we denote the word-length of h ∈ H
over ΓH by |h|H . The definition of the distortion becomes: δ(n) = max{|g|G : g ∈ G, |g|H ≤ n}.
5. Schreier graphs: Let G, H, and F be groups, where F is a subgroup of H. Let ΓH be a
generating set of ΓH , and assume ΓH = Γ
H . We can define the Schreier left coset graph of H/F
over the generating set ΓH , and the distance function dH/F (., .) in this coset graph. By definition,
this Schreier graph has vertex set H/F (i.e., the left cosets, of the form h · F with h ∈ H), and it has
directed edges of the form h · F γ−→ γg · F , for h ∈ H, γ ∈ ΓH . The graph is symmetric; for every
edge as above there is an opposite edge γh ·F γ
−→ h ·F . Because of symmetry the Schreier graph has
a (symmetric) distance function based on path length, dH/F (., .) : H/F ×H/F → N.
Next, assume that G is embedded into H/F by some injective function G →֒ H/F . Such an
embedding happens, e.g., if G and F are subgroups of H such that G∩F = {1}. Indeed, in that case
each coset in H/F contains at most one element of G (since g1F = g2F implies g
2 g1 ∈ F ∩G = {1}).
The group G now inherits a distance function from the path length in the Schreier graph of H/F .
Comparing this distance with other distances in G leads to distortion functions. E.g., if the group G
is also embedded in a monoid M with monoid generating set ΓM , this leads to the following distortion
function: δG(n) = max{dH/F (F, gF ) : g ∈ G, |g|M ≤ n}.
It will turn out that for appropriate choices of G,F,H, ΓH , and ΓM , this last distortion is polyno-
mially related to the computational asymmetry function α of boolean permutations (Theorem 5.10).
6. Asymmetry functions: We already saw the computational asymmetry function of combinational
circuits, and the word-length asymmetry function of a group embedded in a monoid. More generally,
in any quasi-metric space (S, d), where d(., .) is a directed distance function, an asymmetry function
A : R≥0 → R≥0 can be defined by A(n) = max{d(x2, x1) : x1, x2 ∈ S, d(x1, x2) ≤ n}.
This asymmetry function can also be viewed as the distortion of drev with respect to d in S; here
drev denotes the reverse directed distance, defined by drev(x1, x2) = d(x2, x1).
7. Other distortions:
- Distortion can compare lengths of proofs (or lengths of expressions) in various, more or less pow-
erful proof systems (respectively description languages). Distortion can also compare the duration of
computations or of rewriting processes in various models of computation. Hence, many (perhaps all)
notions of complexity are examples of distortion. Distortion is an algebraic or geometric representation
(or cause) of complexity.
- Instead of length and distance, other measures (e.g., volumes in higher dimension, energy, action,
entropy, etc.) could be used.
1.6 Thompson-Higman groups and monoids
The Thompson groups, introduced by Richard J. Thompson [38, 26, 39], are finitely presented infinite
groups that act as bijections between certain subsets of {0, 1}∗. So, the elements of the Thompson
groups are transformations of bitstrings, and hence they are related to input-output maps of boolean
circuits. In this subsection we define the Thompson group G2,1 (also known as “V ”), as well as its
generalization (by Graham Higman [18]) to the group Gk,1 that partially acts on A
∗, for any finite
alphabet A of size k ≥ 2. We will follow the presentation of [6] (see also [8] and [7]); another reference
is [33], which is also based on string transformations but with a different terminology; the classical
references [38, 26, 39, 18, 12] do not describe the Thompson groups by transformations of finite strings.
Because of our interest in strings and in circuits, we also use generalizations of the Thompson groups
to monoids, as introduced in [9].
Some preliminary definitions, all fairly standard, are needed in order to define the Thompson-
Higman group Gk,1. First, we pick any alphabet A of cardinality |A| = k. By A∗ we denote the set
of all finite words (or “strings”) over A; the empty word ε is also in A∗. We denote the length of
w ∈ A∗ by |w| and we let An denote the set of words of length n. We denote the concatenation of
two words u, v ∈ A∗ by uv or by u · v; the concatenation of two subsets B,C ⊆ A∗ is defined by
BC = {uv : u ∈ B, v ∈ C}. A right ideal of A∗ is a subset R ⊆ A∗ such that RA∗ ⊆ R. A generating
set of a right ideal R is, by definition, a set C such that R is equal to the intersection of all right
ideals that contain C; equivalently, C generates R (as a right ideal) iff R = CA∗. A right ideal R is
called essential iff R has a non-empty intersection with every right ideal of A∗. For u, v ∈ A∗, we call
u a prefix of v iff there exists z ∈ A∗ such that uz = v. A prefix code is a subset C ⊆ A∗ such that
no element of C is a prefix of another element of C. A prefix code C over A is maximal iff C is not
a strict subset of any other prefix code over A. It is easy to prove that a right ideal R has a unique
minimal (under inclusion) generating set CR, and that CR is a prefix code; moreover, CR is a maximal
prefix code iff R is an essential right ideal.
For a partial function f : A∗ → A∗ we denote the domain by Dom(f) and the image (range) by
Im(f). A restriction of f is any partial function f1 : A
∗ → A∗ such that Dom(f1) ⊆ Dom(f), and
such that f1(x) = f(x) for all x ∈ Dom(f1). An extension of f is any partial function of which f
is a restriction. An isomorphism between right ideals R1, R2 of A
∗ is a bijection ϕ : R1 → R2 such
that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. The isomorphism ϕ is uniquely determined
by a bijection between the prefix codes that minimally generate R1, respectively R2. One can prove
[39, 33, 6] that every isomorphism ϕ between essential right ideals has a unique maximal extension
(within the category of isomorphisms between essential right ideals of A∗); we denote this unique
maximal extension by max(ϕ).
Now, finally, we define the Thompson-Higman group Gk,1: It consists of all maximally extended
isomorphisms between finitely generated essential right ideals of A∗. The multiplication consists of
composition followed by maximum extension: ϕ · ψ = max(ϕ ◦ ψ). Note that Gk,1 acts partially and
faithfully on A∗ on the left.
Every element ϕ ∈ Gk,1 can be described by a bijection between two finite maximal prefix codes;
this bijection can be described concretely by a finite function table. When ϕ is described by a maximally
extended isomorphism between essential right ideals, ϕ : R1 → R2, we call the minimum generating
set of R1 the domain code of ϕ, and denote it by domC(ϕ); similarly, the minimum generating set of
R2 is called the image code of ϕ, denoted by imC(ϕ).
Thompson and Higman proved that Gk,1 is finitely presented. Also, when k is even Gk,1 is a simple
group, and when k is odd Gk,1 has a simple normal subgroup of index 2. In [6] it was proved that
the word problem of Gk,1 over any finite generating set is in P (in fact, more strongly, in the parallel
complexity class AC1). In [8, 7] it was proved that the word problem of Gk,1 over Γ∪{τi,j : 1 ≤ i < j} is
coNP-complete, where Γ is any finite generating set of Gk,1, and where τi,j is the position transposition
introduced in Subsection 1.1.
Because of connections with circuits we consider the subgroup lpGk,1 of all length-preserving
elements of Gk,1; more precisely, lpGk,1 = {ϕ ∈ Gk,1 : ∀x ∈ Dom(ϕ), |x| = |ϕ(x)|}. See [8] for
a study of lpGk,1 and some of its properties. In particular, it was proved that lpGk,1 is a direct limit
of finite alternating groups, and that lpG2,1 is generated by the set {N,C, T} ∪ {τi,i+1 : 1 ≤ i},
where N : x1w 7→ x1w, C : x1x2w 7→ x1 (x2 ⊕ x1)w, and T : x1x2x3w 7→ x1x2 (x3 ⊕ (x2 ∧ x1))w
(for x1, x2, x3 ∈ {0, 1} and w ∈ {0, 1}∗). Thus (recalling Subsection 1.4), N,C, T are the not, c-not,
cc-not gates, applied to the first (left-most) bits of a binary string. It is known that the gates not,
c-not, cc-not, together with the wire-swappings, form a complete set of gates for bijective circuits (see
[36, 40, 15]); hence, lpG2,1 is closely related to the field of reversible computing.
It is natural to generalize the bijections between finite maximal prefix codes to functions between
finite prefix codes. Following [9] we will define below the Thompson-Higman monoids Mk,1. First,
some preliminary definitions. A right-ideal homomorphism of A∗ is a total function ϕ : R1 → A∗ such
that R1 is a right ideal, and such that for all r1 ∈ R1 and all z ∈ A∗: ϕ(r1z) = ϕ(r1) · z. It is
easy to prove that Im(ϕ) is then also a right ideal of A∗. From now on we will write a right-ideal
homomorphism as a total surjective function ϕ : R1 → R2, where both R1 and R2 are right ideals. The
homomorphism ϕ is uniquely determined by a total surjective function f : P1 → S2, with P1, S2 ⊂ A∗
where P1 is the prefix code (not necessarily maximal) that generates R1 as a right ideal, and where
S2 is a set (not necessarily a prefix code) that generates R2 as a right ideal; f can be described by a
finite function table.
For two sets X,Y , we say that X and Y “intersect” iff X ∩ Y 6= ∅. We say that a right ideal
R′1 is essential in a right ideal R1 iff R
1 intersects every right ideal that R1 intersects. An essential
restriction of a right-ideal homomorphism ϕ : R1 → R2 is a right ideal-homomorphism Φ : R′1 → R′2
such that R′1 is essential in R1, and for all x
1 ∈ R′1: ϕ(x′1) = Φ(x′1). In that case we also say that ϕ
is an essential extension of Φ. If Φ is an essential restriction of ϕ then R′2 = Im(Φ) will automatically
be essential in R2 = Im(ϕ). Indeed, if I is any no-empty right subideal of R1 then I ∩R′1 6= ∅, hence
∅ 6= Φ(I ∩ R′1) ⊆ Φ(I) ∩ Φ(R′1) = Φ(I) ∩ R′2; moreover, any non-empty right subideal J of R2 is of
the form J = Φ(I), where I = Φ−1(J) is a non-empty right subideal of R1; hence, for any non-empty
right subideal J of R2, ∅ 6= J ∩R′2.
The free monoid A∗ can be pictured by its right Cayley graph, which is easily seen to be the
infinite regular k-ary tree with vertex set A∗ and edge set {(v, va) : v ∈ A∗, a ∈ A}. We simply call
this the tree of A∗. It is a directed, rooted tree, with all paths directed away from the root ε (the
empty word); by “path” we will always mean a directed path. Many of the previously defined concepts
can be reformulated more intuitively in the context of the tree of A∗: A word v is a prefix of a word
w iff v is an ancestor of w in the tree. A set P is a prefix code iff no two elements of P are on a
common path. A set R is a right ideal iff any path that starts in R has all its vertices in R. The
prefix code that generates R consists of the elements of R that are maximal (within R) in the prefix
order, i.e., maximally close (along paths) to the root ε. A finitely generated right ideal R is essential
iff every infinite path eventually reaches R (and then stays in it from there on). Similarly, a finite
prefix code P is maximal iff any infinite path starting at the root eventually intersects P . For two
finitely generated right ideals R′, R with R′ ⊂ R we have: R′ is essential in R iff any infinite path
starting in R eventually reaches R′ (and then stays in it from there on).
Assume now that a total order a1 < a2 < . . . < ak has been chosen for the alphabet A; this means
that the tree of A∗ is now an oriented rooted tree, i.e., the children of each vertex v have a total order
va1 < va2 < . . . < vak. The following can be proved (see [9], Prop. 1.4(1)): Φ is an essential restriction
of ϕ iff Φ can be obtained from ϕ by starting from the table of ϕ and applying a finite number of
restriction steps of the following form: “replace (x, y) in a table by {(xa1, ya1), . . . , (xak, yak)}”.
In the tree of A∗ this means that x and y are replaced by their children xa1, . . . , xak, respectively
ya1, . . . , yak, paired according to the order on the children. One can also prove (see [9], Remark after
Prop. 1.4): Every right ideal homomorphism ϕ with table P → S has an essential restriction ϕ′ that
has a table P ′ → Q′ such that both P ′ and Q′ are prefix codes.
An important fact is the following (see [9], Prop. 1.4(2)): Every homomorphism between finitely
generated right ideals of A∗ has a unique maximal essential extension; we call it the maximum essential
extension of Φ and denote it by max(Φ).
Finally here is the definition of the Thompson-Higman monoid: Mk,1 consists of all maximum es-
sential extensions of homomorphisms between finitely generated right ideals of A∗. The multiplication
is composition followed by maximum essential extension.
One can prove the following, which implies associativity: For all right ideal homomorphisms ϕ1, ϕ2 :
max(ϕ2 ◦ ϕ1) = max(max(ϕ2) ◦ ϕ1) = max(ϕ2 ◦max(ϕ1)).
In [9] the following are proved about the Thompson-Higman monoid Mk,1:
• The Thompson-Higman group Gk,1 is the group of invertible elements of the monoid Mk,1.
• Mk,1 is finitely generated.
• The word problem of Mk,1 over any finite generating set is in P.
• The word problem of Mk,1 over a generating set Γ ∪ {τi,j : 1 ≤ i < j}, where Γ is any finite
generating set of Mk,1, is coNP-complete.
2 Boolean functions as elements of Thompson monoids
The input-output functions of digital circuits map bitstrings of some fixed length to bitstrings of a
fixed length (possibly different from the input length). In other words, circuits have input-output maps
that are total functions of the form f : {0, 1}m → {0, 1}n for some m,n > 0. The Thompson-Higman
monoid Mk,1 has an interesting submonoid that corresponds to fixed-length maps, defined as follows.
Definition 2.1 (the submonoid lepMk,1). Let ϕ : PA
∗ → QA∗ be a right-ideal homomorphism,
where P,Q ⊂ A∗ are finite prefix codes, and where P is a maximal prefix code. Then ϕ is called length
equality preserving iff for all x1, x2 ∈ Dom(ϕ) : |x1| = |x2| implies |ϕ(x1)| = |ϕ(x2)|.
The submonoid lepMk,1 of Mk,1 consists of those elements of Mk,1 that can be represented by
length-equality preserving right-ideal homomorphisms.
It is easy to check that an essential restriction of an element of lepMk,1 is again in lepMk,1, so lepMk,1
is well defined as a subset of Mk,1; moreover, one can easily check that lepMk,1 is closed under
composition, so lepMk,1 is indeed a submonoid of Mk,1.
For ϕ ∈ Mk,1 we have ϕ ∈ lepMk,1 iff there exist m > 0 and n > 0 such that Am ⊂ Dom(ϕ)
and ϕ(Am) ⊆ An. So (by means of an essential restriction, if necessary), ϕ can be represented by a
function table Am → Q ⊆ An with a fixed input length and a fixed output length (but the input and
output lengths can be different).
The motivation for studying the monoid lepMk,1 is the following. Every boolean function f :
{0, 1}m → {0, 1}n (for any m,n > 0) determines an element of lepMk,1, and conversely, this element
of lepMk,1 determines f when restricted to {0, 1}m. By considering all boolean functions as elements
of lepMk,1 we gain the ability to compose arbitrary boolean functions, even if their domain and range
“do not match”. Moreover, in lepMk,1 we are able to generate all boolean functions from gates by
using ordinary functional composition (instead of graph-based circuit lay-outs). The following remains
open:
Question: Is lepMk,1 finitely generated?
However we can find nice infinite generating sets, in connection with circuits.
Proposition 2.2 (Generators of lepMk,1). The monoid lepMk,1 has a generating set of the form
Γ ∪ {τi,i+1 : 1 ≤ i}, for some finite subset Γ ⊂ lepMk,1.
Proof. We only prove the result for k = 2; a similar reasoning works for all k (using k-ary logic).
It is a classical fact that any function f : {0, 1}m → {0, 1}n can be implemented by a combinational
circuit that uses copies of and, or, not, fork and wire-crossings. So all we need to do is to express theses
gates, at any place in the circuit, by a finite subset of lepM2,1 and by positions transpositions τi,i+1.
For each gate g ∈ {and, or} we define an element γg ∈ lepMk,1 by
γg : x1x2w ∈ {0, 1}m 7−→ g(x1, x2) w ∈ {0, 1}m−1.
Similarly we define γnot, γfork ∈ lepMk,1 by
γnot : x1w ∈ {0, 1}m 7−→ x1 w ∈ {0, 1}m,
γfork : x1w ∈ {0, 1}m 7−→ x1 x1 w ∈ {0, 1}m+1.
For each g ∈ {and, or, not, fork}, γg transforms only the first one or two boolean variables, and leaves
the other boolean variables unchanged. We also need to simulate the effect of a gate g on any variable
xi or pair of variables xixi+1, i.e., we need to construct the map
uxixi+1v ∈ {0, 1}m 7−→ u g(xi, xi+1) v ∈ {0, 1}m−1
(and similarly in case where g is not or fork). For this, we apply wire-transpositions to move xixi+1 to
the wire-positions 1 and 2, then we apply γg, then we apply more wire-transpositions in order to move
g(x1, x2) back to position i. Thus the effect of any gate anywhere in the circuit can be expressed as a
composition of γg and position transpositions in {τi,i+1 : 1 ≤ i}. ✷
Proposition 2.3 (Change of generators of lepMk,1). Let {τi,i+1 : 1 ≤ i} be denoted by τ . If
Γ,Γ′ ⊂ lepMk,1 are two finite sets such that Γ∪τ and Γ′∪τ generate lepMk,1, then the word-length over
Γ ∪ τ is linearly related to the word-length over Γ′ ∪ τ . In other words, there are constants c′ ≥ c ≥ 1
such that for all m ∈ lepMk,1 : |m|Γ∪τ ≤ c · |m|Γ′∪τ ≤ c′ · |m|Γ∪τ .
Proof. Since Γ is finite, the elements of Γ can be expressed by a finite set of words of bounded length
(≤ c) over Γ′ ∪ τ . Thus, every word of length n over Γ∪ τ is equivalent to a word of length ≤ c n over
Γ′ ∪ τ . This proves the first inequality. A similar reasoning proves the second inequality. ✷
Proposition 2.4 (Circuit size vs. lepM2,1 word-length).
Let ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} be a generating set of lepM2,1 with ΓlepM2,1 finite. Let f : {0, 1}m →
Q (⊆ {0, 1}n) be a function defining an element of lepM2,1, and let |f |lepM2,1 the word-length of f
over the generating set ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j}. Let |Cf | be the circuit size of f (using any
finite universal set of gates and wire-swappings). Then |f |lepM2,1 and |Cf | are linearly related. More
precisely, for some constants c1 ≥ co ≥ 1 :
|Cf | ≤ co · |f |lepM2,1 ≤ c1 · |Cf |.
Proof. For the proof we assume that the set of gates for circuits (not counting the wire-transpositions)
is ΓlepM2,1 . If we make a different choice for the universal set of gates for circuits, and a different choice
for the finite portion ΓlepM2,1 of the generating set of lepM2,1 then the inequalities remain the same,
except for the constants c1, co.
The inequality |Cf | ≤ |f |lepM2,1 is obvious, since a word w over ΓlepM2,1 ∪ {τi,j : 1 ≤ i < j} is
automatically a circuit of size |w|.
For the other inequality, we want to simulate each gate of the circuit Cf by a word over ΓlepM2,1 ∪
{τi,j : 1 ≤ i < j}. The reasoning is the same for every gate, so let us just focus on an or gate. The
essential difference between circuit gates and elements of lepM2,1 is that in a circuit, a gate (with
2 input wires, for example) can be applied to any two wires in the circuit; on the other hand, the
functions in lepM2,1 are applied to the first few wires. However, the circuit gate or, applied to (i, i+1)
can be simulated by an element of ΓlepM2,1 and a few wire transpositions, since we have: ori,i+1(.) =
γor ◦ τ2,i+1 ◦ τ1,i(.).
The output wire of ori,i+1(.) is wire number i, whereas the output wire of γor ◦ τ2,i+1 ◦ τ1,i(.) is wire
number 1. However, instead of permuting all the wires in order to place the output of γor τ2,i+1 τ1,i(.)
on wire i, we just leave the output of γor τ2,i+1 τ1,i(.) on wire 1 for now. The simulation of the
next gate will then use appropriate transpositions τ2,j · τ1,k for fetch the correct input wires for the
next gate. Thus, each gate of Cf is simulated by one function in ΓlepM2,1 and a bounded number of
wire-transpositions in {τi,j : 1 ≤ i < j}.
At the output end of the circuit, a permutation of the n output wires is needed in order to send
the outputs to the correct wires; any permutation of n elements can be realized with < n (≤ |Cf |)
transpositions. (The inequality n ≤ |Cf | holds because since we count the output ports in the circuit
size.) ✷
Remark. The above Proposition motivates our choice of generating set of the form Γ∪{τi,j : 1 ≤ i < j}
(with Γ finite) for lepMk,1; in particular, it motivates the inclusion of all the position transpositions
τi,j in the generating set. The Proposition also motivates the definition of word-length in which τi,j
has word-length 1 for all j > i ≥ 1.
Next we will study the distortion of lepMk,1 in Mk,1. We first need some Lemmas.
Lemma 2.5 (Lemma 3.3 in [6]). If P,Q,R ⊆ A∗ are such that PA∗ ∩QA∗ = RA∗ and R is a prefix
code, then R ⊆ P ∪Q.
Proof. For any r ∈ R there are p ∈ P, q ∈ Q and v,w ∈ A∗ such that r = pv = qw. Hence p is a
prefix of q or q is a prefix of p. Let us assume p is a prefix of q = px, for some x ∈ A∗ (the other case
is similar) Hence q = px ∈ PA∗ ∩QA∗ = RA∗, and q is a prefix of r = qw. Since R is a prefix code,
r = q, hence r ∈ Q. ✷
Lemma 2.6 Let P,Q ⊂ A∗ be finite prefix codes, and let θ : PA∗ → QA∗ be a right-ideal homomor-
phism with domain PA∗ and image QA∗. Let S be a prefix code with S ⊂ QA∗. Then θ−1(S) is a
prefix code and θ−1(SA∗) = θ−1(S) A∗.
Proof. First, θ−1(S) is a prefix code. Indeed, if we had x1 = x2u for some x1, x2 ∈ θ−1(S) with u
non-empty, then θ(x1) = θ(x2) u. This would contradict the assumption that S is a prefix code.
Second, θ−1(S) ⊂ θ−1(SA∗), hence θ−1(S) A∗ ⊆ θ−1(SA∗), since θ−1(SA∗) is a right ideal. (Recall
that the inverse image of a right ideal under a right-ideal homomorphism is a right ideal.)
We also want to show that θ−1(SA∗) ⊆ θ−1(S) A∗. Let x ∈ θ−1(SA∗). So, θ(x) = sv for some
s ∈ S, v ∈ A∗, and s = qu for some q ∈ Q, u ∈ A∗. Since θ(x) = quv, we have x = puv for some
p ∈ P with θ(p) = q. Hence θ(pu) = qu = s. Therefore, x = puv with pu ∈ θ−1(s) ⊆ θ−1(S), hence
x ∈ θ−1(S) A∗. ✷
Notation: For a right-ideal homomorphism ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗, where P,Q ⊂ A∗
are finite prefix codes, we define
ℓ(ϕ) = max{|z| : z ∈ P ∪Q},
For any finite prefix code C ⊂ A∗ we define
ℓ(C) = max{|z| : z ∈ C}.
Lemma 2.7 Let ϕ : Dom(ϕ) = PA∗ → Im(ϕ) = QA∗ be a right-ideal homomorphism, where P and
Q are finite prefix codes. Let R ⊂ A∗ be any finite prefix code. Then we have:
(1) ℓ(ϕ−1(R)) < ℓ(ϕ) + ℓ(R),
(2) ℓ(ϕ(R)) < ℓ(ϕ) + ℓ(R).
Proof. (1) Let r ∈ R ∩ Im(ϕ). Then every element of ϕ−1(r) has the form p1w for some p1 ∈ P
and w ∈ A∗ such that r = q1w for some q1 ∈ Q (with ϕ(p1) = q1). Hence |p1w| = |p1| + |r| − |q1| =
|r|+ |p1| − |q1|. Moreover, |r| ≤ ℓ(R) and |p1| − |q1| < ℓ(ϕ), so |p1w| < ℓ(R) + ℓ(ϕ).
(2) If r ∈ R ∩Dom(ϕ) then ϕ(r) has the form q1v for some q1 ∈ Q and v ∈ A∗ such that r = p1w
for some p1 ∈ P (with ϕ(p1) = q1). Hence |q1v| = |q1| + |r| − |p1| = |r| + |q1| − |p1|. Moreover,
|r| ≤ ℓ(R) and |q1| − |p1| < ℓ(ϕ), so |q1w| < ℓ(R) + ℓ(ϕ). ✷
For any right-ideal homomorphisms ϕi (with i = 1, . . . , N), the composite map ϕN ◦ . . . ◦ ϕ1(.) is a
right-ideal homomorphism. We say that right-ideal homomorphisms Φi (with i = 1, . . . , N) are directly
composable iff Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1. The next Lemma shows that we can replace
composition by direct composition.
Lemma 2.8 Let ϕi : Dom(ϕi) = PiA
∗ → Im(ϕi) = QiA∗ be a right-ideal homomorphism (for
i = 1, . . . , N), where Pi and Qi are finite prefix codes. Then each ϕi has a (not necessarily essential)
restriction to a right-ideal homomorphism Φi with the following properties:
• ΦN ◦ . . . ◦ Φ1(.) = ϕN ◦ . . . ◦ ϕ1(.);
• Dom(Φi+1) = Im(Φi), for i = 1, . . . , N − 1;
• ℓ(Φi) ≤
j=1 ℓ(ϕj) for every i = 1, . . . , N .
Proof. We use induction on N . For N = 1 there is nothing to prove. So we let N > 1 and we
assume that the Lemma holds for ϕi : PiA
∗ → QiA∗ with i = 2, . . . , N , i.e., we assume that each
ϕi (for i = 2, . . . , N) has a restriction ϕ
i : P
∗ → Q′iA∗ such that ϕ′N ◦ . . . ◦ ϕ′2 = ϕN ◦ . . . ◦ ϕ2,
P ′i+1 = Q
i (for i = 2, . . . , N − 1), and ℓ(ϕ′i) ≤
j=2 ℓ(ϕj) for every i = 2, . . . , N . From P
i+1 = Q
(for i = 2, . . . , N − 1) it follows that ℓ(ϕ′N ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′i) : i = 2, . . . , N} ≤
j=2 ℓ(ϕj).
Using the notation ϕ′
[N,2]
for ϕ′N ◦ . . . ◦ ϕ′2 we have Dom(ϕ′[N,2]) = P2A
∗ and Im(ϕ′
[N,2]
) = QNA
When we compose ϕ1 and ϕ
[N,2]
we obtain
ϕ−11 (Q1A
∗ ∩ P2A∗)
Φ1−→ Q1A∗ ∩ P2A∗
[N,2]−→ ϕ′
[N,2]
∗ ∩ P2A∗).
In this diagram, Φ1 is the restriction of ϕ1 to the domain ϕ
1 (Q1A
∗∩P2A∗) and image Q1A∗∩P2A∗;
and Φ′
[N,2]
is the restriction of ϕ′
[N,2]
to the domain Q1A
∗ ∩ P2A∗ and image ϕ′[N,2](Q1A
∗ ∩ P2A∗).
Hence, Φ′
[N,2]
◦Φ1 = ϕ′[N,2] ◦ϕ1, and Dom(Φ
[N,2]
) = Im(Φ1) (= Q1A
∗ ∩P2A∗). So Φ1 and Φ′[N,2] are
directly composable.
By Lemma 2.5 there is a prefix code S ⊂ A∗ such that SA∗ = Q1A∗ ∩ P2A∗ and S ⊆ Q1 ∪ P2.
Hence, ℓ(S) ≤ max{ℓ(Q1), ℓ(P2)} ≤ max{ℓ(ϕ1), ℓ(ϕ′2)} ≤ max{ℓ(ϕ1),
j=2 ℓ(ϕj)} ≤
j=1 ℓ(ϕj).
It follows also that ϕ−11 (Q1A
∗∩P2A∗) = ϕ−11 (SA∗) = ϕ
1 (S) A
∗ (the latter equality is from Lemma
2.6). Since S ⊆ Q1 ∪ P2 implies ϕ−11 (S) ⊆ ϕ
1 (Q1) ∪ ϕ
1 (P2) = P1 ∪ ϕ
1 (P2), we have ℓ(ϕ
1 (S)) ≤
max{ℓ(P1), ℓ(ϕ−11 (P2))}. Obviously, ℓ(P1) ≤ ℓ(ϕ1). Moreover, by Lemma 2.7, ℓ(ϕ
1 (P2)) ≤ ℓ(ϕ1) +
ℓ(P2). Since ℓ(P2) ≤ ℓ(ϕ′2) ≤
j=2 ℓ(ϕj) (the latter “≤” by induction), we have ℓ(ϕ
1 (S)) ≤ ℓ(ϕ1)+
j=2 ℓ(ϕj) =
j=1 ℓ(ϕj).
Since the domain code of Φ1 is ϕ
1 (S) and its image code is S, we conclude that ℓ(Φ1) ≤
j=1 ℓ(ϕj).
Let us now consider any Φ′
[i,2]
, for i = 1, . . . , N . By definition, Φ′
[i,2]
is the restriction of ϕ′i ◦ . . . ◦ϕ′2
to the domain SA∗. So the domain code of Φ′
[i,2]
is S, and we just proved that ℓ(S) ≤
j=1 ℓ(ϕj).
The image code of Φ′
[i,2]
is ϕ′i ◦ . . . ◦ ϕ′2(S). Since S ⊆ Q1 ∪ P2 we have
ϕ′i ◦ . . . ◦ ϕ′2(S) ⊆ ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ ϕ′i ◦ . . . ◦ ϕ′2(P2) = ϕ′i ◦ . . . ◦ ϕ′2(Q1) ∪ Q′i.
Therefore: ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤ max{ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)), ℓ(Q′i)}.
We have ℓ(Q′i) ≤ ℓ(ϕ′i) ≤
j=2 ℓ(ϕj) (the last “≤” by induction).
By Lemma 2.7, ℓ(ϕ′i ◦ . . . ◦ ϕ′2(Q1)) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(Q1) ≤ ℓ(ϕ′i ◦ . . . ◦ ϕ′2) + ℓ(ϕ1). And
ℓ(ϕ′i ◦ . . . ◦ ϕ′2) ≤ max{ℓ(ϕ′j) : j = 2, . . . , i}, because Dom(ϕ′r+1) = Im(ϕ′r) for all r = 2, . . . , N − 1.
And by induction, ℓ(ϕ′j) ≤
j=2 ℓ(ϕj). Hence, ℓ(ϕ
i ◦ . . . ◦ ϕ′2(Q1)) ≤
j=1 ℓ(ϕj).
Thus, ℓ(Φ′
[i,2]
j=1 ℓ(ϕj) for every i = 2, . . . , N .
Finally, we factor Φ′
[N,2]
as Φ′
[N,2]
= ΦN ◦ . . . ◦Φ2, where Φi (for i = 2, . . . , N) is defined to be the
restriction of ϕ′i to the domain ϕ
i−1 ◦ . . . ◦ ϕ′2(SA∗) (= Φ′[i−1,2](SA
∗)). Since Dom(ϕ′r+1) = Im(ϕ
(for all r = 2, . . . , N − 1), the domain of ϕ′i is equal to the image of ϕ′i−1 ◦ . . . ◦ ϕ′2. So, the domain
code of Φi is ϕ
i−1 ◦ . . . ◦ϕ′2(S), and its image code is ϕ′i ◦ϕ′i−1 ◦ . . . ◦ϕ′2(S). Since we already proved
that ℓ(ϕ′i ◦ . . . ◦ ϕ′2(S)) ≤
j=1 ℓ(ϕj) (for all i), it follows that ℓ(Φi) ≤
j=1 ℓ(ϕj). ✷
In the next theorem we show that the distortion of lepMk,1 in Mk,1 is at most quadratic (over the
generators considered so far, which include the bit position transpositions). Combined with Proposi-
tion 2.4, this means the following:
Assume circuits are built with gates that are not constrained to have fixed-length inputs and outputs,
but assume the input-output function has fixed-length inputs and outputs. Then the resulting circuits
are not much more compact than conventional circuits, built from gates that have fixed-length inputs
and outputs (we gain at most a square-root in size).
Theorem 2.9 (Distortion of lepMk,1 in Mk,1). The word-length (or Cayley graph) distortion of
lepMk,1 in Mk,1 has a quadratic upper bound; in other words, for all x ∈ lepMk,1:
|x|lepMk,1 ≤ c · (|x|Mk,1)2
where c ≥ 1 is a constant. Here the generating sets used are ΓMk,1 ∪ {τi,j : 1 ≤ i < j} for Mk,1, and
ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j} for lepMk,1, where ΓMk,1 and ΓlepMk,1 are finite. By |x|Mk,1 and |x|lepMk,1
we denote the word-length of x over ΓMk,1 ∪ {τi,j : 1 ≤ i < j}, respectively ΓlepMk,1 ∪ {τi,j : 1 ≤ i < j}.
Proof. We only prove the result for k = 2; a similar proof applies for any k. We abbreviate the set
{τi,j : 1 ≤ i < j} by τ . The choice of the finite sets ΓMk,1 and ΓlepMk,1 does not matter (it only affects
the constant c in the Theorem. By Corollary 3.6 in [9] we can choose ΓMk,1 so that each γ ∈ ΓMk,1
satisfies the following (recall that ℓ(S) denotes the length of the longest words in a set S):
domC(γ) ∪ imC(γ)
≤ 2, and
∣|γ(x)| − |x|
∣ ≤ 1 for all x ∈ Dom(γ).
Let ϕ ∈ lepMk,1, and let w = αN . . . α1 be a shortest word over the generating set ΓMk,1 ∪ τ of
Mk,1, representing ϕ. So N = |ϕ|Mk,1 . We restrict each partial function αi to a partial function α′i such
that imC(α′i) = domC(α
i+1) for i = 1, . . . , N−1, according to Lemma 2.8. Hence, αN ◦ . . .◦α1(.) =
α′N ◦ . . . ◦ α′1(.), and ℓ(α′i) ≤
j=1 ℓ(αj) for every i = 1, . . . , N . Then αN ◦ . . . ◦ α1(.) is a function
{0, 1}m {0, 1}∗ → Q {0, 1}∗, representing ϕ, and we will identify αN ◦ . . . ◦ α1(.) with ϕ. It follows
that domC(α′1) = domC(ϕ) = {0, 1}m, and imC(α′N ) = imC(ϕ) = Q ⊆ {0, 1}n. More generally, it
follows that imC(α′i ◦ . . . ◦ α′1) = imC(α′i), and domC(α′N ◦ . . . ◦ α′i) = domC(α′i).
Since ℓ(α′i) ≤
j=1 ℓ(αj), and ℓ(αj) ≤ 2 for all j, we have for every i = 1, . . . , N : ℓ(α′i) ≤ 2N .
From here on we will simply denote ℓ(α′i) by ℓi. Now, we will replace each α
i ∈ Mk,1 by βi ∈
lepMk,1, such that domC(βi) = {0, 1}ℓi , and imC(βi) ⊆ {0, 1}ℓi+1 ; so βi is length-equality preserving.
This will be done by artificially lengthening those words in domC(α′i) that have length < ℓi and those
words in imC(α′i) that have length < ℓi+1. Moreover, we make βi defined on all of {0, 1}ℓi . In detail,
βi is defined as follows:
• If ℓi ≤ ℓi+1 :
βi(u z) = v z 0
ℓi+1−ℓi−|v|+|u| for all u ∈ domC(α′i), and z ∈ {0, 1}ℓi−|u|; here v = α′i(u);
βi(x) = x 0
ℓi+1−ℓi for all x 6∈ Dom(α′i), |x| = ℓi.
• If ℓi > ℓi+1 :
βi(u z1 z2) = v z1 for all u ∈ domC(α′i) and all z1, z2 ∈ {0, 1}∗ with
|z1| = ℓi+1 − |v|, |z2| = ℓi − ℓi+1 + |v| − |u|; here, v = α′i(u);
βi(x1 x2) = x1 for all x1, x2 ∈ {0, 1}∗ such that x1x2 6∈ Dom(α′i), with
|x1| = ℓi+1, |x2| = ℓi − ℓi+1.
Claim. βN ◦ . . . ◦ β1(.) = ϕ.
Proof of the Claim: We observe first that domC(β1) = domC(α
1) (= domC(ϕ) = {0, 1}m). Next,
assume by induction that for every x ∈ {0, 1}m : α′i−1 ◦ . . . ◦ α′1(x) = u is a prefix of βi−1 ◦ . . . ◦
β1(x) = u z. Then βi(u z) = v z 0
ℓi+1−ℓi−|v|+|u| (if ℓi ≤ ℓi+1); or βi(u z) = v z1 (if ℓi ≥ ℓi+1, with
|z1| = ℓi+1 − |v| and z = z1z2). In either case we find that α′i(α′i−1 ◦ . . . ◦ α′1(x)) = v is a prefix of
βi(βi−1 ◦ . . . ◦ β1(x)) = βi(u z).
Hence, when i = N we obtain for any x ∈ {0, 1}m: βN ◦ . . . ◦ β1(x) = y s is a prefix of
α′N ◦ . . . ◦ α′1(x) = ϕ(x) = y for some y and s with |y s| = ℓN = n. Since y ∈ imC(ϕ) ⊆ {0, 1}n we
conclude that s is empty, hence βN ◦ . . . ◦ β1(x) = α′N ◦ . . . ◦ α′1(x). [End, proof of Claim.]
At this point we have expressed ϕ as a product of N elements βi ∈ lepMk,1, where N = |ϕ|Mk,1 .
We now want to find the word-length of each βi over ΓlepMk,1 ∪ τ , in order to find an upper bound on
the total word-length of ϕ over ΓlepMk,1 ∪ τ . As we saw above, ℓi ≤ 2N for every i = 1, . . . , N .
We examine each generator in ΓMk,1 ∪ τ .
If αi ∈ τ then βi ∈ τ , so in this case |βi|lepMk,1 = 1.
Suppose now that αi ∈ ΓMk,1 . By Proposition 2.4 it is sufficient to construct a circuit that computes
βi; the circuit can then be immediately translated into a word over ΓlepMk,1 ∪ τ with linear increase in
length.
Since domC(αi) ⊆ {0, 1}≤2, we can restrict αi so that its domain code becomes a subset of {0, 1}2;
next, we extend αi to a map α
i that acts as the identity map on {0, 1}2 where αi was undefined. The
image code of α′′i is a subset of {0, 1}≤3. In order to compute βi we first introduce a circuit C(α′′i ) that
computes α′′i . A difficulty here is that α
i does not produce fixed-length outputs in general, whereas
C(α′′i ) has to work with fixed-length inputs and outputs; so the output of C(α
i ) represents the output
of α′′i indirectly, as follows:
The circuit C(α′′i ) has two input bits u = u1u2 ∈ {0, 1}2, and 5 output bits: First there are 3
output bits 03−|v| v ∈ {0, 1}3, where v = α′′i (u); second, there are two more output bits, c1c2 ∈ {0, 1}2,
defined by c1c2 = bin(3− |v|) (the binary representation of the non-negative integer 3− |v|). Hence,
c1c2 = 00 if |v| = 3, c1c2 = 01 if |v| = 2, c1c2 = 10 if |v| = 1; since |v| > 0, the value c1c2 = 11 will
not occur. Thus c1c2 0
3−|v| v contains the same information as v, but has the advantage of having a
fixed length (always 5). The circuit C(α′′i ) can be built with a small constant number of and, or, not,
fork gates, and we will not need to know the details.
We now build a circuit for βi.
• Circuit for βi if ℓi ≤ ℓi+1:
On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z 0ℓi+1−ℓi−|v|+|u|, where
v = α′′i (u).
We first apply the circuit C(α′′i ), thus obtaining c1c2 0
3−|v| v z. Then we apply two fork operations
(always to the last bit in z) to produce c1c2 0
3−|v| v z b b, where b is the last bit of z. Applying a
negation to the first b and an and operation, we obtain c1c2 0
3−|v| v z 0. Applying ℓi+1 − ℓi − 1 more
fork operations to the last 0 yields c1c2 0
3−|v| v z 0ℓi+1−ℓi−1.
Next, we want to move 03−|v| to the right of the output, in order to obtain c1c2 v z 0
3−|v|+ℓi+1−ℓi−1.
For this effect we introduce a controlled cycle. Let κ : x1x2x3 ∈ {0, 1}3 7−→ x3x1x2 be the usual
cyclic permutations of 3 bit positions. The controlled cycle acts as the identity map when c1c2 = 00
or 11, τ1,2 when c1c2 = 01, and κ when c1c2 = 10. More precisely,
κc : c1c2 x1x2x3 ∈ {0, 1}5 7−→
c1c2 x1x2x3 if c1c2 = 00 or 11,
c1c2 x2x1x3 if c1c2 = 01,
c1c2 x3x1x2 if c1c2 = 10.
We apply ℓi copies of κc(c1, c2, ., ., .) (all controlled by the same value of c1c2) to 0
3−|v| v z. The first
κc(c1, c2, ., ., .) is applied to the 3 bits 0
3−|v| v, producing 3 bits y1y2y3; the second κc(c1, c2, ., ., .) is
applied to y2y3 and the first bit of z, producing 3 bits y
3; the third κc(c1, c2, ., ., .) is applied to
3 and the second bit of z, etc. So, each one of the ℓi copies of κc acts one bit further down than
the previous copy of κc. This will yield c1c2 v z 0
3−|v|+ℓi+1−ℓi−1. Finally, to make c1c2 disappear, we
apply two fork operations to c1, then a negation and an and, to make a 0 appear. We combine this 0
with c1 and c2 by and gates, thus transforming 0c1c2 into 0. Finally, an or operation between this 0
and the first bit of v makes this 0 disappear.
The number of gates used to compute βi is O(ℓi+1 + ℓi), which is ≤ O(N).
• Circuit for βi if ℓi > ℓi+1:
On input u z ∈ {0, 1}ℓi (with u ∈ {0, 1}2), we want to produce the output v z1, where v = α′′i (u).
We first apply the circuit C(α′′i ), which yields the output c1c2 0
3−|v| v z. Now we want to erase
the ℓi− ℓi+1+1 last bits of z. For this we apply two fork operations to the last bit of z (let’s call it b),
then a negation and an and, to make a 0 appear. We combine this 0 with the last ℓi − ℓi+1 bits of z,
using that many and gates, turning all these bits into a single 0; finally, an or operation between this 0
and the bit of the remainder of z makes this 0 disappear. At this point, the output is c1c2 0
3−|v| v Z1,
where Z1 is the prefix of length ℓi+1 − 1 of z.
Next, we apply O(ℓi+1) position transpositions to Z1 in order move the two last bits of Z1 to the
front of Z1. Let b1b2 be the last two bits of Z1; so, Z1 = z0b1b2 (where z0 is the prefix of length ℓi+1−3
of z); at this point, the output of the circuit is c1c2 0
3−|v| v b1b2 z0.
We now introduce a fixed small circuit with 7 input bits and 5 output bits, defined by the following
input-output map:
ωc : c1c2 x1x2x3 b1b2 ∈ {0, 1}7 7−→
c1c2 x1x2x3 if c1c2 = 00 or 11,
c1c2 x1x2 b1 if c1c2 = 01,
c1c2 x3 b1b2 if c1c2 = 10.
When this map is applied to c1c2 0
3−|v| v b1b2 the output is therefore given by
ωc : c1c2 0
3−|v| v b1b2 ∈ {0, 1}7 7−→
c1c2 v if |v| = 3,
c1c2 v b1 if |v| = 2,
c1c2 v b1b2 if |v| = 1.
A circuit for ωc can be built with a small fixed number of and, or, not, fork gates, and we will not need
to know the details.
After applying ωc to c1c2 0
3−|v| v b1b2 z0 the output has length ℓi+1 + 2; the “+2” comes from
c1c2. The output is c1c2 v z0, or c1c2 v b1 z0, or c1c2 v b1b2 z0, depending on whether |v| = 3, 2, or 1.
We need to move b1b2 or b1 (or nothing) back to the right-most positions of z0. We do this by
applying ℓi+1 copies of the controlled cycle κc(c1, c2, ., ., .) (all copies controlled by the same value of
c1c2). We proceed in the same way as when we used κc in the previous case, and we obtain the output
c1c2 v z0 (if |v| = 3), or c1c2 v z0 b1 (if |v| = 2), or c1c2 v z0 b1b2 (if |v| = 1).
Finally, we erase c1c2 in the same way as in the previous case, thus obtaining the final output.
The number of gates used to compute βi is O(ℓi+1 + ℓi) ≤ O(N).
This completes the constuction of a circuit for βi. Through this circuit, βi : {0, 1}ℓi → {0, 1}ℓi+1 is
expressed as a word over the generating set ΓlepMk,1 ∪ τ , of length ≤ O(ℓi+1 + ℓi) ≤ O(N).
Since we have described ϕ as a product of N = |ϕ|Mk,1 elements βi ∈ lepMk,1, each of word-length
O(N), we conclude that ϕ has word-length ≤ O(N2) over the generating set ΓlepMk,1 ∪ τ of lepMk,1.
Question: Does the distortion of lepMk,1 inMk,1 (over the generators of Theorem 2.9) have an upper
bound that is less than quadratic?
3 Wordlength asymmetry vs. computational asymmetry
Proposition 3.1 The word-length asymmetry function λ of the Thompson group lpG2,1 within the
Thompson monoid lepM2,1 is linearly equivalent to the computational asymmetry function α:
α ≃lin λ.
Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j : 0 ≤ i < j}, where ΓlepM2,1 is finite. The
gates used for circuits are any finite universal set of gates, together with the wire-swapping operations
{τi,j : 0 ≤ i < j}.
We can choose ΓlepM2,1 to consist exactly of the gates used in the circuits; then α = λ.
Proof. For any g ∈ lpG2,1 we have
C(g−1) ≤ c0 · |g−1|lepM2,1 ≤ c0 · λ(|g|lepM2,1) ≤ c0 · λ(c1 · C(g)).
The first and last “≤” come from Prop. 2.4 (since lpG2,1 ⊂ lepM2,1), and the middle “≤” comes from
the definition of λ; c0 and c1 are positive constants. Hence,
α(n) ≤ c0 · λ(c1 n) for all n.
In a very similar way we prove that λ(n) ≤ c′0 · α(c′1 n) for some positive constants c′0, c′1. ✷
Proposition 3.2 The word-length asymmetry function λM2,1 of the Thompson group lpG2,1 within
the Thompson monoid M2,1 is polynomially equivalent to the word-length asymmetry function λlepM2,1
of lpG2,1 within the Thompson monoid lepM2,1. More precisely we have for all n :
λM2,1(n) ≤ c0 · λlepM2,1(c1 n2),
λlepM2,1(n) ≤ c′0 ·
λM2,1(c
where c0, c1, c
1 are positive constants. Here the generating set used for lepM2,1 is ΓlepM2,1 ∪ {τi,j :
0 ≤ i < j}, where ΓlepM2,1 is finite. The generating set used for M2,1 is ΓM2,1 ∪ {τi,j : 0 ≤ i < j},
where ΓM2,1 is a finite generating set of M2,1.
Proof. For any g ∈ lpG2,1 we have
|g−1|M2,1 ≤ c0 · |g−1|lepM2,1 ≤ c0 · λlepM2,1(|g|lepM2,1) ≤ c0 · λlepM2,1(c1 · |g|2M2,1).
The first “≤” holds because lpG2,1 ⊂ lepM2,1 ⊂M2,1 and because of the choice of the generating sets.
The second “≤” holds by the definition of λlepM2,1 . The third “≤” comes from the quadratic distortion
of lepM2,1 in M2,1 (Theorem 2.9). For the same reasons we also have the following:
|g−1|lepM2,1 ≤ c′0 · |g−1|2M2,1 ≤ c
0 · (λM2,1(|g|M2,1))2 ≤ c′0 · (λM2,1(c1 · |g|lepM2,1))2
where c′0, c
1 are positive constants. ✷
4 Reversible representation over the Thompson groups
Theorems 4.1 and 4.2 below introduce a representation of elements of the Thompson monoid lepM2,1 by
elements of the Thompson group G2,1, in analogy with the Toffoli representation (Theorem 1.5 above),
and the Fredkin representation (Theorem 1.6 above). Our representation preserves complexity, up to a
polynomial change, and uses only one constant-0 input. Note that although the functions and circuits
considered here use fixed-length inputs and outputs, the representations is over the Thompson group
G2,1, which includes functions with variable-length inputs and outputs.
In the Theorem below, ΓG2,1 is any finite generating set of G2,1. We denote the length of a word
w by |w|, and we denote the size of a circuit C by |C|. The gates and, or, not will also be denoted
respectively by ∧,∨,¬. We distinguish between a word Wf (over a generating set of G2,1) and the
element wf of G2,1 represented by Wf .
Theorem 4.1 (Representation of boolean functions by the Thompson group). Let f :
{0, 1}m → {0, 1}n be any total function and let Cf be a minimum-size circuit (made of ∧,∨,¬, fork-
gates and wire-swappings τi,j) that computes f . Then there exists a word Wf over the generating set
ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1 such that:
• For all x ∈ {0, 1}m: wf (0x) = 0 f(x) x, where wf is the element of G2,1 represented by Wf .
• The length of the word Wf is bounded by |Wf | ≤ O(|Cf |4).
• The largest subscript of any transposition τi,i+1 occurring in Wf has an upper bound ≤ |Cf |2 + 2.
Proof. Wire-swappings in circuits are represented by the position transpositions τi,i+1 ∈ G2,1. The
gates not, or, and and of circuits are represented by the following elements of G2,1:
, ϕ∨ =
0x1x2 1x1x2
(x1 ∨ x2)x1x2 (x1 ∨ x2 )x1x2
, ϕ∧ =
0x1x2 1x1x2
(x1 ∧ x2)x1x2 (x1 ∧ x2 )x1x2
where x1, x2 range over {0, 1}. Hence the domain and image codes of ϕ∨ and ϕ∧ are all equal to
{0, 1}3.
To represent fork we use the following element, in which we recognize σ ∈ F2,1, one of the commonly
used generators of the Thompson group F2,1:
0 10 11
00 01 1
00 01 10 11
000 001 01 1
Note that σ agrees with fork only on input 0, but that is all we will need. By its very essense,
the forking operation cannot be represented by a length-equality preserving element of G2,1, because
G2,1∩ lepM2,1 = lpG2,1 (the group of length-preserving elements of G2,1). A small remark: In [6, 7, 8],
what we call “σ” here, was called “σ−1”.
We will occasionally use the wire-swapping τi,j (1 ≤ i < j); note that τi,j can be expressed in terms
of transpositions of neighboring wires as follows:
τi,j(.) = τi,i+1 τi+1,i+2 . . . τj−2,j−1 τj−1,j τj−2,j−1 . . . τi+1,i+2 τi,i+1(.)
so the word-length of τi,j over {τℓ,ℓ+1 : 1 ≤ ℓ} is ≤ 2(j − i)− 1.
For x = x1 . . . xm ∈ {0, 1}m and f(x) = y = y1 . . . yn ∈ {0, 1}n, we will construct a word Wf over
the generators ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1, such that Wf defines the map wf (.) : 0x 7→ 0 f(x) x.
The circuit Cf is partitioned into slices cℓ (ℓ = 1, . . . , L). Two gates g1 and g2 are in the same slice
iff the length of the longest path from g1 to any input port is the same as the length of the longest
path from g2 to any input port. We assume that Cf is strictly layered, i.e., each gate in slice cℓ only
has in-wires coming from slice cℓ−1, and out-wires going toward slice cℓ+1, for all ℓ. To make a circuit
C strictly layered we need to add at most |C|2 identity gates (see p. 52 in [7]). The input-output map
of slice cℓ has the form
cℓ(.) : y
(ℓ−1) = y
(ℓ−1)
1 . . . y
(ℓ−1)
nℓ−1 ∈ {0, 1}nℓ−1 7−→ y(ℓ) = y
1 . . . y
nℓ ∈ {0, 1}nℓ .
Then y(0) = x and y(L) = y, where x ∈ {0, 1}m is the input and y ∈ {0, 1}n is the output of Cf . Each
slice is a circuit of depth 1.
Before studying in more detail how Cf is built from slices, let us see how a slice is built from gates
(inductively, one gate at a time).
Let C be a depth-1 circuit with k + 1 gates, obtained by adding one gate to a depth-1 circuit K
with k gates. Let K(.) : x1 . . . xm 7−→ y1 . . . yn be the input-output map of the circuit K. Assume
by induction that K is represented by a word WK over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of
G2,1. The input-output map of WK is, by induction hypothesis,
wK(.) : 0x1 . . . xm 7−→ 0 y1 . . . yn x1 . . . xm.
The word WC that represents C over G2,1 is obtained as follows from WK ; there are several cases,
depending on the gate that is added to K to obtain C.
Case 1: An identity-gate (or a not-gate) is added to K to form C, i.e.,
C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1
(or, C(.) : x1 . . . xmxm+1 7−→ y1 . . . ynxm+1).
Then WC is given by
wC : 0x1x2 . . . xmxm+1
σ7−→ 00x1x2 . . . xmxm+1
τ3,m+37−→ 00xm+1 x2 . . . xmx1
ϕ∨7−→
xm+10xm+1 x2 . . . xmx1
τ3,m+37−→ xm+1 0x1x2 . . . xmxm+1
π7−→ 0x1x2 . . . xmxm+1xm+1
wK7−→
0 y1 . . . yn x1 . . . xmxm+1xm+1
π′7−→ 0 y1 . . . yn xm+1 x1 . . . xmxm+1 ,
where π(.) = τm+1,m+2 . . . τ2,3 τ1,2(.) shifts xm+1 from position 1 to position m+ 2, while shifting
0x1 . . . xm one position to the left; and π
′(.) = τm+2,m+3 . . . τn+m+1,n+m+2 τn+m+2,n+m+3(.) shifts
xm+1 from position n+m+ 3 to position n+ 2, while shifting x1 . . . xm one position to the right.
So, WC = π
′ WK π τ3,m+3 ϕ∨ τ3,m+3 σ, noting that functions act on the left. Thus, |WC | =
|WK |+m+n+5 if we use all of {τi,j : 1 ≤ i < j} in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+3 has
length ≤ 2m− 1, hence |WC | ≤ 3m + n + 4. If we denote the maximum index in the transpositions
occurring in WC by JC then we have JC = max{JK , n+m+ 3}.
In case a not-gate is added (instead of an identity gate), ϕ∨ is replaced by ϕ¬ ϕ∨ in WC , and the
result is similar.
Case 2: An and-gate (or an or-gate) is added to K to form C, i.e.,
C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∧ xm+2)
(or, C(.) : x1 . . . xmxm+1xm+2 7−→ y1 . . . yn (xm+1 ∨ xm+2)).
Then WC is given by
wC : 0x1x2 . . . xmxm+1xm+2
σ7−→ 00x1x2 . . . xmxm+1xm+2
τ2,m+37−→
τ3,m+47−→
0xm+1xm+2 x2 . . . xm0x1
ϕ∧7−→ (xm+1 ∧ xm+2) xm+1xm+2 x2 . . . xm0x1
τ2,m+37−→
τ3,m+47−→
(xm+1 ∧ xm+2) 0x1x2 . . . xmxm+1xm+2
π7−→ 0x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2
wK7−→
0 y1 . . . yn x1x2 . . . xm (xm+1 ∧ xm+2) xm+1xm+2
π′7−→
0 y1 . . . yn (xm+1 ∧ xm+2) x1x2 . . . xmxm+1xm+2 ,
where π = τm+1,m+2 . . . τ2,3 τ1,2 shifts (xm+1∧xm+2) from position 1 to positionm+2, while shifting
0x1x2 . . . xm one position to the left; and π
′ = τm+2,m+3 . . . τm+n+1,m+n+2 shifts (xm+1 ∧ xm+2)
from position n+m+ 2 to position m+ 2, while shifting x1 . . . xm one position to the right.
So, WC = π
′ WK π τ3,m+4 τ2,m+3 ϕ∧ τ3,m+4 τ2,m+3 σ, hence |WC | = |WK | + n +m + 7 if all
of {τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ3,m+4 and τ2,m+3 have length
≤ 2(m+ 1)− 1, so |WC | ≤ |WK |+ 5m+ n+ 9. Moreover, JC = max{JK , m+ n+ 2}.
Case 3: A fork-gate is added to K to form C, i.e.,
C(.) : x1 . . . xmxm+1 7−→ y1 . . . yn xm+1xm+1.
Then WC is given by
wC : 0x1x2 . . . xmxm+1
σ27−→ 000x1x2 . . . xmxm+1
τ3,m+47−→ 00xm+1x1x2 . . . xm0
ϕ∨7−→
xm+10xm+1x1x2 . . . xm0
τ1,m+47−→ 00xm+1x1x2 . . . xmxm+1
ϕ∨7−→ xm+10xm+1x1x2 . . . xmxm+1
0x1x2 . . . xmxm+1xm+1xm+1
wK7−→ 0 y1 . . . yn x1x2 . . . xm xm+1xm+1xm+1
π′7−→
0 y1 . . . yn xm+1xm+1 x1x2 . . . xmxm+1 ,
where π = τm+3,m+4 . . . τ1,2 τm+3,m+4 . . . τ3,4 shifts the two copies of xm+1 at the left end from
positions 1 and 3 to positions m+ 3 and m+ 4, while shifting 0 to position 1 and shifting x1 . . . xm
two positions to the left; and π′ = τm+3,m+4 . . . τm+n+2,m+n+3 τm+2,m+3 . . . τm+n+1,m+n+2 shifts
xm+1xm+1 from positions m + n + 2 and m + n + 3 to positions m + 2 and m + 3, while shifting
x1 . . . xm two positions to the right.
So, WC = π
′ WK π ϕ∨ τ1,m+4 ϕ∨ τ3,m+4 σ
2, hence |WC | = |WK | + 2m + n + 10, if all of
{τi,j : 1 ≤ i < j} is used in the generating set; over {τi,i+1 : 1 ≤ i}, τ1,m+4 has length ≤ 2(m+3)−1 and
τ3,m+4 has length ≤ 2m−1. Hence, |WC | ≤ |WK |+6m+n+14. Moreover, JC = max{JK , m+n+3}.
In all cases, |WC | ≤ |WK |+c·(m+n+1) (for some constant c > 1), and JC ≤ max{JK , n+m+3}.
Thus, each slice cℓ, with input-output map cℓ(.) : y
(ℓ−1) 7−→ y(ℓ), is represented by a word Wcℓ with
map wcℓ(.) : 0 y
(ℓ−1) 7−→ 0 y(ℓ) y(ℓ−1), such that |Wcℓ| ≤ c · (n2ℓ−1 + n2ℓ) (for some constant c > 1),
and Jcℓ ≤ nℓ−1 + nℓ + c.
Regarding wire-crossings, we do not include them into other slices; we put the wire-crossings into
pure wire-crossing slices. So we consider two kinds of slices: Slices entirely made of wire-crossings
and identities, slices without any wire-crossings. Wire-crossings in circuits are identical to the group
elements τi,i+1.
We now construct the word Wf from the words Wcℓ (ℓ = 1, . . . , L). First observe that since the
map wcℓ(.) is a right-ideal isomorphism (being an element of G2,1), we not only have
wcℓ(.) : 0 y
(ℓ−1) 7−→ 0 y(ℓ)y(ℓ−1)
but also
wcℓ(.) : 0 y
(ℓ−1)y(ℓ−2) . . . y(1)y(0) 7−→ 0 y(ℓ)y(ℓ−1)y(ℓ−2) . . . y(1)y(0).
Then, by concatenating all Wcℓ (and by recalling that y = y
(L) and x = y(0)) we obtain
wcL wcL−1 . . . wc2 wc1(.) : 0x 7−→ 0 y y(L−1) . . . y(2) y(1) x.
Let πCf be the position permutation that shifts y right to the positions just right of x:
πCf : 0 y y
(L−1) . . . y(2) y(1) x 7−→ 0 y(L−1) . . . y(2) y(1) x y.
Observe that for (WcL−1 . . . Wc2 Wc1)
−1 we have
(wcL−1 . . . wc2 wc1)
−1(.) : 0 y(L−1) . . . y(2) y(1) x y 7−→ 0x y.
Then we have:
wcL wcL−1 . . . wc2 wc1 πCf (wcL−1 . . . wc2 wc1)
−1(.) : 0x 7−→ 0x y .
By using the position permutation πm,n : 0x y 7−→ 0 y x, we now see how to define Wf :
Wf = πm,n WcL WcL−1 . . . Wc2 Wc1 πCf (WcL−1 . . . Wc2 Wc1)
Then we have:
wf (.) : 0x 7−→ 0 y x,
where y = f(x).
Finally, we need to examine the length of the word Wf in terms of the size of the circuit Cf that
computes f : {0, 1}m → {0, 1}n.
The position permutation πm,n shifts the n = |y| letters of y to the left over the m = |x| positions
of x. So, πm,n can be written as the product of nm transpositions in {τi,i+1 : 1 ≤ i}, with maximum
subscript Jπm,n ≤ m+ n+ 1.
The position permutation πCf shifts y to the right from positions in the interval [2, n + 1] within
the string 0 y y(L−1) . . . y(2) y(1) x to positions in the interval [2 +
i=0 ni, 2 +
i=0 ni] within
the string 0 y(L−1) . . . y(2) y(1) x y. Note that
i=0 ni = |Cf | (the size of the circuit Cf ), and
nL = |y| = n, n0 = |x| = m. We shift y starting with the right-most letters of y. This takes
i=0 ni = n (|Cf |−n) transpositions in {τi,i+1 : 1 ≤ i}, with maximum subscript JπCf = |Cf |+2.
We saw already that |Wcℓ | ≤ c (n2ℓ−1 + n2ℓ), and Jcℓ ≤ nℓ−1 + nℓ + c, for some constant c > 1.
Note that
i=0 n
i ≤ (
i=0 ni)
2 = |Cf |2. Hence we have: |Wf | ≤ co |Cf |2, for some constant
co > 1. Moreover, the largest subscript in any transposition occurring in Wf is JWf ≤ |Cf |+ 2.
Recall that we assumed that our circuit Cf was strictly layered, and that the circuit size has to be
squared (at most) in order to make the circuit strictly layered. Thus, if Cf was originally not strictly
layered, our bounds become |Wf | ≤ co |Cf |4, and JWf ≤ |Cf |2 + 2. ✷
The next theorem gives a representation of a boolean permutation by an element of the Thompson
group G2,1; the main point of the theorem is the polynomial bound on the word-length in terms of
circuit size.
Theorem 4.2 (Representation of permutations by the Thompson group). Let g : {0, 1}m →
{0, 1}m be any permutation and let Cg and Cg−1 be minimum-size circuits that compute g, respectively
g−1. Then there exists a word W(g,g−1) over the generating set ΓG2,1 ∪ {τi,i+1 : 1 ≤ i} of G2,1,
representing an element w(g,g−1) ∈ G2,1 such that:
• For all x ∈ Dom(g) and all y ∈ Im(g):
w(g,g−1)(0x) = 0 g(x), and (w(g,g−1))
−1(0 y) = 0 g−1(y),
where (w(g,g−1))
−1 ∈ G2,1 is represented by the free-group inverse (W(g,g−1))−1 of the word W(g,g−1).
• w(g,g−1)(.) and (w(g,g−1))−1 stabilize both 0 {0, 1}∗ and 1 {0, 1}∗.
• We have a length upper bound |W(g,g−1)| = |(W(g,g−1))−1| ≤ O(|Cg|4 + |Cg−1 |4).
• The largest subscript of transpositions τi,i+1 occurring in W(g,g−1) is ≤ max{|Cg|2, |Cg−1 |2} +2.
Note that we distinguish between the word W(g,g−1) (over a generating set of G2,1) and the element
w(g,g−1) of G2,1 represented by W(g,g−1). Also, note that although g is length-preserving (g ∈ lpG2,1),
w(g,g−1) ∈ G2,1 is not length-preserving.
Proof. Consider the position permutation π : 0 y x 7−→ 0x y, for all x, y ∈ {0, 1}m; we express π
as a composition of ≤ m2 position transpositions of the form τi,i+1. Let Wg be the word constructed
in Theorem 4.1 for g, and let Wg−1 be the word constructed for g
−1. We define W(g,g−1) by
W(g,g−1) = (Wg−1)
−1 π Wg.
Then for all x ∈ Dom(g) we have: w(g,g−1) : 0x 7−→ 0 y, where y = g(x). More precisely, for all
x ∈ domC(g),
wg−−→ 0 g(x) x = 0 y x π−→ 0 x y = 0 g−1(y)
−−−−−−→ 0 y = 0 g(x).
Since domC(g) is a maximal prefix code, w(g,g−1) maps 0 {0, 1}∗ into 0 {0, 1}∗ (where defined).
Similarly, for all y ∈ Im(g) = Dom(g−1) we have: (w(g,g−1))−1 : 0 y 7−→ 0x, where x = g−1(y),
y = g(x). Since domC(g−1) is a maximal prefix code, (w(g,g−1))
−1 maps 0 {0, 1}∗ into 0 {0, 1}∗ (where
defined). Hence, elements of 0 {0, 1}∗ are never images of 1 {0, 1}∗. Thus, 1 {0, 1}∗ is also stabilized
by w(g,g−1) and by (w(g,g−1))
The length of the wordW(g,g−1) is bounded as follows: We have |Wg| ≤ co |Cg|4, and |(Wg−1)−1| =
|Wg−1 | ≤ co |Cg−1 |4, by Theorem 4.1. Moreover, π can be expressed as the composition of ≤ m2
(< |Cg|2) transpositions in {τi,i+1 : 1 ≤ i}.
The bound on the subscripts also follows from Theorem 4.1. ✷
5 Distortion vs. computational asymmetry
We show in this Section that the computational asymmetry function α(.) is polynomially related to a
certain distortion of the group lpG2,1.
By Theorem 4.2, for every element g ∈ lpG2,1 there is an element w(g,g−1) ∈ G2,1 which agrees with
g on 0 {0, 1}∗, and which stabilizes 0 {0, 1}∗ and 1 {0, 1}∗ . The main property of W(g,g−1) is that its
length is polynomially bounded by the circuit sizes of g and g−1; that fact will be crucial later. First
we want to study how w(g,g−1) is related to g. Recall that we distinguish between the word W(g,g−1)
(over a generating set of G2,1) and the element w(g,g−1) of G2,1 represented by W(g,g−1).
Theorem 4.2 inspires the following concepts.
Definition 5.1 Let G be a subgroup of G2,1. For any prefix codes P1, . . . , Pk ⊂ {0, 1}∗, the joint
stabilizer (in G) of the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ is defined by
StabG(P1, . . . , Pk) =
g ∈ G : g(Pi{0, 1}∗) ⊆ Pi{0, 1}∗ for every i = 1, . . . , k
The fixator (in G) of P1{0, 1}∗ is defined by
FixG(P1) =
g ∈ G : g(x) = x for all x ∈ P1{0, 1}∗)
The fixator is also called “point-wise stabilizer”.
The following is an easy consequence of the definition: FixG(Pi) is a subgroup of G (⊆ G2,1), for
i = 1, . . . , k. If the prefix codes P1, . . . , Pk are such that the right ideals P1{0, 1}∗, . . . , Pk{0, 1}∗ are
two-by-two disjoint, and such that P1 ∪ . . . ∪ Pk is a maximal prefix code, then StabG(P1, . . . , Pk) is
closed under inverse. Hence in this case StabG(P1, . . . , Pk) is a subgroup of G.
In particular, we will consider the following groups:
• The joint stabilizer of 0 {0, 1}∗ and 1 {0, 1}∗,
StabG(0, 1) =
g ∈ G : g(0 {0, 1}∗) ⊆ 0 {0, 1}∗ and g(1 {0, 1}∗) ⊆ 1 {0, 1}∗
• The fixator of 0 {0, 1}∗,
FixG(0) = {g ∈ G : g(x) = x for all x ∈ 0 {0, 1}∗}.
• The fixator of 1 {0, 1}∗,
FixG(1) = {g ∈ G : g(x) = x for all x ∈ 1 {0, 1}∗}.
Clearly, FixG(0) and FixG(1) are subgroups of StabG(0, 1).
Lemma 5.2 (Self-embeddings of G2,1). Let G be a subgroup of G2,1. Then G is isomorphic to
FixG(1) and to FixG(0) by the following isomorphisms:
Λ0 : g ∈ G 7−→ (g)0 ∈ FixG(1)
Λ1 : g ∈ G 7−→ (g)1 ∈ FixG(0)
where (g)0 and (g)1 defined as follows for any g ∈ G2,1:
(g)0 :
0x ∈ 0 {0, 1}∗ 7−→ 0 g(x)
1x ∈ 1 {0, 1}∗ 7−→ 1x (g)1 :
1x ∈ 1 {0, 1}∗ 7−→ 1 g(x)
0x ∈ 0 {0, 1}∗ 7−→ 0x
Proof. It is straightforward to verify that Λ0 and Λ1 are injective homomorphisms. That Λ0 is onto
FixG(1) can be seen from the fact that every element of FixG(1) has a table of the form
0x1 . . . 0xn 1
0y1 . . . 0yn 1
where {x1, . . . , xn} and {y1, . . . , yn} are two maximal prefix codes, and
x1 . . . xn
y1 . . . yn
is an arbitrary
element of G. ✷
Lemma 5.3 Let G be a subgroup of G2,1. Then the direct product G×G is isomorphic to StabG(0, 1)
by the isomorphism
Λ : (f, g) ∈ G×G 7−→
0x 7→ 0 f(x), 1x 7→ 1 g(x)
∈ StabG(0, 1).
Proof. It is straightforward to verify that Λ is a homomorphism. That Λ is onto StabG(0, 1) and
injective follows from the fact that every element of StabG(0, 1) has a table of the form
0x1 . . . 0xm 1x
1 . . . 1x
0y1 . . . 0ym 1y
1 . . . 1y
where {x1, . . . , xm}, {y1, . . . , ym}, {x′1, . . . , x′n}, and {y′1, . . . , y′n}, are maximal prefix codes, and
x1 . . . xm
y1 . . . ym
x′1 . . . x
y′1 . . . y
are arbitrary elements of G (⊆ G2,1). ✷
Lemmas 5.2 and 5.3 reveal certain self-similarity properties of the Thompson group G2,1. (Self-
similarity of groups with total action on an infinite tree is an important subject, see [27]. However,
the action of G2,1 is partial, so much of the known theory does not apply directly.)
The stabilizer and the fixators above have some interesting properties.
Lemma 5.4 .
(1) For all f, g ∈ G: (f)0 (g)1 = (g)1 (f)0
(i.e., the commutator of FixG(0) and FixG(1) is the identity).
(2) FixG(0) · FixG(1) = StabG(0, 1) and FixG(0) ∩ FixG(1) = 1;
(3) StabG(0, 1) is the internal direct product of FixG(0) and FixG(1).
(This is equivalent to the combination of (1) and (2).)
(4) For all f, g ∈ G: Λ(f, g) = Λ0(f) · Λ1(g), Λ0(f) = Λ(f,1), and Λ1(g) = Λ(1, g).
Moreover, FixG(0) = Λ1(G), FixG(1) = Λ0(G), and StabG(0, 1) = Λ(G×G).
Proof. The proof is a straightforward verification. ✷
Lemma 5.5 For every position transposition τi,j, with 1 ≤ i < j, we have
(τi,j)0 = τ2,i+1 ◦ τ3,j+1◦ (τ1,2)0 ◦ τ3,j+1 ◦ τ2,i+1.
Hence, assuming (τ1,2)0 ∈ ΓG2,1 , and abbreviating {τi,j : 0 < i < j} by τ , we have:
|(τi,j)0|ΓG2,1∪τ ≤ 5.
Proof. Recall that for (τ1,2)0 we have, by definition, (τ1,2)0(1w) = 1w, and (τ1,2)0(0x2x3w) =
0x3x2w, for all w ∈ {0, 1}∗ and x2, x3 ∈ {0, 1}. The proof of the Lemma is a straightforward
verification. ✷
Now we arrive at the relation between w(g,g−1) and g.
Lemma 5.6 For all g ∈ lpG2,1 the following relation holds between g and w(g,g−1) :
w(g,g−1) · (g)−10 , (g)
0 · w(g,g−1) ∈ FixlpG2,1(0).
Equivalently,
(g)0 · FixlpG2,1(0) = w(g,g−1) · FixlpG2,1(0), and
FixlpG2,1(0) · (g)0 = FixlpG2,1(0) · w(g,g−1) .
Proof. By Theorem 4.2 we have w(g,g−1)(0x) = 0 g(x) for all x ∈ Dom(g). So, w(g,g−1) and (g)0
act in the same way on 0 {0, 1}∗ . Also, both w(g,g−1) and (g)0 map 0 {0, 1}∗ into 0 {0, 1}∗, and both
map 1 {0, 1}∗ into 1 {0, 1}∗. The Lemma follows from this. ✷
We abbreviate {τi,j : 0 < i < j} by τ . The element w(g,g−1) of G2,1, represented by the word
W(g,g−1), belongs to StablpG2,1(0, 1) as we saw in Theorem 4.2. However, the word W(g,g−1) itself is a
sequence over the generating set ΓG2,1 ∪ τ of G2,1. Therefore, in order to follow the action of W(g,g−1)
and of its prefixes we need to take Fix(0) as a subgroup of G2,1. This leads us to the Schreier left
coset graph of FixG2,1(0) within G2,1, over the generating set ΓG2,1 ∪ τ . By definition this Schreier
graph has vertex set G2,1/FixG2,1(0), i.e., the left cosets, of the form g · FixG2,1(0) with g ∈ G2,1. And
it has directed edges of the form g ·FixG2,1(0)
γ−→ γg ·FixG2,1(0) for g ∈ G2,1, γ ∈ ΓG2,1 ∪ τ . Lemma
5.6 implies that for all g ∈ lpG2,1,
(g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0).
We assume that ΓG2,1 = Γ
, so the Schreier graph is symmetric, and hence it has a distance function
based on path length; we denote this distance by
dG/F (., .) : G2,1/FixG2,1(0)×G2,1/FixG2,1(0) −→ N.
Lemma 5.7 There are injective morphisms
g ∈ lpG2,1 →֒ g ∈ G2,1
≃−→ (g)0 ∈ FixG2,1(1)
≃−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0),
and an inclusion map
(g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) →֒ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0).
In particular,
g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ G2,1/FixG2,1(0)
is an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier graph.
Proof. Recall that the map Λ0 : g ∈ G2,1 7−→ (g)0 ∈ FixG2,1(1) is a bijective morphism (Lemma
5.2). Also, the map u ∈ FixG2,1(1) 7−→ u · FixG2,1(0) ∈ G2,1/FixG2,1(0) is injective; indeed, if
u · FixG2,1(0) = v · FixG2,1(0) with u, v ∈ FixG2,1(1) then v−1u ∈ FixG2,1(0) ∩ FixG2,1(1) = {1}.
The map g ∈ G2,1 7−→ (g)0 · FixG2,1(0) ∈ StabG2,1(0, 1)/FixG2,1(0) is a surjective group
homomorphism since FixG2,1(0) is a normal subgroup of StabG2,1(0, 1). Since FixG2,1(0)∩FixG(1) = {1},
this homomorphism is injective from FixG2,1(1) onto StabG2,1(0, 1)/FixG2,1(0).
The combination of these maps provides an isomorphism from G2,1 onto StabG2,1(0, 1)/FixG2,1(0).
Hence we also have an embedding of G2,1, as a set, into the vertex set G2,1/FixG2,1(0) of the Schreier
graph. ✷
Since by Lemma 5.7 we can consider G2,1 as a subset of the vertex set G2,1/FixG2,1(0) of the
Schreier graph, the path-distance dG/F (., .) on G2,1/FixG2,1(0) leads to a distance on G2,1, inherited
from dG/F (., .) :
Definition 5.8 For all g, g′ ∈ G2,1 the Schreier graph distance inherited by G2,1 is
D(g, g′) = dG/F
(g)0 · FixG2,1(0), (g′)0 · FixG2,1(0)
The comparison of the Schreier graph distance D(., .) on lpG2,1 with the word-length that lpG2,1
inherits from its embedding into lepM2,1 leads to the following distortion of lpG2,1:
Definition 5.9 In lpG2,1 we consider the distortion
∆(n) = max{D(1, g) : |g|lepM2,1 ≤ n, g ∈ lpG2,1}.
We now state and prove the main theorem relating ∆(.) and α. Recall that α(.) is the computational
asymmetry function of boolean permutations, defined in terms of circuit size.
Theorem 5.10 (Computational asymmetry vs. distortion). The computational asymmetry
function α(.) and the distortion ∆(.) of lpG2,1 are polynomially related. More precisely, for all n ∈ N :
)1/2 ≤ c′ ·∆(n) ≤ c n4 + c ·
α(c n)
where c ≥ c′ ≥ 1 are constants.
Proof. The Theorem follows immediately from Lemmas 5.11 and 5.12. ✷
Lemma 5.11. There is a constant c ≥ 1 such that for all n ∈ N : ∆(n) ≤ c n4 + c ·
α(c n)
Proof. By Lemma 5.6, (g)0 · FixG2,1(0) = w(g,g−1) · FixG2,1(0), hence
FixG2,1(0), (g)0 · FixG2,1(0)
FixG2,1(0), w(g,g−1) · FixG2,1(0)
Since the wordW(g,g−1) and the Schreier graph use the same generating set, namely ΓG2,1 ∪ τ , we have
FixG2,1(0), w(g,g−1) · FixG2,1(0)
≤ |W(g,g−1)|.
By Theorem 4.2, |W(g,g−1)| ≤ O(|Cg|4 + |Cg−1 |4). And by the definition of the computational
asymmetry function, |Cg−1 | ≤ α(|Cg |). Hence
FixG2,1(0), (g)0 · FixG2,1(0)
≤ O(|Cg|4 + |Cg−1 |4) ≤ O
|Cg|4 + α(|Cg|)4
By Proposition 2.4, |Cg| = O(|g|lepM2,1). Hence, for some constants c′′, c′ ≥ 1,
FixG2,1(0), (g)0 · FixG2,1(0)
≤ c′ · |g|4
lepM2,1
+ c′ · α(c′′ · |g|lepM2,1)4.
Thus,
FixG2,1(0), (g)0 · FixG2,1(0)
: |g|lepM2,1 ≤ n, g ∈ lpG2,1
≤ c′ n4 + c′ α(c′′ n)4.
By Definition 5.9 of the distortion function ∆ we have therefore
∆(n) ≤ c′ n4 + c′ α(c′′ n)4.
This proves the Lemma. ✷
Lemma 5.12 There is a constant c ≥ 1 such that for all n ∈ N : α(n) ≤ c ·∆(c n)2.
Proof. We first prove the following.
Claim: For every g ∈ lpG2,1, the inverse permutation g−1 can be computed by a circuit Cg−1 of size
|Cg−1 | ≤ c · d
FixG2,1(0), (g)0 · FixG2,1(0)
, for some constant c ≥ 1.
Proof of the Claim: There is a wordW ′ of length |W ′| = d
FixG2,1(0), (g)0 ·FixG2,1(0)
over ΓG2,1∪ τ
that labels a shortest path from FixG2,1(0) to (g)0·FixG2,1(0) in the Schreier graph ofG2,1/FixG2,1(0). Let
W = (W ′)−1 (the free-group inverse of W ′), so |W | = |W ′|. Let w be the element of G2,1 represented
by W . Then W labels a shortest path from FixG2,1(0) to (g
−1)0 · FixG2,1(0) in the Schreier graph of
G2,1/FixG2,1(0); this path has length |W | = |W ′| = d
FixG2,1(0), (g)0 · FixG2,1(0)
FixG2,1(0),
(g−1)0 · FixG2,1(0)
We have w · FixG2,1(0) = (g−1)0 · FixG2,1(0), thus for all x ∈ {0, 1}∗ : w(0x) = 0 g−1(x). We now
take the word VWU over the generating set ΓM2,1 ∪ τ of the monoid M2,1, where we choose the words
U and V to be U = (and, not, fork, fork), and V = (or). The functions and, not, fork, or were defined
in Subsection 1.1. Then for all x = x1 . . . xn ∈ {0, 1}∗, with x1, . . . , xn ∈ {0, 1}, we have
x1 . . . xn
fork−→ x1 x1 . . . xn
fork−→ not−→ x1 x1 x1 . . . xn
and−→ 0x1 . . . xn = 0x
W−→ 0 g−1(x) or−→ g−1(x).
The last or combines 0 and the first bit of g−1(x), and this makes 0 disappear. Thus overall,
VWU(x) = g−1(x). The length is |V WU | = |W |+ 5.
Since g−1 ∈ lpG2,1 ⊂ lepM2,1, Theorem 2.9 implies that there exists a word Z over the generators
ΓlepM2,1 ∪ τ of lepM2,1 such that
(1) |Z| ≤ c1 · |VWU |2, for some constant c1 ≥ 1, and
(2) Z represents the same element of lepM2,1 as VWU , namely g
Moreover, by Prop. 2.4, the word Z can be transformed into a circuit of size ≤ c2 · |Z| (for some
constant c2 ≥ 1). This proves that there is a circuit Cg−1 for g−1 of size |Cg−1 | ≤ c · |W |2 (for
some constant c ≥ 1). Since we saw that |W | = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)), the Claim follows.
[End, Proof of the Claim.]
By definition, D(1, g) = dG/F (FixG2,1(0), (g)0 · FixG2,1(0)). Hence, by the Claim above:
|Cg−1 | ≤ c ·
D(1, g)
By Prop. 2.4 the word-length in lepM2,1 and the circuit size are linearly related; hence |g|lepM2,1 ≤
c0 |Cg|, for some constant c0 ≥ 1. Therefore,
α(n) = max{|Cg−1 | : |Cg| ≤ n, g ∈ lpG2,1}
≤ max{|Cg−1 | : |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1}
≤ max
D(1, g)
: |g|lepM2,1 ≤ c0 n, g ∈ lpG2,1
≤ c ·
∆(c0 n)
This proves the Lemma. ✷
6 Other bounds and distortions
6.1 Other distortions in the Thompson groups and monoids
The next proposition gives more upper bounds on the computational asymmetry function α.
Proposition 6.1. Assume ΓlepG2,1 ⊂ ΓlepM2,1 ⊂ ΓM2,1 . Let δlpG,lepM = δ
|.|ΓlpG2,1∪τ , |.|ΓlepM2,1∪τ
be the distortion function of lpG2,1 in the Thompson monoid lepM2,1, based on word-length. Similarly,
let δlpG,M = δ
|.|ΓlpG2,1∪τ , |.|ΓM2,1∪τ
be the distortion function of lpG2,1 in the Thompson monoid
M2,1. Then for some constant c ≥ 1 and for all n ∈ N,
α(n) ≤ c · δlpG,lepM (c n) ≤ c · δlpG,M (c n).
Proof. We first prove that δlpG,lepM (n) ≤ δlpG,M (n). Recall that by definition, δlpG,lepM (n) =
max{|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n}, and similarly for δlpG,M(n). Since ΓlepM2,1 ⊂ ΓM2,1 we
have |x|lepM2,1 ≤ |x|M2,1 . Hence, {|g|lpG2,1 : g ∈ lpG2,1, |g|lepM2,1 ≤ n} ⊆ {|g|lpG2,1 : g ∈ lpG2,1,
|g|M2,1 ≤ n}. By taking max over each of these two sets it follows that δlpG,lepM (n) ≤ δlpG,M(n).
Next we prove that α(n) ≤ c·δlpG,lepM (c n). For any g ∈ lpG2,1 we have C(g−1) ≤ O(|g−1|lepM2,1),
by Prop. 3.2. Moreover, |g−1|lepM2,1 ≤ |g−1|lpG2,1 since lpG2,1 is a subgroup of lepM2,1, and since the
generating set used for lpG2,1 (including all τi,j) is a subset of the generating set used for lepM2,1. For
any group with generating set closed under inverse we have |g−1|G = |g|G. And by the definition of
the distortion δlpG,lepM we have |g|lpG2,1 ≤ δlpG,lepM (|g|lepM2,1). And again, by Prop. 3.2, |g|lepM2,1 ≤
O(C(g)). Putting all this together we have
C(g−1) ≤ c1 · |g−1|lepM2,1 ≤ c1 · |g−1|lpG2,1 = c1 · |g|lpG2,1
≤ c1 · δlpG,lepM (|g|lepM2,1) ≤ c1 · δlpG,lepM (c2 C(g)).
Thus, c1 ·δlpG,lepM (c2 C(g)) is an upper bound on C(g−1). Since, by definition, α(C(g)) is the smallest
upper bound on C(g−1), it follows that α(C(g)) ≤ c1 · δlpG,lepM (c2 C(g)). ✷
Recall that in the definition 5.9 of the distortion ∆ we compared D(., .) with the word-length in
lepM2,1. If, instead, we compare D(., .) with the word-length inM2,1 we obtain the following distortion
of lpG2,1 :
δ(n) = max{D(1, g) : |g|M2,1 ≤ n, g ∈ lpG2,1}.
Proposition 6.2 The distortion functions ∆(.) and δ(.) are polynomially related. More precisely,
there are constants c′, c1, c2 ≥ 1 such that for all n ∈ N: ∆(n) ≤ c1 δ(n) ≤ c2 ∆(c′ n2).
Proof. Let’s assume first that ΓlepM2,1 ⊆ ΓM2,1 , from which it follows that |g|M2,1 ≤ |g|M2,1 . Therefore,
{D(1, g) : |g|lepM2,1 ≤ n} ⊆ {D(1, g) : |g|M2,1 ≤ n}. Hence, ∆(n) ≤ δ(n).
By Theorem 2.9, |g|lepM2,1 ≤ c · |g|2M2,1 . So, {D(1, g) : |g|M2,1 ≤ n} ⊆ {D(1, g) : |g|lepM2,1 ≤ c n
Hence, δ(n) ≤ ∆(c n2).
When we do not have ΓlepM2,1 ⊆ ΓM2,1 , the constants in the theorem change, but the statement
remains the same. ✷
6.2 Monotone boolean functions and distortion
On {0, 1}∗ we can define the product order, also called “bit-wise order”. It is a partial order (and in
fact, a lattice order), denoted by “�”, and defined as follows. First, 0 ≺ 1; next, for any u, v ∈ {0, 1}∗
we have u � v iff |u| = |v| and ui � vi for all i = 1, . . . , |u|, where ui (or vi) denotes the ith bit of u
(respectively v).
By definition, a partial function f : {0, 1}∗ → {0, 1}∗ is monotone (also called “product-order
preserving”) iff for all u, v ∈ Dom(f) : u � v implies f(u) � f(v).
The following fact is well known (see e.g., [43] Section 4.5): A function f : {0, 1}m → {0, 1}n is
monotone iff f can be computed by a combinational circuit that only uses gates of type and, or, fork,
and wire-swappings; i.e., not is absent. A circuit of this restricted type is called a monotone circuit.
Razborov [30] proved super-polynomial lower bounds for the size of monotone circuits that solve the
clique problem, and in [31] he proved super-polynomial lower bounds for the size of monotone circuits
that solve the perfect matching problem for bipartite graphs; the latter problem is in P. Tardos [37],
based on work by Alon and Boppana [1], gave an exponential lower bound for the size of monotone
circuits that solve a problem in P; see also [42] (Chapter 14 by Boppana and Sipser). Thus, there exist
problems that can be solved by polynomial-size circuits but for which monotone circuits must have
exponential size. In particular (for some constants b > 1, c > 0), there are infinitely many monotone
functions fn : {0, 1}n → {0, 1}n such that fn has a combinational circuit of size ≤ nc, but fn has no
monotone circuit of size ≤ bn.
Based on an alphabet A = {a1, . . . , ak} with a1 ≺ a2 ≺ . . . ≺ ak we define a partial function
f : A∗ → A∗ to be monotone iff f preserves the product order of A∗. The monotone functions enable
us to define the following submonoid of the Thompson-Higman monoid lepMk,1 :
monMk,1 = {ϕ ∈ lepMk,1 : ϕ can be represented by a monotone function P → Q,
where P and Q are prefix codes, with P maximal }.
An essential extension or restriction of an element of monMk,1 is again in monMk,1, so this set is
well-defined as a subset of lepMk,1. It is easily seen to be closed under composition, so monMk,1 is a
submonoid of lepMk,1.
We saw that all monotone finite functions have circuits made from gates of type and, or, fork.
Hence monM2,1 has the following generating set:
{and, or, fork} ∪ {τi,j : j > i ≥ 1}.
The results about monotone circuit size imply the following distortion result. Again, “exponential”
refers to a function with a lower bound of the form n ∈ N 7−→ exp( c
c′ n), for some constants c′ > 0
and c ≥ 1.
Proposition 6.3 Consider the monoid monM2,1 over the generating set {and, or, fork} ∪ {τi,j : j >
i ≥ 1}, and the monoid lepM2,1 over the generating set ΓlepM2,1 ∪ {τi,j : j > i ≥ 1}, where ΓlepM2,1 is
finite. Then monM2,1 has exponential word-length distortion in lepM2,1.
Proof. Let Γmon = {and, or, fork}. By Prop. 2.4 we have |f |ΓlepM2,1∪τ = |Cf |, where |Cf | denotes the
ordinary circuit size of f . By a similar argument we obtain: |f |Γmon∪τ = |monCf |, where |monCf |
denotes the monotone circuit size of f . We saw that as a consequence of the work of Razborov, Alon,
Boppana, and Tardos, there exists an infinite set of monotone functions that have polynomial-size
circuits but whose monotone circuit-size is exponential. The exponential distortion follows. ✷
Since lepM2,1 has quadratic distortion in M2,1, monM2,1 also has exponential word-length distor-
tion in M2,1.
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Nauk SSSR 281(4) (1985) 798-801. (English transl.: Soviet Mathematical Doklady 31 (1985) 354-357.)
http://www.iacr.org/publications/dl/
[31] A.A. Razborov, “Lower bounds of monotone complexity of the logical permanent function”, Matematich-
eskie Zametki 37(6) (1985) 887-900. (English transl.: Mathematical Notes of the Academy of Sciences of
the USSR 37 (1985) 485-493.)
[32] J.E. Savage, Models of Computation, Addison-Wesley (1998).
[33] E.A. Scott, “A construction which can be used to produce finitely presented infinite simple groups”, J. of
Algebra 90 (1984) 294-322.
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203-221.
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59-98.
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Computer-Aided Design of Integrated Circuits and Systems 22(6) (2003) 710-722.
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torica 7(4) (1987) 141-142.
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Jean-Camille Birget
Dept. of Computer Science
Rutgers University at Camden
Camden, NJ 08102, USA
[email protected]
Introduction
One-way functions and one-way permutations
Computational Asymmetry
Wordlength asymmetry
Computational asymmetry and reversible computing
Distortion
Thompson-Higman groups and monoids
Boolean functions as elements of Thompson monoids
Wordlength asymmetry vs. computational asymmetry
Reversible representation over the Thompson groups
Distortion vs. computational asymmetry
Other bounds and distortions
Other distortions in the Thompson groups and monoids
Monotone boolean functions and distortion
|
0704.1570 | A method for the direct determination of the surface gravities of
transiting extrasolar planets | Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 29 October 2018 (MN LATEX style file v2.2)
A method for the direct determination of the surface
gravities of transiting extrasolar planets
John Southworth⋆, Peter J Wheatley and Giles Sams
Department of Physics, University of Warwick, Coventry, CV4 7AL, UK
29 October 2018
ABSTRACT
We show that the surface gravity of a transiting extrasolar planet can be calculated
from only the spectroscopic orbit of its parent star and the analysis of its transit
light curve. This does not require additional constraints, such as are often inferred
from theoretical stellar models or model atmospheres. The planet’s surface gravity
can therefore be measured precisely and from only directly observable quantities. We
outline the method and apply it to the case of the first known transiting extrasolar
planet, HD 209458b. We find a surface gravity of gp = 9.28± 0.15m s
−2, which is an
order of magnitude more precise than the best available measurements of its mass,
radius and density. This confirms that the planet has a much lower surface gravity
that that predicted by published theoretical models of gas giant planets. We apply
our method to all fourteen known transiting extrasolar planets and find a significant
correlation between surface gravity and orbital period, which is related to the known
correlation between mass and period. This correlation may be the underlying effect
as surface gravity is a fundamental parameter in the evaporation of planetary atmo-
spheres.
Key words: stars: planetary systems — stars: individual: HD 209458 — stars: bina-
ries: eclipsing — stars: binaries: spectroscopic — methods: data analysis
1 INTRODUCTION
Since the discovery that the star HD209458 is eclipsed
by a planet in a short-period orbit (Henry et al. 2000;
Charbonneau et al. 2000) it has become possible to derive
the basic astrophysical properties of extrasolar planets and
compare these quantities with theoretical predictions (e.g.
Baraffe et al. 2003). However, the absolute masses, radii and
density of the transiting planet cannot be calculated directly
from the transit light curve and the velocity variation of
the parent star, so extra information is required in order
to obtain them. Additional constraints can be found from
spectral analysis of the parent star or by imposing a theo-
retical stellar mass–radius relation (Cody & Sasselov 2002),
but this causes a dependence on theoretical stellar models or
model atmospheres. The uncertainties in these constraints
dominate the overall errors in mass, radius and density (e.g.
Konacki et al. 2004), limiting the accuracy with which prop-
erties of the star and planet can be measured.
In this work we show that the surface gravity of the
planet can be measured directly using only the transit light
curve and the radial velocity amplitude of the parent star.
No additional information is required and so accurate and
⋆ E-mail: [email protected]
precise surface gravity values can be obtained. As theoretical
studies often supply predicted values for the surface gravities
of planetary objects (e.g., Baraffe et al. 2003) we propose
that this quantity is very well suited for comparing observa-
tion with theory. In addition, the surface gravity is an im-
portant parameter in constructing theoretical models of the
atmospheres of planets (Marley et al. 1999; Hubbard et al.
2001).
After deriving an equation for surface gravity in terms
of directly observed parameters, we illustrate this concept by
studying HD209458. We then apply it to the other known
transiting extradsolar planets using results available in the
literature. The planet HD209458 b is known to be over-
sized for its mass and to be strongly irradiated by the star
it orbits. An excellent transit light curve was obtained for
HD209458 by Brown et al. (2001), who used the HST/STIS
spectrograph to obtain high-precision photometry cover-
ing several different transit events. Precise radial velocity
studies of HD209458 are also available (Henry et al. 2000;
Mazeh et al. 2000; Naef et al. 2004).
c© 0000 RAS
http://arxiv.org/abs/0704.1570v1
2 Southworth, Wheatley & Sams
Table 1. Results of the modelling of the HST/STIS light curve of HD209458. The upper part of the table gives the optimised parameters
and the lower part gives quantities calculated from these parameters. The midpoint of the transit, TMin I, is expressed in HJD − 2 400 000.
Limb darkening Linear Quadratic Square-root Adopted parameters
r⋆+rp 0.12889± 0.00042 0.12771± 0.00049 0.12799± 0.00050
k 0.12260± 0.00011 0.12097± 0.00025 0.12051± 0.00037
i (deg.) 86.472± 0.040 86.665± 0.054 86.689± 0.060 86.677± 0.060
T0 51659.936716± 0.000021 51659.936712± 0.000021 51659.936711± 0.000021
u⋆ 0.494± 0.004 0.297± 0.027 −0.312± 0.127
v⋆ 0.338± 0.047 1.356± 0.218
r⋆ 0.11481± 0.00036 0.11393± 0.00042 0.11418± 0.00042 0.11405± 0.00042
rp 0.01408± 0.00006 0.01378± 0.00007 0.01376± 0.00008 0.01377± 0.00008
1.146 1.056 1.054
2 SURFACE GRAVITY MEASUREMENT
The fractional radii of the star and the planet in the system
are defined as
where a is the orbital semi-major axis, and R⋆ and Rp are
the absolute radii of the star and planet, respectively. r⋆ and
rp can be directly determined from a transit light curve.
The mass function of a spectroscopic binary is given by
(e.g. Hilditch 2001):
f(Mp) =
(1− e2)
p sin
(M⋆ +Mp)2
whereK⋆ is the velocity amplitude of the star, e is the orbital
eccentricity, P is the orbital period, i is the orbital inclina-
tion, and M⋆ and Mp are the masses of the star and planet
respectively. Including Kepler’s Third Law and solving for
the sum of the masses of the two components gives:
(M⋆ +Mp)
2πGM 3p sin
(1− e2)
2K 3⋆ P
(2π)4a6
G2P 4
By substituting Rp = arp into the definition of surface grav-
ity and replacing a using Eq. 3 we find that the surface grav-
ity of the planet, gp is given by:
(1− e2)
r 2p sin i
Eq. 4 shows that we are able to calculate the surface
gravity of a transiting extrasolar planet from the quantities
P , K⋆, e, i and rp. This can be understood intuitively be-
cause both the radial velocity motion of the star and the
planet’s surface gravity are manifestations of the accelera-
tion due to the gravity of the planet. A discussion in the
context of eclipsing binaries is given by Southworth et al.
(2004b).
To measure gp using Eq. 4, the orbital period, P , can be
obtained from either radial velocities or light curves of the
system, and is typically determined very precisely compared
to the other measurable quantities. The radial velocities also
give e and K⋆, whilst the quantities i and rp can be obtained
directly from the transit light curve. Note that it is also pos-
sible to constrain the orbital eccentricity from observations
of the secondary eclipse of a system.
3 APPLICATION TO HD209458B
In order to measure the surface gravity for HD209458 b we
need to know rp and i. These quantities are standard param-
eters in the analysis of transit light curves. We have chosen
to obtain them by modelling the high-precision HST/STIS
light curve presented by Brown et al. (2001). We followed
Brown et al. by rejecting data from the first HST orbit of
each observed transit.
To model the photometric data we used the jktebop
code1 (Southworth et al. 2004a), which is a modified version
of the ebop program (Popper & Etzel 1981; Etzel 1981).
Giménez (2006) has shown that ebop is very well suited
to the analysis of the light curves of transiting extrasolar
planets. Importantly for this application, jktebop has been
extended to treat limb darkening (LD) using several non-
linear LD laws (Southworth et al. 2007). It also includes
Monte Carlo and bootstrapping simulation algorithms for
error analysis (Southworth et al. 2004a,b). ebop and jkte-
bop model the two components of an eclipsing system using
biaxial ellipsoids (Nelson & Davis 1972; Etzel 1975), so al-
low for the deformation of the bodies from a spherical shape.
When modelling the data we adopted the precise orbital
period of 3.52474859 days given by Knutson et al. (2007).
We fitted for the sum of the fractional radii, r⋆+rp, the
ratio of the radii, k =
, the orbital inclination,
and the midpoint of a transit, T0. We also fitted for the LD
coefficients of the star, rather than fixing them at values
calculated using model atmospheres, to avoid introducing a
dependence on theoretical predictions. The linear and non-
linear LD coefficients are denoted by u⋆ and v⋆, respectively.
We assumed that the planet contributes no light at the
optical wavelengths considered here (see Wittenmyer et al.
2005) and that the orbit is circular (see Laughlin et al. 2005;
Deming et al. 2005; Winn et al. 2005). Given suggestions
that the choice of LD law can influence the derived inclina-
tion (Winn et al. 2005), we obtained solutions for the linear,
quadratic and square-root laws (Southworth et al. 2007). A
mass ratio of 0.00056 was used (Knutson et al. 2007) but
large changes in this parameter have a negligible effect on
the solution.
We have calculated robust 1 σ error estimates us-
jktebop is written in fortran77 and the source code is avail-
able at http://www.astro.keele.ac.uk/∼jkt/
c© 0000 RAS, MNRAS 000, 000–000
http://www.astro.keele.ac.uk/~jkt/
Direct determination of surface gravities of exoplanets 3
Figure 1. Best fit to the HST transit light curve of HD209458
using the quadratic LD law. The residuals of the fit are shown
offset downwards by 0.02 in flux for clarity.
ing Monte Carlo simulations (see Press et al. 1992, p.684;
Southworth et al. 2004a), which we have previously found
to provide very reliable results (Southworth et al. 2005a,b).
This procedure assumes that systematic errors are negligi-
ble. We find no reason to suspect that significant systematic
errors remain in the HST light curve after the processing of
this data described by Brown et al. (2001) (see Fig. 1).
The best-fitting parameters of the transit light curve are
given in Table 1. The best-fitting model using the quadratic
LD law is shown in Fig. 1 along with the residuals of that fit.
The solution using linear LD can be rejected as its reduced
2 is substantially larger than for the other two solutions.
The quadratic and square-root LD laws give very similar
solutions with reduced χ2 values close to one. For our final
results we adopt the mean for each parameter along with
uncertainties from the Monte Carlo simulations (Table 1).
These are in good agreement with the light curve solutions
obtained by Giménez (2006) and Mandel & Agol (2002),
both of which used the approximation that the planet is
spherical.
With the orbital period given by Knutson et al. (2007),
the stellar velocity ampitude K⋆ = 85.1 ± 1.0m s
−2 from
Naef et al. (2004), and the results of our light curve analysis
(Table 1) we find the surface gravity of HD209458 b to be
gp = 9.28±0.15m s
−2. In this case the total uncertainty in gp
is due to almost equal contributions from the uncertainties
in K⋆ and rp.
4 APPLICATION TO ALL KNOWN
TRANSITING EXTRASOLAR PLANETS
We have calculated the surface gravity values for each of the
known transiting extrasolar planets (apart from HD209458),
using data taken from the literature (Table 2). In several
cases (marked with asterisks in Table 2) it was not possible
Table 2. Surface gravity values for the known transiting extra-
solar planets. These have been calculated using Eq. 4 with input
parameters taken from the literature.
System Surface gravity Literature references
m s−2 rp and i K⋆
HD 189733 21.5± 3.5 1 2
HD 209458 9.28± 0.15 3 4
OGLE-TR-10 4.5± 2.1 5 6
OGLE-TR-56 17.9± 1.9 5 2
OGLE-TR-111 13.3± 4.2 7 8
TrES-1 16.1± 1.0 9 10
WASP-1 10.6± 1.7 11 12
∗ HAT-P-1 7.1± 1.1 13 13
∗ XO-1 13.3± 2.5 14 14
∗ HD 149026 16.4± 2.5 15 15
∗ OGLE-TR-113 28.3± 4.4 16 17
∗ OGLE-TR-132 18.0± 6.0 18 17
∗ TrES-2 20.7± 2.6 19 19
∗ WASP-2 20.1± 2.7 20 12
∗ The surface gravity values for these objects have larger error
estimates than are needed, because their fractional radii are not
available in the literature. In these cases we have had to calculate
them from Rp and a, which are less certain than rp because of
the need to adopt an additional constraints to calculate them
(see text).
References: (1) Winn et al. (2007c); (2) Bouchy et al. (2005);
(3) This work; (4) Naef et al. (2004); (5) Pont et al. (2007); (6)
Konacki et al. (2005); (7) Winn et al. (2007a); (8) Pont et al.
(2004); (9) Winn et al. (2007b); (10) Alonso et al. (2004);
(11) Shporer et al. (2007); (12) Cameron et al. (2007); (13)
Bakos et al. (2007); (14) McCullough et al. (2006); (15)
Sato et al. (2005); (16) Gillon et al. (2006); (17) Bouchy et al.
(2004); (18) Gillon et al. (2007); (19) O’Donovan et al. (2006);
(20) Charbonneau et al. (2007).
to obtain rp directly from the results available in the litera-
ture. In these cases it had to be calculated from Rp and a,
resulting in an increased uncertainty. This is because rp is
a parameter obtainable directly from a transit light curve,
whereas additional constraints (for example using theoreti-
cal stellar models) are needed to calculate a and Rp.
The orbital periods and surface gravities of all fourteen
transiting extrasolar planets are plotted in Fig. 2, and show
that these quantities are correlated. The linear Pearson cor-
relation coefficient of these data is r = −0.70, indicating
that the correlation is significant at better than the 0.5%
level. This correlation is certainly related to that found by
Mazeh et al. (2005) between the orbital periods and masses
of the six transiting extrasolar planets then known. How-
ever, it may be that surface gravity, rather than mass or
radius, is the main parameter correlated with orbital period
for these objects. Theoretical calculations have shown that
surface gravity is a fundamental parameter in the evapora-
tion rates of the atmospheres of irradiated gas giant planets
(Lammer et al. 2003).
5 SUMMARY AND DISCUSSION
We have shown that the surface gravity of transiting extraso-
lar planets can be measured from analysis of the light curve
c© 0000 RAS, MNRAS 000, 000–000
4 Southworth, Wheatley & Sams
Figure 2. Comparison between the surface gravities and orbital
periods of the known transiting exoplanets. Filled and open circles
denote the systems in the upper and lower halves of Table 2, re-
spectively. The errorbars for HD209458 (P = 3.52 d) are smaller
than the plotted symbol.
and a spectroscopic orbit of the parent star. We have anal-
ysed the HST/STIS light curve of HD209458 (Brown et al.
2001) with the jktebop code. By combining the results
of the light curve analysis with published spectroscopy
(Naef et al. 2004) we find that the planet has a surface grav-
ity of gp = 9.28 ± 0.15m s
−2. We stress that this measure-
ment does not depend on theoretical stellar evolutionary
models or model atmospheres.
In Fig. 3 we have plotted theoretical isochrones for ages
of 0.5 to 10 Gyr from Baraffe et al. (2003) against the mass
and surface gravity of HD209458 b, adopting a mass of
Mp = 0.64±0.06 MJup from Knutson et al. (2007). The dis-
crepancy between the observed and predicted surface gravity
can clearly be seen. Note that these models do not include
the effects of irradiation from the star.
The density of a transiting extrasolar planet is of-
ten used to compare observation with theory, but is typi-
cally measured with a much lower precision than its sur-
face gravity, given the same dataset. For example, the den-
sity of HD209458 b derived by Knutson et al. (2007) is
345±50 kgm−3. Using the mass and radius given by Knutson
et al. leads to gp = 9.1±0.9m s
−2, where the uncertainty has
been calculated by simple error propagation. These quanti-
ties are both much less precise and require more elaborate
calculations than using Eq. 4 to find the surface gravity:
gp = 9.28± 0.15m s
We have applied Eq. 4 to each of the known transiting
extrasolar planets (Table 2). The resulting surface gravities
show a clear correlation with orbital period (Fig. 2) which is
connected with the known correlation between orbital period
and mass for these objects. We propose that surface gravity
may be the underlying parameter of the correlation due to
its influence on the evaporation rates of the atmospheres of
short-period giant planets.
Figure 3. Plot of mass versus surface gravity for HD209458 b
compared to the theoretical model predictions of Baraffe et al.
(2003) for ages 0.5, 1.0, 5.0 and 10.0 Gyr (from lower to higher
log g).
As gp can be very precisely measured, and can be di-
rectly compared with theoretical models and used to con-
struct model atmospheres of the planet, we propose that it
is an important parameter in our understanding of short-
period extrasolar giant planets. In the near future, the high-
precision light curves obtained by the CoRoT and Kepler
satellites will allow accurate surface gravity values to be ob-
tained for the terretrial-mass transiting extrasolar planets
which these satellites should find.
6 ACKNOWLEDGEMENTS
We are grateful to David Charbonneau for making the
HST/STIS light curve of HD209458 available on his website
(http://cfa-www.harvard.edu/∼dcharbon/frames.html),
to Pierre Maxted for discussions, and to the referee whose
report contributed significantly to improvements in this
work.
JS acknowledges financial support from PPARC in the
form of a postdoctoral research assistant position. The fol-
lowing internet-based resources were used in research for
this paper: the NASA Astrophysics Data System; the SIM-
BAD database operated at CDS, Strasbourg, France; and
the arχiv scientific paper preprint service operated by Cor-
nell University.
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Introduction
Surface gravity measurement
Application to HD209458b
Application to all known transiting extrasolar planets
Summary and discussion
Acknowledgements
|
0704.1571 | On restrictions of balanced 2-interval graphs | On restrictions of balanced 2-interval graphs
On restri
tions of balan
ed 2-interval graphs
Philippe Gambette
LIAFA, UMR CNRS 7089, Université Paris 7, Fran
e
Département Informatique, ENS Ca
han, Fran
e
gambette�liafa.jussieu.fr
Stéphane Vialette
LRI, UMR CNRS 8623, Université Paris-Sud 11, Fran
e
vialette�lri.fr
Abstra
t
The
lass of 2-interval graphs has been introdu
ed for modelling s
heduling and allo
ation
problems, and more re
ently for spe
i�
bioinformati
problems. Some of those appli
ations
imply restri
tions on the 2-interval graphs, and justify the introdu
tion of a hierar
hy of
sub
lasses of 2-interval graphs that generalize line graphs: balan
ed 2-interval graphs, unit
2-interval graphs, and (x,x)-interval graphs. We provide instan
es that show that all the
in
lusions are stri
t. We extend the NP-
ompleteness proof of re
ognizing 2-interval graphs
to the re
ognition of balan
ed 2-interval graphs. Finally we give hints on the
omplexity
of unit 2-interval graphs re
ognition, by studying relationships with other graph
lasses:
proper
ir
ular-ar
, quasi-line graphs, K1,5-free graphs, . . .
Keywords: 2-interval graphs, graph
lasses, line graphs, quasi-line graphs,
law-free
graphs,
ir
ular interval graphs, proper
ir
ular-ar
graphs, bioinformati
s, s
heduling.
1 2-interval graphs and restri
tions
The interval number of a graph, and the
lasses of k-interval graphs have been introdu
ed as a
generalization of the
lass of interval graphs by M
Guigan [M
G77℄ in the
ontext of s
heduling
and allo
ation problems. Re
ently, bioinformati
s problems have renewed interest in the
lass
of 2-interval graphs (ea
h vertex is asso
iated to a pair of disjoint intervals and edges denote
interse
tion between two su
h pairs). Indeed, a pair of intervals
an model two asso
iated
tasks in s
heduling [BYHN
06℄, but also two similar segments of DNA in the
ontext of DNA
omparison [JMT92℄, or two
omplementary segments of RNA for RNA se
ondary stru
ture
predi
tion and
omparison [Via04℄.
(a) (b) (
)
Figure 1: Heli
es in a RNA se
ondary stru
ture (a)
an be modeled as a set of balan
ed 2-
intervals among all 2-intervals
orresponding to
omplementary and inverted pairs of letter
sequen
es (b), or as an independent subset in the balan
ed asso
iated 2-interval graph (
).
http://arxiv.org/abs/0704.1571v2
RNA (ribonu
lei
a
id) are polymers of nu
leotides linked in a
hain through phosphodiester
bonds. Unlike DNA, RNAs are usually single stranded, but many RNAmole
ules have se
ondary
stru
ture in whi
h intramole
ular loops are formed by
omplementary base pairing. RNA se
-
ondary stru
ture is generally divided into heli
es (
ontiguous base pairs), and various kinds of
loops (unpaired nu
leotides surrounded by heli
es). The stru
tural stability and fun
tion of
non-
oding RNA (n
RNA) genes are largely determined by the formation of stable se
ondary
stru
tures through
omplementary bases, and hen
e n
RNA genes a
ross di�erent spe
ies are
most similar in the pattern of nu
leotide
omplementarity rather than in the genomi
sequen
e.
This motivates the use of 2-intervals for modelling RNA se
ondary stru
tures: ea
h helix of
the stru
ture is modeled by a 2-interval. Moreover, the fa
t that these 2-intervals are usually
required to be disjoint in the stru
ture naturally suggests the use of 2-interval graphs. Fur-
thermore, aiming at better modelling RNA se
ondary stru
tures, it was suggested in [CHLV05℄
to fo
us on balan
ed 2-interval sets (ea
h 2-interval is
omposed of two equal length intervals)
and their asso
iated interse
tion graphs referred as balan
ed 2-interval graphs. Indeed, heli
es
in RNA se
ondary stru
tures are most of the time
omposed of equal length
ontiguous base
pairs parts. To the best of our knowledge, nothing is known on the
lass of balan
ed 2-interval
graphs.
Sharper restri
tions have also been introdu
ed in s
heduling, where it is possible to
on-
sider tasks whi
h all have the same duration, that is 2-interval whose intervals have the same
length [BYHN
06, Kar05℄. This motivates the study of the
lasses of unit 2-interval graphs,
and (x, x)-interval graphs. In this paper, we
onsider these sub
lasses of interval graphs, and in
parti
ular we address the problem of re
ognizing them.
A graph G = (V,E) of order n is a 2-interval graph if it is the interse
tion graph of a set of
n unions of two disjoint intervals on the real line, that is ea
h vertex
orresponds to a union of
two disjoint intervals Ik = Ik
∪ Ikr , k ∈ J1, nK (l for � left� and r for �right�), and there is an edge
between Ij and Ik i� Ij ∩ Ik 6= ∅. Note that for the sake of simpli
ity we use the same letter to
denote a vertex and its
orresponding 2-interval. A set of 2-intervals
orresponding to a graph
G is
alled a realization of G. The set of all intervals,
k=1{I
, Ikr }, is
alled the ground set of
G (or the ground set of {I1, . . . , In}).
The
lass of 2-interval graphs is a generalization of interval graphs, and also
ontains all
ir
ular-ar
graphs (interse
tion graphs of ar
s of a
ir
le), outerplanar graphs (have a planar
embedding with all verti
es around one of the fa
es [KW99℄),
ubi
graphs (maximum degree
3 [GW80℄), and line graphs (interse
tion graphs of edges of a graph).
Unfortunately, most
lassi
al graph
ombinatorial problems turn out to be NP-
omplete
for 2-interval graphs: re
ognition [WS84℄, maximum independent set [BNR96, Via01℄,
ol-
oration [Via01℄, . . . Surprisingly enough, the
omplexity of the maximum
lique problem for
2-interval graphs is still open (although it has been re
ently proven to be NP-
omplete for
3-interval graphs [BHLR07℄).
For pra
ti
al appli
ation, restri
ted 2-interval graphs are needed. A 2-interval graph is said
to be balan
ed if it has a 2-interval realization in whi
h ea
h 2-interval is
omposed of two
intervals of the same length [CHLV05℄, unit if it has a 2-interval realization in whi
h all intervals
of the ground set have length 1 [BFV04℄, and is
alled a (x, x)-interval graph if it has a 2-interval
realization in whi
h all intervals of the ground set are open, have integer endpoints, and length
x [BYHN+06, Kar05℄. In the following se
tions, we will study those restri
tions of 2-interval
graphs, and their position in the hierar
hy of graph
lasses illustrated in Figure 2.
Note that all (x, x)-interval graphs are unit 2-interval graphs, and that all unit 2-interval
graphs are balan
ed 2-interval graphs. We
an also noti
e that (1, 1)-interval graphs are exa
tly
line graphs: ea
h interval of length 1 of the ground set
an be
onsidered as the vertex of a
root graph and ea
h 2-interval as an edge in the root graph. This implies for example that the
Figure 2: Graph
lasses related to 2-interval graphs and its restri
tions. A
lass pointing towards
another stri
tly
ontains it, and the dashed lines mean that there is no in
lusion relationship be-
tween the two. Dark
lasses
orrespond to
lasses not yet present in the ISGCI Database [BLS
oloration problem is also NP-
omplete for (2, 2)-interval graphs and wider
lasses of graphs. It
is also known that the
omplexity of the maximum independent set problem is NP-
omplete on
(2, 2)-interval graphs [BNR96℄. Re
ognition of (1, 2)-union graphs, a related
lass (restri
tion of
multitra
k interval graphs), was also re
ently proven NP-
omplete [HK06℄.
2 Useful gadgets for 2-interval graphs and restri
tions
For proving hardness of re
ognizing 2-interval graphs, West and Shmoys
onsidered in [WS84℄
the
omplete bipartite graph K5,3 as a useful 2-interval gadget. Indeed, all realizations of this
graph are
ontiguous, that is, for any realization, the union of all intervals in its ground set is an
interval. Thus, by putting edges between some verti
es of a K5,3 and another vertex v, we
an
for
e one interval of the 2-interval v (or just one extremity of this interval) to be blo
ked inside
the realization of K5,3. It is not di�
ult to see that K5,3 has a balan
ed 2-interval realization,
for example the one in Figure 3.
(a) (b) (
)
Figure 3: The
omplete bipartite graph K5,3 (a,b) has a balan
ed 2-interval realization (
):
verti
es of S5 are asso
iated to balan
ed 2-intervals of length 7, and verti
es of S3 are asso
iated
to balan
ed 2-intervals of length 11. Any realization of this graph is
ontiguous, i.e., the union
of all 2-intervals is an interval.
However, K5,3 is not a unit 2-interval graph. Indeed, ea
h 2-interval I = Il∪Ir
orresponding
to a degree 5 vertex interse
t 5 disjoint 2-intervals, and hen
e one of Il or Ir interse
t at least 3
intervals, whi
h is impossible for unit intervals. Therefore, we introdu
e the new gadget K4,4−e
whi
h is a (2, 2)-interval graph with only
ontiguous realizations.
(a) (b) (
)
Figure 4: The graph K4,4 − e (a), a ni
er representation (b), and a 2-interval realization with
open intervals of length 2 (
).
Property 1. Any 2-interval realization of K4,4 − e is
ontiguous.
Proof. Write G = (V,E) the graph K4,4− e. To study all possible realizations of G, let us study
all possible realizations of G[V − I8].
As 2-intervals I1, I2, I3 and I4 are disjoints, their ground set I
= {[li, ri], 1 ≤ i ≤ 8,
ri < li+1} is a set of eight disjoint intervals. The ground set Imobile of I
, I6 and I7 is a set of
six disjoint intervals. Let xi be the number of intervals of Imobile interse
ting i ≤ 8 intervals of
. We have dire
tly:
x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 = |Imobile| = 6. (1)
As there are 12 edges in G[V \{v8}] whi
h is bipartite, we also have:
x1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 ≥ 12. (2)
Finally, to build a realization of G from a realization of G[V \{v8}] , one must pla
e I
so as
to interse
t three disjoint intervals of I
. Thus one of the intervals of I8 interse
ts at least
two intervals ]lk, rk[ and ]ll, rl[ (k < l) of I�xed. So there is �a hole between those two intervals�,
for example [rk, lk+1], whi
h is in
luded in one of the intervals of I
. So we noti
e that I8 has
to �ll one of the seven holes of I
. Thus, the intervals of I
mobile
an not �ll more than six
holes, and the observation that an interval interse
ting i
onse
utive intervals (for i ≥ 1) �lls
i− 1 holes, we get:
x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 6. (3)
Equations 1, 2 and 3 are ne
essary for any valid realization of G[V \{v8}] whi
h gives a valid
realization of G.
Let's suppose by
ontradi
tion that the union of all intervals of the ground set of G is
not an interval. Then there is a hole, that is an interval in
luded in the
overing interval of
{I1, . . . , I8}, whi
h interse
t no Ii. We pro
eed like for equation 3, with the
onstraint that
another hole
annot be �lled by the intervals of I
mobile
, so we get instead:
x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 7x8 ≤ 5. (4)
By adding 1 and 4, and subtra
ting 2, we get x0 ≤ −1 : impossible! So we have proved that
the union of all intervals of the ground set of any realization of G is indeed an interval.
3 Balan
ed 2-interval graphs
We show in this se
tion that the
lass of balan
ed 2-interval graphs is stri
tly in
luded in the
lass of 2-interval graphs, and stri
tly
ontains
ir
ular-ar
graphs. Moreover, we prove that
re
ognizing balan
ed 2-interval graphs is as hard as re
ognizing (general) 2-interval graphs.
Property 2. The
lass of balan
ed 2-interval graphs is stri
tly in
luded in the
lass of 2-interval
graphs.
Proof. We build a 2-interval graph that has no balan
ed 2-interval realization. Let's
onsider
a
hain of gadgets K5,3 (introdu
ed in previous se
tion) to whi
h we add three verti
es I
, I2,
and I3 as illustrated in Figure 5.
Figure 5: An example of unbalan
ed 2-interval graph (a) : any realization groups intervals of
the seven K5,3 in a blo
k, and the
hain of seven blo
ks
reates six �holes� between them, whi
h
make it impossible to balan
e the lengths of the three 2-intervals I1, I2, and I3.
In any realization, the presen
e of holes showed by
rosses in the Figure gives the following
inequalities for any realization: l(Il
2) < l(Il
1), l(Il
3) < l(Ir
2), and l(Ir
1) < l(Ir
3) (or if the
realization of the
hain of K5,3 appears in the symmetri
al order: l(Il
1) < l(Il
3), l(Ir
3) < l(Il
and l(Ir
2) < l(Ir
1)). If this realization was balan
ed, then we would have l(Il
1) = l(Ir
3) = l(Il
3) < l(Ir
2) = l(Il
2) (or for the symmetri
al
ase: l(Ir
1) = l(Il
1) < l(Il
3) = l(Ir
2) = l(Ir
2)) : impossible! So this graph has no balan
ed 2-interval realization although it
has a 2-interval generalization.
Theorem 1. Re
ognizing balan
ed 2-interval graphs is an NP-
omplete problem.
Proof. We just adapt the proof of West and Shmoys [WS84, GW95℄. The problem of determining
if there is a Hamiltonian
y
le in a 3-regular triangle-free graph is proven NP-
omplete, by
redu
tion from the more general problem without the no triangle restri
tion. So we redu
e the
problem of Hamiltonian
y
le in a 3-regular triangle-free graph to balan
ed 2-interval re
ognition.
Let G = (V,E) be a 3-regular triangle-free graph. We build a graph G′ whi
h has a 2-
interval realization (a spe
ial one, very spe
i�
,
alled H-representation and whi
h we prove to
be balan
ed) i� G has a Hamiltonian
y
le.
The
onstru
tion of G′, illustrated in Figure 6(a) is almost identi
al to the one by West
and Shmoys, so we just prove that G′ has a balan
ed realization, shown in Figure 6 (b), by
omputing lengths for ea
h interval to ensure it. All K5,3 have a balan
ed realization as shown
Figure 6: There is a balan
ed 2-interval of G′ (whi
h has been dilated in the drawing to remain
readable) i� there is an H-representation (that is a realization where the left intervals of all
2-intervals are grouped together in a
ontiguous blo
k) for its indu
ed subgraph G i� there is a
Hamiltonian
y
le in G.
in se
tion 1 of total length 79, in parti
ular H3. We
an thus a�e
t length 83 to the intervals of
v0. The intervals of the other vi
an have length 3, and their M(vi) length 79, so through the
omputation illustrated in Figure 6, intervals of z
an have length 80 + 82 + 2(n − 1) + 3, that
is 163 + 2n. We dilate H1 until a hole between two
onse
utive intervals of its S3
an
ontain
an interval of z, that is until the hole has length 165 + 2n : so after this dilating, H1 has length
79(165 + 2n). Finally if G has a Hamiltonian
y
le, then we have found a balan
ed 2-interval
realization of G of total length 13, 273 + 241n.
It is known that
ir
ular-ar
graphs are 2-interval graphs, they are also balan
ed 2-interval.
Property 3. The
lass of
ir
ular-ar
graphs is stri
tly in
luded in the
lass of balan
ed 2-
interval graphs.
Proof. The transformation is simple: if we have a
ir
ular-ar
representation of a graph G =
(V,E), then we
hoose some point P of the
ir
le. We partition V in V1∪V2, where P interse
ts
all the ar
s
orresponding to verti
es of V1 and none of the ar
s of the verti
es of V2. Then
we
ut the
ir
le at point P to map it to a line segment: every ar
of V2 be
omes an interval,
and every ar
of V1 be
omes a 2-interval. To obtain a balan
ed realization we just
ut in half
the intervals of V2 to obtain two intervals of equal length for ea
h. And for ea
h 2-interval
[g(Il), d(Il)] ∪ [g(Ir), d(Ir)] of V1, as both intervals are lo
ated on one of the extremities of the
realization, we
an in
rease the length of the shortest so that it rea
hes the length of the longest
without
hanging interse
tions with the other intervals. The in
lusion is stri
t be
ause K2,3 is a
balan
ed 2-interval graph (as a subgraph of K5,3 for example) but is not a
ir
ular-ar
graph (we
an �nd two C4 in K2,3, and only one
an be realized with a
ir
ular-ar
representation).
4 Unit 2-interval and (x,x)-interval graphs
Property 4. Let x ∈ N, x ≥ 2. The
lass of (x, x)-interval graphs is stri
tly in
luded in the
lass of (x+ 1, x+ 1)-interval graphs.
Proof. We �rst prove that an interval graph with a representation where all intervals have length
k (and integer open bounds) has a representation where all intervals have length k + 1.
We use the following algorithm. Let S be initialized as the set of all intervals of length k,
and let T be initially the empty set. As long as S is not empty, let I = [a, b] be the left-most
interval of S, remove from S ea
h interval [α, β] su
h that α < b (in
luding I), add [α, β + 1]
to T , and translate by +1 all the remaining intervals in S. When S is empty, the interse
tion
graph of T , where all intervals have length k + 1 is the same as the interse
tion graph for the
original S.
We also build for ea
h x ≥ 2 a (x + 1, x + 1)-interval graph whi
h is not a (x, x)-interval
graph. We
onsider the bipartite graph K2x and a perfe
t mat
hing {(vi, v
i), i ∈ J1, xK}. We
all
K ′x the graph obtained from K2x with the following transformations, illustrated in Figure 7(a):
remove edges (vi, v
i) of the perfe
t mat
hing, add four graphs K4,4−e
alled X1, X2, X3, X4 (for
ea
h Xi, we
all v
and vir the verti
es of degree 3), link v
r and v
, link all vi to v
r and v
, link
all v′i to v
and v3r , and �nally add a vertex a (resp. b) linked to all vi, v
i, and to two adja
ent
verti
es of X1 (resp. X4) of degree 4. We illustrate in Figure 7(b) that K
x has a realization
with intervals of length x+ 1. We
an prove by indu
tion on x that K ′x has no realization with
intervals of length x: it is rather te
hni
al, so we just give the idea. Any realization of K ′x for
es
the blo
k X2 to share an extremity with the blo
k X3, so ea
h 2-interval v
i has one interval
interse
ting the other extremity of X2, and the other interse
ting the other extremity of X3.
Then
onstraints on the position of verti
es vi for
e their intervals to appear as two �stairways�
as shown in Figure 7(b). So v1r must
ontain x extremities of intervals whi
h have to be di�erent,
so it must have length x+ 1.
Figure 7: The graph K ′4 (a) is (5,5)-interval but not (4,4)-interval.
The
omplexity of re
ognizing unit 2-interval graphs and (x, x)-interval graphs remains open,
however the following shows a relationship between those
omplexities.
Lemma 1. {unit 2-interval graphs} =
{(x, x)-interval graphs}.
Proof. The ⊃ part is trivial. To prove ⊂, let G = (V,E) be a unit 2-interval graph. Then it has
a realization with |V | = n 2-intervals, that is 2n intervals of the ground set. So we
onsider the
interval graph of the ground set, whi
h is a unit interval graph. There is a linear time algorithm
based on breadth-�rst sear
h to
ompute a realization of su
h a graph where interval endpoints
are rational, with denominator 2n [CKN+95℄. So by dilating by a fa
tor 2n su
h a realization,
we obtain a realization of G where intervals of the ground set have length 2n.
Theorem 2. If re
ognizing (x, x)-interval graphs is polynomial for any integer x then re
ognizing
unit 2-interval graphs is polynomial.
5 Investigating the
omplexity of unit 2-interval graphs
In this se
tion we show that all proper
ir
ular-ar
graphs (
ir
ular-ar
graphs su
h that no ar
is in
luded in another in the representation) are unit 2-interval graphs, and we study a
lass of
graphs whi
h generalizes quasi-line graphs and
ontains unit 2-interval graphs.
Re
all that, a
ording to Property 3,
ir
ular-ar
graphs are balan
ed 2-interval graphs.
However,
ir
ular-ar
graphs are not ne
essarily unit 2-interval graphs.
Property 5. The
lass of proper
ir
ular-ar
graphs is stri
tly in
luded in the
lass of unit
2-interval graphs.
Proof. As in the proof of Property 3, we
hoose a point P on the
ir
le of the representation of
a proper
ir
ular-ar
graph G, and maps the
ut
ir
le into a line segment. We extend the outer
extremities of intervals that have been
ut so that no interval
ontains another. Thus we obtain
a set of 2-intervals for ar
s
ontaining P , and a set I of intervals for ar
s not
ontaining P . For
ea
h interval of I, we add a new interval disjoint of any other to get a 2-interval. If we
onsider
the interse
tion graph of the ground set of su
h a representation, it is a proper interval graph.
So it is also a unit interval graph [Rob69℄, whi
h provides a unit 2-interval representation of G.
To
omplete the proof, we noti
e that the domino (two
y
les C4 having an edge in
ommon)
is a unit 2-interval graph but not a
ir
ular-ar
graph.
Quasi-line graphs are those graphs whose verti
es are bisimpli
ial, i.e., the
losed neighbor-
hood of ea
h vertex is the union of two
liques. This graph
lass has been introdu
ed as a gener-
alization of line graphs and a useful sub
lass of
law-free graphs [Ben81, FFR97, CS05, KR07℄.
Following the example of quasi-line graphs that generalize line graphs, we introdu
e here a new
lass of graphs for generalizing unit 2-interval graphs. Let k ∈ N∗. A graph G = (V,E) is
all-k-simpli
ial if the neighborhood of ea
h vertex v ∈ V
an be partitioned into at most k
liques. The
lass of quasi-line graphs is thus exa
tly the
lass of all-2-simpli
ial graphs. Noti
e
that this de�nition is equivalent to the following: in the
omplement graph of G, for ea
h vertex
u, the verti
es that are not in the neighborhood of u are k-
olorable.
Property 6. The
lass of unit 2-interval graphs is stri
tly in
luded in the
lass of all-4-simpli
ial
graphs.
Proof. The in
lusion is trivial. What is left is to show that the in
lusion is stri
t. Consider the
following graph whi
h is all-4-simpli
ial but not unit 2-interval: start with the
y
le C4,
all
its verti
es vi, i ∈ J1, 4K, add four K4,4 − e gadgets
alled Xi, and for ea
h i we
onne
t the
vertex vi to two
onne
ted verti
es of degree 4 in Xi. This graph is
ertainly all-4-simpli
ial.
But if we try to build a 2-interval realization of this graph, then ea
h of the 2-intervals vk has
an interval trapped into the blo
k Xk. So ea
h 2-interval vk has only one interval to realize
the interse
tions with the other vi: this is impossible as we have to realize a C4 whi
h has no
interval representation.
Property 7. The
lass of
law-free graphs is not in
luded in the
lass of all-4-simpli
ial graphs.
Proof. The Kneser Graph KG(7, 2) is triangle-free, but not 4-
olorable [Lov78℄. We
onsider
the graph obtained by adding an isolated vertex v and then taking the
omplement graph,
i.e., KG(7, 2) ⊎ {v}. It is
law-free as KG(7, 2) is triangle-free. And if it was all-4-simpli
ial,
then the neighborhood of v in KG(7, 2) ⊎ {v}, that is KG(7, 2), would be a union of at most
four
liques, so KG(7, 2) would be 4-
olorable: impossible so this graph is
law-free but not
all-4-simpli
ial.
Property 8. The
lass of all-k-simpli
ial graphs is stri
tly in
luded in the
lass of K1,k+1-free
graphs.
Proof. If a graph G
ontains K1,k+1, then it has a vertex with k + 1 independent neighbors,
and hen
e G is not all-k-simpli
ial. The wheel W2k+1 is a simple example of a graph whi
h is
K1,k+1-free but in whi
h the
enter
an not have its neighborhood (a C2k+1) partitioned into k
liques or less.
Unfortunately, all-k-simpli
ial graphs do not have a ni
e stru
ture whi
h
ould help unit
2-interval graph re
ognition.
Theorem 3. Re
ognizing all-k-simpli
ial graphs is NP-
omplete for k ≥ 3.
Proof. We redu
e from the Graph k-
olorability problem, whi
h is known to be NP-
omplete for k ≥ 3 [Kar72℄. Let G = (V,E) be a graph, and let G′ be the
omplement graph of
G to whi
h we add a universal vertex v. We
laim that G is k-
olorable i� G′ is all-k-simpli
ial.
If G is k-
olorable, then the non-neighborhood of any vertex in G is k-
olorable, so the
neighborhood of any vertex in G is a union of at most k
liques. And the neighborhood of v is
also a union of at most k
liques, so G′ is all-k-simpli
ial.
Conversely, if G′ is all-k-simpli
ial, then in parti
ular the neighborhood of v is a union of
at most k
liques. Let's partition it into k vertex-disjoint
liques X1, . . . ,Xk. Then,
oloring G
su
h that two verti
es have the same
olor i� they are in the same Xi leads to a valid k-
oloring
of G.
6 Con
lusion
Motivated by pra
ti
al appli
ations in s
heduling and
omputational biology, we fo
used in this
paper on balan
ed 2-interval graphs and unit 2-intervals graphs. Also, we introdu
ed two natural
new
lasses: (x, x)-interval graphs and all-k-simpli
ial graphs.
We mention here some dire
tions for future works. First, the
omplexity of re
ognizing unit
2-interval graphs and (x, x)-interval graphs remains open. Se
ond, the relationships between
quasi-line graphs and sub
lasses of balan
ed 2-intervals graphs still have to be investigated.
Last, sin
e most problems remains NP-hard for balan
ed 2-interval graphs, there is thus a natural
interest in investigating the
omplexity and approximation of
lassi
al optimization problems on
unit 2-interval graphs and (x, x)-interval graphs.
A
knowledgments
We are grateful to Vin
ent Limouzy in parti
ular for bringing to our attention the
lass of
quasi-line graphs, and Mi
hael Rao and Mi
hel Habib for useful dis
ussions.
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7 Appendix
We give the detailed proofs of Theorem 1 and Property 4.
Proof of Theorem 1. Let G = (V,E) be a 3-regular triangle-free graph. We build a graph G′
whi
h has a 2-interval realization (a spe
ial one, very spe
i�
, and whi
h we prove to be balan
ed)
i� G has a Hamiltonian
y
le.
First we will detail how we build G′ starting from the graph G, and adding some verti
es,
in parti
ular K5,3 gadgets. The idea is that the edges of G will partition into a Hamiltonian
y
le and a perfe
t mat
hing i� all 2-intervals of the realization of G′
an have their left inter-
val realizing the Hamiltonian
y
le, and their right interval realizing the perfe
t mat
hing. A
realization with su
h a pla
ement of the intervals is
alled an �H-representation� of G.
We pro
eed as illustrated in Figure 6. We
hoose some vertex of G that we
all v0 (whi
h will
be the �origin� of the Hamiltonian
y
le), and the other are
alled v1, . . . , vn. For ea
h vertex
vi of G we link it to a vertex of the S5 of a K5,3
alled M(vi) (whi
h will blo
k one of the four
extremities of the 2-interval vi). We link all verti
es to a new vertex z, whi
h is linked to no
M(v) ex
ept M(v0) (thus the interval of ea
h vi interse
ting M(vi), for i 6= 0, won't interse
t
z). We add three K5,3, H1, H2 and H3 : two verti
es of the S5 of H1 are linked to z, a third one
is linked to one vertex of the S5 of H2, one vertex of the S5 of H3 is linked to z, and all verti
es
of H3 to v0.
To explain this
onstru
tion in detail, we study the realization of G′, if we suppose it is a
(balan
ed) 2-interval graph, and we prove that it leads us to �nd a Hamiltonian
y
le in G.
As the realization of H1 and H2 are two
ontiguous blo
ks of intervals then one of their
extremities must interse
t. As z is linked to two disjoint verti
es of H1, both intervals of z are
used to realize those interse
tions. But one interval of z that we
all zr, also has to interse
t
one vertex of H3 whi
h is not linked to H1, so zr interse
ts the se
ond extremity of the blo
k
H1 (the �rst extremity being o
upied by the extremity of H2). And as zr interse
ts only one
interval of H3, it must be the extremity of H3. The other interval of z is
ontained in the blo
k
H1, thus
an't interse
t M(v0) neither all the verti
es vi, so all those 2-intervals interse
t zr.
And as none of them interse
t H3 ex
ept v0, all of them ex
ept v0 have an interval
ontained
in zr, that we
all vi,g. The other interval of ea
h vi is linked to a K5,3 so it has one extremity
o
upied by K5,3, and the other one is free.
Conversely, if G has a Hamiltonian
y
le, then it is possible to �nd a H-representation,
su
h that all the
onstraints indu
ed by the edges of G′ are respe
ted, as illustrated with the
realization in Figure 6. We have already proved that this realization
an be balan
ed.
Proof of Property 4. In the following, as we only
onsidering the interval of vi
or vir lo
ated at
one extremity of the blo
k Xi, and not the one inside, we will use v
and vir to denote those
extremity intervals. For ea
h vertex vi, we
all vi,l its left interval and vi,r its right interval. We
do the same for v′i, and
all l(I) the left extremity of any interval I.
We prove by indu
tion that the graph K ′x is (x + 1, x + 1)-interval but not (x, x)-interval,
and that for any unit 2-interval realization, there exists an order σ ∈ Sx su
h that :
• either l(vσ(x),l) < . . . < l(vσ(1),l) < l(v
σ(x),l
) < . . . < l(v′
σ(1),l
) and l(v′
σ(x),r
) < . . . <
σ(1),r
) < l(vσ(x),r) < . . . < l(vσ(1),r),
• or the symmetri
ase: l(vσ(1),l) < . . . < l(vσ(x),l) < l(v
σ(1),l
) < . . . < l(v′
σ(x),l
) and
σ(1),r
) < . . . < l(v′
σ(x),r
) < l(vσ(1),r) < . . . < l(vσ(x),r).
Those two equalities
orrespond in fa
t to the �two stairways stru
ture� whi
h appears in Fig-
ure 7.
Base
ase : we study all possible unit 2-interval realizations of K ′2 to prove that one of the
expe
ted inequalities is always true. We also prove that K ′2 has no (2,2)-interval realization.
First re
all that realizations of Xi subgraphs
an only be blo
ks of
ontiguous intervals. The
edge between v2r and v
for
es the two blo
ks of X2 and X3 to be
ontiguous, with intervals
and v3r at their extremities. Ea
h 2-interval v
i must interse
t both v
and v3r , so one of its
intervals interse
ts v2
and the other interse
ts v3r . Thus, one same interval of v
i
an not interse
t
both a and b whi
h are disjoint, so a interse
ts one interval of v′i (say the one interse
ting v
, the
other
ase being treated symmetri
ally) and b interse
ts the other one (so, the one interse
ting
v3r ). Ea
h vi has to interse
t both a and b, so it has to interse
t a with its �rst interval and
b with the se
ond. But 2-interval vi must also interse
t v
r and v
whi
h are both disjoint and
disjoint to a and b. So one interval of ea
h vi must interse
t v
r and the other one must interse
t
So we have shown that any unit 2-interval realization of K ′2 has the following aspe
t (or the
symmetri
) : the extremity of the blo
k X1 interse
ting all vi whi
h interse
t a (or b) whi
h
interse
ts all v′i, whi
h interse
t the extremity X2 (or X3) whi
h interse
ts the extremity of X3
(or X2), whi
h interse
ts all v
i, whi
h interse
t b (or a), whi
h interse
ts all vi, whi
h interse
t
the extremity of X4.
Now we suppose, by
ontradi
tion, that there exists a (2,2)-interval realization of K ′2. v
an interval of length 2, but one of its two parts of length one has to interse
t an element of X1.
The other has to interse
t both v1 and v2. As neither v1 nor v2
an interse
t other intervals of
X1, then the �rst interval of v1 and v2 is the same interval. By pro
eeding the same way on X4
and v4
, we obtain that the se
ond interval of v1 and v2 is the same interval, so v1 and v2 should
orrespond to the same 2-interval: it
ontradi
ts with the fa
t that verti
es v1 and v2 have a
di�erent neighborhood. So K ′2 has no (2,2)-interval realization.
To obtain the expe
ted inequalities, we have to analyze the possible positions of all vi and
v′i. We only treat the �rst two inequalities as the se
ond
ase is symmetri
.
Suppose that l(v2,l) < l(v1,l). As v1 and v
1 are non adja
ent, then interval v1,l is stri
tly on
the left of v′1,l, so v2,l is stri
tly on the left of v
1,l. Thus those two intervals do not interse
t. But
v2 and v
1 are adja
ent, so v2 and v
1 must have interse
ting right intervals. But then we have
l(v′2,r) < l(v
1,r) < l(v2,r) < l(v1,r), and the right intervals of v
2 and v1
an not interse
t. We
dedu
e their left intervals interse
t, so l(v2,l) < l(v1,l) < l(v
2,l) < l(v
1,l).
If we suppose that l(v1,l) < l(v2,l), we get as well that l(v
1,r) < l(v
2,r) < l(v1,r) < l(v2,r) and
l(v1,l) < l(v2,l) < l(v
1,l) < l(v
2,l). So for any unit 2-interval realization of K
2 there exists an
order σ = 12 or σ = 21 su
h that:
• either l(vσ(2),l) < l(vσ(1),l) < l(v
σ(2),l
) < l(v′
σ(1),l
) and l(v′
σ(2),r
) < l(v′
σ(1),r
) < l(vσ(2),r) <
l(vσ(1),r),
• or the symmetri
inequalities.
Re
ursion: suppose that for some x, K ′x−1 is not (x−1, x−1)-interval but is (x, x)-interval,
and that any (x, x)-interval realization veri�es one of the expe
ted inequalities.
Graph K ′x−1 is an indu
e subgraph of K
x = (V,E) : K
x−1 = K
x[V \ {vx, v
x}]. So by the
indu
tion hypothesis, there exists an order σ ∈ Sx−1 su
h that for any unit 2-interval realization
of K ′x :
• either l(vσ(x−1),l) < . . . < l(vσ(1),l) < l(v
σ(x−1),l
) < . . . < l(v′
σ(1),l
) and l(v′
σ(x−1),r
) < . . . <
σ(1),r
) < l(vσ(x−1),r) < . . . < l(vσ(1),r),
• or the symmetri
ase: l(vσ(1),l) < . . . < l(vσ(x−1),l) < l(v
σ(1),l
) < . . . < l(v′
σ(x−1),l
) and
σ(1),r
) < . . . < l(v′
σ(x−1),r
) < l(vσ(1),r) < . . . < l(vσ(x−1),r).
The position of vx and v
x remains to be determined. We treat only the
ase where the �rst
two inequalities are true, as the se
ond
ase is symmetri
.
As vx and v
r are adja
ent, and v
σ(x−1)
and v1r are not, then l(v
r ) < l(vx,l) < l(v
σ(x−1),l
). So
we de�ne j the following way: vσ(j),l is the leftmost interval su
h that l(vx,l) ≤ l(vσ(j),l). if there
is none, we say j = 0. Then we
all σ′ ∈ Sx the permutation de�ned by:
σ′(i) = σ(i) if i < j,
σ′(j + 1) = x,
σ′(i) = σ(i− 1) if i > j.
Then we dire
tly get inequalities:
• l(v1r ) < l(vσ′(x),l) < . . . < l(vσ′(j+1),l) ≤ l(vx,l) < l(vσ′(j−1),l) < . . . < l(vσ′(1),l) <
σ′(x),l
) < . . . < l(v′
σ′(j+1),l
) < l(v′
σ′(j−1),l
) < . . . < l(v′
σ′(1),l
• l(v′
σ′(x),r
) < . . . < l(v′
σ′(j+1),r
) < l(v′
σ′(j−1),r
) < . . . < l(v′
σ′(1),r
) < l(vσ′(x),r) < . . . <
l(vσ′(j+1),r) < l(vσ′(j−1),r) < . . . < l(vσ′(1),r)
We obtain the expe
ted inequalities by reasoning the same way as in the end of the base
ase.
So in parti
ular we have l(vσ(x),l) < . . . < l(vσ(1),l) and v
r must interse
t all those vi for
i ∈ J1, xK, but also an interval of X1 whi
h interse
ts none of the vi. So it must have length
x+ 1, thus K ′x is not a (x, x)-interval graph
Con
lusion: As the base
ase and the re
ursion has been proved, expe
ted properties of
the graph K ′x are true for any x ≥ 2.
2-interval graphs and restrictions
Useful gadgets for 2-interval graphs and restrictions
Balanced 2-interval graphs
Unit 2-interval and (x,x)-interval graphs
Investigating the complexity of unit 2-interval graphs
Conclusion
Appendix
|
0704.1572 | Exchange parameters from approximate self-interaction correction scheme | Exchange parameters from approximate self-interaction correction scheme
A. Akande and S. Sanvito
School of Physics and CRANN, Trinity College, Dublin 2, Ireland
(Dated: November 28, 2018)
The approximate atomic self-interaction corrections (ASIC) method to density functional theory is
put to the test by calculating the exchange interaction for a number of prototypical materials, critical
to local exchange and correlation functionals. ASIC total energy calculations are mapped onto an
Heisenberg pair-wise interaction and the exchange constants J are compared to those obtained
with other methods. In general the ASIC scheme drastically improves the bandstructure, which
for almost all the cases investigated resemble closely available photo-emission data. In contrast the
results for the exchange parameters are less satisfactory. Although ASIC performs reasonably well
for systems where the magnetism originates from half-filled bands, it suffers from similar problems
than those of LDA for other situations. In particular the exchange constants are still overestimated.
This reflects a subtle interplay between exchange and correlation energy, not captured by the ASIC.
PACS numbers:
I. INTRODUCTION
Theoretical studies based on density functional the-
ory (DFT) [1, 2] have given remarkable insights into
the electronic and magnetic properties of both molecules
and solids [3]. In particular, a number of these studies
attempt to quantitatively describe the magnetic inter-
action in a broad range of systems including transition
metals [4], hypothetical atomic chains [5, 6], ionic solids
[7, 8, 9], transition metal oxides [10, 11] and transition
metal polynuclear complexes [12, 13, 14]. DFT uses an
effective single-particle picture where spin symmetry is
generally broken. For this reason exchange parameters J
are conventionally extracted by using a mapping proce-
dure, where total energy calculations are fitted to a classi-
cal Heisenberg Hamiltonian [4, 15]. This is then used for
evaluating the Curie or Néel temperatures, the magnetic
susceptibility and for interpreting neutron diffraction ex-
periments [16].
Notably, the accuracy and reliability of the numerical
values of the J ’s depend on the functional used for the ex-
change and correlation (XC) energy, being this the only
approximated part of the DFT total energy [17]. Cal-
culations based on well-known local functionals, namely
the local density approximation (LDA) and the gener-
alised gradient approximation (GGA), are successful with
itinerant magnetism in transition metals [4], but largely
over-estimates the Heisenberg exchange parameters in
many other situations [7, 8, 9, 11, 14]. Additional cor-
rections based on the kinetic energy density (metaGGA)
[18] marginally improves the agreement with experiments
[9], although an extensive investigation over several solid
state systems has not been carried out so far.
These failures are usually connected to the local char-
acter of the LDA, which is only weakly modified by con-
structing XC potentials including the gradient, or higher
derivative of the charge density. A direct consequence is
that the charge density is artificially delocalized in space,
leading to an erroneous alignment of the magnetic bands.
These are also artificially broadened. A typical example
is that of NiO, which LDA predicts as Mott-Hubbard in-
stead of charge-transfer insulator. Clearly a qualitative
failure in describing the ground state results in an erro-
neous prediction of the exchange parameters.
One of the reasons behind the inability of LDA and
GGA of describing localized charge densities is attributed
to the presence of the self-interaction error (SIE) [19].
This originates from the spurious Coulomb interaction of
an electron with itself, which is inherent to local func-
tionals. Hartree-Fock (HF) methods, in the unrestricted
or spin polarised form, are SIE free and produce sys-
tematic improved J parameters. However, these meth-
ods lack of correlation and usually overcorrect. A typ-
ical example is the antiferromagnetic insulator KNiF3
for which HF predicts a nearest neighbour J of around
2 meV [7, 20, 21, 22, 23] against an experimental value
of 8.6 meV [24]. Direct SIE subtraction, convention-
ally called self-interaction corrected (SIC) LDA, also im-
proves the results and seems to be less affected by over-
correction [5, 25]. Similarly hybrid-functionals, which
mix portions of HF exchange with the local density ap-
proximation of DFT, perform better than local function-
als and in several situations return values for J in close
agreement with experiments [7, 8].
It is important to note that both methods based non-
local exchange or SIC, are computationally demanding
and thus their application to the solid state remains
rather limited. It is then crucial to develop practical com-
putational schemes able to provide a good estimate of the
exchange parameters for those systems critical to LDA,
which at the same time are not numerically intensive.
Based on the idea that most of the SIE originates from
highly localized states, with a charge distribution resem-
bling those of free atoms, Vogel et al. [26] proposed a sim-
ple SIC scheme where the corrections are approximated
by a simple on-site term. This method was then gener-
alized to fractional occupation by Filippetti and Spaldin
[27] and then implemented in a localized atomic orbital
code for large scaling by Pemmaraju et al. [28]. Despite
its simplicity the method has been successfully applied to
a number of interesting physical systems including , tran-
sition metal monoxides [27, 29], silver halides [30], no-
ble metal oxides [31], ferroelectric materials [27, 32, 33],
high-k materials [34], diluted magnetic semiconductors
[35, 36] and also to quantum transport [37, 38].
The method is strictly speaking not variational, in the
sense that a functional generating the ASIC potential via
variational principle is not available. However, since typ-
ically the LDA energy is a good approximation of the
exact DFT energy, although the LDA potential is rather
different from the exact KS potential, a “practical” def-
inition of total energy can be provided. In this work we
evaluate the ability of this approximated energy in de-
scribing exchange parameters for a variety of magnetic
systems.
II. THE ATOMIC SIC METHOD
The seminal work of Perdew and Zunger [19] pioneered
the modern theory of SIC. The main idea is that of sub-
tracting directly the spurious SI for each Kohn-Sham
(KS) orbital ψn. The SIC-LDA [39] XC energy thus
writes
ESICxc [ρ
↑, ρ↓] = ELDAxc [ρ
↑, ρ↓]−
occupied∑
δSICn , (1)
where ELDAxc [ρ
↑, ρ↓] is the LDA-XC energy and δSICn is the
sum of the self-Hartree and self-XC energy associated to
the charge density ρσn = |ψσn|2 of the fully occupied KS
orbital ψσn
δSICn = U [ρ
n] + E
xc [ρ
n, 0] . (2)
Here U is the Hartree energy and σ is the spin index.
The search for the energy minimum is not trivial, since
ESICxc is not invariant under unitary rotations of the occu-
pied KS orbitals. As a consequence the KS method be-
comes either non-orthogonal or size-inconsistent. These
problems however can be avoided [40, 41, 42] by intro-
ducing a second set of orbitals φσn related to the canonical
KS orbitals by a unitary transformation M
ψσn =
Mσnmφ
m . (3)
The functional can then be minimized by varying both
the orbitals ψ and the unitary transformationM, leading
to a system of equations
n = (H
0 + ∆v
n = �
σ,SIC
n , (4)
ψσn =
Mnmφσm , (5)
∆vSICn =
MnmvSICm
vSICm P̂
m , (6)
where Hσ0 is the LDA Hamiltonian, P̂
n(r) =
φσm(r)〈φσm|ψσn〉 and vSICn = u([ρn]; r) + vσ,LDAxc ([ρ↑n, 0]; r),
with u and vσ,LDAxc the Hatree and LDA-XC potential
respectively.
In equation (4) we have used the fact that at the en-
ergy minimum the matrix of SIC KS-eigenvalues �σ,SICnm
is diagonalized by the KS orbitals ψn. Importantly such
minimization scheme can be readily applied to extended
systems, without loosing the Bloch representation of the
KS orbitals [43, 44].
The ASIC method consists in taking two drastic ap-
proximations in equation (4). First we assume that the
orbitals φm, that minimize the SIC functional are atomic-
like orbitals Φσm (ASIC orbitals) thus∑
vSICm (r)P̂
m → α
ṽσSICm (r)P̂
m , (7)
where ṽσSICm (r) and P̂
m are the SIC potential and the
projector associated to the atomic orbital Φσm. Secondly
we replace the non-local projector P̂Φm with its expecta-
tion value in such a way that the final ASIC potential
reads
vσASIC(r) = α
ṽσSICm (r)p
m , (8)
where pσm is the orbital occupation (essentially the spin-
resolved Mülliken orbital population) of Φm.
Note that in the final expression for the potential a
factor α appears. This is an empirical scaling term that
accounts for the fact that the ASIC orbital Φ in general
do not coincide with those that minimize the SIC func-
tional (1). By construction α = 1 in the single particle
limit, while it vanishes for the homogeneous electron gas.
Although in general 0 < α < 1, extensive testing [28]
demonstrates that a value around 1 describes well ionic
solids and molecules, while a value around 1/2 is enough
for mid- to wide-gap insulators. In the following we will
label with ASIC1/2 and ASIC1 calculations obtained re-
spectively with α = 1/2 and α = 1.
Finally we make a few comments over the total energy.
As pointed out in the introduction the present theory is
not variational since the KS potential cannot be related
to a functional by a variational principle. However, since
typical LDA energies are more accurate than their corre-
sponding KS potentials, we use the expression of equation
(1) as suitable energy. In this case the orbital densities
entering the SIC are those given by the ASIC orbital Φ.
Moreover, in presenting the data, we will distinguish re-
sults obtained by using the SIC energy (1) from those
obtained simply from the LDA functional evaluated at
the ASIC density, i.e. without including the δn correc-
tions (2).
III. RESULTS
All our results have been obtained with an implemen-
tation of the ASIC method [28] based on the DFT code
Siesta [45]. Siesta is an advanced DFT code using pseu-
dopotentials and an efficient numerical atomic orbital ba-
sis set. In order to compare the exchange parameters ob-
tained with different XC functionals we consider the LDA
parameterization of Ceperly and Alder [46], the GGA
functional obtained by combining Becke exchange [47]
with Lee-Yang-Parr correlation [48] (BLYP), the nonem-
pirical Purdew, Burke and Ernzerhof (PBE) GGA [49],
and the ASIC scheme as implemented in reference [28].
Calculations are performed for different systems crit-
ical to LDA and GGA, ranging from molecules to ex-
tended solids. These include hypothetical H-He atomic
chains, the ionic solid KNiF3 and the transition metal
monoxides MnO and NiO. DFT total energy calculations
are mapped onto an effective pairwise Heisenberg Hamil-
tonian
HH = −
Jnm~Sn · ~Sm , (9)
where the sums runs over all the possible pairs of spins.
In doing this we wish to stress that the mapping is a con-
venient way of comparing total energies of different mag-
netic configurations calculated with different function-
als. In this spirit the controversy around using the spin-
projected (Heisenberg mapping) or the non-projected
scheme is immaterial [5, 50, 51].
A. H-He chain
As an example of molecular systems, we consider H-
He monoatomic chains at a inter-atomic separation of
1.625 Å (see figure 1). This is an important benchmark
for DFT since the wave-function is expected to be rather
localized and therefore to be badly described by local
XC functionals. In addition the system is simple enough
to be accessible by accurate quantum chemistry calcula-
tions.
As basis set we use two radial functions (double-ζ)
for the s and p angular momenta of both H and He,
while the density of the real-space grid converges the
self-consistent calculation at 300 Ry. Here we consider
all possible Heisenberg parameters. Thus the triangular
molecule (Fig.1a) has only one nearest neighbour param-
eter Ja12, the 5-atom chain (Fig.1b) has both first J
12 and
second neighbour Jb13 parameters, and the 7-atom chain
(Fig.1c) has three parameters describing respectively the
nearest neighbour interaction with peripheral atoms Jc12,
the nearest neighbour interaction between the two middle
atoms Jc23 and the second neighbour interaction J
Following reference [5], accurate calculations based
on second-order perturbation theory (CASPT2) [12] are
used as comparison. The quality of each particular func-
tionals is measured as the relative mean deviation of the
nearest neighbour exchange parameters only (Ja12, J
FIG. 1: (Color on-line) H-He-H chains at an inter-atomic dis-
tance of 1.625Å.
Method Ja12 J
13 δ (%)
CASPT2 -24 -74 -0.7 -74 -79 -0.7 0
SIC-B3LYP -31 -83 -0.2 -83 -88 -0.3 16
LDA -68 -232 -6 -234 -260 -6 210
PBE -60 -190 -1.8 -190 -194 -1.6 152
BLYP -62 -186 -2 -186 -200 -1 147
ASIC1 -36 -112 -1 -110 -122 -0.6 51
ASIC1/2 -44 -152 -1 -152 -168 -1.4 101
ASIC∗1 -40 -128 -0.6 -128 -142 -1.0 73
ASIC∗1/2 -50 -170 -1.4 -170 -190 -1.8 127
TABLE I: Calculated J values (in meV) for the three different
H–He chains shown in Fig.1. The CASPT2 values are from
reference [12], while the SIC-B3LYP are from reference [5].
The last two rows correspond to J values obtained from the
LDA energy calculated at the ASIC density.
Jc12, J
23), since those are the largest ones
|Ji − JCASPT2i |
|JCASPT2i |
. (10)
Our calculated J values and their relative δ are pre-
sented in table I, where we also include results for a fully
self-consistent SIC calculation over the B3LYP functional
(SIC-B3LYP) [5]. It comes without big surprise that the
LDA systematically overestimates all the exchange pa-
rameters with errors up to a factor 6 for the smaller J
(Jb13 and J
13) and an average error δ for the largest J
of about 200%. Standard GGA corrections considerably
improve the description although the J ’s are still system-
atically larger than those obtained with CASPT2. Note
that the results seem rather independent of the particular
GGA parameterization, with PBE and BLYB producing
similar exchange constants. This is in good agreement
with previous calculations [5].
SIC in general dramatically improves the LDA and
GGA description and our results for ASIC1 are reason-
ably close to those obtained with the full self-consistent
procedure (SIC-B3LYP). This is an interesting result,
considering that our ASIC starts from a local exchange
functional, while B3LYP already contains non-local con-
tributions. We also evaluate the J parameters by using
the LDA energy evaluated at the ASIC density (last two
rows in table I). In general this procedure gives J ’s larger
than those obtained by using the energy of equation (1),
meaning that the δSICn contributions reduce the J values.
It is then clear that the ASIC scheme systematically
improves the J values as compared to local functionals.
The agreement however is not as good as the one ob-
tained by using a fully self-consistent SIC scheme, mean-
ing that for this molecular system the ASIC orbitals are
probably still not localized enough. This can alter the
actual contribution of δSICn to the total energy and there-
fore the exchange parameters.
B. Ionic antiferromagnets: KNiF3
Motivated by the substantial improvement of ASIC
over LDA, we then investigate its performances for real
solid-state systems, starting from KNiF3. This is a pro-
totypical Heisenberg antiferromagnet with strong ionic
character, a material for which our ASIC approxima-
tion is expected to work rather well [28]. It is also a
well studied material, both experimentally [24, 52] and
theoretically [7, 9, 21, 22], allowing us extensive com-
parisons. The KNiF3 has cubic perovskite-like structure
with the nickel atoms at the edges of the cube, flourine
atoms at the sides and potassium atoms at the center (see
Fig.2). At low temperature, KNiF3 is a type II antiferro-
magnetic insulator consisting of ferromagnetic (111) Ni
planes aligned antiparallel to each other. For our calcu-
lations we use a double-ζ polarized basis for the s and p
orbitals of K, Ni and F, a double-ζ for the 3d of K and Ni,
and a single-ζ for the 3d of F. Finally, we use 5×5×5 k-
points in the full Brillouin zone and the real-space mesh
cutoff is 550 Ry. Note that the configuration used to
generate the pseudopotential is that of Ni2+, 4s13d7.
We first consider the band-structure as obtained with
LDA and ASIC. For comparison we also include results
obtained with LDA+U [53, 54] as implemented in Siesta
[55]. In this case we correct only the Ni d shell and we
fix the Hubbard-U and Hund’s exchange-J parameters by
fitting the experimental lattice constant (a0 = 4.014 Å).
The calculated values are U=8 eV and J=1 eV. The
bands obtained with the three methods and the corre-
sponding orbital projected density of states (DOS) are
presented in figures 3 and 4 respectively.
All the three functionals describe KNiF3 as an insula-
tor with bandgaps respectively of 1.68 eV (LDA), 3.19 eV
(ASIC1), and 5.0 eV (LDA+U). An experimental value
for the gap is not available and therefore a comparison
cannot be made. In the case of LDA and ASIC the gap
is formed between Ni states, with conductance band bot-
FIG. 2: (Color on-line) Cubic perovskite structure of KNiF3.
Color code: blue=Ni, red=F, Green=K.
FIG. 3: Band structure for type II antiferromagnetic KNiF3
obtained with a) LDA, b) ASIC1 and c) LDA+U (U=8 eV
and J=1 eV). The valence band top is aligned at E=EF=0 eV
(horizontal line).
tom well described by eg orbitals. These are progressively
moved upwards in energy by the SIC, but still occupy the
gap. Such feature is modified by LDA+U which pushes
the unoccupied eg states above the conductance band
minimum, which is now dominated by K 4s orbitals.
In more detail the valence band is characterized by a
low-lying K 3p band and by a mixed Ni-3d/F 2p. While
the K 3p band is extremely localized and does not present
substantial additional orbital components the amount of
mixing and the broadening of the Ni-3d/F 2p varies with
the functionals used. In particular both LDA and ASIC
predict that the Ni 3d component occupies the high en-
FIG. 4: (Color on-line) DOS for type II antiferromagnetic
KNiF3 obtained with a) LDA, b) ASIC1 and c) LDA+U
(U=8 eV and J=1 eV). The valence band top is aligned at
E=0 eV (vertical line). The experimental UPS spectrum from
reference [56] is also presented (thick green line). The relative
binding energy is shifted in order to match the K 3p peak.
ergy part of the band, while the F 2p the lower. For both
the total bandwidth is rather similar and it is about 9-
10 eV. In contrast LDA+U offers a picture where the
Ni-F hybridization spread across the whole bandwidth,
which is now reduced to less than 7 eV.
Experimentally, ultraviolet photoemission spec-
troscopy (UPS) study of the whole KMF3 (M: Mn, Fe,
Co, Ni, Cu, Zn) series [56] gives us a spectrum dominated
by two main peaks: a low K 3p peak and broad band
mainly attributed to F 2p. These two spectroscopical
features are separated by a binding energy of about
10 eV. In addition the 10 eV wide F 2p band has some
fine structure related to various Ni d multiplets. An
analysis based on averaging the multiplet structure [56]
locates the occupied Ni d states at a bounding energy
about 3 eV smaller than that of the F 2p band. In figure
4 we superimpose the experimental UPS spectrum to
our calculated DOS, with the convention of aligning in
each case the sharp K 3p peak.
It is then clear that ASIC provides in general a better
agreement with the UPS data. In particular both the
Ni-3d/F 2p bandwidth and the position of the Fermi en-
ergy (EF) with respect to the K 3p peak are correctly
predicted. This is an improvement over LDA, which de-
scribes well the Ni-3d/F 2p band, but positions the K 3p
states too close to EF. For this reason, when we align the
Method a0 Jth P
d Jex P
LDA 3.951 46.12 (53.1) 1.829 40.4 1.834
PBE 4.052 33.98 (37.0) 1.813 36.48 1.808
BLYP 4.091 31.10 (37.6) 1.821 36.72 1.812
ASIC1/2 3.960 40.83 1.876 36.14 1.878
ASIC1 3.949 36.22 1.907 30.45 1.914
ASIC∗1/2 3.969 43.44 1.876 38.57 1.878
ASIC∗1 3.949 39.80 1.907 33.56 1.914
LDA+U 4.007 12.55 10.47 1.940
TABLE II: Calculated J parameters (in meV) and the
Mülliken magnetic moment for Ni 3d (Pd) in KNiF3. The ex-
perimental values for J and a0 are 8.2±0.6 meV and 4.014Å
respectively while the values in brackets are those from refer-
ence [9]. In the table we report values evaluated at the theo-
retical (Jth and P
d ) and experimental (Jex and P
d ) lattice
constant. ASIC∗1/2 and ASIC
1 are obtained from the LDA
energies evaluated at the ASIC density.
UPS spectrum at the K 3p position, this extends over EF.
Finally in the case of LDA+U , there is a substantial mis-
alignment between the UPS data and our DOS. LDA+U
in fact erroneously pushes part of the Ni d mainfold below
the F 2p DOS, which now forms a rather narrow band.
We now turn our attention to total energy related
quantities. In table II we present the theoretical equi-
librium lattice constant a0 and the Heisenberg exchange
parameter J for all the functionals used. Experimentally
we have J=8.2± 0.6 meV [24]. The values of a0 and J are
calculated for the type II antiferromagnetic ground state,
by constructing a supercell along the (111) direction. Im-
portantly values of J obtained by considering a supercell
along the (100) direction, i.e. by imposing antiferromag-
netic alignment between ferromagnetic (100) planes (type
I antiferromagnet), yield essentially the same result, con-
firming the fact that the interaction is effectively only
extending to nearest neighbors. Furthermore we report
results obtained both at the theoretical equilibrium lat-
tice constant (Jth) and at the experimental one (Jex).
Also in this case local XC functionals largely overesti-
mate J , with errors for Jex going from a factor 8 (LDA)
to a factor 4.5 (GGA-type). ASIC improves these val-
ues, although only marginally, and our best agreement
is found for ASIC1, while ASIC1/2 is substantially iden-
tical to GGA. Interestingly the ASIC1 performance is
rather similar, if not better, to that of meta-GGA func-
tionals [9]. The situation is however worsened when we
consider J parameters obtained at the theoretical lattice
constant. The ASIC-calculated a0 are essentially identi-
cal to those from LDA and about 2% shorter than those
from GGA. Since J depends rather severely on the lattice
parameter we find that at the theoretical lattice constant
GGA-functionals perform actually better than our ASIC.
Finally, also in this case the J ’s obtained by simply us-
ing the LDA energies are larger than those calculated by
including the SIC corrections (see equation 1).
In general the improvement of the J parameter is cor-
related to an higher degree of electron localization, in
particular of the Ni d shell. In table II the magnetic mo-
ment of the Ni d shell Pd, obtained from the Mülliken
population, is reported. This increases systematically
when going from LDA to GGA to ASIC approaching the
atomic value expected from Ni2+.
Our best result is obtained with LDA+U , which re-
turns an exchange of 10.47 meV for the same U and J
that fit the experimental lattice constant. This is some-
how superior performance of LDA+U with respect to
ASIC should not be surprising and it is partially related
to an increased localization. The Ni ions d shell in oc-
tahedral coordination splits into t2g and eg states, which
further split according to Hund’s rule. The t2g states are
all filled, while for the eg only the majority are. By look-
ing at the LDA DOS one can recognize the occupied t↑2g
orbitals (we indicate majority and minority spins respec-
tively with ↑ and ↓) at -3 eV, the e↑g at -2 eV and the t
at about 0 eV, while the empty e↓g are at between 1 and
3 eV above the valence band maximum.
The local Hund’s split can be estimated from the e↑g-
e↓g separation. The ASIC scheme corrects only occupied
states [57], and therefore it enhances the local exchange
by only a downshift of the valence band. From the DOS
of figure 4 it is clear that this is only a small contri-
bution. In contrast the LDA+U scheme also corrects
empty states, effectively pushing upwards in energy the
e↓g band. The net result is that of a much higher degree
of localization of the d shell with a consequent reduction
of the Ni-Ni exchange. This is similar to the situation de-
scribed by the Hartree-Fock method, which however re-
turns exchange parameters considerably smaller than the
experimental value [20, 21, 22, 23]. Interestingly hybrid
functionals [7] have the right mixture of non-local ex-
change and electron correlation and produce J ’s in close
agreement with the experiments.
We further investigate the magnetic interaction by
evaluating J as a function of the lattice constant. Ex-
perimentally this can be achieved by replacing K with
Rb and Tl, and indeed de Jongh and Block [58] early
suggested a d−α power law with α = 12± 2. Our calcu-
lated J as a function of the lattice constant d for LDA,
GGA, ASIC1 and LDA+U (U=8 eV and J=1 eV) are
presented in figure ??. For all the four functionals in-
vestigated J varies as a power law, although the calcu-
lated exponents are rather different: 8.6 for LDA, 9.1
for GGA, 11.3 for ASIC1 and 14.4 for LDA+U . This
further confirms the strong underestimation of the ex-
change constants from local functionals. Clearly the rel-
ative difference between the J obtained with different
functionals becomes less pronounced for small d, where
the hybridization increases and local functionals perform
better. Note that only ASIC1 is compatible with the ex-
perimental exponent of 12 ± 2, being the one evaluated
from LDA+U too large. Importantly we do not expect
to extrapolate the LDA+U value at any distance, since
FIG. 5: J as a function of the lattice constant for LDA, GGA,
ASIC1 and LDA+U (U=8 eV and J=1 eV). The symbols are
our calculate value while the solid lines represent the best
power-law fit.
the screening of the parameters U and J changes with
the lattice constant.
In conclusion for the critical case of KNiF3 the ASIC
method appears to improve the LDA results. This is es-
sentially due to the better degree of localization achieved
by the ASIC as compared with standard local function-
als. However, while the improvement over the bandstruc-
ture is substantial, it is only marginal for energy-related
quantities. The main contribution to the total energy in
the ASIC scheme comes from the LDA functional, which
is now evaluated at the ASIC density. This is not suf-
ficient for improving the exchange parameters, which in
contrast need at least a portion of non-local exchange.
C. Transition metal monoxides
Another important test for the ASIC method is that of
transition metal monoxides. These have been extensively
studied both experimentally and theoretically and they
are the prototypical materials for which the LDA appears
completely inadequate. In this work we consider MnO
and NiO, which have respectively half-filled and partially-
filled 3d shells. They both crystallize in the rock-salt
structure and in the ground state they are both type-
II antiferromagnetic insulators. The Néel’s temperatures
are 116 K and 525 K respectively for MnO and NiO. In all
our calculations we consider double-ζ polarised basis for
the s and p shell of Ni, Mn and O, double-ζ for the Ni and
Mn 3d orbitals, and single-ζ for the empty 3d of O. We
sample 6×6×6 k-points in the full Brillouin zone of both
the cubic and rhombohedral cell describing respectively
type I and type II antiferromagnetism. Finally the real-
space mesh cutoff is 500 Ry.
The calculated band structures obtained from LDA,
ASIC1/2 and ASIC1 are shown in figures 6 and 7 for MnO
and NiO respectively. These have been already discussed
extensively in the context of the ASIC method [27, 28]
and here we report only the main features. For both the
materials LDA already predicts an insulating behavior,
although the calculated gaps are rather small and the
nature of the gaps is not what experimentally found. In
both cases the valence band top has an almost pure d
component, which suggests these materials to be small
gap Mott-Hubbard insulators. The ASIC downshifts the
occupied d bands which now hybridize with the O-p man-
ifold. The result is a systematic increase of the band-gap
which is more pronounced as the parameter α goes from
1/2 to 1. Importantly, as noted already before [28], the
experimental band-gap is obtained for α ∼ 1/2.
FIG. 6: Calculated band structure for the type II anti-
ferromagnetic MnO obtained from a) LDA, b) ASIC1/2 and
c) ASIC1. The valence band top is aligned at 0 eV (horizontal
line).
We then moved to calculating the exchange parame-
ters. In this case we extend the Heisenberg model to
second nearest neighbors, by introducing the first (J1)
and second (J2) neighbor exchange parameters. These
are evaluated from total energy calculations for a ferro-
magnetic and both type II and type I antiferromagnetic
alignments. Our calculated results, together with a few
selected data available from the literature are presented
in table III.
Let us first focus our attention to MnO. In this case
both the J ’s are rather small and positive (antiferro-
magnetic coupling is favorite), in agreement with the
Goodenough-Kanamori rules [59] and the rather low Néel
temperature. Direct experimental measurements of the
exchange parameters are not available and the com-
monly accepted values are those obtained by fitting the
magnetic susceptibility with semi-empirical methods [60].
Importantly this fit gives almost identical first and second
nearest neighbour exchange constants. In contrast all the
exchange functionals we have investigated offer a picture
where J2 is always approximately twice as large as J1.
FIG. 7: Calculated band structure for the type II anti-
ferromagnetic NiO obtained from a) LDA, b) ASIC1/2 and
c) ASIC1. The valence band top is aligned at 0 eV (horizon-
tal line).
Method MnO NiO
J1 J2 Pd J1 J2 Pd
LDA 1.0 2.5 4.49 (4.38) 13.0 -94.4 1.41 (1.41)
PBE 1.5 2.5 4.55 (4.57) 7.0 -86.8 1.50 (1.59)
ASIC1/2 1.15 2.44 4.72 (4.77) 6.5 -67.3 1.72 (1.77)
ASIC1 0.65 1.81 4.84 (4.86) 3.8 -41.8 1.83 (1.84)
ASIC∗1/2 1.27 2.65 4.72 (4.77) 7.1 -74.6 1.72 (1.77)
ASIC∗1 0.69 2.03 4.84 (4.86) 4.4 -47.9 1.83 (1.84)
SE1a 0.86 0.95
HFb 0.22 0.36
B3LYPc 0.81 1.71
PBE0b 0.96 1.14
B3LYPd 2.4 -26.7
HFd 0.8 -4.6
SIC-LDAe 2.3 -12
Expt.f 1.4 -19.8
Expt.g 1.4 -17.0
TABLE III: Calculated J1 and J2 in meV for MnO and NiO.
Pd is the magnetic moment of the d shell calculated from the
type II antiferromagnetic phase. Values in bracket are for Pd
evaluated from the ferromagnetic ground state. ASIC∗1/2 and
ASIC∗1 are obtained from the LDA energies evaluated at the
ASIC density. a) Ref. [60], b) Ref. [61], c) Ref. [62], d) Ref.
[11], e) Ref. [25], f) Ref. [64], g) Ref. [65]
This gives us a reasonably accurate value of J1 for LDA
and GGA, but J2 is overestimated by approximately a
factor 2, in agreement with previous calculations [10].
ASIC systematically improves the LDA/GGA descrip-
tion, by reducing both J1 and J2. This is related to the
enhanced localization of the Mn d electrons achieved by
the ASIC, as it can be seen by comparing the Mn d mag-
netic moments (Pd) calculated for different functionals
(see table III). Thus ASIC1, which provides the largest
magnetic moment, gives also J ’s in closer agreement with
the experimental values, while ASIC1/2 is not very dif-
ferent from LDA.
Importantly for half-filling, as in MnO, the ASIC
scheme for occupied states is fundamentally analogous
to the LDA+U method, with the advantage that the U
parameter does not need to be evaluated. Finally, at
variance with KNiF3, it does not seem that a portion of
exact exchange is strictly necessary in this case. Hartree-
Fock [61] results in a dramatic underestimation of the J
parameters, while B3LYP [62] is essentially very similar
to ASIC1. Curiously the best results available in the lit-
erature [61] are obtained with the PBE0 functional [63],
which combines 25% of exact-exchange with GGA.
The situation for NiO is rather different. The ex-
perimentally available data [64, 65] show antiferromag-
netic nearest neighbour and ferromagnetic second near-
est neighbour exchange parameters. The magnitude is
also rather different with |J2| > 10 |J1|. Standard lo-
cal functionals (LDA and GGA) fail badly and overes-
timate both the J ’s by more than a factor 5. ASIC
in general reduces the exchange constants and drasti-
cally improves the agreement between theory and exper-
iments. In particular ASIC1 gives exchange parameters
only about twice as large as those measured experimen-
tally.
A better understanding can be obtained by looking at
the orbital-resolved DOS for the Ni d and the O p orbitals
(figure 8) as calculated from LDA and ASIC. There are
two main differences between the LDA and the ASIC
results. First there is an increase of the fundamental
band-gap from 0.54 eV for LDA to 3.86 eV for ASIC1/2 to
6.5 eV for ASIC1. Secondly there is change in the relative
energy positioning of the Ni d and O p contributions to
the valence band. In LDA the top of the valence band is
Ni d in nature, with the O p dominated part of the DOS
lying between 4 eV and 8 eV from the valence band top.
ASIC corrects this feature and for ASIC1/2 the O p and
Ni d states are well mixed across the whole bandwidth. A
further increase of the ASIC corrections (α = 1) leads to
a further downshift of the Ni d band, whose contribution
becomes largely suppressed close to the valence band-top.
Thus, increasing the portion of ASIC pushes NiO further
into the charge transfer regime.
Interestingly, although ASIC1/2 gives the best band-
structure, the J ’s obtained with ASIC1 are in better
agreement with the experiments. This is somehow sim-
ilar to what observed when hybrid functionals are put
to the test. Moreira et al. demonstrated [11] that J ’s
in close agreement with experiments can be obtained by
using 35% Hartree-Fock exchange in LDA. Moreover, in
close analogy to the ASIC behaviour, as the fraction of
exact exchange increases from LDA to purely Hartree-
Fock, the exchange constants decrease while the band-
gap gets larger. However, while the best J ’s are obtained
with 35% exchange, a gap close to the experimental one
FIG. 8: Calculated orbital resolved DOS for type II anti-
ferromagnetic NiO obtained with a) LDA, b) ASIC1/2 and c)
ASIC1. The valence band top is aligned at 0 eV.
is obtained with B3LYP, which in turns overestimates the
J ’s. This remarks the subtile interplay between exchange
and correlations in describing the magnetic interaction
of this complex material. Finally, it is worth remarking
that a fully self-consistent SIC [25] seems to overcorrect
the J ’s, while still presenting the erroneous separation
between the Ni d and O p states.
IV. CONCLUSIONS
In conclusions the approximated expression for the
ASIC total energy is put to the test of calculating ex-
change parameters for a variety of materials, where local
and gradient-corrected XC functionals fail rather badly.
This has produced mixed results. On the one hand, the
general bandstructure and in particular the valence band,
is considerably improved and resembles closely data from
photo-emission. On the other hand, the exchange con-
stants are close to experiments only for the case when the
magnetism originates from half-filled shells. For other
fillings, as in the case of NiO or KNiF3 the ASIC im-
provement over LDA is less satisfactory, suggesting that
a much better XC functional, incorporating a portion at
least of exact exchange, is needed. Importantly ASIC
seems to be affected by the same pathology of hybrid
functional, i.e. the amount of ASIC needed for correct-
ing the J is different from that needed for obtaining a
good bandstructure.
V. ACKNOWLEDGEMENTS
This work is supported by Science Foundation of Ire-
land under the grant SFI05/RFP/PHY0062. Computa-
tional resources have been provided by the HEA IITAC
project managed by the Trinity Center for High Perfor-
mance Computing and by ICHEC.
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Introduction
The atomic SIC method
Results
H-He chain
Ionic antiferromagnets: KNiF3
Transition metal monoxides
Conclusions
Acknowledgements
References
|
0704.1573 | Generalized sqrt(epsilon)-law. The role of unphysical source terms in
resonance line polarization transfer and its importance as an additional test
of NLTE radiative transfer codes | Astronomy & Astrophysics manuscript no. p0603 c© ESO 2018
November 29, 2018
Generalized
ǫ-law
The role of unphysical source terms in resonance line polarization transfer and
its importance as an additional test of NLTE radiative transfer codes.
J. Štěpán1,2 and V. Bommier2
1 Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondřejov, Czech Republic
e-mail: [email protected]
2 LERMA, Observatoire de Paris – Meudon, CNRS UMR 8112, 5, Place Jules Janssen, 92195 Meudon Cedex, France
e-mail: [jiri.stepan;V.Bommier]@obspm.fr
Received 4 October 2006 / Accepted 26 March 2007
ABSTRACT
Context. A derivation of a generalized
ǫ-law for nonthermal collisional rates of excitation by charged perturbers is presented.
Aims. Aim of this paper is to find a more general analytical expression for a surface value of the source function which can be used
as an addtional tool for verification of the non-LTE radiative transfer codes.
Methods. Under the impact approximation hypothesis, static, one-dimensional, plane-parallel atmosphere, constant magnetic field of
arbitrary strength and direction, two-level atom model with unpolarized lower level and stimulated emission neglected, we introduce
the unphysical terms into the equations of statistical equilibrium and solve the appropriate non-LTE integral equations.
Results. We derive a new analytical condition for the surface values of the source function components expressed in the basis of
irreducible spherical tensors.
Key words. line: formation – polarization – radiative transfer
1. Introduction
In the series of papers of Landi Degl’Innocenti et al. (1991a,b),
Landi Degl’Innocenti & Bommier (1994) (from now on refer-
enced as Paper I), the general formalism of resonance line po-
larization scattering for a two-level atom has been developed.
The non-LTE problem of the 2nd kind for an arbitrary magnetic
field, three-dimensional geometry of the medium and arbitrary
irradiation by external sources has been discussed. The effect of
inelastic collisions with charged perturbers has been considered
for the particular case of a relative Maxwellian velocity distribu-
tion.
Paper I analysed the analytical properties of the solutions in
the particular case of a one-dimensional, semi-infinite, static at-
mosphere with a constant magnetic field of arbitrary strength
and direction and assuming zero external irradiation of the at-
mosphere. They derived a generalization of the well known
law (e.g. Avrett & Hummer 1965; Mihalas 1970; Hubený 1987)
for the case of polarized radiation and extended the previous re-
sults of Ivanov (1990) who studied scattering in a non-magnetic
regime.
In most cases of practical interest the polarization degree is
rather small. The purpose of this paper is to find a new analytical
solution of the non-LTE problem in unphysical conditions in or-
der to better verify the accuracy of the polarized radiation trans-
fer codes. This is done by introduction of an unphysical source
term in the polarization into the equations of statistical equilib-
rium. Such a generalization can be useful in testing the accuracy
of the radiative transfer codes whose purpose is to deal with the
Send offprint requests to: J. Štěpán
non-thermal collisional processes (for instance in the impact po-
larization studies of solar flares).
Following the approach of the papers quoted above, we adopt
the formalism of density matrix in the representation of irre-
ducible tensorial operators (e.g. Fano 1957). We consider the
lower level with total angular momentum j to be unpolarized.
This level is completely described by the overall population
which is set to 1 for normalization reasons. The upper level with
angular momentum j′ is described by the multipole components
of the statistical tensor ρKQ. Coherences between different levels
j and j′ are neglected but coherences between Zeeman sublevels
of level j′ are in general taken into account. The calculation is
performed in the Wien limit of line frequency whose assump-
tion makes it possible to neglect stimulated emission effect, and
to preserve the linearity of the non-LTE problem.
2. Equations of statistical equilibrium
The suitable coordinate system Σ0 for atomic state description
is the one with the z-axis directed along the magnetic field (see
Figure 1).
Radiative rate contributions to the evolution of statistical op-
erator ρKQ are given by (Landi Degl’Innocenti 1985)
= −iA j′ jΓQρKQ − A j′ jρ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q. (1)
http://arxiv.org/abs/0704.1573v1
2 J. Štěpán and V. Bommier: Generalized
ǫ-law
In this equation A j′ j (B j j′) is the Einstein coefficient of spon-
taneous emission (absorption) from level j′ ( j) to level j ( j′).
Γ = 2πg j′νL/A j′ j with g j′ being the Landé factor of the level
j′ and νL is the Larmor frequency. The transition-dependent nu-
merical factor w(K)j′ j has been defined by Landi Degl’Innocenti
(1984) as have the irreducible components of the mean radiation
tensor J
Q. Besides the radiative rates, collisional rates have to
be considered in the statistical equilibrium, because the source
of radiation in a semi-infinite atmosphere is the collisional exci-
tation followed by radiative de-excitation. Thus, the source term
of the radiative transfer equation originates in the inelastic col-
lision effect. As the purpose of the present paper is to consider
unphysical source terms in the non-zero ranks (K,Q) of the ir-
reducible tensorial operator basis T KQ , we will introduce an un-
physical (K,Q)-dependence to the inelastic collisional rates of
the statistical equilibrium equation below. The purpose here is
not to thus describe anisotropic collisions, which would require
a proper formalism that is out of the scope of the present paper
(see, for instance, Landi Degl’Innocenti & Landolfi (2004) for
a two-level atom, and Derouich (2006), for polarization trans-
fer rates in a multi-level atom due to isotropic collisions). We
introduce as usual the depolarizing rate due to isotropic elastic
collisions. Thus, the contribution of collisional rates reads
= (C j j′ )
Q − (C
j′ j)
Q − D
Q. (2)
The terms (C j j′ )
Q and (C
j′ j)
Q on the right-hand side of equation
(2) are the multipole components of collisional rates of excita-
tion and relaxation respectively. D(K) is the depolarization rate
due to elastic collisions.1
The radiative and collisional rates can be added under the
impact approximation hypothesis (Bommier & Sahal-Bréchot
1991) dρKQ/dt = [dρ
Q/dt]RAD + [dρ
Q/dt]COLL. Using the equa-
tions (1), (2), and the condition for static atmosphere, dρKQ/dt =
0, we obtain the equations of statistical equilibrium
[iA j′ jΓQ + A j′ j + (C
j′ j)
Q + D
(K)]ρKQ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q + (C j j′)
Q. (3)
By applying the relation between Einstein coefficients for spon-
taneous emission and absorption,
B j j′ =
2 j′ + 1
2 j + 1
2hν30
A j′ j, (4)
and dividing the formula (3) by A j′ j, we obtain the equation
(1 + ǫKQ + δ
+ iΓQ)ρKQ
= (−1)Qw(K)j′ j J
2 j′ + 1
2 j + 1
2hν30
(C j j′ )
A j′ j′
. (5)
One can introduce the dimensionless parameter of the depolar-
ization rate
A j′ j
, (6)
1 This process cannot change a total population of the level.
Therefore it is always D(0) = 0. We take formally into account only
the depolarization rate D(K) to use a formalism coherent with the previ-
ous papers. A general treatment of physically more relevant transfer of
multipole components of the upper level is out of scope of this paper.
Fig. 1. The reference frame Σ1 has its Z-axis oriented vertically
with respect to the atmosphere, while the z-axis of the reference
frame Σ0 is parallel to the direction of magnetic field B. The axes
X and x lie in the same plane defined by Z-axis and B; the axes
Y and y are defined to complement the right-handed orthogonal
coordinate systems.
and the irreducible tensor which plays the role of generalized
photon destruction probability
(CRj′ j)
A j′ j′
. (7)
If the relation (CRj′ j)
Q , 0 is satisfied we may define the quantity
B(KQ) =
2hν30
2 j + 1
2 j′ + 1
(C j j′)
(CRj′ j)
. (8)
It is easy to show (see below) that in the particular case of a
Maxwellian velocity distribution of colliders the relation B(00) =
BP is satisfied, where BP is the Planck function in the Wien limit
at given temperature. Using the definition of irreducible compo-
nents of the two-level source function (cf. Paper I)
S KQ =
2hν30
2 j + 1
2 j′ + 1
Q, (9)
we obtain the statistical equilibrium equations in the compact
(1 + ǫKQ + δ
+ iΓQ)S KQ
= w(K)j′ j (−1)
−Q + ǫ
. (10)
J. Štěpán and V. Bommier: Generalized
ǫ-law 3
3. Solution of the Wiener-Hopf equations
From now on we reduce our analysis to the case of semi-infinite,
plane-parallel geometry with constant magnetic field along the
atmosphere. The velocity distribution and volume density of col-
liders is also constant along the atmosphere but it is in general
non-thermal. The only position coordinate is the common line
optical depth τ.
Following the procedure of Paper I a formal solution of ra-
diative transfer equation is substituted into the definition of ten-
sor J
Q; after that we obtain a set of integral Wiener–Hopf equa-
tions of the 2nd kind,
(1 + ǫKQ + δ
+ iΓQ)S KQ(τ)
K̃KQ,K′Q′ (τ, τ
′)S K
Q′ (τ
′)dτ′ + ǫKQ B
, (11)
which describe coupling of the tensors ρKQ(τ) at differ-
ent optical depths via radiation. Several important prop-
erties of kernels K̃KQ,K′Q′ (τ, τ
′) have been discussed by
Landi Degl’Innocenti et al. (1990) and in Paper I Using their in-
dexing notation one can rewrite the equation (11) in the short-
handed form
aiS i(τ) =
Ki j(|τ − τ′|)S j(τ′)dτ′ + bi, (12)
ai = 1 + ǫ
Q + δ
+ iΓQ, (13)
bi = ǫ
. (14)
The index i in these expressions runs between the limits 1 and
N, where N is the number of KQ-multipoles. In the following
we briefly repeat the derivation performed by Frisch & Frisch
(1975) emphasizing the differences due to presence of bi terms.
Calculation of the derivative of (12) with respect to τ, split-
ting the integral on the right-hand side into two parts, multiplica-
tion of the equation by S i(τ), summation over index i, and finally
integration with respect to τ leads to the set of equations
S i(τ)
dS i(τ)
S j(0)
Ki j(τ)S i(τ)dτ
dτS i(τ)
dτ′Ki j(|τ′ − τ|)
dS j(τ
. (15)
The left-hand side of (15) is easily evaluated as
S i(∞)2 − S i(0)2
The first term on the right-hand side of (15) is evaluated using
the kernels symmetry Ki j(t) = K ji(t) and the equation (12), so
that we obtain
S i(0) [aiS i(0) − bi] , (17)
while the second term equals
S i(∞)2 − S i(0)2
bi [S i(∞) − S i(0)] . (18)
We put these results into (15) to get
aiS i(0)
biS i(∞). (19)
Calculation of the limit τ → ∞ of both sides of the equation
(12) leads to the set of linear algebraic equations for the compo-
nents of source function tensor in the infinite depth:
a jδi j −
Ki j(t)dt
S j(∞) = bi. (20)
We can solve these equations and write
S(∞) = L−1b, (21)
where S is the formal vector of S i components, b is the formal
vector of bi components, and the elements of matrix L are de-
fined by relation
{L}i j = a jδi j −
Ki j(t)dt. (22)
Establishing a new matrix ℓ = L−1 and substituting (21) into
(19) leads to the generalized form of the
ǫ-law
aiS i(0)
bib jℓi j. (23)
4. Particular solutions
Setting the special conditions for magnetic field and collisional
rates, one recovers the less general but more common and ex-
plicit formulations of the
ǫ-law than the one given by (23).
In the following sections we will verify this result in the limit-
ing conditions assumed in recent papers and we will analyse the
simple examples of non-thermal collisional excitation.
4.1. Maxwellian velocity distribution of colliders
In the case of Maxwellian velocity distribution of colliders, re-
laxation rates of all multipole components ρKQ are the same:
(CRj′ j)
Q = C
j′ j, (24)
where CRj′ j is the usual relaxation rate for collisional deexcitation
from j′ to j. For excitation rates one has
(C j j′)
C j j′√
2 j′ + 1
δK0δQ0, (25)
where the factor (2 j′ + 1)−1/2 has been introduced to make a
connection with the usual collisional rate C j j′ of standard unpo-
larized theory. In this isotropic case, there is no collisional exci-
tation of higher ranks of density matrix. From the assumption of
thermodynamic equilibrium one has
C j j′
CRj′ j
2 j′ + 1
2 j + 1
e−hν0/kBT , (26)
where kB stands for the Boltzmann constant and T for a tem-
perature of the atmosphere. From (24) and (7) it is evident that
ǫKQ = ǫ for all possible K and Q, where ǫ is the common photon
destruction probability. Further
B(KQ) = BPδK0δQ0. (27)
4 J. Štěpán and V. Bommier: Generalized
ǫ-law
Substituting the rates (24) and (25) into the formula (22) and
employing the general identity
−∞ Ki1(t)dt = δi1 (see Paper I)
together with bi = δi1, we recover form (23) the formula (16) of
the previously cited paper:
(1 + ǫ + δ(K) + iΓQ)[S KQ(0)]
= ǫB2P. (28)
Assuming that there is zero magnetic field, i.e. Γ = 0, the
source function tensor reduces due to symmetry reasons to the
two non-vanishing components S 00 and S
0 in the reference frame
Σ1. This reference frame is suitable for descriptions of the atomic
system under these conditions, so that we may identify Σ0 ≡ Σ1,
with X and Y axes oriented arbitrary in the plane parallel to atmo-
spheric surface. Further, assuming that there is no depolarization
of the upper level (δ(K) = 0), we realize from (28):
S 00(0)
S 20(0)
1 + ǫ
ǫ′BP, (29)
which is the same result derived in different notation by Ivanov
(1990). For simplicity the common alternative to the photon de-
struction probability has been introduced: ǫ′ = ǫ/(1 + ǫ).
If depolarization of the upper level is high enough to destroy
atomic level polarization (δ(K) → ∞ for K > 0), or the upper
level is unpolarizable, the common
ǫ-law for scalar radiation
is recovered,
S 00(0) =
1 + ǫ
ǫ′BP. (30)
4.2. Anisotropic alignment (de)excitation
The relation ǫKQ = ǫ is not in general satisfied for all the mul-
tipoles because the relaxation of the ρKQ state depends on the
velocity distribution of colliders. In the following text we will
neglect the effects of magnetic field.
Let us assume an example of a relative velocity distribution
of particles that is axially symmetric with the axis of symmetry
parallel to the vertical of the atmosphere (so that it is as in the
former case Σ0 ≡ Σ1) and that the collisional interaction can be
fully described by only the first two even multipole components
of this distribution. Thanks to these assumptions the only non-
vanishing excitation collisional rates are (C j j′ )
0 and (C j j′ )
0, the
relaxation rates (CRj′ j)
0 and (C
j′ j)
0 and for the same reasons the
only non-zero source function components are S 00 and S
An explicit evaluation of the integrals of kernels∫ ∞
−∞ K̃KQ,K′Q′ (τ, τ
′)dτ′ under these conditions shows that
the only non-zero ones are given by (A5) and (A12) of
Landi Degl’Innocenti et al. (1991b). In our notation they read
K̃00,00(τ, τ
′)dτ′ = 1, (31)
K̃20,20(τ, τ
′)dτ′ =
W2, (32)
with W2 = (w
j′ j)
2. Substituting these results into (23) we see that
(1 + ǫ00 )(S
+ (1 + ǫ20 )(S
(00))2 +
(ǫ20 B
(20))2
1 + ǫ20 −
10 W2
. (33)
To check the validity of polarized radiative transfer codes,
it is advantageous if one can verify that the transfer of higher
ranks of the radiation tensor is accurate enough. In the realistic
scattering polarization models the polarization degree does not
exceed a few percent so that |S 00(0)| ≫ |S
Q(0)|. By setting ar-
bitrary (even unphysical) collisional rates it is possible to verify
transfer codes in conditions with |S 00| ≪ |S
To privilege transfer in higher ranks of the radiation tensor
one can artificially suppress the excitation rate (C j j′)
0. In the
extremal case one can set (C j j′)
0 → 0. The easiest way to do
this is the formal interchange of the role of excitation rates of
population and alignment, i.e. (C j j′ )
0 ↔ (C j j′)
0 of the original
Maxwellian velocity distribution:
(C j j′)
0 = 0, (C j j′ )
C j j′√
2 j′ + 1
(no collisional excitation to upper level population) and the re-
laxation rates set to the Maxwellian ones
(CRj′ j)
0 = (C
j′ j)
0 = C
j′ j. (35)
In this case we have
B(00) = 0, B(20) = BP, (36)
and again
0 = ǫ
0 = ǫ. (37)
Substituting this into (33) we find out the
ǫ-law in the form
[S 00(0)]
2 + [S 20(0)]
1 − 710 W2(1 − ǫ
BP. (38)
The particular collisional rates (34) are in fact arbitrary and
have been chosen to obtain a formula similar to the one of the
Maxwellian distribution case.
This relation is useful to test polarized radiative transfer
codes, because in this unphysical case S 20(0) is the largest term,
unlike the physical case where the largest term is S 00(0) and S
is only a few percent of it. By applying Eq. (38) the test is much
more sensitive to the polarization, and the polarization is better
tested. We have thus successfully tested a multilevel non-LTE
radiative transfer code that we are developing, but this code and
its results are the subjects of a forthcoming paper.
5. Conclusions
We have derived a more general formulation of the so-called√
ǫ-law of radiation transfer. This analytical condition couples
the value of source function tensor of a two-level atom with
other physical properties of the atmosphere. The simplest re-
sult obtained in conditions of a non-magnetic, isothermal, plane-
parallel, semi-infinite atmosphere with thermal velocity distri-
bution of particles and unpolarized atomic levels (e.g. Mihalas
1970) has been generalized by Ivanov (1990) to account for scat-
tering of polarized radiation and polarized upper atomic level.
Further generalizations done in Paper I, which account for a
magnetic field of arbitrary strength and direction, has been ex-
tended in the present paper to account for non-thermal colli-
sional interactions. It was done by introducing the tensor of the
photon destruction probability ǫKQ and by defining the function
B(KQ).
The resulting formula (23) reduces to the cases mentioned
above if the physical conditions become more symmetric. On
J. Štěpán and V. Bommier: Generalized
ǫ-law 5
the other hand, situations with a high degree of perturbers ve-
locity distribution anisotropy and especially ones with unphysi-
cal collisional rates result in a wide range of models which can
be calculated both numerically and analytically. Thus they offer
new possibilities for verification of the non-LTE radiation trans-
fer codes.
References
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Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991b, A&A, 244,
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Mihalas, D. 1970, Stellar Atmospheres (W. H. Freeman and Company)
Introduction
Equations of statistical equilibrium
Solution of the Wiener-Hopf equations
Particular solutions
Maxwellian velocity distribution of colliders
Anisotropic alignment (de)excitation
Conclusions
|
0704.1577 | Sharp dark-mode resonances in planar metamaterials with broken
structural symmetry | Sharp dark-mode resonances in planar metamaterials with broken structural
symmetry
V. A. Fedotov,1, ∗ M. Rose,1 S. L. Prosvirnin,2 N. Papasimakis,1 and N. I. Zheludev1, †
Optoelectronics Research Centre, University of Southampton, SO17 1BJ, UK
Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002, Ukraine
(Dated: October 29, 2018)
We report that resonant response with a very high quality factor can be achieved in a planar
metamaterial by introducing symmetry breaking in the shape of its structural elements, which
enables excitation of dark modes, i.e. modes that are weakly coupled to free space.
PACS numbers: 78.67.-n, 42.70.-a, 42.25.Bs
Metamaterials research has attracted tremendous
amount of attention in the recent years. The interest is
mainly driven by the opportunity of achieving new elec-
tromagnetic properties, some with no analog in naturally
available materials. Extraordinary transmission [1], ar-
tificial magnetism and negative refraction [2], invisible
metal [3], magnetic mirror [4], asymmetric transmission
[5] and cloaking [6] are just few examples of the new
phenomena emerged from the development of artificially
structured matter.
The exotic and often dramatic physics predicted for
metamaterials is underpinned by the resonant nature of
their response and therefore achieving resonances with
high quality factors is essential in order to make metama-
terials’ performance efficient. However, resonance quality
factors (that is the resonant frequency over width of the
resonance) demonstrated by conventional metamaterials
are often limited to rather small values. This comes from
the fact that resonating structural elements of metama-
terials are strongly coupled to free-space and therefore
suffer significant losses due to radiation. Furthermore,
conventional metamaterials are often composed of sub-
wavelength particles that are simply unable to provide
large-volume confinement of electromagnetic field neces-
sary to support high-Q resonances. As recent theoretical
analysis showed, high-Q resonances involving dark (or
closed) modes are nevertheless possible in metamaterials
if certain small asymmetries are introduced in the shape
of their structural elements [7].
In this Letter we report the observation of exception-
ally narrow resonant responses in transmission and reflec-
tion of planar metamaterial achieved through introduc-
ing asymmetry into its structural elements. The appear-
ance of narrow resonances is attributed to the excitation
of otherwise forbidden anti-symmetric modes, that are
weakly coupled to free-space (”dark modes”).
Metamaterials that were used in our experiments con-
sisted of identical sub-wavelength metallic ”inclusions”
structured in the form of asymmetrically split rings
(ASR), which were arranged in a periodic array and
placed on a thin dielectric substrate (see Fig. 1). ASR-
patterns were etched from 35 µm copper cladding cov-
ering IS620 PCB substrate of 1.5 mm thickness. Each
copper split ring had the radius of 6 mm and width
of 0.8 mm and occupied a square translation cell of
15×15mm (see Fig. 1). Such periodic structure does not
diffract normal incident electromagnetic radiation for fre-
quencies lower than 20 GHz. The overall size of the sam-
ples used were approximately 220× 220 mm. Transmis-
sion and reflection of a single sheet of this meta-material
were measured in an anechoic chamber under normal in-
cidence conditions using broadband horn antennas.
FIG. 1: (Color online) Fragments of planar metamaterials
with asymmetrically split copper rings. The dashed boxes
indicate elementary translation cells of the structures.
We studied structures with two different types of asym-
metry designated as type A and B in Fig. 1. The rings of
type A had two equal splits dividing them into pairs of
arcs of different length corresponding to 140 and 160 deg
(see Fig. 1A). The rings of type B were split along their
diameter into two equal parts but had splits of different
length corresponding to 10 and 30 deg (see Fig. 1B).
http://arxiv.org/abs/0704.1577v1
Transmission and reflection properties of structures of
both types depended strongly on the polarization state
of incident electromagnetic waves. The most dramatic
spectral selectivity was observed for electrical field being
perpendicular to the mirror line of the asymmetrically
split rings, which corresponded to x-polarization in the
case of structure A and y-polarization for structure B
(as defined in Fig. 1). For the orthogonal polarizations
the ASR-structures did not show any spectral features
originating from asymmetrical structuring.
FIG. 2: (Color online) (a) Normal incidence reflection and
transmission spectra of A-type metamaterial (presented in
Fig. 1A) for x-polarization: solid line - experiment, filled cir-
cles - theory (method of moments), empty circles - theory
for reference structure with symmetrically split rings. (b) x-
Component of the instantaneous current distribution in the
asymmetrically split rings corresponding to resonant features
I, II and III as marked in section (a). Arrows indicate in-
stantaneous directions of the current flow, while their length
corresponds to the current strength.
The results of reflection and transmission measure-
ments of metamaterial A obtained for x-polarization are
presented in Fig. 2a. The reflection spectrum reveals an
ultra-sharp resonance near 6 GHz (marked as II), where
reflectivity losses exceed -10 dB. It is accompanied by
two much weaker resonances (marked as I and III) cor-
responding to reflection peaks at about 5.5 and 7.0 GHz
respectively. The sharp spectral response in reflection
is matched by a very narrow transmission peak reaching
-3 dB and having the width of only 0.27 GHz as mea-
sured at 3 dB below the maximum. The quality factor
Q of such response is 20, which is larger than that of the
most metamaterials based on lossy PCB substrates by at
least one order of magnitude. On both sides of the peak
the transmission decreases resonantly to about -35 dB at
frequencies corresponding to reflection maxima.
Fig. 3a presents transmission and reflection spectra
of B-type metamaterial measured for y-polarization. A
very narrow resonant transmission dip can be seen near
5.5 GHz, where transmission drops to about -5 dB.
The corresponding reflection spectrum shows an usually
sharp roll-off (I-II) between -4 and -14 dB spanning only
0.13 GHz at around the same frequency. At the fre-
quency of about 11.5 GHz the ASR-structure exhibits its
fundamental reflection resonance (marked as III) where
the wavelength of excitation becomes equal to the length
of the arcs.
To understand the resonst nature of the response, the
ASR-structures were modelled using the method of mo-
ments. It is a well established numerical method, which
involves solving the integral equation for the surface cur-
rent induced in the metal pattern by the field of the in-
cident wave. This is followed by calculations of scattered
fields as a superposition of partial spatial waves. The
metal pattern is treated as a perfect conductor, while
the substrate is assumed to be a lossy dielectric. For
both transmission and reflection the theoretical calcula-
tions show a very good agreement with the experimental
results assuming ǫ = 4.07+ i ·0.05 (see Fig. 2 and 3, filled
circles). For comparison we also modelled metamaterial
composed of split rings with no structural asymmetry,
i.e. equally split along their diameter. Our calculations
indicate that for both polarizations the response of such
structure is free form sharp high-Q resonant feature (see
Fig. 2 and 3, open circles).
The origin of the unusually strong and narrow spec-
tral responses of the ASR-structures can be traced to
so-called ”dark modes”, i.e. electromagnetic modes that
are weakly coupled to free-space. It is this property of the
dark modes that allows in principal to achieve high qual-
ity resonances in very thin structures [7]. These modes
are usually forbidden but can be excited in a planar meta-
material if, for example, its particles have certain struc-
tural asymmetry.
Our calculations showed that in the case of structure A
an anti-symmetric current mode can dominate the usual
symmetric one: at the high-Q transmission resonance, as
shown in Fig. 2b (II), two parts of the ring are excited
in anti-phase while currents have almost the same ampli-
tude. The scattered electromagnetic fields produced by
such current configuration are very weak, which dramat-
ically reduces coupling to free-space and therefore radia-
FIG. 3: (Color online) (a) Normal incidence reflection and
transmission spectra of B-type metamaterial (presented in
Fig. 1B) for y-polarization: solid line - experiment, filled cir-
cles - theory (method of moments), empty circles - theory
for reference structure with symmetrically split rings. (b) y-
Component of the instantaneous current distribution in the
asymmetrically split rings corresponding to resonant features
I, II and III as marked in section (a). Arrows indicate in-
stantaneous directions of the current flow, while their length
corresponds to the current strength.
tion losses. As a consequence, the strength of the induced
currents can reach very high values and therefore ensures
high quality factor of the response. At the reflection res-
onances, in contrast to the ”dark mode” regime, currents
in both sections of the asymmetrically split ring oscillate
in phase but excitation of one of the sections dominates
the other (see Fig. 2b (I and III)). Importantly, the ampli-
tudes of the currents in this case are significantly smaller
than in ”dark mode” resonance, which yields lower Q-
factors for this type of the response. If the structural
asymmetry is removed the anti-symmetric current mode
becomes forbidden while two reflection resonances de-
generate to a single low-Q resonance state where both
parts of the ring are excited equally. Thus introduction
of asymmetry in the split-ring structure effectively allows
to create a very narrow pass-band inside its transmission
stop-band. This effect is somewhat analogous to appear-
ance of an allowed state in the bandgap of photonic crys-
tals due to structural defects.
Interpretation of the results obtained for structure B
appears to be slightly more elaborate. From the symme-
try of the split rings it follows that for y-polarized excita-
tion at any frequency current distribution in the opposite
sections of the ring should have equal y-components os-
cillating in phase and equal x-components oscillating in
anti-phase. The net current in the ring has therefore al-
ways zero x-component, while its y-component can not
be fully compensated due to the structural asymmetry.
At low frequencies the net y-component is small but it
increases significantly as the frequency of excitation ap-
proaches 5.5 GHz. At this frequency the wavelength
becomes equal to circumference of the split ring and, as
shown in Fig. 3b (I), the right side of the ring dominates
its left side oscillating in anti-phase. The later results
in a resonant increase of the metamaterial reflection (see
Fig. 3a). Immediately above this resonance contributions
of both sides of the ASR-particle are still in anti-phase
but become nearly identical (see Fig. 3b (II)) making the
y-component of the induced net current almost zero and
therefore dramatically reducing radiation losses (reflec-
tion). Further increase of the excitation frequency leads
to rise of the reflection until the fundamental resonance
of the ASR-structure is reached where the correspond-
ing wavelength is equal to the length of the arcs. In this
case both sides of the ASR-particle oscillate in phase and
equally contribute to electromagnetic field scattering (see
Fig. 3b (III)).
The quality factor of the dark-mode resonances will in-
crease on reducing the degree of asymmetry of metama-
terial particles and in case of low dissipative losses can be
made exceptionally high. In the microwave region met-
als are almost perfect conductors and the main source of
dissipative losses is the substrate material (dielectrics).
Therefore significantly higher resonance quality factors
can be achieved for a free-standing thin metal film, which
is patterned complimentary to ASR-structure, i.e. pe-
riodically perforated with ASR-openings. In the visi-
ble and IR spectral ranges, however, losses in metals
dominate and therefore nano-scaled versions of the orig-
inal metal-dielectric ASR-structures would perform bet-
ter. According to our estimates Q-factor of such ASR-
nanostructures in the near-IR can be as high as 6.
In summary, we experimentally and theoretically
showed that a new type of planar metamaterials com-
posed of asymmetrically split rings exhibit unusually
strong high-Q resonances and provide for extremely nar-
row transmission and reflection pass- and stop-bands.
The metamaterials’ response has a quality factor of about
20, which is one order of magnitude larger than the typ-
ical value for many conventional metamaterials. This is
achieved via weak coupling between ”dark modes” in the
resonant inclusions of the ASR-metamaterial and free-
space, while weak symmetry breaking enables excitation
of so-called ”dark modes”. Achieving the ”dark mode”
resonances will be especially important for metamate-
rials in the optical part of the spectrum, where losses
are significant and unavoidable. In a certain way such
symmetry-breaking resonances in meta-materials resem-
bles the recently identified spectral lines of plasmon ab-
sorbtion of shell nanoparticle appearing due to asymme-
try [8].
The authors would like to acknowledge the financial
support of the EPSRC (UK) and Metamorphose NoE.
∗ Electronic address: [email protected]
† URL: www.nanophotonics.org.uk
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mailto:[email protected]
www.nanophotonics.org.uk
|
0704.1578 | Proper motion L and T dwarf candidate members of the Pleiades | Mon. Not. R. Astron. Soc. 000, 1–11 (2007) Printed 3 December 2018 (MN LATEX style file v2.2)
Proper motion L and T dwarf candidate members of the Pleiades
S. L. Casewell1⋆, P. D. Dobbie1,2, S. T. Hodgkin3, E. Moraux4, R. F. Jameson1,
N. C. Hambly5, J. Irwin3 and N. Lodieu6,1
1Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK
2Anglo-Australian Observatory, PO Box 296, Epping NSW 1710 Australia
3CASU, Institute of Astronomy,University of Cambridge, Maddingley Road, Cambridge, CB3 0HA, UK
4Laboratoire d’Astrophysique, Observatoire de Grenoble, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
5Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh,
Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ
6 Instituto de Astrofı́sica de Canarias, Vı́a Láctea s/n, E-38205 La Laguna, Tenerife, Spain
January 2007
ABSTRACT
We present the results of a deep optical-near-infrared multi-epoch survey covering 2.5 square
degrees of the Pleiades open star cluster to search for new very-low-mass brown dwarf mem-
bers. A significant (∼ 5 year) epoch difference exists between the optical (CFH12k I-, Z-
band) and near infrared (UKIRT WFCAM J-band) observations. We construct I,I-Z and Z,Z-J
colour magnitude diagrams to select candidate cluster members. Proper motions are computed
for all candidate members and compared to the background field objects to further refine the
sample. We recover all known cluster members within the area of our survey. In addition,
we have discovered 9 new candidate brown dwarf cluster members. The 7 faintest candidates
have red Z-J colours and show blue near-infrared colours. These are consistent with being L
and T-type Pleiads. Theoretical models predict their masses to be around 11MJup.
Key words: stars: low-mass, brown dwarfs, open clusters and associa-
tions:individual:Pleiades
1 INTRODUCTION
The initial mass spectrum (IMS), the number of objects manufac-
tured per unit mass interval, is an outcome of the star formation
process which can be constrained via observation. Consequently,
empirical determinations of the form of the IMS can be used to crit-
ically examine our theoretical understanding of the complexities of
star formation. In recent years there has been a particular empha-
sis on building a solid comprehension of the mechanisms by which
very-low-mass stars, brown dwarfs and free-floating planetary mass
objects form (e.g. Boss 2001; Bate 2004; Goodwin, Whitworth &
Ward-Thompson 2004; Whitworth & Goodwin 2005). Neverthe-
less, one key question which remains unanswered is what is the
lowest possible mass of object that can be manufactured by the star
formation process? From a theoretical stance, traditional models
predict that if substellar objects form like stars, via the fragmenta-
tion and collapse of molecular clouds, then there is a strict lower
mass limit to their manufacture of 0.007-0.010 M⊙ (Padoan &
Nordlund, 2002). This is set by the rate at which the gas can radiate
away the heat released by the compression (e.g. Low & Lynden-
Bell, 1976). However, in more elaborate theories, magnetic fields
could cause rebounds in collapsing cloud cores which might lead
⋆ E-mail: [email protected]
to the decompressional cooling of the primordial gas, a lowering of
the Jeans mass and hence the production of gravitationally bound
fragments with masses of only ∼0.001 M⊙ (Boss, 2001). In con-
trast, if feedback from putative winds and outflows driven by the
onset of deuterium burning play a role, the smallest objects which
form via the star formation process may be restricted to masses
equal to or greater than the deuterium burning limit (∼0.013 M⊙;
Adams & Fatuzzo, 1996).
Recent work on very young clusters (τ<10 Myrs) and star
formation regions e.g. σ-Orionis, the Trapezium, IC348 and Up-
per Sco (Béjar et al., 2001; Muench et al., 2002; Muench et al.,
2003; Lodieu et al., 2007a) suggests that the initial mass function
continues slowly rising down to masses of the order M∼0.01 M⊙,
at least in these environments. Indeed, it has been claimed that an
object with a mass as low as 2-3 MJup has been unearthed in σ-
Ori (Zapatero-Osorio et al, 2002). However, the cluster member-
ship of σ-Ori 70 is disputed by Burgasser et al. (2004). Further-
more, mass estimates for such young substellar objects derived by
comparing their observed properties to the predictions of theoreti-
cal evolutionary tracks remain somewhat controversial. Baraffe et
al. (2002) have shown that to robustly model the effective temper-
ature and luminosity of a low mass object with an age less than
∼1 Myr, evolutionary calculations need to be coupled to detailed
simulations of the collapse and accretion phase of star formation.
c© 2007 RAS
http://arxiv.org/abs/0704.1578v2
2 S. L. Casewell et al.
As the current generation of evolutionary models start from arbi-
trary initial conditions, theoretical predictions for ages less than a
few Myrs must be treated with a fair degree of caution. Indeed, the
few available dynamical mass measurements of pre-main sequence
objects indicate that models tend to underestimate mass by a few
tens of percent in the range 0.3<∼M<∼1.0 M⊙ (see Hillenbrand &
White, 2004 for review). A recent dynamical mass measurement of
the 50-125 Myrs old object AB Dor C (spectral type ∼M8), the first
for a pre-main sequence object with M<0.3 M⊙, suggests that the
discrepancy between model predictions and reality might be even
larger at lower masses, with the former underestimating mass by a
factor 2-3 at M∼0.1 M⊙ (Close et al., 2005). However, this conclu-
sion is dependent on the assumed age of AB Dor, which is currently
a matter of great contention (Luhman, Stauffer & Mamajek, 2005;
Janson et al., 2006). On the positive side, Zapatero-Osorio et al.,
(2004) have determined the masses of the brown dwarf binary com-
ponents of GJ 569 Bab and their luminosities and effective temper-
atures are in agreement with theoretical predictions, for an age of
300 Myr. More recently, Stassun, Mathieu & Valenti (2006) discuss
an eclipsing brown dwarf binary in the Orion nebula star forming
region and find the large radii predicted by theory for a very young
dwarf. Surprisingly, they find that the secondary is hotter than the
more massive primary. Clearly further work is still needed to sup-
port the predictions of theoretical models.
It is clearly important to search for the lowest mass objects,
not only in the young clusters, but also in more mature clusters,
such as the Pleiades. The results of previous surveys of the Pleiades
indicate that the present day cluster mass function, across the stel-
lar/substellar boundary and down to M∼0.02 M⊙ (based on the
evolutionary models of the Lyon Group), can be represented by a
slowly rising power law model, dN/dM∝M−α . For example, from
their Canada-France-Hawaii Telescope (CFHT) survey conducted
at R and I and covering 2.5 sq. degrees, Bouvier et al. (1998) identi-
fied 17 candidate brown dwarfs (IC>17.8) and derived a power law
index of α=0.6. From their 1.1 sq degrees Isaac Newton Telescope
(INT) survey conducted at I and Z, with follow-up work undertaken
at K, Dobbie et al. (2002) unearthed 16 candidate substellar mem-
bers and found a power law of index α=0.8 to be compatible with
their data. Jameson et al. (2002) showed that a powerlaw of index
α=0.41±0.08 was consistent with the observed mass function over
the range 0.3>∼M>∼0.035 M⊙. This study used a sample of 49 prob-
able brown dwarf members assembled from the four most extensive
CCD surveys of the cluster available at the time, the International
Time Project survey (Zapatero Osorio et al., 1998), the CFHT sur-
vey (Bouvier et al., 1998; Moraux, Bouvier & Stauffer, 2001), the
Burrell Schmidt survey (Pinfield et al., 2000) and the INT survey
(Dobbie et al., 2002). The CFHT survey was subsequently extended
to an area of 6.4 sq. degrees (at I and Z) and unearthed a total of
40 candidate brown dwarfs. Moraux et al. (2003) applied statistical
arguments to account for non-members in their sample and derived
a power law index of α=0.6. Most recently, Bihain et al. (2006)
have used deep R, I, J and K band photometry and proper motion
measurements to unearth 6 robust L type Pleiades members in an
area of 1.8 sq. degrees with masses in the range 0.04-0.02 M⊙ and
derived a power law index of α=0.5±0.2.
Here we report the results of a new optical/infrared survey of
2.5 sq. degrees of the Pleiades, the aim of which is to extend empir-
ical constraints on the cluster mass function down to the planetary
mass regime (M∼0.01 M⊙). In the next section we describe the ob-
servations acquired/used as part for this study, their reduction, their
calibration and their photometric completeness. In subsequent sec-
tions we describe how we have identified candidate brown dwarf
members on the basis of colours and proper motions. We use our
new results to constrain the form of the cluster mass function and
conclude by briefly discussing our findings in the context of star
formation models.
2 OBSERVATIONS, DATA REDUCTION AND SURVEY
COMPLETENESS
2.1 The J band imaging and its reduction
Approximately 3.0 square degrees of the Pleiades cluster was ob-
served in the J band using the Wide Field Camera (WFCAM)
on the United Kingdom Infrared Telescope (UKIRT) between the
dates of 29/09/2005 and 08/01/2006. WFCAM is a near infrared
imager consisting of 4 Rockwell Hawaii-II (HgCdTe 2048x2048)
arrays with 0.4” pixels, arranged such that 4 separate pointings
(pawprints) can be tiled together to cover a 0.75 sq. degree region
of sky (see http://www.ukidss.org/technical/technical.html for dia-
gram). A total of four tiles were observed in a mixture of photo-
metric and non-photometric conditions but in seeing of typically ≈
1.0 arcsecond or better. To ensure that the images were properly
sampled we employed the 2×2 microstep mode. The locations on
the sky of our tiles (shown in Figure 1) were chosen to provide
maximum overlap with the optical fields surveyed in 2000 by the
Canada-France-Hawaii telescope and CFH12k camera but also to
avoid bright stars and areas of significant interstellar extinction.
The images were reduced at the Cambridge Astronomical Sur-
vey Unit (CASU) using procedures which have been custom writ-
ten for the treatment of WFCAM data. In brief, each frame was de-
biased, dark corrected and then flat fielded. The individual dithered
images were stacked before having an object detection routine run
on them. The detection procedure employs a core radius of 5 pixels,
and identifies objects as islands of more than 4 interconnected pix-
els with flux >1.5σ above the background level. The frames were
astrometrically calibrated using point sources in the Two micron
All Sky Survey (2MASS) catalogue. These solutions, in general,
had a scatter of less than 0.1 arcseconds. The photometric calibra-
tion employed by the CASU pipeline also relies on 2MASS data
(there are typically hundreds of 2MASS calibrators per detector)
and is found to be accurate to ≈2% in good conditions (see Warren
et al., 2007, Hodgkin et al., 2007 for details).
In measuring our photometry we used an aperture of 2”, which
is approximately twice the core radius of point sources. This 2” di-
ameter of the aperture is also twice the seeing FWHM. The reduc-
tion pipeline also attempts to classify each source depending on its
morphology (e.g. galaxy, star, noise). However, at the limit of the
data this classification becomes less reliable. Therefore, in our sub-
sequent analysis we chose to define as stellar all objects which lie
within 3 sigma of the stellar locus, where sigma is defined accord-
ing to Irwin et al. (in prep).
2.2 The far-red optical imaging and a new reduction
As part of this work we have used a subset (2.54 square degrees)
of the far-red optical data obtained in the course of the IZ survey of
the Pleiades conducted in 2000 by Moraux et al. (2003). The rele-
vant CFH12k data were extracted from the Canadian Astrophysical
Data Center archive and were reprocessed at Cambridge University
using the CASU optical imaging pipeline (Irwin & Lewis, 2001).
In brief, these data were bias subtracted and corrected for non-
linearity prior to flat fielding. Fringe maps, which were constructed
c© 2007 RAS, MNRAS 000, 1–11
http://www.ukidss.org/technical/technical.html
Proper motion L and T dwarf candidate members of the Pleiades 3
Figure 1. The regions imaged at I, Z and J with the CFHT and UKIRT. The CFH12k pointings (light rectangular outlines) are labelled alphabetically as in
Moraux et al. (2003), while the WFCAM tiles (bold square outlines) are labelled numerically, ranging from 1 to 4. Note that the observations avoid the region
of high reddening to the south of the Merope and the bright stars in vicinity of the cluster centre.
for each photometric band from images obtained during the observ-
ing run, were used to remove the effects of interference between
night sky lines in the CCD substrate. Subsequently, sources at a
level of significant of 3σ or greater were morphologically classified
and aperture photometry obtained for each. A World Coordinate
System (WCS) was determined for each frame by cross-correlating
these sources with the Automated Plate Measuring (APM) machine
catalogue (Irwin, 1985). The approximately 100 common objects
per CCD chip lead to an internal accuracy of typically better than
0.3 ”. The photometry was calibrated onto a CFH12k I and Z natu-
ral system using stars with near zero colour (B-V-R-I≈0) in Landolt
standard field SA98 (Landolt, 1992) which was observed the same
nights as the science data. The systematic errors in the photome-
try were calculated by comparing the photometry of overlapping
fields as in Moraux et al. (2003). The photometry was found to be
accurate to ≈3%.
2.3 The completeness of datasets
To estimate the completeness of our IR images, we injected fake
stars with magnitudes in the range J=12-22 into each of the 16
chips of every WFCAM frame and re-ran the object detection soft-
ware with the same parameters that were used to detect the real
sources. To avoid significantly increasing the density of all sources
in the data we inserted only 200 fake stars per chip in a given run.
To provide meaningful statistics we repeated this whole procedure
ten times. Subsequently, we calculated percentage completeness at
a given magnitude by taking the ratio of the number of fake stars
recovered to the number of fake stars injected into a given magni-
tude bin (and multiplying by 100). We note that a 100% recovery
rate was never achieved at any magnitude since a small proportion
of the fake stars always fell sufficiently close to other sources to
be overlooked by the object detection algorithm. This method was
also applied to determine the completeness of the I and Z band
CFH12k data. However, the magnitude range of the fake stars was
adjusted to be consistent with the different saturation and faint end
magnitude limits of these data. The results of this procedure for all
3 photometric bands are shown in Table 1.
A glance at this table indicates that the IR data are in general
90% complete to J≈19.7, although Field 3 is slightly less deep,
due to moonlight and poor seeing. In this case the proximity of
the moon led to higher background counts. The I data are typically
90% and 50% complete to I=22.5 and 23.5 respectively. The cor-
responding completeness limits for the Z band data are Z=21.5 and
22.5 respectively.
3 ANALYSIS OF THE DATA
3.1 Photometric selection of candidate cluster members
An initial photometrically culled sample of candidate brown dwarfs
has been obtained from the I,I-Z colour-magnitude diagram (Fig-
c© 2007 RAS, MNRAS 000, 1–11
4 S. L. Casewell et al.
0 1 2 3 4 5
Figure 2. The I,I-Z CMD for the whole of field 1. The solid line is the NEXTGEN model, and the dotted line the DUSTY model. The small points are all
objects that were classed as stellar in both I and Z data. The crosses are all objects that met the following selection criteria: classed as stellar in both I and Z
data, for 16.5 < I < 22.5, they must lie no more than 0.25 magnitudes to the left of the DUSTY isochrone, for I>22.5, they must lie to the right of the line,
I-Z= (I-19.0)/3.5. The filled squares are the previously identified cluster candidate members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al.
(1998), plotted to highlight the cluster sequence.
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 3. The Z,Z-J CMD for the whole of the survey. The solid line is the NEXTGEN model, and the dotted line the DUSTY model. The crosses are all the
objects selected from the I,I-Z (crosses on Figure 2.). The filled diamonds are all objects that met our selection criteria from the I,I-Z,and Z, Z-J CMDs. These
were selected for proper motion analysis, and were found to be non members. The filled squares are our candidate cluster members (objects that remained
after proper motion analysis). The squares are our ZJ only candidates for all four fields that remained after proper motion analysis. The previously identified
probable members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al. (1998) that remained after our proper motion analysis are identified by
open circles around the plotted symbols.
c© 2007 RAS, MNRAS 000, 1–11
Proper motion L and T dwarf candidate members of the Pleiades 5
Table 1. 50 and 90% completeness figures for the optical and infrared fields. The positioning of these fields is shown in Figure 1. Note that while WFCAM
field 1 corresponds to CFHT fields B, C, R and Q, the individual pawprints, do not correspond on a one to one basis - i.e. field1 00 does not correspond to field
Field name I Z WFCAM tile name WFCAM pawprint name J
50% 90% 50% 90% 50% 90%
B 23.2 22.5 22.3 21.5 field1 00 20.9 19.9
C 23.7 22.6 22.6 21.6 field1 01 20.9 20.1
R 24.0 23.0 22.9 21.6 field1 10 20.9 19.8
Q 23.7 22.5 22.7 21.6 field1 11 20.9 19.7
K 23.6 22.5 23.0 21.9 field2 00 20.9 19.7
L 24.0 22.7 23.0 21.8 field2 01 20.9 19.9
D 23.7 22.4 23.0 21.7 field2 10 21.0 19.9
field2 11 20.9 19.7
U 23.5 22.5 22.9 21.7 field3 00 19.5 18.8
V 23.8 22.5 22.7 21.7 field3 01 19.0 17.7
T 23.6 22.5 22.6 21.5 field3 10 19.6 18.6
field3 11 18.9 17.7
I 23.9 22.3 23.1 22.0 field4 00 20.8 19.7
G 23.7 22.7 23.4 22.3 field4 01 20.8 19.7
field4 10 20.8 19.7
field4 11 20.8 19.7
1 2 3 4 5 6 7
Figure 4. The J,I-J CMD for the whole of the survey. The solid line is the NEXTGEN model, and the dotted line the DUSY model. The crosses are all the
objects selected from the I,I-Z (crosses on Figure 2.). The filled diamonds are all objects that met our selection criteria from the I,I-Z,and Z, Z-J CMDs. These
were selected for proper motion analysis, and were found to be non members. The filled squares are our candidate cluster members (objects that remained
after proper motion analysis). The previously identified members from Bihain et al. (2006), Moraux et al. (2003) and Bouvier et al. (1998) that remained after
our proper motion analysis are identified by open circles around the plotted symbols.
ure 2) where the 120 Myr NEXTGEN (Baraffe et al., 1998) and
DUSTY (Chabrier et al., 2000) model isochrones (modified to take
into account the Pleiades distance of 134 pc e.g. Percival, Salaris
& Groenewegen, 2005) served as a guide to the location of the
Pleiades sequence. With the uncertainties in both the photometry
and the age of the cluster in mind, we selected all objects classed
as stellar in both the I and Z data, which in the magnitude range
16.5 < I < 22.5 lay no more than 0.25 magnitudes to the left of
the DUSTY isochrone. All the candidate Pleiads found by Moraux
et al. (2003) and Bihain et al. (2006) lay within ±0.25 magnitudes
of the DUSTY model. Thus our selection criterion is 0.25 magni-
tudes to the left of the DUSTY model. Below I=22.5, the DUSTY
model is not red enough to account for known field stars, and so
is inappropriate in this effective temperature regime. We have cal-
culated an approximate field star sequence from Tinney, Burgasser
& Kirkpatrick (2003) and Hawley et al (2002) and lowered it by
2 magnitudes. This results in the line I-Z= (I-19.0)/3.5. This se-
lection is conservative, and is particularly aimed at removing the
c© 2007 RAS, MNRAS 000, 1–11
6 S. L. Casewell et al.
bulk of the red tail of the background stars. Subsequently, the initial
list of candidates was cross-correlated with our J band photometric
catalogue (using a matching radius of 2 arcseconds) and a refined
photometrically culled sample obtained using the Z,Z-J colour-
magnitude diagram (Figure 3). These objects are also shown on the
J, I-J colour-magnitude diagram (Figure 4). As before, the 120 Myr
model isochrones were used as a guide to the location of the cluster
sequence. With the photometric uncertainties in mind, all candi-
dates with Z620 were retained. All candidates with 20<Z<21 and
Z-J>1.6 were also retained. Finally, all candidates with Z>21 and
Z-J>1.9 were retained. These constraints are conservative and are
based on the field L and T dwarfs sequence (Z-J>3, Chiu et al.,
2006) since the DUSTY models are known to be inappropriate in
this effective temperature regime. Since our survey is limited by the
depth of the I band data, all candidates with Z>20 and no I band
counterpart were also kept.
3.2 Refining the sample using astrometric measurements
To weed out non-members we have measured the proper motion of
each candidate brown dwarf, using the Z and J band data where
the epoch difference was 5 years. In this process, only objects lying
within 2 arcminutes of each candidate were chosen as potential as-
trometric reference stars. This compromise provided a sufficiently
large number of sources but at the same time minimised the effects
of non-linear distortions in the images. Furthermore, objects with
large ellipticity (>0.2), classed by the photometric pipeline as non-
stellar in the Z band data and with Z<16 or Z>20 were rejected.
This ensured that, in the main, the astrometric reference sources
were not of very low S/N in the J band or saturated in the optical
data. These criteria generally provided at least 20 suitable stars per
candidate brown dwarf.
Six coefficient transforms between the epoch 1 Z band im-
ages and the epoch 2 J band images were calculated using routines
drawn from the STARLINK SLALIB package. The iterative fitting
rejects objects having residuals greater than 3σ, where σ is robustly
calculated as the median of absolute deviation of the reference star
residuals, scaled by the appropriate factor (1.48) to yield an equiv-
alent RMS. Once the routine had converged the relative proper mo-
tions in pixels were calculated by dividing the fitting residuals of
each candidate by the epoch difference. For our data the epoch
difference is approximately 5 years. Subsequently, the astrometric
motion in milliarcseconds per year in RA and DEC was derived by
folding these values through the World Coordinate System trans-
form matrix of the relevant WFCAM image.
To estimate the errors on our proper motions measurements,
we have injected fake stars into both the Z and J band data, in a sim-
ilar fashion to that described in section 2.3. However, here we have
determined the difference between the inserted position and the
photometric pipeline estimate of the centroid of each star. Assum-
ing that the differences between these two locations are normally
distributed, we have divided the fake stars into 3 magnitude bins in
each photometric band (Z621, 23>Z>21, 24>Z>23, 21>J>17)
and fit 2d Gaussians to estimate the 1-sigma centroiding uncertainty
as a function of source brightness.
We find that in the Z band data, for objects with magnitudes
Z621, the centroiding uncertainty is equivalent to 3 mas yr−1 in
each axis, while for objects with 23>Z>21 this number increases
to 8 mas yr−1. For our faintest Z band objects, 24>Z>23, the cen-
troiding uncertainty is equivalent to 12 mas yr−1 in each axis. In
the J band data, for objects with magnitudes 21>J>17, the cen-
troiding uncertainty is equivalent to 5 mas yr−1 in each axis. Thus
for our brightest candidates (Z<21, J<19), the quadratic sum of
the Z and J band centroiding errors is less than or comparable to
the RMS of the residuals of the linear transform fit, which is typ-
ically 5-10 mas yr−1 in each axis. We adopt this latter quantity as
the proper motion uncertainty in both the RA and DEC directions
for these objects. It is worth noting at this point that both the stars
and brown dwarfs of the Pleiades appear to be in a state of dynam-
ical relation (e.g. Pinfield et al. 1998, Jameson et al. 2002), where
the velocity dispersion of the members is proportional to 1/M0.5,
where M is mass. Based on an extrapolation of the data in Figure
4 of Pinfield et al. (1998), we would expect our lowest mass brown
dwarf members (0.01-0.02M⊙) to have velocity dispersion of ∼ 7
mas yr−1. This velocity dispersion should be added quadratically
to the above uncertainties. Our final adopted proper motion selec-
tion, effectively a radius of 14 mas yr−1 , is described below, and
the velocity dispersion is small compared to this.
We fitted the proper motions of all of our photometric candi-
dates (excluding the ZJ only candidates) with a 2D Gaussian, which
centred around 1.1, -7 mas yr−1. This Gaussian had a σ of 14.0. We
were not able to fit two Gaussians, one to the background stars and
one to the Pleiades dwarfs, as described in Moraux et al.(2003),
since only ≈ 30 objects have the correct proper motion for cluster
membership. Consequently, we only selected objects to be proper
motion members if they had proper motions that fell within 1σ of
the proper motion of the cluster at +20.0, -40.0 mas yr−1 (Jones
1981; Hambly, Jameson & Hawkins, 1991; Moraux et al., 2001).
We required the selection criteria to be 1σ, as extending this to 2σ,
would seriously overlap with the field stars centred on 0,0. We did
however extend the selection criteria to 1.5σ, which yielded 14 ad-
ditional objects, however all were rejected due to their bright, but
blue (I≈17.0, I-Z<1.0) positions on the I,I-Z CMD, which led us
to believe that they were field objects. We also attempted to tighten
our selection criteria to a circle with radius 10 mas yr−1. This se-
lection meant that we lost as possible members objects PLZJ 78,
9, 77, 23 (see Table 4). PLIZ 79, 9 and 77 have all been identified
and confirmed as proper motion members by Bihain et al. (2006),
Moraux et al. (2001), and Bouvier et al. (1998). Unfortunately, as
we cannot fit two Gaussians to our data, we cannot calculate a prob-
ability of membership for these objects by the standard method as
defined by Sanders (1971). The proper motion measurements may
be found in Table 4, as well as the I, Z, J, H and K magnitudes for
these candidate members to the cluster.
We have attempted to use control data to determine the level
of contamination within our data, however, the numbers involved
are very small, so any calculated probability will be rather uncer-
tain. We used as controls, two circles of radius 14.0 mas yr−1, at
the same distance from 0,0 proper motion as the Pleiades. We then
separated the data into one magnitude bins, and calculated the prob-
ability for each magnitude bin, using equation 1.
Pmembership =
Ncluster −Ncontrol
Ncluster
Where Pmembership is the probability of membership for that mag-
nitude bin, Ncluster is the number of stars and contaminants within
the cluster circle in that magnitude bin. Ncontrol is the number of
dwarfs in the control circle of proper motion space, see Figure 4.
Ncluster - Ncontrol is the number of Pleiads. It can be seen that the
probability depends on where the control circle is located. Thus as
well as using control circles, we use an annulus and scale down the
count to an area equal to that of a control circle. Note that Figure
5 is for all of the magnitude bins together. Figure 6 is the same
as Figure 5, but for the ZJ selected objects only. The statistics are
c© 2007 RAS, MNRAS 000, 1–11
Proper motion L and T dwarf candidate members of the Pleiades 7
-100 -50 0 50 100
µαcos(δ) mas/yr
Figure 5. Proper motion vector diagram of the photometrically selected
candidate members. The filled triangles are candidate and known cluster
members. The filled diamonds and filled circles are the two separate control
clusters used. The annulus used for the radial method is also plotted.
-100 -50 0 50 100
µαcos(δ) mas/yr
Figure 6. Proper motion vector diagram of the photometrically selected
candidate members. The filled triangles are candidate cluster members se-
lected from the Z,Z-J CMD only. The filled diamonds and filled circles
are the two separate control clusters used. The annulus used for the radial
method is also plotted.
much poorer for the individual magnitude bins and the probabili-
ties are correspondingly more uncertain. It can be seen in Figure
4 that there is not a symmetrical distribution of proper motions.
In fact the distribution in the Vector point diagram, is a classical
”velocity ellipsoid” displaced from zero by reflex motion from the
Sun’s peculiar velocity, and happens to be in the direction of the
Pleiades proper motion vector. We have therefore probably under-
estimated the contamination, as the annulus method of calculating
probabilities assumes that the vector point diagram has a circularly
symmetric distribution of objects. These probabilities are shown in
Table 2, and probabilities derived in the same way but for the ZJ
only candidates can be found in Table 3.
An alternative approach to estimating the contamination is
the use the field L and T dwarf luminosity functions. Chabrier
(2005) gives the T dwarf luminosity function as being 10−3
dwarfs/pc3/unit J mag interval. Our 7 L and T dwarf candidates
cover a total of 0.7 mag in the J band. Note PLZJ 323 and 23 may
be late L dwarfs but we include them in this analysis. The volume of
space we use is 836 pc2, based on 2.5 square degrees and a distance
to the Pleiades of 134±30 pc (Percival et al., 2005). This distance
range corresponds to a distance modulus range of ±0.5 magni-
tudes, which is generous, given that the sequence shown in figure 8
is clearly narrower than ±0.5 magnitudes. Thus the expected num-
ber of contaminating field dwarfs is 0.6. In addition to this, field T
dwarfs are unlikely to have the same proper motion as the Pleiades,
thus reducing the 0.6 further. For the field L dwarfs with MJ≈13.0
(i.e. J≈18.5 at the distance of the Pleiades) the luminosity func-
tion is 3×10−4 dwarfs/pc3/unit J mag interval (Chabrier, 2005). A
similar calculation then gives 0.25 contaminating L dwarfs which
should be further reduced by considering proper motions. It is thus
clear that the field luminosity function indicates that contamination
by field L and T dwarfs should be negligible.
4 RESULTS
Most of these objects, except two bright objects and the faintest
seven have been documented before in surveys - Moraux et al
(2003) and Bihain et al (2006). We recovered all of these objects
within our overlapping area, and none were rejected by our IZ pho-
tometric selection. The objects we recovered were BRB 4, 8, 17, 13,
19, 21, 22, 27 and 28 and PLIZ 2, 3, 5, 6, 13, 14, 19, 20, 26, 28, 31,
34, 35 and 36. PLIZ 18, 27 and 39 were found to have no J counter-
part in our catalogues. Of these objects, BRB 19 and PLIZ 14 and
26 met by our selection criteria on the Z, Z-J CMD, however they
were too blue in their Z-J colour for their place on the sequence.
Out of the remaining objects we find that we agree with the proper
motion measurements as calculated by Bihain et al.(2006) for PLIZ
28, which we believe is a member of the cluster. We agree with Bi-
hain et al.(2006) over their candidates BRB 13 and BRB 19 that
they are not proper motion members to the cluster, however we dis-
agree with their proper motion measurement for BRB 19. We also
find that PLIZ 5 is a non member to the cluster - ie its proper mo-
tion measurement is not within 14 mas yr−1 of the cluster proper
motion value. We find that PLIZ 14 and 26 are not proper motion
members to the cluster, as well as not having met our selection cri-
teria. PLIZ 26 was found to have a proper motion measurement of
35.73±9.00, -25.83±6.96, which did not fall within 14 mas y−1
of the cluster, and also missed the selection made with the wider
circle (21 mas yr−1) as well. We find that PLIZ 19, 20, 34 and 36
are not proper motion members to the cluster. However this means
we disagree with Moraux et al. (2003), over their object PLIZ 20.
They find a proper motion of 25.6±7.3, -44.7±7.4 mas yr−1 for it.
Our proper motion measurement is 0.88± 15.86, -0.92±8.42 mas
yr−1. It is possible that this object has been adversely affected by
its position on the edge of one of the WFCAM chips, thus reducing
the number of reference stars used to calculate its proper motion.
An alternative method of measuring the proper motion using all the
objects on the same chip produced a measurement of 19.14±11.06,
-28.989±11.94 mas yr−1. This value does meet our selection cri-
teria, and has been previously accepted as a member. We suggest
PLIZ 20 is likely to be a member because of this.
We find that PLIZ 2, 3, 6, 31 and 35 are all proper motion
members to the cluster. In addition to this, we find 2 brighter new
candidate members to the cluster. These objects are bright enough
to have appeared in previous surveys, and in the UKIDSS Galactic
cluster survey (GCS). We also have 2 fainter new members to the
c© 2007 RAS, MNRAS 000, 1–11
Table 4. Name,coordinates, Z, I, J, H and K magnitudes for our members to the cluster. The errors quoted are internal (from photon counting). The systematic calibration errors are 2% in
the J, H and K wavebands (Warren et al., 2007), and 3% in the I and Z wavebands. The J, H and K magnitudes are on the MKO system. Previously discovered members also also have their
other known names listed from Moraux et al. (2003), Bihain et al. (2006) and Bouvier et al. (1998). The H and K band magnitudes are taken from the UKIDSS Galactic Cluster Survey with
the exceptions of PLZJ 23, 93, 721 and 235 which have their H band magnitudes listed from our H survey. The K band magnitude for PLZJ 93 is from our UFTI photometry, and PLZJ 23 is
from LIRIS service time. The final 5 objects in the table are our candidates selected from the ZJ data only.
Name Alternate RA dec µαcosδ µδ I Z J H K
name J2000.0 mas yr−1
PLZJ 29 BRB4 03 44 23.23 +25 38 45.11 23.40±8.24 -48.51±6.34 17.005 ± 0.001 16.163 ± 0.001 14.732 ± 0.001 14.132±0.004 13.744±0.004
PLZJ 56 03 44 53.51 +25 36 19.46 19.68±7.34 -35.63±5.29 17.012 ± 0.001 16.351 ± 0.001 15.250 ± 0.001 14.650±0.005 14.342±0.006
PLZJ 45 BRB8, CFHT-PL-7 03 52 58.2 +24 17 31.57 19.72±4.95 -42.37±7.44 17.101 ± 0.001 16.417 ± 0.001 15.247 ± 0.001 14.614±0.005 14.251±0.006
PLZJ 50 03 43 55.98 +25 36 25.45 13.48±8.24 -35.65±5.38 17.239 ± 0.001 16.496 ± 0.001 15.268 ± 0.001 14.693±0.006 14.319±0.006
PLZJ 60 CFHT-PL-10 03 44 32.32 +25 25 18.06 16.93±7.76 -43.15±5.72 17.592 ± 0.001 16.810 ± 0.001 15.460 ± 0.001 14.884±0.007 14.465±0.006
PLZJ 78 PLIZ2 03 55 23.07 +24 49 05.18 19.72±10.06 -29.74±10.45 17.719 ± 0.001 16.948 ± 0.001 15.574 ± 0.001 14.963±0.007 14.552±0.007
PLZJ 46 PLIZ3, BRB11 03 52 67.20 +24 16 01.00 19.55±5.15 -42.58±7.57 17.742 ± 0.001 16.945 ± 0.001 15.583 ± 0.001 14.966±0.007 14.503±0.008
PLZJ 9 PLIZ6,BRB9 03 53 55.09 +23 23 36.38 24.13±13.83 -50.10±22.71 17.752 ± 0.001 16.804 ± 0.001 15.222 ± 0.001 14.548±0.005 14.054±0.005
PLZJ 11 PLIZ20 03 54 05.33 +23 33 59.71 9.14±11.06 -28.98±11.94 19.571±0.004 18.563±0.004 16.691±0.005 15.980±0.016 15.436±0.016
PLZJ 77 PLIZ28,BRB18 03 54 10.04 +23 17 52.28 12.01±14.59 -51.60±15.84 20.760 ± 0.010 19.728 ± 0.010 17.647 ± 0.010 16.789±0.031 16.131±0.030
PLZJ 21 PLIZ31 03 51 47.65 +24 39 59.18 17.84±9.41 -44.92±8.03 20.944±0.014 19.762±0.013 17.575±0.012 16.774±0.026 16.089±0.028
PLZJ 10 PLIZ35,BRB15 03 52 31.19 +24 46 29.61 15.84±8.88 -49.34±6.24 21.293±0.018 20.292±0.016 18.181±0.022 17.118±0.041 16.506±0.0416
PLZJ 4 BRB21 03 54 10.25 +23 41 40.67 29.74±13.17 -38.46±8.88 21.322 ± 0.010 20.215 ± 0.013 18.171 ± 0.010 17.141±0.045 16.377±0.039
PLZJ 61 BRB22 03 44 31.27 +25 35 14.97 25.82±7.89 -40.21±8.47 22.043 ± 0.030 20.782 ± 0.026 18.298 ± 0.020 17.393±0.059 16.657±0.04
PLZJ 32 BRB27 03 44 27.27 +25 44 41.99 25.03±11.52 -38.65±23.46 22.235 ± 0.040 20.962 ± 0.029 18.871 ± 0.030 17.793±0.094 16.950±0.070
PLZJ 37 BRB28 03 52 54.92 +24 37 18.85 18.13±11.53 -48.68±11.38 22.452 ± 0.05 21.216 ± 0.041 18.839 ± 0.030 17.742±0.071 16.921±0.058
PLZJ 23 03 51 53.38 +24 38 12.11 20.75±10.51 -50.05±9.96 23.541 ± 0.140 22.187 ± 0.112 19.960 ± 0.100 19.362±0.100 18.510±0.030
PLZJ 93 03 55 13.00 +24 36 15.8 13.11±14.36 -33.77±12.97 24.488 ± 0.370 22.592 ± 0.164 19.968 ± 0.080 19.955±0.100 19.420 ±0.100
PLZJ 323 03 43 55.27 +25 43 26.2 29.87±12.05 -39.37±11.70 - 21.597±0.054 19.613±0.076 - -
PLZJ 721 03 55 07.14 +24 57 22.34 19.18±22.23 -40.70±12.38 - 22.195±0.092 20.248±0.116 20.417±0.123 -
PLZJ 235 03 52 32.57 +24 44 36.64 20.92±12.16 -45.84±11.75 - 22.339±0.115 20.039±0.112 20.245±0.127 -
PLZJ 112 03 53 19.37 +24 53 31.85 8.56±14.08 -34.59±19.99 - 22.532±0.116 20.281±0.143 - -
PLZJ 100 03 47 19.19 +25 50 53.3 20.23±14.27 -37.28±23.82 - 23.563±0.373 20.254±0.114 - -
000,1
Proper motion L and T dwarf candidate members of the Pleiades 9
Table 2. Probability of membership, magnitude range for our methods of calculating probabilities of membership using the annulus as well as the two control
areas.
Probability Probability Probability Magnitude range
annulus µαcosδ=-20 mas yr−1 µδ=-40 mas yr
−1 µαcosδ=+40 mas yr−1 µδ=-20 mas yr
0.67 0.25 0.0 16 - 17
0.82 0.66 0.0 17 - 18
0.88 1.00 0.0 18 - 19
0.84 1.00 0.0 19 - 20
1.00 1.00 1.00 20 - 21
0.88 0.50 1.00 21 - 22
0.61 1.00 0.00 22 - 23
Table 3. Probability of membership, magnitude range for our methods of calculating probabilities of membership using the annulus as well as the two control
areas for our candidates selected from the ZJ data only.
Probability Probability Probability Magnitude range
annulus µαcosδ=-20 mas yr−1 µδ=-40 mas yr
−1 µαcosδ=+40 mas yr−1 µδ=-20 mas yr
0.61 1.00 1.00 21 - 22
0.35 0.67 0.33 22 - 23
-0.16 -2.00 0.00 23 - 24
cluster, and 5 objects selected using the ZJ photometry only. All of
the objects identified as cluster members in this work are presented
in Table 4. Two WFCAM tiles, 1 and 4, (see Figure 1) also had
deep H band photometry. These tiles were observed at the same
time as the J band imaging, and were observed under the same
conditions, but with the exception that microstepping was not used.
These data were reduced using the same pipeline as the J band data,
but the photometry and object detection used a core radius of 2.5
pixels in this case. Fortunately these tiles also covered our faintest,
previously undiscovered Pleiades candidates, PLZJ 23 and PLZJ
93, as well as two of the candidates selected from the ZJ data only,
PLZJ 721 and 235.
The UKIDSS Galactic Cluster survey (GCS) has also covered
the entire area at J, H and K. The UKIDSS data are reduced using
the same pipeline as the WFCAM data (see Dye et al, 2006 for
details of the pipeline).
We also have used UKIRT service time to measure photom-
etry for PLZJ 93 in the K band. This observation was taken on
09/09/2006 in seeing of better than 1.1” using the UKIRT Fast
Track Imager (UFTI), with a five point dither pattern. The data
were reduced using the ORAC-DR pipeline, and the photometry
was calibrated using UKIRT Faint Standard 115.
The K band photometry for PLZJ 23 was obtained on the night
of 05/03/2007 using the long slit intermediate resolution spectro-
graph (LIRIS) on the William Hershel Telescope in service time,
using a nine point dither pattern in seeing of ≈ 0.9”. The data were
reduced using IRAF and astrometrically and photometrically cali-
brated using 2MASS. The colour transforms presented in Carpen-
ter, (2001) were used to calculate the K band magnitude from the
KS magnitude.
Thus we have I, Z, J, H and K band photometry for the major-
ity of our Pleiades candidates. However H or K band photometry
is still needed for PLZJ 323, 721, 235, 112 and 100, (see Table 4).
Figures 7 and 8 show the K, J-K and H, J-H, colour magnitude
diagrams, together with the NEXTGEN (Baraffe et al, 1998) and
DUSTY (Chabrier et al, 2000) models for the Pleiades age of 120
Myrs (Stauffer et al 1998). The candidate members listed in Table
4 are also plotted in Figures 3 and 4 for clarity. In both of these
diagrams the M dwarf tail, the redward L dwarf sequence and the
L to T blueward transition sequence are clear. The L-T transition
sequence of course only has two objects plotted on it on Figure 7
as we have no K band photometry for the ZJ candidates. As ex-
pected the K, J-K diagram gives the best differentiation between
the sequences. The redward L sequence in this diagram agrees with
that found by Lodieu et al,(2007b) derived from a much greater
area of the Pleiades by the UKIDSS GCS. The GCS is not sensi-
tive enough to see the L-T blueward transition sequence however.
The K, J-K diagram also shows the separation between single and
binary dwarfs quite clearly. Note that the DUSTY theoretical track
is too flat compared to our empirical sequence, see figures 7 and 8.
PLZJ 23, 93, 721 and 235 have J-H colours of 0.60, 0.00, -
0.17 and -0.21 respectively. Comparing these colours with the spec-
tral type colour relations of field dwarfs described in Leggett et al.
(2002), yields estimated spectral types of T1.5, T4.5, T6 and T6
respectively. PLZJ 93 has J-K=0.60 which gives a spectral type
of T3 (Leggett et al., 2002), which is consistent with the spectral
type derived from the J-H colour (T4.5), within the errors. We also
can calculate a H-K colour for this dwarf of 0.6, however the H-K
colour is not a good choice for spectral typing, for instance, H-
K=0.6 covers a range of spectral types from L1 to T3 (Chiu et al.,
2006). The Z-J colour is also not a good choice of colour for mea-
suring spectral types until the later T dwarfs (>T2)(Hawley et al.,
2002). PLZJ 23 has J-K=1.45, which gives a spectral type of be-
tween L8 and T1. The H-K colour for this dwarf is 0.85. We may
thus assume that PLZJ 23 has a spectral type between L8 and T1.5,
and likewise that PLZJ 93 has a spectral type of between T3 and
T5 to take into account the photometric errors. It should be noted
that the Z band quoted in Hawley et al., (2002) is for the Sloan
filter system, and so for this reason we have not chosen to use it
to spectral type our objects. We believe that the J-H colour gives
the best estimate available to us of spectral types. Two of the three
candidate members without H band photometry PLZJ 112 and 100
have fainter J magnitudes than PLZJ 23 and 93, and so it is likely
that they are also T dwarfs. PLZJ 323 is brighter and is therefore
c© 2007 RAS, MNRAS 000, 1–11
10 S. L. Casewell et al.
Table 4.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 7. The K,J-K CMD for our candidate cluster members. The solid
line is the NEXTGEN model of Baraffe et al (1998), and the dotted line
is the DUSTY model of Chabrier et al. (2000). The filled squares are the
candidates identified by Moraux et al. (2003), the filled triangles are the
candidates identified by Bihain et al. (2006), the object enclosed by the
open circle is CFHT-PL-10 identified by Bouvier et al. (1998). The objects
marked by small points are our new candidate members. One of our T dwarf
candidates, PLZJ 93, is found to the bottom of the plot, with a J-K of ≈ 0.6.
PLZJ 23 is also present with a J-K of 1.45.
probably a late L dwarf. Indeed our faintest candidate at Z, PLZJ
100, may be a very late T dwarf, however this assumption is made
using its Z-J colour, which is very red. Using J magnitudes and the
COND models of Baraffe et al. (2003) for 120 Myrs (the DUSTY
models are no longer appropriate for calculating masses for objects
this faint in the Pleiades), we calculate masses of ≈ 11 MJup for
PLZJ 23, 93, 323, 721, 235, 112 and 100. More photometry in the
H and K bands is clearly needed to improve and extend these esti-
mates of the spectral types.
5 MASS SPECTRUM
To calculate the mass spectrum, we first divided the sample into
single dwarfs or single dwarfs with possible low mass companions
and dwarfs that are close to 0.75 magnitudes above the single star
sequence in the K, J-K colour magnitude diagram. The latter we
assume to be equal mass binaries and count them as dwarfs with
masses the same as a dwarf on the single dwarf sequence below
them. From Figures 3, 4, 7 and 8 it can be seen that there are
2 such binaries all with J-K ≈ 1. Dwarfs with J-K <1.2 are as-
signed masses using their H magnitudes and the NEXTGEN mod-
els (Baraffe et al. 1998). For 1.2<J-K<2.0 we use the DUSTY
models (Chabrier et al., 2000) and the J-H colour to assign a mass.
Finally the T dwarf masses were calculated from their J magni-
tudes and the COND models (Baraffe et al., 2003). The masses
were binned into three mass intervals, covering the low, medium
and high mass ranges and the numbers per bin are weighted by
the probabilities of membership calculated using the annulus, and
the bin width has been taken into account. The candidate members
with negative probabilities are obviously omitted from the mass
spectrum. The resultant mass spectrum is shown in Figure 9. The
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Figure 8. The H,J-H CMD for our candidate cluster members. The filled
squares are the candidates identified by Moraux et al. (2003), the filled trian-
gles are the candidates identified by Bihain et al. (2006), the object enclosed
by the open circle is CFHT-PL-10 identified by Bouvier et al. (1998). The
objects marked by small points are our new candidate members.The filled
diamonds are the two candidates with H magnitudes selected from the ZJ
data only. The solid line is the NEXTGEN model of Baraffe et al (1998),
and the dotted line is the DUSTY model of Chabrier et al. (2000).
errors are poissonian. Clearly the statistics are very poor, due to
the small number of objects being dealt with. Using linear regres-
sion we have fitted our data to the relationship dN/dM∝M−α , and
calculate α=0.35±0.31. This is lower but still in agreement with
values presented in the literature (within 1σ), however the error on
this value is large, and the statistics are poor due to the small num-
bers involved. If we take into account the fact that the last mass
bin is only 50% complete (using Tables 1 and 4), then the lowest
mass bin can be increased by 50% to compensate. If we then fit
these data, we derive a value for α of 0.62±0.14, which is in agree-
ment with the literature. Alternatively, we can discount this final
low mass bin as being incomplete and simply omit it from the fit.
In this case we calculate a value for α of 0.86. We have only dis-
played the mass spectrum for the cluster in the area and magnitude
surveyed. This is to avoid trying to take into account biases caused
by some areas being more studied than others, and also because we
are only adding a maximum of 9 objects to the mass spectrum, 7
of which have low probabilities of membership and small masses,
and so are not likely to affect previous results a large amount. The
mass spectrum appears to be rising towards the lowest masses, but
this is not statistically significant due to the large error bars.
6 CONCLUSIONS
We have confirmed a number of L dwarf candidates in the Pleiades.
However the main result in this paper is the discovery of seven L
and T dwarf Pleiads of masses ≈ 11 MJup, below the 13 MJup
deuterium burning limit that is often used, somewhat artificially as
the upper bound for planetary masses. Further H and K band pho-
tometry, currently lacking for some of these candidates, will im-
prove confidence in their membership of the cluster. Planetary mass
brown dwarfs have, of course, been claimed in the Orion nebula
c© 2007 RAS, MNRAS 000, 1–11
Proper motion L and T dwarf candidate members of the Pleiades 11
-1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1
log mass (Solar masses)
Figure 9. The mass spectrum for our Pleiades candidate members. The
mass bin is in units of M⊙. The solid line is the fit to the data,
(α=0.35±0.31).
(Lucas & Roche 2000) and in the σ-Ori cluster (Zapatero-Osorio et
al., 2002). These clusters both have very young ages and may also
have a spread of ages (Béjar et al., 2001), making mass determi-
nations somewhat uncertain. Lodieu et al. (2006, 2007a) have also
found planetary mass brown dwarfs in the Upper Scorpius Associ-
ation which has an age of 5 Myrs (Preibisch & Zinnecker, 2002).
At very young ages the theoretical models may have significant er-
rors when used to assign masses (Baraffe et al., 2002). Our result
is the first detection of planetary mass objects in a mature cluster
whose age is well established. It strengthens the case that the star
formation process can produce very low mass objects.
7 ACKNOWLEDGEMENTS
SLC, NL,and PDD acknowledge funding from PPARC. We also
acknowledge the Canadian Astronomy Data Centre, which is oper-
ated by the Dominion Astrophysical Observatory for the National
Research Council of Canada’s Herzberg Institute of Astrophysics.
This work has been based on observations obtained at the Canada-
France-Hawaii Telescope (CFHT) which is operated by the Na-
tional Research Council of Canada, the Institut National des Sci-
ences de l’Univers of the Centre National de la Recherche Scien-
tifique of France, and the University of Hawaii. Observations were
also made at the United Kingdom Infrared Telescope, which is op-
erated by the Joint Astronomy Centre on behalf of the U.K. Particle
Physics and Astronomy Research Council. This publication makes
use of data products from the Two Micron All Sky Survey, which
is a joint project of the University of Massachusetts and the In-
frared Processing and Analysis Center/California Institute of Tech-
nology, funded by the National Aeronautics and Space Administra-
tion and the National Science Foundation. This research has made
use of NASA’s Astrophysics Data System Bibliographic Services,
the WHT service programme, and the UKIRT service programme.
We would like to thank the referee V.J.S. Béjar for his comments
which have improved the paper.
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Mon. Not. R. Astron. Soc. 000, 1–?? (2009) Printed 3 December 2018 (MN LATEX style file v2.2)
Errata:Proper motion candidate members of the Pleiades
S. L. Casewell1⋆, P. D. Dobbie1,2, S. T. Hodgkin3, E. Moraux4, R. F. Jameson1,
N. C. Hambly5, J. Irwin3 and N. Lodieu6,1
1Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK
2Anglo-Australian Observatory, PO Box 296, Epping NSW 1710 Australia
3CASU, Institute of Astronomy,University of Cambridge, Maddingley Road, Cambridge, CB3 0HA, UK
4Laboratoire d’Astrophysique, Observatoire de Grenoble, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
5Scottish Universities Physics Alliance (SUPA), Institute for Astronomy, School of Physics, University of Edinburgh,
Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ
6 Instituto de Astrofı́sica de Canarias, Vı́a Láctea s/n, E-38205 La Laguna, Tenerife, Spain
3 December 2018
The paper “Proper motion L and T dwarf candidate members
of the Pleiades” was published in the Monthly Notices of the Royal
Astronomical Society, 2007, 378, 1131. It has come to our attention
that there were errors in Table 4 as regards the Right Ascension of
the candidate coordinates. Table 4 should read as follows. This has
no impact on the scientific results presented in the paper.
c© 2009 RAS
http://arxiv.org/abs/0704.1578v2
Table 4Name,coordinates, Z, I, J, H and K magnitudes for our members to the cluster. The errors quoted are internal (from photon counting). The systematic calibration errors are 2% in the
J, H and K wavebands (Warren et al., 2007), and 3% in the I and Z wavebands. The J, H and K magnitudes are on the MKO system. Previously discovered members also also have their other
known names listed from Moraux et al. (2003), Bihain et al. (2006) and Bouvier et al. (1998). The H and K band magnitudes are taken from the UKIDSS Galactic Cluster Survey with the
exceptions of PLZJ 23, 93, 721 and 235 which have their H band magnitudes listed from our H survey. The K band magnitude for PLZJ 93 is from our UFTI photometry, and PLZJ 23 is from
LIRIS service time. The final 5 objects in the table are our candidates selected from the ZJ data only.
Name Alternate RA dec µαcosδ µδ I Z J H K
name J2000.0 mas yr−1
PLZJ 29 BRB4 03 44 23.23 +25 38 45.11 23.40±8.24 -48.51±6.34 17.005 ± 0.001 16.163 ± 0.001 14.732 ± 0.001 14.132±0.004 13.744±0.004
PLZJ 56 03 44 53.51 +25 36 19.46 19.68±7.34 -35.63±5.29 17.012 ± 0.001 16.351 ± 0.001 15.250 ± 0.001 14.650±0.005 14.342±0.006
PLZJ 45 BRB8, CFHT-PL-7 03 52 05.82 +24 17 31.57 19.72±4.95 -42.37±7.44 17.101 ± 0.001 16.417 ± 0.001 15.247 ± 0.001 14.614±0.005 14.251±0.006
PLZJ 50 03 43 55.98 +25 36 25.45 13.48±8.24 -35.65±5.38 17.239 ± 0.001 16.496 ± 0.001 15.268 ± 0.001 14.693±0.006 14.319±0.006
PLZJ 60 CFHT-PL-10 03 44 32.32 +25 25 18.06 16.93±7.76 -43.15±5.72 17.592 ± 0.001 16.810 ± 0.001 15.460 ± 0.001 14.884±0.007 14.465±0.006
PLZJ 78 PLIZ2 03 55 23.07 +24 49 05.18 19.72±10.06 -29.74±10.45 17.719 ± 0.001 16.948 ± 0.001 15.574 ± 0.001 14.963±0.007 14.552±0.007
PLZJ 46 PLIZ3, BRB11 03 52 06.71 +24 16 00.99 19.55±5.15 -42.58±7.57 17.742 ± 0.001 16.945 ± 0.001 15.583 ± 0.001 14.966±0.007 14.503±0.008
PLZJ 9 PLIZ6,BRB9 03 53 55.09 +23 23 36.38 24.13±13.83 -50.10±22.71 17.752 ± 0.001 16.804 ± 0.001 15.222 ± 0.001 14.548±0.005 14.054±0.005
PLZJ 11 PLIZ20 03 54 05.33 +23 33 59.71 9.14±11.06 -28.98±11.94 19.571±0.004 18.563±0.004 16.691±0.005 15.980±0.016 15.436±0.016
PLZJ 77 PLIZ28,BRB18 03 54 14.04 +23 17 52.28 12.01±14.59 -51.60±15.84 20.760 ± 0.010 19.728 ± 0.010 17.647 ± 0.010 16.789±0.031 16.131±0.030
PLZJ 21 PLIZ31 03 51 47.65 +24 39 59.18 17.84±9.41 -44.92±8.03 20.944±0.014 19.762±0.013 17.575±0.012 16.774±0.026 16.089±0.028
PLZJ 10 PLIZ35,BRB15 03 52 39.13 +24 46 29.61 15.84±8.88 -49.34±6.24 21.293±0.018 20.292±0.016 18.181±0.022 17.118±0.041 16.506±0.0416
PLZJ 4 BRB21 03 54 10.25 +23 41 40.67 29.74±13.17 -38.46±8.88 21.322 ± 0.010 20.215 ± 0.013 18.171 ± 0.010 17.141±0.045 16.377±0.039
PLZJ 61 BRB22 03 44 31.27 +25 35 14.97 25.82±7.89 -40.21±8.47 22.043 ± 0.030 20.782 ± 0.026 18.298 ± 0.020 17.393±0.059 16.657±0.04
PLZJ 32 BRB27 03 44 27.27 +25 44 41.99 25.03±11.52 -38.65±23.46 22.235 ± 0.040 20.962 ± 0.029 18.871 ± 0.030 17.793±0.094 16.950±0.070
PLZJ 37 BRB28 03 52 54.92 +24 37 18.85 18.13±11.53 -48.68±11.38 22.452 ± 0.05 21.216 ± 0.041 18.839 ± 0.030 17.742±0.071 16.921±0.058
PLZJ 23 03 51 53.38 +24 38 12.11 20.75±10.51 -50.05±9.96 23.541 ± 0.140 22.187 ± 0.112 19.960 ± 0.100 19.362±0.100 18.510±0.030
PLZJ 93 03 55 13.00 +24 36 15.8 13.11±14.36 -33.77±12.97 24.488 ± 0.370 22.592 ± 0.164 19.968 ± 0.080 19.955±0.100 19.420 ±0.100
PLZJ 323 03 43 55.27 +25 43 26.2 29.87±12.05 -39.37±11.70 - 21.597±0.054 19.613±0.076 - -
PLZJ 721 03 55 07.14 +24 57 22.34 19.18±22.23 -40.70±12.38 - 22.195±0.092 20.248±0.116 20.417±0.123 -
PLZJ 235 03 52 32.57 +24 44 36.64 20.92±12.16 -45.84±11.75 - 22.339±0.115 20.039±0.112 20.245±0.127 -
PLZJ 112 03 53 19.37 +24 53 31.85 8.56±14.08 -34.59±19.99 - 22.532±0.116 20.281±0.143 - -
PLZJ 100 03 47 19.19 +25 20 53.3 20.23±14.27 -37.28±23.82 - 23.563±0.373 20.254±0.114 - -
000–000
000,1
Introduction
Observations, Data Reduction and Survey Completeness
The J band imaging and its reduction
The far-red optical imaging and a new reduction
The completeness of datasets
Analysis of the data
Photometric selection of candidate cluster members
Refining the sample using astrometric measurements
Results
Mass spectrum
Conclusions
Acknowledgements
|
0704.1579 | A Study of Catalogued Nearby Galaxy Clusters in the SDSS-DR4: I. Cluster
Global Properties | Astronomy & Astrophysics manuscript no. aguerri˙rv c© ESO 2019
August 20, 2019
A Study of Catalogued Nearby Galaxy Clusters in the SDSS-DR4
I. Cluster Global Properties
J. A. L. Aguerri, R. Sánchez-Janssen & C. Muñoz-Tuñón
Instituto de Astrofı́sica de Canarias C/ Vı́a Láctea s/n, 38200 La Laguna, Spain. e-mail: [email protected], [email protected], [email protected]
Received ; accepted
ABSTRACT
Context. Large surveys as the Sloan Digital Sky Survey have made available large amounts of spectroscopic and photometric data of galaxies,
providing important information for the study of galaxy evolution in dense environments.
Aims. We have selected a sample of 88 nearby (z < 0.1) galaxy clusters from the SDSS-DR4 with redshift information for the cluster members.
In particular, we focus our results on the galaxy morphological distribution, the velocity dispersion profiles, and the fraction of blue galaxies in
clusters.
Methods. Cluster membership was determined using the available velocity information. We have derived global properties for each cluster,
such as their mean recessional velocity, velocity dispersion, and virial radii. Cluster galaxies have been grouped in two families according to
their u − r colours.
Results. The total sample consists of 10865 galaxies. As expected, the highest fraction of galaxies (62%) turned to be early-type (red) ones,
being located at smaller distances from the cluster centre and showing lower velocity dispersions than late-type (blue) ones. The brightest
cluster galaxies are located in the innermost regions and show the smallest velocity dispersions. Early-type galaxies also show constant velocity
dispersion profiles inside the virial radius and a mild decline in the outermost regions. In contrast, late-type galaxies show always decreasing
velocity dispersions profiles. No correlation has been found between the fraction of blue galaxies and cluster global properties,such as cluster
velocity dispersion and galaxy concentration. In contrast, we found correlation between the X-ray luminosity and the fraction of blue galaxies.
Conclusions. These results indicate that early- and late-type galaxies may have had different evolution. Thus, blue galaxies are located in more
anisotropic and radial orbits than early-type ones. Their star formation seems to be independent of the cluster global properties in low
mass clusters, but not for the most massive ones. We consider that it is unlikely that the whole blue population consists of recent arrivals
to the cluster. These observational results suggest that the global environment could be important for driving the evolution of galaxies
in the most massive cluster (σ > 800 km s−1). However, the local environment could play a key role in galaxy evolution for low mass
clusters.
Key words. galaxies: clusters: general
1. Introduction
The large amount of spectroscopic and photometric data ob-
tained during the last years by surveys such as the Sloan Digital
Sky Survey (SDSS; York et al. 2000) or the 2dF Galaxy
Redshift Survey (2dFGRS; Colless et al. 2001) have opened
a new horizon for the study of galaxy evolution, and in partic-
ular in the study of galaxy clusters. It is well known that the
environment plays an important role in the evolution of galax-
ies, and it is one of the keys that a good galaxy evolution the-
ory should address. There are several physical mechanisms, not
present in the field, which can dramatically transform galaxies
in high density environments. Galaxies in clusters can evolve
due to, e.g., dynamical friction, which can slow down the
more massive galaxies, circularise their orbits and enhance
their merger rate (den Hartog & Katgert 1996; Mamon 1992).
Send offprint requests to: J. A. L. Aguerri
Interactions with other galaxies and with the cluster grav-
itational potential can disrupt the outermost regions of the
galaxies and produce galaxy morphological transformations
from late- to early-types (Moore et al. 1996), or even change
massive galaxies into dwarf ones (Mastropietro et al. 2005).
Swept of cold gas produced by ram pressure stripping
(Gunn & Gott 1972 ; Quilis et al. 2000) or swept of the hot
gas reservoirs (Bekki et al. 2002) can alter the star formation
rate (SFR) of galaxies in clusters. But it is still a matter of
debate which of these mechanisms is the main responsible
of the galaxy evolution in galaxy clusters (see Goto 2005).
Nevertheless, it is clear that all of these mechanisms transform
galaxies from late- to early-types, and can produce the different
segregations observed in galaxy clusters.
One of the first segregations discovered in galaxy clus-
ters was the morphological one. The first evidences of such
segregation date from Curtis (1918) and Hubble & Humason
http://arxiv.org/abs/0704.1579v1
2 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
(1931), and was quantified by Oemler (1974) and Melnick
& Sargent (1977). In a thorough work, Dressler (1980)
analysed a sample of 55 nearby galaxy clusters, contain-
ing over 6000 galaxies, and observed that elliptical and
S0 galaxies represent the largest fraction of galaxies lo-
cated in the innermost and denser regions of galaxy clus-
ters. In contrast, the outskirts of the clusters were domi-
nated by spiral galaxies. In more distant clusters the frac-
tion of E galaxies is as large or larger than in low-redshift
clusters, but the S0 fraction is smaller (Dressler et al. 1997;
Fasano et al. 2000). This has been interpreted as an evolution
with redshift, being late-type galaxies transformed into early-
type ones. Segregations in velocity space have also been ob-
served in galaxy clusters. Early observations found that E
and S0 galaxies showed smaller velocity dispersions than spi-
rals and irregulars (Tammann 1972; Melnick & Sargent 1977;
Moss & Dickens 1977). This has also been confirmed by
other authors during the last two decades (Sodre et al. 1989;
Biviano et al. 1992; Andreon et al. 1996; Stein 1997). The data
from the ENACS survey (Katgert et al. 1998) produced a large
sample of galaxies with spectroscopic redshifts and shed more
light to this problem. Thus, Adami et al. (1998) studied a sam-
ple of 2000 galaxies, confirming early findings that the ve-
locity dispersion of galaxies increases along the Hubble se-
quence: E/S0 galaxies show smaller velocity dispersions than
early- and late-type spirals. This segregation was also observed
in the velocity dispersion profiles (VDPs): late-type galaxies
have decreasing VDPs, while E, S0 and early spirals show al-
most flat VDPs (Adami et al. 1998). The different kinematics
shown by the different types of galaxies was analysed in more
detail by Biviano & Katgert (2004) who found that the ve-
locity segregation of the different Hubble types is due to dif-
ferences in orbits. Thus, early-type spirals have isotropic or-
bits, while late-type ones are located in more anisotropic or-
bits. The observed morphological and velocity segregation in
clusters have been usually used to conclude that late-type spi-
ral galaxies in clusters are recent arrivals to the cluster potential
(Stein 1997; Adami et al. 1998).
Star formation in galaxies is also affected by the envi-
ronment. Butcher & Oemler (1984) found that the fraction
of blue galaxies, fb, in clusters is smaller than in the field
and evolves with redshift: more distant clusters show larger
values of fb. This trend was interpreted as an evolutionary
effect of the SFR in galaxy clusters. But the significant
increase of new data has made it clear that the Butcher-Oemler
effect is not only an evolutionary trend. A large scatter
in the values of fb has been observed in narrow redshift
ranges (Smail et al. 1998; Margoniner & de Carvalho 2000;
Goto et al. 2003), which suggests that the variation of fb
is influenced by environmental effects. In the past, many
authors have tried to find correlations of fb with cluster
properties, such as X-ray luminosity (Andreon & Ettori 1999;
Smail et al. 1998; Fairley et al. 2002), luminosity limit
and clustercentric distance (Ellingson et al. 2001;
Goto et al. 2003; De Propris et al. 2004), richness
(Margoniner et al. 2001; De Propris et al. 2004), cluster con-
centration (Butcher & Oemler 1984; De Propris et al. 2004),
presence of substructure (Metevier et al. 2000) or cluster
velocity dispersion (De Propris et al. 2004). Some of these
works found correlations between fb and the cluster envi-
ronment while others did not, being such connection still a
matter of debate. However, these works were usually done
using small and heterogeneous cluster samples (but see e.g.,
De Propris et al. 2004).
Environmental effects have also been invoked in order
to explain the differences between the photometrical compo-
nents of cluster and field spiral galaxies. Thus, it has been
observed that the scale-lengths of the disks of spiral galax-
ies in the Coma cluster are smaller than those of similar
galaxies in the field (Gutiérrez et al. 2004; Aguerri et al. 2004).
Interactions between galaxies or with the cluster potential
can disrupt the disks of spiral galaxies in clusters. They
can be strong enough for transforming bright late-type spi-
ral galaxies in dwarfs (Aguerri et al. 2005a). The disrupted
material would be part of the intracluster light already de-
tected in some nearby galaxy clusters (Arnaboldi et al. 2002;
Arnaboldi et al. 2004 ; Aguerri et al. 2005b) and galaxy groups
(Castro-Rodrı́guez et al. 2003; Aguerri et al. 2006).
The observational results summarised before illustrate the
important role played by environment in galaxy evolution.
They also indicate that late-type and early-type galaxies in
clusters are two different families of objects with differ-
ent properties, which points to different origins or evolution.
Nevertheless, the main mechanisms responsible of this differ-
ent evolution still remain unknown. In the present paper, we
study one of the largest and more homogeneous galaxy cluster
sample available in the literature. We have obtained the cluster
membership, mean velocity, velocity dispersion, virial radius
and positions for a sample of 88 clusters located at z < 0.1.
We have investigated the main properties of a large sample of
early (red) and late (blue) types of galaxies, such as their lo-
cation within the cluster, their mean velocity dispersion, their
VDPs, the LX −σ relation, and the fraction of blue galaxies for
each cluster. This work provides important information about
the properties of galaxies in nearby clusters, which will be
useful in order to put constraints on cosmological models of
cluster formation. This is the first paper of a series in which
we will analyse the properties of the dwarf galaxy population
(Sánchez-Janssen et al. in preparation), substructure in galaxy
clusters (Aguerri et al. in preparation), and composite luminos-
ity function of galaxy clusters (Sánchez-Janssen et al., in prepa-
ration).
The paper is organised as follows. Section 2 shows the dis-
cussion about the galaxy cluster sample. The cluster member-
ship and cluster global parameters are presented in Section 3.
The results obtained about the morphological segregation, ve-
locity dispersion profiles, LX − σ relation, and the fraction of
blue galaxies are given in Sections 4, 5, 6 and 7, respectively.
The discussion and conclusions are presented in Sections 8 and
9, respectively. Throughout this work we have used the cos-
mological parameters: Ho = 75 km s
−1 Mpc−1, Ωm = 0.3 and
ΩΛ = 0.7.
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 3
2. Galaxy cluster Sample
We have used photometric and spectroscopic data of ob-
jects classified as galaxies from the SDSS-DR4, an imag-
ing and spectroscopic survey of a large area in the sky
(York et al. 2000). The imaging survey was carried out
through five broad-band filters, ugriz, spanning the range
from 3000 to 10000 Å, reaching a limiting r-band mag-
nitude ≈ 22.2 with 95% completeness, and covering an
area of 6670 deg2 (Adelman-McCarthy et al. 2006). A se-
ries of pipelines process the imaging data and perfom the
astrometric calibration (Pier et al. 2003), the photometric re-
duction (Lupton et al. 2002) and the photometric calibration
(Hogg et al. 2001). Objects brighter than mr = 17.77 were se-
lected as possible targets for the spectroscopic survey, covering
an area of 4783 deg2 of the sky for the DR4. The spectroscopic
data were obtained with optical fibers with a diameter of 3
the focal plane, resulting in an spectral covering in the wave-
length range 3800–9200 Å with a resolution of λ/∆λ ≈ 2000.
Our sample consists of all clusters with known redshift at
z < 0.1 from the catalogues of Abell et al. (1989), Zwicky et
al. (1961), Böhringer et al.(2000) and Voges et al. (1999) that
have been mapped by the SDSS-DR4. We downloaded only
those galaxies located within a radius of 4.5 Mpc around the
centres of the galaxy clusters. Only those clusters with more
than 30 galaxies with spectroscopic data in the searching radius
were considered, resulting in a sample formed by 240 clusters
following the previous criteria. The SDSS-DR4 spectroscopic
galaxy target selection was done by an automatic algorithm
(see Strauss et al. 2002). The main galaxy sample consists of
galaxies with r-band Petrosian magnitudes brighter than 17.77
and r-band Petrosian half-light surface brightness brighter than
24.5 mag arcsec−2. The completeness of this sample is high,
exceeding 99% (see Strauss et al. 2002). However, some of
the selected spectroscopic targets were not observed at the end.
This incompleteness has several causes, including the fact that
two spectroscopic fibers cannot be placed closer than 55
given plate, possible gaps between the plates, fibers that fall out
of their holes, and so on. According to these reasons, we expect
that the incompleteness of the spectroscopic data will be more
important for bright galaxies in high density environments such
as galaxy clusters. Figure 1 shows the mean completeness1 of
the SDSS-DR4 spectroscopic data as a function of the r-band
magnitude for the selected galaxies, where a fast increment to-
wards faint magnitudes can be observed. In order to avoid pos-
sible effects on the results due to this effect, we have completed
the spectroscopic SDSS-DR4 observations with the data avail-
able at the Nasa Extragalactic Database (NED). Figure 1 also
shows the mean completeness as a function of r-band magni-
tude after the spectroscopic data from NED were included in
the sample. Notice that the new mean completeness is almost
constant (≈ 85%) for all magnitudes brighter than mr = 17.77.
We have made a second selection of the clusters by considering
only those from our original list with completeness larger than
70% for galaxies brighter than 17.77 in the r-band.
1 We have defined the spectroscopic completeness per magnitude
bin as the ratio of the number of galaxies with spectroscopic data to
the number of galaxies with photometric information.
Fig. 1. Mean completeness of the cluster sample as a function
of the r-band magnitude. Diamonds represent the spectroscopic
data from SDSS-DR4 and black circles after the completion
with data from NED.
3. Cluster Membership
Clusters properties such as the mean cluster velocity, the ve-
locity dispersion, the cluster centre or the virial radius can
be significantly affected by projection effects. Several methods
have been developed during decades in order to obtain reliable
galaxy cluster membership and avoid the presence of interlop-
ers. They can be classified in two families. First, those algo-
rithms that use only the information in the velocity space, e.g.
3σ-clipping techniques (Yahil & Vidal 1977), gapping proce-
dures (Beers et al. 1990; Zabludoff et al. 1990, hereafter ZHG
algorithm) or the KMM algorithm (Ashman et al. 1994). The
other family corresponds to those algorithms which use infor-
mation of both position and velocity, such as the methods de-
signed by Fadda et al. (1996), den Hartog & Katgert (1996), or
Rines et al. (2003).
The cluster membership in our sample was obtained using
a combination of two algorithms. A first rough cluster mem-
bership determination was obtained using the ZHG algorithm,
which in a second step was then refined using the KMM al-
gorithm. The ZHG algorithm is a typical gapping procedure
which determines the cluster membership by the exclusion of
those galaxies located at more than a certain velocity distance
(∆v) from the nearest galaxy in the velocity space. Then, the
mean velocity (vm) and velocity dispersion (σ) of the remain-
ing galaxies are calculated. After sorting objects with velocities
greater than vm, any galaxy separated in velocity more than σ
from the previous one is classified as non member. The same is
done for those galaxies with velocities less than vm. The pro-
cess is repeated several times and finally the mean cluster ve-
locity (vc) and the cluster velocity dispersion (σc) are obtained.
Zabludoff et al. (1990) pointed out that this method lacks statis-
tical rigour and tends to give overestimated values of σc. One
of the disadvantages of this method is that the results obtained
strongly depend on the chosen value of ∆v. Large values of ∆v
imply that a large fraction of interlopers are identified as clus-
ter members. On the contrary, small values of ∆v result in the
4 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
lost of cluster galaxies. We have investigated the variation of
σc for different values of ∆v, obtaining that ∆v=500 km s
is an appropriate value for our clusters. This method has also
the advantage that has an easy implementation and does not re-
quire too much computational time. Recently, it has been used
in works involving a large number of clusters, such as those
from the 2dFGRS (De Propris et al. 2003). The ZHG algorithm
splits the velocity histograms in different galaxy groups, be-
ing one of them located at the catalogued redshift of the clus-
ter. That group was taken and analysed in more detail with the
KMM algorithm. In the few cases where there was no galaxy
group located at the catalogued redshift we identified the most
significant groups having z < 0.1 as the cluster itself.
The KMM algorithm (Ashman et al. 1994) estimates the
statistical significance of bi-modality in a dataset. We have run
it to the group of galaxies given by the ZHG algorithm which
contains the catalogued redshift of the cluster. The KMM algo-
rithm gives us the compatibility of the velocity distribution of
such group of galaxies with a single or multiple Gaussian dis-
tribution. We considered three different cases which are sum-
marised in Fig. 2:
– Single cluster: the velocity distribution of the galaxies is
compatible with a single Gaussian, e.g. Abell 757.
– Cluster with substructure: the velocity distribution is com-
patible with multiple groups. We identified the cluster as
the group with the largest number of galaxies plus those
groups which mean velocities lie within 3σ from the mean
velocity of the largest one2, e.g. Abell 1003.
– Cluster with contamination: the velocity distribution is
compatible with the presence of several groups, but the
mean velocities of the smaller groups deviate more than
3σ from the most populated one, which we identify as the
cluster itself, e.g. Abell 168 .
We have explored the differences in the values of vc and σc
if we consider as interlopers those groups of galaxies located at
a velocity distance larger than 1σ or 3σ from the mean velocity
of the main galaxy group. We obtained that the differences in vc
and σc in 90% of the clusters are less than 20%. The remaining
10% of the clusters are those with significant structure in the
velocity distribution, being most of them more than one cluster
along the line of sight. Thus, we have adopted 3σ as the default
except for those clusters with significant differences between
1σ and 3σ, for which we have measured the mean velocity and
velocity dispersion of the cluster adopting the criteria of 1σ.
Through all of this process, the determination of vc and σc was
done using the biweight robust estimator of Beers et al. (1990).
3.1. Cluster global parameters
Once the cluster membership was determined, we obtained the
global parameters of each cluster, i.e., mean velocity (vc), ve-
locity dispersion (σc), cluster centre, and the radius r200. All of
these parameters were computed using only the cluster mem-
bers.
2 In this case σ is the velocity dispersion of the largest group of
galaxies.
Fig. 2. Velocity histograms of three representative clusters of
the sample. The vertical full lines represent the mean velocity
of the different groups of galaxies in which KMM algorithm
has divided the velocity histogram. The dotted vertical lines
represent vc ± 3σc.
The determination of the cluster centre is important in or-
der to accurately compute the other parameters of the clusters.
The centre of the cluster is determined by the potential well,
which can be traced by the position of the peak of the X-ray lu-
minosity of the cluster. That peak was considered as the centre
of those clusters from our sample with X-ray measurements in
the literature. Unfortunately, not all the clusters from the sam-
ple have X-ray data. In that case, the centre of these clusters
was determined by the peak of the galaxy surface density3. For
those clusters with X-ray data we have compared the centres
given by the peaks of X-ray luminosity and galaxy surface den-
sity, obtaining a mean difference of 150 kpc.
Analytic models (Gott 1972) and simulations
(Cole & Lacey 1996) indicate that the virialized mass of
clusters is generally contained inside the surface where the
mean inner density is 200ρc, where ρc is the critical density
of the Universe. The radius of that surface is called r200.
We have computed the r200 for our clusters using the same
approximation as Carlberg et al. (1997):
r200 =
10 H(zc)
, (1)
where H(zc) is the Hubble constant at the cluster redshift
3 The galaxy surface density was computed using the algorithm de-
signed by Pisani (1996).
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 5
The previous global parameters of the clusters (vc, σc, r200
and centre) were obtained as described above but in a recur-
rent way. In a first step, they where determined using all cluster
member galaxies around 4.5 Mpc from the centre of the clus-
ter. After this step we recalculated the parameters using only
those galaxies located inside r200. The method was repeated
several times until the difference in the parameters obtained in
two consecutive steps was less than 5%. Three or four iterations
were usually enough for reaching the convergence. In order to
obtain reliable parameters of the clusters, those with less than
15 galaxies within r200 were removed from our list. This re-
sults in a final sample formed by 110 nearby galaxy clusters.
Table 1 shows the sample of galaxy clusters and their global
parameters. The columns of Table 1 represent: (1) galaxy clus-
ter name, (2, 3) cluster centres (α (J2000), δ (J2000)), (4) mean
radial velocity, (5) cluster velocity dispersion, (6) r200 radius,
(7) number of galaxies within r200, and (8) spectroscopic com-
pleteness.
For 6 clusters (Abell 1003, Abell 1032, Abell 1459, Abell
2023, Abell 2241 and ZwCl1316.4-0044) large differences in
the mean recessional velocity have been found between the val-
ues given in Table 2 and those from NED. These are the clusters
with no significant galaxy group at the catalogued redshift (see
Section 3).
In order to consider the possible influence of neighbouring
clusters on the global properties of the sample we searched in
the surroundings of each cluster for the presence of compan-
ions. Following Biviano & Girardi (2003), we have considered
that two clusters, i and j, are in interaction when:
|vi − vj| < 3(σi + σj) Ri,j < 2(r200,i + r200,j), (2)
where Ri, j is the projected distance between the centres of
the clusters and vi, j, σi, j, r200,i, j their respective mean velocities,
velocity dispersions and r200. We found 16 couples of clusters
in interaction according to the previous criteria. The remaining
sample (88 clusters) followed the isolation criteria, and will be
used in the analysis presented in the following sections. Figure
3 shows the sky distribution of the cluster members and the
galaxy velocities as a function of clustercentric distance for a
sample of 8 clusters. Red points represent the galaxies taken as
cluster members while black points are interlopers. Notice the
large number of interlopers in some of the galaxy clusters, such
as Abell 1291, Abell 1383, Abell 2244. Some of them, Abell
1291 and Abell 1383, were not included in the final isolated
sample due to the presence of companions.
3.2. Corrections to line-of-sight velocities
Line-of-sight velocities of galaxies in clusters were corrected
by two effects: cosmological redshift and global velocity field.
We should take into account that we will compare the veloc-
ity dispersion of clusters located at different redshifts. Thus,
for each galaxy we have 1 + zobs = (1 + zc)(1 + zgal)
(Danese et al. 1980), being zobs the apparent redshift of the
galaxies, zc the cosmological redshift of the cluster, and zgal
the redshift of the galaxy respect to the cluster centre. This cor-
rection can affect up to 10% for the most distant clusters in our
sample.
Galaxy clusters are frequently part of larger cosmological
structures such as filaments, superclusters or multiple systems,
which can affect the velocity field resulting in a modified clus-
ter velocity dispersion. The interaction between galaxy clus-
ters can also produce distorted velocity fields. We have inves-
tigated the importance of these effects in the velocity field of
our clusters by making a least-square fit to the radial velocities
of cluster galaxies with respect to their position in the plane of
the sky (see den Hartog & Katgert 1996; Girardi et al. 1996).
For each fit we computed the coefficient of multiple determi-
nation, R2. In order to test the significance of the fitted veloc-
ity gradients, we run 1000 Monte Carlo simulations for each
cluster for which the correlation between position and veloc-
ity was removed. This was achieved by shuffling the veloc-
ities of the galaxies with respect to their positions. We de-
fined the significance of velocity gradients as the fraction of
Monte Carlo simulations with R2 smaller than the observed
one. This correction of the velocity field was applied to those
cluster in which the significance of velocity gradients is larger
than 99% ( 30% of the total sample). However, this correction
has small effects both in the shape of the velocity dispersion
profiles and on the total velocity dispersion (the mean abso-
lute correction was about 40 ± 15 km s−1). This is in agree-
ment with similar corrections applied in other cluster samples
(den Hartog & Katgert 1996; Girardi et al. 1996).
3.3. Comparison with other methods
Some of the clusters presented in our sample have been previ-
ously studied by other authors. However, we have avoided com-
paring our results with those from the literature given the differ-
ent datasets used. In order to compare our cluster membership
method with others proposed in the literature, we have com-
puted σc of our clusters with two more methods: a 3σ-clipping
and the method proposed by Fadda et al. (1996). The median
absolute difference between our σc and those computed by the
3σ-clipping method is only 17 km s−1. Only 10% of the clus-
ters show important diferences (∆σc > 200 km s
−1) in the com-
putation of the velocity dispersion of the cluster with the two
methods. They correspond to those clusters affected by large
amount of structure along the line of sight. The 3σ-clipping
method gives for these clusters considerably larger values of
σc than ours. Differences were larger when we compared with
Fadda’s method. In this case the mean absolute difference in
σc between the two methods was 84 km s
−1 and 80% of the
clusters show differences smaller than 200 km s−1.
Recently, Popesso et al. (2006) have obtained the values of
σc for a sample of Abell clusters using SDSS-DR4 data, for
which cluster membership was obtained using the selection al-
gorithm of Katgert et al. (2004). The median absolute differ-
ence between our and their σc is 45 km s
−1 for the 28 clusters
in common. Only for 4 clusters (Abell 1750, Abell 1773, Abell
2244 and Abell 2255) the absolute differences in σc is larger
than 200 km s−1.
We have also compared our results with those given in the
cluster catalogue presented by Miller et al. (2005). We found 16
clusters in common, being 74 km s−1 the median absolute dif-
6 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
Fig. 3. Galaxy surface density (images) and radial velocity versus distance to the cluster center for the galaxy cluster member
(red points) of a subsample of 8 clusters. The overplotted circle have a radius equal to r200 for each cluster. The black points
represent interloper galaxies.
ference between our and their σc. In this case, 3 clusters show
an absolute differences in σc larger than 200 km s
We can conclude that in most of the cases our cluster mem-
bership method reported values of σc similar to those given by
other methods. Only for 10-20% of the clusters the absolute
differences in σc between our method and the others is larger
than 200 km s−1. For these clusters the structure along the line
of sight is the responsible of the difference, being our σc values
smaller than the others.
3.4. Lx-σ relation
We can learn about the nature of cluster assembly by studying
the relations between cluster observables. One of the most uni-
versals is the well known relation between the cluster X-ray lu-
minosity and the velocity dispersion of its galaxies (LX ∝ σbc).
Cluster formation models predict that if the only energy source
in the cluster comes from the gravitational collapse, then b ≈ 4.
This relation has been studied in the literature by many authors
using different cluster samples, finding values of b between
2.9 and 5.3 (Edge & Stewart 1991; Quintana & Melnick 1982;
Mulchaey & Zabludoff 1998; Mahdavi & Geller 2001;
Girardi & Mezzetti 2001; Borgani et al. 1999;
Xue & Wu 2000; Ortiz-Gil et al. 2004; Hilton et al. 2005).
The study of the LX − σc relation in our cluster sample will
be also useful as another check for the values of σc we have
derived. We have X-ray data for 48 galaxy clusters from our
sample. The X-ray data have been obtained from Ebeling et
al. (1998), Böhringer et al.(2000), Ebeling et al.(2000) and
Ledlow et al.(2003), and the X-ray luminosities are measured
in the ROSAT band (0.1-2.4 keV).
Figure 4 shows the LX−σ relation for this subset with avail-
able X-ray data in the literature. The Spearman coefficient of
the relation is 0.56 and the significance from zero correlation
is greater than 3σ. This indicates the existence of a correlation
between LX and σc for the clusters of our sample. We used the
bivariate correlated errors and intrinsic scatter (BCES) bisector
method of Akritas & Bershady (1996) to obtain the coefficient
and power-law slope estimates of the relation. This fitting tech-
nique takes into account errors in both variables and intrinsic
scatter. The LX − σc relation for our clusters is given by:
LX(0.1 − 2.4 keV) = 1033.7±1.2σ3.9±0.4 (3)
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 7
Fig. 4. LX−σ relation for the 48 galaxy clusters with X-ray data
in the ROSAT band (0.1-2.4 keV) from our sample. The full
line represents the best fit using the BCES bisector algorithm
(see text for more details).
Fig. 5. Absolute r-band magnitude as a function of redshift for
the galaxies of our cluster sample.
This result is in very good agreement with another mea-
surement of this relation using the same ROSAT band (0.1-2.4
keV) for the X-ray data and the same fitting algorithm (see
Hilton et al. 2005).
3.5. Redshift distribution and sample completeness
The 88 isolated galaxy clusters are located in a redshift range
between 0.02 and 0.1, with an average redshift of 0.071. Figure
5 shows the absolute r-band magnitude (Mr) as a function
of the redshift for the galaxies in our cluster sample4. It is
clear that the completeness magnitude is a function of redshift.
This figure shows that the full sample is complete for galaxies
brighter than Mr = −20.0. The lack of completeness for fainter
galaxies will be taken into account in the subsequent analysis.
4 See section 3 for the explanation of the computation of the abso-
lute magnitudes of the galaxies.
4. Morphological Segregation
Light concentration or colours have been used extensively in
the literature in order to classify galaxies. Shimasaku et al.
(2001) and Strateva et al. (2001) using SDSS data, found that
the ratio of Petrosian 50 percent light radius to Petrosian 90
percent light radius, Cin, measured in the r-band image was a
useful index for quantifying galaxy morphology. Strateva et al.
(2001) also found that the colour u − r = -2.22 efficiently sep-
arates early- and late-type galaxies at z < 0.4. We have used
colours for classifying galaxies, because properties such as ve-
locity dispersion in galaxy clusters are better correlated with
galaxy colours than galaxy morphology (Goto 2005). The mag-
nitude of the galaxies were corrected by two effects: Galactic
absorption and k−correction. The Galactic absorption in the
different filters was obtained from the dust maps of Schlegel
et al. (1998). We applied the k−correction using the kcor-
rect.v4 1 4 code by Blanton et al. (2003) in order to obtain the
rest-frame magnitudes of the galaxies for the different band-
passes. Once these two corrections were done, we classified
the galaxies in red (u − r ≥ 2.22) and blue ones (u − r < 2.22).
The galaxy data was downloaded from the SDSS database
according to a metric criteria: we downloaded the information
of all galaxies located within a radius of 4.5 Mpc at each galaxy
cluster redshift. This means that we are mapping different phys-
ical regions for each cluster. In order to avoid this problem we
have studied the ratio rmax/r200 for each cluster, being rmax the
maximum distance of a galaxy from the cluster centre for each
galaxy cluster. We have obtained that all clusters of our sample
reach rmaxr200 = 2, and 50% of them reach
Our sample of galaxies consists of 6880 galaxies located
within a radius 2 × r200, being 62% of them red galaxies and
38% blue ones. If we consider all galaxies within 5 × r200 then
the sample has 10865 galaxies, being 55% and 45% red and
blue galaxies, respectively. The red and blue galaxies were also
grouped in three categories according to their r-band magni-
tude: Mr < M
r − 1, M
r − 1 < Mr < M
r + 1, and Mr > M
r + 1
The first group contains the brightest members of the clusters,
the third group contains the so-called dwarf population and the
second one is formed by normal bright galaxies. Table 2 shows
the median location, r-band absolute magnitude, velocity dis-
persion and local density6 of the different galaxy groups. In
general, red galaxies are brighter than blue ones, and are also
located closer to the cluster centre at higher local density re-
gions. The two families of galaxies present different kinemat-
ics, in the sense that red galaxies show a smaller velocity dis-
persion than blue ones. This different kinematic between red
and blue galaxies has also been seen in other studies, and have
been interpreted as red and blue galaxies having different kind
of orbits, being the orbits of blue galaxies more anisotropic
than the red ones (Adami et al. 1998; Biviano & Katgert 2004).
Other authors interpret this difference in velocity dispersion as
an evidence that ram pressure is not playing an important role
in galaxy evolution in clusters. In contrast, tidal interactions
5 M∗r − 5log(h) = −20.04, Blanton et al. (2005)
6 The local surface density (Σ) was computed with the 10 nearest
neighbours to each galaxy belonging to the cluster.
8 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
should be the dominant mechanism (Goto 2005). All of these
properties are independent of the sampled area.
It is also interesting that red dwarf galaxies are located
at similar environments as the brightest red ones: close to
the cluster centre in high local density regions (see also
Hogg et al. 2004). But the red dwarf population shows a larger
velocity dispersion than the brightest red galaxies. Biviano &
Katgert (2004) found that the brightest cluster members were
not in equilibrium with the cluster potential. They are especial
galaxies that could have formed close to the cluster centre or
have fallen to this region due to dynamical friction. In contrast,
dynamical friction is not so efficient in the dwarf population, so
that the main presence of these galaxies in the central regions of
the clusters should be due to their origin. The discussion about
the properties and origin of the dwarf population will be given
in another paper (Sánchez-Janssen et al., in preparation).
5. Velocity Dispersion Profiles
The adopted cluster velocity dispersion was calculated with the
galaxies located within the r200 radius of each cluster. But, how
does σ depend on the clustercentric distance in our sample?.
This can be answered by studying the integrated velocity dis-
persion profiles (VDPs) of the clusters. These profiles also pro-
vide information about the dynamical properties of the galax-
ies. Thus, a system with galaxies predominantly in radial or-
bits produces an outwards declining VDP, while the opposite
behaviour suggests instead that the galactic orbits are largely
circular. In contrast, constant VDPs are characteristic of an
isotropic distribution of velocities (Solanes et al. 2001). Figure
6 shows the VDPs for some of the clusters in our sample. They
show the velocity dispersion of the cluster at a given radius
evaluated using all the galaxies within that radius, without any
restriction in their luminosities. The errors showed in Fig. 6
were computed using the approximation given by Danese et al.
(1980).
In order to classify the VDPs of our clusters, we computed
the velocity dispersion (σi, i = 1, 2, 3, 4, 5) of the galaxies in the
clusters located within five different radius: 0.4×r200, 0.6×r200,
2×r200, 3×r200 and 4×r200, respectively. We compared these
values with σc, given in Table 1. The resulting mean ratios
σi/σc were: 1.02± 0.04, 1.01±0.01, 0.97±0.01, 0.94±0.02 and
0.94±0.02, for i = 1, 2, 3, 4, 5, respectively. These values indi-
cate that within r200 the VDPs of the total galaxy cluster popu-
lation are consistent with being flat. The mean variation of the
VDPs inside r200 is only 2%. The values of σi/σc, i = 3, 4, 5
show that, outside r200 the VDPs slowly decrease. The mean
variation of the VDPs outside r200 is −6%. No differences in the
ratios σi/σc have been found when we have divided the galaxy
sample between bright (Mr < M
r +1) and dwarf (Mr > M
r +1)
galaxies. This flat behaviour of the VDPs inside r200 suggests
that galaxies in these areas have an isotropic distribution of ve-
locities. In contrast, the decline with radius of VDPs outside
r200 points to radial orbits (Solanes et al. 2001).
Figure 6 also shows the VDPs of early- (red) and late-type
(blue) galaxies. In most profiles the velocity dispersion of blue
galaxies is larger than the corresponding one for early-type
ones. We have also analysed the shape of VDPs of blue and
Fig. 7. Histograms of the ratios σi/σc, i = 1, 2, 3, 4, 5 for the
galaxies in the clusters. The black full line represent all galax-
ies, the blue and red lines correspond to late- and early-type
ones. See text for more details.
red galaxies as we did for the total sample. For red galaxies,
we obtained that σi/σc,r are 1.04± 0.03, 1.03±0.03, 0.97±0.02,
0.96±0.02 and 0.96±0.03, for i = 1, 2, 3, 4, 5, respectively.
The values of σi/σc,b for the blue galaxies are: 1.15± 0.07,
1.04±0.03, 0.95±0.04 and 0.93±0.04 and 0.92±0.04, respec-
tively. In those computations, σc,r and σc,b represent the ve-
locity dispersion of the red and blue galaxies within a radius
equal to r200, respectively. Figure 7 show the distribution of
σi/σc, i = 1, 2, 3, 4, 5 for the blue, red and the total galaxy sam-
The VDPs have been studied in the literature by sev-
eral authors. Most of them conclude that for large radii
(r > 1 Mpc) the VDPs are flat (Girardi & Mezzetti 2001;
Rines & Diaferio 2006; Fadda et al. 1996; Muriel et al. 2002).
This is consistent with the mild decrease that we have found
in our clusters. The VDPs for red galaxies in our sample are
almost flat outside r200. This is not the case of the VDPs of
blue galaxies which clearly decrease with radius outside r200.
In the inner regions (r < r200) the VDPs of the total sample
and those corresponding to the red galaxies are flat. In con-
trast, the VDPs of blue galaxies decrease with radius. Different
authors show that VDPs can decrease or increase with radius.
den Hartog & Katgert (1996) made a thorough study and found
that the variations of the VDPs in the innermost regions of clus-
ters (r < 0.5 Mpc) are real and not due to noise or bad centre
election. We have re-computedσ1/σc andσ2/σc only for those
clusters with X-ray centres, and our results did not significantly
change. Thus, we can conclude that in our galaxy cluster sam-
ple only blue galaxies show increasing VDPs towards the cen-
tre of the cluster, while red galaxies show flat VDPs.
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 9
Table 1. Main properties of the different types of galaxies
Galaxies within r/r200 < 5 < r/r200 > < Mr > < σ > < log(Σ) > Ngal
u − r < 2.22 1.85±0.02 -19.65±0.01 1.04±0.01 0.46±0.08 4937
u − r < 2.22 & Mr < M∗r − 1 1.79±0.11 -21.54±0.02 1.08±0.09 0.40±0.25 94
u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.90±0.02 -20.00±0.01 1.03±0.01 0.38±0.08 3126
u − r < 2.22 & Mr > M∗r + 1 1.74±0.03 -18.74±0.02 1.05±0.02 0.64±0.14 1717
u − r ≥ 2.22 1.03±0.01 -20.16±0.01 0.90±0.01 0.80±0.05 5928
u − r ≥ 2.22 & Mr < M∗r − 1 0.95±0.05 -21.62±0.02 0.78±0.03 0.89±0.14 537
u − r ≥ 2.22 & M∗r − 1 < Mr < M
r + 1 1.10±0.02 -20.20±0.01 0.91±0.01 0.75±0.06 4592
u − r ≥ 2.22 & Mr > M∗r + 1 0.85±0.04 -18.91±0.02 0.90±0.03 1.06±0.12 729
Galaxies within r/r200 < 2 < r/r200 > < Mr > < σ > < log(Σ) > Ngal
u − r < 2.22 0.97±0.01 -19.61±0.02 1.08±0.02 0.80±0.07 2636
u − r < 2.22 & Mr < M∗r − 1 1.15±0.07 -21.56±0.03 1.18±0.13 0.62±0.24 54
u − r < 2.22 & M∗r − 1 < Mr < M∗r + 1 1.04±0.01 -19.96±0.01 1.08±0.02 0.70±0.07 1648
u − r < 2.22 & Mr > M∗r + 1 0.85±0.01 -18.70±0.02 1.07±0.03 1.02±0.09 934
u − r ≥ 2.22 0.67±0.01 -20.14±0.01 0.91±0.01 0.98±0.04 4244
u − r ≥ 2.22 & Mr < M∗r − 1 0.57±0.03 -21.65±0.02 0.80±0.03 1.02±0.13 397
u − r ≥ 2.22 & M∗r − 1 < Mr < M
r + 1 0.69±0.01 -20.19±0.01 0.92±0.01 0.94±0.05 3239
u − r ≥ 2.22 & Mr > M∗r + 1 0.61±0.02 -18.92±0.02 0.89±0.03 1.19±0.10 608
The previous findings can also be seen in Fig 8. We show
the VDPs of the different galaxy classes for the normalised
cluster, which was obtained by normalising the scales and ve-
locities of each galaxy of the sample. Thus, the radial distance
of each galaxy to the cluster centre was scaled by r200 of the
corresponding cluster, and the relative velocity of each galaxy
cluster was normalised by the velocity dispersion of the clus-
ter. Figure 8 shows the VDPs which correspond to the total,
bright (Mr < M
r + 1) and dwarf (Mr > M
r + 1) galaxy sam-
ples. We have also distinguished between red and blue objects.
The VDPs of the total galaxy sample indicate that blue galax-
ies have always larger velocity dispersion than red ones. They
also show always decreasing VDPs, while red ones have almost
constant and slowly decrease VDPs inside and outside r200, re-
spectively. These features can also be seen in the VDPs of the
bright galaxy sample. In contrast, red and blue dwarfs show
decreasing VDPs inside r200.
The shape of the VDPs can provide information about
the dynamical state of the galaxies. Thus, clusters with
galaxies predominantly in radial orbits produce an outwards
declining VDP. This is the case of the blue galaxies of
our sample, which is in agreement with previous findings
(Biviano & Katgert 2004; Adami et al. 1998). We have also
obtained that the red dwarf galaxies inside r200 has an outwards
declining VDP. This would imply that this kind of galaxies may
also be located in radial orbits. In contrast, constant VDPs im-
ply an isotropic distribution of velocities (Solanes et al. 2001).
This is the case of the red bright galaxy population inside r200.
6. Fraction of blue galaxies
Butcher & Oemler (1984) observed that the fraction of blue
galaxies ( fb) in clusters evolves with redshift, in the sense that
galaxy clusters located at medium redshift have a larger fb than
nearby ones. This has been usually interpreted as an evolution-
ary trend in clusters. But it is a matter of debate which is the
role played by the environment in the change of the fraction
of blue galaxies. We have computed fb in our sample of galaxy
clusters, studying the variation with z and the possible influence
of the environment.
6.1. Adopted aperture and limiting magnitude
The original analysis of Butcher & Oemler (1984) defined blue
galaxies as those within a radius containing 30 per cent of the
cluster population, being brighter than Mv = −20 and bluer by
0.2 mag in B − V than the colour-magnitude relation defined
by the cluster early-type galaxies. It has been noticed by sev-
eral authors that the fraction of blue galaxies strongly depends
on the magnitude limit and the clustercentric distance used
(Ellingson et al. 2001; Goto et al. 2003; De Propris et al. 2004;
Andreon et al. 2006). They observed that fb grows when the
magnitude limit is fainter and the aperture is larger, reflect-
ing the existence of a large fraction of blue faint galaxies in
the outer regions of the clusters. De Propris et al. (2004) con-
sidered appropriate to measure fb in apertures based on clus-
ter physical properties. They used r200 as the aperture radius
where they measured fb for their clusters. We have adopted
also this radius in order to determine fb in our galaxy clus-
ters. As it was previously commented, fb depends also on
the adopted limiting magnitude of the galaxies in clusters.
It should be noticed that as we move to higher redshifts we
systematically lose faint galaxies (see Fig 2). Our clusters
spread in a redshift range 0.02 < z < 0.1, and only galax-
ies brighter than Mr = −20.0 (≈ M∗r + 0.5) can be observed
at all redshifts. For this reason, we have adopted this abso-
lute magnitude as the limiting magnitude for the computa-
tion of fb. This ensures us to work with a complete galaxy
sample at all redshifts. Other authors adopted fainter limiting
magnitudes, e.g. M∗ + 1.5 (De Propris et al. 2004) or M∗ + 3
(Margoniner & de Carvalho 2000). If there is a large number
of blue galaxies at faint magnitudes, we expect that our val-
ues of fb will be smaller than those reported by the previous
authors.
10 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
Fig. 6. Velocity dispersion profiles of some clusters of our sample. The black symbols represent the velocity dispersion profile
taking into account all types of galaxies. Blue and red symbols represent the velocity dispersion profiles corresponding to blue
and red galaxies, respectively.
6.2. Colour-magnitude diagrams
We determined the g − r versus r colour-magnitude diagrams
for all the clusters in our sample. The colour-magnitude rela-
tion was measured by a robust fitting routine by minimising the
absolute deviation in g−r colour, using only early-type galaxies
located within an aperture of radius equal to r200. The galaxy
types were determined according to the u−r colour and the light
galaxy concentration parameter, Cin. These two criteria allow
us to identify the most reliable sample of E/S0 galaxies (see
Shimasaku et al. 2001; Strateva et al. 2001). Thus, we consid-
ered early-type galaxies those with u − r ≥ 2.22 and Cin < 0.4.
Figure 9 shows the colour-magnitude diagrams of four repre-
sentative galaxy clusters. The colour-magnitude relation fitted
in each case is also overploted. Figure 9 (left column) also
shows the histograms of the colour distribution, marginalised
over the fitted colour-magnitude relation.
The average of the slopes of the colour-magnitude relations
of the early-type galaxies of the clusters is -0.014±0.008. This
slope is within the errors in agreement with the slope obtained
by Gallazzi et al. (2006) for a large sample of galaxies using
SDSS data. It is also in agreement with the average B−R slope
obtained by De Propris et al. (2004) for a sample of galaxy
clusters from 2dFGRS.
6.3. Calculation the blue fraction of galaxies
As we explained before, the blue fraction of galaxies was com-
puted using only those galaxies brighter than Mr = −20 and
located within an aperture of radius r200. In the present study
we only used spectroscopically confirmed galaxy cluster mem-
bers. This should not bias our results, especially due to our high
completeness. Figure 10 presents fb as a function of redshift.
The errors of fb were computed according to the prescription
given by De Propris et al. (2004). We observe no evolution of
fb with redshift, which means that our sample is ideal to study
the effects of the environment on fb.
We have considered three cluster properties (concentration,
velocity dispersion and X-ray luminosity) of each cluster in or-
der to analyse the dependence of fb on the environment. The
concentration parameter was computed following the prescrip-
tion of De Propris et al. (2004), i.e. C = log(r60/r20), where r60
and r20 are the radii containing 60 and 20 per cent of the cluster
galaxies, respectively. The velocity dispersion of the clusters
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 11
Fig. 8. Velocity dispersion profiles of the galaxies of the normalised cluster. The total galaxy population is showed in the top
panel. Bright galaxies (Mr < M
r + 1) are in the middle panel, and the bottom panel shows the VDPs corresponding to the dwarf
galaxy sample (Mr > M
r + 1). The VDP of the total, blue and red galaxy samples are represented by black, blue and red colours,
respectively (see text for more details).
Fig. 10. The fraction of blue galaxies ( fb) as a function of red-
shift of the galaxy clusters.
was taken from Table 1. The X-ray luminosities were obtained
from the literature (Ebeling et al. 1998, Böhringer et al.2000,
Ebeling et al. 2000 and Ledlow et al. 2003), being measured in
the ROSAT band (0.1-2.4 keV). We only found X-ray data for
48 clusters of the sample.
Figure 11 shows the dependence of the fraction of blue
galaxies on concentration, cluster velocity dispersion and X-
ray luminosity. The non-parametric Spearman test returns that
fb has a low correlation with concentration and velocity dis-
persion. The fraction of blue galaxies correlates best with the
velocity dispersion, but the significance of the correlation is
2.6σ. In contrast, the Spearman test shows correlation between
fb and X-ray luminosity, being the significance of this correla-
tion just 3σ. Notice that the points are distributed in the LX − fb
plane following a triangular shape. Clusters with large X-ray
luminosity (LX(0.1 − 2.4keV) > 1045ergs−1) show small frac-
tions of blue galaxies (less than 10%). Nevertheless, those clus-
ters with small X-ray lumisosity show small and large fraction
of blue galaxies. This correlation could indicate that there is a
threshold over which cluster environment can affect the galaxy
colours, and play a role in the galaxy evolution. This means
that, according with our LX −σ relation, the evolution of galax-
ies could be driven by the cluster environment for those clusters
12 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
Fig. 9. Colour-magnitude relation (left) and histograms (right) of marginalised colour distribution for four representative clusters
at different redshifts of our cluster sample. The full line in left panels represent the fitted colour-magnitude relation. The vertical
point lines in right panel represent the blue/red separation in the Butcher-Oemler effect. The red points are the galaxies with
u − r ≥ 2.22 and Cin < 0.4 (see text for more details).
with velocity dispersion larger than σ ≈ 800 km s−1. Recently,
(Popesso et al. 2006) found a similar correlation between LX
and fb for a larger cluster sample. The shape of the our LX − fb
correlation is similar to the correlation between cluster veloc-
ity dispersion and the fraction of [OII] emitters for clusters at
low redshift reported by Poggianti et al. (2006). They found
that clusters with σ > 550 km s−1 have a constant low fraction
(less than 30%) of [OII] emiters. In contrast, those clusters with
smaller σ show large and small fractions of [OII] emiters.
We have recomputed fb taking into account those galax-
ies within an aperture of radius equal to r200 and brighter than
Mr = −19.5. We restricted the analysis only to those clus-
ters with z < 0.05, because our sample is complete down to
Mr = −19.5 in this redshift range. In this case the number of
cluster decreases to 13. We have again studied the correlations
of fb with galaxy concentration, velocity dispersion and X-ray
luminosity, obtaining similar correlations as with the full sam-
7. Discussion
From the study presented in this paper, most of the galaxies
(62%) located in the central regions of galaxy clusters (r/r200 <
2) are early-type galaxies (see section 4). In contrast, the field
population is dominated by late-type galaxies. In the literature
it is also well established that the colour of galaxies in clus-
ters and field is different, an indication of the low star forma-
tion activity found in cluster galaxies (e.g. Balogh et al. 1998;
Lewis et al. 2002; Gómez et al. 2003). These differences in
morphology and stellar content between field and cluster galax-
ies suggest different evolutionary processes. The facts that late-
type galaxies show larger velocity dispersions and are located
at larger distances from the cluster centre than early-type ones
have been interpreted as late-type galaxies being recent ar-
rivals to the cluster potential, forming a non-relaxed group of
galaxies moving in more radial orbits than early-type ones (e.g.
Stein 1997; Adami et al. 1998). As late-type galaxies fall into
the cluster potential and encounter denser environments, they
evolve to early-type ones. The results presented in the present
work are in agreement with previous findings. However, as
pointed out by Goto (2005), this would imply that a large frac-
tion of galaxies (≈ 40% according to our sample) should be
recent arrivals to the cluster, a possibility that seems unlikely.
Goto (2005) concluded that the different observational proper-
ties between red and blue galaxies may indicate which is the
main mechanism driving the evolution of galaxies in clusters.
Gas stripping, mergers and interactions with other galaxies and
with the cluster potential are the main mechanisms which are
able to transform galaxies in clusters, making late-type galax-
ies lose their gas content, stop their star formation, circularise
their orbits and transform their morphology from disk-like ob-
jects to spheroids. All of these mechanisms affect galaxies in
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 13
Fig. 11. The fraction of blue galaxies ( fb) as a function of galaxy distribution (top), cluster velocity dispersion (middle), and
X-ray luminosity (bottom) of the galaxy clusters.
clusters but, can we infer from the observational results which
is the dominant one?.
It should be noted that the different mechanisms of galaxy
evolution have very different time-scales. While gas stripping
has a very short time-scale (≈ 50 Myr, Quilis et al. 2000), the
galaxy infall process can take ≈ 1 Gyr. The different mecha-
nisms also have different underlying physics. Thus, ram pres-
sure stripping is proportional to the density of the intracluster
medium (ICM) and to the square of the velocity of the galaxy.
In contrast, dynamical interactions are more efficient when the
relative velocity of galaxies is smaller (Mamon 1992). This
means that gas stripping is stronger in the cluster centres and
for galaxies with high velocities, while dynamical interactions
should be more efficient for galaxies with smaller velocity dis-
persions. Numerical simulations have shown that most of the
galaxies inside the virial radius have already been through the
cluster core more than once (Mamon et al. 2004). If gas strip-
ping were the main mechanism driving galaxy evolution in
clusters, according to the short time scale of this process, only
few blue (late-type) galaxies should be observed in the central
regions of clusters. Moreover, gas stripping is also stronger in
galaxies with larger velocity dispersion which means that late-
type galaxies should be more affected by this mechanism than
early-type ones. Based on these considerations, Goto (2005)
concluded that gas stripping is not the main responsible mech-
anism driving the evolution of galaxies in clusters. Instead,
galaxies in clusters evolve mainly by dynamical interactions.
We can add to Goto’s discussion that if gas stripping were the
main galactic evolution mechanism in clusters, then the frac-
tion of blue galaxies should depend on the cluster mass as the
temperature and density of the gas increases with the cluster
mass. According to our results, this is true for those clusters
with large X-ray luminosities. In contrast, the cluster environ-
ment is not so important in driving the evolution of galax-
ies in low mass clusters. Thus, gas stripping may not be the
main responsible mechanism transforming late-type to early-
type galaxies in low mass clusters, but could be important in
the most massive ones. This does not mean that gas stripping
is absent in the evolution of galaxies in clusters; some clear ex-
amples of gas stripping have been observed in galaxies in Virgo
(Kenney et al. 2004).
Dynamical interactions include both interactions with the
cluster potential and with other galaxies. These effects can trig-
ger temporary star formation in cluster galaxies (Fujita 1998),
which can be analysed by studying their colour distribution.
These interactions can also disrupt stars from galaxies, form-
ing at the beginning long tidal tails that subsequently will
be diluted and will form the diffuse light observed in some
nearby clusters like Virgo (see Aguerri et al. 2005b, and ref-
erences therein). These effects will be more important in those
galaxies with smaller relative velocities. Fujita (1998) con-
clude that if the tidal effects enhance the SFR in the galaxies,
then the bluest galaxies should be located close to the clus-
ter centre (within ≈ 300 kpc), whereas they should be in the
outer parts of the cluster if the SFR is induced by galaxy-
galaxy encounters. We have investigated the fraction of blue
14 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
galaxies in our clusters located within 300 kpc from the cen-
tre of the cluster. The sample has been divided in bright and
dwarf galaxies (Mr < M
r + 1 and Mr > M
r + 1, respec-
tively). We have obtained that 40% of the blue bright galax-
ies and 30% of the blue dwarf ones are located at smaller dis-
tance than 300 kpc from the cluster center. This means that
tidal interactions with the cluster potential are not the respon-
sible mechanism for the formation of most of the blue galax-
ies in our clusters. The lack of blue galaxies in the central re-
gions of clusters has been observed also in nearby clusters like
Coma (Aguerri et al. 2004) as well as in other distant clusters
(Rakos et al. 1997; Abraham et al. 1996; Balogh et al. 1997).
These evidences indicate that the evolution of galaxies in
clusters could be driven by the cluster environment in the most
massive ones, but galaxies in low mass clusters could mainly
evolve due to the local environment.
8. Conclusions
In the present paper we have analysed the main properties of
the galaxies of one of the largest (10865 galaxies) and homo-
geneous sample presented in the literature. The galaxies have
been grouped in two families according to their u − r colour.
Those galaxies with u − r ≥ 2.22 formed the red (early-type)
family, and those with u − r < 2.22 the blue (late-type) one.
We have derived the position, velocity dispersion, and VDPs of
both families of galaxies, obtaining:
– Within 2×r200, 62% and 38% of the galaxies turned to be
red and blue, respectively.
– The median positions and velocity dispersions are smaller
for red galaxies than for blue ones.
– Bright (Mr < M∗r −1) and dwarf (Mr > M
r +1) red galaxies
are located at smaller distances than the blue ones, sharing
the same cluster environment.
– The brightest cluster members (Mr < −21.0) show smaller
velocity dispersions than the remaining.
– The VDPs of the total galaxy cluster population are con-
stant with radius in the central regions of the clusters
(r < r200), while slowly decrease in the outermost regions
(r ≥ r200). The red galaxy population have also flat VDPs in
the central regions (r < r200). In contrast, the VDPs of blue
galaxies grow towards the cluster centre. In the outer re-
gions (r > r200), the VDPs of red galaxies decline smoothly
with radius, while for blue ones the decrement is faster.
This indicates that the galaxies in the outermost regions of
the clusters are dominated by the blue population, and have
more radial and anisotropic orbits than galaxies in the inner
regions dominated by the red population.
– The fraction of blue galaxies in our cluster sample does not
correlate with cluster global properties, such as the concen-
tration of the galaxy distribution and cluster velocity disper-
sion. However, we found a correlation between the X-ray
luminosity and the fraction of blue galaxies. Those clusters
with LX(0.1 − 2.4keV) > 1045 erg s−1 have a low fraction
of blue galaxies (less than 10%). In contrast, clusters with
low of X-ray luminosity show large and small fractions of
blue galaxies. This could indicate that the star formation in
cluster galaxies may be regulated by global cluster proper-
ties for clusters with LX(0.1 − 2.4keV) > 1045 erg s−1, i.e.
those clusters with σc > 800 km s
All these results are in agreement with previous findings
from other cluster samples, indicating that red and blue galax-
ies have different evolution in galaxy clusters. We have dis-
cussed these results according to the different galaxy transfor-
mation mechanisms presented in galaxy clusters, concluding
the local environment plays a key role in galaxy evolution in
low mass clusters, while the evolution of galaxies in massive
clusters could be driven by the global cluster environment.
Acknowledgements. We wish to thank to the anonymous referee
for useful coments which have improved this manuscript. We
also acknowledge financial support by the Spanish Ministerio de
Ciencia y Tecnologı́a grants AYA2004-08260. We would like also
to thank T. Beers for providing us with copy of his code ROSTAT,
and K. M. Ashman and S. Zepf for making their KMM code
available to us. Funding for the SDSS and SDSS-II has been
provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Science Foundation, the U.S. Department
of Energy, the National Aeronautics and Space Administration,
the Japanese Monbukagakusho, the Max Planck Society, and
the Higher Education Funding Council for England. The SDSS
Web Site is http://www.sdss.org/. The SDSS is managed by the
Astrophysical Research Consortium for the Participating Institutions.
The Participating Institutions are the American Museum of Natural
History, Astrophysical Institute Potsdam, University of Basel,
Cambridge University, Case Western Reserve University, University
of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the
Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese
Academy of Sciences (LAMOST), Los Alamos National Laboratory,
the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-
Institute for Astrophysics (MPA), New Mexico State University, Ohio
State University, University of Pittsburgh, University of Portsmouth,
Princeton University, the United States Naval Observatory, and
the University of Washington. This research has made use of the
NASA/IPAC Extragalactic Database (NED) which is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.
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Table 2. Cluster characteristics
Name α (J2000) δ (J2000) vc σc r200 Ngal C
(degrees) (degrees) (km s−1) (km s−1) (Mpc)
Abell0085 10.4571 -9.30694 16633+40−29 979
−39 2.10 273 0.93
Abell0117 14.0100 -10.0022 16568+31−42 531
−27 1.19 60 0.88
Abell0152 17.5229 13.9804 17888+67−34 538
−38 1.12 27 0.85
Abell0168 18.7429 0.365833 13534+23−14 578
−28 1.19 106 0.88
Abell0257 27.3396 14.0372 21060+47−21 381
−44 0.81 26 0.94
Abell0602 118.341 29.3717 18202+34−51 834
−61 1.87 78 0.74
Abell0628 122.543 35.2958 25139+24−89 666
−38 1.47 43 0.83
Abell0671 127.121 30.4169 14599+19−33 610
−33 1.42 72 0.89
Abell0690 129.815 28.9033 23689+44−23 395
−24 0.85 22 0.95
Abell0695 130.309 32.4174 20251+46−37 456
−32 1.04 16 0.86
Abell0699 131.236 27.7508 25375+35−49 438
−37 0.91 19 0.73
Abell0724 134.600 38.5137 28134+25−55 433
−32 1.00 29 0.94
Abell0727 134.976 39.4389 28571+54−14 423
−29 0.96 33 0.96
Abell0757 138.277 47.7036 15402+27−36 409
−30 0.84 30 0.85
Abell0779 139.962 33.7714 6921+13−33 336
−21 0.79 57 0.75
Abell0819 143.076 9.68861 22872+39−50 536
−37 1.19 31 0.94
Abell0883 147.822 5.48799 21750+109−30 523
−58 1.17 18 0.91
Abell0971 154.997 40.9925 27809+54−43 816
−61 1.88 40 0.87
Abell0999 155.842 12.8466 9618+10−54 271
−17 0.60 25 0.90
Abell1003 156.235 47.8442 18762+44−84 617
−34 1.37 29 0.94
Abell1016 156.762 10.9780 9629+34−9 259
−17 0.60 25 0.91
Abell1024 157.096 3.76341 22067+40−20 532
−34 1.26 35 0.90
Abell1032 157.547 4.03417 20008+26−24 355
−32 0.77 25 0.89
Abell1035 158.092 40.1817 20270+36−34 575
−45 1.34 49 0.97
Abell1066 159.911 5.17444 20708+4−81 826
−44 1.71 95 0.92
Abell1142 165.229 10.5477 10601+30−25 557
−38 1.33 59 0.88
Abell1149 165.769 7.57833 21479+2−64 352
−31 0.85 26 0.94
Abell1169 166.967 43.9506 17532+24−35 433
−32 0.91 35 0.94
Abell1173 167.328 41.5624 22789+23−69 611
−41 1.33 35 0.95
Abell1189 167.775 1.09899 28860+109−60 807
−88 1.76 41 0.94
Abell1190 167.902 40.8417 22610+17−39 706
−30 1.50 77 0.92
Abell1205 168.328 2.53867 22852+17−53 890
−53 2.04 83 0.84
Abell1215 170.100 4.34280 14747+25−10 214
−39 0.48 17 0.89
Abell1238 170.711 1.09389 22140+9−54 544
−41 1.30 60 0.91
Abell1270 172.366 54.0428 20728+49−36 569
−39 1.31 43 0.99
Abell1291 173.092 55.9783 17144+43−60 720
−36 1.58 45 0.97
Abell1318 173.883 55.0767 17185+54−22 360
−25 0.81 22 0.96
Abell1346 175.304 5.74613 29523+41−25 790
−62 1.69 66 0.85
Abell1377 176.883 55.7597 15378+35−33 671
−37 1.47 69 0.96
Abell1383 176.973 54.7089 17855+38−35 456
−23 1.02 35 0.95
Abell1385 177.017 11.5864 25337+49−44 609
−45 1.34 22 0.86
Abell1390 177.378 12.3034 25101+48−29 483
−41 1.06 27 0.86
Abell1424 179.361 5.12000 22736+33−39 617
−43 1.46 63 0.95
Abell1436 180.095 56.2314 19432+39−77 712
−44 1.51 66 0.96
Abell1452 180.802 51.6642 18609+56−58 533
−30 1.30 18 0.91
Abell1459 181.108 1.88281 6010+37−21 527
−48 1.18 65 0.95
Abell1507 183.766 59.8947 18009+9−70 379
−36 0.85 23 0.91
Abell1516 184.729 5.24731 23019+24−84 720
−43 1.49 60 0.90
Abell1552 187.392 11.7733 26495+64−42 442
−32 0.93 20 0.88
Abell1564 188.720 1.78056 23763+30−64 641
−62 1.51 46 0.91
Abell1616 191.817 55.0006 24882+65−93 565
−45 1.32 29 0.87
Abell1620 192.510 -1.53764 25400+64−38 829
−43 1.76 58 0.88
Abell1630 192.942 4.59694 19458+36−33 444
−37 0.98 30 0.90
18 Aguerri et al.: Global Properties of Nearby Galaxy Clusters
Table 2. continued.
Name α (J2000) δ (J2000) vc σc r200 Ngal C
(degrees) (degrees) (km s−1) (km s−1) (Mpc)
Abell1650 194.672 -1.76417 25138+86−18 790
−47 1.61 63 0.85
Abell1663 195.717 -2.51782 24953+60−20 729
−40 1.54 72 0.88
Abell1692 198.060 -0.976000 25395+51−47 607
−43 1.32 40 0.89
Abell1728 200.876 11.2960 26977+95−22 824
−62 1.88 50 0.70
Abell1750 202.709 -1.86389 26259+19−14 518
−33 1.15 35 0.95
Abell1767 204.024 59.2042 21174+39−32 885
−40 2.05 109 0.94
Abell1773 205.533 2.24805 23544+31−48 481
−31 1.09 32 0.87
Abell1780 206.149 2.86750 23285+35−22 624
−53 1.45 53 0.90
Abell1783 205.848 55.6261 20550+48−11 383
−32 0.94 33 0.94
Abell1809 208.245 5.16139 23815+39−31 737
−47 1.68 89 0.82
Abell1885 213.431 43.6634 26793+33−50 541
−56 1.23 22 0.92
Abell1999 223.522 54.2682 29841+11−57 463
−54 1.07 24 0.92
Abell2018 225.266 47.2831 26246+93−5 635
−37 1.45 39 0.88
Abell2023 227.496 2.98910 27743+48−18 516
−73 1.12 23 0.86
Abell2026 227.106 -0.267500 27188+62−37 747
−49 1.50 43 0.91
Abell2030 227.844 -0.0857717 27399+38−27 495
−45 1.10 38 0.91
Abell2061 230.317 30.6122 23646+24−19 622
−32 1.43 98 0.83
Abell2067 230.780 30.8703 23039+33−34 917
−46 2.19 118 0.85
Abell2092 233.348 31.1475 20000+42−17 458
−30 0.93 41 0.86
Abell2110 234.953 30.7173 29250+94−28 622
−49 1.25 21 0.83
Abell2122 236.259 36.1161 19793+28−42 826
−47 1.80 91 0.92
Abell2124 236.263 36.1172 19783+41−32 826
−47 1.78 90 0.92
Abell2145 240.094 33.2306 26583+80−23 632
−56 1.40 24 0.87
Abell2149 240.350 53.9061 19564+63−44 459
−27 1.01 20 0.85
Abell2169 243.422 49.1261 17286+30−50 521
−33 1.24 40 0.82
Abell2175 245.132 29.8953 28876+61−64 878
−57 1.76 58 0.87
Abell2199 247.154 39.5244 9118+15−30 747
−19 1.77 269 0.92
Abell2241 254.928 32.6161 29403+70−68 806
−62 1.61 37 0.90
Abell2244 255.663 34.0411 28927+50−57 428
−49 0.99 23 0.94
Abell2245 255.640 33.5056 25686+40−49 535
−39 1.28 39 0.95
Abell2255 258.222 64.0653 24052+19−39 883
−35 1.86 184 0.91
Abell2428 334.065 -9.34139 25207+15−68 433
−37 0.95 26 0.87
Abell2670 358.556 -10.4133 22755+29−20 642
−29 1.46 137 0.94
MACSJ0810.3+4216 122.600 42.2733 19193+44−25 505
−35 1.23 32 0.84
MACSJ1440.0+3707 220.011 37.0839 29402+88−46 587
−49 1.31 18 0.91
NSCJ152902+524945 232.309 52.8433 22063+101−7 652
−43 1.4 45 0.89
NSCJ161123+365846 242.854 36.9700 20221+39−25 485
−39 1.17 30 0.86
RBS1385 215.969 40.2619 24544+40−78 419
−36 0.84 16 0.91
RXCJ0137.2-0912 24.3137 -9.20277 12169+27−30 453
−30 0.93 49 0.92
RXCJ0828.6+3025 127.162 30.4280 14630+50−25 628
−33 1.45 76 0.89
RXCJ0953.6+0142 148.393 1.70550 29450+24−63 584
−59 1.30 22 0.96
RXCJ1115.5+5426 168.887 54.4350 20965+35−66 639
−38 1.35 50 0.94
RXCJ1121.7+0249 170.428 2.81840 14807+25−14 567
−41 1.40 73 0.85
RXCJ1351.7+4622 207.940 46.3668 18937+32−40 531
−27 1.16 40 0.94
RXCJ1424.8+0240 216.159 2.75677 16337+44−57 539
−36 1.19 22 0.91
RXJ1017.7-0002 154.452 -0.0595327 19169+44−28 413
−39 0.90 16 0.91
RXJ1022.1+3830 155.583 38.5308 16300+37−38 591
−33 1.35 51 0.82
RXJ1053.7+5450 163.449 54.8500 21623+23−49 665
−45 1.52 46 0.86
WBL238 146.732 54.4183 13995+51−17 602
−30 1.30 44 0.87
WBL518 220.179 3.45305 8141+19−28 454
−20 1.03 103 0.85
ZwCl0027.0-0036 7.31721 -0.183598 17994+18−44 465
−36 0.99 36 0.85
ZwCl0743.5+3110 116.655 31.0136 17419+49−89 694
−45 1.40 29 0.82
ZwCl1207.5+0542 182.578 5.38500 23137+43−36 580
−40 1.20 40 0.91
Aguerri et al.: Global Properties of Nearby Galaxy Clusters 19
Table 2. continued.
Name α (J2000) δ (J2000) vc σc r200 Ngal C
(degrees) (degrees) (km s−1) (km s−1) (Mpc)
ZwCl1215.1+0400 184.422 3.66040 23229+22−40 955
−39 2.17 130 0.90
ZwCl1316.4-0044 199.816 -0.907816 24972+56−31 557
−20 1.16 38 0.87
ZwCl1730.4+5829 261.856 58.4749 8379+26−36 491
−22 1.02 33 0.86
Introduction
Galaxy cluster Sample
Cluster Membership
Cluster global parameters
Corrections to line-of-sight velocities
Comparison with other methods
Lx- relation
Redshift distribution and sample completeness
Morphological Segregation
Velocity Dispersion Profiles
Fraction of blue galaxies
Adopted aperture and limiting magnitude
Colour-magnitude diagrams
Calculation the blue fraction of galaxies
Discussion
Conclusions
|
0704.1580 | Optical implementation and entanglement distribution in Gaussian valence
bond states | Optical implementation and entanglement distribution in
Gaussian valence bond states
Gerardo Adesso
Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;
Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185
Roma, Italy;
Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via S. Allende,
84081 Baronissi (SA), Italy
Marie Ericsson
Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
PACS numbers: 42.50.Dv, 03.67.Mn, 03.67.Hk, 03.65.Ud.
Abstract. We study Gaussian valence bond states of continuous variable systems, obtained
as the outputs of projection operations from an ancillary space of M infinitely entangled
bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. The
entanglement distribution in Gaussian valence bond states can be controlled by varying the
input amount of entanglement engineered in a (2M + 1)-mode Gaussian state known as
the building block, which is isomorphic to the projector applied at a given site. We show
how this mechanism can be interpreted in terms of multiple entanglement swapping from
the chain of ancillary bonds, through the building blocks. We provide optical schemes to
produce bisymmetric three-mode Gaussian building blocks (which correspond to a single
bond, M = 1), and study the entanglement structure in the output Gaussian valence bond
states. The usefulness of such states for quantum communication protocols with continuous
variables, like telecloning and teleportation networks, is finally discussed.
http://arxiv.org/abs/0704.1580v1
Optical implementation and entanglement distribution in Gaussian valence bond states 2
1. Introduction
Quantum information aims at the treatment and transport of information in ways forbidden by
classical physics. For this goal, continuous variables (CV) of atoms and light have emerged
as a powerful tool [1]. In this context, entanglement is an essential resource. Recently, the
valence bond formalism, originally developed for spin systems [2], has been generalized to
the CV scenario [3, 4] for the special class of Gaussian states, which play a central role in
theoretical and practical CV quantum information and communication [5].
In this work we analyze feasible implementations of Gaussian valence bond states
(GVBS) for quantum communication between many users in a CV setting, as enabled by
their peculiar structure of distributed entanglement [4]. After recalling the necessary notation
(Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss
the characterization of entanglement and its distribution in such states as regulated by the
entanglement properties of simpler states involved in the valence bond construction [4]
(Sec. 4). We then focus on the realization of GVBS by means of quantum optics, provide
a scheme for their state engineering (Sec. 5), and discuss the applications of such resources in
the context of CV telecloning [6, 7] on multimode harmonic rings (Sec. 6).
2. Continuous variable systems and Gaussian states
A CV system [1, 5] is described by a Hilbert space H =
i=1 Hi resulting from the tensor
product of infinite dimensional Fock spaces Hi’s. Let ai and a†i be the annihilation and
creation operators acting on Hi (ladder operators), and q̂i = (ai + a†i ) and p̂i = (ai − a
i )/i
be the related quadrature phase operators. Let R̂ = (x̂1, p̂1, . . . , q̂N , p̂N ) denote the vector of
the operators q̂i and p̂i. The canonical commutation relations for the R̂i can be expressed in
terms of the symplectic form Ω as
[R̂i, R̂j ] = 2iΩij ,
with Ω ≡ ω⊕N , ω ≡
The state of a CV system can be equivalently described by quasi-probability distributions
defined on the 2N -dimensional space associated to the quadratic form Ω, known as quantum
phase space. In the phase space picture, the tensor productH =
i Hi of the Hilbert spaces
Hi’s of the N modes results in the direct sum Λ =
i Λi of the phase spaces Λi’s.
States with Gaussian quasi-probability distributions are referred to as Gaussian states.
Such states are at the heart of information processing in CV systems [1, 5] and are the subject
of our analysis. By definition, a Gaussian state is completely characterized by the first and
second statistical moments of the field operators, which will be denoted, respectively, by the
vector of first moments R̄ ≡
〈R̂1〉, 〈R̂2〉, . . . , 〈R̂2N−1〉, 〈R̂2N 〉
and the covariance matrix
(CM) γ of elements
γij ≡
〈R̂iR̂j + R̂jR̂i〉 − 〈R̂i〉〈R̂j〉 . (1)
Coherent states, resulting from the application of displacement operatorsDY = e
iY TΩR̂
(Y ∈ R2n) to the vacuum state, are Gaussian states with CM γ = 1 and first statistical
moments R̄ = Y . First moments can be arbitrarily adjusted by local unitary operations
(displacements), which cannot affect any property related to entropy or entanglement. They
Optical implementation and entanglement distribution in Gaussian valence bond states 3
can thus be assumed zero without any loss of generality. A N -mode Gaussian state will be
completely characterized by its real, symmetric, 2N × 2N CM γ.
The canonical commutation relations and the positivity of the density matrix ρ of a
Gaussian state imply the bona fide condition
γ + iΩ ≥ 0 , (2)
as a necessary and sufficient constraint the matrix γ has to fulfill to be a CM corresponding
to a physical state [8, 9]. Note that the previous condition is necessary for the CM of any
(generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger
uncertainty relation [10].
A major role in the theoretical and experimental manipulation of Gaussian states is
played by unitary operations which preserve the Gaussian character of the states on which
they act. Such operations are all those generated by terms of the first and second order in the
field operators. As a consequence of the Stone-Von Neumann theorem, any such operation
at the Hilbert space level corresponds, in phase space, to a symplectic transformation, i.e. to
a linear transformation S which preserves the symplectic form Ω, so that Ω = STΩS, i.e. it
preserves the commutators between the different operators. Symplectic transformations on a
2N -dimensional phase space form the (real) symplectic group, denoted by Sp(2N,R). Such
transformations act linearly on first moments and “by congruence” on the CM (i.e. so that
γ 7→ SγST ). One has DetS = 1, ∀S ∈ Sp(2N,R). A crucial symplectic operation is
the one achieving the normal mode decomposition. Due to Williamson theorem [11], any
N -mode Gaussian state can be symplectically diagonalized in phase space, so that its CM
is brought in the form ν, such that SγST = ν, with ν = diag {ν1, ν1, . . . νN , νN}. The
set {νi} of the positive-defined eigenvalues of |iΩγ| constitutes the symplectic spectrum of
γ and its elements, the so-called symplectic eigenvalues, must fulfill the conditions νi ≥ 1,
following from the uncertainty principle Eq. (2) and ensuring positivity of the density matrix
ρ corresponding to γ.
Ideal beam splitters, phase shifters and squeezers are described by symplectic
transformations. In particular, a phase-free two-mode squeezing transformation, which
corresponds to squeezing the first mode (say i) in one quadrature (say momentum, p̂i) and
the second mode (say j) in the orthogonal quadrature (say position, q̂j) with the same degree
of squeezing r, can be represented in phase space by the symplectic transformation
Sij(r) = diag{exp r, exp−r, exp−r, exp r} . (3)
These trasformations occur for instance in parametric down conversions [12]. Another
important example of symplectic operation is the ideal (phase-free) beam splitter, which acts
on a pair of modes i and j as [13]
B̂ij(θ) :
âi 7→ âi cos θ + âj sin θ
âj 7→ âi sin θ − âj cos θ
and corresponds to a rotation in phase space of the form
Bij(θ) =
cos(θ) 0 sin(θ) 0
0 cos(θ) 0 sin(θ)
sin(θ) 0 − cos(θ) 0
0 sin(θ) 0 − cos(θ)
. (4)
The transmittivity τ of the beam splitter is given by τ = cos2(θ) so that a 50:50 beam splitter
(τ = 1/2) amounts to a phase-space rotation of π/4.
Optical implementation and entanglement distribution in Gaussian valence bond states 4
The combined application of a two-mode squeezing and a 50:50 beam splitter realizes
the entangling twin-beam transformation [14]
Tij(r) = Bij(π/4) · Sij(r) , (5)
which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the identity
matrix), results in the production of a pure two-mode squeezed Gaussian state with CM
σi,j(r) = Tij(r)T
ij (r) given by
σi,j(r) =
cosh(2r) 0 sinh(2r) 0
0 cosh(2r) 0 − sinh(2r)
sinh(2r) 0 cosh(2r) 0
0 − sinh(2r) 0 cosh(2r)
. (6)
The CV entanglement in the state σi,j(r) increases unboundedly as a function of r, and
in the limit r → ∞ Eq. (6) approaches the (unnormalizable) Einstein-Podolski-Rosen (EPR)
state [15], simultaneous eigenstate of relative position and total momentum of the two modes
i and j. Concerning entanglement in general, the “positivity of partial transposition” (PPT)
criterion states that a Gaussian CM γ is separable (with respect to a 1×N bipartition) if and
only if the partially transposed CM γ̃ satisfies the uncertainty principle Eq. (2) [9, 16]. In
phase space, partial transposition amounts to a mirror reflection of one quadrature associated
to the single-mode partition. If {ν̃i} is the symplectic spectrum of the partially transposed
CM γ̃, then a (N + 1)-mode Gaussian state with CM γ is separable if and only if ν̃i ≥ 1 ∀ i.
A proper measure of CV entanglement is the logarithmic negativityEN [17], which is readily
computed in terms of the symplectic spectrum ν̃i of γ̃ as
EN = −
i: ν̃i<1
log ν̃i . (7)
Such an entanglement monotone [18] quantifies the extent to which the PPT condition ν̃i ≥ 1
is violated. For 1 × N Gaussian states, only the smallest symplectic eigenvalue ν̃− of the
partially transposed CM can be smaller than one [10], thus simplifying the expression of EN :
then the PPT criterion simply yields that γ is entangled as soon as ν̃− < 1, and infinite
entanglement (accompanied by infinite energy in the state) is reached for ν̃− → 0+.
For 1× 1 Gaussian states γi,j symmetric under mode permutations, the entanglement of
formation EF is computable as well via the formula [19]
EF (γi,j) = max{0, f(ν̃
− )} , (8)
f(x) =
(1 + x)2
(1 + x)2
− (1− x)
(1− x)2
Being a monotonically decreasing function of the smallest symplectic eigenvalue ν̃i,j− of the
partial transpose γ̃i,j of γi,j , the entanglement of formation is completely equivalent to the
logarithmic negativity in this case. For a two-mode state, ν̃i,j can be computed from the
symplectic invariants of the state [20], and experimentally estimated with measures of global
and local purities [21] (the purity µ = Tr ρ2 of a Gaussian state ρ with CM γ is equal to
µ = (Det γ)−1/2).
3. Gaussian valence bond states
Let us review the basic definitions and notations for GVBS, as adopted in Ref. [4]. The
so-called matrix product Gaussian states introduced in Ref. [3] are N -mode states obtained
Optical implementation and entanglement distribution in Gaussian valence bond states 5Optical implementation and entanglement distribution in Gaussian valence bond states
i i+1
Figure 1. Gaussian valence bond states. Γin is the state of N EPR bonds and γ is the three-
mode building block. After the EPR measurements (depicted as curly brackets), the chain
of modes γ
collapses into a Gaussian valence bond state with global state Γout. See also
Ref. [4].
by taking a fixed number, M , of infinitely entangled ancillary bonds (EPR pairs) shared
by adjacent sites, and applying an arbitrary 2M → 1 Gaussian operation on each site
i = 1, . . . , N . Such a construction, more properly definable as a “valence bond” picture
for Gaussian states, can be better understood by resorting to the Jamiolkowski isomorphism
between quantum operations and quantum states [22]. In this framework, one starts with a
chain ofN Gaussian states of 2M +1 modes (the building blocks). The global Gaussian state
of the chain is described by a CM Γ =
i=1 γ
[i]. As the interest in GVBS lies mainly in
their connections with ground states of Hamiltonians invariant under translation [3], we can
focus on pure (Detγ [i] = 1), translationally invariant (γ[i] ≡ γ ∀i) GVBS. Moreover, in this
work we consider single-bonded GVBS, i.e. withM = 1. This is also physically motivated in
view of experimental implementations of GVBS, as more than one EPR bond would result in a
building block with five or more correlated modes, which appears technologically demanding.
Under the considered prescriptions, the building block γ is a pure Gaussian state of three
modes. As we aim to construct a translationally invariant state, it is convenient to consider a
γ whose first two modes, which will be combined with two identical halves of consecutive
EPR bonds (see Fig. 3), have the same reduced CM. This yields a pure, three-mode Gaussian
building block with the property of being bisymmetric [23], that is with a CM invariant under
permutation of the first two modes. This choice of the building block is further justified
by the fact that, among all pure three-mode Gaussian states, bisymmetric states maximize
the genuine tripartite entanglement [24]: no entanglement is thus wasted in the projection
process. The 6 × 6 CM γ of the building block can be written as follows in terms of 2 × 2
submatrices,
γs εss εsx
εTss γs εsx
εTsx ε
sx γx
. (9)
The 4×4 CM of the first two modes (each of them having reduced CM γs) will be denoted by
γss, and will be regarded as the input port of the building block. On the other hand, the CM
γx of mode 3 will play the role of the output port. The intermodal correlations are encoded
in the off-diagonal ε matrices. Without loss of generality, we can assume γ to be, up to local
unitary operations, in the standard form [24] with
γs = diag{s, s} , γx = diag{x, x} , (10)
Optical implementation and entanglement distribution in Gaussian valence bond states 6
εss = diag{t+, t−} , εsx = diag{u+, u−} ;
x2 − 1±
16s4 − 8(x2 + 1)s2 + (x2 − 1)2
x2 − 1
(x − 2s)2 − 1±
(x+ 2s)2 − 1
The valence bond construction works as follows (see Fig. 3). The global CM Γ =
i=1 γ acts as the projector from the state Γ
in of the N ancillary EPR pairs, to the final
N -mode GVBS Γout. This is realized by collapsing the state Γin, transposed in phase space,
with the ‘input port’ Γss =
i γss of Γ, so that the ‘output port’ Γx =
i γx turns into
the desired Γout. Here collapsing means that, at each site, the two two-mode states, each
constituted by one mode (1 or 2) of γss and one half of the EPR bond between site i and
its neighbor (i − 1 or i + 1, respectively), undergo an “EPR measurement” i.e. are projected
onto the infinitely entangled EPR state [22, 3]. An EPR pair between modes i and j can be
described, see Eq. (6), as a two-mode squeezed state σi,j(r) in the limit of infinite squeezing
(r → ∞). The input state is then Γin = limr→∞
i σi,i+1(r), where we have set periodic
boundary conditions so that N + 1 = 1 in labeling the sites. The projection corresponds
mathematically to taking a Schur complement (see Refs. [4, 3, 22] for details), yielding an
output pure GVBS of N modes on a ring with a CM
out = Γx − ΓTsx(Γss + θΓ
θ)−1Γsx , (11)
where Γsx =
i γsx, and θ =
i diag{1, −1, 1, −1} represents transposition in phase
space (q̂i → q̂i, p̂i → −p̂i).
Within the building block picture, the valence bond construction can be in toto
understood as a multiple CV entanglement swapping [25], as shown in Fig. 3: the GVBS
is created as the entanglement in the bonds is swapped to the chain of output modes via
teleportation [26] through the input port of the building blocks. It is thus clear that at a given
initialization of the output port (i.e. at fixed x), changing the properties of the input port
(i.e. varying s), which corresponds to implementing different Gaussian projections from the
ancillary space to the physical one, will affect the structure and entanglement properties of
the target GVBS. This link is explored in the following section.
4. Entanglement distribution
In Ref. [4] the quantum correlations of GVBS of the form Eq. (11) have been studied,
and related to the entanglement properties of the building block γ. Let us first recall the
characterization of entanglement in the latter. As a consequence of the uncertainty principle
Eq. (2), the CM Eq. (9) of the building block describes a physical state if [24]
x ≥ 1 , s ≥ smin ≡
. (12)
Let us keep the output parameter x fixed. Straightforward applications of the PPT separability
conditions, and consequent calculations of the logarithmic negativity Eq. (7), reveal that the
entanglement in the CM γss of the first two modes (input port) is monotonically increasing
as a function of s, ranging from the case s = smin when γss is separable to the limit s → ∞
when the block γss is infinitely entangled. Accordingly, the entanglement between each of
the first two modes γs of γ and the third one γx decreases with s. One can also show that the
genuine tripartite entanglement in the building block increases with the difference s − smin
[24]. The entanglement properties of the building block are summarized in Fig. 4.
Optical implementation and entanglement distribution in Gaussian valence bond states 7
Figure 2. How a Gaussian valence bond state is created via continuous-variable entanglement
swapping. At each step, Alice attempts to teleport her mode 0 (half of an EPR bond, depicted
in yellow) to Bob, exploiting as an entangled resource two of the three modes of the building
block (denoted at each step by 1 and 2). The curly bracket denotes homodyne detection, which
together with classical communication and conditional displacement at Bob’s side achieves
teleportation. The state will be approximately recovered in mode 2, owned by Bob. Since
mode 0, at each step, is entangled with the respective half of an EPR bond, the process swaps
entanglement from the ancillary chain of the EPR bonds to the modes in the building block.
The picture has to be followed column-wise. For ease of clarity, we depict the process as
constituted by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are the
two input modes of the building block (depicted in blue); in the second sequence [frames (5)
to (8)] modes 1 and 2 are respectively an input and an output mode of the building block.
As a result of the multiple entanglement swapping [frame (9)] the chain of the output modes
(depicted in red), initially in a product state, is transformed into a translationally invariant
Gaussian valence bond state, possessing in general multipartite entanglement among all the
modes (depicted in magenta).
The main question addressed in Ref. [4] is how the initial entanglement in the building
block γ redistributes in the Gaussian MPS Γout. The answer is that the more entanglement
one prepares in the input port γss, the longer the range of pairwise quantum correlations in
the output GVBS is, as pictorially shown in Fig. 4.
In more detail, let us consider first a building block γ with s = smin = (x + 1)/2. In
this case, a separability analysis shows that, for an arbitrary numberN of modes in the GVBS
chain, the target state Γout exhibits bipartite entanglement only between nearest neighbor
modes, for any value of x > 1 (for x = 1 we trivially obtain a product state). In fact, each
reduced two-mode block γouti,j is separable for |i− j| > 1.
With increasing s in the choice of the building block, one finds that in the target GVBS
the correlations start to extend smoothly to distant modes. A series of thresholds sk can be
Optical implementation and entanglement distribution in Gaussian valence bond states 8
Figure 3. Entanglement properties of the three-mode building block γ, Eq. (9), of the
Gaussian valence bond construction, as functions of the standard form covariances x and
d ≡ s − smin. (a) Bipartite entanglement, as quantified by the logarithmic negativity,
between the first two input-port modes 1 and 2; (b) Bipartite entanglement, as quantified by
the logarithmic negativity, between each of the first two modes and the output-port mode 3;
(c) Genuine tripartite entanglement, as quantified by the residual Gaussian contangle [27, 24],
among all the three modes.
found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While
trivially s1(x) = smin for anyN (notice that nearest neighbors are entangled also for s = s1),
the entanglement boundaries for k > 1 are in general different functions of x, depending
on the number of modes. We observe however a certain regularity in the process: sk(x,N)
always increases with the integer k. Very remarkably, this means that the maximum range
of bipartite entanglement between two modes, or equivalently the maximum distribution of
multipartite entanglement, in a GVBS on a translationally invariant ring, is monotonically
related to the amount of entanglement in the reduced two-mode input port of the building
block [4]. Moreover, no complete transfer of entanglement to more distant modes occurs:
closer sites remain still entangled even when correlations between farther pairs arise.
The most interesting feature is perhaps obtained when infinite entanglement is fed in
the input port (s → ∞): in this limit, the output GVBS turns out to be a fully symmetric,
permutation-invariant, N -mode Gaussian state. This means that each individual mode is
equally entangled with any other, no matter how distant they are [4]. These states, being thus
Optical implementation and entanglement distribution in Gaussian valence bond states 9
Figure 4. Pictorial representation of the entanglement between a probe (green) mode and its
neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure.
As soon as the parameter s (encoding entanglement in the input port of the valence bond
building block) is increased, pairwise entanglement between the probe mode and its farther
and farther neighbors gradually appears in the corresponding output Gaussian valence bond
states. By translational invariance, each mode exhibits the same entanglement structure with
its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled
with every other single mode on the ring, independently of their relative distance: the Gaussian
valence bond state is in this case fully symmetric.
built by a symmetric distribution of infinite pairwise entanglement among multiple modes,
achieve maximum genuine multiparty entanglement among all Gaussian states (at a given
energy) while keeping the strongest possible bipartite one in any pair, a property known as
monogamous but promiscuous entanglement sharing [27].
Keeping Fig. 3 in mind, we can conclude that having the two input modes initially
entangled in the building blocks, increases the efficiency of the entanglement-swapping
mechanism, inducing correlations between distant modes on the GVBS chain, which enable to
store and distribute joint information. In the asymptotic limit of an infinitely entangled input
port of the building block, the entanglement range in the target GVBS states is engineered to
be maximum, and communication between any two modes, independently of their distance, is
enabled nonclassically. In the next sections, we investigate the possibility of producing GVBS
with linear optics, and discuss with a specific example the usefulness of such resource states
for multiparty CV quantum communication protocols such as telecloning [6] and teleportation
networks [13].
5. Optical implementation of Gaussian valence bond states
The power of describing the production of GVBS in terms of physical states, the building
blocks, rather than in terms of arbitrary non-unitary Gaussian maps, lies not only in the
immediacy of the analytical treatment. From a practical point of view, the recipe of Fig. 3 can
be directly implemented to produce GVBS experimentally in the domain of quantum optics.
We first note that the EPR measurements are realized by the standard toolbox of a beamsplitter
plus homodyne detection [22], as demonstrated in several CV teleportation experiments [28].
The next ingredient to produce aN -mode GVBS is constituted byN copies of the three-
mode building block γ. We provide here an easy scheme (see also Refs. [6, 29]) to realize
bisymmetric three-mode Gaussian states of the form Eq. (9). As shown in Fig. 5(a), one
can start from three vacuum modes and first apply a twin-beam operation to modes 1 and 3,
characterized by a squeezing r13, then apply another twin-beam operation to modes 1 and 2,
Optical implementation and entanglement distribution in Gaussian valence bond states 10Optical implementation and entanglement distribution in Gaussian valence bond states
50:50
γγγγ0000
γγγγ0000
γγγγ0000
Figure 5. Optical production of bisymmetric three-mode Gaussian states, used as buildingFigure 5. Optical production of bisymmetric three-mode Gaussian states, used as building
blocks for the valence bond construction. (a) Three initial vacuum modes are entangled
through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r13)
acting on modes 1 and 3, and the second (parametrized by a squeezing degree r12) acting on
the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose
covariance matrix is equivalent, up to local unitary operations, to the standard form given in
Eq. (9). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed
in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching
arrows→| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of
the same amount (this anti-squeezing transformation is denoted by the corresponding rotated
symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input
modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state,
with entanglement proportional to the degree of squeezing r.
parametrized by r12. The symplectic operation describing the twin-beam transformation (two-
mode squeezing plus balanced beam splitter) is given by Eq. (5) and pictorially represented
in Fig. 5(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian
state with a CM γB = T12(r12)T13(r13)T
13(r13)T
12(r12) of the form Eq. (9), with
γs = diag
e−2r12
e4r12 cosh (2r13) + 1
e−2r12
cosh (2r13) + e
γx = diag {cosh (2r13) , cosh (2r13)} ,
εss = diag
e−2r12
e4r12 cosh (2r13)− 1
e−2r12
cosh (2r13) − e4r12
εsx = diag
2er12 cosh (r13) sinh (r13) , −
2e−r12 cosh (r13) sinh (r13)
By means of local symplectic operations (unitary on the Hilbert space), like additional single-
mode squeezings, the CM γB can be brought in the standard form of Eq. (10), from which
Optical implementation and entanglement distribution in Gaussian valence bond states 11
one has
r13 = arccos
x+ 1√
, r12 = arccos
−x3 + 2x2 + 4s2x− x
For a given r13 (i.e. at fixed x), the quantity r12 is a monotonic function of the standard-form
covariance s, so this squeezing parameter which enters in the production of the building block
(see Fig. 5) directly regulates the entanglement distribution in the target GVBS, as discussed
in Sec. 4.
The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs.
But are infinitely entangled bonds truly necessary? In Ref. [4] the possibility is considered of
using a Γin given by the direct sum of two-mode squeezed states of Eq. (6), but with finite
r. Repeating the previous analysis to investigate the entanglement properties of the resulting
GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the
various partitions is degraded as r decreases, as somehow expected. Crucially, this does not
affect the connection between input entanglement and output correlation length. Numerical
investigations show that, while the thresholds sk for the onset of entanglement between distant
pairs are quantitatively modified – a bigger s is required at a given x to compensate the less
entangled bonds – the overall structure stays untouched. This ensures that the possibility of
engineering the entanglement structure in GVBS via the properties of the building block is
robust against imperfect resources, definitely meaning that the presented scheme is feasible.
Alternatively, one could from the beginning observe that the triples consisting of two
projective measurements and one EPR pair can be replaced by a single projection onto the
EPR state, applied at each site i between the input mode 2 of the building block and the
consecutive input mode 1 of the building block of site i+1 [3]. The output of all the homodyne
measurements will conditionally realize the target GVBS.
6. Telecloning with Gaussian valence bond resources
The protocol of CV quantum telecloning [6] amongN parties is defined as a process in which
one of them (Alice) owns an unknown coherent state, and wants to distribute her state to
all the other N − 1 remote parties. The telecloning is achieved by a succession of standard
two-party CV teleportations [26] between the sender Alice and each of the N − 1 remote
receivers, exploiting each time the corresponding reduced two-mode state shared as resource
by the selected pair of parties. The 1 → 2 CV telecloning of unknown coherent states has
been recently demonstrated experimentally [7].
The no-cloning theorem [30] yields that the N − 1 remote clones can resemble the
original input state only to a certain extent. The fidelity, which quantifies the success of a
teleportation experiment, is defined as F ≡ 〈ψin|ρout|ψin〉, where “in” and “out” denote the
input and the output state. F reaches unity only for a perfect state transfer, ρout = |ψin〉〈ψin|.
Without using entanglement, by purely classical communication, an average fidelity of
Fcl = 1/2 is the best that can be achieved if the alphabet of input states includes all coherent
states with even weight [31]. The sufficient fidelity criterion states that, if teleportation
is performed with F > Fcl, then the two parties exploited an entangled state [31]. The
converse is generally false, i.e. some entangled resources may yield lower-than-classical
fidelities. In Ref. [32] it has been shown, however, that if the fidelity is optimized over all
possible local unitary operations performed on the shared Gaussian resource (which preserve
entanglement by definition), then it becomes equivalent, both qualitatively and quantitatively,
to the entanglement in the resource.
Optical implementation and entanglement distribution in Gaussian valence bond states 12
Let us also recall that the fidelity of CV two-user teleportation [26] of arbitrary single-
mode Gaussian states with CM γin (equal to the identity for coherent states) exploiting two-
mode Gaussian resources with CM γab =
γa εab
εTab γb
, can be computed [33] as
F = 2√
, Σ ≡ 2γin + ξγaξ + γb + ξεab + εTabξ , (14)
with ξ = diag{−1 , 1}. We can now consider the general setting of 1 → N − 1 telecloning,
where N parties share a N -mode GVBS as an entangled resource, and one of them plays the
role of Alice (the sender) distributing imperfect copies of unknown coherent states to all the
N − 1 receivers. For any N , the fidelity can be easily computed from the reduced two-mode
CMs via Eq. (14) and will depend, for translationally invariant states, on the relative distance
between the two considered modes.
In this work we focus on a practical example of a GVBS on a translationally invariant
harmonic ring, with N = 6 modes. As shown in the previous section, these states can be
produced with the current optical technology. They are completely characterized, up to local
unitary operations, by a 12 × 12 CM analytically obtained from Eq. (11) by considering the
building block in standard form Eq. (9), whose elements are algebraic functions of s and x
here omitted for brevity (as no particular insight is gained from their explicit expressions).
First of all we can construct the reduced CMs γouti,i+k of two modes with distance k, and
evaluate for each k the respective symplectic eigenvalue ν̃i,i+k− of the corresponding partial
transpose. The entanglement condition s > sk will correspond to the inequality ν̃
i,i+k
− < 1.
With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s
12(x2+1)s6+(−34x4+28x2−34)s4+(x6−5x4−5x2+1)s2+(x2−1)2(x4−6x2+1) = 0,
while for the next-next-nearest neighbors threshold one has simply s3(x) = x. This enables us
to classify the entanglement distribution and, more specifically, to observe the interaction scale
in the GVBS Γout: as discussed in Sec. 4 and explicitly shown in Ref. [4], by increasing initial
entanglement in γss one can gradually switch on pairwise quantum correlations between more
and more distant sites.
Accordingly, it is now interesting to test whether this entanglement is useful to achieve
nonclassical telecloning towards distant receivers. In this specific instance, Alice will send
two identical (approximate) clones to her nearest neighbors, two other identical clones (with
in principle different fidelity than the previous case) to her next-nearest neighbors, and one
final clone to the most distant site. The fidelities for the three transmissions can be computed
from Eq. (14) and are plotted in Fig. 6(a). For s = smin, obviously, only the two nearest
neighbor clones can be teleported with nonclassical fidelity, as the reduced states of more
distant pairs are separable. With increasing s also the state transfer to more distant sites is
enabled with nonclassical efficiency, but not in the whole region of the space of parameters s
and x in which the corresponding two-mode resources are entangled.
As mentioned before, one can optimize the telecloning fidelity considering resources
prepared in a different way but whose CM can be brought by local unitary operations (single-
mode symplectic transformations) in the standard form of Eq. (11). For GVBS resources,
this local-unitary freedom can be transferred to the preparation of the building block. A
more general γ locally equivalent to the standard form given in Eq. (10), can be realized by
complementing the presented state engineering scheme for the three-mode building block as in
Eq. (13) [see Fig. 5(a)], with additional single-mode rotations and squeezing transformations
aimed at increasing the output fidelity in the target GVBS states, while keeping both the
entanglement in the building block and consequently the entanglement in the final GVBS
unchanged by definition.
Optical implementation and entanglement distribution in Gaussian valence bond states 13
Figure 6. 1 → 5 quantum telecloning of unknown coherent states exploiting a six-mode
translationally invariant Gaussian valence bond state as a shared resource. Alice owns mode
i. Fidelities F for distributing clones to modes j such as k = |i − j| are plotted for k = 1
[(a),(d)]; k = 2 [(b),(e)]; and k = 3 [(c),(f)], as functions of the local invariants s and
x of the building block. In the first row [(a)–(c)] the fidelities are achieved exploiting the
non-optimized Gaussian valence bond resource in standard form. In the second row [(d)–
(f)] fidelities optimized over local unitary operations on the resource are displayed, which
are equivalent to the entanglement in the corresponding reduced two-mode states (see, as a
comparison, Fig. 3 in Ref. [4]). Only nonclassical values of the fidelities (F > 0.5) are
shown.
The optimal telecloning fidelity, obtained in this way exploiting the results of Ref. [32],
is plotted in Fig. 6(b) for the three teleportations between modes i and j with k = |i − j| =
1, 2, 3. In this case, one immediately recovers a non-classical fidelity as soon as the
separability condition s ≤ sk is violated in the corresponding resources. Moreover, the
optimal telecloning fidelity at a given k is itself a quantitative measure of the entanglement in
the reduced two-mode resource, being equal to [32]
Foptk = 1/(1 + ν̃
i,i+k
− ) , (15)
where ν̃i,i+k− is the smallest symplectic eigenvalue of the partially transposed CM in
the corresponding bipartition. The optimal fidelity is thus completely equivalent to the
entanglement of formation Eq. (8) and to the logarithmic negativity Eq. (7).
In the limit s → ∞, as discussed in Sec. 4, the GVBS become fully permutation-
invariant for anyN . Consequently, the (optimized and non-optimized) telecloning fidelity for
distributing coherent states is equal for any pair of sender-receiver parties. These resources are
thus useful for 1 → N−1 symmetric telecloning. However, due to the monogamy constraints
on distribution of CV entanglement [27], this two-party fidelity will decrease with increasing
N , vanishing in the limit N → ∞ where the resources become completely separable. In this
Optical implementation and entanglement distribution in Gaussian valence bond states 14
respect, it is worth pointing out that the fully symmetric GVBS resources are more useful
for teleportation networks [13, 34], where N − 2 parties first perform local measurements
(momentum detections) on their single-mode portion of the entangled resource to concentrate
as much entanglement as possible onto the two-mode state of Alice and Bob, who can
accomplish non-classical teleportation (after the outcomes of the N − 2 measurements are
classically communicated to Bob). In this case, the optimal fidelity of N -user teleportation
network is an estimator of multipartite entanglement in the shared N -mode resource [32],
which is indeed a GVBS obtained from an infinitely entangled building block.
7. Conclusion
The valence bond picture is a valuable framework to study the structure of correlations in
quantum states of harmonic lattices. In fact, the motivation for such a formalism is quite
different from the finite-dimensional case, where valence bond/matrix product states are
useful to efficiently approximate ground states of N -body systems – generally described by
a number of parameters exponential in N – with polynomial resources [2]. In continuous
variable systems, the key feature of GVBS lies in the understanding of their entanglement
distribution as governed by the properties of simpler structures [4]. This has also experimental
implications giving a robust recipe to engineer correlations in many-body Gaussian states
from feasible operations on the building blocks. We have provided a simple scheme to produce
bisymmetric three-mode building blocks with linear optics, and discussed the subsequent
implementation of the valence bond construction. We have also investigated the usefulness
of such GVBS as resources for nonclassical communication, like telecloning of unknown
coherent states to distant receivers on a harmonic ring. It would be interesting to employ the
valence bond picture to describe quantum computation with continuous-variable cluster states
[35], and to devise efficient protocols for its optical implementation.
Acknowledgments
This work is supported by MIUR (Italy) and by the European Union through the
Integrated Project RESQ (IST-2001-37559), QAP (IST-3-015848), SCALA (CT-015714), and
SECOQC.
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http://arxiv.org/abs/quant-ph/0703277
Introduction
Continuous variable systems and Gaussian states
Gaussian valence bond states
Entanglement distribution
Optical implementation of Gaussian valence bond states
Telecloning with Gaussian valence bond resources
Conclusion
|
0704.1581 | Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8 | Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8
Rich magnetic phase diagram in the Kagome-staircase compound Mn3V2O8
E. Morosan,1 J. Fleitman,1 T. Klimczuk,2,3 and R. J. Cava1
1Department of Chemistry, Princeton University, Princeton NJ 08544
2Division of Thermal Physics, Los Alamos National Laboratories, Los Alamos NM 87545
3Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza
11/12, 80-952 Gdansk, Poland
Mn3V2O8 is a magnetic system in which S = 5/2 Mn2+ is found in the
kagomé staircase lattice. Here we report the magnetic phase diagram for
temperatures above 2 K and applied magnetic fields below 9 T, characterized by
measurements of the magnetization and specific heat with field along the three
unique lattice directions. At low applied magnetic fields, the system first orders
magnetically below Tm1 ≈ 21 K, and then shows a second magnetic phase
transition at Tm2 ≈ 15 K. In addition, a phase transition that is apparent in specific
heat but not seen in magnetization is found for all three applied field orientations,
converging towards Tm2 as H → 0. The magnetic behavior is highly anisotropic,
with critical fields for magnetic phase boundaries much higher when the field is
applied perpendicular to the Kagomé staircase plane than when applied in-plane.
The field-temperature (H - T) phase diagrams are quite rich, with 7 distinct phases
observed.
Geometrically frustrated magnetic materials have recently emerged as the focus of
intense study. Among these, compounds based on the kagomé net, a regular planar lattice made
from corner sharing of equilateral triangles, are of particular interest due to the very high
degeneracy of energetically equivalent magnetic ground states. Breaking the ideal triangular
symmetry of the kagomé net typically favors one particular magnetically ordered state above
others. For the particular case of the kagomé-staircase geometry, however, in which the
symmetry breaking occurs via buckling of the kagomé plane (see inset to Fig. 1), an exquisitely
close competition between different magnetically ordered states has been observed, resulting in
complex temperature-applied field magnetic phase diagrams. The kagomé-staircase lattice is
observed in the transition metal vanadates T3V2O8 (with T = Co, Ni, Cu, and Zn) [1-11], and the
Ni [2-4,7-8,11] and Co [1-3,5-6] variants have been widely studied. Simultaneous long-range
ferroelectric and magnetic order have been found in Ni3V2O8 [11], allowing its classification as a
multiferroic compound.
Orthorhombic symmetry Mn3V2O8 (MVO) is isostructural with Co3V2O8 (CVO) and
Ni3V2O8 (NVO), but its physical properties have been only marginally studied [12]. The t2g3eg2,
isotropic spin, S = 5/2 L = 0 configuration of Mn2+ in MVO presents an interesting contrast to the
t2g5eg2 S = 3/2 Co2+ and t2g6eg2 S = 1 Ni2+ cases of CVO and NVO. Here we report the
observation of rich anisotropic magnetic field-temperature (H - T) phase diagrams for MVO, as
determined from magnetization and specific heat measurements on single crystals. Two distinct
magnetic phase transitions, at 21 K and 15 K, are observed for fields applied in all three principal
crystallographic directions. A phase transition that is apparent in specific heat but not seen in
magnetization is found for all three applied field orientations, converging towards the 15 K
transition as H → 0. The magnetic behavior is highly anisotropic, with critical fields for
magnetic phase boundaries much higher when the magnetic field is applied perpendicular to the
kagomé-staircase plane (H parallel to the b crystallographic axis) then when applied in-plane (H
parallel to the a and c crystallographic axes). The field-temperature (H - T) phase diagrams are
quite rich, with 7 distinct magnetic/structural phases observed. The magnetic phase diagrams are
distinctly different from what is observed for CVO and NVO.
Single crystals of MVO were grown out of a MoO3/V2O5/MnO flux as previously
described [12]. The starting oxides (MnO 99% Aldrich, V2O5 99.6% Alfa Aesar, MoO3 99.95%
Alfa Aesar) were packed in an alumina crucible, which was then heated in a vertical tube furnace
under flowing Argon gas. Sacrificial MnO powder was placed in an alumina crucible above the
MoO3/V2O5/MnO flux to create an oxygen partial pressure that would neither oxidize the Mn2+
nor reduce the V5+[ref 12]. The vertical furnace was heated to 1200 oC at 200 oC/hr, held at 1200
oC for 1 hour, cooled to 900 oC at 5 oC /hr, then cooled to room temperature at 300 oC/hr. After
the heat treatment red-brown platelet crystals were extracted from the flux using a bath of 1 part
glacial acetic acid (Fisher) and 3 parts deionized water. The crystals were found to be single
phase by single crystal and powder X-ray diffraction, with the orthorhombic Cmca structure and
lattice parameters a = 6.2672(3) Å, b = 11.7377(8) Å and c = 8.5044(5) Å. Field- and
temperature-magnetization measurements were performed in a Quantum Design Physical
Properties Measurement System (PPMS). The specific heat data were also collected in a PPMS,
using a relaxation technique with fitting of the whole calorimeter (sample with sample platform
and puck).
The H = 0.5 T inverse magnetization data for MVO (Fig.1) indicates the presence of long
range magnetic ordering below Tm1 ≈ 21 K. Previous low-field magnetization data [12] suggest
the existence of an additional magnetic phase transition near 40 K, but the feature observed is
most likely due to the presence of an Mn2V2O7 impurity phase. A high-temperature fit of the
susceptibility (dotted line, Fig.1) to the Curie-Weiss law χ = χ0 + C/(T-θW) yields an effective
moment µeff = 5.94 µB, in excellent agreement with the theoretical value µeff = 5.92 µB expected
for high-spin S = 5/2 Mn2+. The Weiss temperature θW = -320 K indicates the dominance of
antiferromagnetic exchange interactions. Given the kagomé staircase magnetic lattice (inset in
Fig. 1), it is not surprising that |θW / Tm1| ≈ 15, characteristic of a strongly frustrated
antiferromagnetic spin system. Deviations from Curie-Weiss behavior, typical of magnetically
frustrated materials, are observed to begin on cooling at approximately 70 K.
The easy magnetization axis lies close to the kagomé-staircase ac-plane, where the
magnetization is largest (Fig.1). Upon further inspection of the behavior of the magnetization in
different applied fields, two magnetic phase transitions can be identified in the M(T) data for all
field orientations. Fig. 2a and c illustrate the field dependence of these transitions for H || a and
H || c respectively. The competition between the antiferromagnetic spin coupling and the
anisotropy associated with the kagomé staircase structure precludes the system from attaining a
zero net magnetization ground state. This is suggested by the rapidly increasing magnetization as
the system enters the high temperature, low field state (HT1) upon cooling below Tm1 ≈ 21 K. A
net ferromagnetic component can probably be associated with the HT1 phase. Subsequent
cooling of the sample gives rise to a sharp cusp followed by a local minimum around Tm2 =15 K,
where a second magnetic phase transition, from HT1 to a low-T, low-H state (LT1) occurs.
Increasing magnetic field (Fig.2a,c) has almost no effect on the long-range magnetic ordering
temperature Tm1, but it broadens the cusp and slowly drives the second transition down in
temperature. Above H = 0.04 T the H || a low temperature magnetization plateaus at a finite
value, which strongly suggests a canted spin configuration even for the LT1 state, with a smaller
ferromagnetic component along a than in the HT1 state. Very similar behavior occurs for the
other in-plane orientation H || c (Fig.2c), with the two distinct transitions persisting up to slightly
higher field H = 0.1 T. The insets in Fig. 2a,c represent examples of how the critical
temperatures for the magnetic phase transitions at constant field are determined: as shown, the
vertical arrows mark Tm1 and Tm2, H = 0.01 T, and correspond to local minima in the
temperature-derivative of magnetization dM/dT. The two low-field phases that are observed in
the magnetically ordered state are possibly a result of the ordering of the spins on one or both of
the distinct Mn2+ ions (inset Fig. 1) in the ac-plane, similar to the transitions encountered in
NVO [4]. Fewer magnetic phases are distinguishable at low fields in MVO, however, than are
seen in either NVO or CVO.
A more complex scenario is revealed in MVO at finite fields. Fig. 2b shows a selection of
the H || b M(H) isotherms (full symbols), with the T = 2 K field-derivative dM/dH (open
diamonds) as an example of how the critical field values were determined. At T = 2 K, the
magnetization is low and linear with field for H < 2 T, which corresponds to the LT1 phase. This
behavior is consistent with the antiferromagnetic spins slowly rotating from the easy axis in the
ac plane, closer to the direction of the applied field H || b. A sharp step in magnetization around
Hc1 = 2.1 T marks the transition from LT1 to LT2, possibly due to a spin-flop transition on one
or both of the Mn2+ sites. Although the magnetization increases linearly with field above this
transition, as expected for the spin-reorientation subsequent to a spin-flop, another transition
occurs just below 3 T, where M(H) changes slope (full diamonds, Fig.2b) and the system enters
the state LT3. The spin-flop transition yields a sharp peak in dM/dH (open diamonds, Fig.2b);
the higher critical field value Hc2 is determined using an on-set criterion for dM/dH. Both
transitions are marked by small vertical arrows in Fig.2b.
As the temperature is raised, the initial slope of the M(H) curves increases in the
magnetically ordered phase (Fig.2b) such that the magnetization jump at the spin-flop transition
becomes indistinguishable. The two magnetic phase transitions move slightly down with field,
and are hard to identify in the magnetization measurements above 16 K. Specific heat
measurements complement the magnetization data, by confirming the magnetic phase lines, but
also by revealing another phase transition that was not visible in the M(T,H) data. A selection of
the specific heat curves, plotted as Cp/T vs. T, is shown in Fig. 3a, for H || b and applied fields up
to 9 T. For H = 0 (full squares) a sharp peak associated with long range magnetic ordering is seen
around Tm1 = 21 K, with a second peak at the lower phase transition temperature Tm2. After
subtracting the lattice contribution to the specific heat as measured for the non-magnetic
analogous compound Zn3V2O8 [2] (solid line, Fig.3b) one can estimate the magnetic specific
heat Cm for MVO (open symbols, Fig.3b). The temperature dependence of the magnetic entropy
can then be calculated and is shown in the inset in Fig.3b for H = 0 (open circles): only about
50% of the R ln6 entropy expected for a S = 5/2 state is accounted for between 2 and 40 K. This
could be an indication that additional phase transitions may exist below 2 K. Another possible
explanation, given the observed departure from Curie-Weiss behavior below ~ 70 K (Fig.1), is
that more entropy is associated with short range order below 70 K. No additional entropy is
recovered with the application of magnetic field, as the H = 9 T temperature-dependent entropy
(crosses, inset Fig.3b) differs only slightly from the H = 0 data. However, as the field is turned
on, very different behavior is observed for the two peaks in Cp (Fig.3a): the one just below 16 K
is affected little in temperature by the increasing magnetic field, but the higher-temperature one
moves down in field. Concurrently, a third, broader peak emerges above ~ 1.5 T and is driven
higher in temperature with increasing field. It is likely that both phase transitions exist at finite
fields even for H < 1.5 T, and converge at Tm1 for H → 0, but their proximity in temperature
makes it impossible to discern two separate peaks. For H > Hbc1 the lower temperature peak is
not associated with any phase transition observed in the magnetization data. Given its invisibility
in the magnetization, and the relative insensitivity of the transition temperature to applied field,
we speculate that this transition may have a structural component, though the fact that the
amount of entropy in the transition is suppressed by the field indicates that there must be a
magnetic component as well.
Based on our extensive magnetization and specific heat measurements, we present the H
– T magnetic phase diagrams for MVO, for magnetic fields applied along the unique structural
directions, in Figs. 4 and 5. As the temperature is lowered in zero field, MVO orders
magnetically at Tm1 = (20.7 ± 0.2) K, entering first a high temperature phase (HT1) and then a
low temperature phase (LT1) at Tm2 = (15.2 ± 0.5) K. The response of the system to applied
magnetic field is highly anisotropic. For H || a (Fig. 4a), in finite field, two distinct phase
boundaries emerge at Tm2: one represents the lower temperature magnetic phase transition, which
moves down in temperature as H increases, and the second is an almost vertical line, which is
only visible in the specific heat data. The intermediate temperature phase delineated by these two
phase boundaries is LT4, which extends in field up to about 0.04 T. An almost horizontal phase
line cuts across the phase diagram at Hac1 ≈ 0.04 T. It separates the low field, low temperature
(LT1) and a high temperature (HT1) phases from two different states (LT3 and HT3) at higher
fields.
For the other field orientation close to the plane (H || c, Fig. 4b), the low field phase
diagram is similar to that for H || a, with the HT1, LT4 and LT1 phases extending up in field up
to a much higher critical value Hcc1 = 0.3 T. In the T → 0 limit, a second magnetic phase
transition occurs at Hcc2 = 2.6 T, and the critical field value is slowly reduced with temperature.
The two almost horizontal phase lines at Hcc1 and Hcc2 separate a low temperature (LT2) and a
high temperature (HT2) phase at intermediate field values from the high field states LT3 and
HT3.
When field is applied perpendicular to the kagomé planes (Fig. 5) the phase diagram is
analogous to the in-plane ones. The most noticeable difference is that the critical field values are
much higher: Hbc1 = 2.2 T and Hbc2 = 3.0 T respectively for T → 0. This is expected given the
observed anisotropy, which constrains the magnetic moments to lie closer to the ac-plane:
stronger fields are needed to pull the moments towards the “hard” axis b. In addition, the LT4
phase is missing, and the phase line that starts at Hbc1 at T → 0 converges at Tm2 in the H = 0
limit. As a consequence, the HT2 phase merges with HT1 just below the magnetic ordering at
Tm1.
The temperature-field magnetic phase diagram for Mn3V2O8 is quite different from those
seen in Ni3V2O8 and Co3V2O8. In all three compounds, the competition between the crystalline
anisotropy and the antiferromagnetic interactions in the kagomé staircase structure gives rise to
strong geometric frustration. In NVO and CVO, differences in the magnetically ordered states
have been found to involve differences in the ordering of the moments on the two kinds of
magnetic ion sites, the so-called spine and crosstie sites. The same will no doubt prove true for
MVO, with the present measurements revealing that the magnetic moments on the two distinct
Mn2+ sites lie close to the ac-plane when in the H = 0 magnetically ordered states. For magnetic
fields applied in-plane, the magnetic states in MVO are much more sensitive to applied field than
they are in NVO and CVO, with fig. 4 showing for example that the LT1 and HT1 phases
disappear in applied fields in the a direction as low as 0.04 T. The complexity of the anisotropic
H – T phase diagrams in MVO appears to be derived from competition between nearly balanced
magnetic interactions, leading to canted spin configurations or field-induced spin-flop
transitions. An integration of the entropy observed under the H = 0 phase transitions between 2
and 40 K does not yield the expected Rln6 for Mn2+, suggesting that there may be more magnetic
phase transitions below 2 K, or that additional entropy is associated with short-range order below
70 K. Detailed neutron scattering measurements are desirable in order to elucidate the nature of
the different states observed in MVO, and also to clarify whether the almost field independent
phase boundary at Tm2 is associated with a structural phase transition. Investigation of possible
multiferroic phases will also be of considerable interest.
Acknowledgements
This research was supported by the US Department of Energy, Division of Basic Energy
Sciences, grant DE-FG02- 98-ER45706. We thank G. Lawes for providing the specific heat data
for Zn3V2O8.
References
1. N. Krishnamachari, C. Calvo, Canad. J. Chem. 49 (1971) 1629
2. N. Rogado, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava Solid State Commun. 124
(2002) 229
3. G. Balakrishnan, O. A. Petrenko, M. R. Lees, D. M K Paul c J. Phys.: Condens. Matter 16
(2004) L347
4. M. Kenzelmann, A. B. Harris, A. Aharony, O. Entin-Wohlman, T. Yildirim, Q. Huang, S.
Park, G. Lawes, C. Broholm, N. Rogado, R. J. Cava, K. H. Kim, G. Jorge, A. P. Ramirez Phys.
Rev. B 74 14429
5. R. Szymczak, M. Baran, R. Diduszko, J. Fink-Finowicki, M. Gutowska, A. Szewczyk, H.
Szymczak Phys. Rev. B 73 94425
6. Y. Chen, J. W. Lynn, Q. Huang, F. M. Woodward, T. Yildirim, G. Lawes, A. P. Ramirez,
N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, A. B. Harris Phys. Rev. B 74 (2006)
14430
7. R. P. Chaudhury, F. Yen, C. R. dela Cruz, B. Lorenz, Y. Q. Wang, Y. Y. Sun, C. W. Chu
Phys. Rev. B 75 (2007) 12407
8. G. Lawes, M. Kenzelmann, N. Rogado, K. H. Kim, G. A. Jorge, R. J. Cava, A. Aharony,
O. Entin-Wohlman, A. B. Harris, T. Yildirim, Q. Z. Huang, S. Park, C. Broholm, A. P. Ramirez
Phys. Rev. Lett. 93 247201
9. N. Rogado, M. K. Haas, G. Lawes, D. A. Huse, A. P. Ramirez, R. J. Cava J. Phys.:
Condens. Matter 15 (2003) 907
10. E. E. Sauerbrei, R. Faggiani, C. Calvo Acta Cryst. B 29 (1973) 2304
11. G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin-
Wohlman, T. Yildirim, M. Kenzelmann, C. Broholm, A. P. Ramirez Phys. Rev. Lett. 95 (2005)
87205
12. X. Wang, Z. Liu, A. Ambrosini, A. Maignan, C. L. Stern, K. R. Poeppelmeier, V. P.
Dravid Solid State Sciences 2 (2000) 99
Figure Captions
Fig 1. Anisotropic inverse susceptibility data for H = 0.5 T (symbols) and linear fit of the high-
temperature data (dotted line). Insert: kagomé staircase structure of the Mn2+ array in Mn3V2O8.
Crystallographic axes are shown. Spine sites are shown in purple and crosstie sites are shown in
pink.
Fig. 2. (a) H || a M(T) data for H = 0.015 T, 0.025 T, 0.04 T, 0.06 T, 0.08 T, 0.1 T and 5.0 T.
Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of the minima in
the derivative, from which the critical temperature values are determined. (b) H || b M(H)
isotherms for T = 2 K, 15 K, 18 K, 20 K and 30 K (full symbols, left axis); the T = 2 K dM/dH
curve (open symbols, right axis) illustrates how the critical field values Hc1 and Hc2, marked by
vertical arrows, are determined. (c) H || c M(T) data for H = 0.02 T, 0.05 T, 0.5 T, 1.0 T, 2.0 T,
3.0 T and 5.0 T. Inset: the H = 0.01 T dM/dT curve; small vertical arrows indicate the position of
the minima in the derivative, from which the critical temperature values are determined.
Fig 3. (a) H || b Cp/T vs. T data for H = 0, 1.5 T, 2.0 T, 6.0 T and 9.0 T. (b) Cp/T data for MVO
(full symbols) and Zn3V2O8 (solid line) (right axis) used to determine the magnetic specific heat
Cm of MVO (open symbols, left axis) plotted as Cm/T. Inset: the temperature-dependence of the
magnetic entropy Sm for H = 0 and 9 T.
Fig 4. (a) H || a and (b) H || c H – T phase diagrams: points are determined from M(T) data
(orange symbols), M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are
guides connecting the points determined experimentally; extrapolations of these phase
boundaries in regions where measurements were missing or critical H and T values were difficult
to determine are represented by dotted lines.
Fig 5. H || b H – T phase diagrams: points are determined from M(T) data (orange symbols),
M(H) data (wine symbols) or Cp(T)|H data (open symbols). The solid lines are guides connecting
the points determined experimentally; extrapolations of these phase boundaries in regions where
measurements were missing or critical H and T values were difficult to determine are represented
by dotted lines.
Fig.1.
0 50 100 150 200 250
H = 0.5 T
H || c
H || b
0 10 20 30 40 50 60 70
0 1 2 3 4 5
0 10 20 30 40 50 60
H || c
0.02T
0.05T
0.5 T
H || b
H || a
0.025T
0.015T
0.04T
0.06T
0.08T
0 10 20 30
H = 0.01 T
0 10 20 30
H = 0.01 T
Fig. 2.
Fig. 3.
8 12 16 20 24 28 32 36 40
0 10 20 30 40
0 10 20 30 40
10(a) (b)Mn
H = 0
H = 1.5 T
H = 2.0 T
H = 6.0 T
H = 9.0 T
(J/m
H = 0
H = 0
H = 9 T
R ln2
R ln3
Fig.4.
0 4 8 12 16 20 24 28
T(K)
PMLT3 HT3
HT2LT2
H || c
LT1 HT1
H || a
Fig.5.
0 4 8 12 16 20 24
H || b
HT3LT3
LT1 HT1
|
0704.1582 | L^2-Betti numbers of coamenable quantum groups | L2-BETTI NUMBERS OF COAMENABLE QUANTUM
GROUPS
DAVID KYED
Abstra
t. We prove that a
ompa
t quantum group is
oamen-
able if and only if its
orepresentation ring is amenable. We further
propose a Følner
ondition for
ompa
t quantum groups and prove
it to be equivalent to
oamenability. Using this Følner
ondition,
we prove that for a
oamenable
ompa
t quantum group with tra-
ial Haar state, the enveloping von Neumann algebra is dimension
�at over the Hopf algebra of matrix
oe�
ients. This generalizes a
theorem of Lü
k from the group
ase to the quantum group
ase,
and provides examples of
ompa
t quantum groups with vanishing
-Betti numbers.
Introdu
tion
The theory of L2-Betti numbers for dis
rete groups is originally due
to Atiyah and dates ba
k to the seventies [Ati76℄. These L2-Betti num-
bers are de�ned for those dis
rete groups that permit a free, proper
and
o
ompa
t a
tion on some
ontra
tible, Riemannian manifold X .
If Γ is su
h a group, the spa
e of square integrable p-forms on X
be
omes a �nitely generated Hilbert module for the group von Neu-
mann algebra L (Γ). As su
h it has a Murray-von Neumann dimen-
sion whi
h turns out to be independent of the
hoi
e of X and is
alled
the p-th L2-Betti number of Γ, denoted β
p (Γ). More re
ently, Lü
k
[Lü
97, Lü
98a, Lü
98b℄ transported the notion of Murray-von Neu-
mann dimension to the setting of �nitely generated proje
tive (alge-
brai
) L (Γ)-modules and extended thereafter the domain of de�nition
to the
lass of all modules. With this extended dimension fun
tion,
dimL (Γ)(−), it is possible to extend the notion of L2-Betti numbers to
over all dis
rete groups Γ by setting
β(2)p (Γ) = dimL (Γ) Tor
p (L (Γ),C).
For more details on the relations between the di�erent de�nitions of L2-
Betti numbers and the extended dimension fun
tion we refer to Lü
k's
book [Lü
02℄.
2000 Mathemati
s Subje
t Classi�
ation. 16W30,43A07, 46L89, 16E30.
http://arxiv.org/abs/0704.1582v4
2 DAVID KYED
All the ingredients in the homologi
al algebrai
de�nition above have
fully developed analogues in the world of
ompa
t quantum groups, and
using this di
tionary the notion of L2-Betti numbers was generalized
to the quantum group setting in [Kye08℄. Sin
e this generalization is
entral for the work in the present paper, we shall now explain it in
greater detail. Consider a
ompa
t quantum group G = (A,∆) and
assume that its Haar state h is a tra
e. If we denote by A0 the unique
dense Hopf ∗-algebra and by M the enveloping von Neumann algebra
of A in the GNS representation arising from h, then the p-th L2-Betti
number of G is de�ned as
β(2)p (G) = dimM Tor
p (M,C).
Here C is
onsidered an A0-module via the
ounit ε : A0 → C and
dimM(−) is Lü
k's extended dimension fun
tion arising from (the ex-
tension of) the tra
e-state h. This de�nition extends the
lassi
al one
[Kye08, 1.3℄ in the sense that
β(2)p (G) = β
p (Γ)
when G = (C∗red(Γ),∆red).
The aim of this paper is to investigate the L2-Betti numbers of the
lass
of
oamenable,
ompa
t quantum groups. In the
lassi
al
ase we have
that β
p (Γ) = 0 for all p ≥ 1 whenever Γ is an amenable group. This
an be seen as a spe
ial
ase of [Lü
98a, 5.1℄ where it is proved that
the von Neumann algebra L (Γ) is dimension �at over CΓ, meaning
dimL (Γ) Tor
p (L (Γ), Z) = 0 (p ≥ 1)
for any CΓ-module Z � provided, of
ourse, that Γ is still assumed
amenable. We generalize this result to the quantum group setting
in Theorem 6.1. More pre
isely, we prove that if G = (A,∆) is a
ompa
t,
oamenable quantum group with tra
ial Haar state and Z is
any module for the algebra of matrix
oe�
ients A0 then
dimM Tor
p (M,Z) = 0. (p ≥ 1)
Here M is again the enveloping von Neumann algebra in the GNS rep-
resentation arising from the Haar state. In order to prove this result
we need a Følner
ondition for
ompa
t quantum groups. The
lassi
al
Følner
ondition for groups [Føl55℄ is a geometri
al
ondition, on the
a
tion of the group on itself, whi
h is equivalent to amenability of the
group. In order to obtain a quantum analogue of Følner's
ondition a
detailed study of the ring of
orepresentations, asso
iated to a
ompa
t
quantum group, is needed. The ring of
orepresentations is a spe
ial
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 3
ase of a so-
alled fusion algebra and we have therefore devoted a sub-
stantial part of this paper to the study of abstra
t fusion algebras and
their amenability. Amenability for (�nitely generated) fusion algebras
was introdu
ed by Hiai and Izumi in [HI98℄ where they also gave two
equivalent Følner-type
onditions for fusion algebras. We generalize
their results to the non-�nitely generated
ase and prove that a
om-
pa
t quantum group is
oamenable if and only if its
orepresentation
ring is amenable. From this we obtain a Følner
ondition for
ompa
t
quantum groups whi
h is equivalent to
oamenability. Using this Følner
ondition we prove our main result, Theorem 6.1, whi
h implies that
oamenable
ompa
t quantum groups have vanishing L2-Betti numbers
in all positive degrees.
Stru
ture. The paper is organized as follows. In the �rst se
tion we re-
apitulate (parts of) Woronowi
z's theory of
ompa
t quantum groups.
The se
ond and third se
tion is devoted to the study of abstra
t fu-
sion algebras and amenability of su
h. In the fourth se
tion we dis
uss
oamenability of
ompa
t quantum groups and investigate the relation
between
oamenability of a
ompa
t quantum group and amenability of
its
orepresentation ring. The �fth se
tion is an interlude in whi
h the
ne
essary notation
on
erning von Neumann algebrai
ompa
t quan-
tum groups and their dis
rete duals is introdu
ed. The sixth se
tion is
devoted to the proof of our main theorem (6.1) and the seventh, and
�nal, se
tion
onsists of examples.
A
knowledgements. I wish to thank my supervisor Ryszard Nest for
the many dis
ussions about quantum groups and their (
o)amenability,
and Andreas Thom for pointing out to me that the bi
rossed produ
t
onstru
tion
ould be used to generate examples of quantum groups
satisfying Følner's
ondition.
Notation. Throughout the paper, the symbol ⊙ will be used to denote
algebrai
tensor produ
ts while the symbol ⊗̄ will be used to denote
tensor produ
ts in the
ategory of Hilbert spa
es or the
ategory of
von Neumann algebras. All tensor produ
ts between C∗-algebras are
assumed minimal/spatial and these will be denoted by the symbol ⊗.
1. Preliminaries on
ompa
t quantum groups
In this se
tion we brie�y re
all Woronowi
z's theory of
ompa
t
quantum groups. Detailed treatments, and proofs of the results stated,
an be found in [Wor98℄, [MVD98℄ and [KT99℄.
4 DAVID KYED
A
ompa
t quantum group G is a pair (A,∆) where A is a unital C∗-
algebra and ∆: A −→ A ⊗ A is a unital ∗-homomorphism from A to
the minimal tensor produ
t of A with itself satisfying:
(id⊗∆)∆ = (∆⊗ id)∆ (
oasso
iativity)
∆(A)(1⊗ A) = ∆(A)(A⊗ 1) = A⊗A (non-degenera
y)
For su
h a
ompa
t quantum group G = (A,∆), there exists a unique
state h : A → C,
alled the Haar state, whi
h is invariant in the sense
(h⊗ id)∆(a) = (id⊗h)∆(a) = h(a)1,
for all a ∈ A. Let H be a Hilbert spa
e and let u ∈ M(K(H)⊗ A) be
an invertible multiplier. Then u is
alled a
orepresentation if
(id⊗∆)u = u(12)u(13),
where we use the standard leg numbering
onvention; for instan
e
u(12) = u⊗1. Intertwiners, dire
t sums and equivalen
es between
orep-
resentations as well as irredu
ibility are de�ned in a straight forward
manner. See e.g. [MVD98℄ for details. We shall denote by Mor(u, v)
the set of intertwiners from u to v. It is a fa
t that ea
h irredu
ible
orepresentation is �nite dimensional and equivalent to a unitary
o-
representation. Moreover, every unitary
orepresentation is unitarily
equivalent to a dire
t sum of irredu
ible
orepresentations. For two
�nite dimensional unitary
orepresentations u, v their tensor produ
t is
de�ned as
u T©v = u(13)v(23).
This is again a unitary
orepresentation of G. The algebra A0 gener-
ated by all matrix
oe�
ients arising from irredu
ible
orepresentations
be
omes a Hopf ∗-algebra (with the restri
ted
omultipli
ation) whi
h
is dense in A. We denote its antipode by S and its
ounit by ε. We also
re
all that the restri
tion of the Haar state to the ∗-algebra A0 is always
faithful. The quantum group G is
alled a
ompa
t matrix quantum
group if there exists a fundamental unitary
orepresentation; i.e. a �nite
dimensional, unitary
orepresentation whose matrix
oe�
ients gener-
ate A0 as a ∗-algebra. Ea
h �nite dimensional, unitary
orepresenta-
tion u de�nes a
ontragredient
orepresentation uc on the dual Hilbert
spa
e; if u ∈ B(H) ⊙ A0 for some �nite dimensional Hilbert spa
e H
then uc ∈ B(H ′)⊙A0 is given by uc = (( · )′⊗S)u, where for T ∈ B(H)
the operator T ′ ∈ B(H ′) is the natural dual (T ′(y′))(x) = y′(Tx). In
general uc is not a unitary, but it is a
orepresentation; i.e. it is invert-
ible and satis�es (id⊗∆)uc = uc
and is therefore equivalent to a
unitary
orepresentation. By
hoosing an orthonormal basis e1, . . . , en
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 5
for H we get an identi�
ation of B(H) ⊙ A0 with Mn(A0). If, under
this identi�
ation, u be
omes the matrix (uij) then u
is identi�ed with
the matrix ū = (u∗ij), where we identify B(H
′)⊙A0 with Mn(A0) using
the dual basis e′1, . . . , e
n. From this it follows that u
is equivalent to
u. Note also that one has (u⊕ v)c = uc ⊕ vc and (u T©v)c = vc T©uc for
unitary
orepresentations u and v (see e.g. [Wor87℄). If u ∈ B(H)⊙A0
is a �nite dimensional
orepresentation its
hara
ter is de�ned as
χ(u) = (Tr⊗ id)u ∈ A0,
where Tr is the unnormalized tra
e on B(H). The
hara
ter map has
the following properties.
Proposition 1.1 ([Wor87℄). If u and v are �nite dimensional, unitary
orepresentations then
χ(u T©v) = χ(u)χ(v), χ(u⊕v) = χ(u)+χ(v) and χ(uc) = χ(u)∗.
Moreover, if u and v are equivalent then χ(u) = χ(v).
We end this se
tion with the two basi
examples of
ompa
t quantum
groups arising from a
tual groups.
Example 1.2. If G is a
ompa
t, Hausdor� topologi
al group then the
Gelfand dual C(G) be
omes a
ompa
t quantum group with
omulti-
pli
ation ∆c : C(G) −→ C(G)⊗ C(G) = C(G×G) given by
∆c(f)(s, t) = f(st).
The Haar state is in this
ase given by integration against the Haar
probability measure on G, and the �nite dimensional unitary
orepre-
sentations of C(G) are exa
tly the �nite dimensional unitary represen-
tations of G.
Example 1.3. If Γ is a dis
rete,
ountable group then the redu
ed
group C∗-algebra C∗red(Γ) be
omes a
ompa
t quantum group when
endowed with
omultipli
ation given by
∆red(λγ) = λγ ⊗ λγ .
Here λ denotes the left regular representation of Γ. In this
ase, the
Haar state is just the natural tra
e on C∗red(Γ), and a
omplete family of
irredu
ible, unitary
orepresentations is given by the set {λγ | γ ∈ Γ}.
Remark 1.4. All
ompa
t quantum groups to be
onsidered in the
following are assumed to have a separable underlying C∗-algebra. The
quantum Peter-Weyl theorem [KT99, 3.2.3℄ then implies that the GNS
spa
e arising from the Haar state is separable and, in parti
ular, that
there are at most
ountable many (pairwise inequivalent) irredu
ible
orepresentations.
6 DAVID KYED
2. Fusion Algebras
In this se
tion we introdu
e the notion of fusion algebras and amen-
ability of su
h obje
ts. This topi
was treated by Hiai and Izumi in
[HI98℄ and we will follow this referen
e
losely throughout this se
-
tion. Other referen
es on the subje
t are [Yam99℄, [HY00℄ and [Sun92℄.
Throughout the se
tion, N0 will denote the non-negative integers.
De�nition 2.1 ([HI98℄). Let R be a unital ring and assume that R is
free as Z-module with basis I. Then R is
alled a fusion algebra if the
unit e is an element of I and the following holds:
(i) The abelian monoid N0[I] is stable under multipli
ation. That
is, for all ξ, η ∈ I the unique family (Nαξ,η)α∈I of integers satis-
fying
Nαξ,ηα,
onsists of non-negative numbers.
(ii) The ring R has a Z-linear, anti-multipli
ative involution x 7→ x̄
preserving the basis I globally.
(iii) Frobenius re
ipro
ity holds, i.e. for ξ, η, α ∈ I we have
Nαξ,η = N
α,η̄.
(iv) There exists a Z-linear multipli
ative fun
tion d : R → [1,∞[
su
h that d(ξ) = d(ξ̄) for all ξ ∈ I. This fun
tion is
alled the
dimension fun
tion.
Note that the distinguished basis, involution and dimension fun
tion
are all in
luded in the data de�ning a fusion algebra. Ea
h fusion
algebra
omes with a natural tra
e τ given by
τ7−→ ke.
We shall use this tra
e later to de�ne a C∗-envelope of a fusion algebra.
Note also that the multipli
ativity of d implies
d(ξ)d(η)
Nαξ,η,
for all ξ, η ∈ I. For an element r =
α∈I kαα ∈ R, the set {α ∈ I |
kα 6= 0} is
alled the support of r and denoted supp(r). We shall also
onsider the
omplexi�ed fusion algebra C⊗ZZ[I] whi
h will be denoted
C[I] in the following. Note that this be
omes a
omplex ∗-algebra with
the indu
ed algebrai
stru
tures.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 7
Example 2.2. For any dis
rete group Γ the integral group ring Z[Γ] be-
omes a fusion algebra when endowed with (the Z-linear extension of)
inversion as involution and trivial dimension fun
tion given by d(γ) = 1
for all γ ∈ Γ.
The irredu
ible representations of a
ompa
t group
onstitute the
basis in a fusion algebra where the tensor produ
t of representations is
the produ
t. We shall not go into details with this
onstru
tion sin
e
it will be
ontained in the following more general example.
Example 2.3. If G = (A,∆) is a
ompa
t quantum group its irre-
du
ible
orepresentations
onstitute the basis of a fusion algebra with
tensor produ
t as multipli
ation. Sin
e this example will play a promi-
nent role later, we shall now elaborate on the
onstru
tion. Denote
by Irred(G) = (uα)α∈I a
omplete family of representatives for the
equivalen
e
lasses of irredu
ible, unitary
orepresentations of G. As
explained in Se
tion 1, for all uα, uβ ∈ Irred(G) there exists a �nite
subset I0 ⊆ I and a family (Nγα,β)γ∈I0 of positive integers su
h that
uα T©uβ is equivalent to
uγ ⊕ · · · ⊕ uγ
︸ ︷︷ ︸
times
Thus, a produ
t
an be de�ned on the free Z-module Z[Irred(G)] by
setting
uα · uβ =
and the trivial
orepresentation e = 1A ∈ Irred(G) is a unit for this
produ
t. If we denote by uᾱ ∈ Irred(G) the unique representative
equivalent to (uα)c, then the map uα 7→ uᾱ extends to a
onjugation on
the ring Z[Irred(G)] and sin
e ea
h uα is an element ofMnα(A) for some
nα ∈ N we
an also de�ne a dimension fun
tion d : Z[Irred(G)] → [1,∞[
by d(uα) = nα. When endowed with this multipli
ation,
onjugation
and dimension fun
tion Z[Irred(G)] be
omes a fusion algebra. The only
thing that is not
lear at this moment is that Frobenius re
ipro
ity
holds. To see this, we �rst note that for any α ∈ I and any �nite
dimensional
orepresentation v we have (by S
hur's Lemma [MVD98,
6.6℄) that uα o
urs exa
tly
dimC Mor(u
α, v)
8 DAVID KYED
times in the de
omposition of v. Moreover, we have for any two unitary
orepresentations v and w that
dimC Mor(v, w) = dimC((Vw ⊗ V ′v)w T#v
dimCMor(v
cc, w) = dimC((V
v ⊗ Vw)v
Here the right hand side denotes the linear dimension of the spa
e of in-
variant ve
tors under the relevant
oa
tion. These formulas are proved
in [Wor87, 3.4℄ for
ompa
t matrix quantum groups, but the same proof
arries over to the
ase where the
ompa
t quantum group in question
does not ne
essarily possess a fundamental
orepresentation. Using the
�rst formula, we get for α, β, γ ∈ I that
α,β = dimC Mor(u
γ, uα T©uβ)
= dimC(Vα ⊗ Vβ ⊗ V ′γ)u
T#uβ T#(uγ)c
= dimC(Vγ ⊗ V ′β ⊗ V ′α)u
T#(uβ)c T#(uα)c
= dimC Mor(u
α, uγ T©(uβ)c)
The remaining identity in Frobenius re
ipro
ity follows similarly us-
ing the se
ond formula. The fusion algebra Z[Irred(G)] is
alled the
orepresentation ring (or fusion ring) of G and is denoted R(G).
Re
all that the
hara
ter of a
orepresentation u ∈ Mn(A) is de�ned
as χ(u) =
i=1 uii. It follows from Proposition 1.1 that the Z-linear
extension
χ : Z[Irred(G)] −→ A0
is an inje
tive homomorphism of ∗-rings. I.e. χ is additive and mul-
tipli
ative with χ(uᾱ) = (χ(uα))∗. This gives a link between the two
∗-algebras R(G) and A0 whi
h will be of importan
e later.
Other interesting examples of fusion algebras arise from in
lusions
of II1-fa
tors. See [HI98℄ for details.
Remark 2.4. In the following we shall only
onsider fusion algebras
with an at most
ountable basis. This will therefore be assumed with-
out further noti
e throughout the paper. Sin
e we will primarily be
interested in
orepresentation rings of
ompa
t quantum groups, this
is not very restri
tive sin
e the standing separability assumption (Re-
mark 1.4) ensures that the
orepresentation rings always have a
ount-
able basis.
Consider again an abstra
t fusion algebra R = Z[I]. For ξ, η ∈ I we
de�ne the (weighted)
onvolution of the
orresponding Dira
measures,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 9
δξ and δη, as
δξ ∗ δη =
d(ξ)d(η)
Nαξ,ηδα ∈ ℓ1(I).
This extends linearly and
ontinuously to a submultipli
ative produ
t
on ℓ1(I). For f ∈ ℓ∞(I) and ξ ∈ I we de�ne λξ(f), ρξ(f) : I → C by
λξ(f)(η) =
f(α)(δξ̄ ∗ δη)(α)
ρξ(f)(η) =
f(α)(δη ∗ δξ)(α)
Denote by σ the
ounting measure on I s
aled with d2; that is σ(ξ) =
d(ξ)2. Combining Proposition 1.3, Remark 1.4 and Theorem 1.5 in
[HI98℄ we get
Proposition 2.5 ([HI98℄). For ea
h f ∈ ℓ∞(I) we have λξ(f), ρξ(f) ∈
ℓ∞(I) and for ea
h p ∈ N ∪ {∞} the maps λξ, ρξ : ℓ∞(I) → ℓ∞(I) re-
stri
t to bounded operators on ℓp(I, σ) denoted λp,ξ and ρp,ξ respe
tively.
By linear extension, we therefore obtain a map λp,− : Z[I] → B(ℓp(I, σ))
and this map respe
ts the weighted
onvolution produ
t. Moreover, for
p = 2 the operator U : ℓ2(I) → ℓ2(I, σ) given by U(δη) = 1d(η)δη is
unitary and intertwines λ2,ξ with the operator
lξ : δη 7−→
Nαξ,ηδα.
Remark 2.6. Under the natural identi�
ation of ℓ2(I) with the GNS
spa
e L2(C[I], τ), we see that πτ (ξ) = d(ξ)lξ. In parti
ular the GNS
representation
onsists of bounded operators. Here τ is the natural
tra
e de�ned just after De�nition 2.1.
3. Amenability for Fusion Algebras
The notion of amenability for fusion algebras was introdu
ed in
[HI98℄, but only in the slightly restri
ted setting of �nitely generated
fusion algebras; a fusion algebra R = Z[I] is
alled �nitely generated if
there exists a �nitely supported probability measure µ on I su
h that
supp(µ∗n) and µ(ξ̄) = µ(ξ) for all ξ ∈ I.
That is, if the union of the supports of all powers of µ, with respe
t
to
onvolution, is I and µ is invariant under the involution. The �rst
ondition is referred to as non-degenera
y of µ and the se
ond
ondition
is referred to as symmetry of µ.
10 DAVID KYED
In [HI98℄, amenability is de�ned, for a �nitely generated fusion al-
gebra, by requiring that ‖λp,µ‖ = 1 for some 1 < p < ∞ and some
�nitely supported, symmetri
, non-degenerate probability measure µ.
It is then proved that this is independent of the
hoi
e of µ and p,
using the non-degenera
y property of the measure. If we
onsider a
ompa
t quantum group G = (A,∆) it is not di�
ult to prove that its
orepresentation ring R(G) is �nitely generated exa
tly when G is a
ompa
t matrix quantum group. Sin
e we are also interested in quan-
tum groups without a fundamental
orepresentation we will
hoose the
following de�nition of amenability.
De�nition 3.1. A fusion algebra R = Z[I] is
alled amenable if 1 ∈
σ(λ2,µ) for every �nitely supported, symmetri
probability measure µ
on I.
Here σ(λ2,µ) denotes the spe
trum of the operator λ2,µ. From Propo-
sition 1.3 and Corollary 4.4 in [HI98℄ it follows that our de�nition agrees
with the one in [HI98℄ on the
lass of �nitely generated fusion algebras.
The relation between amenability for fusion algebras and the
lassi
al
notion of amenability for groups will be explained later. See e.g. Re-
mark 3.8 and Corollary 4.7.
De�nition 3.2. Let R = Z[I] be a fusion algebra. For two �nite
subsets S, F ⊆ I we de�ne the boundary of F relative to S as the set
∂S(F ) = {α ∈ F | ∃ ξ ∈ S : supp(αξ) * F}
∪ {α ∈ F c | ∃ ξ ∈ S : supp(αξ) * F c}.
Here, and in what follows, F c denotes the set I \ F .
The modi�ed de�nition of amenability allows the following extension
of [HI98, 4.6℄ from where we also adopt some notation.
Theorem 3.3. Let R = Z[I] be a fusion algebra with dimension fun
-
tion d. Then the following are equivalent:
(A) The fusion algebra is amenable.
(FC1) For every �nitely supported, symmetri
probability measure µ
on I with e ∈ supp(µ) and every ε > 0 there exists a �nite
subset F ⊆ I su
h that
ξ∈supp(χF ∗µ)
d(ξ)2 < (1 + ε)
d(ξ)2.
(FC2) For every �nite, non-empty subset S ⊆ I and every ε > 0 there
exists a �nite subset F ⊆ I su
h that
∀ ξ ∈ S : ‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 11
where ρ1,ξ ∈ B(ℓ1(I, σ)) is the operator from Proposition 2.5.
(FC3) For every �nite, non-empty subset S ⊆ I and every ε > 0 there
exists a �nite subset F ⊆ I su
h that
ξ∈∂S(F )
d(ξ)2 < ε
d(ξ)2.
The
ondition (FC3) was not present in [HI98℄. It is to be
onsidered
as a fusion algebra analogue of the Følner
ondition for groups as it is
presented in [BP92, F.6℄. The strategy for the proof of Theorem 3.3 is
to prove the following impli
ations:
(A) ⇔ (FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2).
The proof of the impli
ations (A) ⇔ (FC2) and (FC1) ⇒ (FC2) are
small modi�
ations of the
orresponding proof in [HI98℄. We �rst set
out to prove the
ir
le of impli
ations
(FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2).
For the proof we will need the following simple lemma.
Lemma 3.4. If Nαξ,η > 0 for some ξ, η, α ∈ I then d(α)d(η) ≥ d(ξ).
Proof. By Frobenius re
ipro
ity, we have Nαξ,η = N
α,η̄ > 0 and hen
e
d(α)d(η) = d(α)d(η̄) =
α,η̄d(γ) ≥ N ξα,η̄d(ξ) ≥ d(ξ).
Proof of (FC2) ⇒ (FC3). We �rst note that (FC2), by the triangle in-
equality, implies the following
ondition:
For every �nite, non-empty set S ⊆ I and every ε > 0 there exists a
�nite set F ⊆ I su
h that
‖ρ1,χS(χF )− |S|χF‖1,σ < ε‖χF‖1,σ. (†)
Here |S| denotes the
ardinality of S. Let S and ε > 0 be given
and
hoose F su
h that (†) is satis�ed. De�ne a map ϕ : I → R by
12 DAVID KYED
ϕ(ξ) = ρ1,χS(χF )(ξ)− |S|χF (ξ). We note that
ϕ(ξ) =
χF (α)(δξ ∗ χS)(α)
− |S|χF (ξ)
(δξ ∗ δη)(α)
− |S|χF (ξ)
d(ξ)d(η)
Nαξ,η − |S|χF (ξ).
We now divide into four
ases.
(i) If ξ ∈ F ∩∂S(F )c then supp(ξη) ⊆ F for all η ∈ S and hen
e we
get the relation
d(ξ)d(η)
Nαξ,η = 1. This implies ϕ(ξ) = 0.
(ii) If ξ ∈ F c ∩ ∂S(F )c we see that Nαξ,η = 0 for all α ∈ F and all
η ∈ S and hen
e ϕ(ξ) = 0.
(iii) If ξ ∈ F c ∩ ∂S(F ) we have χF (ξ) = 0 and there exist α0 ∈ F
and η0 ∈ S su
h that Nα0ξ,η0 6= 0. Using Lemma 3.4, we now get
ϕ(ξ) ≥ d(α0)
d(ξ)d(η0)
Nα0ξ,η0 ≥
d(η0)2
Nα0ξ,η0 ≥
d(η0)2
where M = max{d(η)2 | η ∈ S}.
(iv) If ξ ∈ F ∩ ∂S(F ) we have
ϕ(ξ) =
d(ξ)d(η)
Nαξ,η − |S|
= (−1)
d(ξ)d(η)
Nαξ,η
= (−1)
d(ξ)d(η)
Nαξ,η,
and be
ause ξ ∈ ∂S(F )∩ F there exist η0 ∈ S and α0 /∈ F su
h
that Nα0ξ,η0 6= 0. Using Lemma 3.4 again we
on
lude, as in (iii),
that |ϕ(ξ)| ≥ 1
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 13
We now get
d(ξ)2 = ε‖χF‖1,σ
> ‖ρ1,χS(χF )− |S|χF‖1,σ (by (†))
|ϕ(ξ)|d(ξ)2
ξ∈∂S(F )
|ϕ(ξ)|d(ξ)2 (by (i) and (ii))
ξ∈∂S(F )
d(ξ)2, (by (iii) and (iv))
and sin
e ε was arbitrary the
laim follows. �
Proof of (FC3) ⇒ (FC1). Given a �nitely supported, symmetri
pro-
bability measure µ, with µ(e) > 0, and ε > 0 we put S = supp(µ) and
hoose F ⊆ I su
h that (FC3) is ful�lled with respe
t to ε. We have
(χF ∗ µ)(ξ) =
α∈F,β∈S
d(α)d(β)
α,β ,
(χF ∗ µ)(ξ) = 0 ⇔ ∀α ∈ F ∀β ∈ S : N ξα,β = 0
⇔ ∀α ∈ F ∀β ∈ S : Nα
= 0 (Frobenius)
⇔ ∀α ∈ F ∀β ∈ S : Nαξ,β = 0 (S symmetri
)
⇔ ξ ∈ F c ∩ ∂S(F )c. (e ∈ S)
Hen
e supp(χF ∗ µ) = (F c ∩ ∂S(F )c)c = F ∪ ∂S(F ) and we get
ξ∈supp(χF ∗µ)
d(ξ)2 −
d(ξ)2 =
ξ∈F∪∂S(F )
d(ξ)2 −
d(ξ)2
ξ∈∂S(F )∩F
d(ξ)2
ξ∈∂S(F )
d(ξ)2
d(ξ)2. (by (FC3))
Proof of (FC1) ⇒ (FC2). Given ε > 0 and S ⊆ I we de�ne S̃ = S ∪
S̄ ∪{e} and µ = 1
χS̃. Choose F ⊆ I su
h that µ and F satisfy (FC1)
14 DAVID KYED
with respe
t to
. We aim to prove that (FC2) is satis�ed for all ξ ∈ S̃.
For arbitrary ξ ∈ I we have
‖ρ1,ξ(χF )− χF‖1,σ =
|ρ1,ξ(χF )(α)− χF (α)|d(α)2
d(α)d(ξ)
α,ξ)− χF (α)|d(α)2
d(α)d(ξ)
d(α)2
d(α)d(ξ)
d(α)2
d(η)d(α)
α,ξ +
d(η)d(α)
d(η)d(α)
α,ξ +N
d(η)d(α)
+Nαη,ξ). (†)
For ξ ∈ supp(µ) = S̃ and α /∈ F , it is easy to
he
k that (χF ∗µ)(α) > 0
if there exists an η ∈ F su
h that Nα
+Nαη,ξ > 0. Hen
e the
al
ulation
(†) implies that
‖ρ1,ξ(χF )− χF‖1,σ ≤
α∈supp(χF ∗µ)\F
d(η)d(α)
+Nαη,ξ)
α∈supp(χF ∗µ)\F
d(η)d(α)
+Nαη,ξ)
α∈supp(χF ∗µ)\F
d(α)2
α∈supp(χF ∗µ)
d(α)2 −
d(α)2
< ε‖χF‖1,σ,
where the last estimate follows from (FC1). Note that the
ondition e ∈
supp(µ) was used to get the fourth step in the
al
ulation above. �
We now set out to prove the remaining equivalen
e in Theorem 3.3.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 15
Proof of (A) ⇔ (FC2). At the end of this se
tion four formulas are
gathered; these will be used during the proof and referred to as (F1) -
(F4). For the a
tual proof we also need the following de�nitions. Con-
sider a �nitely supported, symmetri
probability measure µ on I and
de�ne pµ : I × I → R by
pµ(ξ, η) = (δξ ∗ µ)(η) =
d(ξ)d(ω)
Note that the fun
tion pµ satis�es the reversibility
ondition:
σ(ξ)pµ(ξ, η) = σ(η)pµ(η, ξ).
For a �nitely supported fun
tion f ∈ c0(I) and r ∈ N we also de�ne
‖f‖Dµ(r) =
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
Although this is referred to as the generalized Diri
hlet r-norm of f ,
one should keep in mind that the fun
tion ‖·‖Dµ(r) is only a semi norm.
We shall now
onsider the following
ondition:
For all �nitely supported, symmetri
, probability measures µ we have
{‖f‖Dµ(r)
‖f‖r,σ
| f ∈ c0(I) \ {0}
= 0. (NWr)
The reason for the name (NWr), whi
h appeared in [HI98℄, is that
the
ondition is the negation of a so-
alled Wirtinger inequality. See
[HI98℄ for more details. To prove (A) ⇔ (FC2) we will a
tually prove
the following equivalen
es
(FC2) ⇔ (NW1) and ∀r : (NW1) ⇔ (NWr) and (A) ⇔ (NW2).
For the latter of these equivalen
es the following lemma will be useful.
Lemma 3.5. For all f ∈ c0(I) we have
‖f‖2Dµ(2) = 〈f |f〉2,σ − 〈ρ2,µ(f)|f〉2,σ,
where 〈·|·〉2,σ denotes the inner produ
t on ℓ2(I, σ).
Proof. This is proven by a dire
t
al
ulation using the reversibility
ondition and the formula (F4) from the end of this se
tion. �
Proof of (A)⇔ (NW2). Let µ be a �nitely supported, symmetri
prob-
ability measure on I. By [HI98, 1.3,1.5℄, we have that ρ2,µ is self-adjoint
16 DAVID KYED
and ‖ρ2,µ‖ ≤ ‖µ‖1 = 1 so that 1− ρ2,µ ≥ 0. We now get
1 ∈ σ(λ2,µ) ⇔ 1 ∈ σ(ρ2,µ) ([HI98, 1.5℄)
⇔ 0 ∈ σ(1− ρ2,µ)
⇔ 0 ∈ σ(
1− ρ2,µ)
⇔ ∃xn ∈ (ℓ2(I, σ))1 : ‖(
1− ρ2,µ)xn‖2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : ‖(
1− ρ2,µ)fn‖2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : 〈(1− ρ2,µ)fn |fn〉2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : ‖fn‖Dµ(2) −→ 0 (Lem. 3.5)
⇔ inf
{‖f‖Dµ(2)
‖f‖2,σ
| f ∈ c0(I) \ {0}
Hen
e (A) ⇔ (NW2) as desired. �
Proof of (NW1) ⇒ (FC2). Given ε > 0 and ξ1, . . . , ξn ∈ I, we
hoose a
�nitely supported, symmetri
probability measure µ with ξ1, . . . , ξn ∈
supp(µ). De�ne
min{µ(ξ) | ξ ∈ I},
and
hoose, a
ording to (NW1), an f ∈ c0(I) su
h that
‖f‖Dµ(1) < ε′‖f‖1,σ. (∗)
Sin
e ‖|f |‖Dµ(1) ≤ ‖f‖Dµ(1) and ‖|f |‖1,σ = ‖f‖1,σ we may assume that f
is positive. Sin
e f
an be approximated by a rational fun
tion we may
a
tually assume that f has integer values. Put N = max{f(ξ) | ξ ∈ I}
and de�ne, for k = 1, . . . , N , Fk = {ξ | f(ξ) ≥ k}. Then f =
k=1 χFk
and the following formulas hold.
‖f‖Dµ(1) =
‖χFk‖Dµ(1) and ‖f‖1,σ =
‖χFk‖1,σ.
The �rst formula is proved by indu
tion on the integer N and the
se
ond follows from a dire
t
al
ulation using only the reversibility
property of pµ. Be
ause of (∗), there must therefore exist some j ∈
{1, . . . , N} su
h that
‖χFj‖Dµ(1) < ε′‖χFj‖1,σ. (∗∗)
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 17
For the sake of simpli
ity we denote this Fj by F in the following. We
now get
‖χF‖Dµ(1) =
σ(ξ)pµ(ξ, η)|χF (ξ)− χF (η)|
ξ∈F,η/∈F
σ(ξ)pµ(ξ, η) (reversibility)
ξ∈F,η/∈F
d(ξ)d(ω)
ξ∈F,η/∈F
d(ξ)d(η)
ξ∈F,η/∈F
d(ξ)d(η)
ξ,ω +N
ξ,ω̄)
µ(ω)‖ρ1,ω(χF )− χF‖1,σ. (‡)
Here the last equality follows from the
omputation (†) in the proof of
(FC1) ⇒ (FC2). The inequality (∗∗) therefore reads
µ(ω)‖ρ1,ω(χF )− χF‖1,σ < ε′‖χF‖1,σ.
For every ω ∈ I we therefore
on
lude, sin
e ε′ = ε
min(µ), that
µ(ω)‖ρ1,ω(χF )− χF‖1,σ < min(µ)ε‖χF‖1,σ.
Sin
e ea
h of the given ξi's are in supp(µ) we get for all i that
‖ρ1,ξi(χF )− χF‖1,σ < ε‖χF‖1,σ,
as desired. �
Proof of (FC2) ⇒ (NW1). Assume now (FC2) and let µ and ε be given.
Choose F su
h that
‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ
18 DAVID KYED
for all ξ ∈ supp(µ). Using the
al
ulation (‡), from the proof of opposite
impli
ation, we get
‖χF‖Dµ(1) =
µ(ω)‖ρ1,ω(χF )− χF‖1,σ
µ(ω)ε‖χF‖1,σ
‖χF‖1,σ
< ε‖χF‖1,σ.
For the proof of the statement (NW1) ⇔ (NWr) we will need the
following lemma.
Lemma 3.6 ([Ger88℄). For r ≥ 2 and f ∈ c0(I)+ we have
‖f r‖Dµ(1) ≤ 2r‖f‖r−1r,σ ‖f‖Dµ(r).
Proof. First note that
‖f r‖Dµ(1) =
σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r|
σ(ξ)pµ(ξ, η)(f(ξ)
r−1 + f(η)r−1)|f(ξ)− f(η)|,
where the inequality follows from (F1). De�ne a measure ν on I × I
by ν(ξ, η) = 1
σ(ξ)pµ(ξ, η) and
onsider the fun
tions ϕ, ψ : I × I → R
given by
ϕ(ξ, η) = f(ξ)r−1 + f(η)r−1 and ψ(ξ, η) = |f(ξ)− f(η)|.
De�ne s > 1 by the equation 1
= 1. Then the inequality above
an be written as ‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν and using Hölder's inequality
we therefore get
‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν
≤ r‖ϕ‖s,ν‖ψ‖r,ν
σ(ξ)pµ(ξ, η)(f(ξ)
r−1 + f(η)r−1)s
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 19
σ(ξ)pµ(ξ, η)(f(ξ)
(r−1)s + f(η)(r−1)s)
s‖f‖Dµ(r)
σ(ξ)pµ(ξ, η)f(ξ)
(r−1)s
s‖f‖Dµ(r)
pµ(ξ, η)
f(ξ)(r−1)s
s‖f‖Dµ(r)
σ(ξ)f(ξ)(r−1)s
s‖f‖Dµ(r)
σ(ξ)f(ξ)r
] r−1
r ‖f‖Dµ(r)
= 2r‖f‖r−1r,σ ‖f‖Dµ(r).
Also the following observation will be useful.
Observation 3.7. Under the assumptions of Lemma 3.6 we have
‖f‖Dµ(r) =
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r|
(by (F3))
= ‖f r‖
Dµ(1)
Having these results, we are now able to prove (NW1) ⇔ (NWr).
Proof of (NW1) ⇒ (NWr). Assume (NW1) and let µ and ε > 0 be
given. Put ε′ = εr and
hoose non-zero f ∈ c0(I)+ su
h that
‖f‖Dµ(1)
‖f‖1,σ
< ε′.
Using Observation 3.7 we get
f‖Dµ(r)
f‖r,σ
Dµ(1)
< (ε′)
r = ε.
by (F2)
by reversibility
20 DAVID KYED
Proof of (NWr) ⇒ (NW1). Given µ and ε > 0 and put ε′ = 12rε. Then
hoose non-zero f ∈ c0(I)+ with
‖f‖Dµ(r)
‖f‖r,σ
< ε′.
Using Lemma 3.6, we get
‖f r‖Dµ(1)
‖f r‖1,σ
2r‖f‖r−1r,σ ‖f‖Dµ(r)
‖f‖rr,σ
< 2rε′ = ε.
Gathering all the results just proven we get (A) ⇔ (FC2). �
This
on
ludes the proof of Theorem 3.3.
Remark 3.8. Consider a
ountable, dis
rete group Γ and the
orre-
sponding fusion algebra Z[Γ]. It is not di�
ult to prove that Z[Γ]
satis�es (FC3) from Theorem 3.3 if and only if Γ satis�es Følner's
ondition (for groups) as presented in [BP92, F.6℄. Sin
e a group is
amenable if and only if it satis�es Følner's
ondition, we see from this
that Γ is amenable if and only if the
orresponding fusion algebra Z[Γ]
is amenable.
3.1. Formulas used in the proof of Theorem 3.3. We
olle
t here
four formulas used in the proof of Theorem 3.3. Let r, s > 1 and assume
= 1. Then for all z, w ∈ C, a, b ≥ 0 and n ∈ N we have
|ar − br| ≤ r(ar−1 + br−1)|a− b| (F1)
(a+ b)r ≤ 2r−1(ar + br) (F2)
|a− b|n ≤ |an − bn| (F3)
|z − w|2 + |w − z|2 = 2(|z|2 − zw̄) + 2(|w|2 − wz̄) (F4)
Proof. The inequality (F1)
an be proved using the mean value theorem
on the fun
tion f(x) = xr and the interval between a and b. To prove
(F2),
onsider a two-point set endowed with
ounting measure. Using
Hölder's inequality, we then get
a + b = 1 · a + 1 · b ≤ (1s + 1s) 1s (ar + br) 1r .
From this the desired inequality follows using the fa
t that
= r−1
The inequality (F3) follows using the binomial theorem. If, for instan
e,
a = b+ k for some k ≥ 0 we have
(a− b)n = kn ≤ (b+ k)n − bn = an − bn.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 21
The formula (F4) follows by splitting w and z into real and imaginary
parts and
al
ulating both sides of the equation. �
4. Coamenable Compa
t Quantum Groups
In this se
tion we introdu
e the notion of
oamenability for
ompa
t
quantum groups and dis
uss the relationship between
oamenability of
a
ompa
t quantum group and amenability of its
orepresentation ring.
The notion of (
o-)amenability has been treated in di�erent quantum
group settings by numerous people. A number of referen
es for this
subje
t are [BMT01℄, [Voi79℄, [Rua96℄, [Ban99a℄, [Ban99b℄, [ES92℄ and
[BS93℄. For our purposes, the approa
h of Bédos, Murphy and Tuset in
[BMT01℄ is the most natural and we are therefore going to follow this
referen
e throughout this se
tion. We will assume that the reader is
familiar with the basi
s on Woronowi
z's theory of
ompa
t quantum
groups. De�nitions, notation and some basi
properties
an be found
in Se
tion 1 and detailed treatments
an be found in [Wor98℄, [MVD98℄
and [KT99℄.
De�nition 4.1 ([BMT01℄). Let G = (A,∆) be a
ompa
t quantum
group and let Ared be the image of A under the GNS representation πh
arising from the Haar state h. Then G is said to be
oamenable if the
ounit ε : A0 → C extends
ontinuously to Ared.
Remark 4.2. It is well known that a dis
rete group Γ is amenable
if and only if the trivial representation of C∗full(Γ) fa
torizes through
C∗red(Γ). This amounts to saying that (C
red(Γ),∆red) is
oamenable if
and only if Γ is amenable. Note also that the abelian
ompa
t quan-
tum groups (C(G),∆c) are automati
ally
oamenable sin
e the
ounit
is given by evaluation at the identity and therefore already globally
de�ned and bounded.
In the following theorem we
olle
t some fa
ts on
oamenable
om-
pa
t quantum groups. For more
oamenability
riteria and a proof of
the theorem below we refer to [BMT01℄.
Theorem 4.3 ([BMT01℄). For a
ompa
t quantum group G = (A,∆)
the following are equivalent.
(i) G is
oamenable.
(ii) The Haar state h is faithful and the
ounit is bounded with re-
spe
t to the norm on A.
(iii) The natural map from the universal representation A
to the
redu
ed representation Ared is an isomorphism.
22 DAVID KYED
If G is a
ompa
t matrix quantum group with fundamental
orepre-
sentation u ∈ Mn(A) the above
onditions are also equivalent to the
following.
(iv) The number n is in σ(πh(Re(χ(u))) where χ(u) =
i=1 uii is
the
hara
ter map from Se
tion 2.
Re
all that σ(T ) denotes the spe
trum of a given operator T . Thus,
when we are dealing with a
oamenable quantum group the Haar state
is automati
ally faithful and hen
e the
orresponding GNS representa-
tion πh is faithful. We therefore
an, and will, identify A and Ared. The
ondition (iv) is Skandalis's quantum analogue of the so-
alled Kesten
ondition for groups (see [Kes59℄ and [Ban99a℄) whi
h is proved by
Bani
a in [Ban99b℄. The next result is a generalization of the Kesten
ondition to the
ase where a fundamental
orepresentation is not (ne-
essarily) present. The proof draws inspiration from the
orresponding
proof in [BMT01℄.
Theorem 4.4. Let G = (A,∆) be a
ompa
t quantum group. Then
the following are equivalent:
(i) G is
oamenable.
(ii) For any �nite dimensional, unitary
orepresentation u ∈ Mnu(A)
we have nu ∈ σ(πh(Re(χ(u)))).
Proof. AssumeG to be
oamenable and let a �nite dimensional, unitary
orepresentation u ∈ Mnu(A) be given. Sin
e the
ounit extends to a
hara
ter ε : Ared → C and sin
e
ε(Re(χ(u))) = ε(
uii + u
) = nu,
we must have nu ∈ σ(πh(Re(χ(u)))). Assume
onversely that the prop-
erty (ii) is satis�ed and de�ne, for a �nite dimensional, unitary
ore-
presentation u, the set
C(u) = {ϕ ∈ S (Ared) | ϕ(πh(Re(χ(u)))) = nu}.
Here S (Ared) denotes the state spa
e of Ared. It is
lear that ea
h C(u)
is
losed in the weak
-topology and we now prove that the family
F = {C(u) | u �nite dimensional, unitary
orepresentation}
has the �nite interse
tion property. We �rst prove that ea
h C(u) is
non-empty. For given u, we put xij = uij − δij and x =
ijxij .
Then x is
learly positive and a dire
t
al
ulation reveals that
x = 2(nu − Re(χ(u))). (†)
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 23
Hen
e, nu ∈ σ(πh(Re(χ(u)))) if and only if there exists [KR83, 4.4.4℄ a
ϕ ∈ S (Ared) with
ϕ(πh(Re(χ(u)))) = nu.
Thus, C(u) 6= ∅. Let now u(1), . . . , u(k) be given and put u = ⊕ki=1u(i).
We aim at proving that
C(u) ⊆
C(u(i)).
Let ϕ ∈ C(u) be given and note that
nu(k) = ϕ(πh(Re(χ(u)))) =
u(i)∑
ϕ(πh(u
jj ) + πh(u
jj )).
Sin
e the matrix u is unitary, we have ‖πh(ust)‖ ≤ 1 for all s, t ∈
{1, . . . , nu} and hen
e
ϕ(πh(u
jj ) + πh(u
jj )) ∈ [−1, 1].
This for
es
ϕ(πh(u
jj )+πh(u
jj )) = 1 and hen
e ϕ(πh(Re(χ(u
(i))))) =
nu(i). Thus ϕ is in ea
h of the sets C(u
(1)), . . . , C(u(k)) and we
on
lude
that F has the �nite interse
tion property. By
ompa
tness of S (Ared),
we may therefore �nd a state ϕ su
h that ϕ(πh(Re(χ(u)))) = nu for ev-
ery unitary
orepresentation u. Denote by H the GNS spa
e asso
iated
with this ϕ, by ξ0 the natural
y
li
ve
tor and by π the
orresponding
GNS representation of Ared. Consider an arbitrary unitary
orepresen-
tation u and form as before the elements xij and x. Then the equation
(†) shows that ϕ(x∗ijxij) = 0 and hen
e π(xij)ξ0 = 0 and
π(uij)ξ0 = δijξ0.
From the Cau
hy-S
hwarz inequality we get
|ϕ(xij)|2 ≤ ϕ(x∗ijxij)ϕ(1) = 0,
and hen
e ϕ(uij) = δij . We therefore have that π(uij)ξ0 = ϕ(uij)ξ0.
Sin
e the matrix
oe�
ients span A0 linearly we get π(a)ξ0 = ϕ(a)ξ0
for all a ∈ A0. By density of A0 in Ared it follows that π(a)ξ0 = ϕ(a)ξ0
for all a ∈ Ared. From this we see that
H = π(Ared)ξ0
= Cξ0,
and it follows that ϕ : Ared → C is a bounded ∗-homomorphism
oin-
iding with ε on A0. Thus, G is
oamenable.
24 DAVID KYED
The following result was mentioned, without proof, in [HI98, p.692℄
in the restri
ted setting of
ompa
t matrix quantum groups whose Haar
state is a tra
e.
Theorem 4.5. A
ompa
t quantum group G = (A,∆) is a
oamenable
if and only if the
orepresentation ring R(G) is amenable.
For the proof we will need the following lemma. For this, re
all from
Se
tion 2 that the ∗-algebra C[Irred(G)]
omes with a tra
e τ given by
u∈Irred(G)
zuu 7−→ ze,
where e ∈ Irred(G) denotes the identity in R(G). In what follows, we
denote by C∗red(R(G)) the enveloping C
-algebra of C[Irred(G)] on the
GNS spa
e L2(C[Irred(G)], τ) arising from τ .
Lemma 4.6. The
hara
ter map χ : R(G) → A0 extends to an isome-
tri
∗-homomorphism χ : C∗red(R(G)) → Ared.
Proof. Put I = Irred(G). For an irredu
ible, �nite dimensional, unitary
orepresentation u we have h(uij) = 0 unless u is the trivial
orepre-
sentation and therefore the following diagram
ommutes
// A0
Hen
e χ extends to an isometri
embedding
K = L2(C[I], τ) −֒→ L2(A0, h) = H.
Denote by S the algebra χ(R(G)) and by S̄ the
losure of πh(S) inside
Ared. Sin
e S is a ∗-algebra that maps K into itself it also maps K⊥
into itself and hen
e πh(χ(a)) takes the form
πh(χ(a))
0 πh(χ(a))
‖πh(χ(a))‖ = max{‖πh(χ(a))
‖, ‖πh(χ(a))
≥ ‖πh(χ(a))
= ‖πτ (a)‖.
This proves that the map κ : πh(S) → πτ (C[I]) given by κ(πh(χ(a))) =
πτ (a) is bounded and it therefore extends to a
ontra
tion κ̄ : S̄ →
C∗red(R(G)). We now prove that κ̄ is inje
tive. Sin
e h is faithful on
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 25
Ared and τ is faithful on C
red(R(G)) we get the following
ommutative
diagram
πh(S)
πτ (C[I])�
C∗red(R(G))�
L2(S̄, h) // L2(C∗red(R(G)), τ)
One easily
he
ks that κ indu
es an isometry L2(S̄, h) → L2(C∗red(G), τ)
and it therefore follows that κ̄ is inje
tive and hen
e an isometry. Thus,
for χ(a) ∈ S we have
‖πh(χ(a))‖ = ‖κ̄(πh(χ(a)))‖ = ‖πτ (a)‖,
as desired.
Proof of Theorem 4.5. Assume �rst that G is
oamenable and put I =
Irred(G). Consider a �nitely supported, symmetri
probability measure
µ on I. We aim to show that 1 ∈ σ(λ2,µ), where λ2,µ is the operator on
ℓ2(I, σ) de�ned in Se
tion 2. Write µ as
ξ∈I tξδξ and re
all (Lemma
4.6) that the
hara
ter map χ : C[I] → A0 extends to an inje
tive ∗-
homomorphism χ : C∗red(R(G)) → Ared. Using this, and Proposition
2.5, we get that
σ(λ2,µ) = σ(lµ)
tξlξ)
πτ (ξ))
= σ(χ(
πτ (ξ)))
πh(ξii)).
Sin
e G is
oamenable, the
ounit extends to a
hara
ter ε : Ared → C
and we have
ξii)) =
nξ = 1.
26 DAVID KYED
Hen
e 1 ∈ σ
i=1 πh(ξii))
= σ(λ2,µ) and we
on
lude that
R(G) is amenable.
Assume,
onversely, that R(G) is amenable. We aim at proving
that G ful�lls the Kesten
ondition from Theorem 4.4. Let therefore
u ∈ Mn(A) be an arbitrary, �nite dimensional, unitary
orepresenta-
tion. Denote by (uα)α∈S ⊆ Irred(G) the irredu
ible
orepresentations
o
urring in the de
omposition of u and by kα the multipli
ity of uα in
u. Now de�ne
µu(uα) =
if α ∈ S;
0 if α /∈ S.
Putting µ = 1
µū we obtain a �nitely supported, symmetri
probability measure and by assumption we have that 1 ∈ σ(λ2,µ). Using
again that the
hara
ter map extends to an inje
tive ∗-homomorphism
χ : C∗red(R(G)) → Ared we obtain
σ(λ2,µ) = σ
λ2,uα +
λ2,uᾱ
luα +
(Prop. 2.5)
πτ (uα) +
πτ (uᾱ)
(Rem. 2.6)
πh(χ(uα)) +
πh(χ(uᾱ))
πh(χ(u)) +
πh(χ(ū))
πh(Re(χ(u)))
1 ∈ σ(λ2,µ) if and only if n ∈ σ(Re(πh(χ(u)))),
and the result now follows from Theorem 4.4. �
In parti
ular we (re-)obtain the following.
Corollary 4.7. A dis
rete group is amenable if and only if the group
ring,
onsidered as a fusion algebra, is amenable.
Corollary 4.8 ([Ban99b℄). The quantum groups SUq(2) are
oamen-
able.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 27
Proof. By Theorem 4.5, SUq(2) is
oamenable if and only if R(SUq(2))
is amenable. But, R(SUq(2)) = R(SU(2)) (see e.g. [Wor88℄) and sin
e
(C(SU(2)),∆c) is a
oamenable quantum group R(SU(2)) is amenable.
As seen from Theorem 4.5, the answer to the question of whether
a
ompa
t quantum group is
oamenable or not
an be determined
using only information about its
orepresentations � a fa
t noted by
Bani
a in the setting of
ompa
t matrix quantum groups in [Ban99a℄
and [Ban99b℄. With this in mind, we now propose the following Følner
ondition for quantum groups.
De�nition 4.9. A
ompa
t quantum group G = (A,∆) is said to satis-
fy Følner's
ondition if for any �nite, non-empty subset S ⊆ Irred(G)
and any ε > 0 there exists a �nite subset F ⊆ Irred(G) su
h that
u∈∂S(F )
n2u < ε
Here nu denotes the dimension of the irredu
ible
orepresentation u
and ∂S(F ) is the boundary of F relative to S as in De�nition 3.2.
We immediately obtain the following.
Corollary 4.10. A
ompa
t quantum group is
oamenable if and only
if it satis�es Følner's
ondition.
Proof. By Theorem 4.5, the
ompa
t quantum group G is
oamenable
if and only if R(G) is amenable. By Theorem 3.3, R(G) is amenable if
and only if it satis�es (FC3) whi
h is exa
tly the same as saying that
G satis�es Følner's
ondition. �
In Se
tion 6 we will use this Følner
ondition to dedu
e a vanishing
result
on
erning L2-Betti numbers of
ompa
t,
oamenable quantum
groups.
5. An Interlude
In this se
tion we gather various notation and minor results whi
h
will be used in the following se
tion to prove our main result, Theorem
6.1. Some generalities on von Neumann algebrai
quantum groups are
stated without proofs; we refer to [KV03℄ for the details.
Consider again a
ompa
t quantum group G = (A,∆) with tra
ial
Haar state h. Denote by {uα | α ∈ I} a
omplete set of representatives
for the equivalen
e
lasses of irredu
ible, unitary
orepresentations of
G. Consider the dense Hopf ∗-algebra
A0 = spanC{uαij | α ∈ I},
28 DAVID KYED
and its dis
rete dual Hopf ∗-algebra Â0. Sin
e h is tra
ial, the dis
rete
quantum group Â0 is unimodular; i.e. the left and right invariant fun
-
tionals are the same. Denote by ϕ̂ the left and right invariant fun
tional
on Â0 normalized su
h ϕ̂(h) = 1. For a ∈ A0 we denote by â ∈ A′0 the
A0 ∋ x7−→h(ax) ∈ C.
Then, by de�nition, we have Â0 = {â | a ∈ A0}. The algebra Â0 is
∗-isomorphi
to
Mnα(C),
and be
ause h is tra
ial the isomorphism has a simple des
ription; if
we denote by Eαij the standard matrix units in Mnα(C) then the map
Φ((̂uαij)
∗) = 1
Eαij ,
extends to a ∗-isomorphism [MVD98℄. Denote by λ the GNS represen-
tation of A on H = L2(A0, h), by η the
anoni
al in
lusion A0 ⊆ H
and by M (or λ(M)) the enveloping von Neumann algebra λ(A0)
The map η̂ : Â0 → H given by â 7→ η(a) makes (H, η̂) a GNS pair for
(Â0, ϕ̂) and the
orresponding GNS representation L is given by
L(â)η(x) = η̂(âx̂).
We denote by M̂ (or L(M̂)) the enveloping von Neumann algebra
L(Â0)
. This is a dis
rete von Neumann algebrai
quantum group
and ϕ̂ gives rise to a left and right invariant, normal, semi�nite, faith-
ful weight on M̂ . Ea
h �nite subset E ⊆ I gives rise to a
entral
proje
tion
PE = Φ
χE(α)1nα
∈ Â0,
where 1nα denotes the unit in Mnα(C) and χE is the
hara
teristi
fun
tion for the set E. A dire
t
omputation shows that L(PE) is the
orthogonal proje
tion onto the �nite dimensional subspa
e
{uᾱij | 1 ≤ i, j ≤ nα, α ∈ E}.
Re
all from Example 2.3 that uβ̄ is the element in {uα | α ∈ I} whi
h
is equivalent to (uβ)c. Be
ause h is tra
ial, the left invariant weight ϕ̂
on Â0 has the parti
ular simple form [VKV
, p.47℄
nαTrnα
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 29
where Trnα is the non-normalized tra
e on Mnα(C). In parti
ular
ϕ̂(PE) =
n2α = ϕ̂(PĒ),
for any �nite subset E ⊆ I. For any m ∈ M and any �nite subset
E ⊆ I we have [VVD03, 2.10℄ that
TrH(m
∗PEm) = h(m
∗m)ϕ̂(PE),
where TrH denotes the standard tra
e on B(H). Here, and in what fol-
lows, we suppress the representations λ and L ofM and M̂ respe
tively
on H . The
ommutant M ′ is the underlying von Neumann algebra of
a
ompa
t, von Neumann algebrai
quantum group whose Haar state
is also given by the ve
tor state h and whose dis
rete dual is given by
(M̂, ∆̂)op; this quantum group has M̂ as its underlying von Neumann
algebra, but is endowed with
omultipli
ation ∆̂op = σ∆̂ where σ de-
notes the �ip-automorphism on M̂⊗̄M̂ . Sin
e (M̂, ∆̂) is unimodular
we see that ϕ̂op = ϕ̂ and hen
e the tra
e-formula above extends in the
following way.
Lemma 5.1 ([VVD03℄). For any m ∈ M or m ∈ M ′ and any �nite
subset E ⊆ I we have TrH(m∗PEm) = h(m∗m)ϕ̂(PE).
With this lemma we
on
lude the interlude and move towards an
appli
ation of the quantum Følner
ondition.
6. A Vanishing Result
In this se
tion we investigate the L2-Betti numbers of
oamenable
quantum groups. The notion of L2-Betti numbers for
ompa
t quantum
groups was introdu
ed in [Kye08℄ and we refer to that paper (and
the introdu
tion) for the de�nitions and basi
results. Throughout
the se
tion, we will freely use Lü
k's extended Murray-von Neumann
dimension, but whenever expli
it properties are used there will be a
referen
e. These referen
es will be to the original work [Lü
97℄ and
[Lü
98a℄, but for the reader who wants to learn the subje
t Lü
k's
book [Lü
02℄ is probably a better general referen
e.
Consider again a
ompa
t quantum group G = (A,∆) with Haar
state h and denote by M the enveloping von Neumann algebra in the
GNS representation arising from h. As promised in the introdu
tion,
we will now prove the following theorem whi
h should be
onsidered as
a quantum group analogue of Theorem 5.1 from [Lü
98a℄.
30 DAVID KYED
Theorem 6.1. If G is
oamenable and h is tra
ial then for any left
A0-module Z and any k ≥ 1 we have
dimM Tor
k (M,Z) = 0,
where dimM(−) is Lü
k's extended dimension fun
tion arising from the
extension of the tra
e-state h.
If M were �at as a module over A0 we would have Tor
k (M,Z) = 0
for any Z and any k ≥ 1, and the property in Theorem 6.1 is therefore
referred to as dimension �atness of the von Neumann algebra over the
algebra of matrix
oe�
ients. The proof of Theorem 6.1, whi
h is a
generalization of the
orresponding proof of [Lü
98a, 5.1℄, is divided
into three parts. Part I
onsists of redu
tions while part II
ontains
the
entral argument
arried out in detail in a spe
ial
ase. Part III
shows how to boost the argument from part II to the general
ase.
Throughout the proof, we will use freely the quantum group notation
developed in the previous se
tions without further referen
e; in par-
ti
ular, {uα | α ∈ I} will denote a �xed,
omplete set of pairwise
inequivalent, irredu
ible, unitary
orepresentations of G.
Proof of Theorem 6.1.
Part I
We begin with some redu
tions. Let an arbitrary A0-module Z be
given and
hoose a free module F that surje
ts onto Z. Then we have
a short exa
t sequen
e
0 −→ K −→ F −→ Z −→ 0,
and sin
e F is free (in parti
ular �at) the
orresponding long exa
t
Tor-sequen
e gives an isomorphism
TorA0k+1(M,Z) ≃ Tor
k (M,K) for k ≥ 1.
It is therefore su�
ient to prove the theorem for arbitrary Z and k =
1. Moreover, we may assume that Z is �nitely generated sin
e Tor
ommutes with dire
t limits, every module is the dire
ted union of
its �nitely generated submodules and dimM(−) is well behaved with
respe
t to dire
t limits [Lü
98a, 2.9℄. A
tually, we
an assume that Z
is �nitely presented sin
e any �nitely generated module Z is a dire
t
limit of �nitely presented modules. To see this,
hoose a short exa
t
sequen
e
0 −→ K −→ F −→ Z −→ 0,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 31
with F �nitely generated and free. Denote by (Kj)j∈J the dire
ted
system of �nitely generated submodules in K. Then F/Kj is �nitely
presented for ea
h j ∈ J and
Z = lim−→
F/Kj .
Be
ause of this and the dire
t limit formula for the dimension fun
tion
[Lü
98a, 2.9℄ we may, and will, therefore assume that Z is �nitely
presented. Choose a �nite presentation
f−→ Am0 −→ Z −→ 0.
Put H = L2(A, h), K = ker(f) ⊆ An0 ⊆ Hn and denote by f (2) : Hn →
Hm the
ontinuous extension of f . Then we have
TorA01 (M,Z) =
ker(idM ⊗f)
and hen
e
dimM Tor
1 (M,Z) = dimM ker(idM ⊗f)− dimM M ⊗
= dimM ker(f
(2))− dimM K
where the se
ond equality follows from [CS05, 2.11℄. See also [Lü
98a,
p.158-159℄. So we need to prove that K
= ker(f (2)).
Part II
We �rst treat the
ase m = n = 1. Then the map f has the form
Ra (right-multipli
ation by a) for some a ∈ A0. If a = 0 we have
= H = ker(f (2)) so we may assume a 6= 0. Sin
e the uαij's
onstitute a linear basis for A0, the element a ∈ A0 has a unique
expansion
i,j=1
tαiju
ij, (t
ij ∈ C)
and we may therefore
onsider the non-empty, �nite set S ⊆ I given
S = {α ∈ I | ∃ 1 ≤ i, j ≤ nα : tαij 6= 0}.
Denote by H0 the kernel of f
and by q0 ∈ M ′ the proje
tion onto
it. Denote by q the proje
tion onto H0 ∩ K⊥; we need to prove that
this subspa
e is trivial and sin
e the ve
tor-state h is faithful on M ′
this is equivalent to proving h(q) = 0. Let ε > 0 be given. Sin
e G is
32 DAVID KYED
assumed
oamenable, the Følner
ondition provides the existen
e of a
�nite, non-empty subset F ⊆ I su
h that
α∈∂S(F )
n2α < ε
Here we identify a subset E ⊆ I with the
orresponding set of
orep-
resentations {uα | α ∈ E}. To simplify notation further we will write
∂ instead of ∂S(F ) in the following and moreover we will suppress the
GNS-representations λ : M → B(H) and L : M̂ → B(H) as in Se
-
tion 5. Sin
e h is tra
ial, Woronowi
z's quantum Peter-Weyl Theorem
[KT99, 3.2.3℄ takes a parti
ular simple form and states that the set
{√nαuαij | 1 ≤ i, j ≤ nα, α ∈ I}
onstitutes an orthonormal basis for H . Hen
e every x ∈ H has an
ℓ2-expansion
i,j=1
ij. (x
ij ∈ C)
Consider a ve
tor x ∈ H and assume that P∂̄(x) = 0 su
h that the
ℓ2-expansion of x has the form
i,j=1 x
ij. For γ ∈ S and
1 ≤ p, q ≤ nγ we then have
PF̄ (x) =
α/∈∂,α∈F
i,j=1
(x) = PF̄
i,j=1
Here R
denotes the L2-extension of Ruγpq . Sin
e u
pq is
ontained
in the linear span of the matrix
oe�
ients of uα T©uγ and sin
e α /∈
∂ = ∂S(F ) and γ ∈ S we see that the two expressions above are equal.
By linearity and
ontinuity we obtain
f (2)PF̄ (x) = PF̄f
(2)(x).
This holds for all x ∈ ker(P∂̄), so if x ∈ H0 ∩ ker(P∂̄) we have
0 = f (2)PF̄ (x) = f(PF̄ (x)),
where the last equality is due to the fa
t that rg(PF̄ ) ⊆ A0 ⊆ H .
This proves that PF̄ (x) ∈ K = ker(f) and sin
e q was de�ned as the
proje
tion onto H0∩K⊥ we get qPF̄ (x) = 0. Sin
e this holds whenever
x ∈ H0 = q0(H) and P∂̄(x) = 0 we get
qPF̄ (q0 ∧ (1− P∂̄)) = 0.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 33
Thus, the restri
tion qPF̄ : H0 → H fa
torizes through H0/H0∩ker(P∂̄)
and we have
dimC(qPF̄ (H0)) ≤ dimC(H0/H0 ∩ ker(P∂̄))
≤ dimC(H/ ker(P∂̄))
= dimC(rg(P∂̄))
= ϕ̂(P∂).
For any �nite rank operator T ∈ B(H) one has
|TrH(T )| ≤ ‖T‖ dimC(T (H))
and using this and Lemma 5.1 we now get
h(q)ϕ̂(PF ) = h(q)ϕ̂(PF̄ )
= TrH(qPF̄ q)
≤ ‖qPF̄ q‖ dimC(qPF̄ q(H))
≤ dimC(qPF̄ (H0))
≤ ϕ̂(P∂).
h(q) ≤ ϕ̂(P∂)
ϕ̂(PF )
and sin
e ε > 0 was arbitrary we
on
lude that q = 0.
Part III
We now treat the general
ase of a �nitely presented A0-module Z with
�nite presentation
f−→ Am0 −→ Z −→ 0.
In this
ase f is given by right multipli
ation by an n × m matrix
T = (tij) with entries in A0. Ea
h tij has a unique linear expansion as
tij =
α,k,l t
(i,j)
α,k,lu
kl and we put
S = {α ∈ I | ∃ i, j, k, l, α : t(i,j)α,k,l 6= 0}.
As in Part II, we may assume that T 6= 0 so that S 6= ∅. Denote by
H0 the spa
e ker(f
(2)) ⊆ Hn, by q0 ∈ Mn(M ′) the proje
tion onto H0
and by q ∈ Mn(M ′) the proje
tion onto H0 ∩ K⊥. We need to show
that q = 0. Denote by Trn the non-normalized tra
e on Mn(C) and
put hn = h ⊗ Trn : B(H) ⊗ Mn(C) → C. We aim at proving that
34 DAVID KYED
hn(q) = 0, whi
h su�
es sin
e h is faithful on M
. For ea
h x ∈ M̂ we
denote by xn the diagonal operator on Hn whi
h has x in ea
h diagonal
entry. Under the identi�
ation B(H) ⊗ Mn(C) = B(Hn) we see that
TrH ⊗Trn
orresponds to TrHn , and Lemma 5.1 together with a dire
t
omputation therefore gives
TrHn(A
∗P nEA) = hn(A
∗A)ϕ̂(PE), (†)
for any �nite subset E ⊆ I and any A in Mn(M) or Mn(M ′). Let ε > 0
be given and
hoose a
ording to the Følner
ondition a �nite subset
F ⊆ I su
h that
α∈∂S(F )
n2α <
and put ∂ = ∂S(F ) for simpli
ity. By repeating the argument from the
beginning of Part II we arrive at the equation
(q0 ∧ (1− P n∂̄ )) = 0,
whi
h in turn yields
dimC(qP
(H0)) ≤ dimC(rg(P n∂̄ )) = n dimC(rg(P∂̄)) = nϕ̂(P∂).
Using the tra
e-formula (†) we
on
lude that
hn(q)ϕ̂(PF ) = TrHn(qP
F̄ q)
≤ ‖qP n
q‖ dimC(qP nF̄ q(H))
≤ dimC(qP nF̄ (H0))
≤ nϕ̂(P∂).
hn(q) ≤ n
ϕ̂(P∂)
ϕ̂(PF )
and sin
e ε > 0 was arbitrary we
on
lude that hn(q) = 0 as desired.
By putting Z = C in Theorem 6.1, we immediately obtain the fol-
lowing
orollary.
Corollary 6.2. Let G = (A,∆) be a
ompa
t,
oamenable quantum
group with tra
ial Haar state. Then β
n (G) = 0 for all n ≥ 1. Here
n (G) is the n-th L2-Betti number of G as de�ned in [Kye08℄.
In parti
ular we obtain the following extension of [Kye08, 3.3℄.
Corollary 6.3. For an abelian,
ompa
t quantum group G we have
n (G) = 0 for n ≥ 1.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 35
Proof. Sin
e G is abelian it is of the form (C(G),∆c) for some
ompa
t
(se
ond
ountable) group G. Sin
e the
ounit, given by evaluation at
the identity, is already globally de�ned and bounded it is
lear that G
is
oamenable and the result now follows from Corollary 6.2. �
We also obtain the
lassi
al result of Lü
k.
Corollary 6.4. [Lü
98a, 5.1℄ If Γ is an amenable,
ountable, dis
rete
group then for all CΓ-modules Z and all n ≥ 1 we have
dimL (Γ) Tor
n (L (Γ), Z) = 0.
In parti
ular, β
n (Γ) = 0 for n ≥ 1.
Proof. Put G = (C∗red(Γ),∆red). Then G is
oamenable if and only if Γ
is amenable and the result now follows from Theorem 6.1 and Corollary
6.2 �
Note, however, that this does not really give a new proof of Lü
k's
result sin
e the proof of Theorem 6.1
oin
ides with Lü
k's proof of the
statement in Corollary 6.4 when G = (C∗red(Γ),∆red).
In [CS05℄, Connes and Shlyakhtenko introdu
ed a notion of L2-Betti
numbers for tra
ial ∗-algebras. From the above results we also obtain
vanishing of these Connes-Shlyakhtenko L2-Betti numbers for
ertain
Hopf ∗-algebras. More pre
isely we get the following.
Corollary 6.5. Let G = (A,∆) be a
ompa
t,
oamenable quantum
group with tra
ial Haar state h. Then β
n (A0, h) = 0 for all n ≥ 1,
where β
n (A0, h) is the n-th Connes-Shlyakhtenko L
-Betti number of
the ∗-algebra A0 with respe
t to the tra
e h.
Proof. By [Kye08, 4.1℄ we have β
n (G) = β
n (A0, h) and the
laim
therefore follows from Corollary 6.2. �
The knowledge of dimension �atness also gives genuine homologi
al
information about the ring extension A0 ⊆ M . More pre
isely, the
following holds.
Corollary 6.6. If G = (A,∆) is
ompa
t and
oamenable with tra
ial
Haar state then the indu
tion fun
torM⊙A0− is an exa
t fun
tor from
the
ategory of �nitely generated, proje
tive A0-modules to the
ategory
of �nitely generated, proje
tive M-modules.
Proof. Let X and Y be �nitely generated, proje
tive A0-modules and
let f : X → Y be an inje
tive homomorphism. Then
0 −→ X f−→ Y −→ Y/rg(f) −→ 0,
36 DAVID KYED
is a proje
tive resolution of Y/rg(f). Thus TorA01 (M,Y/rg(f)) =
ker(idM ⊗f) and from Theorem 6.1 we
on
lude that
dimM(ker(idM ⊗f)) = 0.
Be
ause idM ⊗f is a map of �nitely generated proje
tive M-modules,
it is not di�
ult to prove that
ker(idM ⊗f) = ker(idM ⊗f)
where ker(idM ⊗f)
is de�ned (see [Lü
98a℄) as the interse
tion of all
kernels arising from homomorphisms fromM⊙A0X toM vanishing on
ker(idM ⊗f). By [Lü
98a, 0.6℄, we
on
lude from this that ker(idM ⊗f)
is �nitely generated and proje
tive. But, sin
e the dimension fun
tion
is faithful on the
ategory of �nitely generated, proje
tive M-modules
this for
es ker(idM ⊗f) = {0} and the
laim follows. �
Corollary 6.6, in parti
ular, implies the following result whi
h was
pointed out to us by A. Thom.
Corollary 6.7. Let G = (A,∆) be
ompa
t and
oamenable with tra
ial
Haar state and let x ∈ A0 be a non-zero element su
h that there exists
a non-zero m ∈ M with mx = 0. Then there exists a non-zero y ∈ A0
with yx = 0.
Proof. This follows by using Corollary 6.6 on the map a 7−→ ax. �
An analogous statement about produ
ts in the opposite order follows
by using the involution in M . So, formulated in ring theoreti
al terms,
we obtain the following: Any regular element in A0 stays regular in the
over-ring M .
7. Examples
A
on
rete example of a non-
ommutative, non-
o
ommutative,
o-
amenable (matrix) quantum group with tra
ial Haar state is the or-
thogonal quantum group Ao(2) ≃ SU−1(2). It follows from [Ban99a,
5.1℄ that Ao(2) is
oamenable. To see that the Haar state is tra
ial, one
observes that the orthogonality property of the
anoni
al fundamental
orepresentation implies that the antipode has period two.
7.1. Examples arising from tensor produ
ts. If G1 = (A1,∆1)
and G2 = (A2,∆2) are
ompa
t quantum groups then the (minimal)
tensor produ
t A = A1 ⊗ A2 may be turned into a quantum group G
by de�ning the
omultipli
ation ∆: A −→ A⊗ A to be
∆(a) = (id⊗σ ⊗ id)(∆1 ⊗∆2)(a),
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 37
where σ denotes the �ip-isomorphism from A1 ⊗ A2 to A2 ⊗ A1. The
Haar state is the tensor produ
t of the two Haar states and the
ounit
is the tensor produ
t of the
ounits. Using these fa
ts, it is not di�
ult
to see [BMT01℄ that if both G1 and G2 are
oamenable and have tra
ial
Haar states, then the same is true for G. See e.g. [KR86, 11.3.2℄.
7.2. Examples arising from bi
rossed produ
ts. Another way to
obtain examples of
ompa
t,
oamenable quantum groups is via bi-
rossed produ
ts. We therefore brie�y sket
h the bi
rossed produ
t
onstru
tion following [VV03℄
losely. In [VV03℄, Vaes and Vainerman
onsider the more general notion of
o
y
le bi
rossed produ
ts, but
sin
e we will mainly be interested in the
ase where the
o
y
les are
trivial we will restri
t our attention to this
ase in the following. The
more general situation will be dis
ussed brie�y in Remark 7.5. The
bi
rossed produ
t
onstru
tion is de�ned using the language of von
Neumann algebrai
quantum groups. We will use this language freely
in the following and refer to [KV03℄ for the ba
kground material.
Let (M1,∆1) and (M2,∆2) be lo
ally
ompa
t (l.
.) von Neumann
algebrai
quantum groups. Let τ : M1⊗̄M2 → M1⊗̄M2 be a faith-
ful ∗-homomorphism and denote by σ : M1⊗̄M2 → M2⊗̄M1 the �ip-
isomorphism. Then τ is
alled a mat
hing from M1 to M2 if the fol-
lowing holds.
• The map α : M2 −→ M1⊗̄M2 given by α(y) = τ(1 ⊗ y) is a
(left)
oa
tion of (M1,∆1) on the von Neumann algebra M2.
• De�ning β : M1 −→M1⊗̄M2 as β(x) = τ(x⊗ 1) the map σβ is
a (left)
oa
tion of (M2,∆2) on the von Neumann algebra M1.
• The
oa
tions satisfy the following two mat
hing
onditions:
τ(13)(α⊗ 1)∆2 = (1⊗∆2)α (M1)
τ(23)σ(23)(β ⊗ 1)∆1 = (∆1 ⊗ 1)β (M2)
Here we use the standard leg numbering
onvention (see e.g. [MVD98℄).
If τ : M1⊗̄M2 → M1⊗̄M2 is a mat
hing from M1 to M2 then it is easy
to see that στσ−1 is a mat
hing fromM2 to M1. We will therefore just
refer to the pair (M1,M2) as a mat
hed pair and to τ as a mat
hing
of the pair. Let (M1,∆1) and (M2,∆2) be su
h a mat
hed pair of
l.
. quantum groups and denote by τ the mat
hing. We denote by Hi
the GNS spa
e ofMi with respe
t to the left invariant weight ϕi and by
Wi and Ŵi the natural multipli
ative unitaries on Hi⊗̄Hi for Mi and
M̂i respe
tively. By H we denote H1⊗̄H2 and by Σ the �ip-unitary on
38 DAVID KYED
H⊗̄H . We may now form two
rossed produ
ts:
M =M1 ⋉α M2 = vNa{α(M2), M̂1 ⊗ 1} ⊆ B(H1⊗̄H2)
M̃ =M2 ⋉σβ M1 = vNa{σβ(M1), M̂2 ⊗ 1} ⊆ B(H2⊗̄H1)
Some of the main results in [VV03℄ are summarized in the following:
Theorem 7.1 ([VV03℄). De�ne operators
Ŵ = (β ⊗ 1⊗ 1)(W1 ⊗ 1)(1⊗ 1⊗ α)(1⊗ Ŵ2)
and W = ΣŴ ∗Σ on H⊗̄H. Then W and Ŵ are multipli
ative uni-
taries and the map ∆: M → B(H⊗̄H) given by ∆(a) = W ∗(1⊗1⊗a)W
de�nes a
omultipli
ation on M turning it into a l.
. quantum group.
Denoting by Σ12 the �ip-unitary fromH1⊗̄H2 to H2⊗̄H1, the dual quan-
tum group M̂ be
omes Σ∗12M̃Σ12 with
omultipli
ation implemented by
Thus, up to a �ip the two
rossed produ
ts above are in duality. In
[DQV02℄, Desmedt, Quaegebeur and Vaes studied (
o)amenability of
bi
rossed produ
ts. Combining their Theorem 15 with [VV03, 2.17℄
we obtain the following: If (M1,M2) is a mat
hed pair with M1 dis-
rete and M2
ompa
t then the bi
rossed produ
t M is
ompa
t, and
M is
oamenable if and only if both M2 and M̂1 are. Here a von Neu-
mann algebrai
ompa
t quantum group is said to be
oamenable if the
orresponding C∗-algebrai
quantum group is. Colle
ting the results
dis
ussed above we obtain the following.
Proposition 7.2. If (M1,M2) is a mat
hed pair of l.
. quantum groups
in whi
h M̂1 and M2 are
ompa
t and
oamenable, then the bi
rossed
produ
t M = M1 ⋉α M2 is
oamenable and
ompa
t. So if the Haar
state on M is tra
ial the quantum group (M,∆) has vanishing L2-Betti
numbers in all positive degrees.
In order to produ
e more
on
rete examples, we will now dis
uss a
spe
ial
ase of the bi
rossed produ
t
onstru
tion in whi
h one of the
oa
tions
omes from an a
tual group a
tion. This part of the theory
is due to De Cannière [DC79℄ and is formulated using the language of
Ka
algebras. We remind the reader that a
ompa
t Ka
algebra is
nothing but a von Neumann algebrai
,
ompa
t quantum group with
tra
ial Haar state. A dis
rete,
ountable group Γ a
ts on a
ompa
t
Ka
algebra (M,∆, S, h) if the group a
ts on the von Neumann algebra
M and the a
tion
ommutes with both the
oprodu
t and the antipode.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 39
Denoting the a
tion by ρ, this means that
∆(ργ(x)) = ργ ⊗ ργ(∆(x)),
S(ργ(x)) = ργ(S(x)),
for all γ ∈ Γ and all x ∈ M . In this situation, the a
tion of Γ on M
indu
es a
oa
tion α : M −→ ℓ∞(Γ)⊗̄M . Denoting by H the Hilbert
spa
e on whi
h M a
ts and identifying ℓ2(Γ)⊗̄H with ℓ2(Γ, H), this
oa
tion is given by the formula
α(x)(ξ)(γ) = ργ−1(x)(ξ(γ)),
for ξ ∈ ℓ2(Γ, H). The
rossed produ
t, whi
h is de�ned as
Γ⋉ρ M = {α(M),L (Γ)⊗ 1}′′,
be
omes again a Ka
algebra [DC79, Thm.1℄. One should note at this
point that De Cannière works with the right
rossed produ
t a
ting
on H⊗̄ℓ2(Γ) where we work with the left
rossed produ
t a
ting on
ℓ2(Γ)⊗̄H . But, one
an
ome from one to the other by
onjugation
with the �ip-unitary and we may therefore freely transport all results
from [DC79℄ to the setting of left
rossed produ
ts. We now prove that
De Cannière's
rossed produ
t
an also be
onsidered as a bi
rossed
produ
t. This is probably well known to experts in the �eld, but we
were unable to �nd an expli
it referen
e.
Proposition 7.3. De�ning τ : ℓ∞(Γ)⊗̄M −→ ℓ∞(Γ)⊗̄M by
τ(δγ ⊗ x) = δγ ⊗ ργ−1(x)
we obtain a mat
hing with the above de�ned α as the
orresponding
oa
tion of ℓ∞(Γ) on M and trivial
oa
tion of (M,∆) on ℓ∞(Γ).
Proof. A dire
t
al
ulation shows that α(x) = τ(1 ⊗ x) and β(f) =
τ(f ⊗ 1) = f ⊗ 1. Therefore the two maps x 7→ τ(1 ⊗ x) and f 7→
στ(f ⊗ 1) are
oa
tions as required. We therefore just have to
he
k
that the mat
hing
onditions are ful�lled. Denote the
oprodu
t on
ℓ∞(Γ) by ∆1 and
hoose f ∈ ℓ∞(Γ) su
h that ∆1(f) ∈ ℓ∞(Γ)⊙ ℓ∞(Γ).
Writing ∆1(f) as f(1) ⊗ f(2) we now get
τ(23)σ(23)(β ⊗ 1)∆1f = τ(23)σ(23)(β ⊗ 1)(f(1) ⊗ f(2))
= τ(23)σ(23)(f(1) ⊗ 1⊗ f(2))
= τ(23)(f(1) ⊗ f(2) ⊗ 1)
= f(1) ⊗ f(2) ⊗ 1
= (∆1 ⊗ 1)β(f),
and hen
e (M2) is satis�ed. An analogous, but slightly more
umber-
some,
al
ulation proves that (M1) is also satis�ed. �
40 DAVID KYED
Thus, as von Neumann algebras, we have ℓ∞(Γ) ⋉α M = Γ ⋉ρ M .
Using the fa
t that β is trivial, one
an prove that the elements λγ ⊗ 1
are group-like and it therefore follows from [DC79, 3.3℄ that also the
omultipli
ations agree. Hen
e the two
rossed produ
t
onstru
tions
are identi
al as l.
. quantum groups. In parti
ular, the the bi
rossed
produ
t ℓ∞(Γ) ⋉α M is a Ka
algebra so if (M,∆) is
ompa
t then
ℓ∞(Γ)⋉α M is also
ompa
t [VV03, 2.7℄ and the Haar state is tra
ial.
We therefore have the following.
Proposition 7.4. If G = (M,∆, S, h) is a
ompa
t,
oamenable Ka
algebra and Γ is a
ountable, dis
rete, amenable group a
ting on G then
the
rossed produ
t Γ⋉M is again a
ompa
t,
oamenable Ka
algebra.
Proof. That Γ⋉M is a Ka
algebra follows from the dis
ussion above
and the
oamenability of the
rossed produ
t follows from [DQV02, 15℄
sin
e ℓ̂∞(Γ) = L (Γ) is
oamenable if (and only if) Γ is amenable. �
Remark 7.5. It is also possibly to
onstru
t examples using the more
general notion of
o
y
le
rossed produ
ts introdu
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David Kyed, Mathematis
hes Institut, Georg-August-Universität,
Göttingen, Bunsenstraÿe 3-5, D-37073 Göttingen, Germany
E-mail address : kyed�uni-math.gwdg.de
URL: www.uni-math.gwdg.de/kyed
www.wis.kuleuven.be/analyse/stefaan
Introduction
1. Preliminaries on compact quantum groups
2. Fusion Algebras
3. Amenability for Fusion Algebras
3.1. Formulas used in the proof of Theorem ??
4. Coamenable Compact Quantum Groups
5. An Interlude
6. A Vanishing Result
7. Examples
7.1. Examples arising from tensor products
7.2. Examples arising from bicrossed products
References
|
0704.1583 | OPserver: interactive online-computations of opacities and radiative
accelerations | Mon. Not. R. Astron. Soc. 000, 1–5 (2006) Printed 11 November 2021 (MN LATEX style file v2.2)
OPserver: interactive online-computations of opacities and
radiative accelerations
C. Mendoza,1,2⋆ M. J. Seaton,3 P. Buerger,4 A. Belloŕın,5
M. Meléndez,6† J. González,1,7 L. S. Rodŕıguez,8 F. Delahaye,9
E. Palacios,7 A. K. Pradhan10 and C. J. Zeippen9
1Centro de F́ısica, Instituto Venezolano de Investigaciones Cient́ıficas (IVIC), PO Box 21827, Caracas 1020A, Venezuela
2Centro Nacional de Cálculo Cient́ıfico Universidad de Los Andes (CeCalCULA), Mérida 5101, Venezuela
3Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
4Ohio Supercomputer Center, Columbus, Ohio 43212, USA
5Escuela de F́ısica, Facultad de Ciencias, Universidad Central de Venezuela, PO Box 20513, Caracas 1020-A, Venezuela
6Departamento de F́ısica, Universidad Simón Boĺıvar, PO Box 89000, Caracas 1080-A, Venezuela
7Escuela de Computación, Facultad de Ciencia y Tecnoloǵıa, Universidad de Carabobo, Valencia, Venezuela
8Centro de Qúımica, Instituto Venezolano de Investigaciones Cient́ıficas (IVIC), P.O. Box 21827, Caracas 1020A, Venezuela
9LUTH, Observatoire de Paris, F-92195 Meudon, France
10Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA
Accepted. Received; in original form
ABSTRACT
Codes to compute mean opacities and radiative accelerations for arbitrary chemical
mixtures using the Opacity Project recently revised data have been restructured in a
client–server architecture and transcribed as a subroutine library. This implementation
increases efficiency in stellar modelling where element stratification due to diffusion
processes is depth dependent, and thus requires repeated fast opacity reestimates.
Three user modes are provided to fit different computing environments, namely a web
browser, a local workstation and a distributed grid.
Key words: atomic processes – radiative transfer – stars: interior.
1 INTRODUCTION
Astrophysical opacities from the Opacity Project (OP) have
been updated by Badnell et al. (2005) to include inner-
shell contributions and an improved frequency mesh. The
complete data set of monochromatic opacities and a suite
of codes to compute mean opacities and radiative accel-
erations (OPCD 2.11) have also been publicly released by
Seaton (2005) to make in-house calculations for arbitrary
mixtures more versatile and expedient. Regarding data ac-
curacy, there is excellent overall agreement between the
OPAL (Iglesias & Rogers 1996) and OP results as dis-
cussed by Seaton & Badnell (2004), Badnell et al. (2005)
and Delahaye & Pinsonneault (2005).
Rosseland mean opacities are sensitive to both the
⋆ E-mail: [email protected].
† Present address: Institute for Astrophysics and Computational
Sciences, Department of Physics, The Catholic University of
America, Washington, DC 20064, and Exploration of the Uni-
verse Division, Code 667, NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA.
1 http://cdsweb.u-strasbg.fr/topbase/op.html
basic atomic data used and the assumed abundances of
the chemical elements. What had been a good agree-
ment between theory and the helioseismological data was
found to be less good using revised solar abundances from
Asplund et al. (2005). The revised OP opacities have been
instrumental in discussions of that problem (Antia & Basu
2005; Bahcall et al. 2005a,b,c; Bahcall & Serenelli 2005;
Delahaye & Pinsonneault 2006).
The modelling of stellar interiors, on the other hand, is
being renewed with the solar experience. Present (WIRE,
MOST, CoRoT) and future (Kepler) space probes and
the well established solar methods are giving the field
of asteroseismology remarkable growth and the guaran-
tee of invaluable data (Metcalfe et al. 2004; Kurtz 2005;
Christensen-Dalsgaard 2006). In future work on stellar mod-
els it may be desirable to take account of revisions in abun-
dances similar to those performed for the Sun.
For some types of stars, models must take into ac-
count microscopic diffusion processes, e.g. radiative levita-
tion, gravitational settling and thermal diffusion, as they can
affect the internal and thermal structures, the depth of the
convection zone, pulsations and give rise to surface abun-
dance anomalies (Seaton 1999; Delahaye & Pinsonneault
c© 2006 RAS
http://arxiv.org/abs/0704.1583v1
http://cdsweb.u-strasbg.fr/topbase/op.html
2 C. Mendoza et al.
2005; Bourge & Alecian 2006). As reviewed by Michaud
(2004), such processes are relevant in the description of
chemically peculiar stars, horizontal-branch stars, white
dwarfs and neutron stars, and in globular cluster age de-
terminations from Population II turnoff stars. Furthermore,
in order to solve the outstanding discrepancy of the at-
mospheric Li abundance in old stars with that predicted
in big-bang nucleosynthesis, Richard et al. (2005) have pro-
posed Li sinking deep into the star due to diffusion. This
hypothesis has been recently confirmed in the observations
by Korn et al. (2006).
The OPCD 2.1 release includes data and codes to com-
pute the radiative accelerations required for studies of dif-
fusion processes. It should be noted that the radiative ac-
celerations are summed over ionization stages and that data
for the calculation of diffusion coefficients are calculated as-
suming that the distribution over ionization stages of the
diffusing ions is the same as that in the ambient plasma. The
validity of this approximation is discussed by Gonzalez et al.
(1995).
In some cases, particularly when element stratification
depends on stellar depth, calculations of mean opacities and
radiative accelerations must be repeated at each depth point
of the model and at each time step of the evolution, and
thus the use of codes more efficient than those in OPCD 2.1
may be necessary. This becomes critical in the new dis-
tributed computing grid environments where the network
transfer of large volumes of data is a key issue. In the present
work we have looked into these problems, and, as a solu-
tion, report on the implementation of a general purpose,
interactive server for astrophysical opacities referred to as
OPserver. It has been installed at the Ohio Supercomputer
Center, Columbus, Ohio, USA, from where it can be ac-
cessed through a web page2 or a linkable subroutine library.
It can also be downloaded locally to be run on a stand-alone
basis but it will demand greater computational facilities. In
Section 2 we discuss the computational strategy of the codes
in OPCD 2.1 followed by a description of OPserver in Sec-
tion 3. In Section 4 we include some tests as an indication
of its performance with a final summary in Section 5.
2 OPCD CODES
We highlight here some of the key features of the codes in
OPCD 2.1. For a chemical mixture specified by abundance
fractions fk, they essentially compute two types of data:
Rosseland mean opacities (RMO) and radiative accelera-
tions (RA).
2.1 Rosseland mean opacities
For the frequency variable
u = hν/kBT (1)
where kB is the Boltzmann constant, RMO are given by
the harmonic mean of the opacity cross section σ(u) of the
mixture
2 http://opacities.osc.edu
dv (2)
where µ is the mean atomic weight. The σ(u) is a weighted
sum of the monochromatic opacity cross sections for each of
the chemical constituents
σ(u) =
fkσk(u) , (3)
and is conveniently tabulated on the v-mesh
v(u) =
F (u)
1− exp(−u)
du (4)
where
F (u) =
15u4 exp(−u)
4π4[1− exp(−u)]2
and v∞ = v(u → ∞). The rationale behind the v-mesh is
that it enhances frequency resolution where F (u) is large
(Badnell et al. 2005).
2.2 Radiative accelerations
Similarly, the RA for a selected k element can be expressed
grad =
µκRγk
F (6)
where µk is its atomic weight and c the speed of light. The
function F is given in terms of the effective temperature Teff
and fractional depth r/R⋆ of the star by
F = πB(Teff)(R⋆/r)
B(T ) =
2(πkBT )
15c2h3
. (8)
The dimensionless parameter
σmtak
dv (9)
depends on the cross section for momentum transfer to the
k element
k = σk(u)[1− exp(−u)]− ak(u) (10)
where ak(u) is a correction to remove the contributions of
electron scattering and momentum transfer to the electrons.
Both σk(u) and ak(u), which are hereafter referred to as the
mono data set (∼1 GB), are tabulated in equally spaced v
intervals to facilitate accurate interpolation schemes.
2.3 Computational strategy
The computational strategy adopted in the OPCD 2.1 re-
lease is depicted in the flowcharts of Figure 1 where it may
be seen that calculations of RMO and RA are carried in
two stages. In a time consuming Stage 1, RMO and RA
are computed with the mixv and accv codes, respectively,
on a representative tabulation of the complete temperature–
electron-density (T,Ne) plane. In mixv the chemical mixture
is specified in the input file mixv.in as
{X,Z,N,Zk, fk} (11)
c© 2006 RAS, MNRAS 000, 1–5
http://opacities.osc.edu
OPserver 3
http://vizier.u-strasbg.fr/topbase/opserver/fig1.eps
Figure 1. Flowcharts for the computations of Rosseland mean opacities (RMO) and radiative accelerations (RA) with the codes in the
OPCD 2.1 release. They are carried out in two stages: in a time consuming Stage 1, data are computed for the whole (T,Ne) plane
followed by fast bicubic interpolations in Stage 2. The intermediate files mixv.xx and acc.xx enable communication between these two
steps.
where X and Z are the hydrogen and metal mass-fractions,
N the number of elements, and Zk and fk are the metal nu-
clear charges and fractional abundances. In accv, the input
data (accv.in) are
{N,Zk, fk, Zi, Nχ, χj} (12)
where now k runs over the N elements of the mixture, and
Zi and χj are respectively the nuclear charge and Nχ abun-
dance multipliers of the test i element. Input data formats
in either mixv.in or accv.in give the user flexible control
over chemical mixture specifications.
As shown in Figure 1, the intermediate output files
mixv.xx (∼85 KB) containing
{ρ, κR}(T,Ne) , (13)
where ρ is the mass-density, and acc.xx (∼470 KB) with
(T,Ne, χj) (14)
are written to disk. They are then respectively read by
the codes opfit and accfit in Stage 2 for fast bicubic
interpolations of RMO and RA on stellar depth profiles
{T, ρ, r/R⋆}(i) specified by the user in the opfit.in and
accfit.in input files. The final output files are opfit.xx
containing
log κR,
∂ log κR
∂ log T
∂ log κR
∂ log ρ
(i) (15)
and accfit.xx with
{log κR, log γ, log grad}(i, χj) . (16)
In this computational approach, performance is mainly
limited by the summation in equation (3) which implies disk
reading the mono data set; for instance, in mixv it takes
up to ∼90% of the total elapsed time. OPCD 2.1 also in-
cludes other codes such as mx and ax which respectively
compute RMO and RA for a star depth profile. The chemi-
cal mixture can be fully varied at each depth point using the
specifications in equations (11–12), the RMO and RA being
obtained in a one-step process using bicubic interpolations
without splines. These methods are thus suitable for cases
with multi-mixture depth profiles (Seaton 2005). Further de-
tails of all the OPCD codes are contained in the reference
manual3.
3 OPSERVER
In OPserver the computational capabilities of the codes in
OPCD 2.1 are greatly enhanced by the following innovative
adaptations.
3 http://opacities.osc.edu/publi/OPCD.pdf
(i) The codes are restructured within a client–server net-
work architecture whereby the time consuming Stage 1 is
performed on a powerful processor while the fast Stage 2 is
moved to the client, e.g. a web server or a user workstation.
In this arrangement performance could be affected by the
client–server transfer of the mixv.xx and acc.xx intermedi-
ate files, but since they are never larger than 0.5 MB, it is
not expected to be a deterrent with present-day bandwidths.
In a local installation where both the client and server reside
on the same machine, communication is managed through
shared buffers in main memory; i.e. the corresponding data
in mixv.xx and acc.xx are not written to disk.
(ii) The codes are transcribed as a subroutine library—
to be referred to hereafter as the OPlibrary—which can be
linked by the user stellar modelling code for recurrent sub-
routine calls that avoid data writing on disk. That is, the in-
put data in the mixv.in, accv.in, opfit.in and accfit.in
files and the output tables in the opfit.xx and accfit.xx
files (see Figure 1) are now handled as subroutine param-
eters while the intermediate mixv.xx and acc.xx files are
passed via shared main-memory buffers. Chemical mixtures
are again specified with the formats of equations (11–12)
which allow full variation at each depth point in a single
subroutine call.
(iii) RMO/RA are computed with the complete mono data
set always loaded in main memory thus avoiding lengthy and
repeated disk readings. This is achieved by implementing
OPserver on a dedicated server where mono is permanently
resident in RAM, or in the case of a local installation, by
disk-reading it once at the outset of a modelling calculation.
(iv) When accessing the remote server, client data re-
quests are addressed through the HTTP protocol, i.e. in
terms of a Uniform Resource Locator (URL). This allows
data fetching from the central facility through an interactive
web page or a network access subroutine, the latter being
particularly suitable for a stellar model code that is to be
run in a distributed grid environment.
(v) The do-loop that computes the summation of equa-
tion (3) has been parallelized in OpenMP which provides
a simple, scalable and portable scheme for shared-memory
platforms.
As shown in Figure 2, the current OPserver enterprise is
implemented as a client–server model at the Ohio Supercom-
puter Center (OSC). The web server communicates with the
supercomputer via a socket interface. Earlier versions were
developed on an SGI Origin2000 server with the PowerFor-
tran parallelizing compiler. The current version runs on a
Linux system with Fortran OpenMP directives. OPserver
offers three user modes with full functionality except when
otherwise indicated in the following description.
Mode A In this mode OPserver is set up locally on a
stand-alone basis (see Figure 2). The facilities of the OSC are
c© 2006 RAS, MNRAS 000, 1–5
http://vizier.u-strasbg.fr/topbase/opserver/fig1.eps
http://opacities.osc.edu/publi/OPCD.pdf
4 C. Mendoza et al.
http://vizier.u-strasbg.fr/topbase/opserver/fig2.eps
Figure 2. OPserver enterprise showing the web-server–supercomputer tandem at the Ohio Supercomputer Center (OSC) and the three
available user modes. (A) The OPlibrary and monochromatic opacities (mono) are downloaded locally and linked to the user modelling
code such that RMO/RA are computed locally. (B) The OPlibrary is downloaded locally and linked to the modelling code but RMO/RA
are computed remotely at the OSC. (C) RMO/RA computations at the OSC through a web client.
not used. A new OPCD release (OPCD 3.34) is downloaded,
followed by (i) installation of both the OPlibrary and the
mono data set and (ii) linking of the OPlibrary to the user
modelling code. Computations of RMO/RA are preceded
by the reading of the complete mono data set from disk and
therefore requires at least 1 GB of RAM.
Mode B In this mode, the OPlibrary is downloaded, in-
stalled and linked to the user code, but Stage 1 is performed
remotely at the OSC (see Figure 2). This option has been
customized for stellar modelling in a distributed grid envi-
ronment that would otherwise imply (i.e. Mode A) the net-
work transfer, installation and disk-reading of the mono data
set at runtime. It is also practical when local computer capa-
bilities (RAM and/or disk space) are limited. The functions
provided by the mx and ax codes have not been implemented.
Mode C In this mode RMO/RA computations at the
OSC are requested through an interactive web page5 which
allows both Stage 1 and Stage 2 to be carried out re-
motely or, alternatively, Stage 2 locally by downloading the
mixv.xx/acc.xx intermediate files (see Figure. 1) with the
browser for further processing with local opfit/accfit ex-
ecutables.
4 TESTS
OPserver benchmarks were initially carried out on an SGI
Origin2000 multiprocessor at the OSC with an earlier re-
lease of OPCD. For the standard S92 mixture (Seaton et al.
1994), the mixv code took up to 140 s to compute the
mixv.xx file, of which 126 s were dedicated to disk-reading
and 14 s to the actual computing of the mean opacities.
OPserver took on average 12.0 ± 0.5 s to compute mixv.xx
which was not written to disk unless requested. In Fig-
ure 3 we show the acceleration obtained on the Origin2000
through parallelization where the calculation of mean opac-
ities is reduced to 2 s with 8 processors. Further significant
acceleration is prevented by data transfer overheads.
On more recent workstations, the local performances of
the codes in OPCD 2.1 and OPserver depend on processor
speed and RAM and cache sizes. For instance, on a Pow-
erMac G5 (PowerPC 970fx processor at 2.0 GHz, 1GB of
RAM and L2 cache of 512 KB) the first time mixv is run it
takes for a single S92 mixture 103.8 s to compute the RMO,
but on subsequent runs the elapsed time is reduced to an av-
erage of 28.2±0.2 s. Similarly, OPserver takes 103.3 s which
is then reduced to 31.4 ± 0.4 s on subsequent runs. Once
the mono data set is loaded in RAM by OPserver (Mode
A), calculations of RMO for a single S92 mixture only take
5.29 ± 0.02 s and 6.13 ± 0.01 s for RA for the test element
Ar. In Mode B, where Stage 1 is carried out remotely at the
OSC and the mixv.xx and accv.xx files are transferred at
4 http://cdsweb.u-strasbg.fr/topbase/op.html
5 http://opacities.osc.edu
the relatively low rate of 1.88 KB/s, computations of RMO
and RA take 5.9±0.1 s and 9.3±0.3 s, respectively. The no-
ticeable longer time taken for the latter is due to the transfer
time taken for the larger accv.xx file.
5 SUMMARY
Rosseland mean opacities and radiative accelerations can be
computed from OP data in any one of the following ways.
(i) Download the original OPCD 2.1 package as described
by Seaton (2005) and perform all calculations locally.
(ii) Mode A, download the upgraded OPCD 3.3 package, in-
stall OPserver and perform all calculations locally by link-
ing the subroutines in the OPlibrary. Calculations with
OPserver are more efficient but require large local computer
memory.
(iii) Mode B, as Mode A but with Stage 1 performed re-
motely at the OSC. Mode B is convenient if fast calculations
are required but local computer memory is limited or when
stellar modelling is to be carried out in a grid environment.
(iv) Mode C, perform all calculations remotely at the OSC
through an interactive web page whereby files are down-
loaded locally with the browser.
ACKNOWLEDGMENTS
We acknowledge the invaluable assistance of Juan Luis
Chaves and Gilberto Dı́az of CeCalCULA during the ini-
tial stages of OPserver. We are also much indebted to the
Ohio Supercomputer Center, Columbus, Ohio, USA, for
hosting OPserver and for technical assistance; to the Centre
de Données astronomiques de Strasbourg, France, for host-
ing the OPCD releases; and to Drs Josslen Aray, Manuel
Bautista, Juan Murgich and Fernando Ruette of IVIC for
allowing us to test the OPserver installation on different
platforms. FD would like to thank S. Rouchy for techni-
cal support. AKP and FD have been partly supported by a
grant from the US National Science Foundation.
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This paper has been typeset from a TEX/ L
ATEX file prepared
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Introduction
OPCD codes
Rosseland mean opacities
Radiative accelerations
Computational strategy
OPserver
Tests
Summary
|
0704.1584 | Can One Estimate The Unconditional Distribution of Post-Model-Selection
Estimators? | arXiv:0704.1584v1 [math.ST] 12 Apr 2007
Can One Estimate The Unconditional Distribution
of Post-Model-Selection Estimators?
Hannes Leeb
Department of Statistics, Yale University
Benedikt M. Pötscher
Department of Statistics, University of Vienna
First version: April 2005
Revised version: February 2007
Abstract
We consider the problem of estimating the unconditional distribution of a post-model-selection esti-
mator. The notion of a post-model-selection estimator here refers to the combined procedure resulting
from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing
procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum
likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional
distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator
for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (lo-
cal) minimax lower bounds on the performance of estimators for the distribution; performance is here
measured by the probability that the estimation error exceeds a given threshold. These lower bounds
are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar
impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the
post-model-selection estimator.
AMS Mathematics Subject Classification 2000: 62F10, 62F12, 62J05, 62J07, 62C05.
Keywords: Inference after model selection, Post-model-selection estimator, Pre-test estimator, Selection of re-
gressors, Akaike’s information criterion AIC, Thresholding, Model uncertainty, Consistency, Uniform consistency,
Lower risk bound.
Research of the first author was supported by the Max Kade Foundation and by the Austrian National Science
Foundation (FWF), Grant No. P13868-MAT. A preliminary draft of the material in this paper was already written
in 1999.
1 Introduction and Overview
In many statistical applications a data-based model selection step precedes the final parameter estimation and
inference stage. For example, the specification of the model (choice of functional form, choice of regressors,
http://arxiv.org/abs/0704.1584v1
number of lags, etc.) is often based on the data. In contrast, the traditional theory of statistical inference
is concerned with the properties of estimators and inference procedures under the central assumption of
an a priori given model. That is, it is assumed that the model is known to the researcher prior to the
statistical analysis, except for the value of the true parameter vector. As a consequence, the actual statistical
properties of estimators or inference procedures following a data-driven model selection step are not described
by the traditional theory which assumes an a priori given model; in fact, they may differ substantially
from the properties predicted by this theory, cf., e.g., Danilov and Magnus (2004), Dijkstra and Veldkamp
(1988), Pötscher (1991, Section 3.3), or Rao and Wu (2001, Section 12). Ignoring the additional uncertainty
originating from the data-driven model selection step and (inappropriately) applying traditional theory can
hence result in very misleading conclusions.
Investigations into the distributional properties of post-model-selection estimators, i.e., of estimators
constructed after a data-driven model selection step, are relatively few and of recent vintage. Sen (1979)
obtained the unconditional large-sample limit distribution of a post-model-selection estimator in an i.i.d.
maximum likelihood framework, when selection is between two competing nested models. In Pötscher (1991)
the asymptotic properties of a class of post-model-selection estimators (based on a sequence of hypothesis
tests) were studied in a rather general setting covering non-linear models, dependent processes, and more
than two competing models. In that paper, the large-sample limit distribution of the post-model-selection
estimator was derived, both unconditional as well as conditional on having chosen a correct model, not
necessarily the minimal one. See also Pötscher and Novak (1998) for further discussion and a simulation
study, and Nickl (2003) for extensions. The finite-sample distribution of a post-model-selection estimator,
both unconditional and conditional on having chosen a particular (possibly incorrect) model, was derived
in Leeb and Pötscher (2003) in a normal linear regression framework; this paper also studied asymptotic
approximations that are in a certain sense superior to the asymptotic distribution derived in Pötscher (1991).
The distributions of corresponding linear predictors constructed after model selection were studied in Leeb
(2005, 2006). Related work can also be found in Sen and Saleh (1987), Kabaila (1995), Pötscher (1995),
Ahmed and Basu (2000), Kapetanios (2001), Hjort and Claeskens (2003), Dukić and Peña (2005), and Leeb
and Pötscher (2005a). The latter paper provides a simple exposition of the problems of inference post model
selection and may serve as an entry point to the present paper.
It transpires from the papers mentioned above that the finite-sample distributions (as well as the large-
sample limit distributions) of post-model-selection estimators typically depend on the unknown model pa-
rameters, often in a complicated fashion. For inference purposes, e.g., for the construction of confidence
sets, estimators of these distributions would be desirable. Consistent estimators of these distributions can
typically be constructed quite easily, e.g., by suitably replacing unknown parameters in the large-sample
limit distributions by estimators; cf. Section 2.2.1. However, the merits of such ‘plug-in’ estimators in small
samples are questionable: It is known that the convergence of the finite-sample distributions to their large-
sample limits is typically not uniform with respect to the underlying parameters (see Appendix B below and
Corollary 5.5 in Leeb and Pötscher (2003)), and there is no reason to believe that this non-uniformity will
disappear when unknown parameters in the large-sample limit are replaced by estimators. This observation
is the main motivation for the present paper to investigate in general the performance of estimators of the
distribution of a post-model-selection estimator, where the estimators of the distribution are not necessar-
ily ‘plug-in’ estimators based on the limiting distribution. In particular, we ask whether estimators of the
distribution function of post-model-selection estimators exist that do not suffer from the non-uniformity
phenomenon mentioned above. As we show in this paper the answer in general is ‘No’. We also show that
these negative results extend to the problem of estimating the distribution of linear functions (e.g., linear
predictors) of post-model-selection estimators. Similar negative results apply also to the estimation of the
mean squared error or bias of post-model-selection estimators; cf. Remark 4.7.
To fix ideas consider for the moment the linear regression model
Y = V χ+Wψ + u (1)
where V and W , respectively, represent n× k and n× l non-stochastic regressor matrices (k ≥ 1, l ≥ 1), and
the n× 1 disturbance vector u is normally distributed with mean zero and variance-covariance matrix σ2In.
We also assume for the moment that (V : W )′(V : W )/n converges to a non-singular matrix as the sample
size n goes to infinity and that limn→∞ V
′W/n 6= 0 (for a discussion of the case where this limit is zero see
Example 1 in Section 2.2.2). Now suppose that the vector χ represents the parameters of interest, while
the parameter vector ψ and the associated regressors in W have been entered into the model only to avoid
possible misspecification. Suppose further that the necessity to include ψ or some of its components is then
checked on the basis of the data, i.e., a model selection procedure is used to determine which components
of ψ are to be retained in the model, the inclusion of χ not being disputed. The selected model is then used
to obtain the final (post-model-selection) estimator χ̃ for χ. We are now interested in the unconditional
finite-sample distribution of χ̃ (appropriately scaled and centered). Denote this k-dimensional cumulative
distribution function (cdf) by Gn,θ,σ(t). As indicated in the notation, this distribution function depends on
the true parameters θ = (χ′, ψ′)′ and σ. For the sake of definiteness of discussion assume for the moment
that the model selection procedure used here is the particular ‘general-to-specific’ procedure described at
the beginning of Section 2; we comment on other model selection procedures, including Akaike’s AIC and
thresholding procedures, below.
As mentioned above, it is not difficult to construct a consistent estimator of Gn,θ,σ(t) for any t, i.e., an
estimator Ĝn(t) satisfying
Pn,θ,σ
(∣∣∣Ĝn(t)−Gn,θ,σ(t)
∣∣∣ > δ
n→∞−→ 0 (2)
for each δ > 0 and each θ, σ; see Section 2.2.1. However, it follows from the results in Section 2.2.2 that any
estimator satisfying (2), i.e., any consistent estimator of Gn,θ,σ(t), necessarily also satisfies
||θ||<R
Pn,θ,σ
(∣∣∣Ĝn(t)−Gn,θ,σ(t)
∣∣∣ > δ
n→∞−→ 1 (3)
for suitable positive constants R and δ that do not depend on the estimator. That is, while the probability
in (2) converges to zero for every given θ by consistency, relation (3) shows that it does not do so uniformly
in θ. It follows that Ĝn(t) can never be uniformly consistent (not even when restricting consideration to
uniform consistency over all compact subsets of the parameter space). Hence, a large sample size does not
guarantee a small estimation error with high probability when estimating the distribution function of a post-
model-selection estimator. In this sense, reliably assessing the precision of post-model-selection estimators
is an intrinsically hard problem. Apart from (3), we also provide minimax lower bounds for arbitrary (not
necessarily consistent) estimators of the conditional distribution function Gn,θ,σ(t). For example, we provide
results that imply that
lim inf
Ĝn(t)
||θ||<R
Pn,θ,σ
(∣∣∣Ĝn(t)−Gn,θ,σ(t)
∣∣∣ > δ
> 0 (4)
holds for suitable positive constants R and δ, where the infimum extends over all estimators of Gn,θ,σ(t).
The results in Section 2.2.2 in fact show that the balls ||θ|| < R in (3) and (4) can be replaced by suitable
balls (not necessarily centered at the origin) shrinking at the rate n−1/2. This shows that the non-uniformity
phenomenon described in (3)-(4) is a local, rather than a global, phenomenon. In Section 2.2.2 we further
show that the non-uniformity phenomenon expressed in (3) and (4) typically also arises when the parameter
of interest is not χ, but some other linear transformation of θ = (χ′, ψ′)′. As discussed in Remark 4.3, the
results also hold for randomized estimators of the unconditional distribution function Gn,θ,σ(t). Hence no
resampling procedure whatsoever can alleviate the problem. This explains the anecdotal evidence in the
literature that resampling methods are often unsuccessful in approximating distributional properties of post-
model-selection estimators (e.g., Dijkstra and Veldkamp (1988), or Freedman, Navidi, and Peters (1988)).
See also the discussion on resampling in Section 6.
The results outlined above are presented in Section 2.2 for the particular ‘general-to-specific’ model
selection procedure described at the beginning of Section 2. Analogous results for a large class of model
selection procedures, including Akaike’s AIC and thresholding procedures, are then given in Section 3, based
on the results in Section 2.2. In fact, the non-uniformity phenomenon expressed in (3)-(4) is not specific to
the model selection procedures discussed in Sections 2 and 3 of the present paper, but will occur for most
(if not all) model selection procedures, including consistent ones; cf. Sections 5 and 6 for more discussion.
Section 5 also shows that the results are – as is to be expected – by no means limited to the linear regression
model.
We focus on the unconditional distributions of post-model-selection estimators in the present paper. One
can, however, also envisage a situation where one is more interested in the conditional distribution given
the outcome of the model selection procedure. In line with the literature on conditional inference (see,
e.g., Robinson (1979) or Lehmann and Casella (1998, p. 421)), one may argue that, given the outcome
of the model selection step, the relevant object of interest is the conditional rather than the unconditional
distribution of the post-model-selection estimator. In this case similar results can be obtained and are
reported in Leeb and Pötscher (2006b). We note that on a technical level the results in Leeb and Pötscher
(2006b) and in the present paper require separate treatment.
The plan of the paper is as follows: Post-model-selection estimators based on a ‘general-to-specific’ model
selection procedure are the subject of Section 2. After introducing the basic framework and some notation,
like the family of models Mp from which the ‘general-to-specific’ model selection procedure p̂ selects, as
well as the post-model-selection estimator θ̃, the unconditional cdf Gn,θ,σ(t) of (a linear function of) the
post-model-selection estimator θ̃ is discussed in Section 2.1. Consistent estimators of Gn,θ,σ(t) are given in
Section 2.2.1. The main results of the paper are contained in Section 2.2.2 and Section 3: In Section 2.2.2
we provide a detailed analysis of the non-uniformity phenomenon encountered in (3)-(4). In Section 3 the
‘impossibility’ result from Section 2.2.2 is extended to a large class of model selection procedures including
Akaike’s AIC and to selection from a non-nested collection of models. Some remarks are collected in Section
4, while Section 5 discusses extensions and the scope of the results of the paper. Conclusions are drawn
in Section 6. All proofs as well as some auxiliary results are collected into appendices. Finally a word on
notation: The Euclidean norm is denoted by ‖·‖, and λmax(E) denotes the largest eigenvalue of a symmetric
matrix E. A prime denotes transposition of a matrix. For vectors x and y the relation x ≤ y (x < y,
respectively) denotes xi ≤ yi (xi < yi, respectively) for all i. As usual, Φ denotes the standard normal
distribution function.
2 Results for Post-Model-Selection Estimators Based on a
‘General-to-Specific’ Model Selection Procedure
Consider the linear regression model
Y = Xθ + u, (5)
where X is a non-stochastic n× P matrix with rank(X) = P and u ∼ N(0, σ2In), σ2 > 0. Here n denotes
the sample size and we assume n > P ≥ 1. In addition, we assume that Q = limn→∞X ′X/n exists and is
non-singular. In this section we shall – similar as in Pötscher (1991) – consider model selection from the
collection of nested models MO ⊆ MO+1 ⊆ · · · ⊆ MP , where O is specified by the user, and where for
0 ≤ p ≤ P the model Mp is given by
(θ1, . . . , θP )
′ ∈ RP : θp+1 = · · · = θP = 0
[In Section 3 below also general non-nested families of models will be considered.] Clearly, the model Mp
corresponds to the situation where only the first p regressors in (5) are included. For the most parsimonious
model under consideration, i.e., forMO, we assume that O satisfies 0 ≤ O < P ; if O > 0, this model contains
as free parameters only those components of the parameter vector θ that are not subject to model selection.
[In the notation used in connection with (1) we then have χ = (θ1, . . . , θO)
′ and ψ = (θO+1, . . . , θP )
Furthermore, note that M0 = {(0, . . . , 0)′} and that MP = RP . We call Mp the regression model of order p.
The following notation will prove useful. For matrices B and C of the same row-dimension, the column-
wise concatenation of B and C is denoted by (B : C). If D is an m× P matrix, let D[p] denote the m× p
matrix consisting of the first p columns of D. Similarly, let D[¬p] denote the m× (P − p) matrix consisting
of the last P −p columns of D. If x is a P × 1 vector, we write in abuse of notation x[p] and x[¬p] for (x′[p])′
and (x′[¬p])′, respectively. [We shall use the above notation also in the ‘boundary’ cases p = 0 and p = P .
It will always be clear from the context how expressions containing symbols like D[0], D[¬P ], x[0], or x[¬P ]
are to be interpreted.] As usual, the i-th component of a vector x is denoted by xi, and the entry in the i-th
row and j-th column of a matrix B is denoted by Bi,j .
The restricted least-squares estimator of θ under the restriction θ[¬p] = 0, i.e., under θp+1 = · · · = θP = 0,
will be denoted by θ̃(p), 0 ≤ p ≤ P (in case p = P the restriction being void). Note that θ̃(p) is given by the
P × 1 vector
θ̃(p) =
(X [p]
′X [p])
X [p]′Y
(0, . . . , 0)′
where the expressions θ̃(0) and θ̃(P ), respectively, are to be interpreted as the zero-vector in RP and as the
unrestricted least-squares estimator of θ. Given a parameter vector θ in RP , the order of θ (relative to the
nested sequence of models Mp) is defined as
p0(θ) = min {p : 0 ≤ p ≤ P, θ ∈Mp} .
Hence, if θ is the true parameter vector, a model Mp is a correct model if and only if p ≥ p0(θ). We stress
that p0(θ) is a property of a single parameter, and hence needs to be distinguished from the notion of the
order of the model Mp introduced earlier, which is a property of the set of parameters Mp.
A model selection procedure is now nothing else than a data-driven (measurable) rule p̂ that selects a
value from {O, . . . , P} and thus selects a model from the list of candidate modelsMO, . . . ,MP . In this section
we shall consider as an important leading case a ‘general-to-specific’ model selection procedure based on a
sequence of hypothesis tests. [Results for a larger class of model selection procedures, including Akaike’s AIC,
are provided in Section 3.] This procedure is given as follows: The sequence of hypotheses H
0 : p0(θ) < p is
tested against the alternatives H
1 : p0(θ) = p in decreasing order starting at p = P . If, for some p > O, H
is the first hypothesis in the process that is rejected, we set p̂ = p. If no rejection occurs until even HO+10
is not rejected, we set p̂ = O. Each hypothesis in this sequence is tested by a kind of t-test where the error
variance is always estimated from the overall model (but see the discussion following Theorem 3.1 in Section
3 below for other choices of estimators of the error variance). More formally, we have
p̂ = max {p : |Tp| ≥ cp, 0 ≤ p ≤ P} , (6)
with cO = 0 in order to ensure a well-defined p̂ in the range {O,O + 1, . . . , P}. For O < p ≤ P , the
critical values cp satisfy 0 < cp < ∞ and are independent of sample size (but see also Remark 4.2). The
test-statistics are given by
nθ̃p(p)
σ̂ξn,p
(0 < p ≤ P )
with the convention that T0 = 0. Furthermore,
ξn,p =
X [p]′X [p]
(0 < p ≤ P )
denotes the nonnegative square root of the p-th diagonal element of the matrix indicated, and σ̂2 is given by
σ̂2 = (n− P )−1(Y −Xθ̃(P ))′(Y −Xθ̃(P )).
Note that under the hypothesis H
0 the statistic Tp is t-distributed with n − P degrees of freedom for 0 <
p ≤ P . It is also easy to see that the so-defined model selection procedure p̂ is conservative: The probability
of selecting an incorrect model, i.e., the probability of the event {p̂ < p0(θ)}, converges to zero as the sample
size increases. In contrast, the probability of the event {p̂ = p}, for p satisfying max{p0(θ),O} ≤ p ≤ P ,
converges to a positive limit; cf., for example, Proposition 5.4 and equation (5.6) in Leeb (2006).
The post-model-selection estimator θ̃ can now be defined as follows: On the event p̂ = p, θ̃ is given by
the restricted least-squares estimator θ̃(p), i.e.,
θ̃(p)1(p̂ = p), (7)
where 1(·) denotes the indicator function of the event shown in the argument.
2.1 The Distribution of the Post-Model-Selection Estimator
We now introduce the distribution function of a linear transformation of θ̃ and summarize some of its
properties that will be needed in the subsequent development. To this end, let A be a non-stochastic k × P
matrix of rank k, 1 ≤ k ≤ P , and consider the cdf
Gn,θ,σ(t) = Pn,θ,σ
nA(θ̃ − θ) ≤ t
(t ∈ Rk). (8)
Here Pn,θ,σ(·) denotes the probability measure corresponding to a sample of size n from (5).
Depending on the choice of the matrix A, several important scenarios are covered by (8): The cdf of
n(θ̃ − θ) is obtained by setting A equal to the P × P identity matrix IP . In case O > 0, the cdf of those
components of
n(θ̃− θ) which correspond to the parameter of interest χ in (1) can be studied by setting A
to the O×P matrix (IO : 0) as we then have Aθ = (θ1, . . . , θO)′ = χ. Finally, if A 6= 0 is an 1×P vector, we
obtain the distribution of a linear predictor based on the post-model-selection estimator. See the examples
at the end of Section 2.2.2 for more discussion.
The cdf Gn,θ,σ and its properties have been analyzed in detail in Leeb and Pötscher (2003) and Leeb
(2006). To be able to access these results we need some further notation. Note that on the event p̂ = p the
expression A(θ̃ − θ) equals A(θ̃(p) − θ) in view of (7). The expected value of the restricted least-squares
estimator θ̃(p) will be denoted by ηn(p) and is given by the P × 1 vector
ηn(p) =
θ[p] + (X [p]
′X [p])−1X [p]′X [¬p]θ[¬p]
(0, . . . , 0)′
(9)
with the conventions that ηn(0) = (0, . . . , 0)
′ ∈ RP and that ηn(P ) = θ. Furthermore, let Φn,p denote the
cdf of
nA(θ̃(p) − ηn(p)), i.e., the cdf of
nA times the restricted least-squares estimator based on model
Mp centered at its mean. Hence, Φn,p is the cdf of a k-variate Gaussian random vector with mean zero and
variance-covariance matrix σ2A[p](X [p]′X [p]/n)−1A[p]′ in case p > 0, and it is the cdf of point-mass at zero
in Rk in case p = 0. If p > 0 and if the matrix A[p] has full row rank k, then Φn,p has a density with respect
to Lebesgue measure, and we shall denote this density by φn,p. We note that ηn(p) depends on θ and that
Φn,p depends on σ (in case p > 0), although these dependencies are not shown explicitly in the notation.
For p > 0 we introduce
bn,p = C
n (A[p](X [p]
′X [p]/n)−1A[p]′)−, (10)
ζ2n,p = ξ
n,p − C(p)
n (A[p](X [p]
′X [p]/n)−1A[p]′)−C(p)n , (11)
with ζn,p ≥ 0. Here C
n = A[p](X [p]
′X [p]/n)−1ep, where ep denotes the p-th standard basis vector in R
and B− denotes a generalized inverse of a matrix B. [Observe that ζ2n,p is invariant under the choice of the
generalized inverse. The same is not necessarily true for bn,p, but is true for bn,pz for all z in the column-
space of A[p]. Also note that (12) below depends on bn,p only through bn,pz with z in the column-space of
A[p].] We observe that the vector of covariances between Aθ̃(p) and θ̃p(p) is precisely given by σ
2n−1C
(and hence does not depend on θ). Furthermore, observe that Aθ̃(p) and θ̃p(p) are uncorrelated if and only
if ζ2n,p = ξ
n,p if and only if bn,pz = 0 for all z in the column-space of A[p]; cf. Lemma A.2 in Leeb (2005).
Finally, for a univariate Gaussian random variable N with zero mean and variance s2, s ≥ 0, we write
∆s(a, b) for P(|N− a| < b), a ∈ R∪{−∞,∞}, b ∈ R. Note that ∆s(·, ·) is symmetric around zero in its first
argument, and that ∆s(−∞, b) = ∆s(∞, b) = 0 holds. In case s = 0, N is to be interpreted as being equal
to zero, hence a 7→ ∆0(a, b) reduces to the indicator function of the interval (−b, b).
We are now in a position to present the explicit formula for Gn,θ,σ(t) derived in Leeb (2006):
Gn,θ,σ(t) = Φn,O(t−
nA(ηn(O)− θ))
q=O+1
∆σξn,q (
nηn,q(q), scqσξn,q)h(s)ds
p=O+1
nA(ηn(p)−θ)
[ ∫ ∞
(1−∆σζn,p(
nηn,p(p) + bn,pz, scpσξn,p)) (12)
q=p+1
∆σξn,q(
nηn,q(q), scqσξn,q)h(s)ds
Φn,p(dz).
In the above display, Φn,p(dz) denotes integration with respect to the measure induced by the normal cdf
Φn,p on R
k and h denotes the density of σ̂/σ, i.e., h is the density of (n−P )−1/2 times the square-root of a
chi-square distributed random variable with n−P degrees of freedom. The finite-sample distribution of the
post-model-selection estimator given in (12) is in general not normal, e.g., it can be bimodal; see Figure 2 in
Leeb and Pötscher (2005a) or Figure 1 in Leeb (2006). [An exception where (12) is normal is the somewhat
trivial case where C
n = 0, i.e., where Aθ̃(p) and θ̃p(p) are uncorrelated, for p = O + 1, . . . , P ; see Leeb
(2006, Section 3.3) for more discussion.] We note for later use that Gn,θ,σ(t) =
p=O Gn,θ,σ(t|p)πn,θ,σ(p)
where Gn,θ,σ(t|p) represents the cdf of
nA(θ̃ − θ) conditional on the event {p̂ = p} and where πn,θ,σ(p) is
the probability of this event under Pn,θ,σ. Note that πn,θ,σ(p) is always positive for O ≤ p ≤ P ; cf. Leeb
(2006), Section 3.2.
To describe the large-sample limit of Gn,θ,σ, some further notation is necessary. For p satisfying 0 < p ≤
P , partition the matrix Q = limn→∞X
′X/n as
Q[p : p] Q[p : ¬p]
Q[¬p : p] Q[¬p : ¬p]
where Q[p : p] is a p× p matrix. Let Φ∞,p be the cdf of a k-variate Gaussian random vector with mean zero
and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′, 0 < p ≤ P , and let Φ∞,0 denote the cdf of point-mass
at zero in Rk. Note that Φ∞,p has a Lebesgue density if p > 0 and the matrix A[p] has full row rank k; in
this case, we denote the Lebesgue density of Φ∞,p by φ∞,p. Finally, for p = 1, . . . , P , define
ξ2∞,p = (Q[p : p]
−1)p,p,
ζ2∞,p = ξ
∞,p − C(p)′∞ (A[p]Q[p : p]−1A[p]′)−C(p)∞ , (13)
b∞,p = C
∞ (A[p]Q[p : p]
−1A[p]′)−,
where C
∞ = A[p]Q[p : p]
−1ep, with ep denoting the p-th standard basis vector in R
p; furthermore, take ζ∞,p
and ξ∞,p as the nonnegative square roots of ζ
∞,p and ξ
∞,p, respectively. As the notation suggests, Φ∞,p is
the large-sample limit of Φn,p, and C
∞ , ξ
∞,p, and ζ
∞,p are the limits of C
n , ξ
n,p, and ζ
n,p, respectively;
moreover, bn,pz converges to b∞,pz for each z in the column-space of A[p]. See Lemma A.2 in Leeb (2005).
The next result describes the large-sample limit of the cdf under local alternatives to θ and is taken from
Leeb (2006, Corollary 5.6). Recall that the total variation distance between two cdfs G and G∗ on Rk is
defined as ||G−G∗||TV = supE |G(E)−G∗(E)|, where the supremum is taken over all Borel sets E. Clearly,
the relation |G(t)−G∗(t)| ≤ ||G−G∗||TV holds for all t ∈ Rk. Thus, if G and G∗ are close with respect to
the total variation distance, then G(t) is close to G∗(t), uniformly in t.
Proposition 2.1 Suppose θ ∈ RP and γ ∈ RP and let σ(n) be a sequence of positive real numbers which
converges to a (finite) limit σ > 0 as n → ∞. Then the cdf Gn,θ+γ/√n,σ(n) converges to a limit G∞,θ,σ,γ in
total variation, i.e.,
∣∣∣∣Gn,θ+γ/√n,σ(n) −G∞,θ,σ,γ
n→∞−→ 0. (14)
The large-sample limit cdf G∞,θ,σ,γ(t) is given by
Φ∞,p∗(t− β
(p∗))
q=p∗+1
∆σξ∞,q (νq, cqσξ∞,q)
p=p∗+1
z≤t−β(p)
(1−∆σζ∞,p(νp + b∞,pz, cpσξ∞,p))Φ∞,p(dz)
q=p+1
∆σξ∞,q (νq, cqσξ∞,q) (15)
where p∗ = max{p0(θ),O}. Here for 0 ≤ p ≤ P
Q[p : p]
−1Q[p : ¬p]γ[¬p]
−γ[¬p]
with the convention that β(p) = −Aγ if p = 0 and that β(p) = (0, . . . , 0)′ if p = P . Furthermore, we have
set νp = γp + (Q[p : p]
−1Q[p : ¬p]γ[¬p])p for p > 0. [Note that β(p) = limn→∞
nA(ηn(p) − θ − γ/
for p ≥ p0(θ), and that νp = limn→∞
nηn,p(p) for p > p0(θ). Here ηn(p) is defined as in (9), but with
θ + γ/
n replacing θ.]
If p∗ > 0 and if the matrix A[p∗] has full row rank k, then the Lebesgue density φ∞,p of Φ∞,p exists for
all p ≥ p∗ and hence the density of (15) exists and is given by
φ∞,p∗(t− β
(p∗))
q=p∗+1
∆σξ∞,q (νq, cqσξ∞,q)
p=p∗+1
(1−∆σζ∞,p(νp + b∞,p(t− β
(p)), cpσξ∞,p))φ∞,p(t− β
q=p+1
∆σξ∞,q(νq, cqσξ∞,q).
Like the finite-sample distribution, the limiting distribution of the post-model-selection estimator given
in (15) is in general not normal. An exception is the case where C
∞ = 0 for p > p∗ in which case G∞,θ,σ,γ
reduces to Φ∞,P ; see Remark A.6 in Appendix A. If γ = 0, we write G∞,θ,σ(t) as shorthand for G∞,θ,σ,0(t)
in the following.
2.2 Estimators of the Finite-Sample Distribution
For the purpose of inference after model selection the finite-sample distribution of the post-model-selection-
estimator is an object of particular interest. As we have seen, it depends on unknown parameters in a
complicated manner, and hence one will have to be satisfied with estimators of this cdf. As we shall see,
it is not difficult to construct consistent estimators of Gn,θ,σ(t). However, despite this consistency result,
we shall find in Section 2.2.2 that any estimator of Gn,θ,σ(t) typically performs unsatisfactory, in that the
estimation error can not become small uniformly over (subsets of) the parameter space even as sample size
goes to infinity. In particular, no uniformly consistent estimators exist, not even locally.
2.2.1 Consistent Estimators
We construct a consistent estimator of Gn,θ,σ(t) by commencing from the asymptotic distribution. Spe-
cializing to the case γ = 0 and σ(n) = σ in Proposition 2.1, the large-sample limit of Gn,θ,σ(t) is given
G∞,θ,σ(t) = Φ∞,p∗(t)
q=p∗+1
∆σξ∞,q(0, cqσξ∞,q)
p=p∗+1
(1 −∆σζ∞,p(b∞,pz, cpσξ∞,p))Φ∞,p(dz)
q=p+1
∆σξ∞,q(0, cqσξ∞,q) (16)
with p∗ = max{p0(θ),O}. Note that G∞,θ,σ(t) depends on θ only through p∗. Let Φ̂n,p denote the cdf of a k-
variate Gaussian random vector with mean zero and variance-covariance matrix σ̂2A[p](X [p]′X [p]/n)−1A[p]′,
0 < p ≤ P ; we also adopt the convention that Φ̂n,0 denotes the cdf of point-mass at zero in Rk. [We use the
same convention for Φ̂n,p in case σ̂ = 0, which is a probability zero event.] An estimator Ǧn(t) of Gn,θ,σ(t)
is now defined as follows: We first employ an auxiliary procedure p̄ that consistently estimates p0(θ) (e.g.,
p̄ could be obtained from BIC or from a ‘general-to-specific’ hypothesis testing procedure employing critical
values that go to infinity but are o(n1/2) as n → ∞). The estimator Ǧn(t) is now given by the expression
in (16) but with p∗, σ, b∞,p, ζ∞,p, ξ∞,p, and Φ∞,p replaced by max{p̄,O}, σ̂, bn,p, ζn,p, ξn,p, and Φ̂n,p,
respectively. A little reflection shows that Ǧn is again a cdf. We have the following consistency results.
Proposition 2.2 The estimator Ǧn is consistent (in the total variation distance) for Gn,θ,σ and G∞,θ,σ.
That is, for every δ > 0
Pn,θ,σ
(∣∣∣∣Ǧn(·)−Gn,θ,σ(·)
) n→∞−→ 0, (17)
Pn,θ,σ
(∣∣∣∣Ǧn(·)−G∞,θ,σ(·)
) n→∞−→ 0 (18)
for all θ ∈ RP and all σ > 0.
While the estimator constructed above on the basis of the formula for G∞,θ,σ is consistent, it can be
expected to perform poorly in finite samples since convergence of Gn,θ,σ to G∞,θ,σ is typically not uniform
in θ (cf. Appendix B), and since in case the true θ is ‘close’ to Mp0(θ)−1 the auxiliary decision procedure p̄
(although being consistent for p0(θ)) will then have difficulties making the correct decision in finite samples.
In the next section we show that this poor performance is not particular to the estimator Ǧn constructed
above, but is a genuine feature of the estimation problem under consideration.
2.2.2 Performance Limits and Impossibility Results
We now provide lower bounds for the performance of estimators of the cdf Gn,θ,σ(t) of the post-model-
selection estimator Aθ̃; that is, we give lower bounds on the worst-case probability that the estimation
error exceeds a certain threshold. These lower bounds are large, being 1 or 1/2, depending on the situation
considered; furthermore, they remain lower bounds even if one restricts attention only to certain subsets of
the parameter space that shrink at the rate n−1/2. In this sense the ‘impossibility’ results are of a local
nature. In particular, the lower bounds imply that no uniformly consistent estimator of the cdf Gn,θ,σ(t)
exists, not even locally.
In the following, the asymptotic ‘correlation’ between Aθ̃(p) and θ̃p(p) as measured by C
limn→∞ C
n will play an important rôle. [Recall that θ̃(p) denotes the least-squares estimator of θ based
on model Mp and that Aθ is the parameter vector of interest. Furthermore, the vector of covariances be-
tween Aθ̃(p) and θ̃p(p) is given by σ
2n−1C
n with C
n = A[p](X [p]
′X [p]/n)−1ep.] Note that C
∞ equals
A[p]Q[p : p]−1ep, and hence does not depend on the unknown parameters θ or σ. In the important special
case discussed in the Introduction, cf. (1), the matrix A equals the O×P matrix (IO : 0), and the condition
∞ 6= 0 reduces to the condition that the regressor corresponding to the p-th column of (V :W ) is asymp-
totically correlated with at least one of the regressors corresponding to the columns of V . See Example 1
below for more discussion.
In the result to follow we shall consider performance limits for estimators of Gn,θ,σ(t) at a fixed value of the
argument t. An estimator of Gn,θ,σ(t) is now nothing else than a real-valued random variable Γn = Γn(Y,X).
For mnemonic reasons we shall, however, use the symbol Ĝn(t) instead of Γn to denote an arbitrary estimator
of Gn,θ,σ(t). This notation should not be taken as implying that the estimator is obtained by evaluating
an estimated cdf at the argument t, or that it is a priori constrained to lie between zero and one. We shall
use this notational convention mutatis mutandis also in subsequent sections. Regarding the non-uniformity
phenomenon, we then have a dichotomy which is described in the following two results.
Theorem 2.3 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C
∞ 6= 0, for some q sat-
isfying O < q ≤ P , and let q∗ denote the largest q with this property. Then the following holds for every
θ ∈ Mq∗−1, every σ, 0 < σ < ∞, and every t ∈ Rk: There exist δ0 > 0 and ρ0, 0 < ρ0 < ∞, such that any
estimator Ĝn(t) of Gn,θ,σ(t) satisfying
Pn,θ,σ
(∣∣∣Ĝn(t)−Gn,θ,σ(t)
∣∣∣ > δ
n→∞−→ 0 (19)
for each δ > 0 (in particular, every estimator that is consistent) also satisfies
ϑ∈Mq∗
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣Ĝn(t)−Gn,ϑ,σ(t)
∣∣∣ > δ0
n→∞−→ 1. (20)
The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q, A, σ, and the critical
values cp for O < p ≤ P . Moreover,
lim inf
Ĝn(t)
ϑ∈Mq∗
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣Ĝn(t)−Gn,ϑ,σ(t)
∣∣∣ > δ0
> 0 (21)
lim inf
Ĝn(t)
ϑ∈Mq∗
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣Ĝn(t)−Gn,ϑ,σ(t)
∣∣∣ > δ
, (22)
where the infima in (21) and (22) extend over all estimators Ĝn(t) of Gn,θ,σ(t).
Remark 2.4 Assume that the conditions of the preceding theorem are satisfied. Suppose further that p⊙,
O ≤ p⊙ < q∗, is such that either p⊙ > 0 and some row of A[p⊙] equals zero, or such that p⊙ = 0. Then
there exist δ0 > 0 and 0 < ρ0 <∞ such that the left-hand side of (21) is not less than 1/2 for each θ ∈Mp⊙ .
Theorem 2.3 a fortiori implies a corresponding ‘impossibility’ result for estimation of the functionGn,θ,σ(·)
when the estimation error is measured in the total variation distance or the sup-norm; cf. also Section 5.
It remains to consider the – quite exceptional – case where the assumption of Theorem 2.3 is not satisfied,
i.e., where C
∞ = 0, for all q in the range O < q ≤ P . Under this ‘uncorrelatedness’ condition it is indeed
possible to construct an estimator of Gn,θ,σ which is uniformly consistent: It is not difficult to see that the
asymptotic distribution of Gn,θ,σ reduces to Φ∞,P under this ‘uncorrelatedness’ condition. Furthermore, the
second half of Proposition B.1 in Appendix B shows that then the convergence of Gn,θ,σ to its large-sample
limit is uniform w.r.t. θ, suggesting Φ̂n,P , an estimated version of Φ∞,P , as an estimator for Gn,θ,σ.
Proposition 2.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C
∞ = 0, for all q
satisfying O < q ≤ P . Then
σ∗≤σ≤σ∗
Pn,θ,σ
∣∣∣Φ̂n,P −Gn,θ,σ
n→∞−→ 0 (23)
holds for each δ > 0, and for any constants σ∗ and σ
∗ satisfying 0 < σ∗ ≤ σ∗ <∞.
Inspection of the proof of Proposition 2.5 shows that (23) continues to hold if the estimator Φ̂n,P is
replaced by any of the estimators Φ̂n,p for O ≤ p ≤ P . We also note that in case O = 0 the assumption
of Proposition 2.5 is never satisfied in view of Proposition 4.4 in Leeb and Pötscher (2006b), and hence
Theorem 2.3 always applies in that case. Another consequence of Proposition 4.4 in Leeb and Pötscher
(2006b) is that – under the ‘uncorrelatedness’ assumption of Proposition 2.5 – the restricted least squares
estimators Aθ̃(q) for q ≥ O perform asymptotically as well as the unrestricted estimator Aθ̃(P ); this clearly
shows that the case covered by Proposition 2.5 is highly exceptional.
In summary we see that it is typically impossible to construct an estimator of Gn,θ,σ(t) which performs
reasonably well even asymptotically. Whenever Theorem 2.3 applies, any estimator of Gn,θ,σ(t) suffers
from a non-uniformity defect which is caused by parameters belonging to shrinking ‘tubes’ surrounding
Mq∗−1. For the sake of completeness, we remark that outside a ‘tube’ of fixed positive radius that surrounds
Mq∗−1 the non-uniformity need not be present: Let q
∗ be as in Theorem 2.3 and define the set U as
U = {θ ∈ RP : |θq∗ | ≥ r} for some fixed r > 0. Then Φ̂n,P (t) is an estimator of Gn,θ,σ(t) that is uniformly
consistent over θ ∈ U ; more generally, it can be shown that then the relation (23) holds if the supremum
over θ on the left-hand side is restricted to θ ∈ U .
We conclude this section by illustrating the above results with some important examples.
Example 1: (The distribution of χ̃) Consider the model given in (1) with χ representing the parameter
of interest. Using the general notation of Section 2, this corresponds to the case Aθ = (θ1, . . . , θO)
′ = χ
with A representing the O × P matrix (IO : 0). Here k = O > 0. The cdf Gn,θ,σ then represents the cdf
n (χ̃− χ). Assume first that limn→∞ V ′W/n 6= 0. Then C(q)∞ 6= 0 holds for some q > O. Consequently,
the ‘impossibility’ results for the estimation of Gn,θ,σ given in Theorem 2.3 always apply. Next assume
that limn→∞ V
′W/n = 0. Then C
∞ = 0 for every q > O. In this case Proposition 2.5 applies and a
uniformly consistent estimator of Gn,θ,σ indeed exists. Summarizing we note that any estimator of Gn,θ,σ
suffers from the non-uniformity phenomenon except in the special case where the columns of V and W
are asymptotically orthogonal in the sense that limn→∞ V
′W/n = 0. But this is precisely the situation
where inclusion or exclusion of the regressors in W has no effect on the distribution of the estimator χ̃
asymptotically; hence it is not surprising that also the model selection procedure does not have an effect on
the estimation of the cdf of the post-model-selection estimator χ̃. This observation may tempt one to enforce
orthogonality between the columns of V and W by either replacing the columns of V by their residuals from
the projection on the column space ofW or vice versa. However, this is not helpful for the following reasons:
In the first case one then in fact avoids model selection as all the restricted least-squares estimators for χ
under consideration (and hence also the post-model selection estimator χ̃) in the reparameterized model
coincide with the unrestricted least-squares estimator. In the second case the coefficients of the columns of
V in the reparameterized model no longer coincide with the parameter of interest χ (and again are estimated
by one and the same estimator regardless of inclusion/exclusion of columns of the transformed W -matrix).
Example 2: (The distribution of θ̃) For A equal to IP , the cdf Gn,θ,σ is the cdf of
n(θ̃ − θ). Here,
Aθ̃(q) reduces to θ̃(q), and hence Aθ̃(q) and θ̃q(q) are perfectly correlated for every q > O. Consequently,
the ‘impossibility’ result for estimation of Gn,θ,σ given in Theorem 2.3 applies. [In fact, the slightly stronger
result mentioned in Remark 2.4 always applies here.] We therefore see that estimation of the distribution of
the post-model-selection estimator of the entire parameter vector is always plagued by the non-uniformity
phenomenon.
Example 3: (The distribution of a linear predictor) Suppose A 6= 0 is a 1×P vector and one is interested
in estimating the cdf Gn,θ,σ of the linear predictor Aθ̃. Then Theorem 2.3 and the discussion following
Proposition 2.5 show that the non-uniformity phenomenon always arises in this estimation problem in case
O = 0. In case O > 0, the non-uniformity problem is generically also present, except in the degenerate case
where C
∞ = 0, for all q satisfying O < q ≤ P (in which case Proposition 4.4 in Leeb and Pötscher (2006b)
shows that the least-squares predictors from all models Mp, O ≤ p ≤ P , perform asymptotically equally
well).
3 Extensions to Other Model Selection Procedures Including AIC
In this section we show that the ‘impossibility’ result obtained in the previous section for a ‘general-to-
specific’ model selection procedure carries over to a large class of model selection procedures, including
Akaike’s widely used AIC. Again consider the linear regression model (5) with the same assumptions on the
regressors and the errors as in Section 2. Let {0, 1}P denote the set of all 0-1 sequences of length P . For
each r ∈{0, 1}P let Mr denote the set {θ ∈ RP : θi(1 − ri) = 0 for 1 ≤ i ≤ P} where ri represents the i-th
component of r. I.e., Mr describes a linear submodel with those parameters θi restricted to zero for which
ri = 0. Now let R be a user-supplied subset of {0, 1}P . We consider model selection procedures that select
from the set R, or equivalently from the set of models {Mr : r ∈ R}. Note that there is now no assumption
that the candidate models are nested (for example, if R = {0, 1}P all possible submodels are candidates for
selection). Also cases where the inclusion of a subset of regressors is undisputed on a priori grounds are
obviously covered by this framework upon suitable choice of R.
We shall assume throughout this section that R contains rfull = (1, . . . , 1) and also at least one element r∗
satisfying |r∗| = P − 1, where |r∗| represents the number of non-zero coordinates of r∗. Let r̂ be an arbitrary
model selection procedure, i.e., r̂ = r̂(Y,X) is a random variable taking its values in R. We furthermore
assume throughout this section that the model selection procedure r̂ satisfies the following mild condition:
For every r∗ ∈ R with |r∗| = P − 1 there exists a positive finite constant c (possibly depending on r∗) such
that for every θ ∈Mr∗ which has exactly P − 1 non-zero coordinates
Pn,θ,σ ({r̂ = rfull}N{|Tr∗ | ≥ c}) = lim
Pn,θ,σ ({r̂ = r∗}N{|Tr∗ | < c}) = 0 (24)
holds for every 0 < σ < ∞. Here N denotes the symmetric difference operator and Tr∗ represents the usual
t-statistic for testing the hypothesis θi(r∗) = 0 in the full model, where i(r∗) denotes the index of the unique
coordinate of r∗ that equals zero.
The above condition is quite natural for the following reason: For θ ∈ Mr∗ with exactly P − 1 non-zero
coordinates, every reasonable model selection procedure will – with probability approaching unity – decide
only betweenMr∗ andMrfull ; it is then quite natural that this decision will be based (at least asymptotically)
on the likelihood ratio between these two models, which in turn boils down to the t-statistic. As will be
shown below, condition (24) holds in particular for AIC-like procedures.
Let A be a non-stochastic k×P matrix of full row rank k, 1 ≤ k ≤ P , as in Section 2.1. We then consider
the cdf
Kn,θ,σ(t) = Pn,θ,σ
nA(θ̄ − θ) ≤ t
(t ∈ Rk) (25)
of a linear transformation of the post-model-selection estimator θ̄ obtained from the model selection procedure
r̂, i.e.,
θ̃(r)1(r̂ = r)
where the P × 1 vector θ̃(r) represents the restricted least-squares estimator obtained from model Mr, with
the convention that θ̃(r) = 0 ∈ RP in case r = (0, . . . , 0). We then obtain the following result for estimation
of Kn,θ,σ(t) at a fixed value of the argument t which parallels the corresponding ‘impossibility’ result in
Theorem 2.3.
Theorem 3.1 Let r∗ ∈ R satisfy |r∗| = P − 1, and let i(r∗) denote the index of the unique coordinate of
r∗ that equals zero; furthermore, let c be the constant in (24) corresponding to r∗. Suppose that Aθ̃(rfull)
and θ̃i(r∗)(rfull) are asymptotically correlated, i.e., AQ
i(r∗)
6= 0, where e
i(r∗)
denotes the i(r∗)-th standard
basis vector in RP . Then for every θ ∈ Mr∗ which has exactly P − 1 non-zero coordinates, for every σ,
0 < σ <∞, and for every t ∈ Rk the following holds: There exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that any
estimator K̂n(t) of Kn,θ,σ(t) satisfying
Pn,θ,σ
(∣∣∣K̂n(t)−Kn,θ,σ(t)
∣∣∣ > δ
n→∞−→ 0 (26)
for each δ > 0 (in particular, every estimator that is consistent) also satisfies
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣K̂n(t)−Kn,ϑ,σ(t)
∣∣∣ > δ0
n→∞−→ 1 . (27)
The constants δ0 and ρ0 may be chosen in such a way that they depend only on t, Q,A, σ, and c. Moreover,
lim inf
K̂n(t)
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣K̂n(t)−Kn,ϑ,σ(t)
∣∣∣ > δ0
> 0 (28)
lim inf
K̂n(t)
||ϑ−θ||<ρ0/
Pn,ϑ,σ
(∣∣∣K̂n(t)−Kn,ϑ,σ(t)
∣∣∣ > δ
≥ 1/2 (29)
hold, where the infima in (28) and (29) extend over all estimators K̂n(t) of Kn,θ,σ(t).
The basic condition (24) on the model selection procedure employed in the above result will certainly
hold for any hypothesis testing procedure that (i) asymptotically selects only correct models, (ii) employs a
likelihood ratio test (or an asymptotically equivalent test) for testing Mrfull versus smaller models (at least
versus the models Mr∗ with r∗ as in condition (24)), and (iii) uses a critical value for the likelihood ratio
test that converges to a finite positive constant. In particular, this applies to usual thresholding procedures
as well as to a variant of the ‘general-to-specific’ procedure discussed in Section 2 where the error variance
in the construction of the test statistic for hypothesis H
0 is estimated from the fitted model Mp rather
than from the overall model. We next verify condition (24) for AIC-like procedures. Let RSS(r) denote the
residual sum of squares from the regression employing model Mr and set
IC(r) = log (RSS(r)) + |r|Υn/n (30)
where Υn ≥ 0 denotes a sequence of real numbers satisfying limn→∞ Υn = Υ and Υ is a positive real number.
Of course, IC(r) = AIC(r) if Υn = 2. The model selection procedure r̂IC is then defined as a minimizer
(more precisely, as a measurable selection from the set of minimizers) of IC(r) over R. It is well-known that
the probability that r̂IC selects an incorrect model converges to zero. Hence, elementary calculations show
that condition (24) is satisfied for c = Υ1/2.
The analysis of post-model-selection estimators based on AIC-like model selection procedures given in this
section proceeded by bringing this case under the umbrella of the results obtained in Section 2. Verification of
condition (24) is the key that enables this approach. A complete analysis of post-model-selection estimators
based on AIC-like model selection procedures, similar to the analysis in Section 2 for the ‘general-to-specific’
model selection procedure, is certainly possible but requires a direct and detailed analysis of the distribution
of this post-model-selection estimator. [Even the mild condition that R contains rfull and also at least one
element r∗ satisfying |r∗| = P − 1 can then be relaxed in such an analysis.] We furthermore note that in the
special case where R = {rfull, r∗} and an AIC-like model selection procedure as in (30) is used, the results
in the above theorem in fact hold for all θ ∈Mr∗ .
4 Remarks and Extensions
Remark 4.1 Although not emphasized in the notation, all results in the paper also hold if the elements of
the design matrix X depend on sample size. Furthermore, all results are expressed solely in terms of the
distributions Pn,θ,σ(·) of Y , and hence they also apply if the elements of Y depend on sample size, including
the case where the random vectors Y are defined on different probability spaces for different sample sizes.
Remark 4.2 The model selection procedure considered in Section 2 is based on a sequence of tests which
use critical values cp that do not depend on sample size and satisfy 0 < cp < ∞ for O < p ≤ P . If these
critical values are allowed to depend on sample size such that they now satisfy cn,p → c∞,p as n → ∞ with
0 < c∞,p < ∞ for O < p ≤ P , the results in Leeb and Pötscher (2003) as well as in Leeb (2005, 2006)
continue to hold; see Remark 6.2(i) in Leeb and Pötscher (2003) and Remark 6.1(ii) in Leeb (2005). As a
consequence, the results in the present paper can also be extended to this case quite easily.
Remark 4.3 The ‘impossibility’ results given in Theorems 2.3 and 3.1 (as well as the variants thereof
discussed in the subsequent Remarks 4.4-4.7) also hold for the class of all randomized estimators (with
P ∗n,θ,σ replacing Pn,θ,σ in those results, where P
n,θ,σ denotes the distribution of the randomized sample).
This follows immediately from Lemma 3.6 and the attending discussion in Leeb and Pötscher (2006a).
Remark 4.4 a. Let ψn,θ,σ denote the expectation of θ̃ under Pn,θ,σ, and consider the cdf Hn,θ,σ(t) =
Pn,θ,σ(
nA(θ̃−ψn,θ,σ) ≤ t). Results for the cdf Hn,θ,σ quite similar to the results for Gn,θ,σ obtained
in the present paper can be established. A similar remark applies to the post-model-selection estimator
θ̄ considered in Section 3.
b. In Leeb (2006) also the cdf G∗n,θ,σ is analyzed, which correspond to a (typically infeasible) model
selection procedure that makes use of knowledge of σ. Results completely analogous to the ones in the
present paper can also be obtained for this cdf.
Remark 4.5 Results similar to the ones in Section 2.2.2 can also be obtained for estimation of the asymp-
totic cdf G∞,θ,σ(t) (or of the asymptotic cdfs corresponding to the variants discussed in the previous remark).
Since these results are of limited interest, we omit them. In particular, note that an ‘impossibility’ result
for estimation of G∞,θ,σ(t) per se does not imply a corresponding ‘impossibility’ result for estimation of
Gn,θ,σ(t), since Gn,θ,σ(t) does in general not converge uniformly to G∞,θ,σ(t) over the relevant subsets in
the parameter space; cf. Appendix B. [An analogous remark applies to the model selection procedures
considered in Section 3.]
Remark 4.6 Let πn,θ,σ(p) denote the model selection probability Pn,θ,σ(p̂ = p), O ≤ p ≤ P corresponding
to the model selection procedure discussed in Section 2. The finite-sample properties and the large-sample
limit behavior of these quantities are thoroughly analyzed in Leeb (2006); cf. also Leeb and Pötscher (2003).
For these model selection probabilities the following results can be established which we discuss here only
briefly:
a. The model selection probabilities πn,θ,σ(p) converge to well-defined large-sample limits which we denote
by π∞,θ,σ(p). Similar as in Proposition B.1 in Appendix B, the convergence of πn,θ,σ(p) to π∞,θ,σ(p)
is non-uniform w.r.t. θ. [For the case O = 0, this phenomenon is described in Corollary 5.6 of Leeb
and Pötscher (2003).]
b. The model selection probabilities πn,θ,σ(p) can be estimated consistently. However, uniformly consis-
tent estimation is again not possible. A similar remark applies to the large-sample limits π∞,θ,σ(p).
Remark 4.7 ‘Impossibility’ results similar to the ones given in Theorems 2.3 and 3.1 for the cdf can also be
obtained for other characteristics of the distribution of a linear function of a post-model-selection estimator
like the mean-squared error or the bias of
nAθ̃.
5 On the Scope of the Impossibility Results
The non-uniformity phenomenon described, e.g., in (20) of Theorem 2.3 is caused by a mechanism that can
informally be described as follows. Under the assumptions of that theorem, one can find an appropriate θ
and an appropriate sequence ϑn = θ + γ/
n exhibiting two crucial properties:
a. The probability measures Pn,ϑn,σ corresponding to ϑn are ‘close’ to the measures Pn,θ,σ corresponding
to θ, in the sense of contiguity. This entails that an estimator, that converges to some limit in probability
under Pn,θ,σ, converges to the same limit also under Pn,ϑn,σ.
b. For given t, the estimands Gn,ϑn,σ(t) corresponding to ϑn are ‘far away’ from the estimands Gn,θ,σ(t)
corresponding to θ, in the sense that Gn,ϑn,σ(t) and Gn,θ,σ(t) converge to different limits, i.e.,
G∞,θ,σ,0(t) is different from G∞,θ,σ,γ(t).
In view of Property a, an estimator Ĝn(t) satisfying Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,θ,σ, also
satisfies Ĝn(t)−Gn,θ,σ(t) → 0 in probability under Pn,ϑn,σ. In view of Property b, such an estimator Ĝn(t)
is hence ‘far away’ from the estimand Gn,ϑn,σ(t) with high probability under Pn,ϑn,σ. In other words, an
estimator that is close to Gn,θ,σ(t) under Pn,θ,σ must be far away from Gn,ϑn,σ(t) under Pn,ϑn,σ. Formalized
and refined, this argument leads to (20) and, as a consequence, to the non-existence of uniformly consistent
estimators for Gn,θ,σ(t). [There are a number of technical details in this formalization process that need
careful attention in order to obtain the results in their full strength as given in Sections 2 and 3.]
The above informal argument that derives (20) from Properties a and b can be refined and formalized in
a much more general and abstract framework, see Section 3 of Leeb and Pötscher (2006a) and the references
therein. That paper also provides a general framework for deriving results like (21) and (22) of Theorem
2.3. The mechanism leading to such lower bounds is similar to the one outlined above, where for some of
the results the concept of contiguity of the probability measures involved has to be replaced by closeness of
these measures in total variation distance. We use the results in Section 3 of Leeb and Pötscher (2006a) to
formally convert Properties a and b into the ‘impossibility’ results of the present paper; cf. Appendix C.
Verifying the aforementioned Property a in the context of the present paper is straightforward because
we consider a Gaussian linear model. What is technically more challenging and requires some work is the
verification of Property b; this is done in Appendix A inter alia and rests on results of Leeb (2002, 2005,
2006).
Two important observations on Properties a and b are in order: First, Property a is typically satisfied
in general parametric models under standard regularity conditions; e.g., it is satisfied whenever the model is
locally asymptotically normal. Second, Property b relies on limiting properties only and not on the finite-
sample structure of the underlying statistical model. Now, the limit distributions of post-model-selection
estimators in sufficiently regular parametric or semi-parametric models are typically the same as the limiting
distributions of the corresponding post-model-selection estimators in a Gaussian linear model (see, e.g., Sen
(1979), Pötscher (1991), Nickl (2003), or Hjort and Claeskens (2003)). Hence, establishing Property b for
the Gaussian linear model then typically establishes the same result for a large class of general parametric
or semi-parametric models.1 For example, Property b can be verified for a large class of pre-test estimators
in sufficiently regular parametric models by arguing as in Appendix A and using the results of Nickl (2003)
to reduce to the Gaussian linear case. Hence, the impossibility result given in Theorem 2.3 can be extended
1Some care has to be taken here. In the Gaussian linear case the finite-sample cdfs converge at every value of the argument t,
cf. Propisition 2.1. In a general parametric model, sometimes the asymptotic results (e.g., Hjort and Claeskens (2003, Theorem
4.1)) only guarantee weak convergence. Hence, to ensure convergence of the relevant cdfs at a given argument t as required
in Proberty b, additional considerations have to be employed. [This is, however, of no concern in the context discussed in the
next but one paragraph in this section.]
to more general parametric and semiparametric models with ease. The fact that we use a Gaussian linear
model for the analysis in the present paper is a matter of convenience rather than a necessity.
The non-uniformity results in Theorem 2.3 are for (conservative) ‘general-to-specific’ model selection
from a nested family of models. Theorem 3.1 extends this to more general (conservative) model selection
procedures (including AIC and related procedures) and to more general families of models. The proof of
Theorem 3.1 proceeds by reducing the problem to one where only two nested models are considered, and then
to appeal to the results of Theorem 2.3. The condition on the model selection procedures that enables this
reduction is condition (24). It is apparent from the discussion in Section 3 that this condition is satisfied for
many model selection procedures. Furthermore, for the same reasons as given in the preceding paragraph,
also Theorem 3.1 can easily be extended to sufficiently regular parametric and semi-parametric models.
The ‘impossibility’ results in the present paper are formulated for estimating Gn,θ,σ(t) for a given value
of t. Suppose that we are now asking the question whether the cdf Gn,θ,σ(·) viewed as a function can be
estimated uniformly consistently, where consistency is relative to a metric that metrizes weak convergence.2
Using a similar reasoning as above (which can again be made formal by using, e.g., Lemma 3.1 in Leeb and
Pötscher (2006a)) the key step now is to show that the function G∞,θ,σ,0(·) is different from the function
G∞,θ,σ,γ(·). Obviously, it is a much simpler problem to find a γ such that the functions G∞,θ,σ,0(·) and
G∞,θ,σ,γ(·) differ, than to find a γ such that the values G∞,θ,σ,0(t) and G∞,θ,σ,γ(t) for a given t differ.
Certainly, having solved the latter problem in Appendix A, this also provides an answer to the former. This
then immediately delivers the desired ‘impossibility’ result. [We note that in some special cases simpler
arguments than the ones used in Appendix A can be employed to solve the former problem: For example,
in case A = I the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) can each be shown to be convex combinations of
cdfs that are concentrated on subspaces of different dimensions. This can be exploited to establish without
much difficulty that the functions G∞,θ,σ,0(·) and G∞,θ,σ,γ(·) differ. For purpose of comparison we note that
for general A the distributions G∞,θ,σ,0 and G∞,θ,σ,γ can both be absolutely continuous w.r.t. Lebesgue
measure, not allowing one to use this simple argument.] Again the discussion in this paragraph extends to
more general parametric and semiparametric models without difficulty.
The present paper, including the discussion in this section, has focussed on conservative model selection
procedures. However, the discussion should make it clear that similar ‘impossibility’ results plague consistent
model selection. Section 2.3 in Leeb and Pötscher (2006a) in fact gives such an ‘impossibility’ result in a
simple case.
We close with the following observations. Verification of Property b, whether it is for G∞,θ,σ,0(t) and
G∞,θ,σ,γ(t) (for given t) or for G∞,θ,σ,0(·) and G∞,θ,σ,γ(·), shows that the post-model-selection estimator Aθ̃
is a so-called non-regular estimator for Aθ: Consider an estimator β̃ in a parametric model {Pn,β : β ∈ B}
where the parameter space B is an open subset of Euclidean space Rd. Suppose β̃, properly scaled and
centered, has a limit distribution under local alternatives, in the sense that
n(β̃ − (β + γ/
n)) converges
in law under Pn,β+γ/
n to a limit distribution L∞,β,γ(·) for every γ. The estimator β̃ is called regular if for
every β the limit distribution L∞,β,γ(·) does not depend on γ; cf. van der Vaart (1998, Section 8.5). Suppose
now that the model is, e.g., locally asymptotically normal (hence the contiguity property in Property a is
satisfied). The informal argument outlined at the beginning of this section (and which is formalized in
Lemma 3.1 of Leeb and Pötscher (2006a)) then in fact shows that the cdf of any non-regular estimator
2Or, in fact, any metric w.r.t. which the relevant cdfs converge.
can not be estimated uniformly consistently (where consistency is relative to any metric that metrizes weak
convergence).
6 Conclusions
Despite the fact that we have shown that consistent estimators for the distribution of a post-model-selection
estimator can be constructed with relative ease, we have also demonstrated that no estimator of this distri-
bution can have satisfactory performance (locally) uniformly in the parameter space, even asymptotically.
In particular, no (locally) uniformly consistent estimator of this distribution exists. Hence, the answer to
the question posed in the title has to be negative. The results in the present paper also cover the case of
linear functions (e.g., predictors) of the post-model-selection estimator.
We would like to stress here that resampling procedures like, e.g., the bootstrap or subsampling, do
not solve the problem at all. First note that standard bootstrap techniques will typically not even provide
consistent estimators of the finite-sample distribution of the post-model-selection estimator, as the bootstrap
can be shown to stay random in the limit (Kulperger and Ahmed (1992), Knight (1999, Example 3))3.
Basically the only way one can coerce the bootstrap into delivering a consistent estimator is to resample from
a model that has been selected by an auxiliary consistent model selection procedure. The consistent estimator
constructed in Section 2.2.1 is in fact of this form. In contrast to the standard bootstrap, subsampling will
typically deliver consistent estimators. However, the ‘impossibility’ results given in this paper apply to any
estimator (including randomized estimators) of the cdf of a post-model-selection estimator. Hence, also any
resampling based estimator suffers from the non-uniformity defects described in Theorems 2.3 and 3.1; cf.
also Remark 4.3.
The ‘impossibility’ results in Theorems 2.3 and 3.1 are derived in the framework of a normal linear
regression model (and a fortiori these results continue to hold in any model which includes the normal
linear regression model as a special case), but this is more a matter of convenience than anything else: As
discussed in Section 5, similar results can be obtained in general statistical models allowing for nonlinearity
or dependent data, e.g., as long as standard regularity conditions for maximum likelihood theory are satisfied.
The results in the present paper are derived for a large class of conservative model selection procedures
(i.e., procedures that select overparameterized models with positive probability asymptotically) including
Akaike’s AIC and typical ‘general-to-specific’ hypothesis testing procedures. For consistent model selection
procedures – like BIC or testing procedures with suitably diverging critical values cp (cf. Bauer, Pötscher, and
Hackl (1988)) – the (pointwise) asymptotic distribution is always normal. [This is elementary, cf. Lemma 1
in Pötscher (1991).] However, as discussed at length in Leeb and Pötscher (2005a), this asymptotic nor-
mality result paints a misleading picture of the finite sample distribution which can be far from a normal,
the convergence of the finite-sample distribution to the asymptotic normal distribution not being uniform.
‘Impossibility’ results similar to the ones presented here can also be obtained for post-model-selection esti-
mators based on consistent model selection procedures. These will be discussed in detail elsewhere. For a
3Brownstone (1990) claims the validity of a bootstrap procedure that is based on a conservative model selection procedure
in a linear regression model. Kilian (1998) makes a similar claim in the context of autoregressive models selected by a conser-
vative model selection procedure. Also Hansen (2003) contains such a claim for a stationary bootstrap procedure based on a
conservative model selection procedure. The above discussion intimates that these claims are at least unsubstantiated.
simple special case such an ‘impossibility’ result is given in Section 2.3 of Leeb and Pötscher (2006a).
The ‘impossibility’ of estimating the distribution of the post-model-selection estimator does not per se
preclude the possibility of conducting valid inference after model selection, a topic that deserves further
study. However, it certainly makes this a more challenging task.
A Auxiliary Lemmas
Lemma A.1 Let Z be a random vector with values in Rk and let W be a univariate standard Gaussian
random variable independent of Z. Furthermore, let C ∈ Rk and τ > 0. Then
P(Z ≤ Cx)P(|W − x| < τ ) + P(Z ≤ CW, |W − x| ≥ τ) (31)
is constant as a function of x ∈ R if and only if C = 0 or P(Z ≤ Cx) = 0 for each x ∈ R.
Proof of Lemma A.1: Suppose C = 0 holds. Using independence of Z andW it is then easy to see that
(31) reduces to P(Z ≤ 0), which is constant in x. If P(Z ≤ Cx) = 0 for every x ∈ R, then P(Z ≤ CW ) = 0,
and hence (31) is again constant, namely equal to zero.
To prove the converse, assume that (31) is constant in x ∈ R. Letting x→ ∞, we see that (31) must be
equal to P(Z ≤ CW ). This entails that
P(Z ≤ Cx)P(|W − x| < τ) = P(Z ≤ CW, |W − x| < τ )
holds for every x ∈ R. Write F (x) as shorthand for P(Z ≤ Cx), and let Φ(z) and φ(z) denote the cdf and
density of W , respectively. Then the expression in the above display can be written as
F (x)(Φ(x + τ )− Φ(x− τ)) =
∫ x+τ
F (z)φ(z)dz. (x ∈ R) (32)
We now further assume that C 6= 0 and that F (x) 6= 0 for at least one x ∈ R, and show that this leads to a
contradiction.
Consider first the case where all components of C are non-negative. Since F is not identically zero, it
is then, up to a scale factor, the cdf of a random variable on the real line. But then (32) can not hold for
all x ∈ R as shown in Example 7 in Leeb (2002) (cf. also equation (7) in that paper). The case where
all components of C are non-positive follows similarly by applying the above argument to F (−x) and upon
observing that both Φ(x+ τ )− Φ(x− τ) and φ(x) are symmetric around x = 0.
Finally, consider the case where C has at least one positive and one negative component. In this case
clearly limx→−∞ F (x) = limx→∞ F (x) = 0 holds. Since F (x) is continuous in view of (32), we see that F (x)
attains its (positive) maximum at some point x1 ∈ R. Now note that (32) with x1 replacing x can be written
as ∫ x1+τ
(F (x1)− F (z))φ(z)dz = 0.
This immediately entails that F (x) = F (x1) for each x ∈ [x1 − τ , x1 + τ ] (because F (x) is continuous and
because of the definition of x1). Repeating this argument with x1−τ replacing x1 and proceeding inductively,
we obtain that F (x) = F (x1) for each x satisfying x ≤ x1 + τ , a contradiction with limx→−∞ F (x) = 0. ✷
Lemma A.2 Let M and N be matrices of dimension k × p and k × q, respectively, such that the matrix
(M : N) has rank k (k ≥ 1, p ≥ 1, q ≥ 1). Let t ∈ Rk, and let V be a random vector with values in Rp
whose distribution assigns positive mass to every (non-empty) open subset of Rp (e.g., it possesses an almost
everywhere positive Lebesgue density). Set f(x) = P(MV ≤ t + Nx), x ∈ Rq. If one of the rows of M
consists of zeros only, then f is discontinuous at some point x0. More precisely, there exist x0 ∈ Rq, z ∈ Rq
and a constant c > 0, such that f(x0 + δz) ≥ c and f(x0 − δz) = 0 hold for every sufficiently small δ > 0.
Proof of Lemma A.2: The case where M is the zero-matrix is trivial. Otherwise, let I0 denote the
set of indices i, 1 ≤ i ≤ k, for which the i-th row of M is zero. Let (M0 : N0) denote the matrix consisting
of those rows of (M : N) whose index is in I0, and let (M1 : N1) denote the matrix consisting of the
remaining rows of (M : N). Clearly, M0 is then the zero matrix. Furthermore, note that N0 has full
row-rank. Moreover, let t0 denote the vector consisting of those components of t whose index is in I0 and
let t1 denote the vector containing the remaining components. With this notation, f(x) can be written as
P(0 ≤ t0 +N0x, M1V ≤ t1 +N1x).
For vectors µ ∈ Rp and η ∈ Rq to be specified in a moment, set t∗ = t+Mµ+Nη, and let t∗0 and t∗1 be
defined similarly to t0 and t1. Since the matrix (M : N) has full rank k, we can choose µ and η such that
t∗0 = 0 and t
1 > 0. Choose z ∈ Rq such that N0z > 0, which is possible because N0 has full row-rank. Set
x0 = η. Then for every ǫ ∈ R we have
f(x0 + ǫz) = f(η + ǫz) = P(MV ≤ t+N(η + ǫz))
= P(0 ≤ t0 +N0(η + ǫz), M1V ≤ t1 +N1(η + ǫz))
= P(0 ≤ t∗0 + ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z)
= P(0 ≤ ǫN0z, M1(V + µ) ≤ t∗1 + ǫN1z)
Since t∗1 > 0, we can find a t
1 such that 0 < t
1 < t
1 + ǫN1z holds for every ǫ with |ǫ| small enough. If now
ǫ > 0 then
f(x0 + ǫz) = P(M1(V + µ) ≤ t∗1 + ǫN1z) ≥ P(M1(V + µ) ≤ t∗∗1 ).
The r.h.s. in the above display is positive because t∗∗1 > 0 and because the distribution ofM1(V +µ) assigns
positive mass to any neighborhood of the origin, since the same is true for the distribution of V + µ and
since M1 maps neighborhoods of zero into neighborhoods of zero. Setting c = P(M1(V + µ) ≤ t∗∗1 )/2, we
have f(x0 + ǫz) ≥ c > 0 for each sufficiently small ǫ > 0. Furthermore, for ǫ < 0 we have f(x0 + ǫz) = 0,
since f(x0 + ǫz) ≤ P(0 ≤ ǫN0z) = 0 in view of N0z > 0. ✷
Lemma A.3 Let Z be a random vector with values in Rp, p ≥ 1, with a distribution that is absolutely
continuous with respect to Lebesgue measure on Rp. Let B be a k×p matrix, k ≥ 1. Then the cdf P(BZ ≤ ·)
of BZ, is discontinuous at t ∈ Rk if and only if P(BZ ≤ t) > 0 and if for some i0, 1 ≤ i0 ≤ k, the i0-th row
of B and the i0-th component of t are both zero, i.e., Bi0,· = (0, . . . , 0) and ti0 = 0.
Proof of Lemma A.3: To establish sufficiency of the above condition, let P(BZ ≤ t) > 0, ti0 = 0 and
Bi0,· = (0, . . . , 0) for some i0, 1 ≤ i0 ≤ k. Then, of course, P(Bi0,·Z = 0) = 1. For tn = t−n−1ei0 , where ei0
denotes the i0-th unit vector in R
k, we have P(BZ ≤ tn) ≤ P(Bi0,·Z ≤ tn,i0) = P(Bi0,·Z ≤ −1/n) = 0 for
every n. Consequently, P(BZ ≤ t) is discontinuous at t.
To establish necessity, we first show the following: If tn ∈ Rk is a sequence converging to t ∈ Rk as
n→ ∞, then every accumulation point of the sequence P(BZ ≤ tn) has the form
P(Bi1,·Z ≤ ti1 , . . . , Bim,·Z ≤ tim , Bim+1,·Z < tim+1 , . . . , Bik,·Z < tik) (33)
for somem, 0 ≤ m ≤ k, and for some permutation (i1, . . . , ik) of (1, . . . , k). This can be seen as follows: Let α
be an accumulation point of P(BZ ≤ tn). Then we may find a subsequence such that P(BZ ≤ tn) converges
to α along this subsequence. From this subsequence we may even extract a further subsequence along which
each component of the k × 1 vector tn converges to the corresponding component of t monotonously, that
is, either from above or from below. Without loss of generality, we may also assume that those components
which converge from below are strictly increasing. The resulting subsequence will be denoted by nj in the
sequel. Assume that the components of tnj with indices i1, . . . , im converge from above, while the components
with indices im+1, . . . , ik converge from below. Now
P(BZ ≤ tnj ) =
1(−∞,tnj,s](zs)PBZ(dz), (34)
where PBZ denotes the distribution of BZ. The integrand in (34) now converges to
l=1 1(−∞,til ](zil)
l=m+1 1(−∞,til )(zil) for all z ∈ R
k as nj → ∞. The r.h.s. of (34) converges to the
expression in (33) as nj → ∞ by the Dominated Convergence Theorem, while the l.h.s. of (34) converges to
α by construction. This establishes the claim regarding (33).
Now suppose that P(BZ ≤ t) is discontinuous at t; i.e., there exists a sequence tn converging to t as
n → ∞, such that P(BZ ≤ tn) does not converge to P(BZ ≤ t) as n → ∞. From the sequence tn we
can extract a subsequence tns along which P(BZ ≤ tns) converges to a limit different from P(BZ ≤ t) as
ns → ∞. As shown above, the limit has to be of the form (33) and m < k has to hold. Consequently, the
limit of P(BZ ≤ tns) is smaller than P(BZ ≤ t) = P(Bi,·Z ≤ ti, i = 1, . . . , k). The difference of P(BZ ≤ t)
and the limit of P(BZ ≤ tns) is positive and because of (33) can be written as
P(Bij ,·Z ≤ tij for each j = 1, . . . , k, Bij ,·Z = tij for some j = m+ 1, . . . , k) > 0.
We thus see that P(Bij0 ,·Z = tij0 ) > 0 for some j0 satisfying m + 1 ≤ j0 ≤ k. As Z is absolutely
continuous with respect to Lebesgue measure on Rp, this can only happen if Bij0 ,· = (0, . . . , 0) and tij0 = 0.
Lemma A.4 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C
∞ 6= 0, for some q satisfying
O < q ≤ P , and let q∗ denote the largest q with this property. Moreover let θ ∈ Mq∗−1, let σ satisfy
0 < σ < ∞, and let t ∈ Rk. Then G∞,θ,σ,γ(t) is non-constant as a function of γ ∈ Mq∗\Mq∗−1. More
precisely, there exist δ0 > 0 and ρ0, 0 < ρ0 <∞, such that
γ(1),γ(2)∈Mq∗ \Mq∗−1
||γ(i)||<ρ0,i=1,2
∣∣G∞,θ,σ,γ(1)(t)−G∞,θ,σ,γ(2)(t)
∣∣ > 2δ0 (35)
holds. The constants δ0 and ρ0 can be chosen in such a way that they depend only on t, Q, A, σ, and the
critical values cp for O < p ≤ P .
Lemma A.5 Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C
∞ 6= 0, for some q satisfying
O < q ≤ P , and let q∗ denote the largest q with this property. Suppose further that for some p⊙ satisfying
O ≤ p⊙ < q∗ either p⊙ = 0 holds or that p⊙ > 0 and A[p⊙] has a row of zeros. Then, for every θ ∈ Mp⊙ ,
every σ, 0 < σ < ∞, and every t ∈ Rk the quantity G∞,θ,σ,γ(t) is discontinuous as a function of γ ∈ Mq∗ .
More precisely, for each s = O, . . . , p⊙, there exist vectors β∗ and γ∗ in Mq∗ and constants δ∗ > 0 and ǫ∗ > 0
such that
∣∣G∞,θ,σ,β∗+ǫγ∗(t)−G∞,θ,σ,β∗−ǫγ∗(t)
∣∣ ≥ δ∗ (36)
holds for every θ satisfying max{p0(θ),O} = s and for every ǫ with 0 < ǫ < ǫ∗. The quantities δ∗, ǫ∗, β∗,
and γ∗ can be chosen in such a way that – besides t, Q, A, σ, and the critical values cp for O < p ≤ P –
they depend on θ only through max{p0(θ),O}.
Before we prove the above lemmas, we provide a representation of G∞,θ,σ,γ(t) that will be useful in
the following: For 0 < p ≤ P define Zp =
r=1 ξ
∞ Wr, where C
∞ has been defined after (13) and
the random variables Wr are independent normally distributed with mean zero and variances σ
2ξ2∞,r; for
convenience, let Z0 denote the zero vector in R
k. Observe that Zp, p > 0, is normally distributed with
mean zero and variance-covariance matrix σ2A[p]Q[p : p]−1A[p]′ since it has been shown in the proof of
Proposition 4.4 in Leeb and Pötscher (2006b) that the asymptotic variance-covariance matrix σ2A[p]Q[p :
p]−1A[p]′ of
nAθ̃(p) can be expressed as
r=1 σ
2ξ−2∞,rC
∞ . Also the joint distribution of Zp and the
set of variables Wr, 1 ≤ r ≤ P , is normal, with the covariance vector between Zp and Wr given by σ2C(r)∞ in
case r ≤ p; otherwise Zp and Wr are independent. Define the constants νr = γr+(Q[r : r]−1Q[r : ¬r]γ[¬r])r
for 0 < r ≤ P . It is now easy to see that for p ≥ p∗ = max{p0(θ),O} the quantity β(p) defined in
Proposition 2.1 equals −
r=p+1 ξ
∞ νr. [This is seen as follows: It was noted in Proposition 2.1 that
β(p) = limn→∞
nA(ηn(p) − θ − γ/
n) for p ≥ p0(θ), when ηn(p) is defined as in (9), but with θ + γ/
replacing θ. Using the representation (20) of Leeb (2005) and taking limits, the result follows if we observe
nηn,r(r) −→ νr for r > p ≥ p0(θ).] The cdf in (15) can now be written as
Zp∗ ≤ t+
r=p∗+1
ξ−2∞,rC
q=p∗+1
P(|Wq + νq| < cqσξ∞,q)
p=p∗+1
Zp ≤ t+
r=p+1
ξ−2∞,rC
∞ νr, |Wp + νp| ≥ cpσξ∞,p
q=p+1
P(|Wq + νq| < cqσξ∞,q). (37)
That the terms corresponding to p = p∗ in (37) and (15) agree is obvious. Furthermore, for each p > p∗
the terms under the product sign in (37) and (15) coincide by definition of the function ∆s(a, b). It is also
easy to see that the conditional distribution of Wp given Zp = z is Gaussian with mean b∞,pz and variance
σ2ζ2∞,p. Consequently, the probability of the event {|Wp + νp| ≥ cpσξ∞,p} conditional on Zp = z is given by
the integrand shown in (15). Since Zp has distribution Φ∞,p as noted above, it follows that (37) and (15)
agree.
Remark A.6 If C
∞ = 0 for p > p∗, then in view of the above discussion Zp∗ = Zp = ZP , and hence
Φ∞,p∗ = Φ∞,p = Φ∞,P , holds for all p > p∗. Using the independence of Wr, r > p∗, from Zp∗ , inspection of
(37) shows that G∞,θ,σ,γ reduces to Φ∞,P ; see also Leeb (2006, Remark 5.2).
Proof of Lemma A.4: From (37) (or (15)) it follows that the map γ 7→ G∞,θ,σ,γ(t) depends only on t,
Q, A, σ, the critical values cp for O < p ≤ P , as well as on θ; however, the dependence on θ is only through
p∗ = max{p0(θ),O}. It hence suffices to find, for each possible value of p∗ in the range p∗ = O, . . . , q∗ − 1,
constants 0 < ρ0 < ∞ and δ0 > 0 such that (35) is satisfied for some (and hence all) θ returning this
particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show that for every θ ∈Mq∗−1 the
quantity G∞,θ,σ,γ(t) is non-constant as a function of γ ∈Mq∗\Mq∗−1.
Let θ ∈Mq∗−1 and assume that G∞,θ,σ,γ(t) is constant in γ ∈Mq∗\Mq∗−1. Observe that, by assumption,
∞ is non-zero while C
∞ = 0 for p > q
∗. For γ ∈ Mq∗ , we clearly have νq∗ = γq∗ and νr = 0 for r > q∗.
Letting γq∗−1 → ∞ while γq∗ is held fixed, we see that νq∗−1 → ∞; hence,
P(|Wq∗−1 + νq∗−1| < cq∗−1σξ∞,q∗−1) → 0.
It follows that (37) converges to
Zq∗−1 ≤ t+ ξ−2∞,q∗C(q
∞ γq∗
P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗)
q=q∗+1
P(|Wq| < cqσξ∞,q)
Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗
q=q∗+1
P(|Wq| < cqσξ∞,q) (38)
p=q∗+1
Zp ≤ t, |Wp| ≥ cpσξ∞,p
q=p+1
P(|Wq| < cqσξ∞,q).
By assumption, the expression in the above display is constant in γq∗ ∈ R\{0}. Dropping the terms that do
not depend on γq∗ and observing that P(|Wq| < cqσξ∞,q) is never zero for q > q∗ > O, we see that
Zq∗−1 ≤ t+ ξ−2∞,q∗C(q
∞ γq∗
P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗)
Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗
has to be constant in γq∗ ∈ R\{0}. We now show that the expression in (39) is in fact constant in γq∗ ∈ R:
Observe first that P(|Wq∗ + γq∗ | < cq∗σξ∞,q∗) is positive and continuous in γq∗ ∈ R; also the probability
Zq∗ ≤ t, |Wq∗ + γq∗ | ≥ cq∗σξ∞,q∗
is continuous in γq∗ ∈ R since Wq∗ , being normal with mean zero and
positive variance, is absolutely continuously distributed. Concerning the remaining term in (39), we note
that Zq∗−1 =MV where M = [ξ
∞ , . . . , ξ
∞,q∗−1C
(q∗−1)
∞ ] and V = (W1, . . . ,Wq∗−1)
′. In case no row of
M is identically zero, Lemma A.3 shows that also P
Zq∗−1 ≤ t+ ξ−2∞,q∗C
∞ γq∗
is continuous in γq∗ ∈ R.
Hence, in this case (39) is indeed constant for all γq∗ ∈ R. In case a row of M is identically zero, define
N = ξ
∞,q∗C
∞ and rewrite the probability in question as P
MV ≤ t+Nγq∗
. Note that (M : N) has full
row-rank k, since
(M : N)diag[ξ2∞,1, . . . , ξ
∞,q∗ ](M : N)
ξ−2∞,rC
ξ−2∞,rC
∞ = AQ
−1A′ (40)
by definition of q∗ and since the latter matrix is non-singular in view of rank A = k. Lemma A.2 then shows
that there exists a γ
q∗ ∈ R, z ∈ {−1, 1}, and a constant c > 0 such that P
MV ≤ t+N(γ(0)q∗ − δz)
and P
MV ≤ t+N(γ(0)q∗ + δz)
≥ c holds for arbitrary small δ > 0. Observe that γ(0)q∗ − δz as well as
q∗ − δz are non-zero for sufficiently small δ > 0. But then (39) – being constant for γq∗ ∈ R\{0} – gives
the same value for γq∗ = γ
q∗ − δz and γq∗ = γ
q∗ + δz and all sufficiently small δ > 0. Letting δ go to zero
in this equality and using the continuity properties for the second and third probability in (39) noted above
we obtain that
cP(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) + P
Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗
≤ lim inf
Zq∗−1 ≤ t+ ξ−2∞,q∗C(q
q∗ + δz)
P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗)
Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗
= lim inf
Zq∗−1 ≤ t+ ξ−2∞,q∗C(q
q∗ − δz)
P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗)
Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗
Zq∗ ≤ t, |Wq∗ + γ(0)q∗ | ≥ cq∗σξ∞,q∗
which is impossible since c > 0 and P(|Wq∗ + γ(0)q∗ | < cq∗σξ∞,q∗) > 0. Hence we have shown that (39) is
indeed constant for all γq∗ ∈ R.
Now write Z,W , C, τ , and x for Zq∗−1−t, −Wq∗/σξ∞,q∗ , σξ
∞,q∗C
∞ , cq∗ , and γq∗/σξ∞,q∗ , respectively.
Upon observing that Zq∗ equals Zq∗−1 + ξ
∞,q∗C
∞ Wq∗ , it is easy to see that (39) can be written as in (31).
By our assumptions, this expression is constant in x = γq∗/σξ∞,q∗ ∈ R. Lemma A.1 then entails that either
C = 0 or that P(Z ≤ Cx) = 0 for each x ∈ R. Since C equals σξ−1∞,q∗C
∞ , it is non-zero by assumption.
Hence,
Zq∗−1 ≤ t+ ξ−2∞,q∗C(q
∞ γq∗
must hold for every value of γq∗ . But the above probability is just the conditional probability that Zq∗ ≤ t
given Wq∗ = −γq∗ . It follows that P(Zq∗ ≤ t) equals zero as well. By our assumption C
∞ = 0 for p > q
and hence Zq∗ = ZP . We thus obtain P(ZP ≤ t) = 0, a contradiction with the fact that ZP is a Gaussian
random variable on Rk with non-singular variance-covariance matrix σ2AQ−1A′. ✷
Inspection of the above proof shows that it can be simplified if the claim of non-constancy of G∞,θ,σ,γ(t)
as a function of γ ∈Mq∗\Mq∗−1 in Lemma A.4 is weakened to non-constancy for γ ∈Mq∗ . The strong form
of the lemma as given here is needed in the proof of Proposition B.1.
Proof of Lemma A.5: Let p⊕ be the largest index p, O ≤ p ≤ P , for which A[p] has a row of zeroes, and
set p⊕ = 0 if no such index exists. We first show that p⊕ satisfies p⊕ < q
∗. Suppose p⊕ ≥ q∗ would hold. Since
Zp⊕ is a Gaussian random vector with mean zero and variance-covariance matrix σ
2A[p⊕]Q[p⊕ : p⊕]
−1A[p⊕]
at least one component of Zp⊕ is equal to zero with probability one. However, Zp⊕ equals ZP because of
p⊕ ≥ q∗ and the definition of q∗. This leads to a contradiction since ZP has the non-singular variance-
covariance matrix σ2AQ−1A′. Without loss of generality, we may hence assume that p⊙ = p⊕.
In view of the discussion in the first paragraph of the proof of Lemma A.4, it suffices to establish, for
each possible value s in the range O ≤ s ≤ p⊙, the result (36) for some θ with s = max{p0(θ),O} = p∗. Now
fix such an s and θ (as well as, of course, t, Q, A, σ, and the critical values cp for O < p ≤ P ). Then (37)
expresses the map γ 7→ G∞,θ,σ,γ(t) in terms of ν = (ν1, . . . , νP )′. It is easy to see that the correspondence
between γ and ν is a linear bijection from RP onto itself, and that γ ∈ Mq∗ if and only if ν ∈ Mq∗ . It is
hence sufficient to find a δ∗ > 0 and vectors ν and µ in Mq∗ such that (37) with ν + ǫµ in place of ν and
(37) with ν − ǫµ in place of ν differ by at least δ∗ for sufficiently small ǫ > 0. Note that (37) is the sum of
P − p∗ + 1 terms indexed by p = p∗, . . . , P . We shall now show that ν and µ can be chosen in such a way
that, when replacing ν with ν + ǫµ and ν − ǫµ, respectively, (i) the resulting terms in (37) corresponding to
p = p⊙ differ by some d > 0, while (ii) the difference of the other terms becomes arbitrarily small, provided
that ǫ > 0 is sufficiently small.
Consider first the case where s = p∗ = p⊙. Using the shorthand notation
g(ν) = P
Zp⊙ ≤ t+
r=p⊙+1
ξ−2∞,rC
note that the p⊙-th term in (37) is given by g(ν) multiplied by a product of positive probabilities which are
continuous in ν. To prove property (i) it thus suffices to find a constant c > 0, and vectors ν and µ in Mq∗
such that |g(ν + ǫµ)− g(ν − ǫµ)| ≥ c holds for each sufficiently small ǫ > 0.
In the sub-case p⊙ = 0 choose c = 1, set
ν = −[C(1)∞ , . . . , C(P )∞ ]′
ξ−2∞,rC
µ = [C(1)∞ , . . . , C
ξ−2∞,rC
(1, . . . , 1)′,
observing that the matrix to be inverted is indeed non-singular, since – as discussed after Lemma A.5 – it
is up to a multiplicative factor σ2 identical to the variance-covariance matrix σ2AQ−1A′ of ZP . But then
ν and µ satisfy
r=p⊙+1
ξ−2∞,rC
∞ νr = −t and
r=p⊙+1
ξ−2∞,rC
∞ µr = (1, . . . , 1)
′ if we note that by the
definition of q∗
r=p⊙+1
ξ−2∞,rC
∞ νr =
ξ−2∞,rC
holds and that a similar relation holds with µ replacing ν. Since Zp⊙ = Z0 = 0 ∈ Rk, it is then obvious that
g(ν + ǫµ) and g(ν − ǫµ) differ by 1 for each ǫ > 0.
In the other sub-case p⊙ > 0, define M = [ξ
∞ , . . . , ξ
∞,p⊙C
∞ ], N =
[ξ−2∞,p⊙+1C
(p⊙+1)
∞ , . . . , ξ
∞,q∗C
∞ ], and V = (W1, . . . ,Wp⊙)
′. It is then easy to see that g(ν) equals
f((νp⊙+1, . . . , νq∗)
′), with f defined as in Lemma A.2, and that M has a row of zeros. Furthermore, the
matrix (M : N) has rank k by the same argument as in the proof of Lemma A.4; cf. (40). By Lemma A.2,
we thus obtain vectors x0 and z, and a c > 0 such that |f(x0+ ǫz)−f(x0− ǫz)| ≥ c holds for each sufficiently
small ǫ > 0. Setting (νp⊙+1, . . . , νq∗)
′ = x0, (µp⊙+1, . . . , µq∗)
′ = z, setting ν[¬q∗], and µ[¬q∗] each equal to
zero, and setting ν[p⊙] and µ[p⊙] to arbitrary values, we see that g(ν ± ǫµ) has the desired properties.
To complete the proof in case s = p∗ = p⊙, we need to establish property (ii) for which it suffices to
show that, for p > p⊙, the p-th term in (37) depends continuously on ν. For p > q
∗, the p-th term does not
depend on ν, because C
∞ = 0 for r = q∗, . . . , P . For p satisfying p⊙ < p ≤ q∗, it suffices to show that
h(νp, . . . , νq∗) = P
Zp ≤ t+
r=p+1
∞ νr, |Wp + νp| ≥ cpσξ∞,p
is a continuous function. Suppose that (ν
p , . . . , ν
q∗ ) converges to (νp, . . . , νq∗) as m→ ∞. For arbitrary
α > 0,
r=p+1 ξ
∞ νr and
r=p+1 ξ
r differ by less than α in each coordinate, provided that
m is sufficiently large. This implies
lim sup
h(ν(m)p , . . . , ν
≤ lim sup
P(Zp ≤ t+
r=p+1
∞ νr + α(1, . . . , 1)
′, |Wp + ν(m)p | ≥ cpσξ∞,p)
= P(Zp ≤ t+
r=p+1
ξ−2∞,rC
∞ νr + α(1, . . . , 1)
′, |Wp + νp| ≥ cpσξ∞,p),
observing that the latter probability is obviously continuous in the single variable νp (since Wp has an ab-
solutely continuous distribution). Letting α decrease to zero we obtain lim supm→∞ h(ν
p , . . . , ν
q∗ ) ≤
h(νp, . . . , νq∗). A similar argument establishes lim infm→∞ h(ν
p , . . . , ν
q∗ ) ≥ P(Zp < t +
r=p+1 ξ
∞ νr, |Wp + νp| ≥ cpσξ∞,p). The proof of the continuity of h is then complete if we can show
that P
Zp ≤ ·, |Wp + νp| ≥ cpσξ∞,p
is continuous or, equivalently, that P
Zp ≤ ·
∣∣|Wp + νp| ≥ cpσξ∞,p
is a continuous cdf. Since p > p⊙, the variance-covariance matrix σ
2A[p]Q[p : p]−1A[p]′ of Zp does only
have non-zero diagonal elements. Consequently, when representing Zp as B(W1, . . . ,Wp)
′, the matrix B
cannot have rows that consist entirely of zeros. The conditional distribution of (W1, . . . ,Wp)
′ given the
event {|Wp + νp| ≥ cpσξ∞,p} is clearly absolutely continuous w.r.t. p-dimensional Lebesgue measure. But
then Lemma A.3 delivers the desired result.
The case where s = p∗ < p⊙ is reduced to the previously discussed case as follows: It is easy to see that,
for νp⊙ → ∞, the expression in (37) converges to a limit uniformly w.r.t. all νp with p 6= p⊙. Then observe
that this limit is again of the form (37) but now with p⊙ taking the rôle of p∗. ✷
B Non-Uniformity of the Convergence of the Finite-Sample Cdf
to the Large-Sample Limit
Proposition B.1 a. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically correlated, i.e., C
∞ 6= 0, for some
q satisfying O < q ≤ P , and let q∗ denote the largest q with this property. Then for every θ ∈Mq∗−1,
every σ, 0 < σ <∞, and every t ∈ Rk there exists a ρ, 0 < ρ <∞, such that
lim inf
ϑ∈Mq∗
||ϑ−θ||<ρ/
|Gn,ϑ,σ(t)−G∞,ϑ,σ(t)| > 0 (41)
holds. The constant ρ may be chosen in such a way that it depends only on t, Q, A, σ, and the critical
values cp for O < p ≤ P .
b. Suppose that Aθ̃(q) and θ̃q(q) are asymptotically uncorrelated, i.e., C
∞ = 0, for all q satisfying O <
q ≤ P . Then Gn,θ,σ converges to Φ∞,P in total variation uniformly in θ ∈ RP ; more precisely
σ∗≤σ≤σ∗
||Gn,θ,σ − Φ∞,P ||TV
n→∞−→ 0
holds for any constants σ∗ and σ
∗ satisfying 0 < σ∗ ≤ σ∗ <∞.
Under the assumptions of Proposition B.1(a), we see that convergence of Gn,θ,σ(t) to G∞,θ,σ(t) is non-
uniform over shrinking ‘tubes’ aroundMq∗−1 that are contained in Mq∗ . [On the complement of a tube with
a fixed positive radius, i.e., on the set U = {θ ∈ RP : |θq∗ | ≥ r} with fixed r > 0, convergence of Gn,θ,σ(t)
to G∞,θ,σ(t) is in fact uniform (even with respect to the total variation distance), as can be shown. Note
that for θ ∈ U the cdf G∞,θ,σ(t) reduces to the Gaussian cdf Φ∞,P (t), i.e., to the asymptotic distribution of
the least-squares estimator based on the overall model; cf. Remark A.6.] A precursor to Proposition B.1(a)
is Corollary 5.5 of Leeb and Pötscher (2003) which establishes (41) in the special case where O = 0 and
where A is the P × P identity matrix. Proposition B.1(b) describes an exceptional case where convergence
is uniform. [In this case G∞,θ,σ reduces to the Gaussian cdf Φ∞,P for all θ and Φ∞,P = Φ∞,p, O ≤ p ≤ P ,
holds; cf. Remark A.6.] Recall that under the assumptions of part (b) of Proposition B.1 we necessarily
always have (i) O > 0, and (ii) rank A[O] = k; cf. Proposition 4.4 in Leeb and Pötscher (2006b).
Proof of Proposition B.1: We first prove part (a). As noted at the beginning of the proof of Lemma
A.4, the map γ 7→ G∞,θ,σ,γ(t) depends only on t, Q, A, σ, the critical values cp for O < p ≤ P , as well as on
θ, but the dependence on θ is only through p∗ = max{p0(θ),O}. It hence suffices to find, for each possible
value of p∗ in the range p∗ = O, . . . , q∗ − 1, a constant 0 < ρ < ∞ such that (41) is satisfied for some (and
hence all) θ returning this particular value of p∗ = max{p0(θ),O}. For this in turn it is sufficient to show
that given such a θ we can find a γ ∈Mq∗ such that
lim inf
|Gn,θ+γ/√n,σ(t)−G∞,θ+γ/√n,σ(t)| > 0 (42)
holds. Note that (42) is equivalent to
lim inf
|G∞,θ,σ,γ(t)−G∞,θ+γ/√n,σ(t)| > 0 (43)
in light of Proposition 2.1. To establish (43), we proceed as follows: For each γ ∈ Mq∗ with γq∗ 6= 0,
G∞,θ+γ/
n,σ(t) in (15) reduces to Φ∞,q∗(t) as is easily seen from (37) since p0(θ+γ/
n) = q∗ which in turn
follows from p0(θ) < q
∗ and γq∗ 6= 0. Furthermore, Lemma A.4 entails that G∞,θ,σ,γ(t) is non-constant in
γ ∈Mq∗\Mq∗−1. But this shows that (43) must hold.
To prove part (b), we write
||Gn,θ,σ − Φ∞,P ||TV =
∣∣∣∣∣∣
∣∣∣∣∣∣
Gn,θ,σ(·|p)πn,θ,σ(p)− Φ∞,P (·)
∣∣∣∣∣∣
∣∣∣∣∣∣
||Gn,θ,σ(·|p)− Φ∞,P (·)||TV πn,θ,σ(p),
where the conditional cdfs Gn,θ,σ(·|p) and the model selection probabilities πn,θ,σ(p) have been introduced
after (12). By the ‘uncorrelatedness’ assumption, we have that Φ∞,p = Φ∞,P for all p in the rangeO ≤ p ≤ P ;
cf. Remark A.6. We hence obtain
σ∗≤σ≤σ∗
||Gn,θ,σ − Φ∞,P ||TV ≤
σ∗≤σ≤σ∗
||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p). (44)
Now for every p with O ≤ p ≤ P and for every ρ, 0 < ρ <∞, we can write
σ∗≤σ≤σ∗
||Gn,θ,σ(·|p)− Φ∞,p(·)||TV πn,θ,σ(p)
≤ max
‖θ[¬p]‖<ρ/
σ∗≤σ≤σ∗
||Gn,θ,σ(·|p)− Φ∞,p(·)||TV , sup
‖θ[¬p]‖≥ρ/
σ∗≤σ≤σ∗
πn,θ,σ(p)
. (45)
In case p = P , we use here the convention that the second term in the maximum is absent and that the first
supremum in the first term in the maximum extends over all of RP . Letting first n and then ρ go to infinity
in (45), we may apply Lemmas C.2 and C.3 in Leeb and Pötscher (2005b) to conclude that the l.h.s. of (45),
and hence the l.h.s. of (44), goes to zero as n→ ∞. ✷
C Proofs for Sections 2.1 to 2.2.2
In the proofs below it will be convenient to show the dependence of Φn,p and Φ∞,p on σ in the notation. Thus,
in the following we shall write Φn,p,σ and Φ∞,p,σ, respectively, for the cdf of a k-variate Gaussian random
vector with mean zero and variance-covariancematrix σ2A[p](X [p]′X [p]/n)−1A[p]′ and σ2A[p]Q[p : p]−1A[p]′,
respectively. For convenience, let Φn,0,σ and Φ∞,0,σ denote the cdf of point-mass at zero in R
The following lemma is elementary to prove, if we recall that bn,pz converges to b∞,pz as n → ∞ for
every z ∈ ImA[p], the column space of A[p].
Lemma C.1 Suppose p > O. Define Rn,p(z, σ) = 1 − ∆σζn,p(bn,pz, cpσξn,p) and R∞,p(z, σ) = 1 −
∆σζ∞,p(b∞,pz, cpσξ∞,p) for z ∈ ImA[p], 0 < σ <∞. Let σ
(n) converge to σ, 0 < σ < ∞. If ζ∞,p 6= 0, then
Rn,p(z, σ
(n)) converges to R∞,p(z, σ) for every z ∈ ImA[p]; if ζ∞,p = 0, then convergence holds for every
z ∈ ImA[p], except possibly for z ∈ ImA[p] satisfying |b∞,pz| = cpσξ∞,p. [This exceptional subset of ImA[p]
has rank(A[p])-dimensional Lebesgue measure zero since cpσξ∞,p > 0.]
The following observation is useful in the proof of Proposition 2.2 below: Since the proposition depends
on Y only through its distribution (cf. Remark 4.1), we may assume without loss of generality that the
errors in (5) are given by ut = σεt, t ∈ N, with i.i.d. εt that are standard normal. In particular, all random
variables involved are then defined on the same probability space.
Proof of Proposition 2.2: Since Pn,θ,σ(p̄ = p0(θ)) → 1 by consistency, we may replace max{p̄,O}
by p∗ = max{p0(θ),O} in the formula for Ǧn for the remainder of the proof. Furthermore, since σ̂ → σ
in Pn,θ,σ-probability, each subsequence contains a further subsequence along which σ̂ → σ almost surely
(with respect to the probability measure on the common probability space supporting all random variables
involved), and we restrict ourselves to such a further subsequence for the moment. In particular, we write
{σ̂ → σ} for the event that σ̂ converges to σ along the subsequence under consideration; clearly, the event
{σ̂ → σ} has probability one. Also note that we can assume without loss of generality that σ̂ > 0 holds
on this event (at least from some data-dependent n onwards), since σ > 0 holds. But then obviously
q=p∗+1
∆σ̂ξn,q (0, cqσ̂ξn,q) converges to
q=p∗+1
∆σξ∞,q (0, cqσξ∞,q), and Φ̂n,p∗(t) converges to Φ∞,p∗,σ(t)
in total variation by Lemma A.3 of Leeb (2005) in case p∗ > 0, and trivially so in case p∗ = 0. This proves
that the first term in the formula for Ǧn converges to the corresponding term in the formula for G∞,θ,σ in
total variation.
Next, consider the term in Ǧn that carries the index p > p∗. By Lemma A.3 in Leeb (2005), Φ̂n,p = Φn,p,σ̂
has a density dΦn,p,σ̂/dΦ∞,p,σ with respect to Φ∞,p,σ, which converges to 1 except on a set that has measure
zero under Φ∞,p,σ. By Scheffé’s Lemma (Billingsley (1995), Theorem 16.12), dΦn,p,σ̂/dΦ∞,p,σ converges to 1
also in the L1(Φ∞,p,σ)-sense. By Lemma C.1, Rn,p(z, σ̂) converges to R∞,p(z, σ) except possibly on a set that
has measure zero under Φ∞,p,σ. (Recall that Φ∞,p,σ is concentrated on ImA[p] and is not degenerate there.)
Observing that |Rn,p(z, σ̂)| is uniformly bounded by 1, we obtain that Rn,p(z, σ̂) converges to R∞,p(z, σ)
also in the L1(Φ∞,p,σ)-sense. Hence,
∥∥∥∥Rn,p(z, σ̂)
dΦn,p,σ̂
dΦ∞,p,σ
(z)−R∞,p(z, σ)
∥∥∥∥Rn,p(z, σ̂)
dΦn,p,σ̂
dΦ∞,p,σ
(z)−Rn,p(z, σ̂)
∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖ (46)
dΦn,p,σ̂
dΦ∞,p,σ
(z)− 1
∥∥∥∥+ ‖Rn,p(z, σ̂)−R∞,p(z, σ)‖
n→∞−→ 0
where ‖·‖ denotes the L1(Φ∞,p,σ)-norm. Since
q=p+1 ∆σ̂ξn,q (0, cqσ̂ξn,q) obviously converges to∏P
q=p+1 ∆σξ∞,q (0, cqσξ∞,q), the relation (46) shows that the term in Ǧn carrying the index p converges
to the corresponding term in G∞,θ,σ in the total variation sense. This proves (18) along the subsequence
under consideration. However, since any subsequence contains such a further subsequence, this establishes
(18). Since Gn,θ,σ converges to G∞,θ,σ in total variation by Proposition 2.1, the claim in (17) also follows.✷
Before we prove the main result we observe that the total variation distance between Pn,θ,σ and
Pn,ϑ,σ satisfies ||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ
max(X
′X)/2σ) − 1; furthermore, if θ(n) and ϑ(n) sat-
∥∥∥θ(n) − ϑ(n)
∥∥∥ = O(n−1/2), the sequence Pn,ϑ(n),σ is contiguous with respect to the sequence Pn,θ(n),σ
(and vice versa). This follows exactly in the same way as Lemma A.1 in Leeb and Pötscher (2006a).
Proof of Theorem 2.3: We first prove (20) and (21). For this purpose we make use of Lemma 3.1 in
Leeb and Pötscher (2006a) with α = θ ∈ Mq∗−1, B = Mq∗ , Bn = {ϑ ∈ Mq∗ : ‖ϑ− θ‖ < ρ0n−1/2}, β = ϑ,
ϕn(β) = Gn,ϑ,σ(t), ϕ̂n = Ĝn(t), where ρ0, 0 < ρ0 < ∞, will be chosen shortly (and σ is held fixed). The
contiguity assumption of this lemma (as well as the mutual contiguity assumption used in the corrigendum
to Leeb and Pötscher (2006a)) is satisfied in view of the preparatory remark above. It hence remains only to
show that there exists a value of ρ0, 0 < ρ0 < ∞, such that δ
in Lemma 3.1 of Leeb and Pötscher (2006a)
(which represents the limit inferior of the oscillation of ϕn(·) over Bn) is positive. Applying Lemma 3.5(i)
of Leeb and Pötscher (2006a) with ζn = ρ0n
−1/2 and the set G0 equal to the set G, it remains, in light
of Proposition 2.1, to show that there exists a ρ0, 0 < ρ0 < ∞, such that G∞,θ,σ,γ(t) as a function of γ is
non-constant on the set {γ ∈ Mq∗ : ‖γ‖ < ρ0}. In view of Lemma 3.1 of Leeb and Pötscher (2006a), the
corresponding δ0 can then be chosen as any positive number less than one-half of the oscillation of G∞,θ,σ,γ(t)
over this set. That such a ρ0 indeed exists follows now from Lemma A.4 in Appendix A, where it is also
shown that ρ0 and δ0 can be chosen such that they depend only on t, Q,A, σ, and cp for O < p ≤ P . This
completes the proof of (20) and (21).
To prove (22) we use Corollary 3.4 in Leeb and Pötscher (2006a) with the same identification of
notation as above, with ζn = ρ0n
−1/2, and with V = Mq∗ (viewed as a vector space isomorphic to
). The asymptotic uniform equicontinuity condition in that corollary is then satisfied in view of
||Pn,θ,σ − Pn,ϑ,σ||TV ≤ 2Φ(‖θ − ϑ‖λ
max(X
′X)/2σ) − 1. Given that the positivity of δ∗ has already be es-
tablished in the previous paragraph, applying Corollary 3.4(i) in Leeb and Pötscher (2006a) then establishes
(22). ✷
Proof of Remark 2.4: The proof is similar to the proof of (22) just given, except for using Corol-
lary 3.4(ii) and Lemma 3.5(ii) in Leeb and Pötscher (2006a) instead of Corollary 3.4(i) and Lemma 3.5(i)
from that paper. Furthermore, Lemma A.5 in Appendix A instead of Lemma A.4 is used. ✷
Proof of Proposition 2.5: In view of Proposition B.1(b) and the fact that Φ̂n,P (·) = Φn,P,σ̂(·) holds
(in case σ̂ > 0), it suffices to show that
σ∗≤σ≤σ∗
||Φn,P,σ(·)− Φ∞,P,σ(·)||TV
n→∞−→ 0 (47)
σ∗≤σ≤σ∗
Pn,θ,σ
||Φn,P,σ̂(·)− Φn,P,σ(·)||TV > δ
) n→∞−→ 0 (48)
hold for each δ > 0, and for any constants σ∗ and σ
∗ satisfying 0 < σ∗ ≤ σ∗ <∞. [Note that the probability
in (48) does in fact not depend on θ.] But this has already been established in the proof of Proposition 4.3
of Leeb and Pötscher (2005b). ✷
D Proofs for Section 3
Proof of Theorem 3.1: After rearranging the elements of θ (and hence the regressors) if necessary and
then correspondingly rearranging the rows of the matrix A, we may assume without loss of generality that
r∗ = (1, . . . , 1, 0), and hence that i(r∗) = P . That is, Mr∗ = MP−1 and Mrfull = MP . Furthermore, note
that after this arrangement C
∞ 6= 0. Let p̂ be the model selection procedure introduced in Section 2 with
O = P − 1, cP = c, and cO = 0. Let θ̃ be the corresponding post-model-selection estimator and let Gn,θ,σ(t)
be as defined in Section 2.1. Condition (24) now implies: For every θ ∈ MP−1 which has exactly P − 1
non-zero coordinates
Pn,θ,σ ({r̂ = rfull}N{p̂ = P}) = lim
Pn,θ,σ ({r̂ = r∗}N{p̂ = P − 1}) = 0 (49)
holds for every 0 < σ < ∞. Since the sequences Pn,ϑ(n),σ and Pn,θ,σ are contiguous for ϑ
(n) satisfying∥∥∥θ − ϑ(n)
∥∥∥ = O(n−1/2) as remarked prior to the proof of Theorem 2.3 in Appendix C, it follows that
condition (49) continues to hold with Pn,ϑ(n),σ replacing Pn,θ,σ. This implies that for every sequence of
positive real numbers sn with sn = O(n
−1/2), for every σ, 0 < σ < ∞, and for every θ ∈ MP−1 which has
exactly P − 1 non-zero coordinates
||ϑ−θ||<sn
‖Kn,ϑ,σ −Gn,ϑ,σ‖TV → 0 (50)
holds as n → ∞. From (50) we conclude that the limit of Kn,θ+γ/√n,σ (with respect to total variation
distance) exists and coincides with G∞,θ,σ,γ . Repeating the proof of Theorem 2.3 with q
∗ = P , with
Kn,ϑ,σ(t) replacing Gn,ϑ,σ(t), and with K̂n(t) replacing Ĝn(t) gives the desired result. ✷
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|
0704.1585 | Fixed Phase Quantum Search Algorithm | Fixed Phase Quantum Search Algorithm
Ahmed Younes∗
Department of Math. & Comp. Science
Faculty of Science
Alexandria University
Alexandria, Egypt
October 24, 2018
Abstract
Building quantum devices using fixed operators is a must to simplify the hardware con-
struction. Quantum search engine is not an exception. In this paper, a fixed phase quantum
search algorithm that searches for M matches in an unstructured search space of size N
will be presented. Selecting phase shifts of 1.91684π in the standard amplitude amplification
will make the technique perform better so as to get probability of success at least 99.58% in
better than any know fixed operator quantum search algorithms. The algorithm
will be able to handle either a single match or multiple matches in the search space. The
algorithm will find a match in O
whether the number of matches is known or not
in advance.
1 Introduction
In 1996, Lov Grover [10] presented an algorithm that quantum mechanically searches an unstruc-
tured list assuming that a unique match exists in the list with quadratic speed-up over classical
algorithms. To be able to define the target problem of this paper, we have to organize the ef-
forts done by others in that field. The unstructured search problem targeted by Grover’s original
algorithm is deviated in the literature to the following four major problems:
• Unstructured list with a unique match.
• Unstructured list with one or more matches, where the number of matches is known
• Unstructured list with one or more matches, where the number of matches is unknown.
• Unstructured list with strictly multiple matches.
The efforts done in all the above cases, similar to Grover’s original work, used quantum paral-
lelism by preparing superposition that represents all the items in the list. The superposition could
be uniform or arbitrary. The techniques used in most of the cases to amplify the amplitude(s) of
the required state(s) have been generalized to an amplitude amplification technique that iterates
∗[email protected]
http://arxiv.org/abs/0704.1585v2
the operation URs (φ)U
†Rt (ϕ), on U |s〉 where U is unitary operator, Rs (φ) = I− (1− eiφ) |s〉 〈s|,
Rt (ϕ) = I − (1− eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target state(s)
and I is the identity operator.
Grover’s original algorithm replaces U be W , where W is the Walsh-Hadamard transform, pre-
pares the superpositionW |0〉 (uniform superposition) and iteratesWRs (π)WRt (π) for O
where N is the size of the list, which was shown be optimal to get the highest probability with the
minimum number of iterations [23], such that there is only one match in the search space.
In [11, 15, 9, 17, 1], Grover’s algorithm is generalized by showing that U can be replaced by
almost any arbitrary superposition and the phase shifts φ and ϕ can be generalized to deal with the
arbitrary superposition and/or to increase the probability of success even with a factor increase
in the number of iterations to still run in O(
N). These give a larger class of algorithms for
amplitude amplification using variable operators from which Grover’s algorithm was shown to be
a special case.
In another direction, work has been done trying to generalize Grover’s algorithm with a uniform
superposition for known number of multiple matches in the search space [3, 8, 7, 6], where it was
shown that the required number of iterations is approximately π/4
N/M for small M/N , where
M is the number of matches. The required number of iterations will increase for M > N/2,
i.e. the problem will be harder where it might be excepted to be easier [19]. Another work has
been done for known number of multiple matches with arbitrary superposition and phase shifts
[18, 2, 4, 14, 16] where the same problem for multiple matches occurs. In [5, 18, 4], a hybrid
algorithm was presented to deal with this problem by applying Grover’s fixed operators algorithm
for π/4
N/M times then apply one more step using specific φ and ϕ according to the knowledge
of the number of matches M to get the solution with probability close to certainty. Using this
algorithm will increase the hardware cost since we have to build one more Rs and Rt for each
particular M . For the sake of practicality, the operators should be fixed for any given M and are
able to handle the problem with high probability whether or notM is known in advance. In [21, 22],
Younes et al presented an algorithm that exploits entanglement and partial diffusion operator to
perform the search and can perform in case of either a single match or multiple matches where
the number of matches is known or not [22] covering the whole possible range, i.e. 1 ≤ M ≤ N .
Grover described this algorithm as the best quantum search algorithm [12]. It can be shown that
we can get the same probability of success of [21] using amplitude amplification with phase shifts
φ = ϕ = π/2, although the amplitude amplification mechanism will be different. The mechanism
used to manipulate the amplitudes could be useful in many applications, for example, superposition
preparation and error-correction.
For unknown number of matches, an algorithm for estimating the number of matches (quantum
counting algorithm) was presented [5, 18]. In [3], another algorithm was presented to find a match
even if the number of matches is unknown which will be able to work if M lies within the range
1 ≤M ≤ 3N/4 [22].
For strictly multiple matches, Younes et al [20] presented an algorithm which works very ef-
ficiently only in case of multiple matches within the search space that splits the solution states
over more states, inverts the sign of half of them (phase shift of -1) and keeps the other half
unchanged every iteration. This will keep the mean of the amplitudes to a minimum for multiple
matches. The same result was rediscovered by Grover using amplitude amplification with phase
shifts φ = ϕ = π/3 [13], in both algorithms the behavior will be similar to the classical algorithms
in the worst case.
In this paper, we will propose a fixed phase quantum search algorithm that runs inO
This algorithm is able to handle the range 1 ≤ M ≤ N for both known and unknown number of
matches more reliably than known fixed operator quantum search algorithms that target this case.
The plan of the paper is as follows: Section 2 introduces the general definition of the target
unstructured search problem. Section 3 presents the algorithm for both known and unknown
number of matches. The paper will end up with a general conclusion in Section 4.
2 Unstructured Search Problem
Consider an unstructured list L of N items. For simplicity and without loss of generality we will
assume that N = 2n for some positive integer n. Suppose the items in the list are labeled with the
integers {0, 1, ..., N − 1}, and consider a function (oracle) f which maps an item i ∈ L to either
0 or 1 according to some properties this item should satisfy, i.e. f : L → {0, 1}. The problem
is to find any i ∈ L such that f(i) = 1 assuming that such i exists in the list. In conventional
computers, solving this problem needs O (N/M) calls to the oracle (query),where M is the number
of items that satisfy the oracle.
3 Fixed Phase Algorithm
3.1 Known Number of Matches
Assume that the system is initially in state |s〉 = |0〉. Assume that
denotes a sum over i
which are desired matches, and
denotes a sum over i which are undesired items in the list.
So, Applying U |s〉 we get,
∣ψ(0)
= U |s〉 = 1√
′ |i〉+ 1√
′′ |i〉, (1)
where U =W and the superscript in
∣ψ(0)
represents the iteration number.
Let M be the number of matches, sin(θ) =
M/N and 0 < θ ≤ π/2, then the system can be
re-written as follows,
∣ψ(0)
= sin(θ) |ψ1〉+ cos(θ) |ψ0〉 , (2)
where |ψ1〉 = |t〉 represents the matches subspace and |ψ0〉 represents the non-matches subspace.
Let D = URs (φ)U
†Rt (ϕ), Rs (φ) = I − (1 − eiφ) |s〉 〈s|, Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t| and set
φ = ϕ as the best choice [14]. Applying D on
∣ψ(0)
we get,
∣ψ(1)
∣ψ(0)
= a1 |ψ1〉+ b1 |ψ0〉 , (3)
such that,
a1 = sin(θ)(2 cos (δ) e
iφ + 1), (4)
b1 = e
iφ cos(θ)(2 cos (δ) + 1), (5)
where cos (δ) = 2 sin2(θ) sin2(φ
)− 1.
Let q represents the required number of iterations to get a match with the highest possible
probability. After q applications of D on
∣ψ(0)
we get,
∣ψ(q)
∣ψ(0)
= aq |ψ1〉+ bq |ψ0〉 , (6)
such that,
aq = sin(θ)
eiqφUq (y) + e
i(q−1)φUq−1 (y)
, (7)
bq = cos(θ)e
i(q−1)φ (Uq (y) + Uq−1 (y)) , (8)
where y = cos(δ) and Uq is the Chebyshev polynomial of the second kind defined as follows,
Uq (y) =
sin ((q + 1) δ)
sin (δ)
. (9)
Let P qs represents the probability of success to get a match after q iterations and P
ns is the
probability not to get a match after applying measurement, so P qs = |aq|
and P qns = |bq|
that P qs +P
ns = 1. To calculate the required number of iterations q to get a match with certainty,
one the following two approaches might be followed:
• Analytically. The usual approach used in the literature when the number of matches M is
known in advance is to equate P qs to 1 or P
ns to 0 and then find an algebraic formula that
represents the required number of iterations, as well as, the phase shifts φ and ϕ in terms on
M . Using this approach is not possible for the case that the phase shifts should be fixed for
an arbitrary M such that 1 ≤M ≤ N as shown in the following theorem.
Theorem 3.1 (No Certainty Principle) Let D be an amplitude amplification operator
such that D = URs (φ)U
†Rt (ϕ), where U is unitary operator, Rs (φ) = I − (1 − eiφ) |s〉 〈s|,
Rt (ϕ) = I − (1 − eiϕ) |t〉 〈t|, |s〉 is the initial state of the system, |t〉 represents the target
state(s) and I is the identity operator. Let D performs on a system initially set to U |s〉.
If the phase shifts φ and ϕ should be fixed, then iterating D an arbitrary number of times
will not find a match with certainty for an arbitrary known number of matches M such that
1 ≤M ≤ N .
Proof To prove this theorem, we will use the usual approach, i.e. start with P qs = 1 or
P qns = 0 and calculate the required number of iterations q.
Since P qs = |aq|
and from Eqn.7, we can re-write P qs as follows setting φ = ϕ as the best
choice [14],
P qs =
sin2 (θ)
sin2 (δ)
(1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)) . (10)
Setting P qs = 1 and using simple trigonometric identities we get, q =
, i.e. the required
number of iterations is independent of M , φ and ϕ, and represents an impossible value for a
required number of iterations.
• Direct Search. The alternative approach used in this paper is to empirically assume an
algebraic form for the required number of iterations that satisfy the quadratic speed-up of
the known quantum search algorithms and use a computer program to search for the best
phase shift φ that satisfy the condition,
max (min (P qs (φ))) such that 0 ≤ φ ≤ 2π and, 1 ≤ M ≤ N. (11)
i.e. find the value of φ that maximize the minimum value of P qs over the range 1 ≤M ≤ N .
0 0.2 0.4 0.6 0.8 1
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
Probability of Success Probability lower bound
Figure 1: The probability of success the proposed algorithm after the required number of iterations.
Assume that q =
sin(θ)
. Using this form for q, a computer program has been
written using C language to find the best φ with precision 10−15 that satisfy the conditions shown
in Eqn. 11. The program shows that using φ = 6.021930660106538 ≈ 1.91684π, the minimum
probability of success will be at least 99.58% compared with 87.88 % for Younes et al [22] and
50% for the original Grover’s algorithm [3] as shown in Fig. 2. To prove these results, using
φ = 1.91684π, the lower bound for the probability of success is as follows as shown in Fig. 1.
P qs =
sin2(θ)
sin2(δ)
(1− cos (δ) cos ((2q + 1) δ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ))
sin2(θ)
sin2(δ)
(1− cos (δ) cos ((2q + 1) δ) + cos (φ) cos (δ)− cos (φ) cos ((2q + 1) δ))
≥ sin
sin2(δ)
(1 + cos2 (δ) + 2 cos (φ) cos (δ)) ≥ 0.9958.
where, cos (δ) = 2 sin2(θ) sin2(φ
)− 1, 0 < θ ≤ π/2, and cos ((2q + 1) δ) ≤ −cos(δ).
3.2 Unknown Number of Matches
In case we do not know the number of matches M in advance, we can apply the algorithm shown
in [3] for 1 ≤M ≤ N by replacing Grover’s step with the proposed algorithm. The algorithm can
be summarized as follows,
1- Initialize m = 1 and λ = 8/7. (where λ can take any value between 1 and 4/3)
2- Pick an integer j between 0 and m− 1 in a uniform random manner.
3- Run j iterations of the proposed algorithm on the state
∣ψ(0)
∣ψ(j)
∣ψ(0)
. (13)
0 0.2 0.4 0.6 0.8 1
Grover’s
Younes et al[21]
Fixed Phase
Figure 2: The probability of success of Grover’s algorithm, Younes et al algorithm [21] and the
proposed algorithm after the required number of iterations.
4- Measure the register
∣ψ(j)
and assume i is the output.
5- If f(i) = 1, then we found a solution and exit.
6- Set m = min
and go to step 2.
where m represents the range of random numbers (step 2), j represents the random number of
iterations (step3), and λ is a factor used to increase the range of random numbers after each trial
(step 6).
For the sake of simplicity and to be able to compare the performance of this algorithm with
that shown in [3], we will try to follow the same style of analysis used in [3]. Before we construct
the analysis, we need the following lemmas.
Lemma 3.2 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1,
0 < θ ≤ π/2 where c = 2 sin2(φ
) is a constant,
sin2 ((q + 1) δ) + sin2 (qδ) = m− cos (δ) sin (2mδ)
2 sin (δ)
Proof By mathematical induction.
Lemma 3.3 For any positive integer m and real numbers θ, δ such that cos (δ) = c sin2(θ) − 1,
0 < θ ≤ π/2 where c = 2 sin2(φ
) is a constant,
sin ((q + 1) δ) sin (qδ) =
cos (δ)− sin (2mδ)
4 sin (δ)
Proof By mathematical induction.
Lemma 3.4 AssumeM is the unknown number of matches such that 1 ≤M ≤ N . Let θ, δ be real
numbers such that cos (δ) = 2 sin2(θ) sin2(φ
)− 1, sin2(θ) =M/N , φ = 1.91684π and 0 < θ ≤ π/2.
Let m be any positive integer. Let q be any integer picked in a uniform random manner between
0 and m− 1. Measuring the register after applying q iterations of the proposed algorithm starting
from the initial state, the probability Pm of finding a solution is as follows,
c (1− cos (δ))
1 + cos (δ) cos (φ)− (cos (δ) + cos (φ)) sin (2mδ)
2m sin (δ)
where c = 2 sin2(φ
), then Pm ≥ 1/4 for m ≥ 1/ sin (δ) and small M/N .
Proof The average probability of success when applying q iterations of the proposed algorithm
when 0 ≤ q ≤ m is picked in a uniform random manner is as follows,
sin2(θ)
m sin2(δ)
sin2 ((q + 1) δ) + sin2 (qδ) + 2 cos (φ) sin ((q + 1) δ) sin (qδ)
sin2(θ)
m sin2(δ)
m− cos(δ) sin(2mδ)
2 sin(δ)
+ cos (φ) cos (δ)− cos(φ) sin(2mδ)
2 sin(δ)
c(1−cos(δ))
1 + cos (δ) cos (φ)− (cos(δ)+cos(φ)) sin(2mδ)
2m sin(δ)
If m ≥ 1/ sin (δ) and M ≪ N then cos (δ) ≈ −1, so,
1− cos (φ)− (cos (φ)− 1) sin (2mδ)
1− cos (φ)− (1− cos (φ))
= 0.25
where −1 ≤ sin (2mδ) ≤ 1 for 0 < θ ≤ π/2.
We calculate the total expected number of iterations as done in Theorem 3 in [3]. Assume that
mq ≥ 1/ sin (δ), and vq = ⌈logλmq⌉. Notice that, mq = O
for 1 ≤M ≤ N , then:
1- The total expected number of iterations to reach the critical stage, i.e. when m ≥ mq:
λv−1 ≤ 1
2 (λ− 1)
mq = 3.5mq. (14)
2- The total expected number of iterations after reaching the critical stage:
λvq+u =
2 (1− 0.75λ)
mq = 3.5mq. (15)
The total expected number of iterations whether we reach to the critical stage or not is 7mq
which is in O(
N/M) for 1 ≤M ≤ N .
When this algorithm employed Grover’s algorithm, and based on the conditionmG ≥ 1/ sin (2θG) =
for M ≤ 3N/4,the total expected number of iterations is approximately 8mG for
1 ≤ M ≤ 3N/4. Employing the proposed algorithm instead, and based on the condition
0 0.2 0.4 0.6 0.8 1
s Fixed Phase
Younes et al[22]
Grover’s
Figure 3: The actual behavior of the functions representing the total expected number of iterations
for Grover’s algorithm, Younes et al algorithm [22] and the proposed algorithm taking λ = 8/7,
where the number of iterations is the flooring of the values (step function).
mq ≥ 1/ sin (δ) = O
,the total expected number of iterations is approximately 7mq
for 1 ≤M ≤ N , i.e. the algorithm will be able to handle the whole range, since mq will be able to
act as a lower bound for q over 1 ≤ M ≤ N . Fig. 3 compares between the total expected number
of iterations for Grover’s algorithm, Younes et al algorithm [22] and the Fixed Phase algorithm
taking λ = 8/7.
4 Conclusion
To be able to build a practical search engine, the engine should be constructed from fixed operators
that can handle the whole possible range of the search problem, i.e. whether a single match or
multiple matches exist in the search space. It should also be able to handle the case where the
number of matches is unknown. The engine should perform with the highest possible probability
after performing the required number of iterations.
In this paper, a fixed phase quantum search algorithm is presented. It was shown that selecting
the phase shifts to 1.91684π could enhance the searching process so as to get a solution with
probability at least 99.58%. The algorithm still achieves the quadratic speed up of Grover’s
original algorithm. It was shown that Younes et al algorithm [22] might perform better in case
the number of matches is unknown, although the presented algorithm might scale similar with an
acceptable delay. i.e. both run in O
. In that sense, the Fixed Phase algorithm can act
efficiently in all the possible classes of the unstructured search problem.
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Introduction
Unstructured Search Problem
Fixed Phase Algorithm
Known Number of Matches
Unknown Number of Matches
Conclusion
|
0704.1586 | Core excitation in the elastic scattering and breakup of $^{11}$Be on
protons | Core excitation in the elastic scattering and breakup of 11Be on protons
N. C. Summers1, 2, 3 and F. M. Nunes3, 4
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854
National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824
(Dated: October 25, 2018)
The elastic scattering and breakup of 11Be from a proton target at intermediate energies is studied.
We explore the role of core excitation in the reaction mechanism. Comparison with the data suggests
that there is still missing physics in the description.
PACS numbers: 24.10.Eq, 25.40.Cm, 25.60.Gc, 27.20.+n
I. INTRODUCTION
Nuclear reactions offer the most diverse methods to
study nuclei at the limit of stability. Understanding re-
action mechanisms in nuclear processes involving nuclei
near the driplines is of great importance, particularly at
this time, when there is such a high demand for accuracy
on the structure information to be extracted from the
data. Reaction and structure models are undoubtedly
entangled, therefore improving reaction models often im-
plies incorporating more detailed structure models in the
description [1].
It is generally accepted that, in reactions with loosely
bound nuclei, the coupling to the continuum needs to be
considered. Continuum effects are very much enhanced
in breakup but can also have imprints on other reac-
tion channels, for example elastic and inelastic scatter-
ing. One framework that explicitly includes continuum
effects is the Continuum Discretized Coupled Channel
(CDCC) method [2]. A large amount of work has been
devoted to the analysis of experiments within this frame-
work [3, 4, 5, 6, 7] and in general results are very good.
The eXtended Continuum Discretized Coupled Chan-
nel (XCDCC) method [8, 9] was recently developed. It
brings together a coupled channel description of the pro-
jectile with a coupled channel model of the reaction, en-
abling the description of interference between the multi-
channel components of the projectile as well as dynam-
ical excitation of the core within the projectile, during
the reaction [9]. The model has been applied to the
breakup of 11Be→10Be+n and 17C→16C+n on 9Be at
≈ 60 MeV/nucleon [8, 9]. The projectiles are described
within a two-body core + n multi-channel model, where
the core can be in the ground state but also in an ex-
cited state. This model produces breakup cross sections
to specific final states of the core, given a coupled chan-
nel Hamiltonian for the projectile. Results presented in
Ref. [8, 9] show that core excitation effects in the total
cross section to the ground state of the core are small,
but become very large when considering the total popula-
tion of the core’s excited state. Other differences can be
seen in angular and energy distributions but at present
no such data is available. The effect of core excitation
needs to be studied in other regimes, for very light and
very heavy targets, as well as a variety of energies.
In this work we concentrate on the proton target. In
this case the process is nuclear driven, and recoil effects
are very important. Several reactions of 11Be on protons
have been measured in a number of facilities, namely
elastic scattering at 50 MeV/nucleon [10], quasi-elastic
and breakup at 64 MeV/nucleon [11] as well as transfer
at 35 MeV/nucleon [12].
11Be proton elastic scattering at 49.3 MeV/nucleon
was performed in GANIL [10], at the same time as the
elastic scattering of the core 10Be at 59.4 MeV/nucleon.
Even though the outcoming 11Be measurements corre-
spond to quasi-elastic, these are essentially elastic as
the contribution from the first excited state is negligible.
Standard optical potentials (either density folding as in
JLM [13] or global optical potentials coming from elas-
tic fits as in CH89 [14]) could reproduce the 10Be elastic
reasonably well, requiring small renormalizations of the
real and imaginary parts of the interaction (λV = 0.9 and
λW = 1.1 for CH89) [10] . Larger renormalizations were
required in order to reproduce the distribution of 11Be
(λV = 0.7 or λW = 1.3 for CH89) [10].
It is clear from Ref. [10] that the global optical poten-
tial overestimates the elastic cross section for 11Be. In
Ref. [15] the elastic scattering of 11Be on 12C was success-
fully described using a 10Be+n two-body model, incor-
porating breakup effects. As the 10Be-target interaction
was fixed by the 10Be elastic scattering data, the large
modification in the 11Be+12C elastic data was described,
without renormalization, purely through breakup effects.
Due to the loosely bound nature of the last neutron in
11Be this loss of flux from the elastic channel can be at-
tributed to breakup into 10Be+n. It is thus possible that
the large renormalizations for 11Be scattering on protons
[10] are also due to breakup effects.
In Ref. [11], elastic data is only described after large
renormalizations of both the real and imaginary part of
the 10Be+p interaction (λV = 0.75 and λW = 1.8), much
larger than those used in Ref. [10]. These same renormal-
izations can no longer describe the 10Be+p elastic data
from Ref. [10], and are inconsistent with few-body reac-
tion theory. We will re-examine the elastic scattering of
http://arxiv.org/abs/0704.1586v1
10/11Be+p to see if one can consistently describe both
sets of data using the same interaction for 10Be+p, by
including continuum and core excitation.
In addition to elastic and inelastic measurements,
breakup data from NSCL exist at 63.7 MeV/nucleon [11].
This breakup data is integrated into two wide energy
bins due to statistics. The lower energy bin covers the
1.78 MeV resonance and a reasonable angular distribu-
tion is obtained, which underpredicts the cross section
[11]. The higher energy bin covers resonances that are
thought to be built on excited core states. The cal-
culations presented in Ref. [11] failed to reproduce the
shape of this higher energy bin, and the authors suggest
that the source of the disagreement may be due to an
active core during the reaction. Now that it is possi-
ble to include core excitation in the reaction mechanism
[9] we will re-examine the breakup data using a consis-
tent 10Be+p interaction, and including systematically the
coupling to the 2+ state in 10Be.
Transfer reactions have also been performed with the
11Be beam, at 35.3 MeV/u in GANIL [12] with the aim
of extracting spectroscopic factors for the ground state.
While the reaction mechanisms proved to be more com-
plicated than the 1-step DWBA theory, results for (p,d)
show evidence for a significant core excited component.
The inverse reaction, 10Be(d,p)11Be, has also been stud-
ied in GANIL [16], the main interest being the resonance
structure of 11Be. This illustrates how transfer is being
used beyond the standard application of spectroscopy of
bound states, underlining the need to better understand
the transfer mechanism and its coupling to the contin-
All these different data offer a good testing ground for
theory. A comprehensive theoretical study [17] focusing
on 10Be(d,p) show inconsistencies of the extracted spec-
troscopic factors for data at different energies. Optical
potential uncertainties and core excitation effects could
be at the heart of the problem.
In this work we perform calculations including elastic,
inelastic, and breakup channels of 11Be on protons at in-
termediate energies. We explore explicitly the effect of
the inclusion of core excitation in the reaction mecha-
nism. Comparison to elastic and breakup data will be
presented here. The analysis of the inelastic channel is
presented in [18] and we leave a detailed study of the
transfer channel for a future publication. In section II
we provide the details of the calculations. In section III
we present the results: first for the elastic channel (III A),
then for the breakup (III B). Finally, in section IV, we
draw our conclusions and provide an outlook into the
future.
II. DETAILS OF THE CALCULATIONS
The calculations for breakup of loosely bound systems
on protons have a rather different convergence require-
ment as compared to the breakup on heavier systems.
energy V RV aV W RW aW
40 60.84 1.000 0.7 23.16 0.600 0.6
60 31.64 1.145 0.69 8.78 1.134 0.69
TABLE I: 10Be-proton Woods-Saxon potential parameters.
All energies are in MeV and lengths in fm.
The model space needs to span large excitation energies,
while the radial dependence can be reduced significantly.
For the CDCC calculations at 40 MeV/nucleon, the
continuum was discretized upto 35 MeV, with 10 bins
upto 10 MeV for s-, p-, and d-waves, and 8 bins from
10–35 MeV. We include 12 bins from 0–35 MeV for all
other partial waves up to lmax = 4. The same binning
scheme was used for the XCDCC calculations, except
that the higher bin density upto 10 MeV was only used
for channels with outgoing ground state core components.
Partial waves up to lmax = 4 were used for the coupled
channels projectile states.
For the 60 MeV/nucleon CDCC calculations, a slightly
different binning scheme was adopted to match the ex-
perimental energy bin integrations. From 0–0.5 MeV, 2
bins were used; over the observed energy bins from 0.5–
3.0 and 3.0–5.5 MeV, 3 bins were used in each case; and
from 5.5–30 MeV, 6 bins. For the XCDCC calculations
where the outgoing channel had excited core states, only
1 bin was used from 0–0.5 MeV, 1 bin for each observed
energy range, and 5 bins above.
The radial integrals for the bins were calculated upto
40 fm in steps of 0.1 fm. The radial equations in the
CDCC method were calculated for 30 partial waves with
the lower radial cutoff for the integrals set to 4 fm inside
the point Coulomb radius, and matched to the asymp-
totic Coulomb functions at 150 fm.
The 11Be bound state potential parameters are taken
from Ref. [19], using the Be12-pure interaction for the
CDCC calculations and the Be12-b for the XCDCC cal-
culations. The 10Be-proton interaction is fitted to the
proton elastic data available at the two energies. A good
fit could be obtained from a renormalized CH89 interac-
tion [14]. The parameters are given in Table I. For the
cases including 10Be excitation, the OM potentials were
deformed with the same β2 deformations as used in the
11Be bound state. The coupling matrix elements to the
excited state in 10Be assume a rotational model with the
deformation fitted to the experimental B(E2) strength
[20]. The deformation length is in good agreement with
that obtained from inelastic scattering ot the 2+ state in
10Be [21], and the optical potential used here reproduces
the angular distribution of the inelastic scattering well.
0 20 40 60 80
(deg)
Be OM
Be CF
Be CDCC
Be XCDCC
FIG. 1: (Color online) 10/11Be elastic scattering on a proton
target at ∼40 MeV/nucleon, using an optical model fit to the
10Be elastic and various 11Be reaction models for the 11Be
data. The experimental data are from GANIL [22].
0 20 40 60
(deg)
Be OM
Be CDCC
Be XCDCC
FIG. 2: (Color online) 10/11Be elastic scattering on a proton
target at ∼60 MeV/nucleon, using an optical model fit to the
10Be elastic and various 11Be reaction models for the 11Be
data. The experimental data are from Ref. [10] (10Be) and
Ref. [11] (11Be).
III. RESULTS
A. Elastic channel
The elastic scattering is the first test on the reaction
model. In Fig. 1 we show the 10Be and 11Be elastic data
and theoretical calculations at ∼40 MeV/nucleon. The
optical model for the 10Be (dashed/black line) is fitted to
the 10Be elastic data (open circles). The cluster folding
model (dotted/red line) folds the 10Be+p and n-p interac-
tions over the 11Be ground state wave function to produce
the 11Be+p potential. Also shown in Fig. 1 is the effect of
the 11Be continuum within CDCC (dot-dashed/blue line)
and core excitation within XCDCC (solid line). Even
though there is significant improvement over the simple
optical model when including breakup, results still over-
estimate the 11Be elastic cross section at larger angles,
and no improvement is found by including excited core
contributions.
Calculations were repeated for a higher energy, around
60 MeV/nucleon, where both 10Be and 11Be elastic data
exist. Once again, when the 10Be data is fitted with an
optical model, and the 11Be elastic is described within the
CDCC approach, the cross section is over-estimated (see
Fig. 2). Note that the data at this energy does not span
a large angular range, but it is evident that the pattern
of over-predicting the 11Be cross section remains.
Other reaction calculations have been performed in an
effort to describe this data [23], which consisted of a
transfer to the continuum approach in which the breakup
continuum was described using the deuteron basis. This
also failed to describe the data when the 10Be potential
was fixed to the elastic data. The same pattern of over-
predicting the 11Be elastic was also seen at a lower energy
of ∼40 MeV/nucleon [23].
As pointed out earlier, in [10, 11] large renormaliza-
tion factors were needed to reproduce the elastic cross
section. By including more relevant reaction channels,
one might account for a part of the renormalization re-
quired, corresponding to flux that is being removed from
the elastic channel. This suggests that there are still
channels coupled to the elastic that have not been con-
sidered. Preliminary calculations including the deuteron
transfer channel along with the breakup in the 11Be basis
show improvement at small angles, but the disagreement
still remains at large angles. Due to large non-orthonality
corrections, CDCC calculations including the deuteron
transfer coupling turn out to be numerically challenging.
They will be discussed in a later publication.
B. Breakup: comparison with data at ∼60
MeV/nucleon
Breakup data was also obtained at 63.7 MeV/nucleon
[11], summed into two energy bins. The first covers the
energy range 0.5–3.0 MeV, which spans the 1.78 MeV res-
onance, predominantly a d-wave neutron coupled to the
ground state of the core. The second energy bin is over
the energy range 3.0–5.5 MeV, which spans a resonance
at 3.89 MeV, thought to be predominantly an s-wave
neutron coupled to a 10Be(2+) core [11]. In Ref. [11],
CDCC results were presented which underestimated the
cross section for the lower energy bin, but did not repro-
duce the higher energy bin. It was suggested that since
the higher energy bin spanned a resonance with a pos-
sible excited core component, the disagreement could be
due to the spectator core approximation in the standard
CDCC theory. Since XCDCC can handle excited core
components, this data is re-examined.
The breakup angular distribution data and the asso-
ciated theory prediction for the lower energy bin (0.5–
3.0 MeV) and the higher energy bin (3.0–5.5 MeV) are
presented in Figs. 3 and 4 (the equivalent of Figures 3b
and 3c of Ref. [11]). Firstly, the CDCC calculations of
Ref. [11] were redone, with a higher CDCC bin density.
0 20 40
(deg)
XCDCC
Be+n)p @ 63.7 MeV/nucleon
0.5-3.0 MeV
FIG. 3: 11Be breakup at 60 MeV/nucleon with the relative
energy between breakup fragments in the range 0.5–3.0 MeV.
Experimental data are from Ref. [11].
0 20 40
(deg)
XCDCC
) x10
Be+n)p @ 63.7 MeV/nucleon
3.0-5.5 MeV
FIG. 4: (Color online) 11Be breakup at 60 MeV/nucleon with
the relative energy between breakup fragments in the range
3.0–5.5 MeV. Experimental data are from Ref. [11].
We find that converged CDCC calculations, for the lower
energy bin, do in fact agree well with the data (dashed
line in Fig. 3), contrary to what is presented in Ref. [11].
Results do not change significantly when core excitation
is included with XCDCC (solid line in Fig. 3), as could be
expected expected due to the resonance structure in this
energy region. One can conclude that a single-particle
description for the first d5/2 resonance is adequate.
The data for the higher energy bin is not well described
within the single particle CDCC model (dashed line in
Fig. 4). To see if this discrepancy can be explained by ex-
cited core contributions, we include the excited 10Be(2+)
components in the reaction mechanism, within XCDCC
(solid line). As shown in Fig. 4, core excitation lowers the
cross section but does not significantly change the shape
of the distribution. It becomes clear that core excitation
does not help to reproduce the shape of the higher en-
ergy angular distribution. The main reason for this is
that for the 11Be coupled channel model of [20], most of
the breakup ends up in the 10Be ground state. Fig. 4
also shows the breakup cross section to the 0+ and 2+
states of 10Be (red/dotted line and the dot-dashed/blue
line respectively). We see that whereas the ground state
distribution has the original shape of the CDCC calcu-
lation, the shape of the distribution to the excited state
reproduces the data (to illustrate this fact, we show the
breakup cross section to 10Be(2+) multiplied by 10 by
the dashed/blue line). The reason for this maybe that
the large number of resonances in this region are not re-
produced well by our particle-rotor model for 11Be. The
only resonance that appears in this model is the 3/2+, for
which the width is not narrow enough to attract signifi-
cant cross section. Some suggest that more exotic struc-
tures are responsible for resonances in this region [24].
Without exotic resonances built on excited core compo-
nents in our structure model, the breakup cross section
is still dominated by ground state 10Be fragments.
IV. CONCLUDING REMARKS
A consistent analysis of reactions involving the halo
nucleus 11Be on protons, at two intermediate energies
(∼40 and ∼60 MeV/nucleon) are performed and com-
pared with data. An optical model approach, based on
a cluster folding potential constructed from the 10Be+p
potential fitted to the appropriate elastic data, is un-
able to describe the 11Be elastic data. The inclusion of
breakup effects improve the description, but theoretical
predictions still overestimate the elastic cross section at
larger scattering angles. The inclusion of core excitation
does not affect the elastic distribution significantly. Note
however that these results include no artificial renormal-
ization of the optical potential. Elastic scattering ex-
periments with radioactive beams at large facilities have
repeatedly been undermined. The fact that the best re-
action models are still unable to fully describe the mecha-
nisms for the 11Be case, shows the need for a more varied
and better elastic scattering experimental program.
In this work we also study the breakup channel explic-
itly. Core excitation in the description of the continuum,
within XCDCC, produces a slight modification of the dis-
tribution. These breakup calculations are compared to
the data at 63.7 MeV/nucleon, for two energy bins 0.5–
3.0 MeV and 3.0–5.5 MeV. For the lower energy bin, the
shape of the angular distribution is well reproduced by
the models. The same cannot be said for the higher en-
ergy bin.
The XCDCC calculations predict breakup states to
specific states of the core 10Be. This level of detail is
still not available in the data, but it could be helpful in-
formation, even at an integrated level, to identify possible
causes for the remaining disagreement with the data.
Another important point is related to the basis used
to describe the breakup states. As discussed in Ref. [25],
within CDCC, one can describe the three body final state
continuum 10Be+n+p in the 11Be continuum basis or in
the deuteron continuum basis. In this work we used the
11Be basis. Work in Ref. [25] shows that in practice the
two choices do not provide the same result. Efforts are
underway to tackle this problem within a Faddeev frame-
work [26]. These results may have important implications
to the theory-experiment mismatch.
Acknowledgements
We thank the high performance computing center
(HPCC) at MSU for the use of their facilities. This
work is supported by NSCL, Michigan State University,
the National Science Foundation through grant PHY-
0555893, by NNSA through US DOE Cooperative Agree-
ment DEFC03-03NA00143 at Rutgers University, and by
the US DOE under contract No. DE-FG02-96ER40983
at the University of Tennessee.
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|
0704.1587 | Possible X-ray diagnostic for jet/disk dominance in Type 1 AGN | Possible X-ray diagnostic for jet/disk dominance in Type 1 AGN
Barbara J. Mattson⋆, Kimberly A. Weaver
NASA/Goddard Space Flight Center, Astrophysics Science Division, Greenbelt, MD, 20771
Christopher S. Reynolds
Department of Astronomy, University of Maryland, College Park, MD, 20742
ABSTRACT
Using Rossi X-ray Timing Explorer Seyfert 1 and 1.2 data spanning 9 years,
we study correlations between X-ray spectral features. The sample consists of
350 time-resolved spectra from 12 Seyfert 1 and 1.2 galaxies. Each spectrum
is fitted to a model with an intrinsic powerlaw X-ray spectrum produced close
to the central black hole that is reprocessed and absorbed by material around
the black hole. To test the robustness of our results, we performed Monte Carlo
simulations of the spectral sample. We find a complex relationship between the
iron line equivalent width (EW ) and the underlying power law index (Γ). The
data reveal a correlation between Γ and EW which turns over at Γ . 2, but
finds a weak anti-correlation for steeper photon indices. We propose that this
relationship is driven by dilution of a disk spectrum (which includes the narrow
iron line) by a beamed jet component and, hence, could be used as a diagnostic
of jet-dominance. In addition, our sample shows a strong correlation between
R and Γ, but we find that it is likely the result of modeling degeneracies. We
also see the X-ray Baldwin effect (an anti-correlation between the 2-10 keV X-
ray luminosity and EW ) for the sample as a whole, but not for the individual
galaxies and galaxy types.
Subject headings: galaxies: Seyfert, X-rays: galaxies
also Department of Astronomy, University of Maryland, College Park, MD and Adnet Systems, Inc.,
Rockville, MD
http://arxiv.org/abs/0704.1587v1
– 2 –
1. Introduction
Time-resolved X-ray spectroscopy studies of active galactic nuclei (AGN) offer the op-
portunity to investigate emission regions near the central black hole. In fact, X-ray spec-
troscopy offers the clearest view of processes occurring very close to the black hole itself,
probing matter to its final plunge into the black hole. Armed with such information, we can
unlock the structure of the innermost regions of AGN.
Typical X-ray spectra of AGN show an underlying powerlaw produced near the central
black hole with signatures of reprocessed photons often present. These reprocessed photons
show up as an Fe Kα line at ∼6.4 keV and a “reflection hump” which starts to dominate near
10 keV and is produced by the combined effects of photoelectric absorption and Compton
downscattering in optically-thin cold matter irradiated by the hard X-ray continuum. The
Fe Kα line has been observed in both type 1 (unabsorbed) and type 2 (absorbed) Seyfert
galaxies. It has been attributed to either the broad line region, the accretion disk, the molec-
ular torus of unification models (Antonucci 1993), or some combination of these. Signatures
of reflection have also been observed in both Seyfert 1 and 2 galaxies.
If the unification models are correct, we should see similar spectral correlations be-
tween Seyfert 1 and 2 galaxies, with any differences easily attributable to our viewing
angle. Regardless of the accuracy of the reflection models, we expect changes in the un-
derlying continuum to drive changes in the reprocessing features. However, results from
X-ray spectral studies of AGN have so far produced puzzling results. Samples of Seyfert
1 observations from ASCA (Weaver, Gelbord & Yaqoob 2001) and Rossi X-ray Timing Ex-
plorer (Markowitz, Edelson, & Vaughan 2003) have shown no obvious relationship between
changes in the continuum and iron line. Several galaxies have shown an anticorrelation
between reflection and/or iron line equivalent width and the source flux; e.g. NGC 5548
(Chiang et al. 2000), MCG −6-30-15 (Papadakis et al. 2002), NGC 4051 (Papadakis et al.
2002; Wang et al. 1999), NGC 5506 (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000).
Recent data from Suzaku on MCG −6-30-15, on the other hand, show that the iron line and
reflection remain relatively constant while the powerlaw is highly variable (Miniutti et al.
2006). Zdziarski, Lubiński & Smith (1999) found that Seyfert galaxies and X-ray binaries
show a correlation between the continuum slope and reflection fraction, so those with soft
intrinsic spectra show stronger reflection than those with hard spectra. However, other stud-
ies have found either a shallower relationship than Zdziarski et al. (Perola et al. 2002) or an
anticorrelation (Papadakis et al. 2002; Lamer, Uttley & McHardy 2000).
Here we present the first results of a larger study of the X-ray spectral properties of
Seyfert galaxies observed by the Rossi X-ray Timing Explorer (RXTE ). Our full study
consists of observations of 30 galaxies. In this letter, we focus on the spectral results from
– 3 –
the subset of 12 Seyfert 1 and 1.2 galaxies. In § 2 we present our method of data analysis,
including our sample selection criteria (§ 2.1), a description of our data pipeline (§ 2.2), and
results of our spectral analysis (§ 2.3). We discuss the implications of our results in § 3 and
detail our conclusions in § 4.
2. Data Analysis
2.1. The Sample
The RXTE public archive1 represents one of the largest collections of X-ray data for
AGN, with pointed observations of over 100 AGN spanning 10 years. The RXTE bandpass
allows the study of absorption and iron line properties of AGN spectra, as well as a glimpse
at the Compton reflection hump. We use data from the RXTE proportional counter array
(PCA), which is sensitive to energies from 2 to 60 keV and consists of five Proportional
Counter Units (PCUs). Most of the sources in our sample do not show significant counts in
the RXTE Hard Energy X-ray Timing Experiment (HEXTE), so we do not include HEXTE
data in this study.
To focus this study, we choose only Seyfert galaxies for which the RXTE public archive
contained a minimum of two pointings separated by at least two weeks. We further required
the total observed time be > 40 ks. These selection criteria led to a sample of 40 Seyfert
galaxies. For the analysis presented here, we examine the 18 Seyfert 1 and 1.2 galaxies. Six
galaxies were eliminated after they were put through our data pipeline (see § 2.2 for more),
so the final sample presented here consists of 12 galaxies, listed in Table 1. Because the data
come from the public archive, the sample is not uniform from galaxy to galaxy or even from
observation to observation; however, we use the Standard 2 data, which provides a standard
data mode for these diverse observations.
2.2. Data Pipeline
To ensure consistent data reduction of the large volume of data, we developed a data
pipeline. The Standard 2 data for each observation was reduced using a combination of
FTOOLs and the Pythonr scripting language. The pipeline produces time-resolved spectra,
1Hosted by the High Energy Astrophysics Science Archive and Research Center (HEASARC;
http://heasarc.gsfc.nasa.gov/)
http://heasarc.gsfc.nasa.gov/
– 4 –
each with a minimum of 125,000 net photons, which are extracted using standard PCA
selection criteria and background models (Jahoda et al. 2006). Sources which did not have
sufficient net photons for even one spectrum were eliminated from the final sample (Table 1
shows the final sample with the 6 eliminated sources listed in the table notes). Each spectrum
includes 1% systematic errors. We are confident in the instrument response and background
models up to energies of ∼25 keV, so we ignore channels with higher energies.
2.3. Spectral Fitting and Results
The data pipeline produced 350 spectra for the 12 galaxies in our sample. Each spectrum
was fitted from 3 to 25 keV with an absorbed Compton reflection model plus a Gaussian iron
line. In xspec, the PEXRAV (Magdziarz & Zdziarski 1995) model simulates the effects of an
exponentially cut-off powerlaw reflected by neutral matter and has seven model parameters:
photon index of the intrinsic underlying power-law (Γ), the cutoff energy of the power law
in keV (Ec), the relative amount of reflection (R), the redshift (z), the abundance of heavy
elements in solar units (Z), the disk inclination angle (i), and the photon flux of the power
law at 1 keV in the observer’s frame (A). The relative amount of reflection is normalized to
1 for the case of an isotropic source above a disk of neutral material (Ω = 2π). Adding a
Gaussian line (energy in keV (EFe), physical width (σ) in keV, and normalization in units of
photons cm−2 s−1) and an absorbing column (NH , in cm
−2) yields a total of 12 parameters.
We fixed the following values in PEXRAV: Ec = 500 keV, Z = 1.0, and cos i = 0.95.
This inclination represents an almost face-on disk; however, since we are seeking trends in
the spectral parameters, rather than absolute values, the precise value is not important to
this study. In addition, z is fixed at the appropriate value from the NASA Extragalactic
Database for each galaxy2. After fitting all spectra to this model, we derived the mean
Gaussian width for each source (Table 1), then held σ fixed for a second fit to the model.
Our final model has free parameters: Γ, R, A, EFe, iron line normalization and NH . To
prevent xspec from pursuing unphysical values of the parameters, we set the following hard
limits: 0 ≤ Γ ≤ 5, 0 ≤ R ≤ 5, 5.5 ≤ EFe ≤ 7.5 keV, and 0 ≤ σ ≤ 1.5 keV (for the free-σ
fits).
Looking at the iron line equivalent width (EW ) and Γ, we find a complex relationship
with a “hump” peaking near Γ ∼ 2.0 (Figure 1a). The EW -Γ plot shows a correlation for
Γ . 2.0 and an anti-correlation for Γ & 2.0, with a peak near Γ ∼ 2.0 with EW ∼ 250
eV. We also find a strong correlation between R and Γ (Figure 2a), with a best-fit line of
2http://nedwww.ipac.caltech.edu/
http://nedwww.ipac.caltech.edu/
– 5 –
R = −0.87 + 0.54 Γ (χ2 = 506/349 = 1.46).
We performed a Monte Carlo simulation to determine if our results were an artifact
of modeling degeneracies. Each spectrum in the Monte Carlo sample was simulated with
NH=10
22 cm−2, Γ=2.0, R=1.0, EFe=6.4 keV, and σFe=0.23 keV. The flux and exposure
times were randomly varied for each spectrum. The flux was varied by randomly choosing A
from a uniform distribution between 0.004 and 0.06 photons keV−1 cm−2 s−1. The exposure
time was randomly generated from a uniform distribution between 300 and 11000s. The
ranges for A and the exposure time represent the range of A and exposure for the spectra
in the full sample.
We generated 200 spectra: 100 simulated using an RXTE Epoch 3 response, 50 using an
Epoch 4 response, and 50 using an Epoch 5 response, roughly corresponding to our RXTE
sample. Each spectrum was then fitted to the same model as our full sample. The R over Γ
plot (Figure 2b) clearly shows a strong correlation with a best-fit line of R = −7.3 + 4.1 Γ
(χ2 = 28.96/159 = 0.182), which strongly suggests that the observed R-Γ correlation is a
result of modeling degeneracies. The correlation shows a much steeper relationship than the
Seyfert 1 data, due to the large number of Seyfert 1 spectra showing R ∼ 0.
EW and Γ, however, do not suffer the same degeneracies, which is clear from the Monte
Carlo results (Figure 1b). Based on the lack of correlation in our Monte Carlo results, we
are confident that the shape of the EW -Γ plot for the data sample is real.
To further examine the EW -Γ relationship, we reproduced the EW -Γ plot to show the
contribution from each galaxy (Figure 3). The radio-loud galaxies form the rising leg, with
the quasar, 3C 273, anchoring the low Γ-low EW portion of the plot. The Seyfert 1 (radio
quiet) and 1.2 galaxies tend to congregate at the peak and the falling leg of the plot. The
one narrow-line Seyfert 1 diverges from the main cluster of points.
Finally, we examined EW as a function of the intrinsic 2-10 keV X-ray luminosity (Lx),
using H0 = 70 km s
−1 Mpc−1. We fitted the data for each galaxy, each type, and the sample
as a whole to linear and powerlaw models. The data were well-fit for either model. For
consistency with other publications, we report here the powerlaw results. For the sample
as a whole, we see an anticorrelation, i.e. the X-ray Baldwin effect (Iwasawa & Taniguchi
1993), with EW ∝ L−0.14±0.01x . When examining galaxy types, however, the anticorrelation
does not always hold up (Table 1). We find an anticorrelation in the radio loud galaxies and
the Seyfert 1.2s, but a marginal correlation for the quasar and radio quiet Seyfert 1s.
– 6 –
3. Discussion
3.1. EW -Γ Relationship
The simulations of George & Fabian (1991) for the observed spectrum from an X-ray
source illuminating a half-slab showed that the spectra should include a “Compton hump”
and an iron line. They found that the iron line EW should decrease as the spectrum softens.
This is easy to understand, since as the spectrum softens (Γ increases), there are fewer
photons with energies above the iron photoionization threshold. Our results show that the
relationship between EW and Γ is not quite so simple. We find a correlation between EW
and Γ when Γ . 2 and an anticorrelation when Γ & 2. Other researchers have found a
correlation for Seyfert 1 samples (Perola et al. 2002; Lubiński & Zdziarski 2001), but the
galaxies in their samples primarily fell in the Γ . 2 region. Page et al. (2004) also find that
their data suggest a slight correlation for a sample of radio loud and radio quiet Type 1
A close examination of our EW -Γ plot shows that the data for different galaxy types
progresses across the plot. The plot is anchored at the low-Γ, low-EW end by the quasar,
3C 273, in our sample. The rising arm of the plot, Γ ∼ 1.5− 2.0 and EW ∼ 0 − 300 eV, is
primarily formed by radio loud Seyfert 1 galaxies. The radio-quiet Seyfert 1 galaxies cluster
near the Γ ∼ 2.0, EW ∼ 300 eV peak of the hump, and the radio-quiet Seyfert 1.2 galaxies
form the falling arm of the plot for Γ > 2.0.
Physically, the most obvious difference between these sources is the presence or absence
of a strong jet. We propose that this relationship is driven by the degree of jet-dominance
of the source. The iron line features are associated with the X-ray emission from the disk.
Since the disk is essentially isotropic, it will excite an observable iron line from matter out of
our line-of-sight. On the other hand, the jet is beamed away from the obvious configurations
of matter in the system and, more importantly, is beamed toward us in the quasar and radio-
loud sources. Both of these jet-related phenomena reduce the observed equivalent width of
any iron line emission associated with the jet continuum.
In order for the Γ to increase as the jet-dominance decreases, the jet in these sources must
have a hard X-ray component, which implies that the radio-loud Seyferts in our sample are to
be associated with low-peaked BL Lac objects (LBLs). BL Lac objects show two broad peaks
in their spectral energy density plots (Giommi & Padovani 1994), with the lower-energy peak
due to synchrotron emission and the higher-energy peak due to inverse Compton emission.
BL Lacs are divided into two classes, depending on where the peaks occur: high-peaked BL
Lacs (HBLs) and LBLs. The X-ray continuum in the HBLs is rather soft, since we are seeing
the synchrotron spectrum cutting off in these sources. LBLs, on the other hand, tend to
– 7 –
have a harder X-ray continua, since we are observing well into the inverse Compton part of
the spectrum (Donato, Sambruna, & Gliozzi 2005).
We also note that much of the falling arm of the EW -Γ relationship is formed by MCG
−6-30-15. Recent observations of MCG −6-30-15 by Suzaku have shown that the reflection
component, including the iron line, remains relatively constant Miniutti et al. (2006). We
would expect, then, that as Γ increases, the EW should decrease, which is exactly what we
see in our data.
3.2. R-Γ Relationship
Significant degeneracies between the photon index, absorbing column, and reflection
fraction can easily lead to false conclusions about spectral correlations. These degeneracies
occur as these three parameters trade off against each other in the modeling process, an effect
that is especially strong in the RXTE bandpass. Our R-Γ plot shows a strong correlation
which is mimicked in our Monte Carlo results. The few points that lie under the main
concentration are likely to be outliers, and not indicative of a subclass of galaxy. These
points all come from spectra that have been fitted to have NH = 0, and are primarily
radio-loud galaxies. We conclude that the observed R-Γ correlation in our sample cannot be
trusted as a real correlation.
3.3. EW -Lx Relationship
Looking at the EW -Lx relationship, we do see the X-ray Baldwin effect for our sample
as a whole, with a slighly shallower anticorrelation than reported elsewhere. We find EW ∝
L−0.14x , whereas Iwasawa & Taniguchi (1993) and Jiang, Wang & Wang (2006) find EW ∝
L−0.20x and Page et al. (2004) find EW ∝ L
−0.17
x . However, when Jiang, Wang & Wang
(2006) exclude the radio loud galaxies from their sample, they find EW ∝ L−0.10x .
We find, though, that when we examine our data on a galaxy-by-galaxy or type-by-type
basis, the effect is not consistent from source to source. At this point, we cannot determine
if these variations are real or are simply due to the small number of spectra for some of our
galaxies and types.
– 8 –
4. Conclusions
We have examined time-resolved spectra of 12 Seyfert 1 and 1.2 galaxies observed by
RXTE over seven years. We find a complex relationship between the iron line equivalent
width and the continuum slope, with a correlation for Γ . 2 that turns over to an anticor-
relation for Γ & 2. We propose that this relationship is a possible diagnostic for jet- versus
disk-dominated sources, where jet-dominated sources show a correlation between EW and
Γ, and disk-dominated sources show an anticorrelation. We also see a strong correlation
between Γ and R which is likely an artifact of modeling degeneracies caused by the interplay
of Γ, R, and nH in the RXTE bandpass. Finally, we observe the X-ray Baldwin effect for
the sample as a whole, but not for each galaxy and galaxy type individually.
This research has made use of data obtained from the High Energy Astrophysics Science
Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center.
This research has also made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration.
CSR gratefully acknowledges support from the National Science Foundation under
grants AST0205990 and AST0607428.
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– 10 –
Table 1. Sample of RXTE -observed Seyfert 1 and 1.2 galaxiesa
Galaxy Seyfert Fitted Average EW/Lx correlation
Typeb Spectrac σFeKα
d α WV/Num.
All -0.14+0.01
−0.01 700/350
Quasars +0.09+0.20
−0.25 105/81
3C 273 1 81 0.329 +0.09+0.20
−0.25 105/81
Broadline Seyfert 1s -0.24+0.14
−0.15 48.0/66
3C 111 1 4 0.239 +0.70+2.60
−1.52 0.654/4
3C 120f 1 40 0.261 -0.70+0.63
−0.61 20.9/39
3C 382 1 5 0.328 -0.80+1.69
−1.70 2.54/5
3C 390.3 1 17 0.203 -0.51+0.44
−0.41 2.70/17
Seyfert 1s (Radio quiet) 0.01+0.300.30 23.6/31
Ark 120 1 15 0.197 -0.66+0.58
−0.57 6.62/15
Fairall 9 1 16 0.155 +0.41+0.44
−0.44 11.1/16
Seyfert 1.2s -0.08+0.03
−0.03 192/169
IC 4329A 1.2 41 0.214 -0.55+0.36
−0.37 27.5/41
MCG -6-30-15 1.2 75 0.292 -0.65+0.34
−0.33 89.2/75
Mkn 509 1.2 16 0.102 -0.52+0.91
−0.99 7.57/16
NGC 7469 1.2 37 0.145 -0.58+0.30
−0.31 17.7/37
Narrow Line Seyfert 1 8.80+20.80
−6.08 0.196/3
TON S180 1.2 3 0.379 8.80+20.80
−6.08 0.196/3
aThe following sources were eliminated after running the data pipeline described
in the text, due to having no spectra with at least 125,000 net photons: Mkn 110,
PG 0804+761, PG 1211+143, Mkn 79, Mkn 335, and PG 0052+251.
bSeyfert type based on the NASA Extragalactic Database
cTotal number of spectra extracted using our data pipeline (§ 2.2).
dThe average physical width of the Fe Kα line for all spectra from a source when
fitted to the absorbed powerlaw model with Compton reflection and Gaussian iron
line (§ 2.3).
– 11 –
eResults of fitting the X-ray luminosity over EW plot to a powerlaw model; e.g.
EW ∝ Lαx , where Lx is the 2-10 keV X-ray luminosity in ergs s
−1 and EW is the
iron line equivalent width in eV.
fOne 3C 120 spectrum shows a flare, where Lx jumps by ∼ 6×. The number
quoted above excludes this point from the sample. If we include the flare, we find
EW ∝ L
0.07(+0.18/−0.25)
– 12 –
(a) (b)
Fig. 1.— Iron line equivalent width in eV (EW ) versus powerlaw photon index (Γ) for the
Seyfert 1/1.2 sample (a) and for the Monte Carlo simulations (b).
– 13 –
(a) (b)
Fig. 2.— Reflection fraction (R) versus powerlaw photon index (Γ) for the Seyfert 1/1.2
sample (a) and for the Monte Carlo simulations (b). In both plots, the line shows the
best-fit linear model for the Monte Carlo simulations.
– 14 –
Fig. 3.— The iron line equivalent width in eV versus the powerlaw photon index. This plot
is similar to the left panel in Figure 1, but with each galaxy plotted with a separate symbol.
The open circles are 3C 111, open squares are 3C120, pluses (+) are 3C273, open triangles
are 3C 382, open diamonds 3C 390.3, open stars Akn 120, open crosses Fairall 9, filled circles
IC 4329A, filled squares MCG −6-30-15, filled triangles Mkn 509, filled stars NGC 7469, and
asterisks (*) TON S180.
Introduction
Data Analysis
The Sample
Data Pipeline
Spectral Fitting and Results
Discussion
EW- Relationship
R- Relationship
EW-Lx Relationship
Conclusions
|
0704.1588 | On algebraic automorphisms and their rational invariants | On algebraic automorphisms and
their rational invariants
Philippe Bonnet
Mathematisches Institut, Universität Basel
Rheinsprung 21, 4051 Basel, Switzerland
e-mail: [email protected]
Abstract
Let X be an affine irreducible variety over an algebraically closed field k of char-
acteristic zero. Given an automorphism Φ, we denote by k(X)Φ its field of invariants,
i.e the set of rational functions f on X such that f ◦Φ = f . Let n(Φ) be the transcen-
dence degree of k(X)Φ over k. In this paper, we study the class of automorphisms Φ
of X for which n(Φ) = dimX − 1. More precisely, we show that under some condi-
tions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic
action of a linear algebraic group G of dimension 1 on X, and where g belongs to
G. As an application, we determine the conjugacy classes of automorphisms of the
plane for which n(Φ) = 1.
1 Introduction
Let k be an algebraically closed field of characteristic zero. Let X be an affine irreducible
variety of dimension n over k. We denote by O(X) its ring of regular functions, and by
k(X) its field of rational functions. Given an algebraic automorphism Φ of X , denote by
Φ∗ the field automorphism induced by Φ on k(X), i.e. Φ∗(f) = f ◦Φ for any f ∈ k(X). An
element f of k(X) is invariant for Φ (or simply invariant) if Φ∗(f) = f . Invariant rational
functions form a field denoted k(X)Φ, and we set:
n(Φ) = trdegk k(X)
In this paper, we are going to study the class of automorphisms of X for which n(Φ) =
n − 1. There are natural candidates for such automorphisms, such as exponentials of
locally nilpotent derivations (see [M] or [Da]). More generally, one can construct such
automorphisms by means of algebraic group actions as follows. Let G be a linear algebraic
group over k. An algebraic action of G on X is a regular map:
ϕ : G×X −→ X
http://arxiv.org/abs/0704.1588v1
of affine varieties, such that ϕ(g.g′, x) = ϕ(g, ϕ(g′, x)) for any (g, g′, x) in G × G × X .
Given an element g of G, denote by ϕg the map x 7→ ϕ(g, x). Then ϕg clearly defines an
automorphism of X . Let k(X)G be the field of invariants of G, i.e. the set of rational
functions f on X such that f ◦ ϕg = f for any g ∈ G. If G is an algebraic group of
dimension 1, acting faithfully on X , and if g is an element of G of infinite order, then one
can prove by Rosenlicht’s Theorem (see [Ro]) that:
n(ϕg) = trdegk k(X)
G = n− 1
We are going to see that, under some mild conditions on X , there are no other automor-
phisms with n(Φ) = n−1 than those constructed above. In what follows, denote by O(X)ν
the normalization of O(X), and by G(X) the group of invertible elements of O(X)ν .
Theorem 1.1 Let X be an affine irreducible variety of dimension n over k, such that
char(k) = 0 and G(X)∗ = k∗. Let Φ be an algebraic automorphism of X such that
n(Φ) = n − 1. Then there exist an abelian linear algebraic group G of dimension 1, and
an algebraic action ϕ of G on X such that Φ = ϕg for some g ∈ G of infinite order.
Note that the structure of G is fairly simple. Since every connected linear algebraic group
of dimension 1 is either isomorphic to Ga(k) = (k,+) or Gm(k) = (k
∗,×) (see [Hum], p.
131), there exists a finite abelian group H such that G is either equal to H × Ga(k) or
H × Gm(k). Moreover, the assumption on the group G(X) is essential. Indeed, consider
the automorphism Φ of k∗ × k given by Φ(x, y) = (x, xy). Obviously, its field of invariants
is equal to k(x). However, it is easy to check that Φ cannot have the form given in the
conclusion of Theorem 1.1.
This theorem is analogous to a result given by Van den Essen and Peretz (see [V-P]).
More precisely, they establish a criterion to decide if an automorphism Φ is the exponential
of a locally nilpotent derivation, based on the invariants and on the form of Φ. A similar
result has been developed by Daigle (see [Da]).
We apply these results to the group of automorphisms of the plane. First, we obtain
a classification of the automorphisms Φ of k2 for which n(Φ) = 1. Second, we derive a
criterion on automorphisms of k2 to have no nonconstant rational invariants.
Corollary 1.2 Let Φ be an algebraic automorphism of k2. If n(Φ) = 1, then Φ is conjugate
to one of the following forms:
• Φ1(x, y) = (a
nx, amby), where (n,m) 6= (0, 0), a, b ∈ k, b is a root of unity but a is
• Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0}, a, b ∈ k are roots of unity.
Corollary 1.3 Let Φ be an algebraic automorphism of k2. Assume that Φ has a unique
fixpoint p and that dΦp is unipotent. Then n(Φ) = 0.
We then apply Corollary 1.3 to an automorphism of C3 recently discovered by Pierre-Marie
Poloni and Lucy Moser (see [M-P]).
We may wonder whether Theorem 1.1 still holds if the ground field k is not algebraically
closed or has positive characteristic. The answer is not known for the moment. In fact,
two obstructions appear in the proof of Theorem 1.1 when k is arbitrary. First, the group
Gm(k) needs to be divisible (see Lemma 4.2), which is not always the case if k is not
algebraically closed. Second, the proof uses the fact that every Ga(k)-action on X can be
reconstructed from a locally nilpotent derivation on O(X) (see subsection 4.1), which is
no longer true if k has positive characteristic. This phenomenom is due to the existence
of differents forms for the affine line (see [Ru]). Note that, in case Theorem 1.1 holds and
k is not algebraically closed, the algebraic group G needs not be isomorphic to H ×Ga(k)
or H ×Gm(k), where H is finite. Indeed consider the unit circle X in the plane R
2, given
by the equation x2 + y2 = 1. Let Φ be a rotation in R2 with center at the origin and
angle θ 6∈ 2πQ. Then Φ defines an algebraic automorphism of X with n(Φ) = 0, and the
subgroup spanned by Φ is dense in SO2(R). But SO2(R) is not isomorphic to either Ga(R)
or Gm(R), even though it is a connected linear algebraic group of dimension 1.
We may also wonder what happens to the automorphisms Φ of X for which n(Φ) =
dimX − 2. More precisely, does there exist an action ϕ of a linear algebraic group G
on X , of dimension 2, such that Φ = ϕg for a given g ∈ G? The answer is no. Indeed
consider the automorphism Φ of k2 given by Φ = f ◦ g, where f(x, y) = (x + y2, y) and
g(x, y) = (x, y + x2). Let d(n) denote the maximum of the homogeneous degrees of the
coordinate functions of the iterate Φn. If there existed an action ϕ of a linear algebraic
group G such that Φ = ϕg, then the function d would be bounded, which is impossible
since d(n) = 4n. A similar argument on the length of the iterates also yields the result. But
if we restrict to some specific varieties X , for instance X = k3, one may ask the following
question: If n(Φ) = 1, is Φ birationally conjugate to an automorphism that leaves the first
coordinate of k3 invariant? The answer is still unknown.
2 Reduction to an affine curve C
Let X be an affine irreducible variety of dimension n over k. Let Φ be an algebraic
automorphism of X such that n(Φ) = n − 1. In this section, we are going to construct
an irreducible affine curve on which Φ acts naturally. This will allow us to use some
well-known results on automorphisms of curves. We set:
K = {f ∈ k(X)|∃m > 0, f ◦ Φm = f ◦ Φ ◦ ... ◦ Φ = f}
It is straightforward that K is a subfield of k(X) containing both k and k(X)Φ. We begin
with some properties of this field.
Lemma 2.1 K has transcendence degree (n − 1) over k, and is algebraically closed in
k(X). In particular, the automorphism Φ of X has infinite order.
Proof: First we show that K has transcendence degree (n − 1) over k. Since K contains
the field k(X)Φ, whose transcendence degree is (n − 1), we only need to show that the
extension K/k(X)Φ is algebraic, or in other words that every element of K is algebraic
over k(X)Φ. Let f be any element of K. By definition, there exists an integer m > 0 such
that f ◦ Φm = f . Let P (t) be the polynomial of k(X)[t] defined as:
P (t) =
(t− f ◦ Φi)
By construction, the coefficients of this polynomial are all invariant for Φ, and P (t) belongs
to k(X)Φ[t]. Moreover P (f) = 0, f is algebraic over k(X)Φ and the first assertion follows.
Second we show that K is algebraically closed in k(X). Let f be an element of k(X)
that is algebraic over K. We need to prove that f belongs to K. By the first assertion
of the lemma, f is algebraic over k(X)Φ. Let P (t) = a0 + a1t + ... + apt
p be a nonzero
minimal polynomial of f over k(X)Φ. Since P (f) = 0 and all ai are invariant, we have
P (f ◦ Φ) = P (f) ◦ Φ = 0. In particular, all elements of the form f ◦ Φi, with i ∈ N, are
roots of P . Since P has finitely many roots, there exist two distinct integers m′ < m′′ such
that f ◦ Φm
= f ◦ Φm
. In particular, f ◦ Φm
′′−m′ = f and f belongs to K.
Now if Φ were an automorphism of finite order, then K would be equal to k(X). But
this is impossible since K and k(X) have different transcendence degrees.
Lemma 2.2 There exists an integer m > 0 such that K = k(X)Φ
Proof: By definition, k(X) is a field of finite type over k. Since K is contained in k(X),
K has also finite type over k. Let f1, ..., fr be some elements of k(X) such that K =
k(f1, ..., fr). Let m1, ..., mr be some positive integers such that fi ◦ Φ
mi = fi, and set
m = m1...mr. By construction, all fi are invariant for Φ
m. In particular, K is invariant
for Φm and K ⊆ k(X)Φ
. Since k(X)Φ
⊆ K, the result follows.
Let L be the algebraic closure of k(X), and let A be the K-subalgebra of L spanned by
O(X). By construction, A is an integral K-algebra of finite type of dimension 1. Let m
be an integer satisfying the conditions of lemma 2.2. The automorphism Ψ∗ = (Φm)∗ of
O(X) stabilizes A, hence it defines a K-automorphism of A, of infinite order (see lemma
2.1). Let B be the integral closure of A. Then B is also an integral K-algebra of finite
type, of dimension 1, and the K-automorphism Ψ∗ extends uniquely to B. If K stands for
the algebraic closure of K, we set:
C = B ⊗K K
By construction, C = Spec(C) is an affine curve over the algebraically closed field K.
Moreover the automorphism Ψ∗ acts on C via the operation:
Ψ∗ : C −→ C, x⊗ y 7−→ Ψ∗(x)⊗ y
This makes sense since Ψ∗ fixes the field K. Therefore Ψ∗ induces an algebraic automor-
phism of the curve C. Since K is algebraically closed in k(X) by lemma 2.1, C is integral
(see [Z-S], Chap. VII, §11, Theorem 38). But by construction, B and K are normal rings.
Since C is a domain and char(K) = 0, C is also integrally closed by a result of Bourbaki
(see [Bou], p. 29). So C is a normal domain and C is a smooth irreducible curve.
Lemma 2.3 Let C be the K-algebra constructed above. Then either C = K[t] or C =
K[t, 1/t].
Proof: By lemma 2.1, the automorphism Φ of X has infinite order. Since the fraction field
of B is equal to k(X), Ψ∗ has infinite order on B. But B ⊗ 1 ⊂ C, so Ψ∗ has infinite
order on C. In particular, Ψ acts like an automorphism of infinite order on C. Since C is
affine, it has genus zero (see [Ro2]). Since K is algebraically closed, the curve C is rational
(see [Che], p. 23 ). Since C is smooth, it is isomorphic to P1(K) − E, where E is a finite
set. Moreover, Ψ acts like an automorphism of P1(K) that stabilizes P1(K) − E. Up to
replacing Ψ by one of its iterates, we may assume that Ψ fixes every point of E. But an
automorphism of P1(K) that fixes at least three points is the identity, which is impossible.
Therefore E consists of at most two points, and C is either isomorphic to K or to K
particular, either C = K[t] or C = K[t, 1/t].
3 Normal forms for the automorphism Ψ
Let C and Ψ∗ be the K-algebra and the K-automorphism constructed in the previous
section. In this section, we are going to give normal forms for the couple (C,Ψ∗), in case
the group G(X) is trivial, i.e. G(X) = k∗. We begin with a few lemmas.
Lemma 3.1 Let X be an irreducible affine variety over k. Let Ψ be an automorphism of
X. Let α, f be some elements of k(X)∗ such that (Ψ∗)n(f) = αnf for any n ∈ Z. Then α
belongs to G(X).
Proof: Given an element h of k(X)∗ and a prime divisor D on the normalization Xν , we
consider h as a rational function on Xν , and denote by ordD(h) the multiplicity of h along
D. This makes sense since the variety Xν is normal. Fix any prime divisor D on X . Since
(Ψ∗)n(f) = αnf for any n ∈ Z, we obtain:
ordD((Ψ
∗)n(f)) = nordD(α) + ordD(f)
Since Ψ is an algebraic automorphism of X , it extends uniquely to an algebraic automor-
phism of Xν , which is still denoted Ψ. Moreover, this extension maps every prime divisor
to another prime divisor, does not change the multiplicity and maps distinct prime divisors
into distinct ones. If div(f) =
i niDi, where all Di are prime, then we have:
div((Ψ∗)n(f)) =
∗)n(Di)
where all (Ψ∗)n(Di) are prime and distinct. So the multiplicity of (Ψ
∗)n(f) along D is
equal to zero if D is none of the (Ψ∗)n(Di), and equal to ni if D = (Ψ
∗)n(Di). In all cases,
if R = max{|ni|}, then we find that |ordD((Ψ
∗)n(f))| ≤ R and |ordD(f)| ≤ R, and this
implies for any integer n:
|nordD(α)| ≤ 2R
In particular we find ordD(α) = 0. Since this holds for any prime divisor D, the support
of div(α) in Xν is empty and div(α) = 0. Since Xν is normal, α is an invertible element
of O(X)ν , hence it belongs to G(X).
Lemma 3.2 Let K be a field of characteristic zero and K its algebraic closure. Let C be
either equal to K[t] or to K[t, 1/t]. Let Ψ∗ be a K-automorphism of C such that Ψ∗(t) = at,
where a belongs to K. Let σ1 be a K-automorphism of C, commuting with Ψ
∗, such that
σ1(K) = K. Then σ1(a) is either equal to a or to 1/a.
Proof: We distinguish two cases depending on the ring C. First assume that C = K[t].
Since σ1 is a K-automorphism of C that maps K to itself, we have K[t] = K[σ1(t)]. In
particular σ1(t) = λt + µ, where λ, µ belong to K and λ 6= 0. Since Ψ
∗ and σ1 commute,
we obtain:
Ψ∗ ◦ σ1(t) = λat + µ = σ1 ◦Ψ
∗(t) = σ1(a)(λt + µ)
In particular, we have σ1(a) = a and the lemma follows in this case. Second assume that
C = K[t, 1/t]. Since σ1 is a K-automorphism of C, we find:
σ1(t)σ1(1/t) = σ1(t.1/t) = σ1(1) = 1
Therefore σ1(t) is an invertible element of C, and has the form σ1(t) = a1t
n1 , where
a1 ∈ K
and n1 is an integer. Since σ1 is a K-automorphism of C that maps K to K,
we have K[t, 1/t] = K[σ1(t), 1/σ1(t)]. In particular |n1| = 1 and either σ1(t) = a1t or
σ1(t) = a1/t. If σ1(t) = a1t, the relation Ψ
∗ ◦ σ1(t) = σ1 ◦ Ψ
∗(t) yields σ(a) = a. If
σ1(t) = a1/t, then the same relation yields σ(a) = 1/a.
Lemma 3.3 Let X be an irreducible affine variety of dimension n over k, such that
G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let Ψ∗ be the au-
tomorphism of C constructed in the previous section. If either C = K[t] or C = K[t, 1/t],
and if Ψ∗(t) = at, then a belongs to k∗.
Proof: We are going to prove by contradiction that a belongs to k∗. So assume that a 6∈ k∗.
Let σ be any element of Gal(K/K), and denote by σ1 the K-automorphism of C defined
as follows:
∀(x, y) ∈ B ×K, σ1(x⊗ y) = x⊗ σ1(y)
Since Ψ∗ ◦σ1(x⊗ y) = Ψ
∗(x)⊗σ1(y) = σ1 ◦Ψ
∗(x⊗ y) for any element x⊗ y of B⊗KK, Ψ
and σ1 commute. Moreover if we identify K with 1⊗K, then σ1(K) = K by construction.
By lemma 3.2, we obtain:
∀σ ∈ Gal(K/K), σ(a) = a or σ(a) =
In particular, the element (ai + a−i) is invariant under the action of Gal(K/K) for any i,
and so it belongs to K because char(K) = 0. Now let f be an element of B −K. Since f
belongs to C, we can express f as follows:
Choose an f ∈ B−K such that the difference (s−r) is minimal. We claim that (s−r) = 0,
i.e. f = fst
s. Indeed, assume that s > r. Since f is an element of B, the following
expressions:
Ψ∗(f) + (Ψ∗)−1(f)− (as + a−s)f =
i=r fi(a
i + a−i − as − a−s)ti
Ψ∗(f) + (Ψ∗)−1(f)− (ar + a−r)f =
i=r+1
i + a−i − ar − a−r)ti
also belong to B. By minimality of (s− r), these expressions belong to K. In other words,
i+ a−i− as− a−s) = 0 (resp. fi(a
i+ a−i− ar− a−r) = 0) for any i 6= 0, s (resp. for any
i 6= 0, r). Since k is algebraically closed and a 6∈ k∗ by assumption, (ai + a−i − as − a−s)
(resp. (ai+a−i−ar−a−r)) is nonzero for any i 6= s (resp. for any i 6= r). Therefore fi = 0
for any i 6= 0, and f belongs to K, a contradiction. Therefore s = r and f = fst
s. Since
f belongs to B, it also belongs to k(X). Since Ψ is an automorphism of X , the element
as = Ψ∗(f)/f belongs to k(X). Moreover (Ψ∗)n(f) = ansf for any n ∈ Z. By lemma
3.1, as belongs to G(X) = k∗. Since k is algebraically closed, a belongs to k∗, hence a
contradiction, and the result follows.
Proposition 3.4 Let X be an irreducible affine variety of dimension n over k, such that
G(X) = k∗. Let Φ be an automorphism of X such that n(Φ) = (n− 1). Let C and Ψ∗ be
the K-algebra and the K-automorphism constructed in the previous section. Then up to
conjugation, one of the following three cases occurs:
• C = K[t] and Ψ∗(t) = t + 1,
• C = K[t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity,
• C = K[t, 1/t] and Ψ∗(t) = at, where a ∈ k∗ is not a root of unity.
Proof: By lemma 2.3, we know that either C = K[t] or C = K[t, 1/t]. We are going to
study both cases.
First case: C = K[t].
The automorphism Ψ∗ maps t to at + b, where a ∈ K
and b ∈ K. If a = 1, then
b 6= 0 and up to replacing t with t/b, we may assume that Ψ∗(t) = t+ 1. If a 6= 1, then up
to replacing t with t− c for a suitable c, we may assume that Ψ∗(t) = at. But then lemma
3.3 implies that a belongs to k∗. Since Ψ∗ has infinite order, a cannot be a root of unity.
Second case: C = K[t, 1/t].
Since Ψ∗(t)Ψ∗(1/t) = Ψ∗(1) = 1, Ψ∗(t) is an invertible element of C. So Ψ∗(t) = atn,
where a ∈ K
and n 6= 0. Since Ψ∗ is an automorphism, n is either equal to 1 or to
−1. But if n were equal to −1, then a simple computation shows that (Ψ∗)2 would be the
identity, which is impossible. So Ψ∗(t) = at, where a ∈ K
. By lemma 3.3, a belongs to
k∗. As before, a cannot be a root of unity.
4 Proof of the main theorem
In this section, we are going to establish Theorem 1.1. We will split its proof in two steps
depending on the form of the automorphism Ψ∗ given in Proposition 3.4. But before, we
begin with a few lemmas.
Lemma 4.1 Let Φ be an automorphism of an affine irreducible variety X. Let G be a
linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such
that the group <h> spanned by h is Zariski dense in G. If Φ and ψh commute, then Φ
and ψg commute for any g in G.
Proof: It suffices to check that Φ∗ and ψ∗g commute for any g ∈ G. For any k-algebra
automorphisms α, β of O(X), denote by [α, β] their commutator, i.e. [α, β] = α ◦β ◦α−1 ◦
β−1. For any f ∈ O(X), set:
λ(g, f)(x) = [Φ∗, ψ∗g ](f)(x)− f(x)
Since G is a linear algebraic group acting algebraically on the affine variety X , λ(g, f)(x)
is a regular function on G×X . Since Φ∗ and ψ∗h commute, the automorphisms Φ
∗ and
ψ∗hn commute for any integer n. So the regular function λ(g, f)(x) vanishes on <h> ×X .
Since < h > is dense in G by assumption, < h > ×X is dense in G × X and λ(g, f)(x)
vanishes identically on G × X . In particular, [Φ∗, ψ∗g ](f) = f for any g ∈ G. Since this
holds for any element f of O(X), the bracket [Φ∗, ψ∗g ] coincides with the identity on O(X)
for any g ∈ G, and the result follows.
Lemma 4.2 Let Φ be an automorphism of an affine irreducible variety X. Let G be a
linear algebraic group and ψ be an algebraic G-action on X. Let h be an element of G such
that the group <h> spanned by h is Zariski dense in G. Assume there exists a nonzero
integer r such that Φr = ψh, and that G is divisible. Then there exists an algebraic action
ϕ of G′ = Z/rZ×G such that Φ = ϕg′ for some g
′ in G′.
Proof: Fix an element b in G such that br = h, and set ∆ = Φ◦ψb−1 . This is possible since
G is divisible. By construction, ∆ is an automorphism of X . Since Φr = ψh, Φ and ψh
commute. By lemma 4.1, Φ and ψg commute for any g ∈ G. In particular, we have:
∆r = (Φr) ◦ ψb−r = (Φ
r) ◦ ψh−1 = Id
So ∆ is finite, Φ = ∆ ◦ ψb and ∆ commutes with ψg for any g ∈ G. The group G
′ then
acts on X via the map ϕ defined by:
ϕ(i,g)(x) = ∆
i ◦ ψg(x)
Moreover we have Φ = ϕg′ for g
′ = (1, b).
The proof of Theorem 1.1 will then go as follows. In the following subsections, we are going
to exhibit an algebraic action ψ of Ga(k) (resp. Gm(k)) on X , such that Ψ = Φ
m = ψh for
some h. In both cases, the group G we will consider will be linear algebraic of dimension 1,
and divisible. Moreover the element h will span a Zariski dense set because h 6= 0 (resp. h
is not a root of unity). With these conditions, Theorem 1.1 will become a direct application
of Lemma 4.2.
4.1 The case Ψ∗(t) = t+ 1
Assume that C = K[t] and Ψ∗(t) = t+1. We are going to construct a nontrivial algebraic
Ga(k)-action ψ on X such that Ψ = ψ1. Since O(X) ⊂ C, every element f of O(X) can
be written as f = P (t), where P belongs to K[t]. We set r = degt P (t). Since Ψ
∗ stabilizes
O(X), the expression:
(Ψi)∗(f) = P (t+ i) =
P (j)(t)
belongs to O(X) for any integer i. Since the matrix M = (ij/j!)0≤i,j≤r is invertible in
Mr+1(Q), the polynomial P
(j)(t) belongs to O(X) for any j ≤ r. So the K-derivation
D = ∂/∂t on C stabilizes the k-algebra O(X). Since Dr+1(f) = 0, the operator D,
considered as a k-derivation on O(X), is locally nilpotent (see [Van]). Therefore the
exponential map:
exp uD : O(X) −→ O(X)[u], f 7−→
Dj(f)
is a well-defined k-algebra morphism. But exp uD defines also a K-algebra morphism from
C to C[u]. Since exp uD(t) = t + u, expD coincides with Ψ∗ on C. Since C contains the
ring O(X), we have expD = Ψ∗ on O(X). So the exponential map induces an algebraic
Ga(k)-action ψ on X such that Ψ = ψ1 (see [Van]).
4.2 The case Ψ∗(t) = at
Assume that Ψ∗(t) = at and a is not a root of unity. We are going to construct a nontrivial
algebraic Gm(k)-action ψ on X such that Ψ = ψa. First note that either C = K[t] or
C = K[t, 1/t]. Let f be any element of O(X). Since O(X) ⊂ C, we can write f as:
f = P (t) =
where the fit
i belong a priori to C. Since Ψ∗ stabilizes O(X), the expression:
(Ψj)∗(f) = P (ajt) =
ajifit
belongs to O(X) for any integer j. Since a belongs to k∗ and is not a root of unity, the
Vandermonde matrix M = (aij)0≤i,j≤s−r is invertible in Ms−r+1(k). So the elements fit
all belong to O(X) for any integer i. Consider the map:
ψ∗ : O(X) −→ O(X)[v, 1/v], f 7−→
Then ψ∗ is a well-defined k-algebra morphism, which induces a regular map ψ from k∗×X
to X . Moreover we have ψv ◦ ψv′ = ψvv′ on X for any v, v
′ ∈ k∗. So ψ defines an algebraic
Gm(k)-action on X such that Ψ = ψa.
5 Proof of Corollary 1.2
Let Φ be an automorphism of the affine plane k2, such that n(Φ) = 1. By Theorem 1.1,
there exists an algebraic action ϕ of an abelian linear algebraic group G of dimension 1
such that Φ = ϕg. We will distinguish the cases G = Z/rZ×Gm(k) and G = Z/rZ×Ga(k).
First case: G = Z/rZ×Gm(k).
Then G is linearly reductive and ϕ is conjugate to a representation in Gl2(k) (see [Ka]
or [Kr]). Since G consists solely of semisimple elements, ϕ is even diagonalizable. In par-
ticular, there exists a system (x, y) of polynomial coordinates, some integers n,m and some
r-roots of unity a, b such that:
ϕ(i,u)(x, y) = (a
iunx, biumy)
Note that, since the action is faithful, the couple (n,m) is distinct from (0, 0). Since k is
algebraically closed, we can even reduce Φ = ϕg to the first form given in Corollary 1.2.
Second case: G = Z/rZ×Ga(k).
Let ψ and ∆ be respectively the Ga(k)-action and finite automorphism constructed in
Lemma 4.2. By Rentschler’s theorem (see [Re]), there exists a system (x, y) of polynomial
coordinates and an element P of k[t] such that:
ψu(x, y) = (x, y + uP (x))
For any f ∈ k[x, y], set degψ(f) = degu exp uD(f). It is well-known that this defines a
degree function on k[x, y] (see [Da]). Since ψ and ∆ commute, ∆∗ preserves the space En
of polynomials of degree ≤ n with respect to degψ. In particular, ∆
∗ preserves E0 = k[x].
So ∆∗ induces a finite automorphism of k[x], hence ∆∗(x) = ax + b, where a is a root of
unity. Since ∆ is finite, either a 6= 1 or a = 1 and b = 0. In any case, up to replacing
x by x − µ for a suitable constant µ, we may assume that ∆∗(x) = ax. Moreover ∆∗
preserves the space E1 = k[x]{1, y}. With the same arguments as before, we obtain that
∆∗(y) = cy+ d(x), where c is a root of unity and d(x) belongs to k[x]. Composing ∆ with
ψ1/m then yields the second form given in Corollary 1.2.
6 Proof of Corollary 1.3
Let Φ be an algebraic automorphism of k2. We assume that Φ has a unique fixpoint p and
that dΦp is unipotent. We are going to prove that n(Φ) = 0.
First we check that n(Φ) cannot be equal to 2. Assume that n(Φ) = 2. Then k(x, y)Φ
has transcendence degree 2, and the extension k(x, y)/k(x, y)Φ is algebraic, hence finite.
Moreover Φ∗ acts like an element of the Galois group of this extension. In particular, Φ∗ is
finite. By a result of Kambayashi (see [Ka]), Φ can be written as h◦A◦h−1, where A is an
element of Gl2(k) of finite order and h belongs to Aut(k
2). Since Φ has a unique fixpoint p,
we have h(0, 0) = p. In particular, dΦp is conjugate to A in Gl2(k). Since dΦp is unipotent
and A is finite, A is the identity. Therefore Φ is also the identity, which contradicts the
fact that it has a unique fixpoint.
Second we check that n(Φ) cannot be equal to 1. Assume that n(Φ) = 1. By the
previous corollary, up to conjugacy, we may assume that Φ has one of the following forms:
• Φ1(x, y) = (a
nx, amby), where (n,m) 6= (0, 0), b is a root of unity but a is not,
• Φ2(x, y) = (ax, by + P (x)), where P belongs to k[t]− {0} and a, b are roots of unity.
Assume that Φ is an automorphism of type Φ1. Then dΦp is a diagonal matrix of Gl2(k),
distinct from the identity. But this is impossible since dΦp is unipotent. So assume that Φ
is an automorphism of type Φ2. Then dΦp is a linear map of the form (u, v) 7→ (au, bv+du),
with d ∈ k. Since dΦp is unipotent, we have a = b = 1. So (α, β) is a fixpoint if and only
if P (α) = 0. In particular, the set of fixpoints is either empty or a finite union of parallel
lines. But this is impossible since there is only one fixpoint by assumption. Therefore
n(Φ) = 0.
7 An application of Corollary 1.3
In this section, we are going to see how Corollary 1.3 can be applied to the determination
of invariants for automorphisms of C3. Set Q(x, y, z) = x2y − z2 − xz3 and consider the
following automorphism (see [M-P]):
Φ : C3 −→ C3,
7−→
y(1− xz) + Q
z − Q
We are going to show that:
C(x, y, z)Φ = C(x) and C[x, y, z]Φ = C[x]
Let k be the algebraic closure of C(x). Since Φ∗(x) = x, the morphism Φ∗ induces an
automorphism of k[y, z], which we denote by Ψ∗. The automorphism Ψ has clearly (0, 0)
as a fixpoint, and its differential at this point is unipotent, distinct from the identity.
Indeed, it is given by the matrix:
dΨ(0,0) =
1 −x3/2
Moreover, the set of fixpoints of Ψ is reduced to the origin. Indeed, if (α, β) is a point of
k2 fixed by Ψ, then xQ = 0 and 4β4 − 4xαβ +Q2 = 0. Since x belongs to k∗, we have:
Q = x2α− β2 − xβ3 = 0 and β4 − xαβ = 0
If β = 0, then α = 0 and we find the origin. If β 6= 0, then dividing by β and multiplying
by −x yields the relation:
x2α− xβ3 = 0
This implies β2 = 0 and β = 0, hence a contradiction. By Corollary 1.3, the field of invari-
ants of Ψ has transcendence degree zero. So the field of invariants of Φ has transcendence
degree ≤ 1 over C. Since this field contains C(x) and that C(x) is algebraically closed in
C(x, y, z), we obtain that C(x, y, z)Φ = C(x). As a consequence, the ring of invariants of
Φ is equal to C[x].
References
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Introduction
Reduction to an affine curve C
Normal forms for the automorphism
Proof of the main theorem
The case *(t)=t+1
The case *(t)=at
Proof of Corollary ??
Proof of Corollary ??
An application of Corollary ??
|
0704.1589 | Improving immunization strategies | Improving immunization strategies
Lazaros K. Gallos1, Fredrik Liljeros2, Panos Argyrakis1, Armin Bunde3, and Shlomo Havlin4
Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece
Department of Sociology, Stockholm University 106 91 Stockholm, Sweden
Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen,
Heinrich-Buff-Ring 16, 35392 Giessen, Germany and
Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel
(Dated: November 20, 2018)
We introduce an immunization method where the percentage of required vaccinations for immunity
are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our
strategy retains the advantage of being purely local, without the need of knowledge on the global
network structure or identification of the highest degree nodes. The method consists of selecting
a random node and asking for a neighbor that has more links than himself or more than a given
threshold and immunizing him. We compare this method to other efficient strategies on three real
social networks and on a scale-free network model, and find it to be significantly more effective.
PACS numbers: 89.75.Hc, 87.23.Ge
Immunization of large populations through vaccination
is an extremely important issue with obvious implications
for the public health [1, 2, 3]. The eradication of Small
Pox through a global mass vaccination campaign during
the second part of the 20th century represents, for exam-
ple, a landmark in the history of the medical sciences [4].
Global or national mass vaccination may however not al-
ways be possible. The number of vaccinated people may
need to be minimized due to severe side effects of vacci-
nation such as for Small Pox, or temporary shortage of
vaccine that could be the case for a pandemic influenza.
The cost for a vaccine may also be an important limiting
factor. Improving efficiency of immunization is thus an
urgent task.
Recently [5], developments in the study of population
connectivities helped researchers in the field to present
new ideas on immunization, based on the heterogeneity
in the number of contacts between individuals. A number
of strategies have been proposed for lowering the required
minimum fraction fc of the population to be immunized.
The problem can be mapped to the well-known percola-
tion problem where nodes are immunized (removed) up
to a concentration fc, above which the spanning clus-
ter does not survive. Random immunization of nodes
has been shown incapable of protecting the population
when the contacts distribution is wide, since the perco-
lation threshold is close to fc = 1, i.e. practically all
nodes need to be immunized [6, 7, 8]. The best known
strategy today is believed to be targeted immunization,
where the highest connected nodes in the system are im-
munized in decreasing order of their degree. In this case
fc is less than 10% [7, 9, 10]. For all practical applica-
tions, though, this approach is unrealistic because it is
a ‘global’ strategy and requires a complete knowledge of
the high degree nodes, which is in many cases impossi-
ble. An effective strategy, called acquaintance immuniza-
tion, was recently introduced [11] that combines both ef-
ficiency and somewhat greater ease of applicability. Ac-
cording to this scheme a random individual is selected
who then points to one of his random acquaintances and
this node is the one to be immunized. This method is
more efficient compared to random immunization (fc is
of the order of 20-25%) but less efficient than targeted
immunization.
In this paper we introduce an immunization method,
which is practically as efficient as the accepted as opti-
mum strategy, but at the same time depends on local in-
formation only. The method consists in selecting random
individuals and asking them to direct us to their friend
who is more connected than they are and this acuain-
tance is immunized. If such a friend does not exist we
continue with another random selection. Alternatively,
in a second variation of the method we ask the randomly
chosen individual to point us to a random neighbor that
has a number of neighbors larger than e.g. k = 5 (or
an equally small and easily countable threshold value).
If they point to such an individual it is immunized, oth-
erwise we select another individual. Similar results are
obtained if the chosen individual is asked to estimate his
own number of contacts, rather than of his random neigh-
bor. Although this procedure is simpler, the selection of
a neighbor can also eliminate the bias that may be in-
troduced due to selfish people, lying about their contacts
in order to receive the vaccine themselves. The method
is proposed for social networks, but it is expected that
it can be even more efficient for technological networks,
such as e.g. the Internet, where the number of links for
a given node is exactly known to the local network ad-
ministrator, and need not be estimated.
Our method is local because the decision for immu-
nization of a given node is taken without the need to
know the connectivity of other nodes. This is in contrast
with global strategies where immunization of a node has
to be decided only after we have gathered information for
the entire network. This means that for immunization of
e.g. a city or a country in a global method we have to
http://arxiv.org/abs/0704.1589v1
send special teams to collect this information and trans-
mit it to a central place. This central authority decides
then which nodes should be immunized and transmits
back the outcome to the local authorities which then go
on with vaccinations. For a local method, there is no
need to collect or compare data from other areas of the
network. Based on the answer of each individual the de-
cision is made immediately on whether a node should be
immunized or not.
We study the proposed method on real social networks
with a fat tail in their degree distribution, as well as on
a random scale-free model network. We also compare
this method with several other immunization strategies,
including such that partial knowledge on the global net-
work of contacts is available and we demonstrate the ad-
vantage of the proposed method via the improvement in
The social networks used in this study represent dif-
ferent interactions among the members of an online com-
munity, as described in Ref. [12]. These interactions in-
clude a) exchange of messages, b) signing of guestbooks,
c) flirt requests, and d) established friendships. The first
three networks are directed but we consider only their
undirected projection, by transforming arcs into edges.
No significant difference is observed in the results for
the undirected network and the projections of the di-
rected networks. The size of the networks is of the order
N = 104. The percentage of immunized nodes is denoted
with f , while the percentage of nodes suveyed is denoted
with p. The four strategies that we employ are summa-
rized below. Strategy I: Immunize a node with probabil-
ity proportional to kα, where k is the number of connec-
tions and α tunes the probability of preferentially select-
ing high-connectivity or low-connectivity nodes. Large
positive values of α tend towards mainly selecting the
hubs (α → ∞ is equivalent to targeted immunization),
the value α = 0 represents the random immunization
model, while negative α values lead to selecting the lower-
connected nodes [13]. This parameter can be interpreted
as a measure of the extent of our knowledge on the struc-
ture. Strategy II: Select a node with probability propor-
tional to kα and immunize a random acquaintance of this
node. The value α = 0 corresponds to the acquaintance
immunization scheme [11]. Strategy III: Select a random
node and immunize one of its acquaintances i, with prob-
ability proportional to kαi , where ki represents the degree
of the neighbor. Strategy IV: Select a random node and
ask for an acquaintance, which is immunized if a cer-
tain condition is met. We study two variations: a) The
selected node points randomly to a node which is more
connected than himself. If there are no such neighbors
no node is immunized. b) The selected node is asked
to choose a random neighbor with degree larger than a
threshold value kcut then this acquaintance is immunized.
Equivalently, we can ask the node to estimate its own de-
gree. If it is larger than a threshold value we immunize
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
(b) Guestbook(a) Messages
(c) Flirt
(d) γ=2.5
II II
IV IV
FIG. 1: Critical immunized fraction fc of the population as
a function of α for (a)-(c) Real-life social networks, and (d)
scale-free network model with γ = 2.5. Four different strate-
gies are used as described in the text and indicated in the
plot. The two symbols correspond to the critical fraction
for the strategies of the enhanced acquaintance immuniza-
tion method (the open circle corresponds to asking for an
acquintance with threshold kcut = 7, while the filled circle
corresponds to asking for a better connected node).
the node, otherwise we ignore it. These two variations are
similar when kcut = 〈k〉. We call strategy IV “enhanced
acquaintance immunization” (EAI) method.
In Figs. 1a-c we present the results of fc for the four de-
scribed strategies applied to three of the social networks,
as defined by different types of interactions. All networks
follow similar patterns for a given strategy. In strategy
I we can see the abrupt decrease of fc when increasing
α from α ≤ 0 (random immunization) with fc = 1 to
α = ∞ (targeted immunization) with fc ≪ 1. Strategy
II presents an improvement over the first strategy for val-
ues α . 1. The critical value fc presents a minimum at
α ≃ 1, indicating that identification of large hubs actu-
ally deteriorates the results, since the neighbors of large
hubs, which are chosen to be immunized, are with higher
probability low degree nodes for dissasortative networks,
similarly with the acquaintance immunization method
[11]. Strategy III leads to monotonic decrease in fc and
prevails from the first two methods when we have limited
global network knowledge, i.e. in the range α ∈ [0, 1].
However, in Strategy III we find that when α = ∞ (i.e.
we always immunize the most connected neighbor) it may
be impossible to destroy the spanning cluster, because al-
most all selected nodes point to the same hubs. Finally,
the enhanced acquaintance immunization strategy seems
to be the most efficient method, although it assumes no
1 2 3 4 5 6 7
1 2 3 4 5 6 7
(a) Messages (b) Guestbook
(c) Flirt
(d) γ=2.5
FIG. 2: Critical immunized fraction fc of the population as
a function of the threshold value kcut for the enhanced ac-
quaintance immunization strategy applied to (a)-(c) social in-
teraction networks, (d) random model scale-free network with
γ = 2.5 (of size N = 105 nodes). Filled symbols correspond to
immunizing a random neighbor of the selected node if its de-
gree is ≥ kcut and open symbols to immunizing the selected
node itself. The upper horizontal dotted line is the result
for acquaintance immunization, the dashed line in the middle
corresponds to immunizing a more connected acquaintance,
while the lower line refers to targeted immunization.
knowledge of the underlying structure (the method is in-
dependent of α). The value of fc is lower than an attack
with α = 3 and very close to the results of the targeted
immunization.
To gain more insight into the different immunization
methods we also performed numerical simulations on a
model network. We consider each member of a popula-
tion represented by a node, while the acquaintances of a
person with other people form links. It is well established
that many social networks follow a broad distribution in
the degree of a node, such as the power-law distribution
P (k) ∼ k−γ , where the exponent γ is usually found to
be between 2 < γ < 4 [5, 14, 15, 16]. The above real
networks are scale-free with γ ≃ 2.4[12]. The results in
Fig. 1d correspond to the four strategies in such a model
network (created with the configuration random model
[17]) with exponent γ = 2.5, which is close to the re-
ported exponent γ ≃ 2.4 of the real networks used. All
strategies in this plot follow closely the results for the
real networks.
The two ‘transition’ points for the first three strategies
are located at α = 0 and α = 1. At α = 0, strategies II
and III coincide. In the range α ∈ [0, 1] strategy III is
0 0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4
P∞(f)
P∞(f)
1 2 3 4 5 6 7
1 2 3 4 5 6 7
(b) Guestbook(a) Messages
(c) Flirt (d) γ=2.5
FIG. 3: Size of epidemics, measured via the fraction of nodes
belonging to the largest cluster over the number of not-
immunized nodes P∞(f), as a function of the fraction f of im-
munized nodes. In each plot, from top to bottom, the curves
correspond to acquaintance immunization, EAI redirecting to
a better connected node, the EAI with kcut = 7, and targeted
immunization. (a)-(c): real networks, and (d): random scale-
free network with γ = 2.5. Insets: Ratios for f1/fc of the
critical immnunized fraction fc over the critical fraction f1
for acquaintance immunization (kcut = 1) and pc/p1, i.e. the
number of people surveyed, as a function of kcut for the EAI
method.
more efficient, indicating that in this range it is preferable
to let the nodes choose their neighbors according to their
connectivity, rather than selecting nodes with probabil-
ity proportional to kα and following random links. The
value α ≃ 1 is the optimum value for strategy II. In prac-
tice, the process is equivalent to selecting a random link
and immunizing one of the two nodes attached to the
given link (provided the uncorrelated network hypothe-
sis holds). It is also interesting to note that up to the
value α = 1 the acquaintance immunization strategy is
superior to direct immunization of the initially selected
nodes, but close to this value the two methods yield a
similar value for fc. When α > 1 the direct immuniza-
tion method becomes more efficient than acquaintance
immunization.
The enhanced acquaintance immunization is, however,
found to be superior to all the above methods. The value
of fc for a given kcut value is of course independent of α,
meaning that it works equally well when there is no fur-
ther information on the network structure, i.e. global
knowledge does not offer any significant advantage over
completely random selections. Thus, the strategy is local
and easy to implement. The choice of kcut, though, influ-
ences fc and can further reduce the fc value when more
accurate knowledge on the network structure is available.
The gain of this method for kcut = 7 when compared to
the original acquaintance immunization method is about
a factor of 4, which is for practical purposes a significant
improvement. This striking variation is evident in Fig. 2,
where the critical percentage decays from fc ≃ 0.26 at
kcut = 1 (acquaintance immunization) to fc ≃ 0.06 at
kcut = 7. For kcut = 7 the strategy works comparably
well to the targeted immunization. The fraction fc, how-
ever, remains very low even when the cutoff value kcut
decreases to values close to, but less than 7. This stabil-
ity over the value of kcut offers greater flexibility since the
method seems tolerant to mistakes of lower degree nodes
being pointed at for immunization, without siginificant
loss in the efficiency (even at a value of kcut = 4 the
critical fraction fc remains lower than 10%). The results
are different when we immunize directly the initially se-
lected random node (without asking for an acquaintance)
and only at kcut = 7 the two methods seem to coincide
(Fig. 2). There exists, though, a critical degree above
which this strategy no longer works, simply because the
number of nodes with degree larger than this value is
smaller than the critical number needed for complete im-
munization. Thus, it seems preferable to remain con-
servative on the estimation of kcut and choose a smaller
value over a larger one.
A considerable advantage is gained, even when the
question is posed in a much simpler way, i.e. we ask
a random node to direct us to a friend who is better con-
nected than his and immunize him. This simple approach
already offers a significant improvement over the original
acquaintance method, as is evident in Fig. 2, although it
is not as efficient as when asking for a friend whose degree
exceeds the cutoff value. Since it is, however, much easier
for an individual to estimate an acquaintance who is bet-
ter connected than himself, and practically everyone can
understand and correctly answer this simple question, we
consider this method as a useful strategy which is easy
to apply in real-life situations.
In order to assess the size of the epidemics in the im-
munization process we measure the size of the spanning
cluster (epidemics size) as a function of the immunized
nodes f . In Figs. 3a-c we present the fraction of nodes
belonging to the spanning cluster over the total number
of non-immunized nodes for the real networks described
above and compare the targeted immunization with the
enhanced acquaintance immunization and the original ac-
quaintance immunization methods. The results for the
model scale-free networks (Fig. 3d) are averages over 100
different realizations of networks with exponent γ = 2.5.
In all cases the critical fraction for the targeted immu-
nization and the EAI with the cutoff value are similar,
while acquaintance immunization leads to considerably
higher values of fc. Again, the EAI with an estimation
of a better connected friend yields a result between these
two extremes. However, during the removal process the
targeted immunization yields the faster decomposition of
the spanning cluster, since it first removes the most con-
nected nodes in the system. The results for all the ac-
quaintance immunization methods depend on when these
largest hubs will be selected and the averaging conceals
the fact that during one realization the size of the largest
cluster drops abruptly when the largest hubs are selected.
Despite this, the proposed methods follow closely the re-
sults of targeted immunization, while retaining the ad-
vantage of being local.
In the insets of Fig. 3 we can see that compared to the
acquaintance immunization method (which is the EAI
method with kcut = 1) in general we need to survey more
nodes for their acquaintances as kcut increases, but this
is a small change compared to the improvement in the
number of required immunizations presented in the same
plots.
A work with similar scope was performed by Holme
[18]. Among other methods, an immunization scheme
was introduced, where a random node points to one of its
highest degree neighbors or to its most connected neigh-
bor. This corresponds to strategy III of the current work
with α → ∞ (where we encounter the problem of select-
ing always the same nodes as described above) and the
first variation of Strategy IV. The results in that paper
are consistent with the ones presented above for these
limiting cases.
In summary, we introduced and compared various im-
munization strategies on real and model networks. We
have shown that the fraction of immunized nodes can
be significantly reduced to the almost optimum level of
intentional immunization using a completely local infor-
mation strategy. This simple process is enough to ensure
that the immunization threshold is significantly lowered,
as compared to other local methods.
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NEST/PATHFINDER project DYSONET 012911, by a
project of the Greek GGET in conjunction with ESF and
by the Israel Science Foundation.
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|
0704.1590 | Constraints on the Very Early Universe from Thermal WIMP Dark Matter | April 2007
Constraints on the Very Early Universe from
Thermal WIMP Dark Matter
Manuel Dreesa, ∗, Hoernisa Iminniyaza,b, † and Mitsuru Kakizakia, ‡
aPhysikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany
bPhysics Dept., Univ. of Xinjiang, 830046 Urumqi, P.R. China
Abstract
We investigate the relic density nχ of non–relativistic long–lived or sta-
ble particles χ in non–standard cosmological scenarios. We calculate the
relic abundance starting from arbitrary initial temperatures of the radiation–
dominated epoch, and derive the lower bound on the initial temperature
T0 ≥ mχ/23, assuming that thermally produced χ particles account for the
dark matter energy density in the universe; this bound holds for all χ annihi-
lation cross sections. We also investigate cosmological scenarios with modified
expansion rate. Even in this case an approximate formula similar to the stan-
dard one is capable of predicting the final relic abundance correctly. Choosing
the χ annihilation cross section such that the observed cold dark matter abun-
dance is reproduced in standard cosmology, we constrain possible modifications
of the expansion rate at T ∼ mχ/20, well before Big Bang Nucleosynthesis.
∗[email protected]
†[email protected]
‡[email protected]
http://arxiv.org/abs/0704.1590v2
1 Introduction
One of the most notable recent developments in cosmology is the precise determina-
tion of cosmological parameters from observations of the large–scale structure of the
universe, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP). In
particular, the accurate determination of the non–baryonic cold Dark Matter (DM)
density [1],
0.08 < ΩCDMh
2 < 0.12 (95% C.L.) , (1)
has great influence on particle physics models which possess dark matter candidates
[2, 3]. The requirement that the predicted DM density falls in the range (1) is a
powerful tool for discriminating between various models and for constraining the
parameter space of surviving models.
Many dark matter candidate particles have been proposed. In particular long–
lived or stable weakly interacting massive particles (WIMPs) with weak–scale masses
are excellent candidates. In standard cosmology WIMPs decoupled from the thermal
background during the radiation–dominated epoch after inflation. In this framework
convenient and accurate analytic approximate solutions for the relic abundance have
been derived [4, 5]. One of the best motivated candidates for WIMPs is the lightest
neutralino in supersymmetric (SUSY) models. Assuming that the neutralino is the
lightest supersymmetric particle (LSP) stabilized due to R–parity, its relic abundance
has been extensively discussed [3]. Similar analyses have also been performed for
other WIMPs whose existence is postulated in other extensions of the Standard
Model (SM) of particle physics. In many cases the cosmologically favored parameter
space of WIMP models can be directly tested at the CERN Large Hadron Collider
(LHC) in a few years [6]. The same parameter space often also leads to rates of
WIMP interactions with matter within the sensitivity of near–future direct DM
detection experiments.
This discussion shows that we are now entering an interesting time where the
standard cosmological scenario can be examined by experiments at high–energy col-
liders as well as DM searches [7]. In this respect we should emphasize that the
relic abundance of thermally produced WIMPs depends not only on their annihila-
tion cross section, which can be determined by particle physics experiments, but in
general is also very sensitive to cosmological parameters during the era of WIMP
production and annihilation. Of particular importance are the initial temperature
T0 at which WIMPs began to be thermally produced, and the expansion rate of the
universe H .
In the standard cosmological scenario, the expansion rate is uniquely determined
through the Friedmann equation of general relativity. In this scenario the density of
WIMPs with massmχ followed its equilibrium value until the freeze–out temperature
TF ≃ mχ/20. Below TF , interactions of WIMPs are decoupled, and thus the present
density is independent of T0 as long as T0 > TF .
It should be noted that in non–standard scenarios the relic density can be larger
or smaller than the value in the standard scenario. One example is the case where
T0 is smaller than or comparable to TF , which can be realized in inflationary models
with low reheat temperature. Since in many models the inflationary energy scale
must be much higher than mχ in order to correctly predict the density perturbations
[8], the standard assumption T0 > TF is not unreasonable. On the other hand, the
constraint on the reheat temperature from Big Bang Nucleosynthesis (BBN) is as low
as T0 >∼ MeV [9, 10]. From the purely phenomenological viewpoint, it is therefore
also interesting to investigate the production of WIMPs in low reheat temperature
scenarios [9, 11, 12, 13, 14].
The standard scenario also assumes that entropy per comoving volume is con-
served for all temperatures T ≤ TF . Late entropy production can dilute the predicted
relic density [15, 16]. The reason is that the usual calculation actually predicts the
ratio of the WIMP number density to the entropy density. On the other hand, if
late decays of a heavier particle non–thermally produce WIMPs in addition to the
usual thermal production mechanism, the resulting increase of the WIMP density
competes with the dilution caused by the decay of this particle into radiation, which
increases the entropy density [17, 18, 19, 20, 13, 21].
Another example of a non–standard cosmology changing the WIMP relic density
is a modified expansion rate of the universe. This might be induced by an anisotropic
expansion [16], by a modification of general relativity [16, 22], by additional contri-
butions to the total energy density from quintessence [23], by branes in a warped
geometry [24], or by a superstring dilaton [25].
These examples show that, once the WIMP annihilation cross section is fixed,
with the help of precise measurements of the cold dark matter density we can probe
the very early stage of the universe at temperatures of O(mχ/20) ∼ 10 GeV. This
is reminiscent of constraining the early evolution of the universe at T = O(100) keV
∗Note that TF can be formally defined in the standard way even if T0 < TF . In this case WIMPs
never were in full equilibrium, and correspondingly never “froze out”.
using the primordial abundances of the light elements produced by BBN.
The goal of this paper is to investigate to what extent the constraint (1) on
the WIMP relic abundance might allow us to derive quantitative constraints on
modifications of standard cosmology. So far the history of the universe has been
established by cosmological observations as far back as the BBN era. In this paper
we try to derive bounds on cosmological parameters relevant to the era before BBN.
Rather than studying specific extensions of the standard cosmological scenario, we
simply parameterize deviations from the standard scenario, and attempt to derive
constraints on these new parameters. Since we only have the single constraint (1),
for the most part we only allow a single quantity to differ from its standard value.
We expect that varying two quantities simultaneously will allow to get the right
relic density for almost any WIMP annihilation cross section. This has been shown
explicitly in [13] for the case that both late entropy production and non–thermal
WIMP production are considered, even if both originate from the late decay of a
single scalar field.
We first analyze the dependence of the WIMP abundance on the initial temper-
ature T0 of the conventional radiation–dominated epoch. We showed in [14] that for
fixed T0 the predicted WIMP relic density reaches a maximum as the annihilation
cross section is varied from very small to very large values. A small annihilation cross
section corresponds to a large TF > T0; in this case the relic density increases with
the annihilation cross section, since WIMP production from the thermal plasma is
more important than WIMP annihilation. On the other hand, increasing this cross
section reduces TF ; once TF < T0 a further increase of the cross section leads to
smaller relic densities since in this case WIMPs continue to annihilate even after the
temperature is too low for WIMP production. Here we turn this argument around,
and derive the lower bound on T0 ≥ mχ/23 under the assumption that all WIMPs
are produced thermally. Note that we do not need to know the WIMP annihilation
cross section to derive this bound.
We then examine the dependence of the predicted WIMP relic density on the
expansion rate in the epoch prior to BBN, where we allow the Hubble parameter to
depart from the standard value. The standard method of calculating the thermal
relic density [2, 4] is found to be still applicable in this case. Our working hypothesis
here is that the standard prediction for the Hubble expansion rate is essentially
correct, i.e. that the true expansion rate differs by at most a factor of a few from
the standard prediction. We then simply employ a generic Taylor expansion for the
temperature dependence of this modification factor; note that the success of standard
BBN indicates that this factor cannot deviate by more than ∼ 20% from unity at
low temperatures, T <∼ 1 MeV. Similarly, we assume that the WIMP annihilation
cross section has been determined (from experiments at particle colliders) to have
the value required in standard cosmology. Our approach is thus quite different from
that taken in [7], where present upper bounds on the fluxes of WIMP annihilation
products are used to place upper bounds on the Hubble expansion rate during WIMP
decoupling. The advantage of their approach is that no prior assumption on the
WIMP annihilation cross section needs to be made, whereas we assume a cross
section that reproduces the correct relic density in the standard scenario. On the
other hand, the bounds derived in refs.[7] are still quite weak, allowing the Hubble
parameter to exceed its standard prediction by a factor >∼ 30; moreover, no lower
bound on H can be derived in this fashion.
The remainder of this paper is organized as follows: In Sec. 2 we will briefly review
the calculation of the WIMP relic abundance assuming a conventional radiation–
dominated universe, and derive the lower bound on the initial temperature T0. In
Sec. 3 we discuss the case where the pre–BBN expansion rate is allowed to depart
from the standard one. Using approximate analytic formulae for the predicted WIMP
relic density for this scenario, we derive constraints on the early expansion parameter.
Finally, Sec. 4 is devoted to summary and conclusions.
2 Relic Abundance in the Radiation–Dominated
Universe
We start the discussion of the relic density nχ of stable or long–lived particles χ by
reviewing the structure of the Boltzmann equation which describes their creation and
annihilation. The goal of this Section is to find the lowest possible initial temperature
of the radiation–dominated universe, assuming that the present relic abundance of
cold dark matter is entirely due to thermally produced χ particles.
As usual, we will assume that χ is self–conjugate†, χ = χ̄, and that some symme-
try, for example R–parity, forbids decays of χ into SM particles; the same symmetry
then also forbids single production of χ from the thermal background. However,
†The case χ 6= χ̄ differs in a non–trivial way only in the presence of a χ− χ̄ asymmetry, i.e. if
nχ 6= nχ̄.
the creation and annihilation of χ pairs remains allowed. The time evolution of the
number density nχ of particles χ in the expanding universe is then described by the
Boltzmann equation [2],
+ 3Hnχ = −〈σv〉(n2χ − n2χ,eq) , (2)
where nχ,eq is the equilibrium number density of χ, and 〈σv〉 is the thermally averaged
annihilation cross section multiplied with the relative velocity of the two annihilating
χ particles. Finally, the Hubble parameter H = Ṙ/R is the expansion rate of the
universe, R being the scale factor in the Friedmann–Robertson–Walker metric. The
first (second) term on the right–hand side of Eq.(2) describes the decrease (increase)
of the number density due to annihilation into (production from) lighter particles.
Eq.(2) assumes that χ is in kinetic equilibrium with standard model particles.
It is useful to rewrite Eq.(2) in terms of the scaled inverse temperature x = mχ/T
as well as the dimensionless quantities Yχ = nχ/s and Yχ,eq = nχ,eq/s. The entropy
density is given by s = (2π2/45)g∗sT
3, where
g∗s =
i=bosons
i=fermions
. (3)
Here gi denotes the number of intrinsic degrees of freedom for particle species i
(e.g. due to spin and color), and Ti is the temperature of species i. Assuming that
the universe expands adiabatically, the entropy per comoving volume, sR3, remains
constant, which implies
+ 3Hs = 0 . (4)
The time dependence of the temperature is then given by
. (5)
Therefore the Boltzmann equation (2) can be written as
= −〈σv〉s
(Y 2χ − Y 2χ,eq) . (6)
Thermal production of WIMPs takes place during the radiation–dominated epoch,
when the expansion rate is given by
, (7)
with MPl = 2.4× 1018 GeV being the reduced Planck mass and
i=bosons
i=fermions
. (8)
In the following we use Hst to denote the standard expansion rate (7). If the post–
inflationary reheat temperature was sufficiently high, WIMPs reached full thermal
equilibrium. This remains true for temperatures well below mχ. We can therefore
use the non–relativistic expression for the χ equilibrium number density,
nχ,eq = gχ
e−mχ/T . (9)
In the absence of non–thermal production mechanisms, nχ ≤ nχ,eq at early times.
The annihilation rate Γ = nχ〈σv〉 then depends exponentially on T , and thus drops
more rapidly with decreasing temperature than the expansion rate Hst of Eq.(7)
does. When the annihilation rate falls below the expansion rate, the number density
of WIMPs ceases to follow its equilibrium value and is frozen out.
For T ≪ mχ the annihilation cross section can often (but not always [5]) be
approximated by a non–relativistic expansion in powers of v2. Its thermal average
is then given by
〈σv〉 = a + b〈v2〉+O(〈v4〉) = a+ 6b
. (10)
In this standard scenario, the following approximate formula has been shown [4, 2, 5]
to accurately reproduce the exact (numerically calculated) relic density:
Yχ,∞ ≡ Yχ(x → ∞) ≃
1.3 mχMPl
g∗(xF )(a/xF + 3b/x
, (11)
with xF = mχ/TF , TF being the decoupling temperature. For WIMPs, xF ≃ 22.
Here we assume g∗ ≃ g∗s and dg∗/dx ≃ 0. It is useful to express the χ mass density
as Ωχ = ρχ/ρc, ρc = 3H
Pl being the critical density of the universe. The present
relic mass density is then given by ρχ = mχnχ,∞ = mχs0Yχ,∞; here s0 ≃ 2900 cm−3
is the present entropy density. Eq.(11) then leads to
2 = 2.7× 1010 Yχ,∞
100 GeV
≃ 8.5× 10
−11 xF GeV
g∗(xF )(a+ 3b/xF )
, (12)
where h ≃ 0.7 is the scaled Hubble constant in units of 100 km sec−1 Mpc−1. We
defer further discussions of this expression to Sec. 3, where scenarios with modified
expansion rate are analyzed. Note that in the standard scenario leading to Eq.(12),
the present χ relic density is inversely proportional to its annihilation cross section
and has no dependence on the reheat temperature. Recall that this result depends
on the assumption that the highest temperature in the post–inflationary radiation
dominated epoch, which we denote by T0, exceeded TF significantly.
On the other hand, if T0 was too low to fully thermalize WIMPs, the final result
for Ωχ will depend on T0. In particular, if WIMPs were thermally produced in a com-
pletely out–of–equilibrium manner starting from vanishing initial abundance during
the radiation–dominated era, such that WIMP annihilation remains negligible, the
present relic abundance is given by [14]
Y0(x → ∞) ≃ 0.014 g2χg−3/2∗ mχMPle−2x0x0
. (13)
Note that the final abundance depends exponentially on T0, and increases with
increasing cross section.
In in–between cases where WIMPs are not completely thermalized but WIMP
annihilation can no longer be neglected, we have shown [14] that re–summing the
first correction term δ enables us to reproduce the full temperature dependence of
the density of WIMPs:
1− δ/Y0
≡ Y1,r . (14)
Here δ < 0 describes the annihilation of WIMPs produced according to Eq.(13):
δ(x → ∞) ≃ −2.5 × 10−4 g4χg−5/2∗ m3χM3Ple−4x0x0
. (15)
Since δ is proportional to the third power of the cross section, the re–summed ex-
pression Y1,r is inversely proportional to the cross section for large cross section. In
ref.[14] we have shown that this feature allows the approximation (14) to be smoothly
matched to the standard result (12). Not surprisingly, as long as we only consider
thermal χ production, decreasing T0 can only reduce the final χ relic density.
With the help of these results, we can explore the dependence of the χ relic
density on T0 as well as on the annihilation cross section. Some results are shown in
Fig. 1, where we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0,
mχ = 100 GeV, gχ = 2 and g∗ = 90.
The results depicted in this Figure can be understood as follows. For small T0,
i.e. large x0, Eq.(13) is valid, leading to a very strong dependence of Ωχh
2 on x0.
10−10 10−9 10−8 10−7
a (GeV−2 )
b = 0
10−310−4
10−9 10−8 10−7 10−6
b (GeV−2 )
a = 0
10−310−4
Figure 1: Contour plots of the present relic abundance Ωχh2. Here we take (a) a 6= 0, b = 0, and
(b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2, g∗ = 90. The shaded region
corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh
2 < 0.12
(95% C.L.).
Recall that in this case the relic density is proportional to the cross section. In this
regime one can reproduce the relic density (1) with quite small annihilation cross
section, a + 6b/x0 <∼ 10−9 GeV−2, for some narrow range of initial temperature,
x0 <∼ 22.5. Note that this allows much smaller annihilation cross sections than the
standard result, at the cost of a very strong dependence of the final result on the
initial temperature T0.
In this Section we set out to derive a lower bound on T0. In this regard the
region of parameter space described by Eq.(13) is not optimal. Increasing the χ
annihilation cross section at first allows to obtain the correct relic density for larger
x0, i.e. smaller T0. However, the correction δ then quickly increases in size; as noted
earlier, once |δ| > Y0 a further increase of the cross section will lead to a decrease
of the final relic density. The lower bound on T0 is therefore saturated if Ωχh
2 as a
function of the cross section reaches a maximum. From Fig.1 we read off
T0 ≥ mχ/23 , (16)
if we require Ωχh
2 to fall in the range (1).
We just saw that in the regime where this bound is saturated, the final relic den-
sity is (to first order) independent of the annihilation cross section, ∂(Ωχh
2)/∂〈σv〉 =
0. If T0 is slightly above the absolute lower bound (16), the correct relic density can
therefore be obtained for a rather wide range of cross sections. For example, if
x0 = 22.5, the entire range 3 × 10−10 GeV−2 <∼ a <∼ 2 × 10−9 GeV
−2 is allowed. Of
course, the correct relic density can also be obtained in the standard scenario of
(arbitrarily) high T0, if a + 3b/22 falls within ∼ 20% of 2× 10−9 GeV−2.
3 Relic Abundance for Modified Expansion Rate
In this section we discuss the calculation of the WIMP relic density nχ in modi-
fied cosmological scenarios where the expansion parameter of the pre–BBN universe
differed from the standard value Hst of Eq.(7). For the most part we will assume
that WIMPs have been in full thermal equilibrium. Various cosmological models
predict a non–standard early expansion history [22, 23, 24, 25]. Here we analyze to
what extent the relic density of WIMP Dark Matter can be used to constrain the
Hubble parameter during the epoch of WIMP decoupling. As long as we assume
large T0 we can use a modification of the standard treatment [4, 2] to estimate the
relic density for given annihilation cross section and expansion rate. We will show
that the resulting approximate solutions again accurately reproduce the numerically
evaluated relic abundance.
Let us introduce the modification parameter A(x), which parameterizes the ratio
of the standard value Hst(x) to the assumed H(x):
A(x) =
Hst(x)
. (17)
Note that A > 1 means that the expansion rate is smaller than in standard cos-
mology. Allowing for this modified expansion rate, the Boltzmann equation (6) is
altered to
G(x)mχMPl
〈σv〉A(x)
Y 2χ − Y 2χ,eq
, (18)
where
G(x) =
. (19)
Following refs.[4, 2], we can obtain an approximate solution of this equation by
considering the differential equation for ∆ = Yχ − Yχ,eq. For temperatures higher
than the decoupling temperature, Yχ tracks Yχ,eq very closely and the ∆
2-term can
be ignored:
≃ −dYeq
− 4π√
mχMPl
G(x)〈σv〉A(x)
(2Yχ,eq∆) . (20)
Here dYχ,eq/dx ≃ −Yχ,eq for x ≫ 1. In order to keep |∆| small, the derivative d∆/dx
must also be small, which implies
∆ ≃ x
90)mχMPlG(x)〈σv〉A(x)
. (21)
This solution is used down to the freeze–out temperature TF , defined via
∆(xF ) = ξYχ,eq(xF ) , (22)
where ξ is a constant of order of unity. This leads to the following expression:
xF = ln
ξmχMPlgχ
〈σv〉A(x)
xg∗(x)
, (23)
which can e.g. be solved iteratively. In our numerical calculations we will choose
2− 1 [4, 2].
On the other hand, for low temperatures (T < TF ), the production term ∝ Y 2χ,eq
in Eq.(18) can be ignored. In this limit, Yχ ≃ ∆, and the solution of Eq.(18) is given
∆(xF )
∆(x → ∞)
mχMPlI(xF ) , (24)
where the annihilation integral is defined as
I(xF ) =
G(x)〈σv〉A(x)
. (25)
Assuming ∆(x → ∞) ≪ ∆(xF ), the final relic abundance is
Yχ,∞ ≡ Yχ(x → ∞) =
90)mχMPlI(xF )
. (26)
Plugging in numerical values for the Planck mass and for today’s entropy density,
the present relic density can thus be written as
8.5× 10−11
I(xF ) GeV
. (27)
The constraint (1) therefore corresponds to the allowed range for the annihilation
integral
7.1× 10−10 GeV−2 < I(xF ) < 1.1× 10−9 GeV−2 . (28)
The standard formula (12) for the final relic density is recovered if A(x) is set to
unity and G(x) is replaced by the constant
g∗(xF ).
The further discussion is simplified if we use the normalized temperature z =
T/mχ ≡ 1/x, rather than x. Phenomenologically A(z) can be any function subject
to the condition that A(z) approaches unity at late times in order not to contradict
the successful predictions of BBN. We need to know A(z) only for the interval from
around the freeze-out to BBN: zBBN ∼ 10−5 − 10−4 <∼ z <∼ zF ∼ 1/20. This suggests
a parameterization of A(z) in terms of a power series in (z − zF,st):
A(z) = A(zF,st) + (z − zF,st)A′(zF,st) +
(z − zF,st)2A′′(zF,st) , (29)
where zF,st is the normalized freeze–out temperature in the standard scenario and
a prime denotes a derivative with respect to z. The ansatz (29) should be of quite
general validity, so long as the modification of the expansion rate is relatively modest.
This suits our purpose, since we wish to find out what constraints can be derived on
the expansion history if standard cosmology leads to the correct WIMP relic density.
We further introduce the variable
k = A(z → 0) = A(zF,st)− zF,stA′(zF,st) +
z2F,stA
′′(zF,st) , (30)
which describes the modification parameter at late times. Since zBBN is almost
zero, we treat k as the modification parameter at the era of BBN in this paper.‡
Deviations from k = 1 are conveniently discussed in terms of the equivalent number
of light neutrino degrees of freedom Nν . BBN permits that the number of neutrinos
differs from the standard model value Nν = 3 by δNν = 1.5 or so [26]. We therefore
take the uncertainty of k to be 20%. In the following we treat A(zF,st), A
′(zF,st) and
k as free parameters; A′′(zF,st) is then a derived quantity.
Note that we allow A(z) to cross unity, i.e. to switch from an expansion that
is faster than in standard cosmology to a slower expansion or vice versa. This is
illustrated in Fig. 2, which shows examples of possible evolutions of A(z) as function
of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each case
we consider scenarios with A(zF ) = 1.3 (slower expansion at TF than in standard
cosmology) as well as A(zF ) = 0.7 (faster expansion); moreover, we allow the change
of A at z = zF to be either positive or negative. However, we insist that H remains
positive at all times, i.e. A(z) must not cross zero. This excludes scenarios with
very large positive A′(zF,st), which would lead to A < 0 at some z < zF . Similarly,
‡Presumably the Hubble expansion rate has to approach the standard rate even more closely
for T < TBBN. However, since all WIMP annihilation effectively ceased well before the onset of
BBN, this epoch plays no role in our analysis.
0 0.01 0.02 0.03 0.04 0.05
z = T/mχ
k = 1.2
A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
0 0.01 0.02 0.03 0.04 0.05
z = T/mχ
k = 0.8A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
Figure 2: Examples of possible evolutions of the modification parameter A(z) as function of z
for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each frame we choose
A(zF ) = 1.3, A
′(zF ) = −3 (thick line), A(zF ) = 1.3, A′(zF ) = 9 (dashed), A(zF ) = 0.7, A′(zF ) =
−3 (dotted), A(zF ) = 0.7, A′(zF ) = 9 (dot–dashed).
demanding that our ansatz (29) remains valid for some range of temperatures above
TF excludes scenarios with very large negative A
′(zF,st). We will come back to this
point shortly.
Eq.(23) shows that zF 6= zF,st (xF 6= xF,st) if A(zF ) 6= 1. This is illustrated by
Fig. 3, which shows the difference between xF and xF,st in the (A(zF,st), A
′(zF,st))
plane. Here we take parameters such that Ωχh
2 = 0.099 in the standard cosmology,
which is recovered for A(zF,st) = 1, A
′(zF,st) = 0. Due to the logarithmic dependence
on A, xF (or zF ) differs by at most a few percent from its standard value if A(zF,st) is
O(1). Since TF only depends on the expansion rate at TF , it is essentially insensitive
to the derivative A′(zF,st).
In our treatment the modification of the expansion parameter affects the WIMP
relic density mostly via the annihilation integral (25). In terms of the normalized
temperature z, the latter can be rewritten as
I(zF ) =
dz G(z)〈σv〉A(z) . (31)
One advantage of the expansion (29) is that this integral can be evaluated analyti-
cally:
I(zF ) ≃ G(zF )
k(azF + 3bz
F ) + (A
′(zF,st)− zF,stA′′(zF,st))
z2F + 2bz
A′′(zF,st)
z3F +
. (32)
0.6 0.8 1 1.2 1.4
A(zF,st )
xF − x F,st
a = 2.0*10−9 GeV −2
b = 0
k = 1
xF,st = 22.0
−0.4 −0.2 0 0.2 0.4
Figure 3: Contour plot of xF −xF,st in the (A(zF,st), A′(zF,st)) plane. Here we take a = 2.0×10−9
GeV−2, b = 0, mχ = 100 GeV, gχ = 2, g∗ = 90 (constant) and k = 1. This parameter set produces
xF,st = 22.0 and Ωχh
2 = 0.099 for the standard approximation.
0.6 0.8 1 1.2 1.4
A(zF,st )
2(approx) / Ωχh
2(exact)
0.997
0.998
0.999
1.000
a = 2.0*10−9 GeV−2
b = 0
k = 1
0.6 0.8 1 1.2 1.4
A(zF,st)
2(approx) / Ωχh
2(exact)
0.996
0.998
1.000
1.002
1.004 a = 0
b = 1.5*10−8 GeV−2
k = 1
Figure 4: Ratio of the analytic result of the relic density to the exact value in the (A(zF,st),
A′(zF,st)) plane for a = 2.0× 10−9 GeV−2, b = 0 (left frame) and for a = 0, b = 1.5× 10−8 GeV−2
(right). The other parameters are as in Fig. 3.
Here we have assumed that G(z) varies only slowly.
Before proceeding, we first have to convince ourselves that the analytic treatment
developed in this Section still works for A 6= 1. This is demonstrated by Fig. 4,
which shows the ratio of the analytic solution obtained from Eqs. (27) and (32)
to the exact one, obtained by numerically integrating the Boltzmann equation (18),
assuming constant g∗. We see that our analytical treatment is accurate to better
than 1%, and can thus safely be employed in the subsequent analysis.
0.6 0.8 1 1.2 1.4
A(zF,st )
a = 2.0*10−9 GeV −2
b = 0
k = 1
0.6 0.8 1 1.2 1.4
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
0.6 0.8 1 1.2 1.4
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1.2
0.6 0.8 1 1.2 1.4
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 0.8
Figure 5: Contour plots of the relic abundance in the (A(zF,st), A′(zF,st)) plane. Here we choose
(a) a = 2.0×10−9 GeV−2, b = 0, k = 1; (b) a = 0, b = 1.5×10−8 GeV−2, k = 1; (c) a = 2.0×10−9
GeV−2, b = 0, k = 1.2; (d) a = 2.0× 10−9 GeV−2, b = 0, k = 0.8. The other parameters are as in
Fig. 3.
We are now ready to analyze the impact of the modified expansion rate on the
WIMP relic density. In Fig. 5, we show contour plots of Ωχh
2 in the (A(zF,st),
A′(zF,st)) plane. Recall that large (small) values of A correspond to a small (large)
expansion rate. Since a smaller expansion rate allows the WIMPs more time to
annihilate, A > 1 leads to a reduced WIMP relic density, whereas A < 1 means
larger relic density, if the cross section is kept fixed.
However, unlike the freeze–out temperature, the annihilation integral is sensitive
to A(z) for all z ≤ zF . Note that A′(zF,st) > 0 implies A(z) < A(zF,st) for z <
zF,st ≃ zF . A positive first derivative, A′(zF,st) > 0, can therefore to some extent
compensate for A(zF,st) > 1; analogously, a negative first derivative can compensate
for A(zF,st) < 1. This explains the slopes of the curves in Fig. 5. Recall also that
A′(zF,st) = 0 does not imply a constant modification factor A(z); rather, the term
∝ A′′(zF,st) in Eq.(29) makes sure that A approaches k as z → 0. This explains why
a change of A by some given percentage leads to a smaller relative change of Ωχh
2, as
can be seen in the Figure. This also illustrates the importance of ensuring appropriate
(near–standard) expansion rate in the BBN era. Finally, since the expansion rate
at late times is given by Hst/k, bigger (smaller) values of k imply that the χ relic
density is reduced (enhanced).
Fig. 5 shows that we need additional physical constraints if we want to derive
bounds on A(zF,st) and/or A
′(zF,st). Once the annihilation cross section is known,
the requirement (1) will single out a region in the space spanned by our three new
parameters (including k) which describe the non–standard evolution of the universe,
but this region is not bounded. Such additional constraints can be derived from
the requirement that the Hubble parameter should remain positive throughout the
epoch we are considering. As noted earlier, requiring H > 0 for all T < TF,st leads
to an upper bound on A′(zF,st); explicitly,
A′(zF,st) <
A(zF,st) +
kA(zF,st)
zF,st
. (33)
On the other hand, a lower bound on A′(zF,st) is obtained from the condition that
the modified Hubble parameter is positive between the highest temperature Ti where
the ansatz (29) holds and TF,st:
A′(zF,st) > −
zi − zF,st
2− zi
zF,st
A(zF,st) + k
zF,st
, (34)
for (1− zF,st/zi)2k < A(zF,st), and
A′(zF,st) >
A(zF,st)−
kA(zF,st)
zF,st
, (35)
for A(zF,st) < (1− zF,st/zi)2k, where zi = Ti/mχ.
Evidently the lower bound on A′(zF,st) depends on zi, i.e. on the maximal tem-
perature where we assume our ansatz (29) to be valid. In ref.[14] we have shown that
in standard cosmology (A ≡ 1) essentially full thermalization is already achieved for
xi <∼ xF − 0.5, even if nχ(xi) = 0. However, it seems reasonable to demand that H
should remain positive at least up to xi = xF−(a few). In Fig. 6 we therefore show
0 1 2 3 4 5 6 7
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1
xF,st − x i = 4
xF,st − x i = 10
0 1 2 3 4 5 6 7
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
xF,st − x i = 4
xF,st − x i = 10
Figure 6: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), A′(zF,st)) plane. The
dashed line corresponds to the upper bound on A′(zF,st). The dotted lines correspond to the lower
bounds calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and
a = 0, b = 1.5× 10−8 GeV−2 (right frame). The other parameters are as in Fig 3.
the physical constraints on the modification parameter A(z) for xF,st − xi = 4, 10
and k = 1. The dashed and dotted lines correspond to the upper and lower bounds
on A′(zF,st), described by Eq.(33) and Eqs.(34), (35), respectively. We see that when
xF,st − xi = 4 the allowed region is 0.4 <∼ A(zF,st) <∼ 6.5 with −60 <∼ A′(zF,st) <∼ 400
for b = 0 (left frame), and 0.4 <∼ A(zF,st) <∼ 4.5 with −60 <∼ A′(zF,st) <∼ 300 for a = 0
(right frame). When xF,st − xi = 10, the lower bounds are altered to 0.6 <∼ A(zF,st),
−10 <∼ A′(zF,st) for b = 0 (left frame), and 0.6 <∼ A(zF,st), −20 <∼ A′(zF,st) for a = 0
(right frame). Note that the lower bounds on A(zF,st), which depend only weakly on
xi so long as it is not very close to xF , are almost the same in both cases, which also
lead to very similar relic densities in standard cosmology. However, the two upper
bounds differ significantly. The reason is that large values of A(zF,st), i.e. a strongly
suppressed Hubble expansion, require some degree of finetuning: One also has to
take large positive A′(zF,st), so that A becomes smaller than one for some range of
z values below zF , leading to an annihilation integral of similar size as in standard
cosmology. Since the b−terms show different zF dependence in the annihilation in-
tegral (32), the required tuning between A(zF,st) and A
′(zF,st) is somewhat different
than for the a−terms, leading to a steeper slope of the allowed region. This allowed
region therefore saturates the upper bound (33) on the slope for somewhat smaller
A(zF,st).
The effect of this tuning can be seen by analyzing the special case where A′′(zF,st) =
0 0.5 1 1.5 2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
4 0.08
0 0.5 1 1.5 2
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
Figure 7: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), k) plane for A′′(zF,st) = 0.
The dotted lines correspond to the lower bounds of A(zF,st), calculated for xF,st − xi = 4, 10. We
take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5 × 10−8 GeV−2 (right frame).
The other parameters are as in Fig 3.
0. The modification parameter then reads
A(z) =
A(zF,st)− k
zF,st
z + k . (36)
Note that A is now a monotonic function of z, making large cancellations in the
annihilation integral impossible. Imposing that A(z) remains positive for zF,st ≤
z ≤ zi leads to the lower limit
A(zF,st) >
zF,st
k . (37)
There is no upper bound, since A(z) is now automatically positive for all z ∈ [0, zF,st]
if A(zF,st) and A(0) ≡ k are both positive. Fig. 7 shows constraints on the relic
abundance in the (A(zF,st), k) plane for A
′′(zF,st) = 0. The dotted lines correspond
to the lower bounds (37) on A(zF,st) for xF,st − xi = 4, 10. As noted earlier, k
is constrained by the BBN bound. This leads to the bounds 0.5 <∼ A(zF,st) <∼ 1.8
for b = 0 (left frame), and 0.65 <∼ A(zF,st) <∼ 1.6 for a = 0 (right frame), when
xF,st − xi = 10. Evidently the constraints now only depend weakly on whether the
a− or b−term dominates in the annihilation cross section. As the initial temperature
is lowered, the impact of the constraint (37) disappears.
So far we have assumed in this Section that the reheat temperature was high
enough for WIMPs to have attained full thermal equilibrium. If this was not the
case, the initial temperature as well as the suppression parameter affects the final
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1, A’’(zF,st ) = 0
0.080.12
Figure 8: Contour plot of the relic abundance Ωχh2 in the (A(zF,st), x0) plane. Here we choose
a = 2.0 × 10−9 GeV−2, b = 0, k = 1, A′′(zF,st) = 0. The other parameters are as in Fig. 3.
The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 <
ΩCDMh
2 < 0.12 (95% C.L.).
relic abundance. Here we show that the lower bound on the reheat temperature
derived in the previous Section survives even in scenarios with altered expansion
history as long as WIMPs were only produced thermally.
This can be understood from the observation that the Boltzmann equation with
modified expansion rate is obtained by replacing 〈σv〉 in the radiation–dominated
case by 〈σv〉A. Increasing (decreasing) A therefore has the same effect as an in-
crease (decrease) of the annihilation cross section. Since the lower bound on T0 was
independent of σ (more exactly: we quoted the absolute minimum, for the optimal
choice of σ), we expect it to survive even if A(z) 6= 1 is introduced.
This is borne out by Fig. 8, which shows the relic abundance Ωχh
2 in the (A(zF,st),
x0) plane for the simplified case A
′′(zF,st) = 0; similar results can be obtained for
the more general ansatz (29). The shaded region corresponds to the bound (1) on
the cold dark matter abundance. As expected, this figure looks similar to Fig. 1 if
the annihilation cross section in Fig. 1 is replaced by A(zF,st). The maximal value of
x0 consistent with the WMAP data remains around 23 even in these scenarios with
modified expansion rate. Fig. 8 also shows that A(zF,st) ≪ 1 is allowed for some
narrow range of initial temperature T0 < TF . This is analogous to the low cross
section branch in Fig. 1.
4 Summary and Conclusions
In this paper we have investigated the relic abundance of WIMPs χ, which are non-
relativistic long–lived or stable particles, in non–standard cosmological scenarios.
One motivation for studying such scenarios is that they allow to reproduce the ob-
served Dark Matter density for a large range of WIMP annihilation cross sections.
Our motivation was the opposite: we wanted to quantify the constraints that can
be obtained on parameters describing the early universe, under the assumption that
thermally produced WIMPs form all Dark Matter. Wherever necessary, we fixed
particle physics quantities such that standard cosmology yields the correct relic den-
sity.
Specifically, we first considered scenarios with low post–inflationary reheat tem-
perature, while keeping all other features of standard cosmology (known particle
content and Hubble expansion parameter during WIMP decoupling; no late entropy
production; no non–thermal WIMP production channels). If the temperature was so
low that WIMPs could not achieve full thermal equilibrium, the dependence of the
abundance on the mass and annihilation cross section of the WIMPs is completely
different from that in the standard thermal WIMP scenario. In particular, if the
maximal temperature T0 is much less than the decoupling temperature TF , nχ re-
mains exponentially suppressed. By applying the observed cosmological amount of
cold dark matter to the predicted WIMP abundance, we therefore found the lower
bound of the initial temperature T0 >∼ mχ/23. One might naively think that this
bound could be evaded by choosing a sufficiently large WIMP production (or anni-
hilation) cross section. However, increasing this cross section also reduces TF . For
sufficiently large cross section one therefore has TF ≤ T0 again; in this regime the
relic density drops with increasing cross section. Our lower bound is the minimal
T0 required for any cross section; once the latter is known, the bound on T0 might
be slightly sharpened. As a by–product, we also noted that the final relic density
depends only weakly on the annihilation cross section if T0 is slightly above this
lower bound.
We also investigated the effect of a non–standard expansion rate of the universe
on the WIMP relic abundance. In general the abundance of thermal relics depends
on the ratio of the annihilation cross section to the expansion rate; the latter is
determined unambiguously in standard cosmology. We found that even for non–
standard Hubble parameter the relic abundance can be calculated accurately in
terms of an annihilation integral, very similar to the case of standard cosmology. We
assumed that the WIMP annihilation cross section is such that the standard scenario
yields the observed relic density, and parameterized the modification of the Hubble
parameter as a quadratic function of the temperature. In this analysis it is crucial to
make sure that at low temperatures the Hubble parameter approaches its standard
value to within ∼ 20%, as required for the success of Big Bang Nucleosynthesis
(BBN).
Keeping the annihilation cross section fixed and allowing a 20% variation in the
relic density, roughly corresponding to the present “2σ” band, we found that the
expansion of the universe at T = TF might have been more than two times faster, or
more than six times slower, than in standard cosmology. These large variations of
H(TF ) can only be realized by finetuning of the parameters describing H(T < TF ).
However, even if we forbid such finetuning by choosing a linear parameterization for
the modification of the expansion rate, a 20% variation of Ωχh
2 allows a difference
between H(TF ) and its standard expectation of more than 50%. This relatively weak
sensitivity of the predicted Ωχh
2 on H(TF ) is due to the fact that the relic density
depends on all H(T < TF ); as stressed above, we have to require that H(T ≪ TF )
approaches its standard value to within ∼ 20%. The fact that determining Ωχh2 will
yield relatively poor bounds on H(TF ) remains true even if the annihilation cross
section is such that a non–standard behavior of H(T ) is required for obtaining the
correct χ relic density. Finally, we showed that the absolute lower bound on the
temperature required for thermal χ production is unaltered by allowing H(T ) to
differ from its standard value.
Of course, in order to draw the conclusions derived in this article, we need to
convince ourselves that WIMPs do indeed form (nearly) all Dark Matter. This
requires not only the detection of WIMPs, e.g. in direct search experiments; we
also need to show that their density is in accord with the local Dark Matter density
derived from astronomical observations. To that end, the cross sections appearing in
the calculation of the detection rate need to be known independently. This can only
be done in the framework of a definite theory, using data from collider experiments.
For example, in order to determine the cross section for the direct detection of
supersymmetric WIMPs, one needs to know the parameters of the supersymmetric
neutralino, Higgs and squark sectors [3]. We also saw that inferences about H(TF )
can only be made if the WIMP annihilation cross section is known. This again
requires highly non–trivial analyses of collider data [27], as well as a consistent
overall theory. We thus see that the interplay of accurate cosmological data with
results obtained from dark matter detections and collider experiments can give us
insight into the pre–BBN universe, which to date remains unexplored territory.
Acknowledgments
This work was partially supported by the Marie Curie Training Research Network
“UniverseNet” under contract no. MRTN-CT-2006-035863, as well as by the Euro-
pean Network of Theoretical Astroparticle Physics ENTApP ILIAS/N6 under con-
tract no. RII3-CT-2004-506222.
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Introduction
Relic Abundance in the Radiation–Dominated Universe
Relic Abundance for Modified Expansion Rate
Summary and Conclusions
|
0704.1591 | Flavour-Dependent Type II Leptogenesis | FTUAM 07-07
IFT-UAM/CSIC 07-18
Flavour-Dependent Type II Leptogenesis
S. Antuscha
Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI,
Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
Abstract
We reanalyse leptogenesis via the out-of-equilibrium decay of the lightest right-
handed neutrino in type II seesaw scenarios, taking into account flavour-dependent
effects. In the type II seesaw mechanism, in addition to the type I seesaw con-
tribution, an additional direct mass term for the light neutrinos is present. We
consider type II seesaw scenarios where this additional contribution arises from
the vacuum expectation value of a Higgs triplet, and furthermore an effective
model-independent approach. We investigate bounds on the flavour-specific de-
cay asymmetries, on the mass of the lightest right-handed neutrino and on the
reheat temperature of the early universe, and compare them to the corresponding
bounds in the type I seesaw framework. We show that while flavour-dependent
thermal type II leptogenesis becomes more efficient for larger mass scale of the
light neutrinos, and the bounds become relaxed, the type I seesaw scenario for
leptogenesis becomes more constrained. We also argue that in general, flavour-
dependent effects cannot be ignored when dealing with leptogenesis in type II
seesaw models.
aE-mail: [email protected]
http://arxiv.org/abs/0704.1591v3
1 Introduction
Leptogenesis [1] is one of the most attractive and minimal mechanisms for explaining
the observed baryon asymmetry of the universe nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2].
A lepton asymmetry is dynamically generated and then converted into a baryon asym-
metry due to (B + L)-violating sphaleron interactions [3] which exist in the Standard
Model (SM) and its minimal supersymmetric extension, the MSSM. Leptogenesis can be
implemented within the type I seesaw scenario [4], consisting of the SM (MSSM) plus
three right-handed Majorana neutrinos (and their superpartners) with a hierarchical
spectrum. In thermal leptogenesis [5], the lightest of the right-handed neutrinos is pro-
duced by thermal scattering after inflation, and subsequently decays out-of-equilibrium
in a lepton number and CP-violating way, thus satisfying Sakharov’s constraints [6].
In models with a left-right symmetric particle content like minimal left-right sym-
metric models, Pati-Salam models or Grand Unified Theories (GUTs) based on SO(10),
the type I seesaw mechanism is typically generalised to a type II seesaw [7], where
an additional direct mass term mIILL for the light neutrinos is present. From a model
independent perspective, the type II mass term can be considered as an additional con-
tribution to the lowest dimensional effective neutrino mass operator. In most explicit
models, the type II contribution stems from seesaw suppressed induced vevs of SU(2)L-
triplet Higgs fields. One motivation for considering the type II seesaw is that it allows
to construct unified flavour models for partially degenerate neutrinos in an elegant way,
e.g. via a type II upgrade [8], which is otherwise difficult to achieve in type I models.
For leptogenesis in type II seesaw scenarios with SU(2)L-triplet Higgs fields, there
are in general two possibilities to generate the baryon asymmetry: via decays of the
lightest right-handed neutrinos or via decays of the SU(2)L-triplets [9,10,11,12]. In the
first case, there are additional one-loop diagrams where virtual triplets are running in
the loop [9, 13, 14, 15, 16]. In the following, we focus on this possibility, and assume
hierarchical right-handed neutrino masses (and that the triplets are heavier than ν1R).
In this limit, to a good approximation the decay asymmetry depends mainly on the
low energy neutrino mass matrix mνLL = m
LL + m
LL and on the Yukawa couplings
to the lightest right-handed neutrino and its mass [16]. It has been shown that type
II leptogenesis imposes constraints on the seesaw parameters, which, in the flavour-
independent approximation, differ substantially from the constraints in the type I case.
For instance, the bound on the decay asymmetry increases with increasing neutrino mass
scale [16], in contrast to the type I case where it decreases. As a consequence, the lower
bound on the mass of the lightest right-handed neutrino from leptogenesis decreases for
increasing neutrino mass scale [16]. One interesting application of type II leptogenesis
is the possibility to improve consistency of classes of unified flavour models with respect
to thermal leptogenesis [17]. Finally, since the type II contribution typically does not
effect washout, there is no bound on the absolute neutrino mass scale from type II
leptogenesis, as has been pointed out in [15]. For further applications and realisations
of type II leptogenesis in specific models of fermion masses and mixings, see e.g. [18].
In recent years, the impact of flavour in thermal leptogenesis has merited increasing
attention [19] - [38]. In fact, the one-flavour approximation is only rigorously correct
when the interactions mediated by the charged lepton Yukawa couplings are out of
equilibrium. Below a given temperature (e.g. O(1012GeV) in the SM and (1+tan2 β)×
O(1012GeV) in the MSSM), the tau Yukawa coupling comes into equilibrium (later
followed by the couplings of the muon and electron). Flavour effects are then physical
and become manifest, not only at the level of the generated CP asymmetries, but also
regarding the washout processes that destroy the asymmetries created for each flavour.
In the full computation, the asymmetries in each distinguishable flavour are differently
washed out, and appear with distinct weights in the final baryon asymmetry.
Flavour-dependent leptogenesis in the type I seesaw scenario has recently been ad-
dressed in detail by several authors. In particular, flavour-dependent effects in lepto-
genesis have been studied, and shown to be relevant, in the two right-handed neutrino
models [24] as well as in classes of neutrino mass models with three right-handed neu-
trinos [26]. The quantum oscillations/correlations of the asymmetries in lepton flavour
space have been included in [22, 32, 33, 35] and the treatment has been generalised to
the MSSM [26,29]. Effects of reheating, and constraints on the seesaw parameters from
upper bounds on the reheat temperature, have been investigated in [29]. Leptogenesis
bounds on the reheat temperature [29] and on the mass of the lightest right-handed
neutrino [29, 36] have also been considered including flavour-dependent effects. Strong
connections between the low-energy CP phases of the UMNS matrix and CP violation for
flavour-dependent leptogenesis have been shown to emerge in certain classes of neutrino
mass models [26] or under the hypothesis of no CP violation sources associated with the
right-handed neutrino sector (real R) [25,27,28,31]. Possible effects regarding the decays
of the heavier right-handed neutrinos for leptogenesis have been discussed in this con-
text in [21, 34], and flavour-dependent effects for resonant leptogenesis were addressed
in [38]. Regarding the masses of the light neutrinos, assuming hierarchical right-handed
neutrinos and considering experimentally allowed light neutrino masses (below about 0.4
eV), there is no longer a bound on the neutrino mass scale from thermal leptogenesis if
flavour-dependent effects are included [24].
In view of the importance of flavour-dependent effects on leptogenesis in the type I
seesaw case, it is pertinent to investigate their effects on type II leptogenesis. In this
paper, we therefore reanalyse leptogenesis via the out-of-equilibrium decay of the lightest
right-handed neutrino in type II seesaw scenarios, taking into account flavour-dependent
effects. We investigate bounds on the decay asymmetries, on the mass of the lightest
right-handed neutrino and on the reheat temperature of the early universe, and discuss
how increasing the neutrino mass scale affects thermal leptogenesis in the type I and
type II seesaw frameworks.
2 Type I and type II seesaw mechanisms
Motivated by left-right symmetric unified theories, we consider two generic possibilities
for explaining the smallness of neutrino masses: via heavy SM (MSSM) singlet fermions
(i.e. right-handed neutrinos) [4] and via heavy SU(2)L-triplet Higgs fields [7]. In both
cases, the effective dimension five operator for Majorana neutrino masses in the SM or
p2≪M2
Figure 1: Generation of the dimension 5 neutrino mass operator in the type I seesaw mechanism.
the MSSM, respectively,
κgf (LC
· φ) (Lf · φ) + h.c. , (1a)
κ = −
κgf (L̂
g · Ĥu) (L̂f · Ĥu)
+ h.c. , (1b)
is generated from integrating out the heavy fields. This is illustrated in figures 1 and 2. In
equation (1), the dots indicate the SU(2)L-invariant product, (L̂
f · Ĥu) = L̂fa(iτ2)ab(Ĥu)b,
with τA (A ∈ {1, 2, 3}) being the Pauli matrices. Superfields are marked by hats. After
electroweak symmetry breaking, the operators of equation (1) lead to Majorana mass
terms for the light neutrinos,
Lν = −12m
LLνLν
L , with m
LL = −
(κ)∗ . (2)
In the type I seesaw mechanism, it is assumed that only the singlet (right-handed)
neutrinos νRi contribute to the neutrino masses. With Yν being the neutrino Yukawa
matrix in left-right convention,1 MRR the mass matrix of the right-handed neutrinos
and vu = 〈φ0〉 (= 〈H0u〉) the vacuum expectation value of the Higgs field which couples
to the right-handed neutrinos, the effective mass matrix of the light neutrinos is given
by the conventional type I seesaw formula
mILL = −v2u Yν M−1RR Y
ν . (3)
In the type II seesaw mechanism, the contributions to the neutrino mass matrix from
both, right-handed neutrinos νRi and Higgs triplet(s) ∆L, are considered. The additional
contribution to the neutrino masses from ∆L can be understood in two ways: as another
contribution to the effective neutrino mass operator in the low energy effective theory
or, equivalently, as a direct mass term after the Higgs triplet obtains an induced small
vev after electroweak symmetry breaking (c.f. figure 2). The neutrino mass matrix in
the type II seesaw mechanism has the form
mνLL = m
LL +m
LL = m
LL − v2uYνM−1RRY
ν , (4)
1The neutrino Yukawa matrix corresponds to −(Yν)fi(Lf · φ) νiR in the Lagrangian of the SM and,
analogously, to (Yν)fi(L̂
f · Ĥu) ν̂Ci in the superpotential of the MSSM (see [16] for further details).
p2≪M2
Figure 2: Extra diagram generating the dimension 5 neutrino mass operator in the type II seesaw
mechanism from a SU(2)L-triplet Higgs field.
where mIILL is the additional term from the Higgs triplet(s). In left-right symmetric
unified theories, the generic size of both seesaw contributionsmILL andm
LL isO(v2u/vB−L)
where vB−L is the B-L breaking scale (i.e. the mass scale of the right-handed neutrinos
and of the Higgs triplet(s)).
3 Baryogenesis via flavour-dependent leptogenesis
Flavour-dependent effects can have a strong impact in baryogenesis via thermal lep-
togenesis [19] - [38]. The effects are manifest not only in the flavour-dependent CP
asymmetries, but also in the flavour-dependence of scattering processes in the thermal
bath, which can destroy a previously produced asymmetry.
The relevance of the flavour-dependent effects depends on the temperatures at which
thermal leptogenesis takes place, and thus on which interactions mediated by the charged
lepton Yukawa couplings are in thermal equilibrium. For example, in the MSSM, for
temperatures between circa (1+tan2 β)×105GeV and (1+tan2 β)×109GeV, the µ and τ
Yukawa couplings are in thermal equilibrium and all flavours in the Boltzmann equations
are to be treated separately. For tan β = 30, this applies for temperatures below about
1012 GeV and above 108GeV, a temperature range which is of most interest for thermal
leptogenesis in the MSSM. In the SM, in the temperature range between circa 109 GeV
and 1012 GeV, only the τ Yukawa coupling is in equilibrium and is treated separately in
the Boltzmann equations, whereas µ and e flavours are indistinguishable. A discussion
of the temperature regimes in the SM and MSSM, where flavour is important, can be
found, e.g., in [26].
We now briefly review the estimation of the produced baryon asymmetry in flavour-
dependent leptogenesis.2 For definiteness, we focus on the temperature range where
all flavours are to be treated separately. In the following discussion of thermal type II
leptogenesis, we will assume that the mass M∆L of the triplet(s) is much larger than
MR1. In this limit, the flavour-dependent efficiencies calculated in the type I seesaw
scenario can also be used in the type II framework. The out-of-equilibrium decays of the
heavy right-handed (s)neutrinos ν1R and ν̃
R give rise to flavour-dependent asymmetries
in the (s)lepton sector, which are then partly transformed via sphaleron conversion into
2For a discussion of approximations which typically enter these estimates, and which also apply to
our discussion, see e.g. section 3.1.3 in [29].
a baryon asymmetry YB.
3 The final baryon asymmetry can be calculated as
Y SMB =
Y SM∆f , (5)
Y MSSMB =
Ŷ MSSM∆f , (6)
where Ŷ∆f ≡ YB/3 − YLf are the total (particle and sparticle) B/3 − Lf asymmetries,
with YLf the lepton number densities in the flavour f = e, µ, τ . The asymmetries Ŷ
and Y SM∆f , which are conserved by sphalerons and by the other SM (MSSM) interactions,
are then usually calculated by solving a set of coupled Boltzmann equations, describing
the evolution of the number densities as a function of temperature.
It is convenient to parameterise the produced asymmetries in terms of flavour-specific
efficiency factors ηf and decay asymmetries ε1,f as
Y SM∆f = η
f ε1,f Y
, (7)
Ŷ MSSM∆f = η
(ε1,f + ε1, ef) Y
(εe1,f + εe1, ef) Y
. (8)
and Y
are the number densities of the neutrino and sneutrino for T ≫ M1 if
they were in thermal equilibrium, normalised with respect to the entropy density. In
the Boltzmann approximation, they are given by Y
≈ Y eq
≈ 45/(π4g∗). g∗ is the
effective number of degrees of freedom, which amounts 106.75 in the SM and 228.75 in
the MSSM.
ε1,f , ε1, ef , εe1,f and εe1, ef are the decay asymmetries for the decay of neutrino into Higgs
and lepton, neutrino into Higgsino and slepton, sneutrino into Higgsino and lepton, and
sneutrino into Higgs and slepton, respectively, defined by
ε1,f =
− Γν1
f (Γν1RLf + Γν1RLf
1, ef
f (Γν1
εe1,f =
Γeν∗1
f (Γeν∗1R Lf + Γeν1RLf
, εe1, ef =
f(Γeν1
. (9)
The flavour-dependent efficiency factors ηf in the SM and in the MSSM are defined
by Eqs. (7) and (8), respectively. As stated above, we assume that the mass M∆L of the
triplet(s) is much larger than MR1. In this limit, the efficiencies for flavour-dependent
thermal leptogenesis in the type I and type II frameworks are mainly determined by
the properties of ν1R, which means in particular that the flavour-dependent efficiencies
3In the following, Y will always be used for quantities which are normalised to the entropy density
s. The quantities normalised with respect to the photon density can be obtained using the relation
s/nγ ≈ 7.04k.
-2 -1 0 1 2
ÈAff K f È
�������������������������������
È Aff K f È
�������������������������������
È Aff K f È
�������������������������������
È Aff K f È
= 100
Figure 3: Flavour-dependent efficiency factor η(AffKf ,K) in the MSSM as a function of AffKf , for
fixed values of K/|AffKf | = 2, 5 and 100, obtained from solving the flavour-dependent Boltzmann
equations in the MSSM with zero initial abundance of right-handed (s)neutrinos (figure from [26]).
A is a matrix which appears in the Boltzmann equations (see [19, 24] for A in the SM and [26] for
the MSSM case), and which has diagonal elements |Aff | of O(1). The small off-diagonal entries of A
have been neglected, which is a good approximation in most cases. In general, however, they have to
be included. More relevant than the differences in the flavour-dependent efficiency factors for different
K/|AffKf | is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which
is weighted by the corresponding efficiency factors.
calculated in the type I seesaw scenario can also be used in the type II framework. In
the definition of the efficiency factor, the equilibrium number densities serve as a nor-
malization: A thermal population νR1 (and ν̃R1) decaying completely out of equilibrium
(without washout effects) would lead to ηf = 1.
The efficiency factors can be computed by means of the flavour-dependent Boltzmann
equations, which can be found for the SM in [19,22,23,24] and for the MSSM in [26,29].
In general, the flavour-dependent efficiencies depend strongly on the washout parameters
m̃1,f for each flavour, and on the total washout parameter m̃1, which are defined as
m̃1,f =
v2u |(Yν)f1|2
, m̃1 =
m̃1,f . (10)
Alternatively, one may use the quantities Kf , K, which are related to m̃1,f , m̃1 by
m̃1,f
, K =
Kf , (11)
with m∗SM ≈ 1.08 × 10−3 eV and m∗MSSM ≈ sin2(β) × 1.58 × 10−3 eV. Figure 3 shows
the flavour-specific efficiency factor ηf in the MSSM. Maximal efficiency for a specific
flavour corresponds to Kf ≈ 1 (m̃1,f ≈ m∗).
The most relevant difference between the flavour-independent approximation and
the correct flavour-dependent treatment is the fact that in the latter, the total baryon
asymmetry is the sum of each individual lepton asymmetries, which is weighted by the
corresponding efficiency factor. Therefore, upon summing over the lepton asymmetries,
the total baryon number is generically not proportional to the sum over the CP asymme-
tries, ε1 =
f ε1,f , as in the flavour-independent approximation where the lepton flavour
is neglected in the Boltzmann equations. In other words, in the flavour-independent ap-
proximation the total baryon asymmetry is a function of
f ε1,f
× ηind (
g Kg). In
the correct flavour treatment the baryon asymmetry is (approximately) a function of∑
f ε1,fη (AffKf , K). From this, it is already clear that flavour-dependent effects can
have important consequences also in type II leptogenesis.
The most important quantities for computing the produced baryon asymmetry are
thus the decay asymmetries ε1,f and the efficiency factors ηf (which depend mainly on
m̃1,f and m̃1 (or Kf and K)). While the efficiency factors can be computed similarly
to the type I seesaw case, important differences between leptogenesis in type I and type
II seesaw scenarios arise concerning the decay asymmetries as well as concerning the
connection between leptogenesis and seesaw parameters.
4 Decay asymmetries
4.1 Right-handed neutrinos plus triplets
Regarding the decay asymmetry in the type II seesaw mechanism, where the direct
mass term for the neutrinos stems from the induced vev of a Higgs triplet, there are
new contributions from 1-loop diagrams where virtual SU(2)L-triplet scalar fields (or
their superpartners) are exchanged in the loop. The relevant diagrams for the decay
ν1R → LfaHub in the limit M1 ≪ MR2,MR2,M∆ are shown in figure 4. Compared to
the type I seesaw framework, the new contributions are the diagrams (c) and (f). The
calculation of the corresponding decay asymmetries for each lepton flavour yields
1,f =
j 6=1
Im [(Y †)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− (1 + xj) ln
xj + 1
, (12a)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− xj
, (12b)
1,f = −
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
−1 + y ln
y + 1
, (12c)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
−1 + xj ln
xj + 1
, (12d)
1,f =
j 6=1 Im [(Y
†)1f(Y
ν Yν)1j(Y
T )jf ]
ν Yν)11
1− xj
, (12e)
1,f = −
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
1− (1 + y) ln
y + 1
, (12f)
∆̃1, ∆̃2
Figure 4: Loop diagrams in the MSSM which contribute to the decay ν1
→ LfaHub for the case of a
type II seesaw mechanism where the direct mass term for the neutrinos stems from the induced vev of
a Higgs triplet. In diagram (f), ∆̃1 and ∆̃2 are the mass eigenstates corresponding to the superpartners
of the SU(2)L-triplet scalar fields ∆ and ∆̄. The SM diagrams are the ones where no superpartners
(marked by a tilde) are involved and where Hu is renamed to the SM Higgs φ.
where y := M2∆/M
R1 and xj := M
R1 for j 6= 1 and where we assume hierarchical
right-handed neutrino masses and M∆ ≫ MR1.4
The MSSM results for the type II contributions have been derived in [16]. In the SM,
the results in [16] correct the previous result of [15] by a factor of −3/2. In equation
(12) they have been generalised to the flavour-dependent case. The results for the
contributions to the decay asymmetries from the triplet in the SM and from the triplet
superfield in the MSSM are
SM,II
1,f = ε
1,f , (13a)
MSSM,II
1,f = ε
1,f + ε
1,f . (13b)
In the MSSM, we furthermore obtain
MSSM,II
1,f = ε
MSSM,II
1, ef
MSSM,II
MSSM,II
e1, ef
. (14)
The results corresponding to the diagrams (a), (b), (d) and (e) which contribute to
εI1 in the type I seesaw in the SM and in the MSSM, have been presented first in [39].
The results for the type I contribution to the decay asymmetries in the SM and in the
4Integrating out the heavy particles ν2
,∆ (and their superpartners) in figure 4 leads to an
effective approach involving the dimension 5 neutrino mass operator (c.f. figures 1, 2 and 5), as will be
discussed in section 4.2. We note that there are additional diagrams not shown in figure 4 (since they
are generically suppressed for M1 ≪ MR2,MR2,M∆) which are related to the dimension 6 operator
containing two lepton doublets, two Higgs doublets and a derivative.
MSSM are
1,f = ε
1,f + ε
1,f , (15a)
MSSM,I
1,f = ε
1,f + ε
1,f + ε
1,f + ε
1,f . (15b)
Again, in the MSSM, the remaining decay asymmetries are equal to ε
MSSM,I
1,f :
MSSM,I
1,f = ε
MSSM,I
1, ef
MSSM,I
MSSM,I
e1, ef
. (16)
Finally, the total decay asymmetries from the decay of ν1R in the type II seesaw,
where the direct mass term for the neutrinos stems from the induced vev of a Higgs
triplet, are given by
εSM1,f = ε
1,f + ε
SM,II
1,f , (17)
εMSSM1,f = ε
MSSM,I
1,f + ε
MSSM,II
1,f . (18)
It is interesting to note that the type I results can be brought to a form which
contains the neutrino mass matrix using
j 6=1 Im [(Y
†)1f (Y
ν Yν)1j(Y
T )jf ]
8π (Y
ν Yν)11
= −MR1
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
8π (Y
ν Yν)11
.(19)
In the limit y ≫ 1 and xj ≫ 1 for all j 6= 1, which corresponds to a large gap between
the mass MR1 and the masses MR2, MR3 and M∆, we obtain the simple results for the
flavour-specific decay asymmetries εSM1,f and ε
1,f [16]
εSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL +m
LL)fg]
ν Yν)11
, (20a)
εMSSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL +m
LL)fg]
ν Yν)11
. (20b)
In the presence of such a mass gap, the calculation can also be performed in an effective
approach after integrating out the two heavy right-handed neutrinos and the heavy
triplet, as we now discuss.
4.2 Effective approach to leptogenesis
Let us now explicitly use the assumption that the lepton asymmetry is generated via
the decay of the lightest right-handed neutrino and that all other additional particles, in
particular the ones which generate the type II contribution, are much heavier than MR1.
Furthermore, we assume that we can neglect their population in the early universe, e.g.
that their masses are much larger than the reheat temperature TRH and that they are
Figure 5: Loop diagrams contributing to the decay asymmetry via the decay ν1
→ LfaHub in the
MSSM with a (lightest) right-handed neutrino ν1
and a neutrino mass matrix determined by κ′ [16].
Further contributions to the generated baryon asymmetry stem from the decay of ν1
into slepton and
Higgsino and from the decays of the sneutrino ν̃1
. With Hu renamed to the SM Higgs, the first diagram
contributes in the extended SM.
not produced non-thermally in a large amount. Under these assumptions we can apply
an effective approach to leptogenesis, which is independent of the mechanism which
generates the additional (type II) contribution to the neutrino mass matrix [16].
For this minimal effective approach, it is convenient to isolate the type I contribution
from the lightest right-handed neutrino as follows:
(mνLL)fg = −
2(Yν)f1M
R1 (Y
ν )1g + (κ
′∗)fg
. (21)
κ′ includes type I contributions from the heavier right-handed neutrinos, plus any ad-
ditional (type II) contributions from heavier particles. Examples for realisations of the
neutrino mass operator can be found, e.g., in [40].
At MR1, the minimal effective field theory extension of the SM (MSSM) for lepto-
genesis includes the effective neutrino mass operator κ′ plus one right-handed neutrino
ν1R with mass MR1 and Yukawa couplings (Yν)f1 to the lepton doublets L
f , defined as
−(Yν)f1(Lf · φ) ν1R in the Lagrangian of the SM and, analogously, as (Yν)f1(L̂f · Ĥu) ν̂C1
in the superpotential of the MSSM.
The contributions to the decay asymmetries in the effective approach stem from the
interference of the diagram(s) for the tree-level decay of νR1 (and ν̃R1) with the loop
diagrams containing the effective operator, shown in figure 5. In the SM, we obtain the
simple result [16] for the flavour-specific effective decay asymmetries (corresponding to
diagram (a) of figure 5)
εSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
. (22)
For the supersymmetric case, diagram (a) and diagram (b) contribute to εMSSM1,f and we
obtain [16]:
εMSSM1,f =
g Im [(Y
ν )f1(Y
ν )g1(m
LL)fg]
ν Yν)11
. (23)
Explicit calculation furthermore yields
εMSSM1,f = ε
1, ef
= εMSSM
= εMSSM
e1, ef
. (24)
The results are independent of the details of the realisation of the neutrino mass operator
κ′. Note that, since the diagrams where the lightest right-handed neutrino runs in the
loop do not contribute to leptogenesis, we have written mνLL = −v2u(κ)∗/2 instead of
m′νLL := −v2u(κ′)∗/2 in the formulae in equations (22) - (23). The decay asymmetries
are directly related to the neutrino mass matrix mνLL.
For neutrino masses via the type I seesaw mechanism, the results are in agreement
with the known results [39], in the limit MR2,MR3 ≫ MR1. The results obtained in the
effective approach are also in agreement with our full theory calculation in the type II
scenarios with SU(2)L-triplets in equation (12) [16], in the limit M∆ ≫ MR1.
5 Type II bounds on decay asymmetries and on MR1
In the limit MR2,MR3,M∆ ≫ MR1 (or alternatively in the effective approach), upper
bounds for the total decay asymmetries in type II leptgenesis, i.e. for the sums |εSM1 | =
1,f | and |εMSSM1 | = |
1,f |, have been derived in [16]. For the flavour-specific
decay asymmetries εSM1,f and ε
1,f , the bounds can readily be obtained as
|εSM1,f | ≤
mνmax , |εMSSM1,f | ≤
mνmax . (25)
They are thus identical to the bounds for the total asymmetries. In particular, they also
increase with increasing mass scale of the light neutrinos. Note that, compared to the
low energy value, the neutrino masses at the scale MR1 are enlarged by renormalization
group running by ≈ +20% in the MSSM and ≈ +30% in the SM, which raises the
bounds on the decay asymmetries by the same values (see e.g. figure 4 of [41]).
A situation where an almost maximal baryon asymmetry is generated by thermal
leptogenesis can be realised, for example, if the total decay asymmetry nearly saturates
its upper bound and if, in addition, the washout parameters m̃1,f for all three flavours
approximately take its optimal value. Classes of type II seesaw models, where this
can be accommodated, have been considered in [8, 42, 17]. In these so-called “type-
II-upgraded” seesaw models, the type II contribution to the neutrino mass matrix is
proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry or by one
of its non-Abelian discrete subgroups). From equation (20), one can readily see that if
the type II contribution (∝ 1) dominates the neutrino mass matrix mνLL, and if (Yν)f1
are approximately equal for all flavours f = 1, 2, 3 and chosen such that the resulting
m̃1,f are approximately equal to m
∗, we have realised ηf ≈ ηmax for all flavours and
simultaneously nearly saturated the bound for the total decay asymmetry.5
5We further note that the bound for one of the flavour-specific decay asymmetries can be nearly
saturated in this scenario if, for instance, (Yν)21 ≈ (Yν)31 ≈ 0.
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
-0.75
-0.25
type I seesaw
type II seesaw
Figure 6: Bound on the decay asymmetry ε1,f in type II leptogenesis (solid blue line) and type I lepto-
genesis (dotted red line) as a function of the mass of the lightest neutrino mν
:= min (mν1 ,mν3 ,mν3)
in type I and type II seesaw scenarios (see also [29]). The washout parameter |Aff |m̃1,f is fixed to
m∗ (close to optimal), and the asymmetry is normalised to ε
max,0
= 3MR1 (∆m
)1/2/(16π v
), where
≈ 2.5× 10−3 eV2 is the atmospheric neutrino mass squared difference. We have considered the
MSSM with tanβ = 30 as an explicit example.
Assuming a maximal efficiency factor ηmax for all flavours in a given scenario, and
taking an upper bound for the masses of the light neutrinos mνmax as well as the observed
value nB/nγ ≈ (6.0965 ± 0.2055) × 10−10 [2] for the baryon asymmetry, equation (25)
can be transformed into lower type II bounds for the mass of the lightest right-handed
neutrino [16]:
MSMR1 ≥
mνmax
nB/nγ
0.99 · 10−2 ηmax
, MMSSMR1 ≥
mνmax
nB/nγ
0.92 · 10−2 ηmax
. (26)
The bound on MR1 is lower for a larger neutrino mass scale.
The situation in the type II framework differs from the type I seesaw case: In the
latter, the flavour-specific decay asymmetries are constrained by [24]
|εI,SM1,f | ≤
mνmax
m̃1,f
, |εI,MSSM1,f | ≤
mνmax
m̃1,f
. (27)
Note that compared to the type II bounds, there is an extra factor of (m̃1,f/m̃1)
which depends on the washout parameters. As we shall now discuss, this factor implies
that it is not possible to have a maximal decay asymmetry ε1,f and an optimal washout
parameter m̃1,f simultaneously. Let us recall first that in the type I seesaw, in contrast
to the type II case, the flavour-independent washout parameter has the lower bound [43]
m̃1 ≥ mνmin , (28)
with mνmin = min (mν1 , mν3 , mν3). On the contrary, in the type I and type II seesaw, the
flavour-dependent washout parameters m̃1,f are generically not constrained. Note that in
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
type I seesaw
type II seesaw
Figure 7: Lower bound on MR1 in type II leptogenesis (solid blue line) and type I leptogenesis (dotted
red line) as a function of the mass of the lightest neutrinomν
:= min (mν1 ,mν3 ,mν3). For definiteness,
the MSSM with tanβ = 30 has been considered as an example.
the flavour-independent approximation, Eq. (28) leads to a dramatically more restrictive
bound on ε1 =
f ε1,f [44] for quasi-degenerate light neutrino masses, and finally even
to a bound on the neutrino mass scale [43]. This can be understood from the fact that
for m̃1 ≫ m∗ in the flavour-independent approximation, washout effects strongly reduce
the efficiency of thermal leptogenesis. Similarly, in the flavour-dependent treatment,
m̃1,f ≫ m∗ would lead to a strongly reduced efficiency for this specific flavour. This
strong washout for quasi-degenerate light neutrinos can be avoided in flavour-dependent
type I leptogenesis, and m̃1,f ≈ m∗ can realise a nearly optimal scenario regarding
washout (c.f. figure 3). However, we see from equation (27) that the decay asymmetries
in this case are reduced by a factor of (m∗/mνmin)
1/2 when compared to the optimal value,
leading to a reduced baryon asymmetry. On the other hand, realizing nearly optimal
ε1,f requires m̃1,f ≈ m̃1 ≥ mνmin, leading to large washout effects for quasi-degenerate
light neutrinos and even to a more strongly suppressed generation of baryon asymmetry
(c.f. figure 3). As a consequence, increasing the neutrino mass scale increases the lower
bound on MR1 (also in the presence of flavour-dependent effects), in contrast to the type
II seesaw case.
Comparing the type II and type I seesaw cases, in the latter the baryon asymmetry
is suppressed for quasi-degenerate light neutrino masses either by a factor (m∗/mνmin)
in the decay asymmetries or by a non-optimal washout parameter much larger than m∗
(or Kf ≫ 1, c.f. figure 3). The bounds on the decay asymmetries in type I and type
II leptogenesis are compared in figure 6, where m̃1,f has been fixed to m
∗, close to its
optimal value. From figure 6 we see that in the type I case the maximal baryon asymme-
try is obtained for hierarchical neutrino masses, whereas in the type II case, increasing
the neutrino mass scale increases the produced baryon asymmetry and therefore allows
to relax the bound on MR1, as shown in figure 7. In addition, for the same reason,
increasing the neutrino mass scale also relaxes the lower bound on the reheat tempera-
ture TRH from the requirement of successful type II leptogenesis. Including reheating in
the flavour-dependent Boltzmann equations as in Ref. [29] (for the flavour-independent
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
2´109
4´109
6´109
8´109
type I seesaw
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
2´109
4´109
6´109
8´109
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
5´108
1´109
1.5´109
2´109
type II seesaw
0 0.1 0.2 0.3 0.4
log10 Hm
min �eVL
5´108
1´109
1.5´109
2´109
Figure 8: Lower bound on the reheat temperature TRH in type I leptogenesis (left panel) and in type II
leptogenesis (right panel) as a function of the mass of the lightest neutrino mν
= min (mν1 ,mν3 ,mν3),
in the MSSM with tanβ = 30. In the grey regions, values of TRH are incompatible with thermal
leptogenesis for the corresponding mν
case, see [45]), we obtain the mνmin-dependent lower bounds on TRH in type I and type
II scenarios shown in figure 8. While the bound decreases in type II leptogenesis by
about an order of magnitude when the neutrino mass scale increases to 0.4 eV, it in-
creases in the type I seesaw case. In the presence of upper bounds on TRH, this can
lead to constraints on the neutrino mass scale , i.e. on mνmin = min (mν1 , mν3, mν3). For
instance, with an upper bound TRH ≤ 5× 109 GeV, values of mνmin in the approximate
range [0.01 eV, 0.32 eV] would be incompatible with leptogenesis in the type I seesaw
framework (c.f. figure 8).
6 Summary, discussion and conclusions
We have analysed flavour-dependent leptogenesis via the out-of-equilibrium decay of
the lightest right-handed neutrino in type II seesaw scenarios, where, in addition to
the type I seesaw, an additional direct mass term for the light neutrinos is present.
We have considered type II seesaw scenarios where this additional contribution stems
from the vacuum expectation value of a Higgs triplet, and furthermore an effective
approach, which is independent of the mechanism which generates the additional (type
II) contribution to neutrino masses. We have taken into account flavour-dependent
effects, which are relevant if thermal leptogenesis takes place at temperatures below
circa 1012 GeV in the SM and below circa (1 + tan2 β) × 1012 GeV in the MSSM. As
in type I leptogenesis, in the flavour-dependent regime the decays of the right-handed
(s)neutrinos generate asymmetries in in each distinguishable flavour (proportional to the
flavour-specific decay asymmetries ε1,f), which are differently washed out by scattering
processes in the thermal bath, and thus appear with distinct weights (efficiency factors
ηf ) in the final baryon asymmetry.
The most important quantities for computing the produced baryon asymmetry are
the decay asymmetries ε1,f and the efficiency factors ηf (which mainly depend on
washout parameters m̃1,f and m̃1 =
f m̃1,f). With respect to the flavour-specific
efficiency factors ηf , in the limit that the mass M∆L of the triplet is much larger than
MR1 (and MR1 ≪ MR2,MR3), they can be estimated from the same Boltzmann equa-
tions as in the type I seesaw framework. Regarding the decay asymmetries ε1,f , in the
type II seesaw case there are additional contributions where virtual Higgs triplets (and
their superpartners) run in the 1-loop diagrams. Here, we have generalised the results
of [16] to the flavour-dependent case. The most important effects of flavour in leptoge-
nesis are a consequence of the fact that in the flavour-independent approximation the
total baryon asymmetry is a function of
f ε1,f
× ηind (
g m̃1,g), whereas in the cor-
rect flavour-dependent treatment the baryon asymmetry is (approximately) a function
f ε1,fη (Affm̃1,f , m̃1).
We have then investigated the bounds on the flavour-specific decay asymmetries ε1,f .
In the type I seesaw case, it is known that the bound on the flavour-specific asymmetries
εI1,f is substantially relaxed [24] compared to the bound on ε
1,f [44] in the case
of a quasi-degenerate spectrum of light neutrinos. For experimentally allowed light
neutrino masses below about 0.4 eV, there is no longer a bound on the neutrino mass
scale from the requirement of successful thermal leptogenesis. In the type II seesaw case,
we have derived the bound on the flavour-specific decay asymmetries ε1,f = ε
1,f + ε
1,f ,
which turns out to be identical to the bound on the total decay asymmetry ε1 =
f ε1,f .
We have compared the bound on the flavour-specific decay asymmetries in type I and
type II scenarios, and found that while the type II bound increases with the neutrino
mass scale, the type I bound decreases (for experimentally allowed light neutrino masses
below about 0.4 eV). The relaxed bound on ε1,f (figure 6) leads to a lower bound on the
mass of the lightest right-handed neutrino MR1 in the type II seesaw scenario (figure
7), which decreases when the neutrino mass scale increases. Furthermore, it leads to
a relaxed lower bound on the reheat temperature TRH of the early universe (figure 8),
which helps to improve consistency of thermal leptogenesis with upper bounds on TRH
in some supergravity models. This is in contrast to the type I seesaw scenario, where
the lower bound on TRH from thermal leptogenesis increases with increasing neutrino
mass scale. Constraints on TRH can therefore imply constraints on the mass scale of the
light neutrinos also in flavour-dependent type I leptogenesis, although a general bound
is absent.
We have furthermore argued that these relaxed bounds on ε1,f MR1 and TRH in the
type II case can be nearly saturated in an elegant way in classes of so-called “type-
II-upgraded” seesaw models [8], where the type II contribution to the neutrino mass
matrix is proportional to the unit matrix (enforced e.g. by an SO(3) flavour symmetry
or by one of its non-Abelian subgroups). One interesting application of these type II
seesaw scenarios is that the consistency of thermal leptogenesis with unified theories
of flavour is improved compared to the type I seesaw case. This effect, investigated in
the flavour-independent approximation in [17], is also present analogously in the flavour-
dependent treatment of leptogenesis. The reason is that if the type II contribution (∝ 1)
dominates, the decay asymmetries ε1,f become approximately equal and the estimate for
the produced baryon asymmetry is similar to the flavour-independent case. Nevertheless,
an accurate analysis of leptogenesis in this scenario requires careful inclusion of the
flavour-dependent effects. In many applications and realisations of type II leptogenesis
in specific models of fermion masses and mixings (see e.g. [18]), flavour-dependent effects
may substantially change the results and they therefore have to be taken into account.
In summary, type II leptogenesis provides a well-motivated generalisation of the
conventional scenario of leptogenesis in the type I seesaw framework. We have argued
that flavour-dependent effects have to be included in type II leptogenesis, and can change
predictions of existing models as well as open up new possibilities for for successful
models of leptogenesis. Comparing bounds on ε1,f MR1 and TRH in flavour-dependent
thermal type I and type II leptogenesis scenarios, we have shown that while type II
leptogenesis becomes more efficient for larger mass scale of the light neutrinos, and the
bounds become relaxed, leptogenesis within the type I seesaw framework becomes more
constrained.
Acknowledgments
I would like to thank Steve F. King, Antonio Riotto and Ana M. Teixeira for useful
discussions and for their collaboration on leptogenesis issues. This work was supported
by the EU 6th Framework Program MRTN-CT-2004-503369 “The Quest for Unification:
Theory Confronts Experiment”.
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Introduction
Type I and type II seesaw mechanisms
Baryogenesis via flavour-dependent leptogenesis
Decay asymmetries
Right-handed neutrinos plus triplets
Effective approach to leptogenesis
Type II bounds on decay asymmetries and on bold0mu mumu MR1MR13.91663pt plus 1.95831pt minus 1.30554ptMR1MR1MR1MR1
Summary, discussion and conclusions
|
0704.1592 | Multi-wavelength Analysis of a Quiet Solar Region | The Physics of Chromospheric Plasmas
ASP Conference Series, Vol. 368, 2007
Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds.
Multi-wavelength Analysis of a Quiet Solar Region
G. Tsiropoula1, K. Tziotziou1, J. Giannikakis1, P. Young2, U. Schühle3
and P. Heinzel4
1Institute for Space Applications and Remote Sensing, Athens, Greece
2CCLRC Rutherford Appleton Laboratory, United Kingdom
3MPI für Sonnensystemforschung, Katlenburg-Lindau, Germany
4Astronomical Institute AS, Ondřejov, Czech Republic
Abstract. We present observations of a solar quiet region obtained by the
ground-based Dutch Open Telescope (DOT), and by instruments on the space-
craft SOHO and TRACE. The observations were obtained during a coordinated
observing campaign on October 2005. The aim of this work is to present the rich
diversity of fine-scale structures that are found at the network boundaries and
their appearance in different instruments and different spectral lines that span
the photosphere to the corona. Detailed studies of these structures are crucial
to understanding their dynamics in different solar layers, as well as the role such
structures play in the mass balance and heating of the solar atmosphere.
1. Introduction
In the quiet regions of the solar surface the magnetic field is mainly concen-
trated at the boundaries of the network cells. Over the past decade, apart from
the well-known mottles and spicules, several other structures residing at the
network boundaries such as explosive events, blinkers, network flares, upflow
events have been mentioned in the literature. However, their interpretation,
inter-relationship and their relation to the underlying photospheric magnetic
concentrations remain ambiguous, because the same feature has a different ap-
pearance when observed in different spectral lines and by different instruments.
For most of the events mentioned above magnetic reconnection is suggested as
the driving mechanism. This is not surprising, since it is now well established
from investigations of high resolution magnetograms, that new bipolar elements
emerge continuously inside the cell interiors and are, subsequently, swept at the
network boundaries by the supergranular flow (Wang et al. 2006; Schrijver et al.
1997). Interactions of the magnetic fields have as a result either the enhance-
ment of the flux concentration in the case of same polarities or its cancellation in
the case of opposite polarities. Observations support the idea that flux cancel-
lation most likely invokes magnetic reconnection. In this context, the study and
comprehension of the dynamical behaviour of the different fine-scale structures
is crucial to the understanding of the dynamics of the solar atmosphere.
In this work we present observations of a solar quiet region and some of the
properties of several different structures appearing at the network boundaries
and observed in different wavelengths by the different instruments involved in a
coordinated campaign.
http://arxiv.org/abs/0704.1592v1
172 Tsiropoula et al.
Figure 1. Left : C IV TRACE image. Right : MDI magnetogram. The white
rectangle inside the images marks the DOT’s field-of-view.
2. Observations and Data Reduction
In October 2005 we ran a 12 days observational campaign. The aim of that
campaign was the collection of multi-wavelength observations both from the
ground and space that could be used for the study of the dynamical behaviour
of mottles/spicules and other fine structures, observed in different layers of the
solar atmosphere.
Three ground-based telescopes were involved in that campaign: DOT on La
Palma, THEMIS on Tenerife and SOLIS at Kitt Peak. From space telescopes two
spacecraft were involved: SOHO (with CDS, SUMER, and MDI) and TRACE.
The analysed data were obtained on October 14 and consist of time se-
quences of observations of a quiet region found at the solar disk center recorded
by different instruments. Sequences recorded by the DOT were obtained be-
tween 10:15:43 – 10:30:42 UT and consist of 26 speckle reconstructed images
taken simultaneously at a cadence of 35 s with a pixel size of 0.071′′ in 5 wave-
lengths along the Hα line profile (i.e. at −0.7 Å, −0.35 Å, line centre, 0.35 Å
and 0.7 Å), in the G band with a 10 Å filter, in the Ca IIH line taken with a
narrow band filter and in the blue and red continuum. TRACE obtained high
cadence filter images at 1550 Å, 1600 Å and 1700 Å. SUMER obtained raster
scans and sit-and-stare observations from 8:15 to 10:30 UT. CDS obtained sit-
and-stare observations from 6:44 to 10:46 UT and six 154′′ × 240′′ raster scans
(each one having a duration of 30min) from 10:46 to 13:52 UT. Both SOHO
instruments (i.e., CDS and SUMER) observed in several spectral lines spanning
the upper solar atmosphere. Using the standard software the raw measurements
were corrected for flat field, cosmic rays and other instrumental effects. A single
Gaussian with a linear background and Poisson statistics were used for fitting
each spectral line profile. MDI obtained high cadence images at its high resolu-
tion mode.
Multi-wavelength Analysis of a Quiet Region 173
Figure 2. DOT images of a rosette region. Left : Hα -0.7 Å (first row),
Hα+0.7 Å (second row). Right : Ca IIH (first row), G band (second row).
Extensive work went into collecting, scaling and co-aligning the various
data sets to a common coordinate system (see Fig. 1 showing the coalignment
of TRACE, MDI, and DOT images).
3. Analysis and Results
3.1. DOT observations
DOT’s field-of-view (FOV) is 80′′ in the X direction and 63′′ in the Y direction.
For the present study we selected a smaller region which contains a rosette
with several mottles pointing to a common center (see Fig. 2) and is found at
the middle upper part of the DOT’s FOV. In the G-band image (Fig. 2, right,
second row) isolated bright points show up in the regions of strong magnetic
field as can be seen in the MDI image. These seem to be passively advected
with the general granular flow field in the intergranular lanes. While we have
not conducted an exhaustive study of bright point lifetimes we find that bright
points can be visible from some minutes up to almost the entire length of the
time series. In the contemporal Ca IIH images (Fig. 2, right, first row bright
points are less sharp due to strong scattering in this line and possibly due to
increasing flux tube with height. Reversed granulation caused by convection
reversal is obvious in this image.
In the Hα − 0.7 Å (Fig. 2, left, first row) the dark streaks are part of the
elongated Hα mottles seen better at Hα line center. Some mottle endings appear
174 Tsiropoula et al.
Figure 3. Cross correlation function vs time for Left : the intensity at Hα−
0.7 Å, Middle: the intensity at Hα line center. Right : the velocity at Hα −
0.7 Å.
extra dark in the blue wing image through Doppler blueshift. Near the mottle
endings one can see bright points. These are sharper in the G-band but stand
out much clearer in the Hα wing. Thus Hα wing represents a promising proxy
magnetometer to locate and track isolated intermittent magnetic elements. This
is because the Hα wing has a strong photospheric contribution, as it is shown
by Leenaarts et al. (2006) who arrived to this conclusion by using radiative
transfer calculations and convective simulations. In the Hα+0.7 Å (Fig. 2, left,
second row) dark streaks around the rosette’s center are signatures of redshifts.
Blueshifts in the outer endings and redshifts in the inner endings of mottles
provide evidence for the presence of bi-directional flows along these structures.
An important parameter for the study of the dynamics of mottles is their
velocity. For its determination, when filtergrams at two wavelengths of equal
intensity at the blue and the red side of the line are available, a technique
based on the subtraction of images can be used. In this technique, by using the
well known representation of the line intensity profile and assuming a Gaussian
wavelength dependence of the optical thickness, we can define the parameter
ΣI − 2I0λ
, (1)
where ∆I = I(−∆λ) − I(+∆λ), ΣI = I(−∆λ) + I(+∆λ) and I0(∆λ) is the
reference profile emitted by the background. DS is called Doppler signal, has
the same sign as the velocity and can be used for a qualitative description of the
velocity field (for a description of the method see Tsiropoula 2000). When an
optical depth less than one is assumed then quantitative values of the velocity
can be obtained from the relation:
1 +DS
, (2)
since in that case the velocity depends only on DS (obtained from the obser-
vations) and the Doppler width, ∆λD (obtained from the literature). By using
these relations we have constructed 2-D intensity and velocity images for the
whole time series. We found out that it is difficult to follow each one mottle for
more than two or three frames and that the general appearance of the region
Multi-wavelength Analysis of a Quiet Region 175
Figure 4. CDS raster image obtained at the OV 629.7 Å line with an over-
plotted SUMER image at NeVIII 770 Å.
seems to change quite rapidly with time. For a quantitative estimate of the
temporal changes we computed the value of the cross correlation (CC) function
over the 2-D FOV both for the intensity and velocity. Figure 3 shows the inten-
sity CC curve at Hα− 0.7 Å (left) and Hα line center (middle) and the velocity
CC curve at Hα± 0.7 Å (right). The decay of the CC curve is a measure of the
lifetime of the structures. The e-folding time for the left curve is found equal to
2min, the middle curve equal to 5min and the right curve is of the order of the
cadence.
3.2. CDS and SUMER observations
In Fig. 4 we show the CDS raster image obtained in the OV 629.7 Å line with the
SUMER image at NeVIII 770 Å overplotted. Although there is a time difference
of 3 hours between the two images the network is constant enough to allowed the
coalignement of the two images. In CDS intensity maps several brightenings are
observed which are called blinkers (Harrison et al. 1999). These events are best
observed in transition region lines and show an intensity increase of 60 - 80%.
Most of them have a repetitive character and reappear at the same position
several times.
In Fig. 5 (left, up) we show an integrated (over the spectral line with the
background included) intensity image in the Ne VIII 770 Å line produced by sit-
and-stare observations of a network region. The image is produced by binning
over 6 spectra in order to improve the signal-to-noise ratio. The Doppler shift
map was derived by applying a single Gaussian fitting (Fig. 5, left, bottom). The
Doppler shift map and the spectral line profiles were visually inspected for any
non-Gaussian profiles with enhancements in both the blue and red wings that
are the main characteristics of the presence of bi-directional jets. Large numbers
of such profiles were found at the network boundaries (Fig. 5, right).
176 Tsiropoula et al.
Figure 5. SUMER sit-and-stare observations in the NeVIII line. Left : in-
tensities (up), Doppler velocities (bottom) (network boundaries are bright in
the intensity image). Right : non-Gaussian profiles in the positions marked by
“x” inside the images in the left.
4. Conclusions
In this work we present observations of a quiet solar region obtained by different
instruments in different spectral lines. Network boundaries are found to be the
locus of several structures which have different appearances when observed by
different instruments e.g., blinkers (when observed with CDS), mottles (when
observed with DOT), jets (when observed with SUMER). Their interrelationship
is to be further explored.
Regarding flows no-clear pattern is found in blinkers, while bi-directional
flows are found in jets. In dark mottles downward velocities are found at their
footpoints and upwards velocities at their upper parts and very fast changes in
their appearance.
The network shows a remarkable constancy when observed in low resolution
images. However when seen in high resolution images several fine structures are
observed which change so fast that it is very hard to follow.
Acknowledgments. K. Tziotziou acknowledges support by Marie Curie European
Reintegration Grant MERG-CT-2004-021626. This work has been partly supported by
a Greek-Czech programme of cooperation.
References
Harrison, A., Lang, J., Brooks, D.H., & Innes, D.E. 1999, A&A, 351, 1115
Leenarts, J., Rutten, R.J., Sütterlin, P., Carlsson, M., & Uitenbroek H. 2006, A&A,
449, 1209
Schrijver, C.J., Title, A.M., Van Ballegooijen, A.A., Hagenaar, H.J., & Shine, R.A.
1997, ApJ, 487, 424
Tsiropoula, G. 2000, New Astronomy, 5, 1
Wang, H., Tang, F., Zirin, H., & Wang, J. 1996, Solar Phys. , 165, 223
|
0704.1593 | Orbital currents in the Colle-Salvetti correlation energy functional and
the degeneracy problem | Orbital currents in the Colle-Salvetti correlation energy functional and
the degeneracy problem
S. Pittalis1, S. Kurth1, S. Sharma1,2 and E.K.U. Gross1
1 Institut für Theoretische Physik, Freie Universität Berlin,
Arnimallee 14, D-14195 Berlin, Germany and
2 Fritz Haber Institute of the Max Planck Society,
Faradayweg 4-6, D-14195 Berlin, Germany.
(Dated: October 30, 2018)
Abstract
Popular density functionals for the exchange-correlation energy typically fail to reproduce the
degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explic-
itly depend on the current density have been suggested. We present an analysis of this problem
by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the
exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional
we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is
evaluated. A very good description of the degeneracy of ground-states for atoms of the first and
second row of the periodic table is obtained.
http://arxiv.org/abs/0704.1593v2
I. INTRODUCTION
Common approximations to the exchange-correlation functional of density functional
theory (DFT) [1, 2] and spin-DFT (SDFT) [3] often fail to reproduce the degeneracy of dif-
ferent ground states. An illustrative example are ground states of open-shell atoms where
one usually and erroneously obtains different total energies for states with zero and non-
vanishing current density. The local spin density approximation (LSDA) gives rather
small splittings (of the order of 1 kcal/mole), but generalized gradient approximations
(GGAs) and meta-GGAs can introduce splittings of 10 kcal/mol [4, 5, 6, 7, 8, 9, 10, 11].
These spurious energy splittings would vanish if the exact exchange-correlation func-
tional could be used. The exchange-correlation energy functional can be represented
in terms of the exchange-correlation hole function. Considering current-carrying states,
Dobson showed how the expression for the exchange-hole curvature has to be changed
by including current-dependent terms [12]. Later, Becke found the same kind of terms
in the short-range behavior of the exchange-correlation hole, and observed that they
also enter the spin-like correlation-hole function [13]. For open-shell atoms, inclusion
of these current-dependent terms results in spurious energy splittings of less than 1
kcal/mole [9]. Along these lines, Maximoff at al. [10] worked out a correction for the
system-averaged exchange hole of the Perdew-Burke-Ernzerhof (PBE) GGA [14], which
improves the corresponding spurious splittings. Alternatively, Tao and Perdew [11, 15]
proposed a scheme for the extension of existing functionals using ideas of current density
functional theory (CDFT) [16, 17] which, again, improves the description of the degener-
The performance of the exact-exchange (EXX) energy functional - which, by definition,
describes the exchange hole correctly - has been evaluated for the spurious splittings in
DFT and SDFT [18]. In the EXX-DFT (i.e., spin-restricted calculations using one and the
same Kohn-Sham potential for spin-up and spin-down orbitals) the degeneracy is well re-
produced to within 0.6 kcal/mole but, surprisingly, in EXX-SDFT (i.e., spin-unrestricted
calculations using two Kohn-Sham potentials, one for spin-up and one for spin-down or-
bitals) spurious splittings up to 3 kcal/mole are obtained. In particular, current-carrying
states always have higher total energies than states without current. This observation
motivated the applications of the optimized-effective-potential (OEP) method [19, 20, 21]
generalized to current-spin-density functional theory (CSDFT) to these current-carrying
states [22]. As expected, EXX-CSDFT total energies for current-carrying states are lower
than those of EXX-SDFT. However, this lowering is too small to give a substantial im-
provement of the spurious energy splittings. These studies lead to the conclusion that
correlation is needed for any further improvement.
The construction of a correlation energy functional compatible with EXX is a difficult
task [23, 24, 25], but for spherical atoms it was found that EXX combined with the Colle-
Salvetti (CS) functional for correlation [26, 27, 28] leads to very accurate total energies
[29]. The CS functional has been used to derive the popular Lee-Yang-Parr (LYP) func-
tional [30], which is most commonly used together with Becke’s exchange functional [31]
(BLYP) and in hybrid schemes such as B3LYP [32, 33]. On the other hand, the CS correla-
tion energy functional also has its limitations [34, 35, 36]. In particular, while short-range
correlations are well described [35], very important long-range correlations are missing.
These correlations often cannot be ignored in molecules and solids, but are negligible in
atoms. This fact, together with the encouraging results for spherical atoms [29], indicates
that it is appropriate to employ the CS functional to analyze the degeneracy problem for
open-shell atoms beyond EXX. Furthermore, the expression of the CS functional also al-
lows a reconsideration of the relevance of the orbital currents as ingredient of correlation
functionals.
Although the general density functional formalism to deal with degenerate ground
states includes densities which can only be obtained by a weighted sum of several deter-
minantal densities [37, 38], as in many previous investigations [8, 9, 10, 11] we only con-
sider densities which may be represented by a single Slater determinant of Kohn-Sham
orbitals.
II. THEORY
Going beyond the EXX approximation, we here consider the correlation-energy func-
tional of Colle and Salvetti [26]. This expression relies on the assumption that the corre-
lated two-body reduced density matrix may be approximated by the Hartree-Fock (HF)
two-body reduced density matrix ρHF2 (r1, r2), multiplied by a Jastrow-type correlation
factor. After a series of approximations, the following expression is obtained for the cor-
relation energy
Ec = −4a
ρHF2 (r, r)
1 + bρ−
3 (r)
, r− s
3 (r)
1 + dρ−
3 (r)
where ρHF2 (r, s) is expressed in terms of the average and relative coordinates r =
(r1+r2)
and s = r1 − r2. Here, ρ(r) is the electron density and the constants a = 0.049, b = 0.132,
c = 0.2533, d = 0.349 are determined by a fitting procedure using the Hartree-Fock (HF)
orbitals for the Helium atom.
Following Lee, Yang and Parr, this expression can be restated as a formula involving
only the total charge-density, the charge-density of each Hartree-Fock orbital and their
gradient and Laplacian [30]. In this derivation, the single-particle orbitals are tacitly as-
sumed to be real. We denote the resulting expression as CSLYP. In the following, we
relax this restriction and consider complex orbitals. We then proceed in analogy to the
inclusion of current-dependent terms in the Fermi-hole curvature [9, 12] and in the ex-
tension of the electron-localization-function (ELF) [39] for time-dependent states [40]. As
a consequence, in addition to the term already present in CSLYP expression, the current
densities of the single-particle orbitals appear in the final formula. In order to obtain this
expression, which in the following will be denoted as JCSLYP, we rewrite the Laplacian of
the Hartree-Fock (HF) two-body reduced density matrix in Eq.(1) in terms of the original
particle coordinates
∇21 +
∇22 −
∇1 · ∇2
2 (r1, r2)|r1=r2 . (2)
where
2 (r1, r2) =
ρ(r1)ρ(r2)−
1,σ (r1, r2)ρ
1,σ (r1, r2) . (3)
Here,
1,σ (r1, r2) =
ψk,σ(r1)ψ
k,σ(r2), (4)
is the first-order HF density matrix (for a single Slater determinant) expressed in terms
of the single-particle orbitals ψk,σ(r). The corresponding spin-density is simply given by
ρσ(r) = ρ
1,σ (r, r). (5)
Allowing the single-particle orbitals ψk,σ(r) to be complex, a given orbital not only gives
the contribution ρk,σ(r) = |ψk,σ(r)|
2 to the density, but also the contribution jp k,σ(r) =
ψk,σ(r)∇ψ
k,σ(r)
to the paramagnetic current density which is given by
jp,σ(r) =
jp k,σ(r) . (6)
After some straightforward algebra, the Laplacian of the second-order HF reduced den-
sity matrix takes the final form
ρ(r)∇2ρ(r)−
(∇ρ(r))
ρσ(r)∇
ρσ(r) (7)
ρσ(r)
(∇ρk,σ(r))
ρk,σ(r)
+ J(r)
where
J(r) =
ρσ(r)
j2pσ(r)
ρσ(r)
j2p k,σ(r)
ρk,σ(r)
contains all the current-dependent terms. Alternatively, Eq. (7) may also be expressed in
terms of the non-interacting kinetic energy density
τσ(r) =
|∇ψk,σ(r)|
(∇ρk,σ(r))
ρk,σ(r)
j2p k,σ(r)
ρk,σ(r)
ρ(r)∇2ρ(r)−
(∇ρ(r))
ρσ(r)∇
ρσ(r)
2ρσ(r)τσ(r)− jpσ(r)
. (10)
Comparison of Eq.(7) and Eq.(10) shows that J(r), as defined in Eq.(8), also contains
current-dependent terms coming from the kinetic energy density. Thus, this would also
suggest to reconsider the gradient expansion of τ for current-carrying states. Impor-
tant consequences maybe expected for all approximated exchange-correlation functional
involving the kinetic energy density as ingredient, such as the CS functional, and meta-
GGAs. This issue will be specifically considered in a future work. In the next section, we
assess the performance of the CS functional, and, in particular, the relevance of J(r), in
reproducing the degeneracy of atomic states.
III. RESULTS AND DISCUSSION
We consider ground states of open-shell atoms having densities that can be repre-
sented by a single Slater determinant of Kohn-Sham orbitals. Due to the symmetry of the
problem, these Slater determinants are eigenstates of the z-component of both spin and
orbital angular momentum. The single-particle orbitals are generally complex-valued
and states with different total magnetic quantum numbers, ML, correspond to differ-
ent current densities. By means of an accurate exchange-correlation functional the same
total energies would be obtained. In the previous section, we have shown how current-
dependent terms enter the expression of the CS functional when complex-valued orbitals
are considered. Here, we evaluate the performance of EXX plus the CS functional in re-
producing the degeneracy and study the effect of the current-dependent terms, in SDFT
and DFT calculations.
We consider atoms of the first and second row of the periodic table: these are the ref-
erence cases for which a vast amount of numerical data is available [8, 9, 10, 11, 18, 22].
In analogy to the procedure where Hartree-Fock orbitals are used as input to the CS for-
mula, we have evaluated the correlation energies in a post-hoc fashion using Kohn-Sham
(KS) orbitals. We expand the KS orbitals in Slater-type basis functions (QZ4P of Ref. [42])
for the radial part, multiplied with spherical harmonics for the angular part. We obtain
the KS orbitals from self-consistent EXX-only calculations employing the approximation
of Krieger, Li, and Iafrate (KLI) [41], which has been shown to be extremely good at least
for small systems [21]. In principle a functional should be evaluated with KS orbitals
obtained from self-consistent calculations, and this is certainly possible for the CS func-
tional [29]. However, thanks to the variational nature of DFT, it is common experience to
observe only minor quantitative differences between post-hoc and self-consistent evalu-
ation of total energies. This is the reason why the functionals designed for solving the
degeneracy problem are typically evaluated in a post-hoc manner [9, 10, 11].
Table I shows the spurious energy splittings (difference in the total energies) between
Kohn-Sham Slater determinants with total magnetic quantum number |ML| = 1 and
ML = 0, from our SDFT and DFT calculations. The deviation of these total energies from
the exact values is plotted in Fig. (1), where again the spurious energy splittings are visi-
SDFT (DFT)
Atom ∆JCSLY P ∆CSLY P
B 0.8 (-0.3) 2.4 (1.4)
C 0.9 (-0.1) -3.2 (-4.3)
O -0.6 (-1.9) 0.9 (-0.4)
F -0.1 (-1.5) -3.5 (-5.1)
Al 0.4 (-0.5) 1.1 (0.2)
Si 0.5 (-0.4) -1.2 (-2.2)
S 0.1 (-1.6) 1.1 (-0.7)
Cl 0.7 (-1.3) -1.1 (-3.2)
me 0.3 (-1.0) -0.4 (-1.8)
mae 0.5 (1.0) 1.8 (2.2)
TABLE I: Spurios energy splittings, ∆ = E(|ML| = 1) − E(ML = 0) in kcal/mol for open-shell
atoms, computed in SDFT (DFT results in parenthesis for comparison). Correlation energy has
been added to the KLI-EXX energies including (JCSLYP) and neglecting (CSLYP) the current terms
of Eq. (7). The last row shows the mean error (me) and mean absolute (mae) of the spurious
splittings.
ble. These results highlight the importance of including J(r) in Eq. (7). In particular, it is
remarkable to observe that SDFT splittings are within 0.9 kcal/mol (with a mean error of
0.3 kcal/mol and a mean absolute error of 0.5 kcal/mol), and the corresponding DFT spu-
rious energy splittings are less than 1.9 kcal/mol (with mean errors of 1.0 kcal/mol). It is
worthwhile to note that in several cases inclusion of correlation leads to current-carrying
states (|ML| = 1) with lower total energy than zero-current states (ML = 0). These results
are in contrast to EXX-only [18] cases where: (a) the zero-current states are always lowest
in energy and (b) the spurious energy splittings are always smaller in DFT than in SDFT.
Going beyond EXX by including correlation in the form of CS functional accurate to-
tal energies can be obtained within the OEP method [29]. Figure (1) and Table II show
the deviations from exact total energies for the states with different magnetic quantum
B C O F Al Si S Cl
CSLYP M
= 0
JCSLYP M
CSLYP |M
|= 1
JCSLYP |M
B C O F Al Si S Cl
SDFT DFT
FIG. 1: Deviation from exact total energies for SDFT and DFT calculations employing the CS
functional, including (JCSLYP) and not including (CSLYP) the current-dependent term J in Eq. (7).
States with different magnetic quantum numbers ML are plotted. Exact total energies are taken
from Ref. [29] and references therein.
numbers, i.e. different current-carrying states. This further emphasizes the importance
of proper inclusion of J(r) in Eq. (7).
IV. CONCLUSIONS
We have shown that going beyond the exact-exchange approximation by including
correlation energy in the form of the Colle-Salvetti functional leads to a very good de-
scription of the degeneracy of open-shell atoms in both SDFT and DFT calculations.
Comparing DFT and SDFT results for the first and second row of the periodic table, on
average we observe a reduction of the spurious energy splittings and better total ener-
gies for SDFT. Furthermore, we have also shown how current-dependent terms enter the
expression of the Colle-Salvetti functional. If these terms are neglected, the degeneracy
is not well described and the total energies are also less accurate. Thus, this analysis re-
confirms the advantage of properly including the orbital current dependent terms as an
JCSLYP (CSLYP)
SDFT DFT SDFT DFT
|ML| 0 0 1 1
me 2.3 (16.1) 4.7 (18.5) 2.6 (15.7) 3.7 (16.8)
mae 4.8 (16.4) 5.4 (18.5) 4.7 (15.7) 4.7 (16.8)
TABLE II: Mean error (me) and mean absolute error (mae) in the total energies for the CS
functional, including (JCSLYP) and not including (CSLYP results in parenthesis) the current-
dependent term J in Eq. (7), in kcal/mol. Exact total energies are taken from Ref. [29] and ref-
erences therein.
ingredient in correlation functionals.
Acknowledgements
We acknowledge the Deutsche Forschungsgemeinschaft (SPP-1145) and NoE
NANOQUANTA Network (NMP4-CT-2004-50019) for financial support.
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Introduction
Theory
Results and discussion
Conclusions
Acknowledgements
References
|
0704.1594 | Cooper pairs in atomic nuclei | Cooper pairs in atomic nuclei
G. G. Dussel1, S. Pittel2, J. Dukelsky3, and P. Sarriguren3
Departamento de Fisica J. J. Giambiagi, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Bartol Research Institute and Department of Physics and Astronomy,
University of Delaware, Newark, DE 19716 USA
Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain
(Dated: Received October 26, 2018)
We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the
even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS
treatment and an exact treatment using the Richardson ansatz. While projected BCS captures
much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation
energy, typically of order 1 MeV . Furthermore, because it does not average over the properties of
the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper
wave function and of the fraction of pairs that are collective.
PACS numbers: 21.60.-n, 03.75.Ss, 02.30.Ik, 74.20.Fg
The first breakthrough in the derivation of a micro-
scopic theory of superconductivity was the demonstra-
tion by Cooper [1] in 1956 that bound pairs could be
produced in the vicinity of the Fermi surface for an ar-
bitrarily small attractive interaction. This was followed
soon thereafter by the development of the BCS theory
[2], in which superconductivity was described as the con-
densation of a set of correlated pairs averaged over the
whole system. Soon after the BCS paper, Bohr, Mottel-
son and Pines [3] suggested that a similar phenomenon
could explain the large gaps in the spectra of even-even
nuclei. Since then, the BCS theory has been widely used
to describe superconductivity in condensed matter and
nuclear systems. Moreover, the concept of Cooper pairs
as strongly overlapping objects that go through a con-
densation process at the superconducting transition is
central in the interpretation of the superconducting phe-
nomenon. However, it is not easy to define the Cooper
pair wave function from the mean field BCS theory, and
most frequently it has been related to the pair correlator.
By using the exact solution of the BCS Hamiltonian
given by Richardson in the sixties [4], it was recently
shown [5] that the Cooper pair wave function in a super-
conducting medium has a precise definition. The unique
form of its wave function transforms from a Cooper reso-
nance in the weak coupling BCS region to a quasi bound
pair in the Bose-Einstein condensed (BEC) phase. More-
over, the Richardson solution gives a clear prescription
for evaluating the fraction of correlated pairs as compared
with Yang’s definition [6], providing a more accurate de-
scription of the condensation phenomenon.
The subject of Cooper pairing in atomic nuclei has
come under renewed focus recently in the context of the
mean-field Hartree-Fock-Bogolyubov approach [7, 8]. In
this work, we also explore the role of Cooper pairs in
mean-field treatments of atomic nuclei, comparing the
traditional number-nonconserving BCS approach with a
projected BCS approach and the exact Richardson treat-
ment. We show that substantial differences in correlation
energies arise when pairing is treated exactly for the same
pairing strength, and that interesting differences emerge
in some conceptual properties of the paired system.
We begin by detailing the differences between the three
approaches, focusing on a pairing Hamiltonian with con-
stant strength G acting in a space of doubly-degenerate
time-reversed states (k, k̄),
ck −G
′ck′ , (1)
where ǫk are the single-particle energies for the doubly-
degenerate orbits k, k̄.
Cooper studied the problem of adding a pair of
fermions with an attractive pairing interaction on top of
an inert Fermi sea (FS). He showed that the pair eigen-
state is
|ΨCooper〉 =
2ǫk − E
|FS〉 , (2)
where E is the energy eigenvalue. It turns out that E is
negative for any attractive value of G, implying that the
Cooper pair is bound and that the FS is unstable against
the formation of bound pairs. Cooper suggested [1] that
a theory considering a collection of bound pairs on top
of an effective FS could explain superconductivity.
The BCS approach follows a somewhat different path,
defining instead a variational wave function as a coherent
state of pairs properly averaged over the whole system,
|ΨBCS〉 = e
Γ† |0〉 , (3)
where Γ† =
is the coherent pair. The
BCS wave function breaks particle-number conservation.
Though errors due to the nonconservation of particle
number are negligible in the thermodynamic limit, they
can be important in finite systems such as atomic nuclei.
Indeed, Bohr, Mottelson and Pines [3] noted already in
1958 the importance of taking into account finite size ef-
fects in its application to nuclei. To accommodate these
http://arxiv.org/abs/0704.1594v1
effects, the number-projected BCS formalism (PBCS) [9]
assumes a condensed state of pairs of the form
|ΨPBCS〉 =
|0〉 , (4)
where M is the number of pairs and Γ† has the same
form as in BCS. We would like to emphasize here that Γ†
should not be confused with the operator that creates a
Cooper pair since its structure contains an average over
the correlated pairs close to the Fermi energy and the
free fermions deep inside the Fermi sphere.
The Richardson ansatz [4] for the exact solution of the
pairing Hamiltonian (1) follows closely Cooper’s original
idea. For a system with 2M particles, it involves (in the
ν = 0 sector) a product of M distinct pairs of the form
|Φ >=
Γ†α| 0〉 , Γ
2ǫk − eα
. (5)
The eα, called pair energies in analogy with the Cooper
wave function (2), are in general complex parameters,
which are obtained by solving the set of coupled non-
linear Richardson equations
2ǫk − eα
β( 6=α)=1,M
eβ − eα
= 0 . (6)
The energy eigenvalues are obtained by summing the
lowest M pair energies of each independent solution
The key point to note upon inspection of the Richard-
son pair (5) is that a pair energy close to a particular
2ǫk, i.e. close to the energy of an unperturbed pair, is
dominated by this particular configuration and thus de-
fines an uncorrelated pair. In contrast, a pair energy
lying sufficiently far away in the complex plane produces
a correlated Cooper pair.
As mentioned before, the BCS coherent pairs, with
amplitudes zk = vk/uk, cannot be interpreted as Cooper
pairs since they mix correlated and uncorrelated pairs
over the whole system. Indeed, it has been shown [5] that
only in the extreme BEC limit are all pairs bound and
condensed, and amenable to description by the two ap-
proaches. Usually the structure of the Cooper pair is as-
signed to the pair correlator 〈BCS| c
|BCS〉 = ukvk.
However, if the BCS state represents a fraction of cor-
related pairs within a Fermi sea of free uncorrelated
fermions, the pair correlator cannot guarantee that it
picks up the two fermions from the same pair. The pair
correlator is another averaged property over the set of
correlated pairs.
In what follows we explore the structure of pairing cor-
relations in the even Sm isotopes, from 144Sm through
158Sm. The results are based on a series of self-consistent
deformed Hartree Fock+BCS calculations. The calcula-
tions make use of the density-dependent Skyrme force,
SLy4, and treat pairing correlations using a pairing force
with constant strength G.
The calculations are carried out in an axially sym-
metric harmonic oscillator space of 11 major shells (286
doubly-degenerate single-particle states). This basis in-
volves oscillator parameters b0 and axis ratio q, opti-
mized in order to minimize the energy in the given space.
The strength of the pairing force for protons and neu-
trons is chosen in such a way as to reproduce the ex-
perimental pairing gaps in 154Sm (∆n = 0.98 MeV ,
∆p = 0.94 MeV ), extracted from the binding energies
in neighboring nuclei. We obtain Gn = 0.106 MeV and
Gp = 0.117 MeV . Once we have fitted this reference
strength, we determine the pairing strengths appropri-
ate to the 142−158Sm isotopic chain by assuming a 1/A
dependence. These calculations provide an excellent de-
scription of the properties of the even Sm isotopes.
We then use the results at self-consistency to define
the HF mean field and consider the alternative number-
conserving PBCS and exact Richardson approach to treat
the pairing correlations within this mean field. We ig-
nore the issue of whether the mean field should be self-
consistently modified in these other approaches. In this
way we are able to directly compare the three approaches
to pairing with the same pairing Hamiltonian, which is
the focus of this investigation.
As is well known that the numerical solution of the
Richardson equations (6) involves instabilities due to sin-
gularities arising at some critical values of the pairing
strength G. There have been two recent works that
study these critical regions of parameter space [10] and
propose ways to overcome the singularities [11]. While
these methods alleviate the numerical divergences, thus
allowing for an interpolation method to cross the criti-
cal regions, some problems still persist and we have thus
chosen to use a different approach. Since the singulari-
ties arise as crossings of real pair energies eα with the
unperturbed single-pair energies 2ǫk in the denomina-
tors of (6), we start the numerical procedure at strong
coupling (G = 1 MeV ) with complex single-particle en-
ergies, obtained by adding a small arbitrary imaginary
component. In this way, the singularities are avoided in
the evolution of the system from strong coupling almost
to the G = 0 limit. To obtain the exact solution at the
physical value of G, we then let the imaginary parts go to
zero starting with the solution already obtained for that
G value. The method seems to work for any distribution
of single-particle energies.
A principal focus of our investigation is on the pairing
correlation energy, defined as
EC = 〈Φcorr|H | Φcorr〉 − 〈Φuncorr|H | Φuncorr〉 , (7)
where |Φcorr〉 is the correlated ground-state wave func-
tion and |Φuncorr〉 is the uncorrelated Hartree Fock Slater
determinant obtained by filling all levels up to the Fermi
energy. This quantity reflects the additional energy that
derives from the inclusion of pairing.
Table 1 summarizes our results for the pairing corre-
lation energy in table 1 for all the even Sm isotopes un-
der consideration. Note that the calculations include the
semi-magic nucleus 144Sm, for which the BCS calculation
leads to a normal solution with no pairing correlation en-
ergy. In contrast, the projected BCS calculation leads to
substantial pairing correlations in the ground state. That
number projection is critical in mean-field treatments of
semi-magic nuclei is well known from other calculations
[12]. The exact treatment of pairing leads to a further
lowering of the energy of the ground state of the system,
by 0.3 MeV .
In the calculations other than 144Sm, the effect on the
pairing correlation energy of the exact solution is more
pronounced. While PBCS gives a significant lowering of
the energy of the system due to number projection, it
misses about 1 MeV of the full correlation energy of an
exact treatment. Considering the extensive recent efforts
to carry out systematic microscopic calculations of nu-
clear masses using mean-field methods [13], we feel that
this effect may be quite meaningful. It is not clear that a
renormalization of the strength of the pairing interaction
can accommodate these important corrections.
The results obtained for the Sm isotopes are consis-
tent with studies performed in ultrasmall superconduct-
ing grains [14]. The quantum phase transition from a
superconducting to a normal metal predicted by BCS
and PBCS completely disappears after fully including the
pairing fluctuations by means of the exact solution of the
BCS model. Moreover, the PBCS wave function displays
a strange behavior in the transitional region as compared
with the smooth behavior of the exact wave function [15].
Table I: Pairing correlation energies associated with the
BCS, PBCS and exact Richardson treatments of pairing
for the even Sm isotopes. All energies are given in MeV
Mass EC(Exact) EC(PBCS) EC(BCS)
142 -4.146 -3.096 -1.107
144 -2.960 -2.677 0.
146 -4.340 -3.140 -1.384
148 -4.221 -3.014 -1.075
150 -3.761 -2.932 -0.386
152 -3.922 -2.957 -0.637
154 -3.678 -2.859 -0.390
156 -3.716 -2.832 -0.515
158 -3.832 -2.824 -0.717
A second important feature of Cooper pairing is the
condensate fraction, namely the fraction of pairs of the
whole system that are correlated. Analysis of the off-
diagonal long-range order (ODLRO) that characterizes
superconductors and superfluids led Yang [6] to a defini-
tion of the condensate fraction, λ, in terms of the single
macroscopic eigenvalue of the two-body density matrix.
For a homogeneous system of two spin fermion species in
the thermodynamic limit, λ is given by
d3r1d
r2 |〈ψ↓ (r1)ψ↑ (r2)〉| =
k . (8)
This definition is not appropriate for finite Fermi sys-
tems, however, where several eigenvalues of the two-body
density matrix are of the same order. We modify it,
therefore, by excluding from the two-body density ma-
trix the amplitude of finding two uncorrelated fermions.
More specifically, our prescription for finite systems is to
evaluate the matrix elements of the operator
M(1−M/L)
k,k′=1
ck̄′ck〉 − 〈c
ck〉〈c
ck̄′〉 ,
where L is the total number of doubly-degenerate, canon-
ically conjugate pair states k, k̄.
In BCS approximation, the modified Yang prescription
leads to a condensate fraction
λBCS =
M(1−M/L)
. (10)
We have calculated this quantity for the BCS solutions
obtained for 154Sm as a function of the pairing strength
G and plot the results as the smooth curve in figure 1.
An alternative prescription for the condensate fraction
from the exact Richardson solution was proposed in [5]
and shown to more properly reflect the properties of a su-
perfluid system as it undergoes the crossover from BCS
to BEC. In particular, this new prescription gives a fully
condensed state at the change of sign of the chemical
potential where the whole system becomes bound. This
prescription, however, requires knowledge of the proper-
ties of the precise Cooper pairs in the problem, not an
average over the whole system as provided by the BCS
or PBCS approximations (3,4). The Richardson ansatz
(5) is ideally suited for this as it provides an exact wave
function for each individual Cooper pair. One has to sim-
ply distinguish which pairs are correlated and which are
not. As previously discussed, a correlated pair is char-
acterized by a pair energy eα that is far enough away in
the complex plane from any particular 2ǫk. We therefore
propose the following practical definition for the conden-
sate fraction. It is the fraction of pair energies which in
the complex energy plane lie further from any unperturbed
single-pair energy, 2ǫk, than the mean single-particle level
spacing.
We now return to a discussion of the condensate frac-
tion, as plotted in figure 1 for 154Sm as a function of
the pairing strength G. In addition to the results based
on the pair correlator, as discussed earlier, we also plot
(in the sawtooth curve) the results that derive from the
exact Richardson solution using the prescription just de-
scribed. To illustrate how these latter results emerge,
we show in figure 2 the associated pair energies for four
0.0 0.2 0.4 0.6 0.8
G (MeV)
Exact
FIG. 1: The modified Yang prescription for the BCS treat-
ment of pairing (smooth curve) and the alternative prescrip-
tion discussed in the text (sawtooth curve) for the exact
Richardson treatment. G0 = 0.106 MeV denotes the physical
value of the pairing strength and ǫ1 = µ denotes the strength
at which the whole system binds.
values of G in 154Sm, ranging from the physical value
of G = 0.106 MeV to a fairly strong pairing strength
of G = 0.4 MeV . In 154Sm the mean level spacing be-
tween the Hartree Fock single-particle levels is roughly
0.5 MeV , both around the Fermi surface and far from
it. For G = 0.106 MeV , most of the pair energies lie
very near the real axis and quite close to at least one un-
perturbed single-pair energy, 2ǫk. Two of them (which
form a complex conjugate pair) extend about 1 MeV
in the complex plane, while another two are marginally
collective, lying roughly 0.5 MeV from the closest 2ǫk.
The two most collective pairs, denoted C1 in the figure,
each have a real energy of −15.55MeV , which is roughly
twice the energy of the single-particle levels just below
the Fermi surface. This suggests that the first pairs that
become collective are indeed those built out of the va-
lence orbits. As G increases, we see a gradual increase
in the number of collective pairs, which form an arc in
the complex plane. As can be seen from figure 1, by a
pairing strength of roughly 0.5 MeV all of the pairs of
the system are correlated giving a condensate fraction of
1, even though the BEC regime has not yet been reached.
The BEC limit is realized when the chemical potential µ
crosses the lowest single-particle energy ǫ1 at G = 0.788
for 154Sm. At this point all pairs are bound. However,
the revised Yang prescription (9) fails to predict a com-
plete condensate at this point, in the same way as it fails
to do so in the homogeneous case [5].
The Richardson prescription for Cooper pairs also
gives rise to a different interpretation of their internal
structure. In figure 3, we compare the square of the wave
function for the most correlated Cooper pairs in 154Sm,
i.e. those whose pair energies lie farthest from any un-
-80 -60 -40 -20 0 20 40 60 80
-1.0 -0.5 0.0 0.5 1.0
-40 -20 0 20 40
-20 -15 -10 -5 0 5 10 15 20
Imaginary Part
G=0.4
2C2C1 C1
G=0.106
Imaginary Part
G=0.3
G=0.2
FIG. 2: Pair energies (in MeV ) for the exact Cooper pairs
that emerge from four calculations of the 154Sm isotope. G =
0.106 MeV is the physical value of the pairing strength. In
that panel, we denote the most collective pairs as Ci, for
subsequent notational purposes.
38 40 42 44 46 48 50 52
FIG. 3: Square of the wave function of the most collective
Cooper pairs in 154Sm (denoted C1, C2, C3, C4, and C5) and
the pair correlator (BCS) versus the single-particle levels.
perturbed single-pair energy, with the square of the pair
correlator wave function obtained from the BCS calcula-
tion. All wave functions are plotted versus the order of
the single-particle states to make clear the relevant mix-
ing of configurations in each pair. The pair labels in the
figure (C1 through C5) refer to corresponding labels in
the upper left panel of figure 2. C1 refers to the two most
collective pairs, namely those that are farthest from any
unperturbed single-particle pair. Being complex conju-
gate pairs, both have exactly the same absolute square
of their wave function and thus we only show one in the
figure. C2 refers to the next two most collective pairs,
which as noted earlier are marginally collective according
to our prescription. C3 refers to the next two most col-
lective pairs after C2, which according to the prescription
given above involve perturbative mixing of configurations
and are not collective. C4 and C5, the following pairs in
descending collective order, have real pair energies and
involve almost pure single-particle configurations.
¿From the figure, we see that the pair correlator wave
function is quite spread over several single-particle con-
figurations and is peaked between at the 47th single-
particle level, just beyond the Fermi energy (154Sm has
46 neutron pairs). In contrast, the most highly corre-
lated Cooper pair wave function C1 is somewhat narrower
(less collective) and is peaked slightly within the Fermi
sphere. The less-collective Cooper pairs, C2 through C3,
are peaked progressively further inside the Fermi sphere
and are progressively narrower. From this figure, we con-
clude that the size of even the most collective Cooper
pairs in coordinate space will be larger than the size of
the pair correlator, as was already demonstrated in the
weak coupling BCS regime of cold atomic gases [5]. Re-
cent investigations [7, 8] on the size of the pair correlator
in spherical nuclei have concluded that it is unexpectedly
small in the nuclear surface (2− 3 fm). The present cal-
culations would suggest that the actual size of the few
highly collective Cooper pairs is larger than the typical
size of the pair correlations in the nuclear medium. Fur-
thermore, as is also evident from the figure, less bound
pairs get progressively closer to a particular 2ǫk and the
corresponding Cooper pair wave function is less collec-
tive, i.e. more narrow in energy space, and peaked at
this particular configuration.
In this work, we have studied the role of Cooper pair-
ing in atomic nuclei, focusing on a realistic description
of the even Sm isotopes. We assume that the mean field
is given by the self-consistent HF solution from coupled
HF+BCS calculations, and then consider how the effects
of pairing on that mean field would be modified at several
levels of improved treatment. We consider both the pro-
jected BCS approximation and an exact treatment based
on Richardson’s solution of the pairing problem. Sev-
eral important points emerged. On the one hand, even
though PBCS approximation gives a significant gain in
binding energy over ordinary BCS, it still fails to capture
a sizable component, typically of order 1 MeV . This
might have important implications in efforts to derive
nuclear masses from a microscopic approach. Second, we
discussed a new and improved prescription for identify-
ing the fraction of the pairs in a nucleus that are collec-
tive, which can only be realized when the properties of
the various Cooper pairs in the problem are treated sep-
arately. This new prescription suggests that a slightly
larger number of pairs are collective when compared to
the more usual prescription based on Yang’s definition
of the condensate fraction. Furthermore, it suggests that
the few collective Cooper pairs that arise in real nuclei,
being individually less collective than the pair correlator,
would be spatially more spread out.
The Richardson solution, as generalized in ref. [16],
can be obtained for integrable pairing hamiltonians only.
It is possible, however, to use the Richardson ansatz (5)
in a variational treatment of general non-integrable pair-
ing hamiltonians. The pair energies would play the role
of variational parameters within a generalized Pfaffian
pairing wave function [17], making it possible to treat
pair correlations in a more precise manner for realistic
nuclear systems.
We acknowledge fruitful discussions with N. Sand-
ulescu, P. Schuck and W. Nazarewicz. This work was
supported in part by the Spanish DGI under grants
FIS2005-00640 and FIS2006-12783-C03-01, in part by the
US National Science Foundation under grant # 0553127,
and in part by UBACYT X-053, Carrera del Investigador
Cient́ıfico and PIP-5287 (CONICET-Argentina). One of
the authors (SP) wishes to acknowledge the generous sup-
port and hospitality of the CSIC in Madrid where much
of his contribution to the work was carried out
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Seizawa, Phys. Rev. C 71, 064326 (2005).
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[12] M.V. Stoitsov, J. Dobaczewski, R. Kirchner, W.
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http://arxiv.org/abs/nucl-th/0701086
http://arxiv.org/abs/nucl-th/0610061
|
0704.1595 | An adaptive numerical method for the Vlasov equation based on a
multiresolution analysis | An adaptive numerical method for the Vlasov
equation based on a multiresolution analysis
N. Besse1, F. Filbet2, M. Gutnic2, I. Paun2, E. Sonnendrücker2
1 C.E.A, BP 12, 91680 Bruyères-le-Châtel, France, [email protected]
2 IRMA, Université Louis Pasteur, 67084 Strasbourg cedex, France,
filbet,gutnic,ipaun,[email protected]
1 Introduction
Plasmas, which are gases of charged particles, and charged particle beams
can be described by a distribution function f(t, x, v) dependent on time t,
on position x and on velocity v. The function f represents the probability of
presence of a particle at position (x, v) in phase space at time t. It satisfies
the so-called Vlasov equation
+ v · ∇xf + F (t, x, v) · ∇vf = 0. (1)
The force field F (t, x, v) consists of applied and self-consistent electric and
magnetic fields:
(Eself + Eapp + v × (Bself +Bapp)),
wherem represents the mass of a particle and q its charge. The self-consistent
part of the force field is solution of Maxwell’s equations
+∇×B = µ0j, ∇ · E =
+∇×E = 0, ∇ ·B = 0.
The coupling with the Vlasov equation results from the source terms ρ
and j such that:
ρ(t, x) = q
f(t, x, v) dv, j = q
f(t, x, v)v dv.
We then obtain the nonlinear Vlasov-Maxwell equations. In some cases, when
the field are slowly varying the magnetic field becomes negligible and the
Maxwell equations can be replaced by the Poisson equation where:
Eself (t, x) = −∇xφ(t, x), −ε0∆xφ = ρ. (2)
The numerical resolution of the Vlasov equation is usually performed
by particle methods (PIC) which consist in approximating the plasma by a
http://arxiv.org/abs/0704.1595v1
2 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker
finite number of particles. The trajectories of these particles are computed
from the characteristic curves given by the Vlasov equation, whereas self-
consistent fields are computed on a mesh of the physical space. This method
allows to obtain satisfying results with a few number of particles. However,
it is well known that, in some cases, the numerical noise inherent to the
particle method becomes too important to have an accurate description of
the distribution function in phase space. Moreover, the numerical noise only
decreases in
N , when the number of particles N is increased. To remedy to
this problem, methods discretizing the Vlasov equation on a mesh of phase
space have been proposed. A review of the main methods for the resolution
of the Vlasov equation is given in these proceedings [5].
The major drawback of methods using a uniform and fixed mesh is that
their numerical cost is high, which makes them rather inefficient when the
dimension of phase-space grows. For this reason we are investigating here
a method using an adaptive mesh. The adaptive method is overlayed to a
classical semi-Lagrangian method which is based on the conservation of the
distribution function along characteristics. Indeed, this method uses two steps
to update the value of the distribution function at a given mesh point. The
first one consists in following the characteristic ending at this mesh point
backward in time, and the second one in interpolating its value there from
the old values at the surrounding mesh points. Using the conservation of the
distribution function along the characteristics this will yield its new value
at the given mesh point. This idea was originally introduced by Cheng and
Knorr [2] along with a time splitting technique enabling to compute exactly
the origin of the characteristics at each fractional step. In the original method,
the interpolation was performed using cubic splines. This method has since
been used extensively by plasma physicists (see for example [4, 6] and the ref-
erences therein). It has then been generalized to the frame of semi-Lagrangian
methods by E. Sonnendrücker et al. [8]. This method has also been used to
investigate problems linked to the propagation of strongly nonlinear heavy
ion beams [9].
In the present work, we have chosen to introduce a phase-space mesh
which can be refined or derefined adaptively in time. For this purpose, we
use a technique based on multiresolution analysis which is in the same spirit
as the methods developed in particular by S. Bertoluzza [1], A. Cohen et al.
[3] and M. Griebel and F. Koster [7]. We represent the distribution function
on a wavelet basis at different scales. We can then compress it by eliminat-
ing coefficients which are small and accordingly remove the associated mesh
points. Another specific feature of our method is that we use an advection in
physical and velocity space forward in time to predict the useful grid points
for the next time step, rather than restrict ourselves to the neighboring points.
This enables us to use a much larger time step, as in the semi-Lagrangian
method the time step is not limited by a Courant condition. Once the new
mesh is predicted, the semi-Lagrangian methodology is used to compute the
An adaptive numerical method for the Vlasov equation 3
new values of the distribution function at the predicted mesh points, using
an interpolation based on the wavelet decomposition of the old distribution
function. The mesh is then refined again by performing a wavelet transform,
and eliminating the points associated to small coefficients.
This paper is organized as follows. In section 2, we recall the tools of
multiresolution analysis which will be needed for our method, precizing what
kind of wavelets seem to be the most appropriate in our case. Then, we
describe in section 3 the algorithm used in our method, first for the non
adaptive mesh case and then for the adaptive mesh case. Finally we present
a few preliminary numerical results.
2 Multiresolution analysis
The semi-Lagrangian method consists mainly of two steps, an advection step
and an interpolation step. The interpolation part is performed using for ex-
ample a Lagrange interpolating polynomial on a uniform grid. Thus interpo-
lating wavelets provide a natural way to extend this procedure to an adaptive
grid in the way we shall now shortly describe.
For simplicity, we shall restrict our description to the 1D case of the whole
real line. It is straightforward to extend it to periodic boundary conditions
and it can also be extended to an interval with Dirichlet boundary conditions.
The extension to higher dimension is performed using a tensor product of
wavelets and will be addressed at the end of the section.
For any value of j ∈ Z, we consider a uniform grid Gj of step 2−j . The
grid points are located at x
k = k2
−j . This defines an infinite sequence of
grids that we denote by (Gj)j∈Z, and j will be called the level of the grid.
In order to go from one level to the next or the previous, we define a pro-
jection operator and a prediction operator. Consider two grid levels Gj and
Gj+1 and discrete values (of a function) denoted by (c
k)k∈Z and (c
k )k∈Z.
Even though we use the same index k for the grid points in the two cases,
there are of course twice as many points in any given interval on Gj+1 as on
Gj . Using the terminology in [3], we then define the projection operator
j+1 : Gj+1 → Gj ,
2k 7→ c
which is merely a restriction operator, as well as the prediction operator
j : Gj → Gj+1,
such that c
2k = c
2k+1 = P2N+1(x
2k+1),
4 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker
where P2N+1 stands for the Lagrange interpolation polynomial of odd degree
2N + 1 centered at the point (x
2k+1).
Using the just defined prediction operator, we can construct on Gj a
subspace of L2(R) that we shall denote by Vj , a basis of which being given
by (ϕ
k)k∈Z such that ϕ
k′ ) = δkk′ where δkk′ is the Kronecker symbol. The
value of ϕ
k at any point of the real line is then obtained by applying, possibly
an infinite number of times, the prediction operator.
In the wavelets terminology the ϕ
k are called scaling functions. We shall
also denote by ϕ = ϕ0
. Let us notice that
k(x) = ϕ(2
jx− k).
It can be easily verified that the scaling functions satisfy the following prop-
erties:
– Compact support: the support of ϕ is included in [−2N − 1, 2N + 1].
– Interpolation: by construction ϕ(x) is interpolating in the sense that ϕ(0) =
1 and ϕ(k) = 0 if k 6= 0.
– Polynomial representation: all polynomials of degree less or equal to 2N+1
can be expressed exactly as linear combinations of the ϕ
– Change of scale: the ϕ at a given scale can be expressed as a linear combi-
nation of the ϕ at the scale immediately below:
ϕ(x) =
−2N−1
hlϕ(2x− l).
Moreover the sequence of spaces (Vj)j∈Z defines a multiresolution analysis
of L2(R), i.e. it satisfies the following properties:
– . . . ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . . ⊂ Vn ⊂ . . . ⊂ L2(R).
– ∩Vj = {0}, ∪Vj = L2(R).
– f ∈ Vj ↔ f(2 ·)Vj+1.
– ∃ϕ (scaling function) such that {ϕ(x− k)}k∈Z is a basis of V0 and {ϕjk =
2j/2ϕ(2j x− k)}k∈Z is a basis of Vj .
As Vj ⊂ Vj+1, there exists a supplementary of Vj in Vj+1 that we shall
call the detail space and denote by Wj :
Vj+1 = Vj ⊕Wj .
The construction of Wj can be made in the following way: an element of
Vj+1 is characterized by the sequence(c
k )k∈Z and by construction we have
k = c
2k . Thus, if we define d
k = c
2k+1 − P2N+1(x
2k+1), where P2N+1 is
the Lagrange interpolation polynomial by which the value of an element of
Vj at the point (x
2k+1) can be computed, d
k represents exactly the difference
between the value in Vj+1 and the value predicted in Vj . Finally, any element
An adaptive numerical method for the Vlasov equation 5
of Vj+1 can be characterized by the two sequences (c
k)k of values in Vj
and (d
k)k of details in Wj . Moreover this strategy for constructing Wj is
particularly interesting for adaptive refinement as d
k will be small at places
where the prediction from Vj is good and large elsewhere, which gives us a
natural refinement criterion. Besides, there exists a function ψ, called wavelet
such that {ψjk = 2j/2ψ(2j x− k)}k∈Z is a basis of Wj .
In practise, for adaptive refinement we set the coarsest level j0 and the
finest level j1, j0 < j1, and we decompose the space corresponding to the
finest level on all the levels in between:
Vj1 = Vj0 ⊕Wj0 ⊕Wj0+1 ⊕ · · · ⊕Wj1−1.
A function f ∈ Vj1 can then be decomposed as follows
f(x) =
l (x) +
l (x),
where the (c
l )l are the coefficients on the coarse mesh and the (d
l )l the
details at the different level in between.
2k1+2,2k2+1
k1,k2
2k1,2k2+1
2k1+1,2k2+1
2k1+1,2k2+2
k1,k2+1
k1+1,k2+1
2k1+1,2k2
k1+1,k2
Fig. 1. Mesh refinement in 2D.
In two dimensions, the prediction operator which defines the multireso-
lution analysis is constructed by tensor product from the 1D operator. In
practise three different cases must be considered (see figure 1 for notations):
1. Refinement in x (corresponding to points c
2k1+1,2k2
and c
2k1+1,2k2+2
): we
use the 1D prediction operator in x for fixed k2.
2. Refinement in v (corresponding to points c
2k1,2k2+1
and c
2k1+2,2k2+1
): we
use the 1D prediction operator in v for fixed k1.
3. Refinement in v (corresponding to point c
2k1+1,2k2+1
): we first use the
1D prediction operator in v for fixed k1 to determine the points which
are necessary for applying the 1D prediction operator in x for fixed k2
which we then apply.
6 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker
The corresponding wavelet bases are respectively of type ψ(x)ϕ(v), ϕ(x)ψ(v)
and ψ(x)ψ(v) where ϕ and ψ are respectively the scaling function and the 1D
wavelet. We then obtain a 2D wavelet decomposition of the following form:
f(x, v) =
k1,k2
k1,k2
(v) +
row,j
k1,k2
col,j
k1,k2
(v) + d
mid,j
k1,k2
. (3)
3 The algorithms
We want to numerically solve the Vlasov equation (1) given an initial value
of the distribution function f0.
We start by describing the method based on an interpolation using the
wavelet decomposition of f in the non adaptive case. Then we overlay an
adaptive algorithm to this method.
For those two algorithms, we first pick the resolution levels for the phase-
space meshes, from the coarsest j0 to the finest j1. Although these levels
could be different in x and v, we consider here for the sake of conciseness and
clarity that they are identical.
We also compute our scaling function on a very fine grid so that we can
obtain with enough precision its value at any point.
3.1 The non adaptive algorithm
We are working in this case on the finest level corresponding to j1 keeping
all the points.
Initialization: We decompose the initial condition in the wavelet basis
by computing the coefficients ck1,k2 of the decomposition in Vj0 for the coarse
mesh, and then adding the details d
k1,k2
in the detail spaces Wj for all the
other levels j = j0, . . . , j1 − 1. We then compute the initial electric field.
Time iterations:
– Advection in x: We start by computing for each mesh point the origin
of the corresponding characteristic exactly, the displacement being vj∆t.
As we do not necessarily land on a mesh point, we compute the values of
the distribution function at the intermediate time level, denoted by f∗, at
the origin of the characteristics by interpolation from fn. We use for this
the wavelet decomposition (3) applied to fn from which we can compute
fn at any point in phase space.
An adaptive numerical method for the Vlasov equation 7
– Computation of the electric field: We compute the charge density
by integrating f∗ with respect to v, then the electric field by solving the
Poisson equation (this step vanishes for the linear case of the rotating
cylinder where the advection field is exactly known).
– Advection in v: We start by computing exactly the origin of the char-
acteristic for each mesh point, the displacement being E(tn, xi)∆t. As we
do not necessarily land on a mesh point, we compute the values of the
distribution function at the intermediate time level, denoted by fn+1, at
the origin of the characteristics by interpolation from f∗. We use for this
the wavelet decomposition of f∗ given by (3) used at the previous step.
3.2 The adaptive algorithm
In the initialization phase, we first compute the wavelet decomposition of
the initial condition f0, and then proceed by compressing it, i.e. eliminating
the details which are smaller than a threshold that we impose. We then
construct an adaptive mesh which, from all the possible points at all the
levels between our coarsest and finest, contains only those of the coarsest
and those corresponding to details which are above the threshold. We denote
by G̃ this mesh.
– Prediction in x: We predict the positions of points where the details
should be important at the next time split step by advancing in x the
characteristics originating from the points of the mesh G̃. For this we use an
explicit Euler scheme for the numerical integration of the characteristics.
Then we retain the grid points, at one level finer as the starting point,
surrounding the end point the characteristic.
– Construction of mesh Ĝ: From the predicted mesh G̃, we construct
the mesh Ĝ where the values of the distribution at the next time step
shall be computed. This mesh Ĝ contains exactly the points necessary for
computing the wavelet transform of f∗ at the points of G̃.
– Advection in x: As in the non adaptive case.
– Wavelet transform of f∗: We compute the ck and dk coefficients at the
points of G̃ from the values of f∗ at the points of Ĝ.
– Compression:We eliminate the points of G̃ where the details dk are lower
than the fixed threshold.
– Computation of the electric field: As in the non adaptive case.
– Prediction in v: As for the prediction in x.
– Construction of mesh Ĝ: As previously. This mesh Ĝ contains exactly
the points necessary for computing the wavelet transform of fn+1 at the
points of G̃ determined in the prediction in v step.
– Advection in v: As in the non adaptive case.
– Wavelet transform of fn+1: We compute the ck and dk at the points of
G̃ from the values of fn+1 at the points of Ĝ.
– Compression:We eliminate the points of G̃ where the details dk are lower
than the fixed threshold.
8 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker
4 Numerical results
We show here our first results obtained with the adaptive method. We con-
sider first a linear problem, namely the test case of the rotating cylinder
introduced by Zalesak [10] to test advection schemes. Then we consider a
classical nonlinear Vlasov-Poisson test case, namely the two stream instabil-
4.1 The slit rotating cylinder
We consider the following initial condition:
f(0, x, v) =
x2 + v2 < 0.5 and if x < 0 or |v| > 0.125,
0 else.
The computational domain is [−0.5, 0.5]× [−0.5, 0.5].
The advection field is (v,−x), which corresponds to the Vlasov equation
with an applied electric field Eapp(x, t) = −x and without self-consistent field.
Figure 2 represents the evolution of the rotating cylinder on a half turn with
a coarse mesh of 16× 16 points and 4 adaptive refinement levels. We notice
that the cylinder is well represented and that the mesh points concentrate
along the discontinuities.
4.2 The two-stream instability
We consider two streams symmetric with respect to v = 0 and represented
by the initial distribution function
f(0, x, v) =
v2 exp(−v2/2)(1 + α cos(k0 x)),
with α = 0.25, k0 = 0.5, and L = 2 π/k0. We use a maximum of Nx = 128
points in the x direction, and Nv = 128 points in the v direction with vmax =
7, and a time step∆t = 1/8. The solution varies first very slowly and then fine
scales are generated. Between times of around t ≃ 20 ω−1p and t ≃ 40 ω−1p ,
the instability increases rapidly and a hole appears in the middle of the
computational domain. After t = 45 ω−1p until the end of the simulation,
particles inside the hole are trapped. On figure 3 we show a snapshot of the
distribution function at times t = 5 ω−1p and t = 30 ω
p for a coarse mesh of
16 × 16 points and 3 levels of refinement. The adaptive method reproduces
well the results obtained in the non adaptive case.
5 Conclusion
In this paper we have described a new method for the numerical resolution of
the Vlasov equation using an adaptive mesh of phase-space. The adaptive al-
gorithm is based on a multiresolution analysis. It performs qualitatively well.
An adaptive numerical method for the Vlasov equation 9
Fig. 2. Rotating cylinder: evolution for a coarse mesh of 24×24 points and 4 adap-
tive refinement levels. Snapshots of the cylinder and the corresponding adaptive
mesh: (upper) after one time step, (lower) after 1/2 turn.
Fig. 3. Two stream instability for a coarse mesh of 24 × 24, and 3 adaptive refine-
ment levels, (left) at time t = 5ω−1p , (right) at time t = 30ω
10 N. Besse, F. Filbet, M. Gutnic, I. Paun, E. Sonnendrücker
However, there is a large overhead due to the handling of the adaptive mesh
which has not been optimized yet. The performance of the code needs to be
improved before we can recommend this technique for actual computations.
We are currently working on optimizing the code and trying different kinds
of wavelets, as well as obtaining error estimates for the adaptive method.
References
1. S. Bertoluzza, An adaptive collocation method based on interpolating wavelets.
Multiscale wavelet methods for partial differential equations, pp. 109–135,
Wavelet Anal. Appl., 6, Academic Press, San Diego, CA, 1997.
2. C.Z. Cheng, G. Knorr, The integration of the Vlasov equation in configuration
space. J. Comput. Phys., 22 (1976), pp. 330–348.
3. A. Cohen, S.M. Kaber, S. Mueller and M. Postel, Fully adaptive multiresolu-
tion finite volume schemes for conservation laws, to appear in Mathematics of
Computation.
4. M.R. Feix, P. Bertrand, A. Ghizzo, Eulerian codes for the Vlasov equation,
Series on Advances in Mathematics for Applied Sciences, 22, Kinetic Theory
and Computing (1994), pp. 45–81.
5. F. Filbet, E. Sonnendrücker, Numerical methods for the Vlasov equation, these
proceedings.
6. A. Ghizzo, P. Bertrand, M. Shoucri, T.W. Johnston, E. Filjakow, M.R. Feix,
A Vlasov code for the numerical simulation of stimulated Raman scattering, J.
Comput. Phys., 90 (1990), no. 2, pp. 431–457.
7. M. Griebel, F. Koster, Adaptive wavelet solvers for the unsteady incompressible
Navier-Stokes equations, Advances in Mathematical Fluid Mechanics J. Malek
and J. Necas and M. Rokyta eds., Springer Verlag, (2000).
8. E. Sonnendrücker, J. Roche, P. Bertrand, A. Ghizzo, The Semi-Lagrangian
Method for the Numerical Resolution of Vlasov Equations, J. Comput. Phys.,
149 (1999), no. 2, pp. 201–220.
9. E. Sonnendrücker, J.J. Barnard, A. Friedman, D.P. Grote, S.M. Lund, Sim-
ulation of heavy ion beams with a semi-Lagrangian Vlasov solver, Nuclear
Instruments and Methods in Physics Research, Section A, 464, no. 1-3, (2001),
pp. 653–661.
10. S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for
fluids, J. Comput. Phys., 31 (1979), no. 3, pp. 335–362.
An adaptive numerical method for the Vlasov equation based on a multiresolution analysis
N. Besse, F. Filbet, M. Gutnic, I. Paun , E. Sonnendrücker
|
0704.1596 | Turbulence and the Navier-Stokes equations | Turbulence and the Navier-Stokes Equations
R. M. Kiehn
Emeritus Professor of Physics, University of Houston
Retired to Mazan, France
http://www.cartan.pair.com
Abstract: The concept of continuous topological evolution, based upon Car-
tan’s methods of exterior differential systems, is used to develop a topological
theory of non-equilibrium thermodynamics, within which there exist processes
that exhibit continuous topological change and thermodynamic irreversibility.
The technique furnishes a universal, topological foundation for the partial differ-
ential equations of hydrodynamics and electrodynamics; the technique does not
depend upon a metric, connection or a variational principle. Certain topological
classes of solutions to the Navier-Stokes equations are shown to be equivalent to
thermodynamically irreversible processes.
Prologue
THE POINT OF DEPARTURE
This presentation summarizes a portion of 40 years of research interests1 in applied
physics from a perspective of continuous topological evolution. The motivation for the
past and present effort continues to be based on the recognition that topological evolution
(not geometrical evolution) is required if non-equilibrium thermodynamic systems and irre-
versible turbulent processes are to be understood without the use of statistics. This essay
is written (by an applied physicist) as an alternative response to the (more mathematical)
challenge of the Clay Institute regarding the properties of the Navier-Stokes equations and
their relationship to hydrodynamic turbulence. To replicate a statement made by the Clay
Institute:
”The challenge is to make substantial progress toward a mathematical theory
which will unlock the secrets hidden in the Navier-Stokes equations.”
1This work is summarized in a series of reference monographs [40], [41], [42], [43], [44] which have been
constructed and updated from numerous publications. These volumes contain many examples and proofs of
the basic concepts.
http://arxiv.org/abs/0704.1596v1
http://www.cartan.pair.com
The point of departure starts with a topological (not statistical) formulation of Thermo-
dynamics, which furnishes a universal foundation for the Partial Differential Equations of
classical hydrodynamics and electrodynamics [40]. The topology that is of significance is
defined in terms of Cartan’s topological structure [34], which can be constructed from an ex-
terior differential 1-form, A, defined on a pre-geometric domain of base variables. The topo-
logical method extends the classical geometrical approach to the study of non-equilibrium
thermodynamic systems.
Claim 1 The topological method permits the conclusion that among the solutions to the
Navier-Stokes equations there are C2 smooth, thermodynamically irreversible processes which
permit description of topological change and the decay of turbulence.
In addition, the method permits examples to be constructed showing the difference be-
tween certain piecewise-linear processes which are reversible, but which are different from
certain smooth processes which are irreversible. Such concepts (of smoothness) seem to be
of direct interest to the challenge of the Clay Institute, and are to be associated with the
fact that there are differences between piecewise linear, smooth, and topological manifolds
(see p. 106, [38]).
However, the methods of topological thermodynamics go well beyond these types of
questions. In particular, the methods permit non-statistical engineering design criteria to
be developed for non-equilibrium systems. The theory of Topological Thermodynamics,
based upon Continuous Topological Evolution [35] of Cartan’s topological structure, can
explain why topologically coherent, compact structures, far from equilibrium, will emerge
as long-lived artifacts of thermodynamically irreversible, turbulent, continuous processes. I
want to present the idea that:
Theorem 2 The Pfaff Topological Dimension (PTD) of a Thermodynamic System can
change dynamically and continuously via irreversible dissipative processes from a non-equilibrium
turbulent state of PTD = 4 to an excited “topologically stationary, but excited,” state of PTD
= 3, which is still far from equilibrium! The PTD=3 state admits an extremal Hamiltonian
evolutionary process which, if dominant, produces a relatively long lifetime.
There exist C2 smooth processes that can describe the topological evolution from an
Open non-equilibrium turbulent domain of Pfaff Topological Dimension 4 to Closed, but
non-equilibrium, domains of Pfaff Topological Dimension 3, and ultimately to equilibrium
domains of Pfaff dimension 2 or less. The Topological domains of Pfaff Topological Di-
mension 3 emerge via thermodynamically irreversible, dissipative processes as topologically
coherent, deformable defects, embedded in the turbulent environment of Pfaff Topological
Dimension 4.
Now I am well aware of the fact that Thermodynamics (much less Topological Thermo-
dynamics) is a topic often treated with apprehension. In addition, I must confess, that as
undergraduates at MIT we used to call the required physics course in Thermodynamics, The
Hour of Mystery! Let me present a few quotations (taken from Uffink, [39]) that describe
the apprehensive views of several very famous scientists:
Any mathematician knows it is impossible to understand an elementary course
in thermodynamics ....... V. Arnold 1990.
It is always emphasized that thermodynamics is concerned with reversible pro-
cesses and equilibrium states, and that it can have nothing to do with irreversible
processes or systems out of equilibrium ......Bridgman 1941
No one knows what entropy really is, so in a debate (if you use the term entropy)
you will always have an advantage ...... Von Neumann (1971)
On the other hand Uffink states:
Einstein, ..., remained convinced throughout his life that thermodynamics is the
only universal physical theory that will never be overthrown.
I wish to demonstrate that from the point of view of Continuous Topological Evolution
(which is based upon Cartan’s theory of exterior differential forms) many of the mysteries of
non-equilibrium thermodynamics, irreversible processes, and turbulent flows, can be resolved.
In addition, the non-equilibrium methods can lead to many new processes and patentable
devices and concepts.
There are many intuitive, yet disputed, definitions of what is meant by turbulence, but
the one property of turbulence that everyone agrees upon is that turbulent evolution in a
fluid is a thermodynamic irreversible process. Isolated or equilibrium thermodynamics can
be defined on a 4D space-time variety in terms of a connected Cartan topology of Pfaff
Topological Dimension 2 or less. Non-equilibrium thermodynamics can be constructed in
terms of disconnected Cartan topology of Pfaff Topological Dimension of 3 or more. As
irreversibility requires a change in topology, the point of departure for this article will be to
use the thermodynamic theory of continuous topological evolution in 4D space-time. It will
be demonstrated, by example, that the non-equilibrium component of the Cartan topology
can support topological change, thermodynamic irreversible processes and turbulent solu-
tions to the Navier-Stokes equations, while the equilibrium topological component cannot.
In addition, it will be demonstrated that complex isotropic macroscopic Spinors are the
source of topological fluctuations and irreversible processes in the topological dynamics of
non-equilibrium systems. This, perhaps surprising, fact has been ignored by almost all re-
searchers in classical hydrodynamics who use classic real vector analysis and symmetries to
produce conservation laws, which do not require Spinor components. The flaw in such sym-
metrical based theories is that they describe evolutionary processes that are time reversible.
Time irreversibility requires topological change.
EXTERIOR DIFFERENTIAL FORMS
overcomes the
LIMITATIONS of REAL VECTOR ANALYSIS
During the period 1965-1992 it became apparent that new theoretical foundations were
needed to describe non-equilibrium systems and continuous irreversible processes - which
require topological (not geometrical) evolution. I selected Cartan’s methods of exterior
differential topology to encode Continuous Topological Evolution. The reason for this
choice is that many years of teaching experience indicated that such methods were rapidly
learned by all research scientists and engineers. In short:
1. Vector and Tensor analysis is not adequate to study the evolution of topology. The
tensor constraint of diffeomorphic equivalences implies that the topology of the initial
state must be equal to the topology of the final state. Turbulence is a thermodynamic,
irreversible process which can not be described by tensor fields alone.
2. However, Cartan’s methods of exterior differential systems and the topological perspec-
tive of Continuous Topological Evolution (not geometrical evolution) CAN be used to
construct a theory of non-equilibrium thermodynamic systems and irreversible pro-
cesses.
3. Bottom Line: Exterior differential forms carry topological information and can be
used to describe topological change induced by processes. Real ”Vector” direction-
fields alone cannot describe processes that cause topological change; but Spinor direc-
tionfields can.
A cornerstone of classic Vector (tensor) analysis is the constraint of functional equivalence
with respect to diffeomorphisms. However, diffeomorphisms are a subset of a homeomor-
phisms, and homeomorphisms preserve topology. Hence to study topological change, Vector
(tensor) analysis is not adequate. In topological thermodynamics, processes are defined in
terms of directionfields which may or may not be tensors. The ubiquitous concepts of 1-1 dif-
feomorphic equivalence, and non-zero congruences, for the eigen directionfields of symmetric
matrices do not apply to the eigen directionfields of antisymmetric matrices. The eigen di-
rection fields of antisymmetric matrices (which are equivalent to Cartan’s isotropic Spinors)
may be used to define components of a thermodynamic process, but such Spinors have a null
congruence (zero valued quadratic form), admit chirality, and are not 1-1. Where classic
geometric evolution is described in terms of symmetries and conservation laws, topological
evolution is described in terms of antisymmetries.
Cartan’s theory of exterior differential forms is built over completely antisymmetric struc-
tures, and therefore is the method of choice for studying topological evolution. The exterior
differential defines limit sets; the Lie differential defines continuous topological evolution.
The concept of Spinors arise naturally in theories using Cartan’s methods of exterior dif-
ferential forms; i.e., Spinors are not added to the theory ad hoc. The Cartan theory of
extended differential forms can be used to study topological change. The word extended is
used to emphasize the fact that differential forms are functionally well defined with respect a
larger class of transformations than those used to define tensors. Extended differential forms
behave as scalars with respect to C1 maps which do not have an inverse, much less an inverse
Jacobian. Both the inverse map and the inverse Jacobian are required by a diffeomorphism.
The exterior differential form on the final state of such C1 non-invertible maps permits the
functional form of the differential form on the initial state to be functionally well defined in
a retrodictive, pullback sense - not just at a point, but over a neighborhood.
Theorem 3 Tensor fields can be neither retrodicted nor predicted in functional form by
maps that are not diffeomorphisms [14].
CONTINUOUS TOPOLOGICAL EVOLUTION
Objectives of CTE The objectives of the theory of Continuous Topological Evolution
are to:
1. Establish the long sought for connection between irreversible thermodynamic processes
and dynamical systems – without statistics!
2. Demonstrate the connection between thermodynamic irreversibility and Pfaff Topolog-
ical Dimension equal to 4. The result suggests that “2-D Turbulence is a myth” for it
is a thermodynamic system of Pfaff Topological Dimension equal to 3 [21].
3. Demonstrate that topological thermodynamics leads to universal topological equiva-
lences between Electromagnetism, Hydrodynamics, Cosmology, and Topological Quan-
tum Mechanics.
4. Demonstrate that Cartan’s methods of exterior differential forms permits important
topological concepts to be displayed in a useful, engineering format.
New Concepts deduced from CTE The theory of Continuous Topological Evolution
introduces several new important concepts that are not apparent in a geometric equilibrium
analysis.
1. Continuous Topological Evolution is the dynamical equivalent of the FIRST LAW OF
THERMODYNAMICS.
2. The Pfaff Topological Dimension, PTD, is a topological property associated with any
Cartan exterior differential 1-form, A. The PTD can change via topologically contin-
uous processes.
3. Topological Torsion is a 3-form (on any 4D geometrical domain) that can be used to
describe irreversible processes. As a 4D non-equilibrium direction field it is completely
determined by the coefficient functions that encode the thermodynamic system. Other
process directionfields are determined by the system topology based upon the 1-form
of Action, A, and the refinement based on the topology of the 1-form of work, W .
4. Closed thermodynamic topological defects of Pfaff Topological Dimension 3 can emerge
from Open thermodynamic systems of Pfaff Topological Dimension 4 by means of irre-
versible dissipative processes that represent topological evolution and change. When
the topologically coherent defect structures emerge, their evolution can be dominated
by a Hamiltonian component (modulo topological fluctuations), which maintains the
topological deformation invariance, and yields hydrodynamic wakes [20] and other Soli-
ton structures. These objects are of Pfaff Topological Dimension 3 and are far from
equilibrium. They behave as if they were ”stationary excited” states above the equi-
librium ground state. Falaco Solitons are an easily reproduced hydrodynamic example
that came to my attention in 1986 [41] [33] .
PRESENTATION OUTLINE
The essay is constructed in several sections:
Section 1. Topological Thermodynamics In Section 1, the concepts of topological
thermodynamics in a space-time variety are reviewed (briefly) in terms of Cartan’s method
of exterior differential forms. A thermodynamic system is encoded in terms of a 1-form
of Action, A. Thermodynamic processes are encoded in terms of the Lie differential with
respect to a directionfield, V , acting on the 1-form, A, to produce a 1-form, Q. The process
directionfield can have Vector and Spinor components. The definition of the Lie differential
is a statement of cohomology and defines Q as the composite of a 1-form, W , and a perfect
differential, dU . The formula abstractly represents a dynamical version of the First Law of
Thermodynamics.
The existence of a 1-form on a 4D space-time variety generates a Cartan topology. If
the Pfaff Topological (not geometrical) Dimension of the 1-form of Action, A, is 2 or less,
then the thermodynamic system is an isolated or equilibrium system on the 4D variety. If
the Pfaff Topological Dimension of A is greater than 3, then the system is a non-equilibrium
system on the 4D variety. Examples of systems of Pfaff Topological Dimension 4 which admit
processes which are thermodynamically irreversible are given in the reference monographs
(see footnote page 1).
Section 2. Applications In Section 2, the abstract formalism will be given a specific real-
ization appropriate for fluids in general. First, an electromagnetic format will be described
because my teaching experience has demonstrated that the concepts of non-equilibrium
phenomena are more readily recognized in an electromagnetic format. Then it will be
demonstrated how the PDE’s representing the Hamiltonian version of the hydrodynamic
Lagrange-Euler equations arise from the constraint that the work 1-form, W , should vanish
(Pfaff Topological Dimension of W = 0). The Bernoulli flow will be obtained by constrain-
ing the thermodynamic Work 1-form to be exact, W = dΘ (Pfaff Topological Dimension
1), and the Helmholtz flow will follow from the constraint that the thermodynamic Work
1-form be closed, but not necessarily exact, dW = 0. Such reversible dynamical processes
belong to the connected component of the Work 1-form; irreversible processes belong to the
disconnected component of the Work 1-form.
Section 3. The Navier-Stokes system In Section 3, the topological constraints of
isolated equilibrium systems will be relaxed to produce more general PDE’s defining the
topological evolution of the system relative to an applied process. These relaxed topolog-
ical constraints will include the partial differential equations known as the Navier-Stokes
equations. The method used will be to augment the topology induced by the 1-form of
Action, A, by studying the topological refinements induced by the 1-form of Work, W . It
will be demonstrated that when the Pfaff Topological Dimension of A and W and Q are 4,
there exist C2 solutions (processes) to the Navier-Stokes equations which are thermodynam-
ically irreversible (the most significant property of turbulent flow). An interesting result is
the set of conditions on solutions of the Navier-Stokes equations that produce an adiabatic
irreversible flow.
Those topological refinements of the Work 1-form, required to include the Navier-Stokes
equations, can be related directly to the concept of macroscopic Spinors. Macroscopic,
complex Spinor solutions occur naturally in terms of the eigendirection fields of (real) anti-
symmetric matrices with non-zero eigenvalues, whenever the thermodynamic Work 1-form
is not zero. Spinors can also be associated with topological fluctuations of position and
velocity about kinematic perfection generated by 1-parameter groups. These topological
fluctuations are presumed to be representations of pressure and temperature.
Section 4. Closed States of Topological Coherence embedded as deformable
defects in Turbulent Domains One of the key interests of the Clay problem has to
do with the smoothness of the solutions to the Navier-Stokes equations. In Section 4, the
problem will be attacked from the point of view of thermodynamics. First, the properties
of the different species of topological defects will be discussed. These defects are non-
equilibrium closed domains (of PTD = 3) which can emerge by C2 smooth irreversible
process in open domains (of PTD = 4), as excited states far from equilibrium, yet with long
relative lifetimes. Falaco Solitons are an easily reproduced experimental example of such
topological defects, and are discussed in detail in [41].
The properties of two different species of PTD = 3 defect domains will be given in detail.
In addition, an analytic solution of a thermodynamically irreversible process that causes the
defect domain to emerge will be displayed.
Finally, an example will be given where by combinations of Spinor solutions produce piece-
wise linear processes. These piecewise linear processes are thermodynamically reversible,
while the Spinor solutions of which they are composed are not.
Section 5. Topological Fluctuations and Spinors In Section 5, a few concluding
remarks will be made about the ongoing research concerning topological fluctuations, as
generated by Spinors. The methods of fiber bundle theory are used extend the 4D thermo-
dynamic domain. Such topological fluctuations can be associated with fluid pressure and
temperature.
1 Topological Thermodynamics
1.1 The Axioms of Topological Thermodynamics
The topological methods used herein are based upon Cartan’s theory of exterior differential
forms. The thermodynamic view assumes that the physical systems to be studied can
be encoded in terms of a 1-form of Action Potentials (per unit source, or, per mole), A,
on a four-dimensional variety of ordered independent variables, {ξ1, ξ2, ξ3, ξ4}. The variety
supports a differential volume element Ω4 = dξ
1ˆdξ2ˆdξ3ˆdξ4. This statement implies that
the differentials of the µ = 4 base variables are functionally independent. No metric, no
connection, no constraint of gauge symmetry is imposed upon the four-dimensional pre-
geometric variety. Topological constraints can be expressed in terms of exterior differential
systems placed upon this set of base variables [1].
In order to make the equations more suggestive to the reader, the symbolism for the
variety of independent variables will be changed to the format {x, y, z, t}, but be aware that
no constraints of metric or connection are imposed upon this variety, at this, thermodynamic,
level. For instance, it is NOT assumed that the variety is spatially Euclidean.
With this notation, the Axioms of Topological Thermodynamics can be summarized as:
Axiom 1. Thermodynamic physical systems can be encoded in terms of a 1-
form of covariant Action Potentials, Aµ(x, y, z, t...), on a four-dimensional ab-
stract variety of ordered independent variables, {x, y, z, t}. The variety supports
differential volume element Ω4 = dxˆdyˆdzˆdt.
Axiom 2. Thermodynamic processes are assumed to be encoded, to within a
factor, ρ(x, y, z, t...), in terms of a contravariant Vector and/or complex Spinor
directionfields, symbolized as V4(x, y, z, t).
Axiom 3. Continuous Topological Evolution of the thermodynamic system can
be encoded in terms of Cartan’s magic formula (see p. 122 in [10]). The Lie
differential with respect to the process, ρV4, when applied to an exterior differen-
tial 1-form of Action, A = Aµdx
µ, is equivalent, abstractly, to the first law of
thermodynamics.
Cartan’s Magic Formula L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (1)
First Law : W + dU = Q, (2)
Inexact Heat 1-form Q = W + dU = L(ρV4)A, (3)
Inexact Work 1-form W = i(ρV4)dA, (4)
Internal Energy U = i(ρV4)A. (5)
Axiom 4. Equivalence classes of systems and continuous processes can be de-
fined in terms of the Pfaff Topological Dimension and topological structure gen-
erated by of the 1-forms of Action, A, Work, W , and Heat, Q.
Axiom 5. If QˆdQ 6= 0, then the thermodynamic process is irreversible.
1.2 Cartan’s Magic Formula ≈ First Law of Thermodynamics
The Lie differential (not Lie derivative) is the fundamental generator of Continuous Topo-
logical Evolution. When acting on an exterior differential 1-form of Action, A = Aµdx
Cartan’s magic (algebraic) formula is equivalent abstractly to the first law of thermodynam-
L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A), (6)
= W + dU = Q. (7)
In effect, Cartan’s magic formula leads to a topological basis of thermodynamics, where
the thermodynamic Work, W , thermodynamic Heat, Q, and the thermodynamic internal
energy, U , are defined dynamically in terms of Continuous Topological Evolution. In effect,
the First Law is a statement of Continuous Topological Evolution in terms of deRham
cohomology theory; the difference between two non-exact differential forms is equal to an
exact differential, Q−W = dU .
My recognition (some 30 years ago) of this correspondence between the Lie differential
and the First Law of thermodynamics has been the corner stone of my research efforts in
applied topology.
It is important to realize that the Cartan formula is to be interpreted algebraically. Many
textbook presentations of the Cartan-Lie differential formula presume a dynamic constraint,
such that the vector field V4(x, y, z, t) be the generator of a single parameter group. If
true, then the topological constraint of Kinematic Perfection cn be established as an exterior
differential system of the format:
Kinematic Perfection : dxk −Vkdt⇒ 0. (8)
The topological constraint of Kinematic Perfection, in effect, defines (or presumes) a limit
process. This constraint leads to the concept of the Lie derivative2 of the 1-form A. The
evolution then is represented by the infinitesimal propagation of the 1-form, A, down the
2Professor Zbigniew Oziewicz told me that Slebodzinsky was the first to formulate the idea of the Lie
derivative in his thesis (in Polish).
flow lines generated by the 1-parameter group. Cartan called this set of flow lines ”the tube
of trajectories”.
However, such a topological, kinematic constraint is not imposed in the presentation found
in this essay; the directionfield, V4, may have multiple parameters. This observation leads
to the important concept of topological fluctuations (about Kinematic Perfection), such as
given by the expressions:
Topological : Fluctuations
(dxk −Vkdt) = ∆xk 6= 0, ( ∼ Pressure) (9)
(dV k −Akdt) = (∆Vk) 6= 0, ( ∼ Temperature) (10)
d(∆xk) = −(dVk −Akdt)ˆdt = −(∆Vk)ˆdt, (11)
In this context it is interesting to note that in Felix Klein’s discussions [4] of the development
of calculus, he says
”The primary thing for him (Leibniz) was not the differential quotient (the deriva-
tive) thought of as a limit. The differential, dx, of the variable x had for him
(Leibniz) actual existence...”
The Leibniz concept is followed throughout this presentation. It is important for the
reader to remember that the concept of a differential form is different from the concept of a
derivative, where a (topological) limit has been defined, thereby constraining the topological
evolution.
The topological methods to be described below go beyond the notion of processes which
are confined to equilibrium systems of kinematic perfection. Non-equilibrium systems and
processes which are thermodynamically irreversible, as well as many other classical ther-
modynamic ideas, can be formulated in precise mathematical terms using the topological
structure and refinements generated by the three thermodynamic 1-forms, A, W, and Q.
1.3 The Pfaff Sequence and the Pfaff Topological Dimension
1.3.1 The Pfaff Topological Dimension of the System 1-form, A
It is important to realize that the Pfaff Topological Dimension of the system 1-form of Action,
A, determines whether the thermodynamic system is Open, Closed, Isolated or Equilibrium.
Also, it is important to realize that the Pfaff Topological Dimension of the thermodynamic
Work 1-form, W , determines a specific category of reversible and/or irreversible processes.
It is therefore of some importance to understand the meaning of the Pfaff Topological Di-
mension of a 1-form. Given the functional format of a general 1-form, A, on a 4D variety
it is an easy step to compute the Pfaff Sequence, using one exterior differential operation,
and several algebraic exterior products. For a differential 1-form, A, defined on a geometric
domain of 4 base variables, the Pfaff Sequence is defined as:
Pfaff Sequence {A, dA,AˆdA, dAˆdA...} (12)
It is possible that over some domains, as the elements of the sequence are computed, one of
the elements (and subsequent elements) of the Pfaff Sequence will vanish. The number of
non-zero elements in the Pfaff Sequence (PS) defines the Pfaff Topological Dimension (PTD)
of the specified 1-form3. Modulo singularities, the Pfaff Topological Dimension determines
the minimum number M of N functions of base variables (N ≥ M) required to define the
topological properties of the connected component of the 1-form A.
The Pfaff Topological Dimension of the 1-form of Action, A, can be put into correspon-
dence with the four classic topological structures of thermodynamics. Equilibrium, Isolated,
Closed, and Open systems. The classic thermodynamic interpretation is that the first two
structures do not exchange mass (mole numbers) or radiation with their environment. The
Closed structure can exchange radiation with its environment but not mass (mole numbers).
The Open structure can exchange both mass and radiation with its environment. The fol-
lowing table summarizes these properties. For reference purposes, I have given the various
elements of the Pfaff sequence specific names:
Topological
p-form name
element
Nulls PTD
Thermodynamic
system
Action A dA = 0 1 Equilibrium
Vorticity dA AˆdA = 0 2 Isolated
Torsion AˆdA dAˆdA = 0 3 Closed
Parity dAˆdA − 4 Open
Table 1 Applications of the Pfaff Topological Dimension.
The four thermodynamic systems can be placed into two disconnected topological cate-
gories. If the Pfaff Topological Dimension of A is 2 or less, the first category is determined
by the closure (or differential ideal) of the 1-form of Action, A∪ dA. This Cartan topology
is a connected topology. In the case that the Pfaff Topological Dimension is greater than
2, the Cartan topology is based on the union of two closures, {A ∪ dA ∪ AˆdA ∪ dAˆdA},
and is a disconnected topology.
3The Pfaff Topological dimension has been called the ”class” of a 1-form in the old literature. I prefer
the more suggestive name.
It is a topological fact that there exists a (topologically) continuous C2 process from a
disconnected topology to a connected topology, but there does not exist a C2 continuous
process from a connected topology to a disconnected topology. This fact implies that
topological change can occur continuously by a ”pasting” processes representing the decay
of turbulence by ”condensations” from non-equilibrium to equilibrium systems. On the other
hand, the creation of Turbulence involves a discontinuous (non C2) process of ”cutting” into
parts. This warning was given long ago [19] to prove that computer analyses that smoothly
match value and slope will not replicate the creation of turbulence, but can faithfully replicate
the decay of turbulence.
1.3.2 The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W
The topological structure of the thermodynamic Work 1-form, W , can be used to refine the
topology of the physical system; recall that the physical system is encoded by the Action
1-form, A.
Claim 4 The PDE’s that represent the system dynamics are determined by the Pfaff Topo-
logical Dimension of the 1-form of Work, W , and the 1-form of Action, A, that encodes the
physical system.
The Pfaff Topological Dimension of the thermodynamic Work 1-form depends upon both
the physical system, A, and the process, V4. In particular if the Pfaff Dimension of the
thermodynamic Work 1-form is zero, (W = 0), then system dynamics is generated by an
extremal vector field which admits a Hamiltonian realization. However, such extremal
direction fields can occur only when the Pfaff Topological Dimension of the system encoded
by A is odd, and equal or less than the geometric dimension of the base variables.
For example, if the geometric dimension is 3, and the Pfaff Topological Dimension of
A is 3, then there exists a unique extremal field on the Contact manifold defined by dA.
This unique directionfield is the unique eigen directionfield of the 3x3 antisymmetric matrix
(created by the 2-form F = dA) with eigenvalue equal to zero.
If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 3, then
there exists a two extremal fields on the geometric manifold. These directionfields are those
generated as the eigen directionfields of the 4x4 antisymmetric matrix (created by the 2-form
F = dA) with eigenvalue equal to zero.
If the geometric dimension is 4, and the Pfaff Topological Dimension of A is 4, then there
do not exist extremal fields on the Symplectic manifold defined by dA. All of the eigen
directionfields of the 4x4 antisymmetric matrix (created by the 2-form F = dA) are complex
isotropic spinors with pure imaginary eigenvalues not equal to zero.
1.4 Topological Torsion and other Continuous Processes.
1.4.1 Reversible Processes
Physical Processes are determined by directionfields4 with the symbol, V4, to within an
arbitrary function, ρ. There are several classes of direction fields that are defined as follows
Associated Class:i(ρV4)A = 0, (13)
Extremal Class:i(ρV4)dA = 0, (14)
Characteristic Class:i(ρV4)A = 0, (15)
and : i(ρV4)dA = 0, (16)
Helmholtz Class: d(i(ρV4)dA) = 0, (17)
Extremal Vectors (relative to the 1-form of Action, A) produce zero thermodynamic work,
W = i(ρV4)dA = 0, and admit a Hamiltonian representation. Associated Vectors (relative
to the 1-form of Action, A) can be adiabatic if the process remains orthogonal to the 1-
form, A. Helmholtz processes (which include Hamiltonian processes, Bernoulli processes
and Stokes flow) conserve the 2-form of Topological vorticity, dA. All such processes are
thermodynamically reversible. Many examples of these systems are detailed in the reference
monographs (see footnote on page 1).
1.4.2 Irreversible Processes
There is one directionfield that is uniquely defined by the coefficient functions of the 1-form,
A, that encodes the thermodynamic system on a 4D geometric variety. This vector exists
only in non-equilibrium systems, for which the Pfaff Topological Dimension of A is 3 or 4.
This 4 vector is defined herein as the topological Torsion vector, T4. To within a factor, this
directionfield5 has the four coefficients of the 3-form AˆdA, with the following properties:
4Which include both vector and spinor fields.
5A direction field is defined by the components of a vector field which establish the ”line of action” of
the vector in a projective sense. An arbitrary factor times the direction field defines the same projective
line of action, just reparameterized. In metric based situations, the arbitrary factor can be interpreted as a
renormalization factor.
Properties of :Topological Torsion T4 on Ω4 (18)
i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt = AˆdA, (19)
W = i(T4)dA = σ A, (20)
dW = dσˆA+ σdA = dQ (21)
U = i(T4)A = 0, T4 is associative (22)
i(T4)dU = 0 (23)
i(T4)Q = 0 T4 is adiabatic (24)
L(T4)A = σ A, T4 is homogeneous (25)
L(T4)dA = dσˆA+ σdA = dQ, (26)
QˆdQ = L(T4)AˆL(T4)dA = σ
2AˆdA 6= 0, (27)
dAˆdA = d(AˆdA) = d{(i(T4)Ω4} = (div4T4)Ω4, (28)
L(T4)Ω4 = d{(i(T4)Ω4} = (2σ)Ω4, (29)
If the Pfaff Topological Dimension of A is 4 (an Open thermodynamic system), then T4
has a non-zero 4 divergence, (2σ), representing an expansion or a contraction of the 4D
volume element Ω4. The Heat 1-form, Q, generated by the process, T4, is NOT integrable.
Q is of Pfaff Topological Dimension greater that 2, when σ 6= 0. Furthermore the T4 process
is locally adiabatic as the change of internal energy in the direction of the process path is
zero. Therefore, in the Pfaff Topological Dimension 4 case, where dAˆdA 6= 0, the T4
direction field represents an irreversible, adiabatic process.
When σ is zero and dσ = 0, but AˆdA 6= 0, the Pfaff Topological Dimension of the
system is 3 (a Closed thermodynamic system). In this case, the T4 direction field becomes
a characteristic vector field which is both extremal and associative, and induces a Hamilton-
Jacobi representation (the ground state of the system for which dQ = 0).
For any process and any system, equation ( 27) can be used as a test for irreversibility.
It seems a pity, that the concept of the Topological Torsion vector and its association
with non-equilibrium systems, where it can be used to establish design criteria to minimize
energy dissipation, has been ignored by the engineering community.
1.4.3 The Spinor class
It is rather remarkable (and only fully appreciated by me in February, 2005) that there is a
large class of direction fields useful to the topological dynamics of thermodynamic systems
(given herein the symbol ρS4) that do not behave as vectors (with respect to rotations).
They are isotropic complex vectors of zero length, defined as Spinors by E. Cartan [2], but
which are most easily recognized as the eigen directionfields relative to the antisymmetric
matrix, [F ], generated by the component of the 2-form F = dA:
The Spinor Class [F ] ◦ |ρS4〉 = λ |ρS4〉 6= 0, (30)
〈ρS4| ◦ |ρS4〉 = 0, λ 6= 0 (31)
In the language of exterior differential forms, if the Work 1-form is not zero, the process
must contain Spinor components:
W = i(ρS4)dA 6= 0 (32)
As mentioned above, Spinors have metric properties, behave as vectors with respect to
transitive maps, but do not behave as vectors with respect to rotations (see p. 3, [2]).
Spinors generate harmonic forms and are related to conjugate pairs of minimal surfaces.
The notation that a Spinor is a complex isotropic directionfield is preferred over the names
”complex isotropic vector”, or ”null vector” that appear in the literature. As shown below,
the familiar formats of Hamiltonian mechanical systems exclude the concept of Spinor process
directionfields, for the processes permitted are restricted to be represented by direction fields
of the extremal class, which have zero eigenvalues.
Remark 5 Spinors are normal consequences of antisymmetric matrices, and, as topological
artifacts, they are not restricted to physical microscopic or quantum constraints. According
to the topological thermodynamic arguments, Spinors are implicitly involved in all processes
for which the 1-form of thermodynamic Work is not zero. Spinors play a role in topological
fluctuations and irreversible processes.
The thermodynamic Work 1-form, W , is generated by a completely antisymmetric 2-
form, F , and therefore, if not zero, must have Spinor components. In the odd dimensional
Contact manifold case there is one eigen Vector, with eigenvalue zero, which generates the
extremal processes that can be associated with a Hamiltonian representation. The other
two eigendirection fields are Spinors. In the even dimensional Symplectic manifold case,
any non-zero component of work requires that the evolutionary directionfields must contain
Spinor components. All eigen directionfields on symplectic spaces are Spinors.
The fundamental problem of Spinor components is that there can be more than one Spinor
direction field that generates the same geometric path. For example, there can be Spinors
of left or right handed polarizations and Spinors of expansion or contraction that produce
the same optical (null congruence) path. This result does not fit with the classic arguments
of mechanics, which require unique initial data to yield unique paths. Furthermore, the
concept of Spinor processes can annihilate the concept of time reversal symmetry, inherent
in classical hydrodynamics. The requirement of uniqueness is not a requirement of non-
equilibrium thermodynamics, where Spinor ”entanglement” has to be taken into account.
1.5 Emergent Topological Defects
Suppose an evolutionary process starts in a domain of Pfaff Topological Dimension 4, for
which a process in the direction of the Topological Torsion vector, T4 , is known to represent
an irreversible process. Examples can demonstrate that the irreversible process can proceed
to a domain of the geometric variety for which the dissipation coefficient, σ, becomes zero.
Physical examples [41] such as the skidding bowling ball proceed with irreversible dissipation
(PTD = 6) until the ”no-slip” condition is reached (PTD = 5). In fluid systems the
topological defects can emerge as long lived states far from equilibrium. The process is
most simply visualized as a ”condensation” from a turbulent gas, such as the creation of
a star in the model which presumes the universe is a very dilute, turbulent van der Waals
gas near its critical point. The red spot of Jupiter, a hurricane, the ionized plasma ring
in a nuclear explosion, Falaco Solitons, the wake behind an aircraft are all exhibitions of
the emergence process to long lived topological structures far from equilibrium. It is most
remarkable that the emergence of these experimental defect structures occurs in finite time.
The idea is that a subdomain of the original system of Pfaff Topological Dimension 4 can
evolve continuously with a change of topology to a region of Pfaff Topological Dimension
3. The emergent subdomain of Pfaff Topological Dimension 3 is a topological defect, with
topological coherence, and often with an extended lifetime (as a soliton structure with a
dominant Hamiltonian evolutionary path), embedded in the Pfaff dimension 4 turbulent
background.
The Topological Torsion vector in a region of Pfaff Topological Dimension 3 is an extremal
vector direction field in systems of Pfaff Topological Dimension 3; it then has a zero 4D
divergence, and leaves the volume element invariant. Moreover the existence of an extremal
direction field implies that the 1-form of Action can be given a Hamiltonian representation,
k + H(P, q, t)dt. In the domain of Pfaff dimension 3 for the Action, A, the subse-
quent continuous evolution of the system, A, relative to the process T4, can proceed in an
energy conserving, Hamiltonian manner, representing a ”stationary” or ”excited” state far
from equilibrium (the ground state). This argument is based on the assumption that the
Hamiltonian component of the direction field is dominant, and any Spinor components in
the PTD = 3 domain, representing topological fluctuations, can be ignored. These excited
states, far from equilibrium, can be interpreted as the evolutionary topological defects that
emerge and self-organize due to irreversible processes in the turbulent dissipative system of
Pfaff dimension 4.
The descriptive words of self-organized states far from equilibrium have been abstracted
from the intuition and conjectures of I. Prigogine [9]. The methods of Continuous Topolog-
ical Evolution correct the Prigogine conjecture that ”dissipative structures” can be caused
by dissipative processes and fluctuations. The long-lived excited state structures created
by irreversible processes are non-equilibrium, deformable topological defects almost void
of irreversible dissipation. The topological theory presented herein presents for the first
time a solid, formal, mathematical justification (with examples) for the Prigogine conjec-
tures. Precise definitions of equilibrium and non-equilibrium systems, as well as reversible
and irreversible processes can be made in terms of the topological features of Cartan’s ex-
terior calculus. Using Cartan’s methods of exterior differential systems, thermodynamic
irreversibility and the arrow of time can be well defined in a topological sense, a technique
that goes beyond (and without) statistical analysis [23]. Thermodynamic irreversibility and
the arrow of time requires that the evolutionary process produce topological change.
2 Applications
2.1 An Electromagnetic format
The thermodynamic identification of the terms in Cartan’s magic formula are not whimsical.
To establish an initial level of credence in the terminology, consider the 1-form of Action, A,
where the component functions are the symbols representing the familiar vector and scalar
potentials in electromagnetic theory. The coefficient functions have arguments over the four
independent variables {x, y, z, t},
A = Aµ(x, y, z, t)dx
µ = A ◦ dr− φ dt. (33)
Construct the 2-form of field intensities as the exterior differential of the 1-form of Action,
F = dA = (∂Ak/∂x
j − ∂Aj/∂xk)dxjˆdxk (34)
= Fjkdx
jˆdxk = +Bzdxˆdy...+ Exdxˆdt... . (35)
The engineering variables are defined as electric and magnetic field intensities:
E = −∂A/∂t− grad φ, B = curl A. (36)
Relative to the ordered set of base variables, {x, y, z, t}, define a process directionfield,
ρV4, as a 4-vector with components, [V, 1], with a scaling factor, ρ.
ρ[V4] = ρ[V, 1]. (37)
Note that this direction field can be used to construct a useful 3-form of (matter) current,
C, in terms of the 4-volume element, Ω4 = dxˆdyˆdzˆdt :
C = i(ρV4)dxˆdyˆdzˆdt = i(C4)Ω4. (38)
The process 3-form, C, is not necessarily the same as electromagnetic charge current density
3-form of electromagnetic theory, J . The 4-divergence of C, need not be zero: dC 6= 0.
Using the above expressions, the evaluation of the thermodynamic work 1-form in terms
of 3-vector engineering components becomes:
The thermodynamic Work 1-form: W = i(ρV4)dA = i(ρV4)F, (39)
⇒ −ρ{E+V ×B} ◦ dr+ ρ{V ◦ E}dt (40)
= −ρ{fLorentz} ◦ dr+ ρ{V ◦ E}dt. (41)
The Lorentz force = −{fLorentz} ◦ dr (spatial component) (42)
The dissipative power = +{V ◦ E}dt (time component). (43)
For those with experience in electromagnetism, note that the construction yields the
format, automatically and naturally, for the ”Lorentz force” as a derivation consequence,
without further ad hoc assumptions. The dot product of a 3 component force, fLorentz, and
a differential spatial displacement, dr, defines the elementary classic concept of ”differential
work”. The 4-component thermodynamic Work 1-form,W , includes the spatial component
and a differential time component, Pdt, with a coefficient which is recognized to be the
”dissipative power”, P = {V ◦ E}. The thermodynamic Work 1-form, W , is not necessarily
a perfect differential, and therefore can be path dependent. Closed cycles of Work need not
be zero.
Next compute the Internal Energy term, relative to the process defined as ρV4:
Internal Energy: U = i(ρV4)A = ρ(V ◦A− φ). (44)
The result is to be recognized as the ”interaction” energy density in electromagnetic
plasma systems. It is apparent that the internal energy, U , corresponds to the interaction
energy of the physical system; that is, U is the internal stress energy of system deformation.
Therefore, the electromagnetic terminology can be used to demonstrate the premise that
Cartan’s magic formula is not just another way to state that the first law of thermodynamics
makes practical sense. The topological methods permit the long sought for integration of
mechanical and thermodynamic concepts, without the constraints of equilibrium systems,
and/or statistical analysis.
It is remarkable that although the symbols are different, the same basic constructions and
conclusions apply to many classical physical systems. The correspondence so established
between the Cartan magic formula acting on a 1-form of Action, and the first law of ther-
modynamics is taken both literally and seriously in this essay. The methods yield explicit
constructions for testing when a process acting on a physical system is irreversible. The
methods permit irreversible adiabatic processes to be distinguished from reversible adiabatic
processes, analytically. Adiabatic processes need not be ”slow” or quasi-static.
Given any 1-form, A, W, and/or Q, the concept of Pfaff Topological Dimension (for each
of the three 1-forms, A, W , Q) permits separation of processes and systems into equiva-
lence classes. For example, dynamical processes can be classified in terms of the topological
Pfaff dimension of the thermodynamic Work 1-form, W . All extremal Hamiltonian systems
have a thermodynamic Work 1-form, W , of topological Pfaff dimension of 1, (dW = 0).
Hamiltonian systems can describe reversible processes in non-equilibrium systems for which
the topological Pfaff dimension is 3. Such systems are topological defects whose topology is
preserved by the Hamiltonian dynamics, but all processes which preserve topology are re-
versible. In non-equilibrium systems, topological fluctuations can be associated with Spinors
of the 2-form, F = dA. Even if the dominant component of the process is Hamiltonian,
Spinor fluctuations can cause the system (ultimately) to decay.
2.1.1 Topological 3-forms and 4-forms in EM format
Construct the elements of the Pfaff Sequence for the EM notation,
{A, F = dA,AˆF, FˆF}, (45)
and note that the algebraic expressions of Topological Torsion, AˆF , can be evaluated in
terms of 4-component engineering variables T4 as:
Topological Torsion vector (46)
AˆF = i(T4)Ω4 = i(T4)dxˆdyˆdzˆdt (47)
T4 = [T, h] = −[E×A+Bφ,A ◦B]. (48)
The exterior 3-form, AˆF , with physical units of (~/unit mole)2, is not found (usually) in
classical discussions of electromagnetism6.
If T4 is used as to define the direction field of a process, then
L(T4)A = σA, i(T4)A = 0. (49)
where 2σ = {div4(T4)} = 2(E ◦B). (50)
The important (universal) result is that if the acceleration associated with the direction
field, E, is parallel to the vorticity associated with the direction field, B, then according to
the equations starting with eq. (18) et. seq. the process is dissipatively irreversible. This
result establishes the design criteria for engineering applications to minimize dissipation from
turbulent processes.
The Topological Torsion vector has had almost no utilization in applications of classical
electromagnetic theory.
2.1.2 Topological Torsion quanta
The 4-form of Topological Parity, FˆF , can be evaluated in terms of 4-component engineering
variables as:
Topological Parity
d(AˆF ) = FˆF = {div4(T4)}Ω4 = {2E ◦B} Ω4. (51)
6The unit mole number is charge, e, in EM theory.
This 4-form is also known as the second Poincare Invariant of Electromagnetic Theory.
The fact that FˆF need not be zero implies that the Pfaff Topological Dimension of the
1-form of Action, A, must be 4, and therefore A represents a non-equilibrium Open thermo-
dynamic system. Similarly, if FˆF = 0, but AˆF 6= 0, then the Pfaff Topological Dimension
of the 1-form of Action, A, must be 3, and the physical system is a non-equilibrium Closed
thermodynamic system. When FˆF = 0, the corresponding three-dimensional integral of
the closed 3-form, AˆF , when integrated over a closed 3D-cycle, becomes a deRham period
integral, defined as the Torsion quantum. In other words, the closed integral of the (closed)
3-form of Topological Torsion becomes a deformation (Hopf) invariant with integral values
proportional to the integers.
Torsion quantum =
3D cycle
AˆF. (52)
On the other hand, topological evolution and transitions between ”quantized” states of
Torsion require that the respective Parity 4-form is are not zero. As,
L(T4)Ω4 = d{i(T4)Ω4} = (2σ) Ω4 = 2(E ◦B) Ω4 6= 0, (53)
it is apparent that the evolution of the differential volume element, Ω4, depends upon the
existence and colinearity of both the electric field, E, and the magnetic field, B. It is here
that contact is made with the phenomenological concept of ”4D bulk” viscosity = 2σ. It is
tempting to identify σ2 with the concept of entropy production. Note that the Topological
Torsion directionfield appears only in non-equilibrium systems. These results are universal
and can be used in hydrodynamic systems discussed in that which follows.
2.2 A Hydrodynamic format
2.2.1 The Topological Continuum vs. the Geometrical Continuum
In many treatments of fluid mechanics the (geometrical) continuum hypothesis is invoked
from the start. The idea is ”matter” occupies all points of the space of interest, and that
properties of the fluid can be represented by piecewise continuous functions of space and
time, as long as length and time scales are not too small. The problem is that at very small
scales, one has been led to believe the molecular or atomic structure of particles will become
evident, and the ”macroscopic” theory will breakdown. However, these problems of scale
are geometric issues, important to many applications, but not pertinent to a topological
perspective, where shape and size are unimportant. Suppose that the dynamics can be
formulated in terms of topological concepts which are independent from sizes and shapes.
Then such a theory of a Topological Continuum would be valid at all scales. Such is the
goal of this monograph.
Remark 6 However, one instance where ”scale” many have topological importance is as-
sociated with the example of a surface with a ”teeny” hole. If the hole, no matter what
its size, has a twisted ear (Moebius band) then the whole surface is non-orientable, no mat-
ter how ”small” the hole. Could it be that the world of the quantum is, in effect, that of
non-orientable defects embedded in an otherwise orientable manifold that originally had no
such defects. Note further the strong correspondence with Fermions with non-oriented (half-
integer) multiplet ribbons, and Bosons with oriented (integer) multiplet ribbons of both right
and left twists [44].
As will be developed below, the fundamental equations of exterior differential systems can
lead to field equations in terms of systems of Partial Differential Equations (PDE’s). The
format of the fundamental theory will be in terms of objects (exterior differential forms)
which, although composed of algebraic constructions of tensorial7 things, are in a sense
scalars (or pseudo scalars) that are homogeneous with respect to concepts of scale. The
theory then developed is applicable to hydrodynamics at all scales, from the microworld
to the cosmological arena. The ”breakdown” of the continuum model is not relevant.
The topological system may consist of many disconnected parts when the system is not in
thermodynamic equilibrium or isolation, and the parts can have topological obstructions
or defects, some of which can be used to construct period integrals that are topologically
”quantized”. Hence the ”quantization” of the micro-scaled geometric systems can have it
genesis in the non-equilibrium theory of thermodynamics. However, from the topological
perspective, the rational topological quantum values can also occur at all scales.
2.2.2 Topological Hydrodynamics
The axioms of Topological Thermodynamics are summarized in Section 1.1. For hydrody-
namics (or electrodynamics the axioms are essentially the same. Just exchange the word,
hydrodynamics (or electrodynamics) for the word, thermodynamics, in the formats of Section
By 1969 it had become evident to me that electromagnetism (without geometric con-
straints), when written in terms of differential forms, was a topological theory, and that the
concept of dissipation and irreversible processes required more than that offered by Hamil-
tonian mechanics. At that time I was interested in possible interactions of the gravitational
field and the polarizations of an electromagnetic signal. One of the first ideas discovered
7Relative to diffeomorphisms.
about topological electrodynamics was that there existed an intrinsic transport theorem [11]
that introduced the concept of what is now called Topological Spin, AˆG, into electromag-
netic theory [43]. As a transport theorem not recognized by classical electromagnetism,
the first publication was as a letter to Physics of Fluids. That started my interest in a
topological formulation of fluids.
It was not until 1974 that the Lie differential acting on exterior differential forms was es-
tablished as the key to the problem of intrinsically describing dissipation and the production
of topological defects in physical systems; but methods of visualization of such topological
defects in classical electrodynamics were not known [12]. It was hoped that something in
the more visible fluid mechanics arena would lend credence to the concepts of topological
defects. The first formulations of the PDE’s of fluid dynamics in terms of differential forms
and Cartan’s Magic formula followed quickly [13].
In 1976 it was argued that topological evolution was at the cause of turbulence in fluid
dynamics, and the notion of what is now called Topological Torsion, AˆF, became recognized
as an important concept. It was apparent that streamline flow imposed the constraint that
AˆF = 0 on the equations of hydrodynamics. Turbulent flow, being the antithesis of
streamline flow, must admit AˆF 6= 0. In 1977 it was recognized that topological defect
structures could become ”quantized” in terms of deRham period integrals [15], forming a
possible link between topology and both macroscopic and microscopic quantum physics.
The research effort then turned back to a study of topological electrodynamics in terms of
the dual polarized ring laser, where it was experimentally determined that the speed of an
electromagnetic signal outbound could be different from the speed of an electromagnetic
signal inbound: a topological result not within the realm of classical theory.
Then in 1986 the long sought for creation and visualization of topological defects in
fluids [16] became evident. The creation of Falaco Solitons in a swimming pool was the
experiment that established credence in the ideas of what had, by that time, become a
theory of continuous topological evolution. It was at the Cambridge conference in 1989
[17] that the notions of topological evolution, hydrodynamics and thermodynamics were put
together in a rudimentary form, but it was a year later at the Permb conference in 1990
[18] that the ideas were well established. The Permb presentation also suggested that the
ambiguous (at that time) notion of coherent structures in fluids could be made precise in
terms of topological coherence. A number of conference presentations followed in which
the ideas of continuous thermodynamic irreversible topological evolution in hydrodynamics
were described [19], but the idea that the topological methods of thermodynamics could be
used to distinguish non-equilibrium processes and non-equilibrium systems and irreversible
processes with out the use of statistics slowly came into being in the period 1985-2005 [22].
These efforts have been summarized in [40], and a collection of the old publications appears
in [45].
2.3 Classical Hydrodynamic Theory
There are two classical techniques for describing the evolutionary motion of a fluid: the
Lagrangian method and the Eulerian method. Both methods treat the fluid relative to a
Euclidean 3D manifold, with time as a parameter. The first (Lagrangian) technique treats
a fluid as a collection of ”particles, or parcels” and the flow is computed in terms of ”initial”
data {a, b, c, τ} imposed upon solutions to Newtonian equations of motion for ”particles,
or parcels”. The solution functions describe a map from an initial state {a, b, c; τ} to a
final state {x, y, z, t}. This method is related to solutions of kinematic equations, and is
contravariant in the sense of an immersion to velocities (the tangent space). The kinematic
basis for the Lagrangian motion draws heavily from the Frenet-Serret analysis of a point
moving along a space curve.
The second technique treats a fluid as ”field”, and is representative of a ”wave” point of
view of a Hamiltonian system. The functions that define the field (the covariant momenta)
depend on ”final data” {x, y, z, t}, and are covariant in the sense of a submersion. Each
method has its preimage in the form of an exterior differential 1-form of Action. The
primitive classical Lagrangian Action concept is written in the form
AL = L(x
k, V m, t)dt, (54)
and the primitive Eulerian Action is written as
AE = pkdx
k + Hdt. (55)
As both Actions supposedly describe the same fluid, are they equivalent? That is,
does AE ⇔ AL ? (56)
Note that AL is composed from only two functions (L and t) such that at most AL is of
Pfaff Topological Dimension 2. On the other hand, the Pfaff Topological Dimension of
AE (as written) could be as high as 8, if all functions and differentials are presumed to be
independent. So the two formulations are NOT equivalent, unless constraints reduce the
topological dimension of AE to 2, or if additions are made to the 1-form AL to increase its
Pfaff dimension.
2.3.1 The Lagrange-Hilbert Action
The classic addition to AL is of the form, pk(dx
k − V kdt), where the pk are presumed to be
Lagrange multipliers of the fluctuations in kinematic velocity. The result is defined as the
Cartan - Hilbert 1-form of Action:
ACH = L(x
k, V k, t)dt+ pk(dx
k − V kdt). (57)
Note that at first glance it appears that there are 10=3N+1 independent geometric variables
{xk, V k, pk, t} in the formula for ACH , but if the Pfaff Sequence is constructed, the Pfaff
Topological Dimension turns out to be 8. So with this addition of Lagrange multipliers
to AL, the topological dimensions of the two actions are the same. However, note that by
rearranging variables,
ACH = pkdx
k + (L(xk, V m, t)− pkV k)dt, (58)
= pkdx
k + Hdt = AE . (59)
For a fluid the Eulerian ”momenta” per unit parcel of mass is usually defined as
pk/m = vk, (60)
such that the Eulerian Action per unit mass becomes
AE ⇒ vkdxk + Hdt. (61)
The bottom line is that the Lagrangian and Eulerian point of view can be made compatible
if fluctuations in Kinematic Perfection are allowed.
Recall that the development of elasticity theory (and its emphasis on symmetrical tensors)
also spawned the development of hydrodynamics. Much of the theory of classical hydro-
dynamics was phrased geometrically in the language of vector analysis. The development
followed the phenomenological concepts of an extended Newtonian theory of elasticity. The
classical theory was developed from ”balance” equations for a bounded sample, or parcel, of
matter (mass), which express assumptions (defined as the conservation of mass, momenta
and energy) in terms of integrals over the bounded sample, or parcel, of matter. The
classical integrals are performed usually over three-dimensional volumes (and not over 4D
space-time). The classical Cauchy result (for the momentum equations) is
ρ{∂v/∂t + v ◦ ∇v} = ∇◦T+ ρf , (62)
ρ{∂v/∂t + grad(v ◦ v/2)− v×curlv} = ∇◦T+ ρf , (63)
Constitutive assumptions are then made for the 3D stress tensor T , such that (in matrix
format)
[T] = (−P + λ(∇ ◦ v) [I] + ν{[∇v] + [∇v]T }, (64)
where P is the Pressure, ν the affine ”shear” viscosity and λ the ”expansion” viscosity. The
antisymmetric components {[∇v]− [∇v]T } have been ignored.
It will be demonstrated how the Axioms of Hydrodynamics yield topological information
about the classic Cauchy development, and goes beyond the symmetrical formulations by
recognizing that antisymmetries can introduce complex spinor contributions to the dynamics.
2.4 Euler flows and Hamiltonian fluids
Consider the Action 1-form per unit source (in thermodynamics, the unit source is mole
number, or sometimes mass), constructed from a covariant 3D velocity field, v = vk(x, y, z, t),
and a scalar potential function, φ:
A = v ◦ dr−φdt = vk(x, y, z, t)dxk − φdt. (65)
Compute the exterior differential dA and define the following (3D vector) functions as,
ω = curl v and a = +{∂v/∂t + grad(φ)}, (66)
such that,
F = dA = {∂Ak/∂xj − ∂Aj/∂xk}dxjˆdxk = Fjkdxjˆdxk (67)
= ωzdxˆdy + ωxdyˆdz + ωydzˆdx− axdxˆdt− aydyˆdt− azdzˆdt.
These vector fields always satisfy the Poincare-Faraday induction equations, dF = ddA =
0, or,
curl (−a) + ∂ω/∂t = 0, div ω = 0. (68)
The Eulerian Fluid Consider a process created by the contravariant vector directionfield,
V4 = [V
x,Vy,Vz, 1] and use Cartan’s magic formula,
L(ρV4)A = i(ρV4)dA+ d(i(ρV4)A) =W + dU = Q, (69)
to compute the thermodynamic Work 1-form,W . The expressions for Work,W, and internal
energy, U, become:
W = i(ρV4)dA = −ρ{−∂v/∂t − grad(φ) +V × ω} ◦ dr
− ρV ◦ {∂v/∂t + grad(φ)}dt, (70)
= ρ{a−V × ω} ◦ dr−ρV ◦ {a)}dt (71)
U = i(V4)A = ρ(V · v−φ). (72)
At first, topologically constrain the thermodynamic Work 1-form to be of the Bernoulli
class in terms of the exterior differential system:
W = −dP (73)
ρ{+a−V × ω} ◦ dr−ρV ◦ {a)}dt = −grad P ◦ dr−∂P/∂t dt (74)
Assume that formally V = v, and the potential is equal to φ = v · v/2. Compare the
coefficients of dr to deduce the classic equations of motion for the Eulerian fluid.
{∂v/∂t + grad(v · v/2)− v× ω} = −grad(P )/ρ. (75)
This formula should be compared to the derivation of the Lorentz force term for Work
in electromagnetic systems. The functional format of the hydrodynamic 1-form of Action,
A, is the same as that specified above for the electromagnetic system. All that is changed
is the notation. In essence, the two topological theories are equivalent to the extent that
there is a correspondence between functions:
A ⇔ v, φ ⇔ v · v/2, (76)
E ⇔ −a, B ⇔ ω. (77)
All the results of the preceding section using an electromagnetic format can be translated
into the hydrodynamic format.
Note that the Bernoulli ”pressure”, P , is an evolutionary invariant along a trajectory,
L(ρV4)P = i(V4)dP = i(V4)i(V4)dA = 0. (78)
If the Pressure is barotropic, then the Bernoulli function becomes, dΘ = dP/ρ. The function
Θ can be amalgamated with the potential, φ, such that the thermodynamic Work 1-form
becomes equal to zero. The system then admits an extremal Hamiltonian direction field
such that the thermodynamic Work 1-form is zero.
W = i(VH)dA = 0. (79)
For a process defined in terms of an extremal directionfield, the First Law indicates that the
1-form of Heat, Q, is exact, dQ = 0, and equal to the change in internal energy, Q = dU .
Any Hamiltonian process is reversible, as QˆdQ = 0.
The time-like component of the exterior differential system W + dP = 0 leads to the
equation,
∂P/∂t = −ρv ◦ {∂v/∂t + grad(v · v/2) = ρ(v ◦ a). (80)
It is apparent that if the velocity, v, and the acceleration, a, are orthogonal, then the time
rate of change of the Bernoulli pressure is zero.
It also follows that the ”Master” equation is valid, with the only difference being that
curl v is defined as ω, the vorticity of the hydrodynamic flow. The master equation becomes,
curl(v× ω) = ∂ω/∂t, (81)
and this equation is to be recognized as the equivalent of Helmholtz’ equation for the con-
servation of vorticity. When the Pfaff Topological Dimension of the Work 1-form is 1, it is
possible to show that the ”Master” leads to a diffusion equation.
In the hydrodynamic sense, conservation of vorticity implies uniform continuity. In other
words, the Eulerian flow is not only Hamiltonian, it is also uniformly continuous, and satisfies
both the master equation and the conservation of vorticity constraints. In addition, it may
be demonstrated that such systems are at most of Pfaff dimension 3, and admit a relative
integral invariant which generalizes the hydrodynamic concept of invariant helicity. In the
electromagnetic topology, the Hamiltonian constraint is equivalent to the statement that
the Lorentz force vanishes, a condition that has been used to define the ”ideal” plasma or
”force-free” plasma state [46].
3 The Navier-Stokes fluid
3.1 The classic Navier-Stokes equations
It can be demonstrated that the ”ideal fluid” has a Hamiltonian representation, for which the
dynamics preserves a ”Hamiltonian” energy. This result is in disagreement with experiment
in that it is observed that motions of ”non-ideal” fluids exhibit decay to a stationary state.
The Lagrange Euler equations must be modified to accommodate dissipation of kinetic energy
and angular momentum. In fact, the ideal fluid constraint of zero affine shear stresses
should be replaced by dissipative terms related to both affine shears and a new phenomena
of rotational and expansion shears which have a fixed point. The classical phenomenological
outcome is the Navier-Stokes PDE’s,
∂v/∂t + grad(v · v/2)− v× curl v = −gradϕ− (1/ρ) gradP
+ ν∆v
− (µB + ν) grad div v, (82)
where µB is the ”bulk” viscosity coefficient and ν is the ”shear” viscosity coefficient. If the
fluid is ”incompressible” then the last term, which includes corrections due to bulk viscosity,
vanishes; the incompressible constraint requires that div v ⇒ 0.
In that which follows, the basic momentum equation (82) will be deduced from the per-
spective of Continuous Topological Evolution. Different topological equivalence classes of
thermodynamic processes depend upon the Pfaff Topological Dimension of the Work 1-form.
The different classes of thermodynamic processes are related to the velocity field in a Hy-
drodynamic system. The phenomenological (geometrical) derivation of the equations of
hydrodynamics will be replaced by determining the format of the PDE’s that agree with the
constraint required to satisfy the various PTD equivalence classes the Work 1-form. The
1-form of Work (for barotropic flows as in eq. (75)) will be of Pfaff Topological Dimension 1.
The Pfaff Topological Dimension of the 1-form of Work for the Navier-Stokes fluid can be as
high as 4, and is required to be 4 if the flow is fully turbulent. Various intermediate classes
of the work 1-form are of interest, as well. In particular, the Pfaff Topological Dimension
of the work 1-form must be 3 for a baroclinic system, a result that admits frontal systems
with propagating tangential discontinuities as found in weather systems.
3.2 The Navier-Stokes equations embedded in a non-equilibrium
thermodynamic system
In this subsection, the topological refinement due to the Pfaff Topological Dimension of the
Work 1-form will be employed to demonstrate that processes in a non-equilibrium thermody-
namic system can be put into correspondence with solutions of the Navier-Stokes equations.
In order to go beyond extremal (Hamiltonian) processes, it is necessary that the Pfaff
Topological Dimension of the Work 1-form must be greater than 1. Recall that for any
process, the Work done is transverse to the process trajectory,
(i(ρV4)W = (i(ρV4)(i(ρV4)dA = 0. (83)
Hence, if the PTD of the Work 1-form, W , is to be greater than 1, it must have the format,
W = i(ρV4)dA = −dP +̟j(dxj − vjdt) = −dP +̟j∆xj , (84)
where the ”Bernoulli function”, P , if it exists, must be a first integral (a process invariant),
L(ρV4)P = (i(ρV4)dP = 0. (85)
It is also important to remember that such non-zero contributions to the work 1-form
are due to the complex, isotropic Cartan Spinors, which are the eigen direction fields of the
2-form, F .
The coefficients, ̟j, of the topological fluctuations, ∆x
j , act in the manner of Lagrange
multipliers, and mimic the concept of system forces. If ̟j/ρ is defined (arbitrarily
8) as
υ curl curl v then the spatial components of the thermodynamic Work 1-form, W , are con-
strained to yield the partial differential equations for a constant density Navier-Stokes fluid:
{∂v/∂t + grad(v · v/2)− v × ω} = −grad(P )/ρ+ υ curl curl v. (86)
Density variations can be included by adding a term λdiv(V) to the potential {v · v/2} to
yield:
∂v/∂t + grad{v.v/2} − v × curl v = −gradP/ρ (87)
+ λ{grad(div v)} (88)
+ υ{curl curl v}. (89)
Classically, v can be identified with the geometric kinematic shear viscosity, and λ = µB −υ.
The coefficient µB can be identified with the topological (space-time) bulk viscosity.
8This is one of many formal choices, but the choice demonstrates that the Navier-Stokes equations reside
within the domain of non-equilibrium thermodynamics. QED
3.3 The Topological Torsion process for the Navier-Stokes fluid
The Navier-Stokes constraint implies that the thermodynamic Work 1-form need not be
closed. Then there are thermodynamic processes represented by solutions to the Navier-
Stokes equations that are thermodynamically irreversible. In this subsection, the Topological
Torsion vector will be expressed in terms of the solutions of the Navier-Stokes equations.
From the work in section 2, the 1-form of Action will generate a 3-form of Topological
Torsion, AˆdA = i(T4)Ω4, and leads to the 4 Vector of Topological Torsion (written in
hydrodynamic notation):
T4 = [−a × v + {v.v/2} curl v, (v ◦ curl v)], (90)
= [−a × v + {v.v/2} ω,(v ◦ ω)] = [T, h]. (91)
Use the Navier-Stokes equations (82) to solve for a,
a = [grad{v.v/2}+ ∂v/∂t] (92)
= v × curl v − gradP/ρ
+ λ{grad(div v)}+ υ{curl curl v}, (93)
and then substitute this result into the expression for T4, to yield:
T = [hv − (v ◦ v/2)curl v− v × (gradP lρ)
+ λ{v× grad(div v)} − υ{v× (curl curl v)}], (94)
h = v · curl v, (95)
Note that T4 exists even for Euler flows, where υ = 0, if the flow is baroclinic. The
measurement of the components of the Torsion vector, T4, have been completely ignored by
experimentalists in hydrodynamics.
By a similar substitution, the topological parity 4-form, FˆF, becomes expressible in
terms of engineering quantities as,
K = {2(−a ◦ ω)}Ω4 = {2(σ)} ⇔ 2(E ◦B)
σ = {gradP/ρ ◦ curl v (96)
− λ{ grad(div v) ◦ curl v}
− υ{curl v ◦ (curl curl v)}}Ω4. (97)
The coefficient σ is a measure of the space-time bulk dissipation coefficient (not λ), and it is
the square of this number which must not be zero if the process is irreversible (see eq (27)).
Recall that a turbulent dissipative irreversible flow is defined when the Pfaff dimension of
the Action 1-form is equal to 4, which implies that K 6= 0.
From the expression for σ, it is apparent that if the 3D vector of vorticity is of Pfaff
dimension 2, such that ω◦curl ω = 0, then the last term vanishes, and there is no irreversible
dissipation due to shear viscosity, υ (a result useful in the theory of wakes).
Other useful situations and design criteria for dissipation, or the lack thereof, can be
gleaned from the formula. If the vector field is harmonic, then an irreversible process
requires that,
σ = (−a ◦ ω) = {(gradP/ρ− µBgrad(div v)) ◦ curl v} 6= 0. (98)
(Recall that harmonic vector fields are generators of minimal surfaces.) For fluids where
(µB) ⇒ 0, if the pressure gradient is orthogonal to the vorticity and the flow field is harmonic,
then there is no irreversible dissipation as σ = 0, and the flow is not turbulent. Note that
for many fluids the bulk viscosity is much greater than the shear viscosity. When σ = 0,
no topological torsion defects are created; the acceleration, a, and the vorticity, ω, of the
Navier-Stokes fluid are colinear.
Theorem 7 It is thereby demonstrated that solutions to the Navier-Stokes equations corre-
spond to processes of a non-equilibrium thermodynamic system of PTD(A) = 4, and Work
1-forms of PTD(W ) > 2. Such processes include Spinor direction fields generated by the
Topological Torsion vector. The Topological Torsion vector generates processes, and hence
solutions to the Navier-Stokes equations, that are thermodynamically irreversible..
These results should be compared to those generated by Lamb and Eckart [3] for the
fluid dissipation function, which is defined by the requirement that the dissipative flow has
a (geometric) entropy production rate greater than or equal to zero. More examples can be
found in, ”Wakes, Coherent Structures, and Turbulence” [42].
4 Closed States of Topological Coherence embedded as
deformable defects in Turbulent Domains
In this section 4, the problem of C2 smoothness will be attacked from the point of view of
topological thermodynamics. First, two distinct examples will be given demonstrating two
different emergent PTD = 3 states, that emerge from different 4D rotations (see p. 108,
[38]). Then, an example demonstrating the decay of a PTD = 4 state into a PTD = 3
state will be given in detail.
4.1 Examples of PTD = 3 domains and their Emergence
In section 3, it was demonstrated that there are solutions (thermodynamic processes) to
the Navier Stokes equations in non-equilibrium thermodynamic domains. The properties
of those PTD = 3 domains which emerge by C2 irreversible solutions from domains of
PTD = 4 are of particular interest. From section 2, it is apparent that the key feature of
PTD = 3 domains is that the electric E field (acceleration field a in hydrodynamics) must
be orthogonal to the magnetic B field (vorticity field ω in hydrodynamics). There are 8
cases to consider (including chirality),
Pfaff Topological Dimension 3
E = 0, ±B 6= 0, (99)
B = 0, ±E 6= 0, (100)
E ◦B = 0, with chirality choices, ±E = ±B 6= 0, (101)
of which two will be discussed in detail.
The Finite Helicity case (both E and B finite) PTD = 3 Start with the 4D ther-
modynamic domain, and first consider the 1-form of Action, A, with the format9:
A = Ax(z)dx+ Ay(z)dy − φ(z)dt, (102)
and its induced 2-form, F = dA,
F = dA = (∂Ax(z)/∂z)dzˆdx + (∂Ay(z)/∂z)dzˆdy − (∂φ(z)/∂z)dzˆdt, (103)
= Bx(z)dzˆdx−By(z)dzˆdy + Ez(z)dzˆdt. (104)
9The +E, +B chirality has been selected.
The 3-form of Topological Torsion 3-form becomes
i(T4)Ω4 = AˆF where (105)
T4(z) = [EzAy + φBx, −EzAx + φBy, 0, AxBx + AyBy] (106)
with div4(T4(z)) = 2(E ◦B) = 0, A ◦B 6= 0. (107)
The Zero Helicity case (both E and B finite) PTD = 3 Start with the 4D thermo-
dynamic domain, and consider the 1-form of Action, A, with the format:
A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y)dt, (108)
and its induced 2-form, F = dA,
F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy}
− (∂φ(x, y)/∂x)dxˆdt− (∂φ(x, y)/∂y)dyˆdt, (109)
= Bz(x, y)dxˆdy + Ex(x, y)dxˆdt+ Ey(x, y)dyˆdt. (110)
The 3-form of Topological Torsion 3-form becomes,
i(T4)Ω4 = AˆF where (111)
T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (112)
with div4(T4(x, y)) = 2(E ◦B) = 0, A ◦B = 0. (113)
This case of zero helicity (A ◦B = 0), has the Topological Torsion vector, T4(x, y), colinear
with the B field.
Zero Helicity case: PTD = 4 decays to PTD = 3 The two distinct cases, modulo
chirality, are suggestive of the idea (see p. 108 [38]) that the rotation group of a 4D domain
is not simple. The example,immediately above, is particularly useful because the algebra of
the decay from Pfaff dimension 4 to 3 is transparent.
Start with the 4D thermodynamic domain, and consider the 1-form of Action, A, with
the format:
A = Ax(x, y)dx+ Ay(x, y)dy − φ(x, y, z, t)dt, (114)
and its induced 2-form, F = dA,
F = dA = {(∂Ay(x, y)/∂x)− (∂Ax(x, y)/∂x)dxˆdy}
− (∂φ(x, y, z)/∂x)dxˆdt− (∂φ(x, y, z)/∂y)dyˆdt− (∂φ(x, y, z)/∂z)dzˆdt, (115)
= Bz(x, y)dxˆdy (116)
+ Ex(x, y, z, t)dxˆdt+ Ey(x, y, z, t)dyˆdt+ Ez(x, y, z, t)dyˆdt. (117)
The 3-form of Topological Torsion 3-form becomes,
i(T4)Ω4 = AˆF with PTD(A) = 4 (118)
T4(x, y, z, t) = [−EzAy,+EzAx, (ExAy − EyAx) + φBz, 0] (119)
with div4(T4(x, y, z, t)) = 2{Ez(x, y, z, t)Bz(x, y)} 6= 0. (120)
In this case, the helicity (A ◦B = 0) is still zero, but now the Topological Torsion vector,
T4(x, y, z, t), has three spatial components. Moreover, the Process generated by T4(x, y, z)
is thermodynamically irreversible, as (E ◦B) 6= 0. The example 1-form is of PTD = 4.
To demonstrate the emergence of the PTD = 3 state, suppose the potential function in
this example has the format,
φ = ψ(x, y) + ϕ(z)e−αt (121)
Ez(z, t) = −(∂ϕ(z)/∂z)e−αt = Ez(z)e−αt. (122)
Then the irreversible dissipation function decays as {Ez(z)Bz}e−αt. By addition of Spinor
fluctuation terms to represent the very small components of irreversible dissipation at late
times, the PTD = 3 solution,
T4(x, y) = [0, 0, (ExAy − EyAx) + φBz, 0] (123)
becomes dominant, and represents a long lived ”stationary” state far from equilibrium,
modulo the small Spinor decay terms10.
4.2 Piecewise Linear Vector Processes vs. C2 Spinor processes
It will be demonstrated on thermodynamic spaces of Pfaff Topological Dimension 3, that
there exist piecewise continuous processes (solutions to the Navier-Stokes equations) which
are thermodynamically reversible. These Vector processes can be fabricated by combinations
of Spinor processes, each of which is irreversible. This topological result demonstrates, by
example, the difference between piecewise linear 3-manifolds and smooth complex manifolds.
It appears that the key feature of the irreversible processes is that they have a fixed point
of ”rotation or expansion”.
Consider those abstract physical systems that are represented by 1-forms, A, of Pfaff
Topological Dimension 3. The concept implies that the topological features can be described
10The experimental fact that the defect structures emerge in finite time is still an open topological problem,
although some geomtric success has been achieved through Ricci flows.
in terms of 3 functions (of perhaps many geometrical coordinates and parameters) and their
differentials. For example, if one presumes the fundamental independent base variables
are the set {x, y, z}, with an exterior differential oriented volume element consisting of a
product11 of exact 1-forms Ω3 = +dxˆdyˆdz, (then a local) Darboux representation for a
physical system could have the appearance,
A = xdy + dz. (124)
The objective is to use the features of Cartan’s magic formula to compute the possible
evolutionary features of such a system. The evolutionary dynamics is essentially the first
law of thermodynamics:
LρVA = i(ρV)dA+ di(ρV)A) = W + dU = Q. (125)
The elements of the Pfaff sequence for this Action become,
A = xdy + dz., (126)
dA = dxˆdy, (127)
AˆdA = dxˆdyˆdz, (128)
dAˆdA = 0. (129)
Note that for this example the coefficient of the 3-form of Topological Torsion is not zero,
and depends upon the Enstrophy (square of the Vorticity) of the fluid flow.
4.3 The Vector Processes
Relative to the position vector R = [x, y, z] of ordered topological coordinates {x, y, z},
consider the 3 abstract, linearly independent, orthogonal (supposedly) vector direction fields:
, (130)
, (131)
. (132)
11More abstract systems could be constructed from differential forms which are not exact.
These direction fields can be used to define a class of (real) Vector processes, but these real
vectors do not exhibit the complex Spinor class of eigendirection fields for the 2-form, dA.
The Spinor eigendirection fields are missing from this basis frame. The important fact is
that thermodynamic processes defined in terms of a real basis frame (and its connection) are
incomplete, as such processes ignore the complex spinor direction fields.
For each of the real direction fields, deform the (assumed) process by an arbitrary function,
ρ. Then construct the terms that make up the First Law of topological thermodynamics.
First construct the contractions to form the internal energy for each process,
UVx = i(ρVx)A = 0, dUVx = 0, (133)
UVy = i(ρVy)A = ρx, dUVy = d(ρx), (134)
UE = i(ρE)A = ρ, dUE = dρ. (135)
The extremal vector E is the unique eigenvector with eigenvalue zero relative to the maximal
rank antisymmetric matrix generated by the 2-form, dA. The associated vector Vx (relative
to the 1-form of Action, A, is orthogonal to the y, z plane. Recall that any associated vector
represents a local adiabatic process, as the Heat flow is transverse to the process. The
linearly independent thermodynamic Work 1-forms for evolution in the direction of the 3
basis vectors are determined to be,
WVx = i(ρVx)dA = +ρdy, (136)
WVy = i(ρVy)dA = −ρdx, (137)
WE = i(ρE)dA = 0. (138)
From Cartan’s Magic Formula representing the First Law as a description of topological
evolution,
L(V)A = i(ρV)dA+ d(i(ρV)A) ≡ Q, (139)
it becomes apparent that,
QVx = −ρdy, dQVx = −dρˆdy, (140)
QVy = +xdρ, dQVy = −dρˆdx, (141)
QE = dρ dQE = 0, (142)
All processes in the extremal direction satisfy the conditions that QEˆdQE = 0. Hence, all
extremal processes are reversible. It is also true that evolutionary processes in the direction
of the other basis vectors, separately, are reversible, as the 3-form QˆdQ vanishes forVx, Vy,
or E. Hence all such piecewise continuous, transitive, processes are thermodynamically
reversible.
Note further that the ”rotation” induced by the antisymmetric matrix [dA] acting on Vx
yields Vy and the 4th power of the matrix yields the identity rotation,
[dA] ◦ |Vx〉 = |Vy〉 , (143)
2 ◦ |Vx〉 = − |Vx〉 , (144)
4 ◦ |Vx〉 = + |Vx〉 . (145)
This concept is a signature of Spinor phenomena.
4.4 The Spinor Processes
Now consider processes defined in terms of the Spinors. The eigendirection fields of the
antisymmetric matrix representation of F = dA,
[F ] =
0 1 0
−1 0 0
0 0 0
, (146)
are given by the equations:
EigenSpinor1 |Sp1〉 =
Eigenvalue = +
-1, (147)
EigenSpinor2 |Sp2〉 =
Eigenvalue = -
-1 (148)
EigenVector1 |E〉 =
Eigenvalue = 0 (149)
Now consider the processes defined by ρ times the Spinor eigendirection fields. Compute
the change in internal energy, dU , the Work, W and the Heat, Q, for each Spinor eigendi-
rection field:
= i(ρSp1)A =
−1ρx d(UρSp
−1d(ρx), (150)
= i(ρSp2)A = −
−1ρx d(UρSp
) = −
−1d(ρx), (151)
UρE = i(ρE)A = ρ, d(UρE) = dρ, . (152)
= i(ρSp1)dA = ρ(dy −
-1dx), (153)
= i(ρSp2)dA = +ρ(dy +
-1dx) (154)
WρV1 = i(ρV1)dA = 0, . (155)
= Li(ρSp
)A = ρ(dy −
-1dx) +
-1d(ρx), (156)
= Li(ρSp
)A = ρ(dy +
-1dx)−
-1d(ρx), (157)
QρV1 = Li(ρV1)A = dρ. (158)
4.5 Irreversible Spinor processes
Next compute the 3-forms of QˆdQ for each direction field, including the spinors:
QρV1ˆdQρV1 = 0, (159)
ˆdQρSp
-1ρdρˆdxˆdy, (160)
ˆdQρSp
-1ρdρˆdxˆdy. (161)
It is apparent that evolution in the direction of the Spinor fields can be irreversible in a
thermodynamic sense, if dρˆdxˆdy is not zero. This is not true for the ”piecewise linear”
combinations of the complex Spinors that produce the real vectors, V and V⊥.
Evolution in the direction of ”smooth” combinations of the base vectors may not satisfy
the reversibility conditions, QˆdQ = 0, when the combination involves a fixed point in the
x, y plane. For example, it is possible to consider smooth rotations (polarization chirality)
in the x, y plane:
Vrotation right = V⊥ +
-1V = Sp1, (162)
QˆdQ = −
-1ρdρˆdxˆdy. (163)
Vrotation left = V −
-1V⊥= Sp2, (164)
QˆdQ = +
-1ρdρˆdxˆdy. (165)
The non-zero value ofQˆdQ for the continuous rotations are related to the non-zero Godbillon-
Vey class [7]. A key feature of the rotations is that they have a fixed point in the plane;
the motions are not transitive. If the physical system admits an equation of state of the
form, θ = θ(x, y, ρ) = 0, then the rotation or expansion processes are not irreversible.
Note that the (supposedly) Vector processes of the preceding subsection are combinations
of the Spinor processes,
Vx = (a · Sp1 + b · Sp2)/2 (166)
Vy = −
-1(a · Sp1− b · Sp2)/2. (167)
Almost always, a process defined in terms a linear combinations of the Spinor direction fields
will generate a Heat 1-form, Q, that does not satisfy the Frobenius integrability theorem,
and therefore all such processes are thermodynamically irreversible: QˆdQ 6= 0. However,
with the requirement that a2 is precisely the same as b2, then either piecewise linear process
is reversible, for QˆdQ = 0.
If the coefficients, and therefore the Spinor contributions, have slight fluctuations, the
cancellation of the complex terms is not precise. Then either of the (now approximately)
piecewise continuous process will NOT be reversible due to Spinor fluctuations.
Remark 8 The facts that piecewise (sequential) C1 transitive evolution along a set of di-
rection fields in odd (3) dimensions can be thermodynamically reversible, QˆdQ = 0, while
(smooth) C2 evolution processes composed from complex Spinors can be thermodynamically
irreversible, QˆdQ 6= 0, is a remarkable result which appears to have a relationship to Nash’s
theorem on C1 embedding. Physically, the results are related to tangential discontinuities
such as hydrodynamic wakes.
For systems of Pfaff dimension 4, all of the eigendirection fields are Spinors. The Spinors
occur as two conjugate pairs. If the conjugate variables are taken to be x,y and z,t then the
z,t spinor pair can be interpreted in terms of a chirality of expansion or contraction, where
the x,y pair can be interpreted as a chirality of polarization. In this sense it may be said
that thermodynamic time irreversibility is an artifact of dimension 4.
It is remarkable that a rotation and an expansion can be combined (eliminating the fixed
point) to produce a thermodynamically reversible process.
Ian Stewart points out that there are three types of manifold structure: piecewise linear,
smooth, topological. Theorems on piecewise-linear manifolds may not be true on smooth
manifolds. The work above seems to describe such an effect. Piecewise continuous processes
are reversible, where smooth continuous processes are not (see page 106, [38])!
5 Epilogue: Topological Fluctuations and Spinors
This Section 5 goes beyond the original objective of demonstrating that the Navier-Stokes
equations, based upon continuous topological evolution, can describe the irreversible decay
of turbulence, but not its creation. However, the key features of process irreversibility and
turbulence are entwined with the concept of Topological Torsion and Spinors. Hence this
epilogue calls attention to the fact that the Cartan topological methods permit the analysis
of Spinor entanglement, as well as the analysis of fluctuations about kinematic perfection.
This research area is in its infancy, and extends the thermodynamic approach to the realm
of fiber bundles. A few of the introductory ideas are presented below.
Remark 9 These concepts go beyond the scope of this essay which has the objective of
presenting the important topological ideas in a manner palatable (if not recognizable) to the
engineering community of hydrodynamics.
5.1 The Cartan-Hilbert Action 1-form
To start, consider those physical systems that can be described by a function L(q,v,t) and
a 1-form of Action given by Cartan-Hilbert format,
A = L(qk,vk,t)dt + pk·(dqk − vkdt). (168)
The classic Lagrange function, L(qk,vk,t)dt, is extended to include fluctuations in the
kinematic variables, (dqk−vkdt) 6= 0. It is no longer assumed that the equation of Kinematic
Perfection is satisfied. Fluctuations of the topological constraint of Kinematic Perfection
are permitted;
Topological Fluctuations in position: ∆q = (dqk − vkdt) 6= 0. (169)
As the fluctuations are 1-forms, it is some interest to compute their Pfaff Topological Di-
mension. The first step in the construction of the Pfaff Sequence is to compute the exterior
differential of the fluctuation 1-form:
Fluctuation 2-form: d(∆q) = −(dvk − akdt)ˆdt (170)
= −∆vˆdt, (171)
Topological Fluctuations in velocity:∆v =(dvk − akdt) 6= 0. (172)
It is apparent that the Pfaff Topological Dimension of the fluctuations is at most 3, as
∆qˆ∆vˆdt 6= 0, and has a Heisenberg component,
When dealing with fluctuations in this prologue, the geometric dimension of independent
base variables will not be constrained to the 4 independent base variables of the Thermody-
namic model. At first glance it appears that the domain of definition is a (3n+1)-dimensional
variety of independent base variables, {pk,qk,vk,t}. Do not make the assumption that the
pk are constrained to be canonically defined. Instead, consider pk to be a (set of) Lagrange
multiplier(s) to be determined later. Also, do not assume at this stage that v is a kinematic
velocity function, such that (dqk−vkdt) ⇒ 0. The classical idea is to assert that topological
fluctuations in position are related to pressure, and topological fluctuations in velocity are
related to temperature.
For the given Action, construct the Pfaff Sequence (12) in order to determine the Pfaff
dimension or class [8] of the Cartan-Hilbert 1-form of Action. The top Pfaffian is defined
as the non-zero p-form of largest degree p in the sequence. The top Pfaffian for the Cartan-
Hilbert Action is given by the formula,
Top Pfaffian is 2n+2
(dA)n+1 = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}ˆΩ2n+1, (173)
Ω2n+1 = dp1ˆ...dpnˆdq
1ˆ..dqnˆdt. (174)
The formula is a bit surprising in that it indicates that the Pfaff Topological Dimension of
the Cartan-Hilbert 1-form is 2n+2, and not the geometrical dimension 3n + 1. For n =
3 ”degrees of freedom”, the top Pfaffian indicates that the Pfaff Topological Dimension of
the 2-form, dA is 2n + 2 = 8. The value 3n + 1 = 10 might be expected as the 1-form
was defined initially on a space of 3n+ 1 ”independent” base variables. The implication is
that there exists an irreducible number of independent variables equal to 2n + 2 = 8 which
completely characterize the differential topology of the first order system described by the
Cartan-Hilbert Action. It follows that the exact 2-form, dA, satisfies the equations
(dA)n+1 6= 0, but Aˆ(dA)n+1 = 0. (175)
Remark 10 The idea that the 2-form, dA, is a symplectic generator of even maximal rank,
2n+2, implies that ALL eigendirection fields of the 2-form, F = dA, are complex isotropic
Spinors, and all processes on such domains have Spinor components.
The format of the top Pfaffian requires that the bracketed factor in the expression above,
{Σnk=1(∂L/∂vk − pk)dvk}, can be represented (to within a factor) by a perfect differential,
dS = (n+ 1)!{Σnk=1(∂L/∂vk − pk)dvk}. (176)
The result is also true for any closed addition γ added to A; e.g., the result is ”gauge
invariant”. Addition of a closed 1-form does not change the Pfaff dimension from even to
odd. On the other hand the result is not renormalizable, for multiplication of the Action
1-form by a function can change the algebraic Pfaff dimension from even to odd.
On the 2n+2 domain, the components of (2n+1)-form T = Aˆ(dA)n generate what
has been defined herein as the Topological Torsion vector, to within a factor equal to the
Torsion Current. The coefficients of the (2n+1)-form are components of a contravariant
vector density Tm defined as the Topological Torsion vector, the same concept as defined
previously on a 4D thermodynamic domain, but now extended to (2n+2)-dimensions. This
vector is orthogonal (transversal) to the 2n+2 components of the covector, Am. In other
words,
AˆT = Aˆ(Aˆ(dA)n) = 0 ⇒ i(T)(A) =
TmAm = 0. (177)
This result demonstrates that the extended Topological Torsion vector represents an adia-
batic process. This topological result does not depend upon geometric ideas such as metric.
It was demonstrated above that, on a space of 4 independent variables, evolution in the
direction of the Topological Torsion vector is irreversible in a thermodynamic sense, subject
to the symplectic condition of non-zero divergence, d(AˆdA) 6= 0. The same concept holds
on dimension 2n+2.
The 2n+2 symplectic domain so constructed can not be compact without boundary for it
has a volume element which is exact. By Stokes theorem, if the boundary is empty, then the
surface integral is zero, which would require that the volume element vanishes; but that is
in contradiction to the assumption that the volume element is finite. For the 2n+2 domain
to be symplectic, the top Pfaffian can never vanish. The domain is therefore orientable,
but has two components, of opposite orientation. Examination of the constraint that the
symplectic space be of dimension 2n+2 implies that the Lagrange multipliers, pk, cannot be
used to define momenta in the classical ”conjugate or canonical” manner.
Define the non-canonical components of the momentum, ℏkj, as,
non-canonical momentum: ℏkj = (pj − ∂L/∂vj), (178)
such that the top Pfaffian can be written as,
(dA)n+1 = (n+ 1)!{Σnj=1ℏkjdvj}ˆΩ2n+1, (179)
Ω2n+1 = dp1ˆ...dpnˆdq
1ˆ..dqnˆdt. (180)
For the Cartan-Hilbert Action to be of Pfaff Topological Dimension 2n+2, the factor
{Σnj=1ℏkjdvj} 6= 0. It is important to note, however, that as (dA)n+1 is a volume ele-
ment of geometric dimension 2n+2, the 1-form Σnj=1ℏkjdv
j is exact (to within a factor, say
T (qk, t, pk,Sv)); hence,
Σnj=1ℏkjdv
j = TdSv. (181)
Tentatively, this 1-form, dSv, will be defined as the Topological Entropy production relative
to topological fluctuations of momentum, kinematic differential position and velocity. If
ℏkj is defined as the deviation about the canonical definition of momentum, ℏkj = ∆pj , and
noting the the expression for the top Pfaffian can be written as (n+1)!{Σnj=1ℏkj∆vj}ˆΩ2n+1,
leads to an expression for the entropy production rate in the suggestive ”Heisenberg” format:
TdSv = ∆pj∆v
j . (182)
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Topological Thermodynamics
The Axioms of Topological Thermodynamics
Cartan's Magic Formula First Law of Thermodynamics
The Pfaff Sequence and the Pfaff Topological Dimension
The Pfaff Topological Dimension of the System 1-form, A
The Pfaff Topological Dimension of the Thermodynamic Work 1-form, W
Topological Torsion and other Continuous Processes.
Reversible Processes
Irreversible Processes
The Spinor class
Emergent Topological Defects
Applications
An Electromagnetic format
Topological 3-forms and 4-forms in EM format
Topological Torsion quanta
A Hydrodynamic format
The Topological Continuum vs. the Geometrical Continuum
Topological Hydrodynamics
Classical Hydrodynamic Theory
The Lagrange-Hilbert Action
Euler flows and Hamiltonian fluids
The Navier-Stokes fluid
The classic Navier-Stokes equations
The Navier-Stokes equations embedded in a non-equilibrium thermodynamic system
The Topological Torsion process for the Navier-Stokes fluid
Closed States of Topological Coherence embedded as deformable defects in Turbulent Domains
Examples of PTD = 3 domains and their Emergence
Piecewise Linear Vector Processes vs. C2 Spinor processes
The Vector Processes
The Spinor Processes
Irreversible Spinor processes
Epilogue: Topological Fluctuations and Spinors
The Cartan-Hilbert Action 1-form
|
0704.1597 | Numerical estimation of critical parameters using the Bond entropy | Numerical estimation of critical parameters using the Bond entropy
Rafael A. Molina1 and Peter Schmitteckert2
Instituto de Estructura de la Materia - CSIC, Serrano 123, 28006, Madrid, Spain
Institut für Theorie der Kondensierten Materie,
Universität Karlsruhe, 76128 Karlsruhe, Germany
Using a model of spinless fermions in a lattice with nearest neighbor and next-nearest neighbor
interaction we show that the entropy of the reduced two site density matrix (the bond entropy) can
be used as an extremely accurate and easy to calculate numerical indicator for the critical parameters
of the quantum phase transition when the basic ordering pattern has a two-site periodicity. The
actual behavior of the bond entropy depends on the particular characteristics of the transition under
study. For the Kosterlitz-Thouless type phase transition from a Luttinger liquid phase to a charge
density wave state the bond entropy has a local maximum while in the transition from the Luttinger
liquid to the phase separated state the derivative of the bond entropy has a divergence due to the
cancelation of the third eigenvalue of the two-site reduced density matrix.
PACS numbers: 03.67.Mn,75.10.Jm
Keywords:
I. INTRODUCTION
A Quantum Phase Transitions (QPT) is a qualitative
change in the ground state of a quantum system as some
parameter is varied [1, 2]. Contrary to classical phase
transitions, QPTs occur at zero temperature and are due
to the effect of quantum fluctuations and not of ther-
mal fluctuations. The previous definition is very gen-
eral, however, the abrupt change in the structure of the
ground state that define the phase transition can have
different consequences depending on the different cases.
The ground state energy may become non-analytic when
approaching the critical parameter. The energy gap be-
tween the ground state and the first excited state may go
to zero in the critical point. The correlations at the crit-
ical point may decay as power laws instead of exponen-
tially indicating a diverging correlation length. It is pos-
sible to find quantum systems which have some of these
indications of the QPT and not others [3, 4]. For this rea-
son, alternative ways for the classification of QPTs and
for the numerical investigation of the critical parameters
in a QPT can be very helpful.
In recent years quantum information concepts have
started to be applied to the study of QPTs. One cen-
tral concept in quantum information theory is the con-
cept of entanglement [5]. Two quantum systems in a
pure state are entangled if their state cannot be writ-
ten as the product of two separate pure states for each
of the quantum systems. Entanglement measures quan-
tum correlations and as correlations are typically maxi-
mum at the critical points of QPTs it was realized that
some entanglement measures may have a singularity or
a maximum at the critical point. The amount of en-
tanglement has been shown to be a very sensitive quan-
tity to the value of the critical parameter governing the
phase transition [6, 7, 8, 9, 10]. In particular, concur-
rence [11] has been used to investigate spin models and
this quantity shows an extreme or singular behavior at
the corresponding critical points [7]. The block-block
entanglement between two parts of the system has also
been used, establishing connections with conformal field
theory [8, 12, 13].
In a recent work Gu et al. analyzed the local en-
tanglement and its relationship with phase transitions
in the one-dimensional and two-dimensional XXZ spin
models [14]. The local entanglement was measured with
the Von-Neumann entropy of the two-site density matrix.
It is more convenient numerically than the block-block
entanglement as the size of the density matrix needed
for the latter quantity grows exponentially with the size
of the block. Using Bethe ansatz results for the one-
dimensional XXZ model Gu and coworkers showed that
the local information obtained from the entanglement en-
tropy of the two-site density matrix is enough to study
the phase transitions that occur in this model. Consider-
ing blocks larger than the characteristic length scales of
the system (that in the critical point diverge) was shown
to be unnecessary as the entanglement between a block
of two spins and the rest of the system is sufficient to re-
veal the most important information about the system.
Their results hint to the possibility of using these prop-
erties for the numerical study of phase transitions with
small systems.
It is the purpose of this work to study numerically
the local entanglement as a function of the size of the
system. In particular, we will concentrate in the crit-
ical points of the phase transitions. We will study a
one-dimensional model of spinless fermions with nearest-
neighbor interactions that can be transformed through
the Jordan-Wigner transformation to the XXZ model
with spins at each site Sj = 1/2 [15]. For repulsive inter-
action V1 = 2.0 the ground state performs a Kosterlitz-
Thouless type phase transition from a Luttinger Liquid
to a charge density wave, which corresponds to an antifer-
romagnetically ordered state in the spin picture. For at-
tractive interaction V1 = −2.0 there is a phase transition
to a phase separated state, which corresponds to a ferro-
magnet in the spin picture. We will show that the bond
http://arxiv.org/abs/0704.1597v1
entropy Sbond enables us to determine the critical points
of both quantum phase transitions with an astonishing
accuracy, albeit they present very different characteris-
tics and symmetries. We will show how these differences
are reflected in the behavior of the bond entropy. Finally
we include next-nearest-neighbor interaction to test the
generality of our finding. The effect of longer ranged in-
teraction was studied before in the context of multiple
umklapp scattering [16]. In this case the interaction can
now lead to phase transitions at fillings different from 1/2
and to ordering patterns with increased unit cell.
II. BOND-ENTROPY FOR THE
NEAREST-NEIGHBOR INTERACTION MODEL
We shall consider the one-dimensional spinless
fermions model with next-neighbor interactions.
Ĥ = −t
i ĉi−1 + ĉ
i−1ĉi
V1(n̂i−
)(n̂i−1−
where the operators appearing in the formula are the
usual fermionic creation, annihilation, and number oper-
ator at site i, L is the total number of sites, t is the hop-
ping matrix element between neighboring sites, and V1 is
the nearest-neighbor interaction strength. An important
property of this model is that it can be transformed into
the XXZ spin S = 1/2 model through the Jordan-Wigner
transformation [15, 17].
Ŝ−j = exp
cj , (2)
Ŝzj = n̂j − 1/2. (3)
For an even number of particles a phase term appears
in the boundary condtion when we apply the Jordan-
Wigner transformation to Hamiltonian (1). As we are
not interested in this even-odd effect we will consider
periodic (anti-periodic) boundary conditions, c0 ≡ cM
(c0 ≡ −cM ), for N odd (even) and both models will be
equivalent in our examples. Although we will mainly
consider the spinless fermion model we will make com-
ments regarding the equivalent behavior of both models
when we believe it will be useful to clarify some situation
(specially in the “ferromagnetic” phase). The Hamilto-
nian (1) commutes with the total N̂ =
n̂i operator
so the total number of particles is a conserved quantity
and we will consider subspaces with a definite number of
fermions, equivalent to consider subspaces with a definite
value of Sz in the XXZ model.
This model has an interesting phase diagram depend-
ing on the value of the ratio of the interaction parameter
and the hopping term V1/t and also on the number of
particles N . Without loss of generality we can consider
t = 1 and consider the phases as we change V1. For
V1 < −2 the equivalent spin system is ferromagnetic and
the ground state is fully spin polarized. When we cross
the first transition point Vca = −2 the ground state of
the system can be shown to be non-degenerate and with
spin S = 0 [18]. Only in the half-filled case, N = L/2,
there is another transition point at Vcb = 2. For V1 > 2
the system is a charge density wave type insulator, the
transition is of the Kosterlitz-Thouless type and the or-
der parameter depends exponentially on the difference
V − Vcb making an accurate numerical determination of
the transition point notoriously hard.
We will define the bond-entropy as the Von Neumann
entropy of the reduced density matrix of two-neighboring
sites ρ̂ii+1. As a result of the conservation of N the re-
duced density matrix can be written as a 4 × 4 matrix
with three sectors of N = 0, N = 1, and N = 2. In the
two-site basis |00〉, |01〉, |10〉, |11〉 it can be represented
ρ̂ii+1 =
u− 0 0 0
0 ω z 0
0 z∗ ω 0
0 0 0 u+
. (4)
For this particular model using its invariance under
translations it can be shown that we can write this matrix
elements in terms of certain correlation functions [19, 20],
u− = 1+ 〈n̂in̂i+1〉 −
〈n̂i〉 −
〈n̂i+1〉 . (5)
u+ = 〈n̂in̂i+1〉+
〈n̂i〉 −
〈n̂i+1〉 . (6)
〈n̂i〉+
〈n̂i+1〉 − 〈n̂in̂i+1〉 . (7)
i ĉi+1
i+1ĉi
. (8)
We define the bond entropy Si
as the Von Neu-
mann entropy of the two-site density matrix ρ̂i i+1. In
the model under study, the invariance under translations
also implies that Si
does not depend on the site i.
Sbond = −
λj lnλj , (9)
where λj are the four eigenvalues of the reduced two-site
density matrix ρ̂i i+1.
III. NUMERICAL RESULTS
In this section we will show the numerical results for
the bond entropy as a function of the size of the system
using the DMRG algorithm [21].
In Figure 1 we show the results of Sbond at half fill-
ing for different number of sites L. The behavior of the
-2 -1 0 1 2 3 4 5
L = 10 N = 5
L = 14 N = 7
L = 18 N = 9
L = 22 N = 11
L = 26 N = 13
L = 30 N = 15
FIG. 1: (Color online) Bond entropy Sbond as a function of
the interaction for different sizes of the spinless fermions ring
L at half filling N = L/2. The results for L = 26 and L = 30
are hardly distinguishable. We have a maximum of Sbond at
V1 = 2 with extremely good approximation, see next figure,
that marks the CDW insulator transition. The slope of the
entropy diverges at V1 = −2 marking the appearance of the
ferromagnetic transition.
bond entropy around the two critical points is very dif-
ferent, reflecting the different changes in the symmetries
and correlations of the ground state. The slope of Sbond
diverges around Vca = −2 (we will explain that in more
detail in the next section), while Sbond is continuous but
has a local maximum in the proximity of the second crit-
ical point Vcb = 2. In both cases one can understand the
behavior of Sbond from the behavior of the correlation
functions in the different phases [14]. More importantly,
one can estimate with extraordinary precision the value
of the critical parameter from very small system sizes
even in the case of the transition to the CDW phase. In
addition, Sbond has a minimum for V1 = 0.
In Figure 2 we see a zoom of the last figure 1 for one
particular case with parameter L = 30, N = 15 in the
region around V1 = 2. When we calculate numerically
the position of the maximum with DMRG we obtain
the value of V1 with a precision better than 10
−5. Of
course, in order to take full advantage of these proper-
ties of Sbond we need a very accurate algorithm such as
DMRG. We used at least 500 states per block for the
L ≤ 30 sites leading to an discarded entropy below 10−9,
and 1400 states per block for the 36 site system (see be-
low), including the five lowest lying states in order to
treat the degeneracies correctly, leading to a discarded
entropy typically below 10−10, and up to 2 · 10−8 close
to the phase transitions and applied always eleven finite
lattice sweeps. Close to the CDW-I – Luttinger liquid
transition (see below) we used at least 2050 states per
block and only two low lying states to check our results.
This resulted in a discarded entropy below 10−12. We’d
like to note that despite the large number of states the
DMRG runs are much cheaper as compared to the cal-
culations in [16], since no resolvent has to be computed,
e.g. the largest run took about a hundred CPU minutes.
0.9519
0.951902
0.951904
0.951906
0.951908
0.95191
0.951912
0.951914
1.95 1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05
FIG. 2: (Color online) Behavior of Sbond as a function of V1 in
the neighborhood of the phase transition at V1 = 2 in the case
L = 30, N = 15. We can numerically pinpoint the maximum
of the curve at V1 = 2 with a precision better than 10
The function f(x) is the second order polynomial fit used to
obtain the maximum of the curve.
-2 0 2 4
L = 26 N = 13
L = 26 N = 9
L = 22 N = 9
FIG. 3: (Color online) Behavior of Sbond as a function of V for
L = 26, N = 13 (half-filling) and L = 26, N = 9, and L = 22,
N = 9 (outside half-filling). We observe the same behavior in
the ferromagnetic transition around Vca = −2 but a complete
different one in the transition around Vcb = 2.
In Figure 3 we see some examples comparing results
at half-filling and outside half-filling. The qualitative be-
havior of the bond entropy is exactly the same around
the first critical point Vca = −2, but the maximum of
the bond entropy around Vcb = 2 disappears as soon as
we move outside half-filling reflecting the absence of the
CDW transition for fillings different from 1/2.
IV. FERROMAGNETISM AND THE TWO-SITE
DENSITY MATRIX
As we have mentioned before the slope of Sbond di-
verges at V1 = Vca. In Fig. 4 we show the results for
the value of the third eigenvalue of the two-site density
matrix λ3 for different sizes. We can reach a very high
numerical precision in the determination of the ferromag-
-0.0001
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
-2.05 -2.04 -2.03 -2.02 -2.01 -2 -1.99 -1.98 -1.97 -1.96 -1.95
FIG. 4: (Color online) Behavior of the third eigenvalue of the
two-site density matrix λ3 as a function of V in the neighbor-
hood of the phase transition at V1 = −2 for different sizes at
half filling. λ3 vanishes at V1 = −2 with very high precision.
netic critical point studying the cancellation of the third
eigenvalue of the two-site density matrix.
In the thermodynamic limit at V1 = −2, ω = z = 1/4
in Eq. (4). One can immediately see that the equality of
both matrix elements implies that one of the eigenvalues
of the density matrix is zero, which leads to the singular-
ity of Sbond. If one tries a direct numerical examination
of the values of the correlation functions independently
one does not get a very accurate estimation of the critical
parameter. However, the examination of the particular
combination appearing in the Von Neumann entropy of
the two-site density matrix allows a very accurate calcu-
lation even with very small system sizes due to the fact
that although ω and z converge slowly to the thermo-
dynamic limit value 1/4, their difference converges very
quickly to zero at the critical value of the interaction.
V. NEXT NEAREST-NEIGHBOR
INTERACTION MODEL
In order to test the generality of our conclusions and
to obtain a critical parameter of a phase transition in
a model not solvable with Bethe ansatz we add next-
nearest neighbors interaction,
Ĥ = −t
i ĉi−1 + ĉ
i−1ĉi
n̂i −
n̂i−1 −
n̂i −
n̂i−2 −
, (10)
where V2 is the strength of the interaction between sites
separated by two lattice spacings. This Hamiltonian has
been used to study the physics of materials that exhibit
0 0,25 0,5 0,75 1
0,951
0,952
0,953
FIG. 5: Bond entropy as a function of V2 in the line V2 =
5− 2V1 for L = 36 and N = 18. The maximum can be used
to estimate the critical point at V2,c = 0.280 with very high
precision.
multiple phase transitions. Usually one considers V2 < V1
as one expects the interaction to reduce with distance.
However, there can be exceptions if the nearest-neighbor
interaction is supressed by the lattice geometry.
The phase diagram of the model represented by the
Hamiltonian (10) has been studied as a function of V1 and
V2 by Schmitteckert and Werner [16]. In this paper the
authors used DMRG to calculate the ground state curva-
ture. The phase diagram depends on the filling, commen-
surability effects are extremely important due to the mul-
tiple umklapp scattering. If we concentrate on half-filling
and repulsive interactions we have a charge density wave
(CDW) phase in which the ground state is twofold de-
generate with ordering pattern (•◦•◦) and (◦•◦•). Here ◦
denotes a vacant and • denotes an occupied site. In phase
CDW II the ground state is fourfold degenerate with or-
dering pattern (••◦◦), (◦••◦), (◦◦••), and (•◦◦•). We
will follow reference [16] and study the critical parameters
along the line V2 = 5 − 2V . For example, studying sys-
tems of sizes up to L = 60 they obtained a critical point
for the transition between the CDW I phase and the Lut-
tinger Liquid phase as (V1,c, V2,c) = (2.4±0.05, 0.2±0.1)}.
In Fig. (5) we show results for the ground state bond
entropy as a function of V2 for along the previous men-
tioned line for L = 36 and N = 18. Even from the small
size used we can accurately determine the critical V2 as
V2,c = 0.280, which is within the error bars previously
given by Schmitteckert and Werner [16]. The determina-
tion of the critical parameter for the transition between
CDW I and the Luttinger liquid is done with much less
numerical work as compared to the finite size analysis
of excitation gaps and the ground state curvate in [16].
Notably, the finite size corrections are smaller, e.g. the
results of the same analysis with L = 18 already gives a
critical parameter of V2,c = 0.277. In Fig. 6 we show the
numerical results for the value of the next-neighbor inter-
action V2 in which we have a local maximum of Sbond (
along the same line as before) as a function of the inverse
0 0,025 0,05 0,075
0,275
0 0,04 0,08
0,945
FIG. 6: Finite size scaling for the position of the maximum of
the bond entropy V2,c as a function of 1/L. The continuous
line is a second order polynomial fit. It is used to extrapolate
the value for L = ∞. In the inset we show the value of the
local maximum of Sbond as a function of 1/L, a second order
polynomial also fits very well the numerical results.
of the total length of the system 1/L. We have calculated
numerically the bond entropy at each size with an inter-
val of 0.001 in V2 in the region around the maximum of
Sbond, except in the case of L = 36 where we have used
an interval of = 0.0002. The values of V2 in the maxi-
mum where obtained through a second-order polynomial
fit of the numerical results for Sbond. The actual value
of the interval used was not very critical as the fits were
very good. With another second-order polynomial fit we
can extrapolate the calculated values to obtain the result
for the thermodynamic limit V2,c = 0.2814± 0.0001. In
the inset of the figure we can see the numerical results
used in the extrapolation of the value of the maximum of
the bond entropy in the critical point. The extrapolated
value being Smax = 0.95385± 0.00001.
In this case the bond entropy and the two-site den-
sity matrix does not give very useful information about
the phase transitions to charge density wave phases with
ordering patterns with basic sizes bigger than two. We
obtain no clear signature in the bond entropy for the
quantum phase transition to CDW II. One may have to
study entropies of density matrices of blocks with at least
the size of the basic ordered block of the phases we are
looking at. Also, one could try to study the bond entropy
for the excited states. The numerical determination of
critical parameters in this case is out of the scope of this
study of the bond entropy and will be subject of future
work. We note that in the case of CDW II the ground
state is fourfold degenerate in the thermodynamic limit.
However, the degeneracy of the two lowest lying states is
lifted by finite size effects.
VI. CONCLUSIONS
The bond entropy defined as the Von Neumann en-
tropy of the two-site density matrix can be a very effec-
tive tool for the study of phase transitions and critical
parameters. Its behavior depends on the correlations in
the ground state of the system.
We have studied the bond entropy for a model of spin-
less fermions with nearest-neighbor interactions and pe-
riodic boundary conditions. The size dependence of its
behavior near the two critical points in the model has
been studied in detail, showing an amazing precision in
the estimation of the critical parameter. We have also
studied a model with next-nearest neighbor interactions.
We could determine the critical point of the phase tran-
sition from the Luttinger liquid to the CDW I state with
an ordering pattern of period two. If the fundamental
block contains only two sites we show that the bond en-
tropy displays a clear signature of the quantum phase
transitons and allows for the determination of the crit-
ical parameters. The bond entropy of the ground state
could not be used for the transition to CDW II with or-
dering pattern of period four. In this case we may have
to turn to a block entropy of higher size. In general, we
can say that the bond entropy can be used as a numerical
indicator for phase transitions but the actual behavior of
the bond entropy is not universal and will depend on the
QPT under study. Our results should open the way to
the numerical study of phase transitions with small sized
systems.
Acknowledgments
RAM wishes to acknowledge useful discussions with
J. Dukelsky. He also acknowledges finantial support at
the Instituto de Estructura de la Materia-CSIC by an
I3P contract funded by the European Social Fund. This
work is supported in part by Spanish Government Grant
FIS2006-12783-C03-01.
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|
0704.1598 | Radiative transitions of the helium atom in highly magnetized neutron
star atmospheres | Mon. Not. R. Astron. Soc. 000, 1–14 (2008) Printed 10 November 2018 (MN LATEX style file v2.2)
Radiative transitions of the helium atom in highly
magnetized neutron star atmospheres
Z. Medin1, D. Lai1, and A. Y. Potekhin1,2
1Center for Radiophysics and Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853
2Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia
Accepted 2007 September 26. Received 2007 May 21; in original form 2007 April 12
ABSTRACT
Recent observations of thermally emitting isolated neutron stars revealed spectral fea-
tures that could be interpreted as radiative transitions of He in a magnetized neutron
star atmosphere. We present Hartree–Fock calculations of the polarization-dependent
photoionization cross sections of the He atom in strong magnetic fields ranging from
1012 G to 1014 G. Convenient fitting formulae for the cross sections are given as
well as related oscillator strengths for various bound-bound transitions. The effects
of finite nucleus mass on the radiative absorption cross sections are examined using
perturbation theory.
Key words: atomic processes – magnetic fields – stars: atmospheres – stars: neutron
1 INTRODUCTION
An important advance in neutron star astrophysics in the
last few years has been the detection and detailed studies
of surface emission from a large number of isolated neutron
stars (NSs), including radio pulsars, magnetars, and radio-
quiet NSs (e.g., Kaspi et al. 2006; Harding & Lai 2006).
This was made possible by X-ray telescopes such as Chan-
dra and XMM-Newton. Such studies can potentially provide
invaluable information on the physical properties and evolu-
tion of NSs (e.g., equation of state at super-nuclear densities,
cooling history, surface magnetic field and composition). Of
great interest are the radio-quiet, thermally emitting NSs
(e.g., Haberl 2006): they share the common property that
their spectra appear to be entirely thermal, indicating that
the emission arises directly from the NS surfaces, uncon-
taminated by magnetospheric emission. The true nature of
these sources, however, is unclear at present: they could be
young cooling NSs, or NSs kept hot by accretion from the
ISM, or magnetar descendants. While some of these NSs
(e.g., RX J1856.5−3754) have featureless X-ray spectrum
remarkably well described by blackbody (e.g., Burwitz et al
2003) or by emission from a condensed surface covered by
a thin atmosphere (Ho et al. 2007), a single or multiple ab-
sorption features at E ≃ 0.2–1 keV have been detected from
several sources (see van Kerkwijk & Kaplan 2007): e.g., 1E
1207.4−5209 (0.7 and 1.4 keV, possibly also 2.1, 2.8 keV;
Sanwal et al. 2002; De Luca et al. 2004; Mori et al. 2005),
RX J1308.6+2127 (0.2–0.3 keV; Haberl et al. 2003), RX
J1605.3+3249 (0.45 keV; van Kerkwijk et al. 2004), RX
J0720.4−3125 (0.27 keV; Haberl et al. 2006), and possibly
RBS 1774 (∼ 0.7 keV; Zane et al. 2005). The identifications
of these features, however, remain uncertain, with sugges-
tions ranging from proton cyclotron lines to atomic transi-
tions of H, He, or mid-Z atoms in a strong magnetic field (see
Sanwal et al. 2002; Ho & Lai 2004; Pavlov & Bezchastnov
2005; Mori & Ho 2007). Clearly, understanding these ab-
sorption lines is very important as it would lead to direct
measurement of the NS surface magnetic fields and composi-
tions, shedding light on the nature of these objects. Multiple
lines also have the potential of constraining the mass-radius
relation of NSs (through measurement of gravitational red-
shift).
Since the thermal radiation from a NS is mediated by
its atmosphere (if T is sufficiently high so that the surface
does not condense into a solid; see, e.g., van Adelsberg et al.
2005; Medin & Lai 2006, 2007), detailed modelling of ra-
diative transfer in magnetized NS atmospheres is impor-
tant. The atmosphere composition of the NS is unknown
a priori. Because of the efficient gravitational separation
of light and heavy elements, a pure H atmosphere is ex-
pected even if a small amount of fallback or accretion oc-
curs after NS formation. A pure He atmosphere results if
H is completely burnt out, and a heavy-element (e.g., Fe)
atmosphere may be possible if no fallback/accretion oc-
curs. The atmosphere composition may also be affected by
(slow) diffusive nuclear burning in the outer NS envelope
(Chang, Arras & Bildsten 2004), as well as by the bombard-
ment on the surface by fast particles from NS magneto-
spheres (e.g., Beloborodov & Thompson 2007). Fully ion-
ized atmosphere models in various magnetic field regimes
have been extensively studied (e.g., Shibanov et al. 1992;
Zane et al. 2001; Ho & Lai 2001), including the effect of vac-
uum polarization (see Ho & Lai 2003; Lai & Ho 2002, 2003;
c© 2008 RAS
http://arxiv.org/abs/0704.1598v2
2 Z. Medin, D. Lai, and A. Y. Potekhin
van Adelsberg & Lai 2006). Because a strong magnetic field
greatly increases the binding energies of atoms, molecules,
and other bound species (for a review, see Lai 2001), these
bound states may have appreciable abundances in the NS
atmosphere, as guessed by Cohen, Lodenquai, & Ruderman
(1970) and confirmed by calculations of Lai & Salpeter
(1997) and Potekhin, Chabrier & Shibanov (1999). Early
considerations of partially ionized and strongly magnetized
atmospheres (e.g., Rajagopal, Romani & Miller 1997) relied
on oversimplified treatments of atomic physics and plasma
thermodynamics (ionization equilibrium, equation of state,
and nonideal plasma effects). Recently, a thermodynami-
cally consistent equation of state and opacities for mag-
netized (B = 1012 − 1015 G), partially ionized H plasma
have been obtained (Potekhin & Chabrier 2003, 2004), and
the effect of bound atoms on the dielectric tensor of the
plasma has also been studied (Potekhin et al. 2004). These
improvements have been incorporated into partially ion-
ized, magnetic NS atmosphere models (Ho et al. 2003, 2007;
Potekhin et al. 2004, 2006). Mid-Z element atmospheres for
B ∼ 1012 − 1013 G were recently studied by Mori & Ho
(2007).
In this paper we focus on He atoms and their radiative
transitions in magnetic NS atmospheres. It is well known
that for B ≫ Z2B0, where Z is the charge number
of the nucleus and B0 = e
3m2e/h̄
3c = 2.35 × 109 G,
the binding energy of an atom is significantly increased
over its zero-field value. In this strong-field regime the
electrons are confined to the ground Landau level, and
one may apply the adiabatic approximation, in which
electron motions along and across the field are assumed
to be decoupled from each other (see Sect. 2.1). Using
this approximation in combination with the Hartree–Fock
method (“1DHF approximation”), several groups calcu-
lated binding energies for the helium atom (Pröschel et al.
1982; Thurner et al. 1993) and also for some other atoms
and molecules (Neuhauser, Langanke & Koonin 1986;
Neuhauser, Koonin & Langanke 1987; Miller & Neuhauser
1991; Lai, Salpeter & Shapiro 1992). Mori & Hailey
(2002) developed a “multiconfigurational perturbative
hybrid Hartree–Fock” approach, which is a perturba-
tive improvement of the 1DHF method. Other methods
of calculation include Thomas–Fermi-like models (e.g.,
Abrahams & Shapiro 1991), the density functional the-
ory (e.g., Relovsky & Ruder 1996; Medin & Lai 2006),
variational methods (e.g., Müller 1984; Vincke & Baye
1989; Jones et al. 1999; Turbiner & Guevara 2006),
and 2D Hartree–Fock mesh calculations (Ivanov 1994;
Ivanov & Schmelcher 2000) which do not directly employ
the adiabatic approximation.
In strong magnetic fields, the finite nuclear mass and
centre-of-mass motion affect the atomic structure in a non-
trivial way (e.g., Lai 2001; see Sect. 5). The stronger B
is, the more important the effects of finite nuclear mass
are. Apart from the H atom, these effects have been cal-
culated only for the He atom which rests as a whole, but
has a moving nucleus (Al-Hujaj & Schmelcher 2003a,b),
and for the He+ ion (Bezchastnov, Pavlov & Ventura 1998;
Pavlov & Bezchastnov 2005).
There were relatively few publications devoted to ra-
diative transitions of non-hydrogenic atoms in strong mag-
netic fields. Several authors (Miller & Neuhauser 1991;
Thurner et al. 1993; Jones et al. 1999; Mori & Hailey
2002; Al-Hujaj & Schmelcher 2003b) calculated oscillator
strengths for bound-bound transitions; Miller & Neuhauser
(1991) presented also a few integrated bound-free oscilla-
tor strengths. Rajagopal et al. (1997) calculated opacities
of strongly magnetized iron, using photoionization cross
sections obtained by M. C. Miller (unpublished). To the
best of our knowledge, there were no published calculations
of polarization-dependent photoionization cross sections for
the He atom in the strong-field regime, as well as the calcu-
lations of the atomic motion effect on the photoabsorption
coefficients for He in this regime. Moreover, the subtle effect
of exchange interaction involving free electrons and the pos-
sible role of two-electron transitions (see Sect. 3.2) were not
discussed before.
In this paper we perform detailed calculations of radia-
tive transitions of the He atom using the 1DHF approx-
imation. The total error introduced into our calculations
by the use of these two approximations, the Hartree-Fock
method and the adiabatic approximation, is of order 1%
or less, as can be seen by the following considerations: The
Hartree-Fock method is approximate because electron corre-
lations are neglected. Due to their mutual repulsion, any pair
of electrons tend to be more distant from each other than
the Hartree-Fock wave function would indicate. In zero-field,
this correlation effect is especially pronounced for the spin-
singlet states of electrons for which the spatial wave function
is symmetrical. In strong magnetic fields (B ≫ B0), the elec-
tron spins (in the ground state) are all aligned antiparallel
to the magnetic field, and the multielectron spatial wave
function is antisymmetric with respect to the interchange of
two electrons. Thus the error in the Hartree-Fock approach
is expected to be less than the 1% accuracy characteristic of
zero-field Hartree-Fock calculations (Neuhauser et al. 1987;
Schmelcher, Ivanov & Becken 1999; for B = 0 see Scrinzi
1998). The adiabatic approximation is also very accurate at
B ≫ Z2B0. Indeed, a comparision of the ground-state en-
ergy values calculated here to those of Ivanov (1994) (who
did not use the adiabatic approximation) shows an agree-
ment to within 1% for B = 1012 G and to within 0.1% for
B = 1013 G.
The paper is organized as follows. Section 2 describes
our calculations of the bound states and continuum states
of the He atom, and section 3 contains relevant equations
for radiative transitions. We present our numerical results
and fitting formulae in section 4 and examine the effects of
finite nucleus mass on the photoabsorption cross sections in
section 5.
2 BOUND STATES AND SINGLY-IONIZED
STATES OF HELIUM ATOMS IN STRONG
MAGNETIC FIELDS
2.1 Bound states of the helium atom
To define the notation, we briefly describe 1DHF calcula-
tions for He atoms in strong magnetic fields. Each electron
in the atom is described by a one-electron wave function
(orbital). If the magnetic field is sufficiently strong (e.g.,
B ≫ 1010 G for He ground state), the motion of an electron
perpendicular to the magnetic field lines is mainly governed
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 3
by the Lorentz force, which is, on the average, stronger than
the Coulomb force. In this case, the adiabatic approximation
can be employed – i.e., the wave function can be separated
into a transverse (perpendicular to the external magnetic
field) component and a longitudinal (along the magnetic
field) component:
φmν(r) = fmν(z)Wm(r⊥) . (1)
Here Wm is the ground-state Landau wave function (e.g.,
Landau & Lifshitz 1977) given by
Wm(r⊥) =
, (2)
where (ρ, ϕ) are the polar coordinates of r⊥, ρ0 =
(h̄c/eB)1/2 is the magnetic length and fmν is the longitudi-
nal wave function which can be calculated numerically. The
quantum number m (> 0 for the considered ground Landau
state) specifies the negative of the z-projection of the elec-
tron orbital angular momentum. We restrict our considera-
tion to electrons in the ground Landau level; for these elec-
trons, m specifies also the (transverse) distance of the guid-
ing centre of the electron from the ion, ρm = (2m+1)
1/2ρ0.
The quantum number ν specifies the number of nodes in
the longitudinal wave function. The spins of the electrons
are taken to be aligned anti-parallel with the magnetic field,
and so do not enter into any of our equations. In addition,
we assume that the ion is completely stationary (the ‘infinite
ion mass’ approximation). In general, the latter assumption
is not necessary for the applicability of the adiabatic ap-
proximation (see, e.g., Potekhin 1994). The accuracy of the
infinite ion mass approximation will be discussed in Sect. 5.
Note that we use non-relativistic quantum mechanics
in our calculations, even when h̄ωBe & mec
2 or B & BQ =
2 = 4.414 × 1013 G. This is valid for two reasons: (i)
The free-electron energy in relativistic theory is
1 + 2nL
)]1/2
. (3)
For electrons in the ground Landau level (nL = 0), Eq. (3)
reduces to E ≃ mec2 + p2z/(2me) for pzc ≪ mec2; the elec-
tron remains non-relativistic in the z direction as long as
the electron energy is much less than mec
2; (ii) it is well
known (e.g., Sokolov & Ternov 1986) that Eq. (2) describes
the transverse motion of an electron with nL = 0 at any field
strength, and thus Eq. (2) is valid in the relativistic theory.
Our calculations assume that the longitudinal motion of the
electron is non-relativistic. This is valid for helium at all field
strengths considered in this paper. Thus relativistic correc-
tions to our calculated electron wave functions, binding ener-
gies, and transition cross sections are all small. Our approx-
imation is justified in part by Chen & Goldman (1992), who
find that the relativistic corrections to the binding energy of
the hydrogen atom are of order ∆E/E ∼ 10−5.5−10−4.5 for
the range of field strengths we are considering in this work
(B = 1012 − 1014 G).
A bound state of the He atom, in which
one electron occupies the (m1ν1) orbital, and the
other occupies the (m2ν2) orbital, is denoted by
|m1ν1,m2ν2〉 = |Wm1fm1ν1 ,Wm2fm2ν2〉 (clearly,
|m1ν1,m2ν2〉 = |m2ν2,m1ν1〉). The two-electron wave
function is
Ψm1ν1,m2ν2(r1, r2) =
Wm1(r1⊥)fm1ν1(z1)
×Wm2(r2⊥)fm2ν2(z2)
−Wm2(r1⊥)fm2ν2(z1)Wm1(r2⊥)fm1ν1(z2)
. (4)
The one-electron wave functions are found using
Hartree–Fock theory, by varying the total energy with re-
spect to the wave functions. The total energy is given by
(see, e.g., Neuhauser et al. 1987):
E = EK + EeZ + Edir + Eexc , (5)
where
dz |f ′mν (z)|2 , (6)
EeZ = −Ze2
dz |fmν (z)|2Vm(z) , (7)
Edir =
mν,m′ν′
′ |fmν (z)|2 |fm′ν′(z′)|2
×Dmm′(z − z′) , (8)
Eexc = −
mν,m′ν′
m′ν′(z)fmν(z)
×f∗mν(z′)fm′ν′(z′)Emm′(z − z′) ; (9)
Vm(z) =
|Wm(r⊥)|2
, (10)
Dmm′ (z − z′) =
dr⊥dr
|Wm(r⊥)|2|Wm′(r′⊥)|2
|r′ − r|
, (11)
Emm′(z − z′) =
dr⊥dr
|r′ − r|
×W ∗m′(r⊥)Wm(r⊥)W
⊥)Wm′(r
⊥) . (12)
Variation of Eq. (5) with respect to fmν(z) yields
− Ze2Vm(z)
′ |fm′ν′(z′)|2Dmm′ (z − z′)− εmν
fmν (z)
mν (z
)fm′ν′(z
)Emm′(z − z′)fm′ν′(z) .
In these equations, asterisks denote complex conjugates, and
f ′mν(z) ≡ dfmν/dz. The wave functions fmν(z) must satisfy
appropriate boundary conditions, i.e., fmν → 0 as z → ±∞,
and must have the required symmetry [fmν(z) = ±fmν(−z)]
and the required number of nodes (ν). The equations are
solved iteratively until self-consistency is reached for each
wave function fmν and energy εmν . The total energy of the
bound He state |m1ν1,m2ν2〉 can then be found, using either
Eq. (5) or
εmν − Edir − Eexc . (14)
c© 2008 RAS, MNRAS 000, 1–14
4 Z. Medin, D. Lai, and A. Y. Potekhin
2.2 Continuum states of the helium atom
The He state in which one electron occupies the bound
(m3ν3) orbital, and other occupies the continuum state
(m4k) is denoted by |m3ν3,m4k〉 = |Wm3fm3ν3 ,Wm4fm4k〉.
The corresponding two-electron wave function is
Ψm3ν3,m4k(r1, r2) =
[Wm3(r1⊥)fm3ν3(z1)
×Wm4(r2⊥)fm4k(z2)
−Wm4(r1⊥)fm4k(z1)Wm3(r2⊥)fm3ν3(z2)] . (15)
Here fm4k(z) is the longitudinal wave function of the con-
tinuum electron, and k is the z-wavenumber of the electron
at |z| → ∞ (far away from the He nucleus).
We can use Hartree–Fock theory to solve for the ion-
ized He states as we did for the bound He states. Since the
continuum electron wave function fm4k(z) is non-localized
in z, while the bound electron wave function fm3ν3(z) is lo-
calized around z = 0, it is a good approximation to neglect
the continuum electron’s influence on the bound electron.
We therefore solve for the bound electron orbital using the
equation
− Ze2Vm3(z)
fm3ν3(z) = εm3ν3fm3ν3(z) . (16)
The continuum electron, however, is influenced by the bound
electron, and its longitudinal wave function is determined
− Ze2Vm4(z)
′ |fm3ν3(z
)|2Dm3m4(z − z
)− εf
fm4k(z)
)fm3ν3(z
)Em3m4(z − z
)fm3ν4(z) .
where εf = εm4k = h̄
2k2/(2me). Here, the bound electron
orbital |m3ν3〉 satisfies the same boundary conditions as dis-
cussed in Sect. 2.1. The shape of the free electron wave func-
tion is determined by the energy of the incoming photon and
the direction the electron is emitted from the ion. We will
discuss this boundary condition in the next section. The to-
tal energy of the ionized He state |m3ν3,m4k〉 is simply
E = εm3ν3 + εf . (18)
Note that the correction terms Edir and Eexc that appear
in Eq. (14) do not also appear in Eq. (18). The direct and
exchange energies depend on the local overlap of the elec-
tron wave functions, but the non-localized nature of the free
electron ensures that these terms are zero for the continuum
states.
3 RADIATIVE TRANSITIONS
We will be considering transitions of helium atoms from two
initial states: the ground state, |00, 10〉, and the first excited
state, |00, 20〉.
In the approximation of an infinitely massive, pointlike
nucleus, the Hamiltonian of the He atom in electromagnetic
field is (see, e.g., Landau & Lifshitz 1977)
j=1,2
Atot(rj)
j=1,2
|r1 − r2|
, (19)
where pj = −ih̄∇j is the canonical momentum operator,
acting on the jth electron, rj is the jth electron radius vec-
tor, measured from the nucleus, andAtot(r) is the vector po-
tential of the field. In our case, Atot(r) = AB(r) +Aem(r),
where AB(r) and Aem(r) are vector potentials of the sta-
tionary magnetic field and electromagnetic wave, respec-
tively. The interaction operator is Hint = H − H0, where
H0 is obtained from H by setting Aem(r) = 0. The un-
perturbed Hamiltonian H0 is responsible for the stationary
states of He, discussed in Sect. 2. The vector potential and
the wave functions may be subject to gauge transformations;
the wave functions presented in Sect. 2 correspond to the
cylindrical gauge AB(r) =
B × r. Neglecting non-linear
(quadratic in Aem) term, we have
Hint ≈
j=1,2
[πj ·Aem(rj) +Aem(rj) · πj ], (20)
where
π = p +
AB(r). (21)
is the non-perturbed kinetic momentum operator: π =
meṙ = me(i/h̄)[H0 r − rH0].
For a monochromatic wave of the form Aem(r) ∝ ǫ eiq·r ,
where ǫ is the unit polarization vector, applying the Fermi’s
Golden Rule and assuming the transverse polarization (ǫ ·
q = 0), one obtains the following general formula for the
cross section of absorption of radiation from a given initial
state |a〉 (see, e.g., Armstrong & Nicholls 1972):
σ(ω, ǫ) =
∣ǫ · 〈b|eiq·rj|a〉
δ(ω − ωba), (22)
where |b〉 is the final state, ω = qc is the photon frequency,
ωba = (Eb − Ea)/h̄, and j is the electric current operator.
In our case, j = (−e/me)(π1 + π2).
We shall calculate the cross sections in the dipole ap-
proximation – i.e., drop eiq·r from Eq. (22). This approxima-
tion is sufficiently accurate for calculation of the total cross
section as long as h̄ω ≪ mec2 (cf., e.g., Potekhin & Pavlov
1993, 1997 for the case of H atom). In the dipole approxi-
mation, Eq. (22) can be written as
σ(ω, ǫ) =
2π2e2
fbaδ(ω − ωba), (23)
where
fba =
h̄ωbame
|〈b|ǫ · π|a〉|2 = 2meωba
|〈b|ǫ · r|a〉|2 (24)
is the oscillator strength. In the second equality we have
passed from the ‘velocity form’ to the ‘length form’ of the
matrix element (cf., e.g., Chandrasekhar 1945). These rep-
resentations are identical for the exact wave functions, but
it is not so for approximate ones. In the adiabatic ap-
proximation, the length representation [i.e., the right-hand
side of Eq. (24)] is preferable (see Potekhin & Pavlov 1993;
Potekhin, Pavlov, & Ventura 1997).
To evaluate the matrix element, we decompose the unit
polarization vector ǫ into three cyclic components,
ǫ = ǫ−ê+ + ǫ+ê− + ǫ0ê0, (25)
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 5
with ê0 = êz along the external magnetic field direction
(the z-axis), ê± = (êx ± iêy)/
2, and ǫα = êα · ǫ (with
α = ±, 0). Then we can write the cross section as the sum
of three components,
σ(ω, ǫ) = σ+(ω)|ǫ+|2 + σ−(ω)|ǫ−|2 + σ0(ω)|ǫ0|2, (26)
where σα has the same form as Eq. (23), with the corre-
sponding oscillator strength given by
2meωbaρ
|Mba|2 =
|Mba|2, (27)
Mba = 〈b|ê∗α · r̄|a〉, (28)
where r̄ = r/ρ0 and ωc = eB/(mec) is the electron cyclotron
frequency.
3.1 Bound-bound transitions
Consider the electronic transition
|a〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉
−→ |b〉 = |m′ν′,m2ν2〉 = |Wm′gm′ν′ ,Wm2gm2ν2〉. (29)
The selection rules for allowed transitions and the related
matrix elements are
σ0 : ∆m = 0, ∆ν = odd,
Mba = 〈gmν′ |z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (30)
σ+ : ∆m = 1, ∆ν = even,
Mba =
m+ 1 〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉, (31)
σ− : ∆m = −1, ∆ν = even,
Mba =
m 〈gm′ν′ |fmν〉〈gm2ν2 |fm2ν2〉, (32)
where ∆m = m′ −m, ∆ν = ν′ − ν. The oscillator strengths
for bound-bound transitions from the states |00, 10〉 and
|00, 20〉 are given in Table 1.
The selection rules (30) – (32) are exact in the dipole
approximation. The selection rules in m follow from the con-
servation of the z-projection of total (for the photon and two
electrons) angular momentum. Technically, in the adiabatic
approximation, they follow from the properties of the Lan-
dau functions (e.g., Potekhin & Pavlov 1993). The selection
rules in ν follow from the fact that the functions gm′ν′ and
fmν have the same parity for even ν
′−ν and opposite parity
for odd ν′ − ν.
In addition to these selection rules, there are approxi-
mate selection rules which rely on the approximate orthog-
onality of functions gm′ν′ and fmν (for general ν 6= ν′). Be-
cause of this approximate orthogonality, which holds better
the larger B is, we have
〈gm′ν′ |fmν 〉〈gm2ν2 |fm2ν2〉 = δν,ν′ + ε, (33)
where |ε| ≪ 1 and ε → 0 as ∆ν → ±∞. Therefore, the oscil-
lator strengths for transitions with α = ± and ∆ν = 2, 4, . . .
are small compared to those with ∆ν = 0. The latter oscilla-
tor strengths can be approximated, according to Eqs. (27),
(31), (32) and (33), by
ba ≈ 2(m+ 1)ωba/ωc, f
ba ≈ 2mωba/ωc (34)
(α = ∆m = ±1, ν′ = ν).
The same approximate orthogonality leads to the
smallness of matrix elements for transitions of the type
|mν,m2ν2〉 −→ |m′ν′,m2ν′2〉 with ν′2 6= ν2 for α = ± and the
smallness of cross terms in the matrix elements of the form
〈gm2ν2 |fmν〉〈gm′ν′ |fm2ν2〉 when m′ = m2 (i.e., the so-called
“one-electron jump rule”); we have therefore excluded such
terms from the selection rule equations above [Eqs. (30) –
(32)].
3.2 Photoionization
The bound-free absorption cross section for the transition
from the bound state |b〉 to the continuum state |f〉 is given
by Eq. (22) with obvious substitutions |a〉 → |b〉, |b〉 → |f〉,
→ (Lz/2π)
dk, (35)
where Lz is the normalization length of the continuum elec-
tron [
∫ Lz/2
−Lz/2
dz |gmk(z)|2 = 1] and k is the wave number of
the outgoing electron (Sect. 2.2). Therefore we have
σbf(ω, ǫ) =
2πe2Lz
mech̄
2ωfbk
∣〈fk|eiq·rǫ · π|b〉
∣〈f−k|eiq·rǫ · π|b〉
, (36)
where k =
2meεf/h̄ and |f±k〉 represents the final state
where the free electron has wave number ±k (here and here-
after we assume k > 0). The asymptotic conditions for
these outgoing free electrons are (cf., e.g., Potekhin et al.
1997) gmk(z) ∼ exp[iϕk(z)] at z → ±∞, where ϕk(z) =
|kz|+ (ka0)−1 ln |kz| and a0 = h̄2/mee2 is the Bohr radius.
Since we do not care about direction of the outgoing elec-
tron, we can use for calculations a basis of symmetric and
antisymmetric wave functions of the continuum – that is,
in Eq. (36) we can replace 〈fk| and 〈f−k| by 〈feven| and
〈fodd|. The symmetric state |feven〉 is determined by the
free electron boundary condition g′mk,even(0) = 0 and the
antisymmetric state |fodd〉 is determined by gmk,odd(0) = 0.
Since the coefficients in Eq. (17) are real, gmk,even(z) and
gmk,odd(z) can be chosen real. At z → ±∞, they behave as
gmk,(even,odd)(z) ∼ sin[ϕ(z) + constant] (where the value of
constant depends on all quantum numbers, including k). We
still have the normalization
∫ Lz/2
−Lz/2
dz |gmk,(even,odd)(z)|2 =
Similar to bound-bound transitions, we can decompose
the bound-free cross section into three components, Eq. (26).
Thus, using the dipole approximation and the length form
of the matrix elements, as discussed above, we have for (α =
±, 0)-components of the bound-free cross section
σbf,α(ω) =
× |〈f |ê∗α · r̄|b〉|
, (37)
where |f〉 = |feven〉 or |f〉 = |fodd〉 depending on the parity
of the initial state and according to the selection rules, and
σTh = (8π/3) (e
2/mec
2)2 is the Thomson cross section. The
selection rules and related matrix elements for the bound-
free transitions
|b〉 = |mν,m2ν2〉 = |Wmfmν ,Wm2fm2ν2〉
c© 2008 RAS, MNRAS 000, 1–14
6 Z. Medin, D. Lai, and A. Y. Potekhin
−→ |f〉 = |m′k,m2ν2〉 = |Wm′gm′k,Wm2gm2ν2〉 (38)
are similar to those for the bound-bound transitions [see
Eqs. (30) – (32)]:
σ0 : ∆m = 0, ∆ν = odd,
Mfb = 〈gmk|z̄|fmν 〉〈gm2ν2 |fm2ν2〉, (39)
σ+ : ∆m = 1, ∆ν = even,
Mfb =
m+ 1 (〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉
−δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (40)
σ− : ∆m = −1, ∆ν = even,
Mfb =
m (〈gm′k|fmν〉〈gm2ν2 |fm2ν2〉
−δm′ν,m2ν2〈gm2ν2 |fmν 〉〈gm′k|fm2ν2〉) , (41)
In this case, the condition ∆ν = odd means that gm′k and
fmν must have opposite parity, and the condition ∆ν = even
means that gm′k and fmν must have the same parity. The
oscillator strengths for bound-free transitions from the states
|00, 10〉 and |00, 20〉 are given in Table 2.
Note that in Eqs. (40) and (41), the second term in
the matrix element (of the form 〈gm2ν2 |fmν〉〈gm′k|fm2ν2〉)
corresponds to transitions of both electrons. This appears
to violate the “one-electron jump rule” and other approxi-
mate selection rules discussed in Sect. 3.1 [see Eq. (33)]. In
fact, these approximate rules are not directly relevant for
bound-free transitions, since the matrix elements involving
a continuuum state are always small: 〈gm′k|fmν 〉 → 0 as the
normalization length Lz → ∞. Rather, we use a different set
of selection rules to determine which of these ‘small’ matrix
elements are smaller than the rest. The first is that
〈gm′k|fmν 〉〈gm2ν2 |fm2ν2〉 ≫ 〈gm′k|fmν 〉〈gm2ν′2 |fm2ν2〉,
when ν′2 6= ν2. This selection rule is similar to the bound-
bound transition case as 〈gm2ν′2 |fm2ν2〉 involves a bound
electron transition, not a free electron transition. The second
approximate selection rule that applies here is more compli-
cated: terms of the form 〈gm′ν |fmν 〉〈gm2k|fm2ν2〉 are small,
unless m′ = m2 and ν2 = ν. This exception for m
′ = m2
and ν2 = ν is due to the exchange term in the differential
equation for the free electron wave function [Eq. (17)], which
strongly (anti)correlates the two final wave functions |gm′ν〉
and |gm2k〉. If m′ = m2 and ν = ν2, then since 〈gm′ν |fm2ν2〉
is not small (in fact, it is of order 1), 〈gm2k|fm2ν2〉 will not
be small but will be of the same order as other terms involv-
ing the free electron wave function. In particular, the second
selection rule means, e.g., that the matrix element for the
transition from |00, 10〉 to |00, 0k〉 is
M00,10→00,0k = 〈g0k|f10〉〈g00|f00〉 − 〈g00|f10〉〈g0k|f00〉, (43)
where the second term is non-negligible, but that the matrix
element for the transition from |00, 10〉 to |0k, 20〉, which is
M00,10→0k,20 = 〈g20|f10〉〈g0k|f00〉, (44)
is small compared to the other matrix elements and can be
ignored (see Fig. 1).
We make one final comment here about the effect of
exchange interaction on the free electron state. If the ex-
change term [the right-hand side of Eq. (17)] is neglected in
the calculation of the free electron wave function, then the
cross terms (i.e., those involving two-electron transitions) in
the matrix elements of Eqs. (40) and (41) are small and can
be neglected. One then obtains approximate photoionization
cross sections which are within a factor of two of the true val-
ues in most cases and much better for σ0 transitions. If the
exchange term is included in Eq. (17) but the cross terms in
the matrix elements are ignored, significant errors in the σ±
photoionization cross sections will result. To obtain reliable
cross sections for all cases, both the exchange effect on the
free electron and the contribution of two-electron transitions
must be included.
4 RESULTS
Tables 1 and 2 give results for transitions of helium atoms
from the ground state (|00, 10〉) and the first excited state
(|00, 20〉). Table 1 gives results (photon energies and oscilla-
tor strengths) for all possible bound-bound transitions with
∆ν 6 1, for the field strengths B12 = 1, 5, 10, 50, 100, where
B12 = B/(10
12 G). Transitions |a〉 → |b〉 for α = − are not
listed separately, being equivalent to transitions |b〉 → |a〉
for α = +. One can check that the oscillator strengths fba
presented in Table 1 for α = + are well described by the
approximation (34).
Table 2 gives results (threshold photon energies and
cross section fitting formulas, see below) for all possible
bound-free transitions. Figure 1 shows partial cross section
curves for all bound-free transitions from the ground state
of helium for B12 = 1. The transition |00, 10〉 → |0k, 20〉 is
an example of a ‘weak’ transition, whose oscillator strength
is small because of the approximate orthogonality of one-
electron wave functions, as discussed at the end of Sect. 3.1.
It is included in this figure to confirm the accuracy of our
assumption. Figures 2 and 3 show total cross section curves
for a photon polarized along the magnetic field, for B12 = 1
and 100 respectively. Figures 4 and 5 show total cross sec-
tions for the circular polarizations, α = ±, for B12 = 1.
Finally, Figs. 6 and 7 show total cross sections for α = ±
and B12 = 100.
4.1 Fitting Formula
The high-energy cross section scaling relations from
Potekhin & Pavlov (1993), which were derived for hydro-
gen photoionization in strong magnetic fields, also hold for
helium:
σbf,0 ∝
)2mi+9/2
σbf,± ∝
)2mi+7/2
, (46)
where mi is the m value of the initial electron that tran-
sitions to the free state. In addition, we use similar fitting
formulae for our numerical cross sections:
σbf,0 ≃
(1 +Ay)2.5(1 + B(
1 + y − 1))4(mi+1)
σTh (47)
σbf,± ≃
C(1 + y)
(1 +Ay)2.5(1 + B(
1 + y − 1))4(mi+1)
σTh (48)
where y = εf/h̄ωthr and h̄ωthr is the threshold photon en-
ergy for photoionization. These formulas have been fit to the
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 7
1 10 100 1000 10000
εf (eV)
00,10->00,1k
σbf,0
1 10 100 1000 10000
εf (eV)
σbf,0
00,10->0k,10
1000
10000
1 10 100 1000 10000
εf (eV)
00,10->00,2k
σbf,+
1000
10000
1 10 100 1000 10000
εf (eV)
σbf,+
00,10->10,1k
0.001
0.01
1 10 100
εf (eV)
00,10->0k,20
σbf,+
1000
10000
1 10 100 1000 10000
εf (eV)
σbf,-
00,10->00,0k
Figure 1. Partial cross sections σ(0,+,−) versus final ionized electron energy for photoionization of the ground state helium atom
((m1,m2) = (1, 0)). The field strength is 10
12 G. The transition |00, 10〉 → |0k,20〉 in the bottom left panel is an example of a ‘weak’
transition. We have ignored these transitions in our calculations of the total cross sections.
cross section curves with respect to the free electron energy
εf in approximately the 1 – 10
4 eV range (the curves are fit
up to 105 eV for strong magnetic fields B12 = 50 − 100,
in order to obtain the appropriate high-energy factor). The
data points to be fit are weighted proportional to their cross
section values plus a slight weight toward low-energy values,
according to the formula (error in σ) ∝ σ εf0.25.
Results for the three fitting parameters, A, B, and C,
are given in Table 2 for various partial cross sections over
a range of magnetic field strengths. For photoionization in
strong magnetic fields (B12 & 50) the cross section curves
we generate for the σ+ and σ− transitions have a slight
deficiency at low electron energies, such that the curves
peak at εf ≃ 10 eV, rather than at threshold as expected.
These peaks do not represent a real effect, but rather re-
flect the limits on the accuracy of our code (the overlap
of the wave function of the transitioning electron pre- and
post-ionization is extremely small under these conditions).
Because the cross section values are not correct at low ener-
gies, our fits are not as accurate for these curves. In Table 2
we have marked with a ‘∗’ those transitions which are most
inaccurately fit by our fitting formula, determined by cross
section curves with low-energy dips greater than 5% of the
threshold cross section value.
c© 2008 RAS, MNRAS 000, 1–14
8 Z. Medin, D. Lai, and A. Y. Potekhin
Table 1. Bound-bound transitions |a〉 → |b〉: The photon energy h̄ωba = Eb − Ea (in eV) and
the oscillator strength fα
for different polarization components α [see Eq. (27)]. All transitions
∆ν 6 1 from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths
B12 = B/(10
12 G). The last two columns list the transition energies h̄ω∗
and oscillator strengths
, corrected for the finite mass of the nucleus, according to Sect. 5.1.
B12 σ |a〉 → |b〉 h̄ωba fba h̄ω
1 0 |00, 10〉 → |00, 11〉 147.5 0.234 – –
→ |10, 01〉 271.8 0.124 – –
+ → |00, 20〉 43.11 0.0147 44.70 0.0153
0 |00, 20〉 → |00, 21〉 104.4 0.312 – –
→ |20, 01〉 277.7 0.115 – –
+ → |00, 30〉 18.01 0.00930 19.60 0.0101
→ |20, 10〉 100.7 0.0170 102.3 0.0172
5 0 |00, 10〉 → |00, 11〉 256.2 0.127 – –
→ |10, 01〉 444.8 0.0603 – –
+ → |00, 20〉 66.95 0.00459 74.89 0.00512
0 |00, 20〉 → |00, 21〉 189.2 0.176 – –
→ |20, 01〉 455.0 0.0537 – –
+ → |00, 30〉 28.94 0.00299 36.88 0.00381
→ |20, 10〉 151.1 0.00512 159.0 0.00539
10 0 |00, 10〉 → |00, 11〉 318.9 0.0974 – –
→ |10, 01〉 540.8 0.0457 – –
+ → |00, 20〉 79.54 0.00273 95.42 0.00327
0 |00, 20〉 → |00, 21〉 239.4 0.136 – –
→ |20, 01〉 553.3 0.0405 – –
+ → |00, 30〉 34.84 0.00179 50.72 0.00261
→ |20, 10〉 177.0 0.00301 192.9 0.00328
50 0 |00, 10〉 → |00, 11〉 510.9 0.0557 – –
→ |10, 01〉 822.2 0.0266 – –
+ → |00, 20〉 114.2 7.85e−4 193.6 0.00133
0 |00, 20〉 → |00, 21〉 396.7 0.0776 – –
→ |20, 01〉 841.1 0.0235 – –
+ → |00, 30〉 51.92 5.37e−4 131.3 0.00136
→ |20, 10〉 246.5 8.41e−4 325.9 0.00111
100 0 |00, 10〉 → |00, 11〉 616.4 0.0452 – –
→ |10, 01〉 971.4 0.0221 – –
+ → |00, 20〉 131.4 4.52e−4 290.2 9.98e−4
0 |00, 20〉 → |00, 21〉 485.0 0.0626 – –
→ |20, 01〉 993.4 0.0195 – –
+ → |00, 30〉 60.57 3.13e−4 219.4 0.00114
→ |20, 10〉 280.7 4.80e−4 439.5 7.51e−4
5 FINITE NUCLEUS MASS EFFECTS
So far we have used the infinite ion mass approximation.
In this section we shall evaluate the validity range of this
approximation and suggest possible corrections.
It is convenient to use the coordinate system which con-
tains the centre-of-mass coordinate Rcm and the relative
coordinates {rj} of the electrons with respect to the nu-
cleus. Using a suitable canonical transformation, the Hamil-
tonian H of an arbitrary atom or ion can be separated
into three terms (Vincke & Baye 1988; Baye & Vincke 1990;
Schmelcher & Cederbaum 1991): H1 which describes the
motion of a free pseudo-particle with net charge Q and total
mass M of the ion (atom), the coupling term H2 between
the collective and internal motion, and H3 which describes
the internal relative motion of the electrons and the nu-
cleus. H1 and H2 are proportional to M
−1, so they van-
ish in the infinite mass approximation. It is important to
note, however, that H3 (the only non-zero term in the in-
finite mass approximation) also contains a term that de-
pends on M−10 , where M0 ≈ M is the mass of the nucleus.
Thus, there are two kinds of non-trivial finite-mass effects:
the effects due to H1 + H2, which can be interpreted as
caused by the electric field induced in the co-moving refer-
ence frame, and the effects due to H3, which arise irrespec-
tive of the atomic motion. Both kinds of effects have been
included in calculations only for the H atom (Potekhin 1994;
Potekhin & Pavlov 1997, and references therein) and He+
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 9
Table 2. Bound-free transitions |b〉 → |f〉: The threshold photon energy h̄ωthr (in eV) and
the fitting parameters A, B, and C used in the cross section fitting formulas [Eq. (48)]. All
transitions from the initial states |00, 10〉 and |00, 20〉 are listed, for several magnetic field strengths
B12 = B/(10
12 G).
B12 σ |b〉 → |f〉 mi h̄ωthr A B C
1 0 |00, 10〉 → |00, 1k〉 1 159.2 0.96 0.093 1.43e6
→ |10, 0k〉 0 283.2 0.89 0.20 8.83e5
+ → |00, 2k〉 1 159.2 0.70 0.061 7.95e2
→ |10, 1k〉 0 283.2 0.86 0.094 1.30e3
– → |00, 0k〉 1 159.2 0.62 0.030 8.89e2
0 |00, 20〉 → |00, 2k〉 2 116.0 1.00 0.062 1.78e6
→ |20, 0k〉 0 289.2 0.88 0.22 8.71e5
+ → |00, 3k〉 2 116.0 0.66 0.038 3.94e2
→ |20, 1k〉 0 289.2 0.54 0.14 6.48e2
– → |00, 1k〉 2 116.0 0.62 0.029 5.82e2
5 0 |00, 10〉 → |00, 1k〉 1 268.2 0.86 0.061 8.39e5
→ |10, 0k〉 0 456.4 0.69 0.16 4.60e5
+ → |00, 2k〉 1 268.2 0.68 0.036 1.14e2
→ |10, 1k〉 0 456.4 0.83 0.057 1.93e2
– → |00, 0k〉 1 268.2 0.60 0.020 1.36e2
0 |00, 20〉 → |00, 2k〉 2 201.2 0.92 0.039 1.11e6
→ |20, 0k〉 0 466.5 0.65 0.18 4.39e5
+ → |00, 3k〉 2 201.2 0.65 0.021 5.95e1
→ |20, 1k〉 0 466.5 0.54 0.084 9.13e1
– → |00, 1k〉 2 201.2 0.61 0.015 7.82e1
10 0 |00, 10〉 → |00, 1k〉 1 331.1 0.82 0.051 6.58e5
→ |10, 0k〉 0 552.5 0.63 0.15 3.51e5
+ → |00, 2k〉 1 331.1 0.67 0.029 4.94e1
→ |10, 1k〉 0 552.5 0.81 0.046 8.43e1
– → |00, 0k〉 1 331.1 0.59 0.016 6.00e1
0 |00, 20〉 → |00, 2k〉 2 251.6 0.88 0.033 8.77e5
→ |20, 0k〉 0 564.9 0.59 0.16 3.31e5
+ → |00, 3k〉 2 251.6 0.64 0.017 2.64e1
→ |20, 1k〉 0 564.9 0.53 0.069 3.97e1
– → |00, 1k〉 2 251.6 0.61 0.012 3.25e1
50 0 |00, 10〉 → |00, 1k〉 1 523.3 0.73 0.034 3.74e5
→ |10, 0k〉 0 834.2 0.54 0.11 1.96e5
+ → |00, 2k〉 1 523.3 0.63 0.020 7.15e0
→ |10, 1k〉 0 834.2 0.77 0.033 1.22e1
– → |00, 0k〉 1 523.3 0.57 0.012 8.94e0
0 |00, 20〉 → |00, 2k〉 2 409.1 0.79 0.021 5.02e5
→ |20, 0k〉 0 853.0 0.50 0.13 1.83e5
+ → |00, 3k〉 2 409.1 0.62 0.0104 4.04e0
→ |20, 1k〉 0 853.0 *0.52 0.052 5.88e0
– → |00, 1k〉 2 409.1 0.59 0.0058 4.13e0
100 0 |00, 10〉 → |00, 1k〉 1 628.8 0.69 0.029 2.96e5
→ |10, 0k〉 0 983.4 0.51 0.101 1.56e5
+ → |00, 2k〉 1 628.8 0.62 0.019 3.12e0
→ |10, 1k〉 0 983.4 0.75 0.031 5.33e0
– → |00, 0k〉 1 628.8 0.56 0.012 3.94e0
0 |00, 20〉 → |00, 2k〉 2 498.0 0.75 0.018 3.96e5
→ |20, 0k〉 0 1008 0.47 0.12 1.45e5
+ → |00, 3k〉 2 498.0 0.60 0.0092 1.81e0
→ |20, 1k〉 0 1008 *0.50 0.050 2.60e0
– → |00, 1k〉 2 498.0 0.58 0.0042 1.69e0
c© 2008 RAS, MNRAS 000, 1–14
10 Z. Medin, D. Lai, and A. Y. Potekhin
Figure 2. Total cross section σ0 versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1012 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 104.5 K (steeper lines) and 106 K (flatter
lines).
Figure 3. Total cross section σ0 versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1014 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 105.5 K (steeper lines) and 106 K (flatter
lines).
Figure 4. Total cross section σ+ versus photon energy for helium
photoionization, from initial states (m1,m2) = (1, 0) (solid lines)
and (2, 0) (dashed lines). The field strength is 1012 G. The dotted
lines extending from each cross section curve represent the effect
of magnetic broadening on these cross sections, as approximated
in Eq. (55), for T = 106 K.
Figure 5. The same as in Fig. 4, but for σ−.
ion (Bezchastnov et al. 1998; Pavlov & Bezchastnov 2005).
For the He atom, only the second kind of effects have been
studied (Al-Hujaj & Schmelcher 2003a,b).
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 11
Figure 6. The same as in Fig. 4, but for B = 1014 G.
Figure 7. The same as in Fig. 6, but for σ−.
5.1 Non-moving helium atom
The state of motion of an atom can be described by pseu-
domomentum K , which is a conserved vector since Q = 0
(e.g., Vincke & Baye 1988; Schmelcher & Cederbaum 1991).
Let us consider first the non-moving helium atom: K = 0.
According to Al-Hujaj & Schmelcher (2003a), there are
trivial normal mass corrections, which consist in the ap-
pearance of reduced masses me/(1 ± me/M0) in H3, and
non-trivial specific mass corrections, which originate from
the mass polarization operator.
The normal mass corrections for the total energy E of
the He state |m1ν1,m2ν2〉 can be described as follows:
E(M0, B) =
E(∞, (1 +me/M0)2B)
1 +me/M0
+ h̄Ωc
mj , (49)
where Ωc = (me/M0)ωc (for He, h̄Ωc = 1.588B12 eV). The
first term on the right-hand side describes the reduced mass
transformation. The second term represents the energy shift
due to conservation of the total z component of the angu-
lar momentum. Because of this shift, the states with suffi-
ciently large values of m1 +m2 become unbound (autoion-
izing, in analogy with the case of the H atom considered by
Potekhin et al. 1997). This shift is also important for radia-
tive transitions which change (m1 + m2) by ∆m 6= 0: the
transition energy h̄ωba is changed by h̄Ωc∆m. The dipole
matrix elements Mba are only slightly affected by the normal
mass corrections, but the oscillator strengths are changed
with changing ωba according to Eq. (27). The energy shift
also leads to the splitting of the photoionization thresh-
old by the same quantity h̄Ωc∆m, with ∆m = 0,±1 de-
pending on the polarization (in the dipole approximation).
Clearly, these corrections must be taken into account, unless
Ωc ≪ ωba or ∆m = 0, as illustrated in the last two columns
of Table 1.
The specific mass corrections are more difficult to evalu-
ate, but they can be neglected in the considered B range. In-
deed, calculations by Al-Hujaj & Schmelcher (2003a) show
that these corrections do not exceed 0.003 eV at B 6 104B0.
5.2 Moving helium atom
Eigenenergies and wave functions of a moving atom depend
on its pseudomomentum K perpendicular to the magnetic
field. This dependence can be described by Hamiltonian
components (e.g., Schmelcher & Cederbaum 1991)
H1 +H2 =
K · (B × rj), (50)
where
is the sum over all electrons. The dependence
on Kz is trivial, but the dependence on the perpendicular
component K⊥ is not. The energies depend on the abso-
lute value K⊥. For calculation of radiative transitions, it
is important to take into account that the pseudomomen-
tum of the atom in the initial and final state differ due to
recoil: K ′ = K + h̄q. Effectively the recoil adds a term
∝ q into the interaction operator (cf. Potekhin et al. 1997;
Potekhin & Pavlov 1997). The recoil should be neglected in
the dipole approximation.
The atomic energy E depends on K⊥ differently for
different quantum states of the atom. In a real neutron star
atmosphere, one should integrate the binding energies and
cross sections over the K⊥-distribution of the atoms, in or-
der to obtain the opacities.1 Such integration leads to the
specific magnetic broadening of spectral lines and ioniza-
tion edges. Under the conditions typical for neutron star at-
mospheres, the magnetic broadening turns out to be much
1 For the hydrogen atom, this has been done by
Pavlov & Potekhin (1995) for bound-bound transitions and
by Potekhin & Pavlov (1997) for bound-free transitions.
c© 2008 RAS, MNRAS 000, 1–14
12 Z. Medin, D. Lai, and A. Y. Potekhin
larger than the conventional Doppler and collisional broad-
enings (Pavlov & Potekhin 1995).
At present the binding energies and cross sections of
a moving helium atom have not been calculated. However,
we can approximately estimate the magnetic broadening for
T ≪ |(∆E)min|/kB, where (∆E)min is the energy difference
from a considered atomic level to the nearest level admixed
by the perturbation due to atomic motion, and kB is the
Boltzmann constant. In this case, the K⊥-dependence of E
can be approximated by the formula
E(K⊥) = E(0) +
, (51)
where E(0) is the energy in the infinite mass approxima-
tion and M⊥ = K⊥(∂E/∂K⊥)
−1 is an effective ‘transverse’
mass, whose value (M⊥ > M) depends on the quantum state
considered (e.g., Vincke & Baye 1988; Pavlov & Mészáros
1993).
Generally, at every value of K⊥ one has a different
cross section σ(ω,K⊥). Assuming the equilibrium (Maxwell–
Boltzmann) distribution of atomic velocities, the K⊥-
averaged cross section can be written as
σ(ω) =
E(0)− E(K⊥)
σ(ω,K⊥)
dE(K⊥)
, (52)
where Emin = −h̄ω.
The transitions that were dipole-forbidden for an atom
at rest due to the conservation of the total z-projection of an-
gular momentum become allowed for a moving atom. There-
fore, the selection rule ∆m = α [Eqs. (30)–(32)] does not
strictly hold, and we must write
σ(ω,K⊥) =
σm′(ω,K⊥), (53)
where the sum of partial cross sections is over all final quan-
tum numbers m′ (with m′ > 0 and m′ 6= m2 for ∆ν = 0)
which are energetically allowed. For bound-bound transi-
tions, this results in the splitting of an absorption line at a
frequency ωba in a multiplet at frequencies ωba + δmΩc +
−M−1⊥ )K
⊥/2h̄, where δm ≡ m′−m−α and M⊥,m′
is the transverse mass of final states. For photoionization, we
have the analogous splitting of the threshold. In particular,
there appear bound-free transitions at frequencies ω < ωthr
– they correspond to δm < K2⊥/(2M⊥h̄Ωc). Here, ωthr is
the threshold in the infinite ion mass approximation, and
one should keep in mind that the considered perturbation
theory is valid for K2⊥/2M⊥ ≪ |(∆E)min| < h̄ωthr. Accord-
ing to Eq. (53), σ(ω,K⊥) is notched at ω < ωthr, with the
cogs at partial thresholds ωthr + δmΩc − K2⊥/(2M⊥h̄) (cf.
Fig. 2 in Potekhin & Pavlov 1997).
Let us approximately evaluate the resulting envelope
of the notched photoionization cross section (53), assum-
ing that the ‘longitudinal’ matrix elements [〈. . .〉 con-
structions in Eqs. (30)–(32)] do not depend on K⊥. The
‘transverse’ matrix elements can be evaluated following
Potekhin & Pavlov (1997): in the perturbation approxima-
tion, they are proportional to |ξ||δm|e−|ξ|
2/2, where |ξ|2 =
0/(2h̄
2). Then
σ(ω < ωthr,K⊥) ≈ σ(ωthr, 0) exp
ωthr − ω
− h̄(ωthr − ω)
, (54)
where θ(x) is the step function. A comparison of this approx-
imation with numerical calculations for the hydrogen atom
(Potekhin & Pavlov 1997) shows that it gives the correct
qualitative behaviour of σ(ω,K⊥). For a quantitative agree-
ment, one should multiply the exponential argument by a
numerical factor ∼ 0.5–2, depending on the state and po-
larization. This numerical correction is likely due to the ne-
glected K⊥-dependence of the longitudinal matrix elements.
We assume that this approximation can be used also for the
helium atom. Using Eq. (52), we obtain
σ(ω) ≈ σ(ωthr) exp
ωthr − ω
− h̄(ωthr − ω)
for ω < ωthr. Here the transverse mass M⊥ can be evaluated
by treating the coupling Hamiltonian H2 as a perturbation,
as was done by Pavlov & Mészáros (1993) for the H atom.
Following this approach, retaining only the main perturba-
tion terms according to the approximate orthogonality rela-
tion (33) and neglecting the difference between M and M0,
we obtain an estimate
b(∆m=α)
ba/(2ωba)
1 + ωba/Ωc
, (56)
where |a〉 is the considered bound state (|00, 10〉 or |00, 20〉
for the examples in Figs. 2–7) and |b〉 are the final bound
states to which α = ± transitions |a〉 → |b〉 are allowed.
According to Eq. (34), the numerator in Eq. (56) is close to
m+ 1 for α = + and to m for α = −.
For the transitions from the ground state with po-
larization α = −, which are strictly forbidden in the
infinite ion mass approximation, using the same approx-
imations as above we obtain the estimate σ−(ω) ∝
σ+(ω)h̄ΩckBT/(kBT + h̄Ωc)
Examples of the photoionization envelope approxima-
tion, as described in Eq. (55) above, are shown in Figs. 2–7.
In Figs. 6 and 7 (for B = 1014 G), in addition to the mag-
netic broadening, we see a significant shift of the maximum,
which originates from the last term in Eq. (49). Such shift
is negligible in Figs. 4 and 5 because of the relatively small
Ωc value for B = 10
12 G.
Finally, let us note that the Doppler and collisional
broadening of spectral features in a strong magnetic field
can be estimated, following Pavlov & Mészáros (1993),
Pavlov & Potekhin (1995) and Rajagopal et al. (1997). The
Doppler spectral broadening profile is
φD(ω) =
− (ω − ω0)
, (57)
∆ωD =
]−1/2
, (58)
where θB is the angle between the wave vector and B. The
collisional broadening is given by
φcoll(ω) =
Λcoll
(ω − ω0)2 + (Λcoll/2)2
, (59)
c© 2008 RAS, MNRAS 000, 1–14
Radiative transitions of the helium atom 13
h̄Λcoll = 4.8nea0r
= 41.5
1024 cm−3
eV, (60)
where ne is the electron number density and reff is an
effective electron-atom interaction radius, which is about
the quantum-mechanical size of the atom. The convolution
of the Doppler, collisional and magnetic broadening pro-
files gives the total shape of the cross section. For bound-
free transitions, the Doppler and collisional factors can
be neglected, but for the bound-bound transitions they
give the correct blue wings of the spectral features (cf.
Pavlov & Potekhin 1995).
6 CONCLUSION
We have presented detailed numerical results and fitting for-
mulae for the dominant radiative transitions (both bound-
bound and bound-free) of He atoms in strong magnetic fields
in the range of 1012 − 1014 G. These field strengths may
be most appropriate for the identification of spectral lines
observed in thermally emitting isolated neutron stars (see
Sect. 1).
While most of our calculations are based on the infinite-
nucleus-mass approximation, we have examined the effects
of finite nucleus mass and atomic motion on the opacities.
We found that for the field strengths considered in this paper
(B . 1014 G), these effects can be incorporated into the
infinite-mass results to obtain acceptable He opacities for
neutron star atmosphere modelling. For large field strengths,
more accurate calculations of the energy levels and radiative
transitions of a moving He atom will be needed in order to
obtain reliable opacities.
ACKNOWLEDGMENTS
This work has been supported in part by NSF grant AST
0307252 and Chandra grant TM6-7004X (Smithsonian As-
trophysical Observatory). The work of A.P. is supported in
part by FASI (Rosnauka) grant NSh-9879.2006.2 and RFBR
grants 05-02-16245 and 05-02-22003.
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Introduction
Bound states and singly-ionized states of helium atoms in strong magnetic fields
Bound states of the helium atom
Continuum states of the helium atom
Radiative transitions
Bound-bound transitions
Photoionization
Results
Fitting Formula
Finite nucleus mass effects
Non-moving helium atom
Moving helium atom
Conclusion
|
0704.1600 | The VIMOS VLT Deep Survey. The Assembly History of the Stellar Mass in
Galaxies: from the Young to the Old Universe | Astronomy & Astrophysics manuscript no. VVDS˙mf˙Pozzetti˙accepted c© ESO 2018
November 12, 2018
The VIMOS VLT Deep Survey ⋆
The Assembly History of the Stellar Mass in Galaxies: from the Young to the
Old Universe
L. Pozzetti1 , M. Bolzonella1 , F. Lamareille1,9,6, G. Zamorani1, P. Franzetti2, O. Le Fèvre3, A. Iovino4, S.
Temporin4, O. Ilbert5, S. Arnouts3, S. Charlot6,7, J. Brinchmann8, E. Zucca1, L. Tresse3, M. Scodeggio2, L.
Guzzo4, D. Bottini2, B. Garilli2, V. Le Brun3, D. Maccagni2, J.P. Picat9, R. Scaramella10,11, G. Vettolani10,
A. Zanichelli10, C. Adami3, S. Bardelli1, A. Cappi1, P. Ciliegi1, T. Contini9, S. Foucaud12, I. Gavignaud13,
H.J. McCracken7,14, B. Marano15, C. Marinoni16, A. Mazure3, B. Meneux2,4, R. Merighi1, S. Paltani17,18, R.
Pellò9, A. Pollo3,19, M. Radovich20, M. Bondi10, A. Bongiorno15, O. Cucciati4,21, S. de la Torre3, L.
Gregorini22,10, Y. Mellier7,14, P. Merluzzi20, D. Vergani2, and C.J. Walcher3
(Affiliations can be found after the references)
Received 04 04 2007; accepted 18 08 2007
ABSTRACT
We present a detailed analysis of the Galaxy Stellar Mass Function (GSMF) of galaxies up to z = 2.5 as obtained from the
VIMOS VLT Deep Survey (VVDS). Our survey offers the possibility to investigate it using two different samples: (1) an optical
(I-selected 17.5 < IAB < 24) main spectroscopic sample of about 6500 galaxies over 1750 arcmin
2 and (2) a near-IR (K-selected
KAB < 22.34 & KAB < 22.84) sample of about 10200 galaxies, with photometric redshifts accurately calibrated on the VVDS
spectroscopic sample, over 610 arcmin2. We apply and compare two different methods to estimate the stellar mass Mstars from
broad-band photometry based on different assumptions on the galaxy star-formation history. We find that the accuracy of the
photometric stellar mass is overall satisfactory, and show that the addition of secondary bursts to a continuous star formation
history produces systematically higher (up to 40%) stellar masses. We derive the cosmic evolution of the GSMF, the galaxy number
density and the stellar mass density in different mass ranges. At low redshift (z ≃ 0.2) we find a substantial population of low-mass
galaxies (< 109M⊙) composed by faint blue galaxies (MI −MK ≃ 0.3). In general the stellar mass function evolves slowly up to
z ∼ 0.9 and more significantly above this redshift, in particular for low mass systems. Conversely, a massive population is present
up to z = 2.5 and have extremely red colours (MI−MK ≃ 0.7−0.8). We find a decline with redshift of the overall number density of
galaxies for all masses (59± 5% for Mstars > 10
8M⊙ at z = 1), and a mild mass-dependent average evolution (‘mass-downsizing’).
In particular our data are consistent with mild/negligible (< 30%) evolution up to z ∼ 0.7 for massive galaxies (> 6× 1010M⊙).
For less massive systems the no-evolution scenario is excluded. Specifically, a large fraction (≥ 50%) of massive galaxies have been
already assembled and converted most of their gas into stars at z ∼ 1, ruling out the ‘dry mergers’ as the major mechanism of
their assembly history below z ≃ 1. This fraction decreases to ∼ 33% at z ∼ 2. Low-mass systems have decreased continuously in
number density (by a factor up to 4.1 ± 0.9) from the present age to z = 2, consistently with a prolonged mass assembly also at
z < 1. The evolution of the stellar mass density is relatively slow with redshift, with a decrease of a factor 2.3 ± 0.1 at z = 1 and
about 4.5± 0.3 at z = 2.5.
Key words. galaxies: evolution – galaxies: luminosity function, mass function – galaxies: statistics – surveys
1. Introduction
One of the main and still open question of modern cos-
mology is how and when galaxies formed and in particular
when they assembled their stellar mass. There are growing
but still controversial evidences in near-IR (NIR) surveys
that luminous and rather massive old galaxies were quite
common already at z ∼ 1 (Pozzetti et al. 2003, Fontana
et al. 2004, Saracco et al. 2004, 2005, Caputi et al. 2006a)
and up to z ∼ 2 (Cimatti et al. 2004, Glazebrook et al
Send offprint requests to: Lucia Pozzetti e-mail:
[email protected]
⋆ based on data obtained with the European Southern
Observatory Very Large Telescope, Paranal, Chile, program
070.A-9007(A), and on data obtained at the Canada-France-
Hawaii Telescope, operated by the CNRS of France, CNRC in
Canada and the University of Hawaii
2004). These surveys indicate that a significant fraction of
early-type massive galaxies were already in place at least
up to z ∼ 1. Therefore they should have formed their stars
and assembled their stellar mass at higher redshifts. As in
the local universe, at z ≃ 1.5 these galaxies still dominate
the near-IR luminosity function and stellar mass density
of the universe (Pozzetti et al. 2003, Fontana et al. 2004,
Strazzullo et al. 2006). These results favour a high-z mass
assembly, in particular for massive galaxies, in apparent
contradiction with the hierarchical scenario of galaxy for-
mation, applied to both dark and baryonic matter, which
predicts that galaxies form through merging at later cosmic
time. In these models massive galaxies, in particular, assem-
bled most of their stellar mass via merging only at z < 1
(De Lucia et al. 2006). From several observations it seems
that baryonic matter has a mass-dependent assembly his-
http://arxiv.org/abs/0704.1600v2
2 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
tory, from massive to small objects, (i.e. the ‘downsizing’
scenario in star formation, firstly defined by Cowie et al.
1996, is valid also for mass assembly), opposite to the dark
matter (DM) halos assembly. The continuous merging of
DM halos in the hierarchical models, indeed, should result
in an ‘upsizing’ in mass assembly, with the most massive
galaxies being the last to be fully assembled. If we trust
the hierarchical ΛCDM universe, the source of this discrep-
ancy between observations and simple basic models could
be due to the difficult physical treatment of the baryonic
component, such as the star formation history/timescale,
feedback, dust content, AGN feedback or to a missing in-
gredient in the hierarchical models of galaxy formation (for
the inclusion of AGN feedback, see Bower et al. 2006, Kang
et al. 2006, De Lucia & Blaizot 2007, Menci et al. 2006,
Monaco et al. 2006; and see Neistein et al. 2006 for the de-
scription of a natural downsizing in star formation in the
hierarchical galaxy formation models and a recent review
by Renzini 2007).
Considering optically selected surveys, a strong number
density evolution of early type galaxies has been recently
reported from the COMBO17 and DEEP2 surveys (Bell et
al. 2004, Faber et al. 2005), with a corresponding increase
by a factor 2 of their stellar mass since z ∼ 1, possibly
due to so called ‘dry-mergers’ (even if the observational
results on major merging and dry-merging are still contra-
dictory, see Bell et al. 2006, van Dokkum 2005, Lin et al.
2004 and Renzini 2007 for a summary). This is at variance
with results from the VIMOS-VLT Deep Survey (VVDS,
Le Fèvre et al. 2003b), conducted at greater depth and us-
ing spectroscopic redshifts in a large contiguous area. From
the VVDS, Zucca et al. (2006) found that the B-band lu-
minosity function of early type galaxies is consistent with
passive evolution up to z ∼ 1.1, while the number of bright
(MBAB < −20) early type galaxies has decreased only by
∼ 40% from z ∼ 0.3 to z ∼ 1.1. Similarly, Brown et al.
(2007), in the NOAO Deep Wide Field survey over ∼ 10
deg2, found that the B-band luminosity density of L∗ galax-
ies increases by only 36 ± 13% from z = 0 to z = 1 and
conclude that mergers do not produce rapid growth of lu-
minous red galaxy stellar masses between z = 1 and the
present day.
The VVDS is very well suited for this kind of stud-
ies, thanks to its depth and wide area, covered by multi-
wavelength photometry and deep spectroscopy. The simple
17.5 < IAB < 24 VVDS magnitude limit selection is signif-
icantly fainter than other complete spectroscopic surveys
and allows the determination of the faint and low mass
population with unprecedented accuracy. Most of the pre-
vious existing surveys are instead very small and/or not
deep enough, or based only on photometric redshifts.
Given the still controversial results based on morphol-
ogy or colour-selected early-type galaxies (see Franzetti
et al. 2007 for a discussion on colour-selected contamina-
tion), we prefer to study the total galaxy population us-
ing the stellar mass content. Here we present results on
the cosmic evolution of the Galaxy Stellar Mass Function
(GSMF) and mass density to z = 2.5 in the deep VVDS
spectroscopic survey, limited to 17.5 < IAB < 24, over
∼ 1750 arcmin2 and based on about 6500 galaxies with
secure spectroscopic redshifts and multiband (from UV to
near-IR) photometry. In addition, we derive the GSMF
also for a K-selected sample based on about 6600 galax-
ies (KAB < 22.34) in an area of 442 arcmin
2 and about
3600 galaxies in a deeper (KAB < 22.84) smaller area of
168 arcmin2, making use of photometric redshifts, accu-
rately calibrated on the VVDS spectroscopic sample, and
spectroscopic redshifts when available.
Throughout the paper we adopt the cosmology Ωm =
0.3 and ΩΛ = 0.7, with h70 = H0/70 km s
−1 Mpc−1.
Magnitudes are given in the AB system and the suffix AB
will be dropped from now on.
2. The First Epoch VVDS Sample
The VVDS is an ongoing program aiming to map the evo-
lution of galaxies, large scale structures and AGN through
redshift measurements of ∼ 105 objects, obtained with the
VIsible Multi-Object Spectrograph (VIMOS, Le Fèvre et
al. 2003a), mounted on the ESO Very Large Telescope
(UT3), in combination with a multi-wavelength dataset
from radio to X-rays. The VVDS is described in detail in
Le Fèvre et al. (2005). Here we summarize only the main
characteristics of the survey.
The VVDS is made of a wide part, with spectroscopy
in the range 17.5 ≤ I ≤ 22.5 on 4 fields (∼ 2 × 2 deg2
each), and a deep part, with spectroscopy in the range
17.5 ≤ I ≤ 24 on the field 0226-04 (F02 hereafter).
Multicolour photometry is available for each field (Le Fèvre
et al. 2004). In particular, the B, V , R, I photometry for
the 0226-04 deep field, covering ∼ 1 deg2, has been ob-
tained at CFHT and is described in detail in McCracken
et al. (2003). The photometric depth reached in this field
is 26.5, 26.2, 25.9, 25.0 (50% completeness for point-like
sources), respectively in the B, V , R, I bands. Moreover,
U < 25.4 (50% completeness) photometry obtained with
the WFI at the ESO-2.2m telescope (Radovich et al. 2004)
and Ks band (hereafter K) photometry with NTT+SOFI
at the depth (50% completeness) of 23.34 (Temporin et al.
in preparation) are available for wide sub-areas of this field.
Moreover, an area of about 170 arcmin2 has been covered
by deeper J and K band observations with NTT+SOFI
at the depth (50% completeness) of 24.15 and 23.84, re-
spectively (Iovino et al. 2005). The deep F02 field has been
observed also by the CFHT Legacy Survey (CFHTLS1) in
several optical bands (u∗, g′, r′, i′, z′) at very faint depth
(u∗ = 26.4, g′ = 26.3, r′ = 26.1, i′ = 25.9, z′ = 24.9, 50%
completeness).
Spectroscopic observations of a randomly selected sub-
sample of objects in an area of ∼ 0.5 deg2, with an average
sampling rate of about 25%, were performed in the F02
field with VIMOS at the VLT.
Spectroscopic data were reduced with the VIMOS
Interactive Pipeline Graphical Interface (VIPGI, Scodeggio
et al. 2005, Zanichelli et al. 2005) and redshift mea-
surements were performed with an automatic package
1 Based on observations obtained with
MegaPrime/MegaCam, a joint project of CFHT and
CEA/DAPNIA, at the Canada-France-Hawaii Telescope
(CFHT) which is operated by the National Research Council
(NRC) of Canada, the Institut National des Science de l’Univers
of the Centre National de la Recherche Scientifique (CNRS) of
France, and the University of Hawaii. This work is based in part
on data products produced at TERAPIX and the Canadian
Astronomy Data Centre as part of the Canada-France-Hawaii
Telescope Legacy Survey, a collaborative project of NRC and
CNRS.
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 3
Fig. 1. Upper panel: Comparison of photometric and spec-
troscopic redshifts in theK-selected sample for objects with
highly reliable (confidence level > 97% , i.e. about 1400
galaxies with flag=3, 4) spectroscopic redshifts. The accu-
racy obtained is σ∆z = 0.02(1 + z)
2 (shown as solid lines)
with only 3.7% of outliers, defined as the objects outside
the region limited by the 2 dotted lines (zphoto = zspec ±
0.15(1 + zspec)) in the figure. Lower panel: Spectroscopic
(solid line) and photometric (dotted line) redshift distribu-
tion for the same comparison sample.
(KBRED) and then visually checked. Each redshift mea-
surement was assigned a quality flag, ranging from 0 (failed
measurement) to 4 (100% confidence level); flag 9 indicates
spectra with a single emission line, for which multiple red-
shift solutions are possible. Further details on the quality
flags are given in Le Fèvre et al. (2005).
The analysis presented in this paper is based on the first
epoch VVDS deep sample, which has been obtained from
the first spectroscopic observations (fall 2002) on the field
VVDS-02h, which cover 1750 arcmin2.
2.1. The I-selected Spectroscopic Sample
In this study we use the F02-VVDS deep spectroscopic sam-
ple, purely magnitude limited (17.5 ≤ I ≤ 24), in combi-
nation with the multi-wavelength optical/near-IR dataset.
From the total sample of 8281 objects with measured red-
shift, we removed the spectroscopically confirmed stars and
broad line AGN, as well the galaxies with low quality red-
shift flag (i.e. flag 1), remaining with 6419 galaxy spec-
tra with secure spectroscopic measurement (flags 2, 3, 4,
9), corresponding to a confidence level higher than 80%.
Galaxies with redshift flags 0 and 1 are taken into account
statistically (see Section 4 and Ilbert et al. 2005 and Zucca
et al. 2006 for details). This spectroscopic sample has a
median redshift of ∼ 0.76. Compared to previous optically
selected samples, the VVDS has not only the advantage
of having an unprecedented high fraction of spectroscopic
redshifts (compared, for example, to the purely photomet-
ric redshifts as in COMBO17, Wolf et al. 2003 and Borch
et al. 2006 for the MF), but also of being purely magnitude
selected (17.5 < I < 24), differently, for example, from the
DEEP2 (Bundy et al. 2006 for the MF) survey, which has
a colour-colour selection. Moreover, the VVDS covers an
area from 10 to 40 times wider than the GOODS-MUSIC
field (Fontana et al. 2006) and the FORS Deep Field (FDF,
Drory et al. 2005), respectively.
2.2. The K-selected Photometric Sample
A wide part of the VVDS-02h field (about 623 arcmin2)
has been observed also in the near-IR (Iovino et al. 2005,
Temporin et al. in preparation). This allows us to build a
K-selected sample with a total area of 610 arcmin2 (after
excluding low-S/N borders): 442 arcmin2 are 90% complete
to K < 22.34, while 168 arcmin2 are 90% complete to K <
22.84 (equivalent to KVega = 21).
This sample consists of 11221 objects, of which 2882
have VVDS spectroscopy. In particular, the deep sample
(K < 22.84) consists of 3821 objects, of which 749 have
VVDS spectroscopy, and 596 of them are galaxies with a
secure spectroscopic identification (flags 2, 3, 4, 9). This
latter deep sample is more than one magnitude deeper than
the samples from the K20 spectroscopic survey (Cimatti
et al. 2002) and the MUNICS survey (Drory et al. 2001).
Additionally, the total K-selected sample covers an area
more than 10 times wider than the K20 and the GOODS-
CDFS sample used by Drory et al. (2005) and 4 times wider
than the GOODS-MUSIC field (Fontana et al. 2006).
Since the spectroscopic sampling of the K-selected sam-
ple is less than satisfactory, we take advantage of the
high quality photometric redshifts (zphoto). The method
and the calibration are presented and discussed in Ilbert
et al. (2006). The comparison sample contains 3241 ac-
curate spectroscopic redshifts (confidence level > 97% ,
i.e. flag=3, 4) up to I = 24 obtaining a global accu-
racy of σ∆z/(1+z) = 0.037 with only 3.7% of outliers. Also
in the K-selected photometric sample the agreement be-
tween photometric and highly reliable spectroscopic red-
shifts (about 1400) is excellent (Figure 1). We note, how-
ever, a non-negligible number of catastrophic solutions with
zphoto
∼ 1.2 and zspec
∼ 1 which could introduce a bias at
high-redshift (see also discussion in Section 2.3). Even if we
cannot rely on a wide spectroscopic comparison sample at
high-z, the number of galaxies with zphoto > 1.2 is simi-
lar or only slightly higher (about 20%) than the number
of galaxies with zspec > 1.2 (63 vs. 51, see Figure 1) and
have very similar fluxes and colors. For this reason we do
not expect that our results on the mass function and the
mass density will be strongly biased by the effect of catas-
trophic redshifts (see Section 4). Furthermore, at high-z the
dispersion between photometric and spectroscopic redshifts
increases, but not drammatically, to σ∆z/(1+z) ≃ 0.05, 0.06
4 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
at z > 1.2, 1.4. Over the whole redshift range it can be
represented with σ∆z ≃ 0.02(1 + z)
2 (shown in Figure 1).
For the whole K-selected sample, the median errors on
photometric redshifts, based on χ2 statistics, are σzphoto =
0.06 (0.04 at z < 1, increasing to 0.14 at z > 1.5). As ex-
pected, there is also an increase of σzphoto for the faintest
objects, but this increase is only about 0.02 in the faintest
magnitude bin. As previously noted by Ilbert et al. (2006)
the statistical errors are consistent with σ∆z and could be
used as an indication of their accuracy. We will discuss in
the following sections the effects of these uncertainties and
conclude that they do not affect significantly our conclu-
sions.
We note, moreover, that the K-selected sample se-
lects a different population, in particular of Extremely Red
Objects (EROs) at zphoto > 1 (see Section 2.3 and Fig. 3),
compared to the I-selected sample used to calibrate the de-
rived photometric redshift. Actually, photometric redshifts
greater than 0.8-1.0 for the EROs population have been
confirmed spectroscopically with very low contamination of
low-z objects (Cimatti et al. 2002). Moreover, the near-IR
bands are crucial to constrain photometric redshifts in the
redshift desert since the J-band is sensitive to the Balmer
break up to z = 2.5. Indeed Ilbert et al. (2006) obtain for
the deep sample at K < 23 the most reliable photometric
redshifts on this sub-sample with only 2.1% of outliers and
σ∆z/(1+z) = 0.035 (see their figure 13).
In this paper we therefore use photometric redshifts for
the whole K-selected photometric sample and the highly
reliable spectroscopic redshifts when available.
In order to select galaxies from the total K-selected
photometric sample, we have used a number of photomet-
ric methods to remove candidate stars, as described be-
low. Some of the possible criteria to select stars are: (i)
the CLASS STAR parameter given by SExtractor (Bertin
& Arnouts 1996), providing the “stellarity-index” for each
object, reliable up to I ≃ 21; (ii) the FLUX RADIUS K
parameter, computed by SExtractor from the K band im-
ages, which gives an estimate of the radius containing half
of the flux for each object; this can be considered a good
criterion to isolate point-like sources up to K ≃ 19 (see
Iovino et al. 2005); (iii) the BzK criterion, proposed by
Daddi et al. (2004a), with stars characterized by colours
z − K < 0.3(B − z) − 0.5; (iv) the χ2 of the SED fitting
carried out during the photometric redshift estimate (Ilbert
et al. 2006), with template SEDs of both stars and galaxies.
To efficiently remove stars in the whole magnitude range
of our sample, avoiding as much as possible to lose galaxies,
we decided to use the intersection of the first three crite-
ria. We therefore selected as stars the objects fulfilling all
the constraints (i) CLASS STAR ≥ 0.95 for I < 22.5 or
CLASS STAR ≥ 0.90 if I > 22.5, (ii) FLUX RADIUS K
< 3.4 and (iii) z − K < 0.3(B − z) − 0.5. When it was
not possible to apply criterion (iii), because of non detec-
tion either in the B or z filters, we used criterion (iv) in its
place. Furthermore, we added to the sample of candidate
stars also the objects with K < 16 and FLUX RADIUS K
< 4, to be sure to exclude from the galaxy sample these sat-
urated point-like objects. The final sample consists of 653
candidate stars, which we have removed from the sample
in the following analysis. Comparing to the spectroscopic
subsample (we remind that stars were not excluded from
the spectroscopic targets of VVDS), we found about 87%
of efficiency to photometrically select stars, i.e. only 28 out
Fig. 2. Redshift distributions for the K-selected photomet-
ric sample (filled histogram) and for the I-selected spectro-
scopic sample (empty histogram).
of the 214 spectroscopic stars have not been selected in this
way, and only 3 (1.4%) with highly reliable spectroscopic
flag (3, 4), whereas 21 spectroscopic extragalactic objects
(less than 1%) fall inside the candidate star sample. Three
of them are broad line AGN and the others are all compact
objects, most of them with redshift flags 1 or 2 and only one
with flag 3. This latter object has not been eliminated from
the galaxy sample. We have furthermore removed from the
galaxy sample the spectroscopically confirmed AGNs and
the three secure spectroscopic stars which were not removed
with the photometric criteria.
The final K-selected sample consists of 10160 galaxies
with either photometric redshifts or highly reliable spec-
troscopic redshifts, when available, in the range between 0
and 2.5: 6720 galaxies in the shallow K < 22.34 area of 442
arcmin2 and 3440 galaxies in the deeper area (K < 22.84)
of 168 arcmin2.
2.3. Comparison of the Two Samples
As shown in Fig. 2, the redshift distribution in the K-
selected sample peaks at higher redshift than in the I-
selected spectroscopic sample, with the two median red-
shifts being 0.91 and 0.76, respectively. Even if we cannot
rely on a wide spectroscopic comparison sample at high-z,
we have better investigated the reliability of the high-z tail
in the K-selected sample in term of fraction and colors. We
found indeed that at K < 22 the fraction of objects with
z > 1, 1.5 (35, 13% respectively) is similar to previous spec-
troscopic (K20 survey, see Cimatti et al. 2002) or photomet-
ric studies (Somerville et al. 2004). Moreover, we have used
the BzK color-color diagnostic proposed and calibrated on
a spectroscopic sample to cull galaxies at 1.4 < z < 2.5
(Daddi et al. 2004). We found that most (92%) of the galax-
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 5
Fig. 3. Colour-magnitude diagram and colour distribution
at different redshifts for the K-selected photometric sam-
ple (filled squares and histogram) and for the I-selected
spectroscopic sample (open circles and empty histogram)
ies at 1.5 < zphoto < 2.5 lie in the high-z region of the BzK
diagram.
We conclude that our K-selected sample shows no indi-
cation of significant bias in its high-redshift tail. The global
(I −K) colour distributions of the two samples are similar
to each other up to z ∼ 1.2 (see right panels in Fig. 3), but
they are significantly different at higher redshift. At z > 1.2
the I-selected sample misses many red galaxies fainter than
the I limit, most of them being Extremely Red Objects
(EROs: defined as objects with colours I−K > 2.6), which
are instead included in the K sample (∼ 81% of EROs in
the deep K < 22.84 sample have I > 24). For this rea-
son the K-selected sample is more adequate to study the
massive tail of the GSMF at high-z. Vice versa, the K-
selected sample misses at all redshifts a number of faint
blue galaxies, which are included in the I-selected spectro-
scopic sample (see left panels in Fig. 3). These faint blue
galaxies are important in the estimate of the low-mass tail
of the GSMF.
3. Estimate of the Stellar Masses
The rest-frame near-IR light has been widely used as a
tracer of the galaxy stellar mass, in particular for lo-
cal galaxies (e.g. Gavazzi et al. 1996; Madau, Pozzetti &
Dickinson 1998, Bell & de Jong 2001). However, an accurate
estimate of the galaxy stellar mass at high z, where galax-
ies are observed at widely different evolutionary stages, is
more uncertain because of the variation of the Mstars/LK
ratio as a function of age and other parameters of the stel-
lar population, such as the star formation history and the
metallicity. The use of multiband imaging from UV to near-
IR bands is a way to take into account the contribution to
the observed light of both the old and the young stellar
populations in order to obtain a more reliable estimate of
the stellar mass.
However, even stellar masses estimated using the fit
to the multicolour spectral energy distribution (SED) are
model dependent (e.g. changing with different assumptions
on the initial mass function, IMF) and subject to various
degeneracies (age – metallicity – extinction). In order to re-
duce such degeneracies we have used a large grid of stellar
population synthesis models, covering a wide range of pa-
rameters, in particular in star formation histories (SFH).
Indeed, in the case of real galaxies the possibly complex
star-formation histories and the presence of major and/or
minor bursts of star formation can affect the derived mass
estimate (see Fontana et al. 2004).
We have applied and compared two different methods to
estimate the stellar masses from the observed magnitudes
(using 12 photometric bands from u∗ to K), that are based
on different assumptions on the star-formation history. For
both of them we have adopted the Bruzual & Charlot (2003;
BC03 hereafter) code for spectral synthesis models, in its
more recent rendition, using its low resolution version with
the “Padova 1994” tracks. Different models have also been
considered, e.g. Maraston 2005, and Pégase models (Fioc
and Rocca-Volmerange 1997). The results obtained with
these models are compared with those obtained with the
BC03 models at the end of Section 3.1.
Since most of previous studies at high-z assumed models
with exponentially decreasing SFHs, we have used the same
simple smooth SFHs (see Section 3.1) in order to compare
our results with those of previous surveys. In addition, to
further test the uncertainties in mass determination, we
have used models with complex SFHs (see Section 3.2), in
which secondary bursts have been added to exponentially
decreasing SFHs. These models have been widely used in
studies of SDSS galaxies (see Kauffmann et al. 2003 and
Salim et al. 2005 for further details). Table 1 summarizes
the model parameters used in the 2 methods described in
the following sections.
In our analysis we have adopted the Chabrier IMF
(Chabrier et al. 2003), with lower and upper cutoffs of
0.1 and 100 M⊙. Indeed, all empirical determinations of
the IMF indicate that its slope flattens below ∼ 0.5 M⊙
(Kroupa 2001, Gould et al. 1996, Zoccali et al. 2000) and
a similar flattening is required to reproduce the observed
Mstars/LB ratio in local elliptical galaxies (see e.g. Renzini
2005). As discussed extensively by Bell et al. (2003), the
Salpeter IMF (Salpeter 1955) is too rich in low mass stars
to satisfy dynamical constraints (Kauffmann et al. 2003,
Kranz et al. 2003). Moreover, di Serego Alighieri et al.
(2005) show a rather good agreement between dynamical
masses and stellar masses estimated with the Chabrier IMF
at z ∼ 1. Specifically, this is true at least for high-mass el-
liptical galaxies, less affected than lower-mass galaxies by
uncertainties in the estimate of their dynamical mass due to
possibly substantial rotational contribution to the observed
velocity dispersion.
At fixed age the masses obtained with the Chabrier IMF
are smaller by a factor ∼ 1.7, roughly independent of the
age of the population, than those derived with the classical
Salpeter IMF, used in several previous works that we shall
compare with (e.g. Brinchmann & Ellis 2000; Cole et al.
2001; Dickinson et al. 2003; Fontana et al. 2004). We have
checked this statement in our sample, finding a systematic
median offset of a factor 1.7 and a very small dispersion
6 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
(σ = 0.082 dex) in the masses derived with the two different
IMFs. Since this ratio is approximately constant for a wide
range of star formation histories (SFH), the uncertainty
in the IMF does not introduce a fundamental limitation
with respect to the results we will discuss in the following
Sections. Even if the absolute value of the mass estimate is
uncertain, the use of Salpeter or Chabrier IMFs does not
introduce any significant difference in the relative evolution
with redshift of the mass function and mass density.
One possible limitation of our approach to derive stellar
masses in our sample is the contamination by narrow-line
AGN (broad line AGN have been already excluded, see
Section 2.1). From the available spectroscopic diagnostic,
in the I-selected spectroscopic sample at the mean red-
shift z ≃ 0.7, we found that the contamination due to
type II AGN is less than 10%. Recently, several studies
(Papovich et al. 2006, Kriek et al. 2007, Daddi et al. 2007)
suggest that the fraction of type II AGN increases with
redshift and stellar mass. According to Kriek et al. (2007)
at 2 < z < 2.7 and K < 21.5 the fraction is about 20%
for massive (2.9 × 1011M⊙ for a Salpeter IMF) galaxies.
To derive the contribution of type II AGN to the massive
tail of the MF is beyond the scope of this paper. However
we note, as shown also by the above studies, that for most
of these objects the optical light is dominated by the in-
tegrated stellar emission. Therefore, both our photometric
redshift and mass estimates are likely to be approximately
correct also for them.
3.1. Smooth SFHs
Consistently with previous studies, we have used synthetic
models with smooth SFH models (exponentially decreas-
ing SFH with time scale τ : SFR(t) ∝ exp(−t/τ)) and a
best-fit technique to derive stellar masses from multicolour
photometry.
To this purpose we have developed the code
HyperZmass, a modified version of the public photomet-
ric redshift code HyperZ (Bolzonella et al. 2000): like the
public version, HyperZmass uses the SED fitting technique,
computing the best fit SED by minimizing the χ2 between
observed and model fluxes. We used models built with the
Bruzual & Charlot (2003) synthetic library. When the red-
shift is known, either spectroscopic or photometric, the best
fit SED and its normalization provide an estimate of the
stellar mass contained in the observed galaxy. In particu-
lar, we estimate the stellar mass content of the galaxies,
derived by BC03 code, by integrating the star formation
history over the galaxy age and subtracting from it the
“Return fraction” (R) due to mass loss during the stellar
evolution. For a Chabrier IMF, this fraction is already as
high as ∼ 40% at an age of the order of 1 Gyr and ap-
proaches asymptotically about 50% at older ages.
The parameters used to define the library of synthetic
models are listed in Table 1. Similar parameters have been
used in Fontana et al. (2004). The Calzetti (2000) extinction
law has been used. Following that paper, we have excluded
from the grid some models which may be not physical (e.g.
those implying large dust extinctions, AV > 0.6, in absence
of a significant star-formation rate, Age/τ > 4, see Table
1). To better match the ages of early-type galaxies in the
local universe and following SDSS studies, we also removed
models with τ < 1 Gyr and with star formation starting at
z < 1.
Table 1. Parameters Used for the Library of Template
Method Smooth SFHs Complex SFHs
IMF Chabrier Chabrier
SFR τ (Gyr) [0.1,∞]a [1,∞]
log(Age)b (yr) [8, 10.2] [8, 10]
burst age (yr) − [0, 1010]
burst fraction − [0, 0.9]
Metallicities Z⊙ [0.1Z⊙, 2Z⊙]
Extinction Calzetti law Charlot&Fall model
(n = 0.7, µ ∈ [0.1, 1])
Dust content AcV ∈ [0, 2.4] τV ∈ [0, 6]
a τ < 1 if star formation starts at z < 1.
b At each redshift, galaxies are forced to have ages smaller
than the Hubble time at that redshift.
c AV < 0.6 if Age/τ > 4.
We find that the “formal” typical 1σ statistical errors
(defined as the 68% range as derived from the χ2 statis-
tics) on the estimated masses, not taking into account the
error on the estimate of the photometric redshift for the
K-selected sample, are of the order of 0.04 dex for the
K-selected sample and 0.05 for the I-selected sample. A
more reliable estimate of the errors has been obtained us-
ing HyperZ to simulate catalogs to the same depth of our
sample (see Bolzonella et al. 2000). Using all 12 photo-
metric bands (from u∗ to K), available for a subset of our
data, and realistic photometric errors, the recovered stellar
masses reproduce the input masses with no significant offset
and a dispersion of 0.12 dex up to z ∼ 3. For comparison,
using only the optical bands (from u∗ to z′) the disper-
sion increases to ∼ 0.49 dex at z > 1. The best fit masses
obtained from input simulations built using randomly all
available metallicities and analyzed only with solar metal-
licity models are not significantly shifted from the input
masses, but the dispersion increases from 0.12 dex to 0.21
dex. These dispersions, computed using a 4σ clipping, pro-
vide an estimate of the minimum, intrinsic uncertainties of
this method at our depth. For the K-selected photometric
sample further uncertainties in the fitting technique are due
to the photometric redshift accuracy (σ∆z ≃ 0.02(1 + z)
up to z = 2.5) which corresponds on average to about 0.12
dex of uncertainty in mass, being larger at low redshift (∼
0.2 dex at z < 0.4) than at high-z (∼ 0.10 dex at z = 2).
Although in principle the best-fitting technique pro-
vides estimates also for age, metallicity, dust content and
SFH timescale, our simulations show that on average all
these quantities are much more affected by degeneracies
and therefore less constrained than the stellar mass.
In addition, we have compared our derived masses with
those obtained by using different population synthesis mod-
els, such as Pégase and Maraston (2005) models. In partic-
ular, Maraston (2005) models include the thermally pulsing
asymptotic giant branch (TP-AGB) phase, calibrated with
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 7
local stellar populations. This stellar phase is the dominant
source of bolometric and near-IR energy for a simple stellar
population in the age range 0.2 to 2 Gyr. We have tested
the differences with BC03 models using the I-selected spec-
troscopic sample and we found only a small but systematic
shift (∼ −0.14 dex and a similar dispersion) up to z ∼ 1.2
both with and without the use of near-IR photometry. On
the contrary, masses derived using Pégase models and sim-
ilar SFHs have instead no significant offset.
At higher redshifts the differences between our esti-
mated masses and those obtained with Maraston models in
the K-selected spectroscopic subsample are slightly smaller
and even smaller in the K-selected photometric sample
(∼ −0.11,−0.08, respectively). This differences are smaller
than that found by Maraston et al. (2006), ∼ −0.2, in their
SED fitting (from B up to Spitzer IRAC and MIPS bands)
of a few high redshift passive galaxies with typical ages
in the range 0.5 – 2.0 Gyr, selected in the Hubble Ultra
Deep Field (HUDF). This difference between our results
and those of Maraston et al. could be due to a combina-
tion of effects, such as the absence in our photometric data
of mid-IR Spitzer photometry, which at these redshifts is
sampling the rest frame near-IR part of the SED, mostly in-
fluenced by the TP-AGB phase, and also to the wide range
of complex stellar populations in our sample, in which the
effect of the TP-AGB phase may be diluted by the SFH.
3.2. Complex SFHs
Real galaxies could have undergone a more complex SFH,
in particular with the possible presence of bursts of star
formation on the top of a smooth SFH. Thus, we have com-
puted masses also following a different approach, which has
been intensively used in previous studies of SDSS galaxies
(e.g. Kauffmann et al. 2003, Brinchmann et al. 2004, Salim
et al. 2005, Gallazzi et al. 2005). In this approach we pa-
rameterize each SF history in terms of two components:
an underlying continuous model, with an exponentially de-
clining SF law (SFR(t) ∝ exp(−t/τ)), and random bursts
superimposed on it. We assume that random bursts occur
with equal probability at all times up to galaxy age. They
are parameterized in terms of the ratio between the mass
of stars formed in the burst and the total mass of stars
formed by the continuous model over the age. This ratio is
taken to be distributed between 0.0 and 0.9. During a burst,
stars are assumed to form at a constant rate for a time dis-
tributed uniformly in the range 30 – 300 Myr. The burst
probability is set so that 50% of the galaxies in the library
have experienced a burst in the past 2 Gyr. Attenuation by
dust is described by a two-component model (see Charlot
& Fall 2000), defined by two parameters: the effective V -
band absorption optical depth τV affecting stars younger
than 10 Myr and arising from giant molecular clouds and
the diffuse ISM, and the fraction µ of it contributed by the
diffuse ISM, that also affects older stars. We take τV to be
distributed between 0 and 6 with a broad peak around 1
and µ to be distributed between 0.1 and 1 with a broad
peak around 0.3. Finally, our model galaxies have metallic-
ities uniformly distributed between 0.1 and 2 Z⊙.
The model spectra are computed at the galaxy redshift
and in each of them we measure the k-shifted model magni-
tudes for each VVDS photometric band. We also force the
age of all models in a specific redshift range to be smaller
than the Hubble time at that redshift. The model SEDs
Fig. 4. Effect of NIR photometry in the mass determina-
tion: ratio between masses estimated without and with NIR
photometry vs. mass determined without NIR photometry.
The data have been splitted into different redshift ranges.
Left: masses determined using smooth SFHs. Right: The
same, but using complex SFHs
are then scaled to each observed SED with a least squares
method and the same scaling factor is applied to the model
stellar mass. We compare the observed to the model fluxes
in each photometric band and the χ2 goodness of fit of
each model determines the weight (∝ exp[−χ2/2]) to be
assigned to the physical parameters of that model when
building the probability distributions for each parameter
of any given galaxy. The probability distribution function
(PDF) of a given physical parameter is thus obtained from
the distribution of the weights of all models in the library
at the specified redshift. We characterize the PDF using its
median and the 16 – 84 percentile range (equivalent to ±1σ
range for Gaussian distributions), and also record the χ2 of
the best-fitting model.
Similarly to what has been done for the models with
smooth SFH (see Section 3.1), also in this case the stel-
lar mass content of galaxies is derived by subtracting the
return fraction R from the total formed stellar mass. We
find that the average “formal” 1σ error (defined as half of
the 16 – 84 percentile range) on the estimated masses is of
the order of 0.09 dex for the K-selected sample. The av-
erage error increases with redshift from ∼ 0.06 dex at low
redshift to ∼ 0.11 dex at 1 < z < 2 and decreases with in-
creasing mass from ∼ 0.08 dex for logM < 10 to ∼ 0.05 for
logM > 10 at z ≃ 0.7. In the K-selected sample the photo-
metric redshift accuracy induced a further uncertainty on
the mass of the order of 30% up to z > 1.5. In the I-selected
sample, where near-IR photometry is not always available,
the typical error on the mass is larger and is of the order
of ∼ 0.13 dex.
8 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
3.3. Effect of NIR Photometry
For about half of the sources in the I-selected sample only
optical photometry is available. We have therefore used the
results obtained for theK-selected spectroscopic subsample
to better understand the reliability of the mass estimates in
the whole I-selected sample and quantify potential system-
atic effects. We found that the mass estimates derived using
only optical bands are on average in rather good agreement
with those obtained using also NIR bands up to z ∼ 1.2. In
absence of NIR bands the galaxy stellar masses tend to be
only slightly overestimated, with a median shift < 0.1 dex;
this is due to the fact that already at z = 0.4, for example,
the z′-band (the reddest band used in the fit in absence of
NIR) samples the R-band rest-frame and therefore the SED
fitting is less reliable for the estimate of the stellar masses.
There is however a significant fraction of the galaxies for
which the ratio between the two masses is higher than a
factor of three (see upper panels of Fig.4). This fraction of
galaxies with significantly discrepant mass estimates is ∼
5% for the models with smooth SFH and ∼ 9 % for the
models with complex SFH.
At higher redshifts, where our reddest optical band, i.e.
the z-band, is sampling the rest-frame spectrum bluewards
of the 4000 Å break, the comparison of the two sets of
mass estimates (i.e. with and without near-IR photometry)
is significantly worse. Not only the median shift increases
significantly, but also the ratio of the two sets of masses
is significantly correlated with the mass derived without
using NIR photometry (see lower panels of Fig. 4). For this
reason, we have decided to use the whole I-selected VVDS
spectroscopic sample only up to z ∼ 1.2, whereas at higher
redshifts we use as reference the K-selected photometric
sample.
As shown in the upper panels of Fig.4, the ratio between
the masses computed without and with NIR photometry
has a non-negligible dispersion also for z < 1.2, with the
masses computed without NIR photometry being higher on
average. In order to statistically correct for this effect, we
have performed the following Monte Carlo simulation. For
each galaxy without near-IR photometry in the I-selected
spectroscopic sample we have applied a correction factor
to its estimated mass. This correction factor has been de-
rived randomly from the observed distribution, at the mass
of each galaxy, of the ratios of the masses with and with-
out NIR photometry. The effect on the mass function of
using these “statistically corrected” masses is shown and
discussed in Sect. 4.
3.4. Comparison of the Masses Obtained with the Two
Methods
In this section we compare the mass estimates we obtained
using the two different methods described above for the
VVDS galaxies. Since the ratio of the two estimates is al-
most independent of the mass, in Fig. 5 we show the his-
tograms of this ratio, integrated over all masses, for two dif-
ferent redshift bins. In the upper panel (z < 1.2), both data
from the I-selected and the K-selected samples are shown;
in the lower panel only data from the K-selected sample
are shown, since the I-selected sample is not used to derive
the mass function in this redshift range. Gaussian curves,
representing the bulk of the population, are drawn on the
Fig. 5. Histograms of the ratio of the masses derived with
the smooth and the complex star formation histories. In
the upper panel (z < 1.2) both data from the I-selected
and the K-selected samples are shown; in the lower panel
(z > 1.2) only data from the K-selected sample are shown,
since the I-selected sample is not used to derive the mass
function in this redshift range.
top of each histogram. The parameters of each Gaussian
are reported in the figure.
The global comparison of the two sets of masses is rather
satisfactory, even if it shows a systematic shift, larger for
theK-selected sample, between the two sets of masses, with
the ‘smooth SFH’ masses being on average smaller than the
‘complex SFH’ masses. The values of σ of these Gaussians
are similar, of the order of 0.13 dex. However, in two of the
three cases (i.e. the I-selected sample at low redshift and
the K-selected sample at high redshift) the distributions of
the ratio of the masses appear to be asymmetric, with tails
which are not well represented by a Gaussian distribution.
The fractions of these “outliers” are given in Fig. 5. This
tail is particularly significant for the I-selected sample, for
which there are galaxies with the ratio between the two
masses higher than 3 and in a few cases reaching a value of
We have analyzed the effect of the different parameters
used in the two methods. The impact of different extinction
curves on the mass estimates has already been investigated
by Papovich et al. (2001), Dickinson et al. (2003), Fontana
et al. (2004), and found to be small. We have repeated the
same exercise for the two different dust attenuation models
adopted, finding that for a given SFH the mass estimated
with different extinction laws are similar, with an average
shift of 0.02 dex.
Analyzing in some detail the properties of the galaxies
which are in the extended tail of large mass ratios for the I-
selected sample at low redshift (see the upper panel in Fig.
5), we found that, even if many of them do not have near-
IR photometry and therefore their mass is more uncertain
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 9
(see previous section), on average they are characterized by
blue colours in the bluer bands (B − I) and red colours in
the redder bands (I − z or I − K). Because of this, they
have been fitted typically with low values of the popula-
tion age by HyperZmass with a smooth SFH, while the
estimate obtained with a complex SFH corresponds to an
older, and therefore more massive, population plus a more
recent burst with a mass fraction in the burst of the order
of fburst < 0.15. Vice versa the low ratio outliers in the K-
selected sample at high redshift have been fitted typically
with moderately higher dust content by HyperZmass than
with complex SFHs. Finally, we conclude that the main dif-
ferences between the two methods to determine the masses
are largely due to different assumed SFHs and, in particu-
lar, to the secondary burst component allowed in the model
with complex SFHs.
To summarize, we have explored in detail two differ-
ent methods and a wide parameter space (see Table 1)
to estimate the stellar mass content in galaxies in order
to better understand the uncertainties in the photomet-
ric stellar mass determinations. Indeed, a good estimate of
the intrinsic errors may be critical for the GSMF measure-
ment and interpretation. We found that, within a given
assumption on the SFH, the accuracy of the photomet-
ric stellar mass is overall satisfactory, with intrinsic un-
certainties in the fitting technique of the order of ∼ 30%,
in agreement with similar results in the K20 (Fontana
et al. 2004), HDFN (Dickinson et al. 2003) and HDFS
(Fontana et al. 2003) at the same redshifts. These errors
are smaller than the estimates at higher redshift (z ≃ 3)
(Papovich et al. 2001, Shapley et al. 2001), since at our av-
erage redshifts (z ≃ 0.7−1) we can rely on a better sampling
of the rest-frame near-IR part of the spectrum, while are
similar to the uncertainties estimated at high-z using IRAC
data by Shapley et al. (2005). For theK-selected photomet-
ric sample the uncertainties in the stellar mass due to the
photometric redshift errors are on average of the order of
30% up to z > 1.5, giving a total fitting uncertainty up
to 45% at z > 1.5. Finally, systematic shifts, mainly due
to different assumptions on the SFHs, can be as large as
∼ 40% over the entire redshift range 0.05 < z < 2.5 when
NIR photometry is available. The uncertainty in the de-
rived masses is obviously higher also at low redshift when
NIR photometry is not available and in this case it becomes
extremely large at z > 1.2 (see Fig.4).
For what concerns the absolute value of the mass, its
uncertainty is mainly due to the assumptions on the IMF
and it is within a factor of 2 for the typical IMFs usually
adopted in the literature.
In the following sections we discuss in some detail the
effects of the two methods on the derivation of the GSMF.
3.5. Massive Galaxies at z > 1
Figure 6 shows the stellar masses for the 2 samples (I and
K-selected) derived using the smooth SFHs. It is inter-
esting to note the presence of numerous massive objects
(logM > 11) at all redshifts and up to z = 2.5. High-
z massive galaxies have been already observed in previous
surveys (Fontana et al. 2004, Saracco et al. 2005, Cimatti et
al. 2004, Glazebrook et al 2004, Fontana et al. 2006, Trujillo
et al. 2006). Here the relatively wide area (the K-selected
sample is more than 10 times wider and from 0.5 to 1 mag-
nitude deeper than the K20 survey) allows to better sam-
Fig. 6. Stellar Mass as a function of redshift for the I-
selected spectroscopic (left) and for the K-selected photo-
metric (right) samples for smooth SFHs.
ple the massive tail of the population. We note that mas-
sive galaxies have typically redder optical-NIR rest-frame
colours (〈MI −MK〉 ≃ 0.7) compared to the whole popu-
lation (〈MI −MK〉 ≃ 0.5), consistently with the idea that
massive galaxies host the oldest stellar population. Further
analysis of the stellar population properties and spectral
features of massive galaxies, as well as of red objects will
be presented in forthcoming papers (Lamareille et al. in
preparation, Vergani et al. 2007, Temporin et al. in prepa-
ration).
4. Mass Function Estimate
Once the stellar mass has been estimated for each galaxy
in the sample, the derivation of the corresponding Galaxy
Stellar Mass Function (MF) follows the traditional tech-
niques used for the computation of the luminosity function.
Here we apply both the classical non-parametric 1/Vmax
formalism (Schmidt 1968, Felten 1976) and the parametric
STY (Sandage, Tammann & Yahil 1979) method to esti-
mate best-fit Schechter (1976) parameters (α,M∗stars, φ
In the case of the K-selected sample, in order to take into
account the two different magnitude limits, we perform a
“Coherent Analysis of independent samples” as described
by Avni & Bahcall (1980).
Ilbert et al. (2004) have shown that the estimate of the
faint end of the global luminosity function can be biased,
because, due to different k-corrections, different galaxy
types have different absolute magnitude limits for the same
apparent magnitude limit. The same bias is present also for
the low mass end of the mass function. This is due to the
fact that, because of the existing dispersion in the mass-to-
light ratio of different galaxy types, at small masses the ob-
jects with the largest mass-to-light ratio are not included in
a magnitude limited sample (see Appendix B in Fontana et
10 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
Fig. 7. Effect on the I-selected MF of the use of the statistically corrected masses, which take into account the effect of
near-IR photometry on the mass determination (see text for details). Complex SFHs have been used to derive masses.
Empty circles represent the MF obtained using the original uncorrected masses, filled squares show the MF obtained
using the statistically corrected masses. For comparison the STY Schechter MF is shown for the subsample where near-IR
photometry is available. The vertical dashed lines represent the completeness limit of the sample as defined in the text.
al. 2004 for an extensive discussion). For this reason, when
computing the global mass functions with the STY method,
in order to fully avoid this bias, we should use in each red-
shift range only galaxies above the stellar mass limit where
all the SEDs are potentially observable. In both the K- and
I-selected samples these limits derive from the conversion
between photometry and stellar mass for early-type galax-
ies and are very restrictive. However, recent results show
that the faint-end and the low-mass end of the luminosity
and mass function is dominated by late-type galaxies up to
z ∼ 2 (Fontana et al. 2004, Zucca et al. 2006, Bundy et al.
2006). Therefore for the STY estimate we will use as lower
limit of the mass range the minimum mass above which
late-type SEDs (defined by rest-frame optical/NIR colours
MI −MK < 0.4) are potentially observable (see Fig. 9).
We have estimated the MF for both the deep I-selected
(17.5 < I < 24) spectroscopic sample and the photomet-
ric K-selected (K < 22.34 & K < 22.84) sample. For
each sample the MF has been estimated using masses com-
puted with both methods described in Section 2.3. In the
case of the spectroscopic sample, in order to correct for
both the non-targeted sources in spectroscopy and those
for which the spectroscopic measurement failed, we use a
statistical weight wi, associated with each galaxy i with
a secure redshift measurement (see Ilbert et al. 2005 for
details). This weight is the inverse of the product of the
Target Sampling Rate times the Spectroscopic Success Rate.
Accurate weights have been derived by Ilbert et al. (2005)
for all objects with secure spectroscopic redshifts, tak-
ing into account all the parameters involved (magnitudes,
galaxy size and redshift).
For the K-selected sample, we have tested the effect of
catastrophic photometric redshifts (see discussion in Sec.
2.2) on the evolution of the mass function and mass den-
sity. We have used the I-selected spectroscopic sample, re-
placing spectroscopic redshifts with photometric redshifts.
The two MFs (with either spectroscopic or photometric red-
shifts) are very similar in the whole mass and redshift range
(0.05 < z < 2.5) analyzed and even at z > 1.2, where we
note a not negligible number of catastrophic photometric
redshifts (see discussion in Sec. 2.2). There is no evidence
of a strong bias in the normalization and in the shape of the
MF; also the massive tails of the MFs are similar, within
the statistical errors. We conclude that the catastrophic
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 11
Fig. 8.K-selected MF derived from the 2 subsamples, deep (filled circles) and shallow (empty circles), separately (smooth
SFHs have been used to derive masses). For comparison the STY Schechter MF for the globalK-selected sample is shown.
The vertical dashed lines represent the completeness limit of the 2 K-selected subsamples.
solutions at high photometric redshifts (i.e. masses) do not
strongly affect our results.
4.1. The VVDS Galaxy Stellar Mass Function
The resulting stellar mass functions of the VVDS sample
are derived in the following redshift ranges: (a) 0.05 < z <
1.2 for the I-selected sample, because at higher redshift the
mass estimate becomes very uncertain (see figure 4) and
(b) 0.05 < z < 2.5 for the K-selected sample. We have
furthermore divided the 2 samples into different redshift
bins in order to sample evolution with similar numbers of
sources in each bin.
For the galaxies in the I-selected sample not covered
by near-IR data, we have used the statistically corrected
masses derived through a Monte Carlo simulation, to take
into account the effect of the near-IR photometry in the
mass determination (see Sec. 3.3). Figure 7 shows the ef-
fect in the MF for complex SFHs. The high mass tail is
significantly reduced if we use statistically corrected masses
when near-IR is not available. Consistent MFs have been
12 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
Fig. 9. Galaxy Stellar Mass Functions in the I-selected (squares) and K-selected (triangles) using both methods to
estimate the stellar masses (empty symbols for smooth SFHs and filled for complex SFHs). The STY Schechter fits for
the 2 methods limit the hatched regions (horizontal hatched for the K-selected and vertical hatched for the I-selected
samples). Vertical hatched regions represents the completeness limit of the 2 samples. The local MFs by Cole et al.
(2001), both original and “rescaled” version (Fontana et al. 2004), and by Bell et al. (2003) are reported in each panel
as dotted lines.
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 13
Table 2. STY parameters in the different redshift ranges
sample method z range mean redshift α logM∗stars(h
70 M⊙) φ
∗(10−3h370Mpc
I smooth 0.05 - 0.4 0.27 -1.26+0.01
−0.02 11 1.90
+0.08
−0.16
I smooth 0.4 - 0.7 0.58 -1.23+0.04
−0.04 11
+0.08
−0.08 1.72
+0.33
−0.28
I smooth 0.7 - 0.9 0.8 -1.23+0.12
−0.11 10.88
+0.15
−0.13 1.6
+0.55
−0.45
I smooth 0.9 - 1.2 1.05 -1.09+0.19
−0.17 10.85
+0.14
−0.14 1.3
+0.43
−0.46
I complex 0.05 - 0.4 0.27 -1.28+0.02
−0.01 11.15 1.75
+0.16
−0.08
I complex 0.4 - 0.7 0.58 -1.22+0.04
−0.04 11.15
+0.08
−0.08 1.58
+0.30
−0.26
I complex 0.7 - 0.9 0.81 -1.04+0.08
−0.07 10.83
+0.07
−0.07 3.02
+0.57
−0.51
I complex 0.9 - 1.2 1.04 -1.16+0.1
−0.09 10.89
+0.08
−0.07 1.80
+0.44
−0.39
K smooth 0.05 - 0.4 0.26 -1.38+0.02
−0.01 10.93 1.29
+0.10
−0.05
K smooth 0.4 - 0.7 0.57 -1.14+0.04
−0.04 10.93
+0.06
−0.06 1.83
+0.27
−0.24
K smooth 0.7 - 0.9 0.81 -1.01+0.07
−0.08 10.67
+0.07
−0.05 2.6
+0.38
−0.44
K smooth 0.9 - 1.2 1.05 -1.1+0.07
−0.08 10.78
+0.06
−0.05 1.83
+0.28
−0.30
K smooth 1.2 - 1.6 1.4 -1.15+0.12
−0.12 10.72
+0.07
−0.06 1.48
+0.30
−0.30
K smooth 1.6 - 2.5 1.96 -1.15 10.96+0.01
−0.02 0.9
+0.30
−0.30
K complex 0.05 - 0.4 0.26 -1.39+0.01
−0.02 11.12 1.17
+0.05
−0.09
K complex 0.4 - 0.7 0.57 -1.16+0.04
−0.04 11.12
+0.06
−0.06 1.58
+0.24
−0.22
K complex 0.7 - 0.9 0.81 -1.16+0.07
−0.07 10.98
+0.07
−0.07 1.74
+0.36
−0.30
K complex 0.9 - 1.2 1.05 -1.2+0.07
−0.06 11.07
+0.06
−0.06 1.34
+0.26
−0.21
K complex 1.2 - 1.6 1.4 -1.17+0.12
−0.12 10.93
+0.07
−0.06 1.39
+0.29
−0.28
K complex 1.6 - 2.5 1.96 -1.17 10.97+0.01
−0.02 1.25
+0.09
−0.04
obtained in the sub-area of I-selected sample where near-IR
photometry is available (see Fig. 7).
For the K-selected sample, we have analyzed the ef-
fect of the cosmic variance on small areas, deriving the MF
for the two K-selected subsamples, deep and shallow, sepa-
rately (see Figure 8). We find a significantly lower MF (by
a factor 1.8 and 1.6) in the redshift range 0.4 < z < 0.7 and
0.7 < z < 0.9 in the deep K-selected sample (K < 22.84
over 168 arcmin2) compared to the shallow K-band sam-
ple (K < 22.34 over 442 arcmin2). The significance of such
differences in the MF, is of ∼ 2 − 3σ in each mass bin at
0.4 < z < 0.7 and ∼ 1 − 2σ at 0.7 < z < 0.9. Globally,
i.e. for the total number densities over the complete mass
range, the differences are significant at about 5-3 σ level
in the two redshift ranges, respectively. This problem leads
to a clear warning on the results based on small fields, as
covered by most of the previous existing surveys.
In Figure 9 we show the MFs derived using the I-
selected spectroscopic sample and the K-selected photo-
metric sample for both methods (smooth and complex
SFHs) to derive the masses. The resulting mass functions
are quite well fitted by Schechter functions. The best-fit
Schechter parameters are summarized in Table 2, with the
uncertainties derived from the projection of the 68% confi-
dence ellipse. Since in the lowest and highest redshift bins
(z ≃ 0.2 and 2) the values of M∗stars and the low-mass-end
slope (α), respectively, are poorly constrained, they have
been fixed to the values measured in the following and pre-
vious redshift bins respectively.
We note, first, that the overall agreement between the
MF derived with the different methods for masses determi-
nation is fairly satisfactory, albeit complex SFHs estimates
provide typically larger masses. The systematic shift be-
tween the 2 methods (Section 3.4) is reflected in most of
the redshift bins also in the characteristic mass (M∗stars)
of the MF while the best fit slopes (α) and φ∗ Schechter
parameters agree within the errors between the different
methods in most of the redshift bins.
We note, furthermore, an overall agreement between the
2 samples (I- and K-selected), and in most of the redshift
bins the Schechter parameters agree within the errors, even
if some differences exist. More in detail, the I-selected sam-
ple in the range 0.4 < z < 0.9 has a higher low-mass
(< 9.5 dex) end and a slightly steeper MF (α ∼ −1.23)
than the K-selected one (α > −1.15). These differences are
probably due to the population of blue K-faint galaxies,
that are missed in the K-sample, as discussed in Section
3. These galaxies have, indeed, median colours in the I-
selected sample that are bluer than in the K-selected sam-
ple (I−K ≃ 0.45 compared to ≃ 0.89). A similar behaviour
has been noted in the local MF derived using an optically
selected sample (g band) compared to the local MF from
the near-IR (2MASS) sample (Bell et al. 2003). On the
contrary, at even lower masses (< 8.5 dex) at z < 0.4 the
K-selected MF is slightly steeper than the I-selected one,
but no significant differences in the colour of the two pop-
ulations is found.
4.2. Comparison with Previous Surveys
In general, previous efforts to derive MF have relied on
smaller or more limited samples, or often based mainly
on photometric redshifts (Drory et al. 2004, 2005). We
have compared our MF determination with literature re-
sults based on different surveys (K20, COMBO17, MUSIC,
DEEP2, FDF+CDFS), rescaled to Chabrier IMF (see
Figure 10). Our MFs rely on a higher statistics at inter-
mediate to high-mass ranges, and therefore present lower
statistical errors. At z < 0.2 we sample unprecedented mass
ranges, more than one order of magnitude lower than previ-
ous surveys, while at z > 0.4 the FDF and MUSIC surveys
reach lower mass limits even if on significantly smaller area.
14 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
Fig. 10. Comparison between the I-selected and the K-selected MFs in the VVDS (hatched vertical and horizontal STY
regions, respectively, see caption Figure 9) and the literature data (K20, COMBO17, MUSIC, DEEP2, FDF+CDFS; for
each the band of selection is indicated in the parenthesis). The vertical hatched regions represent the completeness limits
of the VVDS samples.
Our MFs are in fairly good agreement with previous studies
over the whole mass range up to z ∼ 1.2. However, some
differences exist, in particular at the massive end, which is
more sensitive to the different selections, methods, statis-
tics and to cosmic variance due to large scale structures: for
example, in the MUSIC-GOODS survey there are two sig-
nificant overdensities at z ∼ 0.7. The MFs from COMBO17
(Borch et al. 2006) and also from DEEP2 (Bundy et al.
2006) are systematically higher than previous surveys at
the massive-end, in particular in the range 0.7 < z < 1.2.
The MFs in the FDF+CDFS are instead systematically
lower than ours at the massive-end and higher than our
extrapolation to masses lower than our completeness limit.
At z > 1.2 our MF is systematically higher than previ-
ous studies. Given the area sampled (more than a factor
4 wider compared to FDF+CDFS and to MUSIC) and
the consistency at these redshifts of our MF in the 2 K-
selected separated areas (see Figure 8), we are confident
in our results. However at high-z the uncertainties on the
stellar masses estimate increase (up to 0.16 dex including
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 15
also the photometric redshift errors) and could produce a
partially spurios excess in the number densities of galax-
ies, in particular in the massive tail of the MF. This ef-
fect is discussed in Kitzbichler & White (2007) which take
into account in the hierarchical formation Millenium simu-
lation the effect of the dispersion in the mass determination
(0.25 dex, i.e. 78%). We have performed a similar analysis,
taking into account the uncertainty on the mass, due to
the fitting technique (∼ 30%) and to the uncertainty of
the photometric redshifts (both its dependence on redshift
and magnitudes as described in previous sections, i.e. up to
σz ≃ 0.2 at z = 2 and K > 21.5). We found that the effect
on the MF is always small, and only the very massive tail
(M > 2 × 1011M⊙) is systematically overestimated (up to
0.2 dex). This effect can not completely explain the excess
found compared to previous surveys, which are affected in
a similar way by the same bias.
4.3. The Evolution of the Galaxy Stellar Mass Function
The VVDS allows us to follow the evolution of the
MF within a single sample over a wide redshift range.
Difficulties in the interpretation of the evolution are, in-
deed, due to the comparison with the local MFs, which
have been determined with different methods and sample
selection. For example, no local MF has been derived using
complex SFHs for mass determinations. In our analysis we
use, as reference, the local MF by Bell et al. (2003) and Cole
et al. (2001) rescaled to Chabrier IMF. In particular, the
Cole et al. (2001) local mass function, derived with smooth
SFHs but with formation redshift fixed at z = 20, has been
rescaled to smooth SFHs method with free formation red-
shift by Fontana et al. (2004).
The first important result is that, thanks to our very
deep samples, both I- and K-selected, the low-mass end of
the MF is even better determined than in the local sam-
ple up to z < 0.4, probing for the first time masses down
to about 3 × 107M⊙. The low mass-end is rather steep
(−1.38 < α < −1.26), and could even be described by a
double Schechter function, and is steeper than the local es-
timates (α = −1.18±0.03 Cole et al. 2001, α = −1.10±0.02
Bell et al. 2003), possibly due to the fact they are not prob-
ing masses smaller than 109 M⊙ (more than one order of
magnitude more massive than in our sample). As evident
from Figure 9, we find a substantial population of low-mass
(< 109M⊙) galaxies at low redshifts (z < 0.4). This popu-
lation is composed by faint blue galaxies with similar prop-
erties in the 2 samples (I- and K-selected): I,K ≃ 22− 23,
MI ,MK ≃ −16,−17 with median MI − MK ≃ 0.3, and
median z ≃ 0.1− 0.2. This is a very strong result from our
survey which can rely on a wider area and a deeper sam-
ple than previous surveys at low redshifts. At z > 0.4 the
low-mass slope is on the contrary always consistent with
the local values. Even if we are not probing masses smaller
than 108M⊙ at z > 0.4, we found that the MF remains
quite flat (−1.23 < α < −1.04) at all redshifts, similar to
that of Fontana et al. (2006) which probe lower masses (see
figure 10).
From a visual inspection of Figure 9, we see that up to
z ∼ 0.9 there is only a weak evolution of the MF, as sug-
gested by previous results (Fontana et al. 2004, 2006, Drory
et al. 2005), while at higher redshifts there appears to begin
a decrease in the normalization of the MF, even if a massive
tail remains present up to z = 2.5. At intermediate masses
Fig. 11. Cosmological evolution of the galaxy number den-
sity as a function of redshift, as observed from the VVDS
in various mass ranges (> 108M⊙, > 10
9.77M⊙ and >
1010.77M⊙ from top to bottom). Observed data from Vmax
(shown as lower limit in the top panel) have been corrected,
when necessary, for incompleteness integrating the mass
function using the best fit Schechter parameters. VVDS
data (big filled circles), averaged over the I- and the K-
selected samples and the 2 methods to derive the mass, are
plotted along with their statistical errors (solid error bars)
and the scatter between the 2 different samples and meth-
ods (dotted error bars). The solid lines show the best-fit
power laws ∝ (1 + z)β, while the dashed lines correspond
to the no-evolution solution normalized at z = 0. Results
from previous surveys (small points and dot-dashed lines)
are also shown.
(9.5 < logM < 10.5), our VVDS MF is very well defined
and shows a clear evolution, i.e. the number density de-
creases with increasing redshifts compared to both the first
VVDS redshift bin and the local MF. This evolution is quite
mild up to z ≃ 0.9, while it becomes faster at higher z. At
larger masses the high mass end of the MF (> 1011 M⊙)
shows a small evolution up to z ≃ 2.5. However, its evolu-
tion is extremely dependent on the assumed local MF and
on the uncertainties in the mass determination, which pro-
duce a larger dispersion between the different methods and
samples compared to the intermediate-mass range.
In order to quantify the MF evolution, and its mass
dependency independently from the local MF, in the next
section we derive number densities for different mass limits.
16 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
Table 3. Number Density and Stellar Mass Density
zinf zsup Log(ρN ) Scatter Error Log(ρstars)Scatter Error
log(Mstars)> 8
0.05 0.40 -1.40 0.04 0.01 8.45 0.09 0.01
0.40 0.70 -1.63 0.08 0.02 8.34 0.08 0.02
0.70 0.90 -1.70 0.08 0.04 8.22 0.11 0.01
0.90 1.20 -1.78 0.13 0.05 8.14 0.12 0.01
1.20 1.60 -1.81 0.05 0.11 8.04 0.14 0.02
1.60 2.50 -1.90 0.13 0.01 8.05 0.11 0.01
log(Mstars)> 9.77
0.05 0.40 -2.20 0.06 0.01
0.40 0.70 -2.29 0.03 0.01
0.70 0.90 -2.33 0.04 0.01
0.90 1.20 -2.45 0.05 0.01
1.20 1.60 -2.54 0.02 0.01
1.60 2.50 -2.65 0.01 0.01
log(Mstars)> 10.77
0.05 0.40 -3.07 0.18 0.05 8.00 0.22 0.04
0.40 0.70 -3.04 0.10 0.02 8.04 0.14 0.02
0.70 0.90 -3.22 0.16 0.02 7.80 0.19 0.02
0.90 1.20 -3.28 0.15 0.02 7.76 0.18 0.02
1.20 1.60 -3.42 0.23 0.02 7.59 0.28 0.02
1.60 2.50 -3.35 0.11 0.01 7.70 0.11 0.01
4.4. Galaxy Number Density
Here we derive the number density of galaxies as a function
of redshift, using different lower limits in mass (Mmin). We
have estimated the number density from the observed data
(from Vmax), as well as from the incompleteness-corrected
MFs, i.e. integrating the best-fit Schechter functions over
the considered mass range. The corrections due to faint
galaxies dominate for Mmin = 10
8M⊙, while they are neg-
ligible for the other mass limits considered. A formal uncer-
tainty in this procedure was estimated by considering the
Vmax statistical errors and the range of acceptable Schechter
parameters values. In Figure 11 we plot our VVDS determi-
nations, averaged over the I- and K-selected samples and
the 2 methods for mass determination (listed in Table 3),
along with their statistical errors (always less than 10%)
and the scatter between the 2 different samples and meth-
ods (ranging between 10 and 45% and due mainly to the dif-
ferent methods rather than to the different samples). With
the two methods we find similar trends with redshift of the
number densities of galaxies, but with a systematic shift
which is significant only for the highest mass limit (for com-
plex SFHs the galaxy number densities are ∼ 50% higher
than for smooth SFHs). The effect of photometric redshift
and mass uncertainty on the number densities is always
small (< 15%) for the mass range shown in Figure 11, ex-
cept for the very massive galaxies (> 2 × 1011 M⊙, not
shown in the figure because of the small number of galaxies
in this mass range) where the intrinsic values could be up
to a factor ∼ 2 lower (see discussion in section 4.2). The
decrease in number density with redshift for all the adopted
mass limits is evident.
We have compared VVDS results to previous surveys
and with different local determinations. For the total num-
ber density (108 < Mstars < 10
13M⊙) VVDS data are very
well consistent with the evolutionary STY fit determined by
Fontana et al. (2006) in the GOODS-MUSIC survey. At in-
Fig. 12. Cosmological evolution of the stellar mass density
as a function of redshift as observed from the VVDS for
2 mass ranges: integrated over the whole range 108M⊙ ≤
Mstars ≤ 10
13M⊙ (upper panels) and for massive galax-
ies (> 1010.77M⊙) (lower panels). Symbols and lines as in
Figure 11.
termediate masses (> 109.77M⊙, corresponding to 10
for Salpeter IMF) our VVDS data have a better determina-
tion and smaller uncertainties than previous ones and are
consistent with most of them at z < 1.2 and in the upper en-
velope at higher z. For the high mass range (> 1010.77M⊙,
corresponding to 1011M⊙ for Salpeter IMF) we are quite
consistent with previous results, and even if our VVDS have
lower errors than previous ones, the dispersion within the
various VVDS measurements reflect the uncertainties for
massive galaxies.
If we represent the average number density evolution
by a power law ρN ∝ (1 + z)
β , we find that β(Mstars >
108) = −1.28 ± 0.15, β(Mstars > 10
9.77) = −1.26 ± 0.10,
and β(Mstars > 10
10.77) = −1.01 ± 0.05 (the errors on β
represent the uncertainties due to the 2 different methods).
We find on average a similar evolution for the 2 methods
analyzed and a slightly milder evolution with increasing
mass limit (‘downsizing’ in mass assembly). The average
evolution from z = 0 to z = 1 is a factor 2.4 ± 0.3 and
2.0± 0.1 from low to high-mass galaxies, respectively, and
increases to a factor 4.0 ± 0.9 and 3.0 ± 0.2, respectively,
at z = 2. We note moreover that for the highest mass limit
(Mstars > 10
10.77) at low redshift (z < 0.7) the number
density observed is consistent with no-evolution (fixing the
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 17
value to z = 0 we found an evolution < 30%), excluded
instead for the intermediate and low-mass limit. At high-z
(z > 1.5) for intermediate and high-mass range we note a
flattening in the number density from the VVDS data, sim-
ilar to Drory et al. (2005), but higher than Fontana et al.
(2006). This flattening is due to the high mass tail observed
in the range 1.6 < z < 2.5. This population shows ex-
tremely red colours (MI −MK ≃ 0.8) and could be related
to the appearance of a population of massive star forming
dusty galaxies, observed in previous surveys (Fontana et
al. 2004, Daddi et al. 2004b). The small excess induced by
uncertainties on the mass and photometric redshifts (see
discussion in section 4.2) can not completely explain the
difference with Fontana et al. (2006) survey, which is af-
fected by the same bias in a similar way.
The mass dependent evolution (“mass downsizing”) is
very debated and the results from different surveys are still
controversial. Deep surveys, such as the FDF & CDFS (an-
alyzed by Drory et al. 2005) find an evolution consistent
with ours (a decrease of about a factor 2.5 – 4 at z = 1− 2
in the number density of galaxies> 1010.77 M⊙). Fontana et
al. (2006) suggest a similar mild evolution up to z = 1, for
massive galaxies (> 1010.77M⊙) and a stronger evolution
at z > 1.5, reaching a factor about 10 at z = 3. Similarly,
Cimatti et al. (2006) show that the number density of lu-
minous (massive, Mstars > 10
11M⊙) early-type galaxies is
nearly constant up to z ∼ 0.8, while Bundy et al. (2006)
find a slight decrease, consistent with no evolution, only for
even more massive system (> 3×1011M⊙) and a more sig-
nificant decline for Mstars < 3× 10
11M⊙. Vice versa, data
from the MUNICS survey (Drory et al. 2004) show a faster
evolution of massive galaxies, even faster than for the less
massive systems (see also Figure 4 in Drory et al. 2005).
To summarize, our accurate results show that the MF
evolves mildly up to z ≃ 1 (about a factor 2.5 in the total
number density) and that a high-mass tail is still present
up to z = 2.5. Moreover, we find that massive systems
show an evolution that is on average milder (< 50% at
z < 1) than intermediate and low-mass galaxies and con-
sistent with a mild/negligible evolution (< 30%) up to
z ∼ 0.8. Conversely, a no-evolution scenario in the same
redshift range is definitely excluded for intermediate- and
low-mass galaxies.
This behaviour suggests that the assembly of the stellar
mass in objects with mass smaller than the localM∗stars was
quite significant between z = 2 and z = 0. Qualitatively,
this behaviour is expected for galaxies with SFHs prolonged
over cosmic time, which therefore continue to grow in terms
of stellar mass after z ∼ 1. Conversely, our results fur-
ther strengthen the fact that the number density of mas-
sive galaxies is roughly constant up to z ≃ 0.8, consistently
with a SFH peaked at higher redshifts, with the conversion
of most of their gas into stars happening at z > 1.5−2, rul-
ing out the ‘dry mergers’ as the major mechanism of their
assembly history, below z < 1.
5. Mass Density
Various attempts to reconstruct the cosmic evolution of the
stellar mass density have been previously made, mainly us-
ing NIR-selected samples (Dickinson et al. 2003, Fontana et
al. 2004, Drory et al. 2005). Our survey offers the possibil-
ity to investigate it using the MF derived from two different
optical- and NIR-selected samples, taking advantage of our
depth, and relatively wide area covered. Furthermore, the
different methods analyzed here to derive the stellar mass
content give us a direct measure of the uncertainties in-
volved.
We have estimated the stellar mass density from the
observed data, as well as from the incompleteness-corrected
MFs. Up to z < 1 the corrections due to faint galaxies are
relatively small. A formal uncertainty in this procedure was
estimated by considering the Vmax statistical errors and the
range of acceptable Schechter parameters values.
Figure 12 shows our results (averaged over the I- andK-
selected samples and the 2 methods with a typical scatter
of about 30-50% and statistical errors always less than 5%
see Table 3), for the total mass density and for the density
in massive galaxies (> 1010.77), along with their represen-
tative power laws (ρstars ∝ (1+z)
β), and compared to liter-
ature data (see references in the figure). For the total mass
density, even if the results from our survey cover a range
of values with some significant differences between the two
different methods (up to ∼ 40%), the general behaviour
and evolutionary trend is well defined by β = −1.19± 0.05.
We find that the evolution of the stellar mass density is
relatively slow with redshift, with a decrease of a factor
2.3 ± 0.1 up to z ≃ 1, up to a factor 4.5 ± 0.3 at z = 2.5.
The agreement of average total mass density with previous
surveys is reasonably good, and the range covered by VVDS
data reflect the different selection techniques and methods
used in different surveys. The average total mass density
evolution is milder than in the MUSIC sample (Fontana et
al. 2006) already at z > 0.5. Our evolutionary trend is con-
sistent with the upper envelope of previous surveys, even if
our highest-redshift value is uncertain because the low-mass
slope is poorly constrained. For comparison the analysis of
VVDS data using a IRAC-selected sample (see Arnouts et
al. 2007) finds similar values for the mass density, except
that the highest redshift point is lower than ours. Given the
present uncertainties on the low-mass slope of the GSMF,
the total mass density at z ≃ 2 remains poorly constrained.
The mass density of high-mass objects (> 1011M⊙ with
Salpeter IMF) varies by a factor up to 1.8 within the 2
methods adopted, but the evolutionary trend is similar
(β = −1.13± 0.01) and consistent with a decrease of about
a factor 2.18 ± 0.02 to z = 1 and 3.44 ± 0.04 to z = 2.0.
Moreover at low redshift (z < 0.7) the VVDS observed data
are consistent with a mild/negligible evolution (< 30%), as
indicated by the number density of massive galaxies (see
previous Section). Our data are roughly consistent with
Fontana et al. (2006) up to z = 1.5 (even if the slope of the
evolutionary trend is shallower), while at z > 1.5 the VVDS
mass-density of massive galaxies is significantly higher than
that in Fontana et al. (2006), reflecting the excess in MF at
high-z noted in the VVDS MF compared to previous ones
(see Section 4.2). This results, therefore, in a flatter evolu-
tionary trend over the total redshift range. Given the wider
area and completeness for high-mass objects, our samples
guarantee a higher statistical accuracy and confidence level
than before. However some caveat remains due to the effect
of photometric redshift and mass uncertainty on the mass
densities, which is anyhow always small (< 15%) except for
the massive galaxies (> 6 × 1010 M⊙) where the intrinsic
values could be up to 20-30% lower than our estimates (see
discussion in Section 4.2).
18 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
6. Summary and Discussion
We have investigated the evolution of the Galaxy Stellar
Mass Function up to to z = 2.5 using the VVDS survey
covered by deep VIMOS spectroscopy (17.5 < I < 24)
and multiband photometry (from U to K-band). For our
analysis we have used two different samples: (1) the opti-
cal (I-selected, 17.5 < I < 24) main spectroscopic sam-
ple, based on about 6500 secure redshifts over about 1750
arcmin2, and (2) a near-IR sample (K-selected, K < 22.84
& K < 22.34), in a sub-area of about 610 arcmin2 and
based on about 10200 galaxies with accurate photomet-
ric and spectroscopic redshifts. For the first time we have
probed masses down to a very low limit, in particular at
low-z (down to ∼ 3 × 107M⊙ at z ∼ 0.2), while the rela-
tively wide area has allowed us to determine the MF with
much higher statistical accuracy than previous samples.
In order to better understand uncertainties we have ap-
plied and compared two methods to estimate the stellar
mass content in galaxies from multiband SED fitting. The
2 methods differ in the explored parameter space (metal-
licity, dust law and content) and are based on different as-
sumptions on previous star formation history. The main
results from the stellar mass estimate can be summarized
as follows:
– The agreement between the 2 methods is fairly good
even if masses estimated with ‘complex SFHs’ are
systematically higher than ‘smooth SFHs’ masses.
For the K-selected sample the mean difference is
〈d logMstars〉 ≃ 0.12 dex, and the dispersion is σ =
0.13. The differences are mainly due to the secondary
burst component (complex SFHs) compared to smooth
SFHs.
– We found that mass estimates using only optical bands
are in rather good agreement with those using also NIR
bands up to z ∼ 1.2. We have used this information to
statistically correct masses for objects without near-IR
photometry. At higher redshifts the shift and dispersion
dramatically increase and the mass estimates become
unreliable if near-IR photometry is not available.
We have, thus, derived the MF using the VVDS I-
selected sample and extended it up to z = 2.5 thanks to
the K-selected sample. From a detailed analysis of the MF,
galaxy number density and mass density, in different mass
ranges, through cosmic time, we found evidences for:
– a substantial population of low-mass galaxies (<
109M⊙) at z ≃ 0.2 composed by faint (I,K ≃ 22, 23)
blue galaxies with median MI−MK ≃ 0.3, and absolute
magnitudes MI ,MK ≃ −16,−17;
– a slow evolution of the stellar mass function with red-
shift up to z ∼ 0.9 and a faster evolution at higher-z,
in particular for less massive systems. A massive popu-
lation is present up to z = 2.5 and have extremely red
colours (MI −MK ≃ 0.7− 0.8).
– at z > 0.4 the low-mass slope of the GSMF does not
evolve significantly and remains quite flat (−1.23 < α <
−1.04).
– the number density shows, on average, a mild differen-
tial evolution with mass, which is slower with increas-
ing mass limit. Such evolution can be described by a
power law ∝ (1 + z)β(>M). Within the VVDS redshift
range we found that β(> 108M⊙) = −1.28 ± 0.15,
β(> 109.77M⊙) = −1.26± 0.10 and β(> 10
10.77M⊙) =
−1.01± 0.05. For massive galaxies at low redshift (z <
0.7) the evolution is consistent with mild/negligible-
evolution (< 30%), which is excluded for low-mass sys-
tems.
– the evolution of the stellar mass density is relatively slow
with redshift, with a decrease of about a factor 2.3±0.1
to z ≃ 1, while at z ≃ 2.5 the decrease amounts to a fac-
tor up to 4.5± 0.3, milder than in previous surveys. For
massive galaxies the evolution at low redshift (z < 0.7)
is consistent with a mild/negligible evolution(< 30%),
and shows a flattening compared to previous results at
z > 1.5 due to a population with extremely red colours.
Our results provide new clues on the controversial ques-
tion of when galaxy formed and assembled their stellar
mass. Most of the massive galaxies seem to be in place up to
z = 1 and have, therefore, formed their stellar mass at high
redshift (z > 1), rather than assembled it mainly through
continuous galaxy merging of small galaxies at z < 1. On
the contrary, less massive systems have assembled their
mass (through merging or prolonged star formation his-
tory) later in cosmic time. In agreement with our results, a
substantial population of high-z (z ∼ 2−3) dusty and mas-
sive objects have been discovered in near-IR surveys (Daddi
et al. 2004b) and detected by Spitzer in the far-IR (Daddi
et al. 2005, Caputi et al. 2006b). This population could
be related to the initial phase of massive galaxy formation
during their strong star forming and dusty phase.
Finally, our results are not completely accounted for by
most of theoretical models of galaxy formation (see Fontana
et al. 2004, 2006 and Caputi et al. 2006a for a detailed com-
parison with models). For instance, models by De Lucia et
al. (2006) predict that the most massive galaxies generally
form their stars earlier, but assemble them later, mainly
at z < 1 via merging, than the less massive galaxies (i.e.
’downsizing in star formation but ’upsizing’ in mass assem-
bly, see Renzini 2007 for a recent discussion). Furthermore,
the stronger decrease with redshift of the low-mass popu-
lation, with a low-mass end of the GSMF which remains
substantially flat up to high redshift, is not reproduced by
most of the theoretical galaxy assembly models, which tend,
indeed, to overpredict the low-mass end of the MF (see
Fontana et al. 2006).
Understanding the mass assembly of less massive ob-
jects and disentangling merging processes from prolonged
star formation history is more complicated. In this respect
for a better comprehension of galaxy formation the VVDS
will allow us to further investigate the evolution of the stel-
lar mass function up to high-z also for different galaxy types
(spectral and morphological) and in different environments.
For example Arnouts et al. (2007) study the mass density
evolution of different galaxy population. Further analysis
of galaxy mass dependent evolution, using stellar popula-
tion properties, as well as observed spectral features, will
be presented in forthcoming papers (Lamareille et al., in
preparation, Vergani et al. 2007). Furthermore, it will be
possible to push the study of the galaxy stellar mass func-
tion at higher redshifts using SPITZER mid-IR observa-
tions. While most of present studies (Dickinson et al. 2003,
Drory et al. 2005) do not use rest-frame near-IR photome-
try to estimate stellar masses, our VVDS-SWIRE collabo-
ration will allow to combine the deep VVDS spectroscopic
sample with SPITZER-IRAC photometry.
L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History 19
Acknowledgements. This research has been developed within the
framework of the VVDS consortium.
This work has been partially supported by the CNRS-INSU and
its Programme National de Cosmologie (France), and by Italian
Ministry (MIUR) grants COFIN2000 (MM02037133) and COFIN2003
(num.2003020150).
The VLT-VIMOS observations have been carried out on guaranteed
time (GTO) allocated by the European Southern Observatory (ESO)
to the VIRMOS consortium, under a contractual agreement between
the Centre National de la Recherche Scientifique of France, heading
a consortium of French and Italian institutes, and ESO, to design,
manufacture and test the VIMOS instrument. We are in debt with
E. Bell, S. Salimbeni, and E. Fontana for providing the data from
their survey in electronic format, and to C. Maraston for her galaxy
evolution models in BC format.
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1 INAF-Osservatorio Astronomico di Bologna - Via Ranzani,1,
I-40127, Bologna, Italy
2 INAF-IASF - via Bassini 15, I-20133, Milano, Italy
3 Laboratoire d’Astrophysique de Marseille, UMR 6110
CNRS-Université de Provence, BP8, 13376 Marseille Cedex
12, France
4 INAF-Osservatorio Astronomico di Brera - Via Brera 28,
Milan, Italy
http://arxiv.org/abs/astro-ph/0506044
http://arxiv.org/abs/astro-ph/0611724
http://arxiv.org/abs/astro-ph/0311475
http://arxiv.org/abs/astro-ph/0410295
http://arxiv.org/abs/astro-ph/0702148
20 L.Pozzetti et al.: The VVDS: The Galaxy Stellar Mass Assembly History
5 Institute for Astronomy, 2680 Woodlawn Dr., University of
Hawaii, Honolulu, Hawaii, 96822
6 Max Planck Institut fur Astrophysik, 85741, Garching,
Germany
7 Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd
Arago, 75014 Paris, France
8 Centro de Astrofsica da Universidade do Porto, Rua das
Estrelas, 4150-762 Porto, Portugal
9 Laboratoire d’Astrophysique de Toulouse/Tabres
(UMR5572), CNRS, Université Paul Sabatier - Toulouse
III, Observatoire Midi-Pyriénées, 14 av. E. Belin, F-31400
Toulouse, France
10 INAF-IRA - Via Gobetti,101, I-40129, Bologna, Italy
11 INAF-Osservatorio Astronomico di Roma - Via di Frascati
33, I-00040, Monte Porzio Catone, Italy
12 School of Physics & Astronomy, University of Nottingham,
University Park, Nottingham, NG72RD, UK
13 Astrophysical Institute Potsdam, An der Sternwarte 16, D-
14482 Potsdam, Germany
14 Observatoire de Paris, LERMA, 61 Avenue de l’Observatoire,
75014 Paris, France
15 Università di Bologna, Dipartimento di Astronomia - Via
Ranzani,1, I-40127, Bologna, Italy
16 Centre de Physique Théorique, UMR 6207 CNRS-Université
de Provence, F-13288 Marseille France
17 Integral Science Data Centre, ch. d’Écogia 16, CH-1290
Versoix
18 Geneva Observatory, ch. des Maillettes 51, CH-1290
Sauverny, Switzerland
19 Astronomical Observatory of the Jagiellonian University, ul
Orla 171, 30-244 Kraków, Poland
20 INAF-Osservatorio Astronomico di Capodimonte - Via
Moiariello 16, I-80131, Napoli, Italy
21 Universitá di Milano-Bicocca, Dipartimento di Fisica -
Piazza delle Scienze, 3, I-20126 Milano, Italy
22 Università di Bologna, Dipartimento di Fisica - Via Irnerio
46, I-40126, Bologna, Italy
Introduction
The First Epoch VVDS Sample
The I-selected Spectroscopic Sample
The K-selected Photometric Sample
Comparison of the Two Samples
Estimate of the Stellar Masses
Smooth SFHs
Complex SFHs
Effect of NIR Photometry
Comparison of the Masses Obtained with the Two Methods
Massive Galaxies at z>1
Mass Function Estimate
The VVDS Galaxy Stellar Mass Function
Comparison with Previous Surveys
The Evolution of the Galaxy Stellar Mass Function
Galaxy Number Density
Mass Density
Summary and Discussion
|
0704.1602 | What does Hirsch index evolution explain us? A case study: Turkish
Journal of Chemistry | Año 8, No.27, Ene – Mar. 2007
What does Hirsch index evolution explain us?
A case study: Turkish Journal of Chemistry
Metin Orbay, Orhan Karamustafaoğlu and Feda Öner
Amasya University (Turkey)
[email protected], [email protected], [email protected]
Abstract
The evolution of Turkish Journal of Chemistry’s (TURK J CHEM) Hirsch index (h-index)
over the period 1995-2005 is studied and determined in the case of the self and without
self-citations. It is seen that the effect of Hirsch index of TURK J CHEM has a highly
positive trend during the last five years. It proves that TURK J CHEM is improving
itself both in quantity and quality since h-index reflects peer review, and peer review
reflects research quality of a journal.
Key words
Bibliographic citations, citation index, h-index, Hirsch number, Scientometric analysis,
Bibliometry
Resumen
Presenta la evolución del índice Hirsch (h-index) de la Turkish Journal of Chemistry’s
(TURK J CHEM) durante el período 1995-2005; el caso es estudiado y precisado tanto
con auto-citaciones como sin auto-citaciones. El indice Hirsch permite apreciar una alta
tendencia positiva de los indicadores de la TURK J CHEM durante los últimos cinco
años. Se comprueba así que la TURK J CHEM va mejorando tanto en cantidad como en
calidad de contenidos dado que el h-index refleja la revisión por pares, y la revisión por
pares refleja la calidad de las investigaciones de una revista.
Palabras clave
Citas bibliográficas, índice de citación, indice-h, NúmeroHirsch, Cienciometría,
Análisis cienciométrico, Bibliometría
Introduction
Hitherto, several citation-based indicators have been used to measure research
performance (e.g. the number of citations to each of the q most cited papers, the total
number of citations, the citations per paper, the number of highly cited published
papers). There are valid reservations about using above mentioned indicators to
measure performance because some papers are cited for reasons that are unrelated to
the quality or utility of a study (see: Kelly & Jennions, 2006; Miller, 2007 and references
therein).
essay
Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us?
Recently, taking into account above citation-based indicators with advantages and
disadvantages, Jorge E. Hirsch has suggested a new indicator called h-index, which
means that one single index for the assessment of the research performance of an
individual scientist. According to the definition by Hirsch, “A scientist has index h if
his/her N papers have at least h citations each, and the other (N-h) papers have fewer than h
citations each” (Hirsch, 2005). Hirsch’s article has generated considerable interest and
almost immediately provoked reactions in the scientific community (Ball, 2005; Braun,
Glanzel & Schubert, 2005; Glanzel & Persson, 2005; Glanzel, 2006a; Egghe & Rousseau,
2006; Egghe, 2006; Cronin & Meho, 2006; Burrell, 2007; Rousseau, 2007a). The h-index
has generally well received by the research group. Of course, the h-index has also a
number of disadvantages as point out by some authors (Kelly & Jennions, 2006; Van
Raan, 2006). After all these beneficial arguments, W. Glanzel has summarized some
pros and cons of h-index in his excellent recent paper (Glanzel, 2006b).
After a short time, the h-index definition has been adapted into journals and article
citations, as a h-type index-equal to h if you have published h papers, each of which has
at least h citations (Braun, Glanzel & Schubert, 2006). T. Braun and co-workers stressed
that the h-type index for journals would advantageously supplement journal impact
factor (IF), the total number of citations divided by the number of articles (Garfield,
1955), at least two aspects: respectively,
i. It is robust in the sense that it is insensitive to an accidental excess of uncited
articles, and to one or several highly cited articles,
ii. It combines the effect of “quantity” and “quality” in a rather specific.
Naturally, the journal h-index would not be calculated for a “lifetime contributions”, as
defined by Hirsch for the scientific output of a researcher, but for a definite period-in
the simplest case for a given year. Using this procedure, R. Rousseau studied the
evolution of the Journal of American Society of Information Sciences’ Hirsch index and
introduced relative h-index (Rousseau, 2007b).
In this opinion article, the evolution of Hirsch index of Turkish Journal of Chemistry
(TURK J CHEM) over period 1995-2005 is studied and determined in the case of the self
and elimination self-citation (or without self citation) of the journal.
Method and results
As is well known, Web of Science database offers a very simple way to determine the
annual h-index of a journal. Retrieving all source items of a given journal from a given
period and shorting them by the number of “times cited”, it is easy to find the h-index
of the journal for the given year (http://isiknowledge.com).
In this study, we conduct a case study for h-index of TURK J CHEM over period 1995-
2005. Meanwhile, we consider a fixed moment in time when citations are collected
from Web of Science (http://isiknowledge.com, retrieved date 24.12.2006). The h-index
of TURK J CHEM over the period 1995-2005 is determined in the case of the self and
without self-citations, as shown in Figure 1.
Año 8, No.27, Ene – Mar. 2007
h-index h-index(w ithout self citations)
Figure 1. h-index of TURK J CHEM.
However, besides the period over which a volume can collect citations, also the
number of articles published in that volume influences the h-index. For this reason, the
h-index must be divided by the number of articles published, leading to a normalized
(or relative) h-index (Rousseau, 2007b). In this case, the results are shown with self and
without self citations for the journal in Figure 2.a. and Figure 2.b., respectively.
Figure 2a. Normalized h index with self citiations of TURK J CHEM.
Orbay, Karamustafaoğlu &, Öner - What does Hirsch index evolution explain us?
Figure 2b. Normalized h index with eliminated self citations of TURK J CHEM.
As can be clearly seen from Figure 2, using the normalized h-index leads to a linear
increase when going backward in time (or decrease when going forward in time). The
Pearson correlation coefficients of the regression lines of this journal are 0.686 for
normalized h-index (continuous line in Figure 2.a) with self citations and 0.689 for
normalized h-index without self-citations (dot line in Figure 2.b), which are moderate,
but statistically significant (1% level). It is not surprising that these two correlation
coefficients are very close to each other because of the fact that the self-citations over
this period are limited by approximately 20%. On the other hand, we encounter that
this value is high in other twenty randomly selected journals published in the same
field.
It is obvious from the Figure 1 and Figure 2 that h-index and normalized h-index are
extremely different trend between 1995-2000 and 2000-2005 periods. So, we focus on
two periods. In the former period, the Pearson correlation coefficients are 0.336 for
normalized h-index with self citations and 0.192 for without self-citation, which are
very low, but statistically significant (1% level). It is not surprising that TURK J CHEM
has started to be scanned in Web of Science newly in this period. For this reason, it can
be thought that a few researchers were aware of this journal. On the other hand, in the
latter period, the Pearson correlation coefficients are 0.858 for normalized h-index with
self citations and 0.941 for without self-citation, which are very high, and statistically
significant (1% level). From these interesting results, we conclude that a lot of
published papers in this journal have been very high impact with respect to “quantity”
(number of publications) and “quality” (citation rate), recently. Furthermore, it is
known that TURK J CHEM has started open access in the latter period. Thus, we think
that open access contributes h-index of this journal.
References
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Braun, T., Glanzel W. & Schubert, A. (2005). A Hirsch-type index for journals, The
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Braun, T., Glanzel, W. & Schubert, A. (2006). A Hirsch-type index for journals,
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|
0704.1603 | Measuring energy dependent polarization in soft gamma-rays using Compton
scattering in PoGOLite | Measuring energy dependent polarization in soft gamma-rays using Compton
scattering in PoGOLite
M. Axelsson a,∗, O. Engdeg̊ard b,a, F. Ryde b,a, S. Larsson a, M. Pearce b, L. Hjalmarsdotter c,a,
M. Kiss b, C. Marini Bettolo b, M. Arimoto d, C.-I. Björnsson a, P. Carlson b, Y. Fukazawa e,
T. Kamae f,g, Y. Kanai d, J. Kataoka d, N. Kawai d, W. Klamra b, G. Madejski f,g, T. Mizuno e, J. Ng f ,
H. Tajima f,g, T. Takahashi h, T. Tanaka e, M. Ueno d, G. Varner i, K. Yamamoto e
aStockholm Observatory, AlbaNova, SE-106 91 Stockholm, Sweden
bPhysics Department, Royal Institute of Technology, AlbaNova, SE-106 91 Stockholm, Sweden
cObservatory, PO Box 14, FIN-00014 University of Helsinki, Finland
dTokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
eHiroshima University, Physics Department, Higashi-Hiroshima 739-8526, Japan
fStanford Linear Accelerator Center, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
gKavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA
hInstitute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara 229-8510, Japan
iDepartment of Physics and Astronomy, University of Hawaii, 2505 Correa Road, Honolulu, HI 96822, USA
Abstract
Linear polarization in X- and gamma-rays is an important diagnostic of many astrophysical sources, foremost giving information
about their geometry, magnetic fields, and radiation mechanisms. However, very few X-ray polarization measurements have been
made, and then only mono-energetic detections, whilst several objects are assumed to have energy dependent polarization signatures.
In this paper we investigate whether detection of energy dependent polarization from cosmic sources is possible using the Compton
technique, in particular with the proposed PoGOLite balloon-experiment, in the 25–100 keV range. We use Geant4 simulations of a
PoGOLite model and input photon spectra based on Cygnus X-1 and accreting magnetic pulsars (100mCrab). Effective observing
times of 6 and 35 hours were simulated, corresponding to a standard and a long duration flight respectively. Both smooth and sharp
energy variations of the polarization are investigated and compared to constant polarization signals using chi-square statistics.
We can reject constant polarization, with energy, for the Cygnus X-1 spectrum (in the hard state), if the reflected component
is assumed to be completely polarized, whereas the distinction cannot be made for weaker polarization. For the accreting pulsar,
constant polarization can be rejected in the case of polarization in a narrow energy band with at least 50% polarization, and
similarly for a negative step distribution from 30% to 0% polarization.
Key words: Polarization, X-rays, Gamma-rays, Compton technique, PoGOLite, Geant4, Simulations
PACS: 95.55.Ka, 95.55.Qf, 95.75.Hi, 98.70.Qy
1. Introduction
In the areas of spectral and temporal studies, X-ray and
gamma-ray astronomers have been given a wealth of data
on a wide range of objects. Polarization has long been pre-
dicted to play a crucial role in determining physical and geo-
metrical parameters in many astrophysical sources, thereby
discriminating among current models. However, there have
so far been very few measurements of polarization at these
∗ Corresponding author.
E-mail: [email protected]
energies. In light of this, the possibility to detect energy
dependent polarization has hardly been discussed at all in
the literature. In this paper, we present the results from
simulations of a dedicated soft gamma-ray polarimeter us-
ing Compton scattering, and study the response when the
degree of polarization varies with the energy of the emit-
ted photons. While energy-dependent polarization is ex-
pected from many sources, its detection requires an instru-
ment of sufficiently good energy response. The Compton
polarimeter presented in this paper utilizes plastic scintilla-
tors, which are relatively inefficient for energy depositions
Preprint submitted to Elsevier 15 September 2021
http://arxiv.org/abs/0704.1603v2
below a few keV. Thus, simulations are necessary to deter-
mine how sensitive the instrument is and how large varia-
tions must be for detection.
We begin by describing the organisation of the pa-
per. The remainder of this section is devoted to giving
a background of polarimetry in the X/γ-ray regime, and
an overview of the scientific motivation for such measure-
ments. In Section 2 we focus on polarimetry using Comp-
ton scattering and describe an instrument design based on
this technique. We then present the set-up of our simula-
tion of the instrument in Section 3, and the results of the
simulations in Section 4. Finally, in Sections 5 and 6, we
discuss and summarise our results.
1.1. Measurement of polarization
The aim of any polarimetric measurement is to deter-
mine the degree and direction of polarization of incident
radiation. When combined with the traditionally measured
quantities of energy and time, polarimetry has the poten-
tial to double the parameter space available. As such, it can
be a powerful tool to discriminate between physical models
proposed for a given source.
Historically, polarimetry has proven very successful at
optical and radio wavelengths. In these bands, it has been
extensively used to probe both radiation physics and ge-
ometry of sources (see, e.g., [1]). In the X-ray regime, how-
ever, the results are more meagre. Early rocket observa-
tions measured X-ray polarization from the Crab Nebula
[2]. This result was later confirmed by the Orbiting Solar
Observatory 8 (OSO-8, measuring a polarization degree of
19.2% ± 1.0%, [3,4]), the only satellite mission carrying a
dedicated polarimeter to date. As the design was based on
Bragg reflection on graphite crystals, the energies probed
were constrained to 2.6 keV and 5.2 keV.
A number of new polarimetric instruments, designed to
work in the X/γ-ray regime, have recently been proposed.
These include POLAR (10–300 keV, [5]), GRAPE (50–300
keV, [6]), PHENEX (40–300 keV, [7]), CIPHER (10 keV –
1MeV, [8]), and POLARIX (1.5–10 keV, [9]). In this paper
we present PoGOLite, a Compton polarimeter currently
under construction [10].
1.2. Expected objects of interest
The lack of polarimetric measurements in X-rays is not
due to a lack of potential targets. Indeed, from a theoreti-
cal point of view there are many sources that are expected
to display detectable degrees of polarization. Over the past
decades, there have been publications discussing the po-
tential for polarization in sources such as X-ray binary
(XRB) systems, active galactic nuclei (AGN), accretion
and rotation powered pulsars as well as cataclysmic vari-
ables (CVs); see e.g., [11,12,13,14,15]. Other work has fo-
cused on the processes producing polarized radiation, either
the radiative processes themselves (e.g., synchrotron and
Fig. 1. Likely geometry in the hard state of Cyg X-1. Mass being
accreted forms an accretion disc around the compact object. In the
inner regions, there is a hot inner flow/corona. Soft seed photons
from the disc may be Comptonized in the hot flow. A fraction of
the resulting hard photons can then be reflected off the disc, giving
a net polarization.
non-thermal bremsstrahlung, [16,17]), or processes such as
reflection/asymmetric scattering (e.g., [18,19,20]), strong-
field gravity [21,22] and vacuum birefringence in strong
magnetic fields [23].
In most sources, polarization is not expected to re-
main constant with energy. An example is radiation from
strongly magnetized plasmas where the polarization may
change dramatically near the cyclotron resonance energy.
It is therefore important to understand not only what
degree of polarization is needed for detection, but also
how sensitive a given instrument will be to the changes
of polarization with energy. To study such effects we have
chosen to simulate two example sources: Cygnus X-1 and
an accreting magnetic neutron star.
1.2.1. Cygnus X-1
Cygnus X-1 is a high-mass XRB where the compact ob-
ject is believed to be a black hole. The source exhibits two
main spectral states, commonly referred to as hard and soft.
Most of the time is spent in the hard state. Several mod-
els have been proposed to explain the observed states and
transitions. The two main components of such models are
usually a geometrically thin, optically thick accretion disc
and a hot inner flow or corona [24]. A schematic picture of
a likely geometry in the hard state is shown in Fig. 1. Soft
X-rays are produced in the accretion disc, and may then
be Comptonized in the hot inner flow/corona. A fraction
of the hard radiation can be reflected off the accretion disc
before reaching the observer. Polarization from this system
may arise through several processes. In this paper, we will
focus on the polarization introduced by the reflection (for
more details, see, e.g., [19,25,26]). In Cygnus X-1, this con-
tribution is strongest in the energy range of ∼ 20–100keV.
The polarization degree is expected to vary with energy,
following the relative strength of the reflection component.
1.2.2. Accreting magnetic neutron stars
In many high-mass XRBs the accreting object is a
highly magnetic neutron star. The strong magnetic field,
∼ 1013 gauss at the surface, directs the accretion flow to-
wards the magnetic poles of the star. Most of the accretion
energy is released just above the polar cap where the emis-
sion and propagation of radiation is directly connected to
the magnetic field as well as the local properties of the
plasma. For a number of sources cyclotron spectral features
have been observed in hard X-rays, and from these, mag-
netic field strengths have been deduced. The X-rays are
expected to be polarized and the degree, angle and energy
dependence of the polarization will depend on the physical
conditions in the emission region [27]. Measurements of the
detailed polarization properties would therefore provide a
new and very powerful probe of the radiating plasma near
the surface of the neutron star.
2. The Compton technique
Apart from the special case of Bragg reflection, all three
main physical processes of photon-matter interaction in
the X/γ-ray regime may be used in polarimetry: photoab-
sorption, Compton scattering and pair production. Each of
these preserves information on the polarization of the in-
coming radiation. For photon energies between ∼ 100 keV
and 1MeV, Compton scattering is the dominant process. In
this section, we will briefly outline the theoretical basis for
a polarimeter based on Compton scattering, and present a
design for a dedicated polarimeter based on this technique.
2.1. Basic principle
The differential cross section for Compton scattering is
given by:
− 2 sin2 θ cos2 φ
, (1)
where re is the classical electron radius, E0 and E are the
photon frequency before and after scattering, θ is the an-
gle between incident and scattered direction, and φ is the
azimuthal scattering angle relative to the plane of polar-
ization. When projected on a plane, the angle of scattering
will thus be modulated as cos2 φ.
To measure the scattering angles, it is necessary to de-
tect both the site of scattering and that of photoabsorp-
tion. If more than two scattering sites are identified, the
relative energy depositions can be used to help distinguish
between Compton scattering and photoelectric absorption
sites. Some form of segmentation of the detector is neces-
sary to provide spatial resolution, required to determine
the positions of the signals.
Fig. 2. The design of the PoGOLite instrument. The side anticoinci-
dence shield has been partially cut away for clarity. The total length
of the instrument will be ∼ 100 cm.
2.2. PoGOLite
The Polarized Gamma-ray Observer - Light weight ver-
sion (PoGOLite) is a balloon-borne polarimeter, planned
for launch with a stratospheric balloon in 2009. Figure 2
shows the design of the instrument. The instrument con-
sists of 217 phoswich detector cells (PDCs) arranged in
a hexagonal pattern. Each PDC is made up of a hollow
slow scintillator tube, a fast scintillator detector, a bottom
bismuth germanate (BGO) crystal, and a photomultiplier
tube (PMT). Signals from the different optical components
are distinguished using a pulse shape discrimination tech-
nique based on the different scintillation decay times of the
materials [28]. The configuration is surrounded by an an-
ticoincidence shield made of BGO crystals. Together with
the bottom BGO crystals, this allows side and back enter-
ing photons and cosmic rays to be rejected.
The hollow slow scintillator tube acts as an active col-
limator. Photons or charged particles entering the instru-
ment off-axis will be registered in the slow scintillator and
can be rejected. The desired events are from photons that
enter cleanly through the slow scintillator and scatter in
the fast scintillator. After scattering, the photon may be
absorbed in one of the neighbouring fast scintillator cells,
allowing the azimuthal scattering angle to be determined.
The well-type design of PoGOLite allows for efficient
background rejection [29,30], and gives a field of view of
∼ 5 deg2. This allows the instrument to be accurately
pointed at specific sources. As both the initial Compton
scattering and subsequent photon absorption occur in the
same material (the plastic fast scintillator), the effective
energy range is determined by the cross-sections for both
these processes, as well as the background. PoGOLite will
have an energy range of ∼ 25–100keV, which is lower than
the range where Compton scattering dominates. A more
detailed description of the instrument may be found in
[28,31].
The capability of PoGOLite to measure the energy de-
Azimuth angle
Fig. 3. Simplified sketch of the distribution of scattering angles,
used to determine the modulation factor. The maximum (Cmax),
minimum (Cmin) and average (T = [Cmax + Cmin]/2) values of the
distribution are indicated.
pendence of polarization is limited both by the signal-to-
background ratio and the energy resolution. Due to redis-
tribution, some of the higher energy photons will produce
events at lower energies. The flux and polarization in the
low energy band will therefore be affected by the spectrum
at higher energies but not vice-versa. The energy response
has been carefully simulated using Geant4 [32].
Figure 3 shows a hypothetical distribution of azimuthal
scattering angles. The maximum (Cmax) and minimum
(Cmin) values of the distribution and the average (T =
[Cmax+Cmin]/2) can be used to define a modulation factor:
Cmax − Cmin
Cmax + Cmin
Cmax − Cmin
. (2)
The modulation factor is determined by fitting the follow-
ing function to the distribution of azimuthal scattering an-
f(x) = T (1 +M cos(2x+ 2α)) , (3)
with angle x (a function variable, not a fitting parameter),
average T , modulation factorM , and polarization angle α.
In this work, the modulation factor is the discriminator
between different polarizationmodels. If the response of the
instrument to a 100% polarized source is known, the mod-
ulation factor can be used to determine the polarization of
the incoming photon beam [33].
3. Simulations
In this section we will describe the setup of our simula-
tions. The source models used as input are also presented,
as well as the background considered.
3.1. Geant4
Geant4 1 is a multi-purpose software package for sim-
ulating particles travelling through and interacting with
1 http://geant4.cern.ch
matter, using Monte Carlo techniques [34]. The standard
Geant4 package was earlier found [35] to have incorrect im-
plementations concerning photon polarization in Compton
and Rayleigh scattering; the Geant4 version used here is a
corrected version of 4.8.0.p01.
3.2. Simulation setup
The Geant4 implementation includes the essential parts
of PoGOLite: 217 PDCs with slow and fast plastic scintil-
lators and bottom BGO crystals together with a BGO side
shield. The model has no PMTs, and uses a solid BGO side
shield instead of discrete pentagonal bars (cf. Fig. 2). Lay-
ers of tin (50µm) and lead (50µm) surrounding each slow
scintillator and the BaSO4 coating (200µm) of the BGO
crystals are included. The mechanical support structure is
not represented.
During the simulation, separate photons are generated
with random energies from a spectral model. An event is
triggered by a hit in two or three of the fast plastic scintil-
lators. The following is saved as output data for each event:
information about the original gamma momentum, the ID-
number of the cells that had an interaction (ranging from
1 to 217) and the energy deposited in each cell. These data
are preprocessed to simulate the resolution of the PMTs,
as described in Sect. 4.1.
3.3. Source Models
As stated in Sect. 1.2, two sources were considered:
Cygnus X-1 (in the hard state) and an accreting neutron
star. Below we describe the model used for the incident
radiation and polarization in each case. As shown in [10],
PoGOLite is expected to detect polarization in both these
sources; what we are investigating is the sensitivity to
changes in polarization degree with energy.
3.3.1. Cygnus X-1
For our simulations of Cygnus X-1 we used an input spec-
trum of a power-law, with photon index α = −1.2 and an
exponential cutoff at energy Ecut = 120 keV. It was nor-
malised to match the observed spectrum of Cygnus X-1.
The spectrum of the reflection was approximated by the
logarithmic quadratic curve
EFE = 10
−c(logE−log a)(logE−log b), (4)
with a = 24, b = 98 and c = 1.89. Figure 4 shows the
observed radiation of Cygnus X-1, and our model of the
total spectrum as well as that assumed for the reflection
component.
In our simulations, the polarization is assumed to arise
due to the reflection component. Two scenarios were tested:
100% and 20% polarization for the reflection component,
with unpolarized direct emission. This corresponds to a to-
tal average polarization around 17% and 3% respectively.
Fig. 4. Observed radiation spectrum and input model used in the
simulations of observations of Cygnus X-1. Gray lines: Typical ra-
diation spectrum of Cygnus X-1 in the hard state. Black lines: As-
sumed input spectrum: a cut-off power law with index α = −1.2 and
cut-off Ecut = 120 keV, normalized to match the measured flux. The
reflection component is shown, and the energy range of PoGOLite
is indicated by vertical lines.
The energy dependent polarization Π(E) used as input was
set to the relative strength of the reflection component com-
pared to the total flux, scaled down in the case of 20% po-
larization. Simulations were performed for effective observ-
ing times of 6 hours and 35 hours. These times are chosen
as realistic estimates for short and long duration balloon
flights, respectively.
3.3.2. Accreting Magnetic Neutron Star
In the case of the neutron star, we study the observability
of energy dependent effects by simulations of three different
idealized polarized spectra:
– Polarization in a narrow band.
– Polarization only at low energies.
– Polarization only at high energies.
The neutron star spectrum was in all cases approxi-
mated with an exponentially cut-off power law, with index
α = −1.1 and energy cut-off at Ecut = 70keV. It was
normalized to correspond to a 100mCrab source.
Assuming a cyclotron energy Ec at 50 keV, we use three
toy models of the polarization energy dependence Π(E),
with Πmax ≡ p%:
– A Gaussian peak centred at 50 keV, Gp, modelling a
rise in polarization from 0% to maximum p%, using the
Gaussian curve
Π(E) = pe
−(E−50)2
2σ2 % (5)
with E measured in keV and σ = 5 keV.
– Two step functions, Sp and S−p, with polarization
0% if E < 50 keV
p% if E ≥ 50 keV
for Sp, and
10 20 30 40 50 60 70 80 90 100
E (keV)
Fig. 5. Example of assumed energy dependence of the polarization
fraction in the case of an accreting magnetic neutron star. The figure
shows a Gaussian curve with Πmax = 20% (G20), and a positive step
with Πmax = 10% (S10).
p% if E < 50 keV
0% if E ≥ 50 keV
for S−p. Simply put, p is the jump in polarization that
occurs at E = 50 keV.
Figure 5 illustrates examples of G20 and S10, the
Gaussian and positive steps with maxima 20% and
10% respectively. In the simulations, the values p =
{10, 20, 30, 40, 50}% were used, each assuming an observa-
tion time of 35 hours.
3.4. Background
Balloon-borne gamma-ray polarimetry measurements
are subject to several significant sources of background.
Through the use of the well-type phoswich detector tech-
nique, the PoGOLite instrument has been designed to
reduce these backgrounds, allowing 10% polarization of a
100 mCrab source to be measured in one 6 hour balloon
observation in the 25–100keV energy range. The basic
phoswhich design was used in the WELCOME series of
balloon-borne observations and allowed effective back-
ground suppression [36,37,38,39,40,41,42]. The concept
was subsequently improved and effectively used in a satel-
lite instrument, the Suzaku Hard X-ray Detector (HXD)
[29,43,44,45].
The background to PoGOLite measurements can arise
from charged cosmic rays, neutrons (atmospheric and in-
strumental) and gamma-rays (primary and atmospheric).
The background from charged cosmic rays (predominantly
protons, ∼90%, and helium nuclei, ∼10%) is rejected by
the BGO anticoincidence shields and slow plastic collima-
tors. Cosmic rays are minimum ionizing particles and can
be identified through their relatively large energy deposits.
The background presented by atmospheric neutrons and
neutrons produced in the PoGOLite instrument and sur-
rounding structures is currently being studied in detail [46].
Fig. 6. Estimated contribution to the gamma-ray background in the
PoGOLite energy range (∼ 25–100 keV). The radiation from the two
considered sources, Cygnus X-1 and an accreting neutron star, are
also shown.
For the purposes of the study presented in this paper, par-
ticular attention has been paid to what is expected to be the
dominant background: primary and atmospheric gamma-
rays. The gamma-ray background rate is estimated from a
model derived from measurements taken in Texas with the
GLAST Balloon Flight Engineering Model [47].
The primary gamma-ray component originates outside
the atmosphere, i.e., above PoGOLite. The angular distri-
bution of the radiation is uniform within the hemisphere
above PoGOLite. The energy spectrum is modeled by
a doubly-broken power-law with breaks at 50 keV and
1 MeV [48].
Secondary gamma-rays are created in the Earth’s at-
mosphere through bremsstrahlung interactions of charged
cosmic-rays. Two separate components are considered, one
directed upwards and one downwards. The upward flux is
dependent on the zenith angle [49], and the energy spec-
trum consists of a doubly-broken power-law with breaks
at 10 MeV and 1 GeV, and a 511 keV line from electron-
positron annihilation. The downward component is simi-
lar, but with breaks at 1 MeV and 1 GeV. Energies up to
100 GeV were generated for all components. These models
are based on data from satellite- and balloon-borne instru-
ments ([50] and [51], and references therein).
Figure 6 shows the estimated gamma-ray backgrounds
compared to the accreting pulsar and Cygnus X-1 models.
The total gamma-ray background is at the 10 mCrab level.
4. Analysis and Results
4.1. Data Processing
In the first data processing step, the resolution of the
scintillator-PMT assembly is simulated by fluctuating the
number of photo-electrons generated in the scintillating
materials. It is assumed that when the energyE is deposited
in a cell, the average number of photo-electrons generated
is En, with n set to 0.5 photo-electrons per keV. Now, we
fluctuate En to (En)fluct by applying a Gaussian spread.
If En ≤ 10, we do it in two steps: First we subject it to a
Poissonian spread, and thereafter a Gaussian spread with
variance Enσ2, with σ set to 0.4. If En > 10, only a Gaus-
sian spread with variance En is used. If (En)fluct < 0, then
it is set to 0. Finally, we take
Emes =
(En)fluct
as the energy actually measured by the PMT in the cell
of interest. We reject all fast scintillator interactions with
Emes below a certain measurement threshold (2 keV). For
the analysis described in this paper, only events with two
or three hits in the fast scintillators are retained (the veto
logic is not considered at this stage). At PoGOLite energies,
more than 80% of the events are from photons interacting
in no more than three detector cells [52].
For two-site events, the chronological order of the two
cells does not matter for angle calculation, as the distribu-
tion is periodic over the angle π. In the case of three hits,
we calculate the scattering angle by ignoring the hit with
the lowest energy measured, assuming that a low-energy
interaction does not affect direction much, and derive an
angle from the positions of the two cells with highest en-
ergy deposits.
Most photons do not scatter very far; about half will only
go from one cell to its neighbour. As the range of possible
scattering angles resulting in detection in a given adjacent
cell is large, this causes strong peaks in each of the six direc-
tions corresponding to the neighbouring cells. The PoGO-
Lite instrument will rotate about its axis, causing the range
for a given cell to smoothly vary and thereby creating a
continuous distribution over angles. In the simulations, the
uncertainty of the angle determination is instead approxi-
mated by introducing a Gaussian spread to the measured
scattering angles.
In the last steps of data processing we take into account
the mass of air in the atmosphere above the balloon, filter-
ing out roughly half of our incident source radiation, as-
suming the atmospheric overburden 4 g/cm2 at 40 km alti-
tude. We also apply the veto logic, rejecting all events with
detection in any slow plastic scintillator or BGO crystal.
4.2. A χ2 measure
Tomeasure the polarization energy dependence, one can-
not simply calculate the polarization at certain energies and
construct Π(E), since the photon energy Eγ always will be
unknown due to the response of the instrument. Instead we
calculate the modulation factor at different measured en-
ergies, obtaining a curve M(Emes). This curve can then be
compared with theoretic curves resulting from other mod-
els, possibly from the same family of curves, enabling us to
reject complete families of energy dependencies. One such
family, which we will be concerned with here, is the set of
constant polarizations.
Fig. 7. Example of the measured distribution of events as a function
of scattering angle (histogram). In this example, the simulated results
of a six hour observation of Cygnus X-1 are shown, assuming a
completely polarized reflection component. The data are from the
measured energy range of 30–35 keV. A sinusoidal function is fit to
the data (solid line, cf. Eq. 3) and a modulation factor is calculated.
For a given source model (polarization energy depen-
dence) A, the modulation factor M (Eq. 2) was fitted
at different measured energies, yielding a curve MA =
MA(Emes). Figure 7 shows an example of a modulation
curve in the 30–35keV band, generated for a six hour
observation of Cygnus X-1 in the case of a completely
polarized reflection component. The resulting modulation
factor is 2.69± 0.30. The modulation factor is in this way
calculated for each energy band. The result is a curve
showing how the modulation factor varies with energy,
which can then be compared to the corresponding curves
for various models of polarization.
To test the results against constant polarization, we also
generated curves MΠ with constant levels of polarization
Π. Since these latter curves should not be thought of as
measured, but fluctuation-free theoretical constructs, they
were generated by much longer simulations than the obser-
vational curves.
A measure of how much two curves differ is defined as
χ2Π =
(MA,Emes −MΠ,Emes)
σ2Emes
, (9)
with σi as the sum of the two errors in fitting MA,Emes and
MΠ,Emes . When this is calculated for all reasonable values
of Π, we can reject the hypothesis of constant polarization
if the minimum of χ2Π is high enough. For 16 degrees of
freedom, corresponding to data points up to 100 keV, the
95% certainty level requires χ2 > 26.3.
4.3. Cygnus X-1
Table 1 summarises the results for the simulations of
Cygnus X-1. For a 100% polarized reflection component,
the energy dependence is detected both after 6 hours of
observation and after 35 hours. Figure 8 shows the expected
modulation factors and χ2 values for a 35h observation. In
Reflection 100% Reflection 20%
Obs. time 6h 35h 6h 35h
Significance 99.4% >99.99% 19.1% 51.2%
Table 1
The significance in rejecting constant polarization models, shown for
different reflection polarization strengths and observation times in
the case of Cygnus X-1.
Gaussian peaks
Πmax (%) 10 20 30 40 50
Significance 24.3% 24.3% 35.3% 79.9% 99.7%
Positive steps
Πmax (%) 10 20 30 40 50
Significance 46.3% 70.0% 96.8% 91.8% 83.5%
Negative steps
Πmax (%) 10 20 30 40 50
Significance 51.0% 99.3% 95.8% 99.6% >99.99%
Table 2
The significance in rejecting constant polarization models, shown for
different polarization shapes and maxima in the case of a 100mCrab
neutron star.
the case of 20% polarization of the reflection component,
shown in Fig. 9, constant polarization cannot be ruled out
at any higher significance level.
4.4. Magnetic NS
The results for the simulations of the accreting neutron
star are summarised in Table 2. For a 100mCrab source,
the background will start to become significant already in
the higher end of the PoGOLite energy range. To be con-
servative, only M(E) data points up to 60 keV were used
for the χ2 analysis in order to limit the errors due to un-
certainties in the background flux. This is equivalent to 8
degrees of freedom, demanding χ2 > 15.5 for 95% certainty
in rejecting constant polarization.
4.4.1. Gaussian peaks
In Fig. 10 we seeM(E) curves forG50 and a few constant
polarizations (left panel), clearly illustrating their different
characteristics. The resulting χ2 curve (right panel) con-
firms that these models are significantly (99% level) differ-
ent. However, this was not the case for any lower value of
Πmax.
4.4.2. Positive steps
Figure 11 shows the result for S50 when compared to
cases of constant polarization. In this case, the difference in
characteristics between the constant models and the energy
dependent model is not large enough, and constant polar-
ization cannot be ruled out. The only value of Πmax yielding
a significant (95% level) difference was Πmax = 30%. What
is interesting to note is that a higher Πmax, which implies
20 30 40 50 60 70 80 90 100
Modulation factors, 35h
E (keV)
Cygx1 P(E), ref=100
P=15%
P=17%
Fig. 8. Results from simulations of a 35h observation of Cygnus X-1, with the reflection component assumed to be 100% polarized. Left panel:
Expected modulation factor M(E), together with models of constant polarization at 15% and 17%. The modulation is fitted in intervals
of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed line
marks the limit for 99.9% significance in rejecting constant polarization. Constant polarization is rejected with high significance.
20 30 40 50 60 70 80 90 100
Modulation factors, 35h
E (keV)
Cygx1 P(E), ref=20
Fig. 9. Same as Fig. 8, but with the reflection component assumed to be 20% polarized. Left panel: Expected modulation factor M(E),
together with models of constant polarization at 2% and 5%. Right panel: χ2 values when comparing M(E) for the simulation with different
degrees of constant polarization. The dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization
cannot be rejected with high significance.
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, G50
Const. 0%
Const. 10%
Const. 20%
Fig. 10. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a Gaussian polarization curve. Left
panel: Expected modulation factor M(E), together with models of constant polarization at 0%, 10% and 20%. The modulation is fitted in
intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The
dashed line marks the limit for 95% significance in rejecting constant polarization. Constant polarization is rejected with high significance.
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, S50
Const. 20%
Const. 30%
Const. 40%
Fig. 11. Results from simulations of a 35h observation of an accreting magnetic neutron star, assuming a positive step with Πmax = 50%. Left
panel: Expected modulation factor M(E), together with models of constant polarization at 20%, 30% and 40%. The modulation is fitted in
intervals of 5 keV. Right panel: χ2 values when comparing M(E) for the simulation with different degrees of constant polarization. The dashed
line marks the limit for 95% significance in rejecting constant polarization. Constant polarization cannot be rejected with high significance.
a higher total polarization, does not necessarily make the
energy dependence easier to measure.
4.4.3. Negative steps
The result of the S−50 model is shown in Fig. 12, where
the χ2 plot (right panel) allows us to reject constant polar-
ization with a certainty much higher than 99.99%. Other
models yielding significant detections were S−20 (99%),
S−30 (95%), and S−40 (99%). One reason these models are
so easily measurable stems from the fact that, due to the
shape of the neutron star spectrum, data and statistics are
much poorer at higher energies. If the polarized part only
lies at high energies, the total polarization will be much
lower or even undetectable.
5. Discussion
Although no measurements have been made of polariza-
tion in the energy range covered by PoGOLite, some theo-
retical models predict changes in polarization with energy.
The results of the simulations clearly show that it is possible
for the PoGOLite instrument to detect the energy depen-
dence of polarization for several of the investigated cases.
The highest significance is found for Cygnus X-1 assuming
a fully polarized reflection component and a neutron star
in the case of a negative step; in the remaining cases, con-
stant polarization can not be rejected. Our results there-
fore show that PoGOLite has the potential to discriminate
among these models.
As described above, in Cygnus X-1 the hard X-rays are
believed to originate from Comptonization of soft seed pho-
tons in a predominantly thermal electron distribution. Al-
though this process involves Compton scattering – which
could introduce a net polarization – multiple scatterings
are required, making our assumption of the direct compo-
nent being unpolarized a reasonable one. The degree of po-
larization of the reflected component is however more un-
certain. Our idealized case of 100% polarization is certainly
an overestimation. Calculations [19] show that the degree
of polarization in the reflected component varies with in-
clination, with a maximum of ∼ 30% expected at high in-
clination. The inclination of the Cygnus X-1 system is not
well known, but estimates put it at 30◦–50◦ [53], making
our assumption of 20% polarization in the reflected compo-
nent reasonable. The relative size of the reflected compo-
nent compared to the direct emission is in turn dependent
on both inclination, where the dependence is the opposite
one, and system geometry. We note that in other sources
the reflected component may be much stronger, or even
dominate the radiation spectrum (e.g., Cygnus X-3, [54]).
Another issue which may complicate measurements of
energy dependent polarization is the behaviour of the angle
of polarization. In our simulations, we have implicitly as-
sumed that the angle does not vary with energy. However,
for both black hole and neutron star systems, this assump-
tion may be an oversimplification. It is certainly true that
emission originating from the region close to a black hole
will be affected by the strong gravity, affecting the polar-
ization angle [21]. It is not clear how large this effect would
be on the reflected component in, for instance, Cygnus X-1,
but results from both temporal and spectral analysis show
that the accretion disc – assumed to be the reflector – is pre-
sumably truncated at a large distance (Rin & 30Rg, [55,56])
from the black hole in the hard state. We therefore do not
expect this effect to alter the outcome of our simulations.
Our results from simulating Cygnus X-1 indicate that
long observations with PoGOLite are required to search for
energy dependence of polarization. The first flights of the
instrument will likely be shorter flights, covering several
targets. While these observations should be long enough to
detect polarization down to the level of a few per cent, we
do not expect to detect any changes in polarizationwith en-
ergy. However, long duration flights spanning several days
are also planned, and such flights would provide the obser-
20 30 40 50 60 70 80
Modulation factors, 35h
E (keV)
NS, S−50
Const. 20%
Const. 30%
Const. 40%
Fig. 12. Same as in Fig. 11, but using the negative step. In this case, constant polarization is rejected with high significance.
vation time needed to search for variations of polarization
degree with energy.
A point to note from the neutron star simulations is the
result that it is easier to rule out constant polarization in
the case of a negative step than for a positive step. As noted
in Sect. 4, the energy response of the instrument is such
that the energy of the incoming photon cannot be uniquely
determined. The result will be a redistribution of energy,
with a possibility for higher energy photons to be detected
with lower energies. The reverse is however not true – a
low energy photon will not be detected as having a higher
energy. In the case of a positive step, some polarized high
energy photons will be detected at lower energies. This will
give a false polarization signal at lower energies, and act to
“smooth” the detected energy dependance of polarization.
For a negative step, the polarization is introduced at lower
energies and will not “spread” to higher energies. The low
energy polarization will be somewhat diluted by redistribu-
tion of high energy photons but the polarization contrast
will still be higher than in the positive case.
By excluding energies above 60 keV in the χ2 analysis of
accreting X-ray pulsars we have been fairly conservative in
our estimate of significances. The restriction of the energy
range was motivated by a potentially high sensitivity to
systematic errors in the background level at high energies.
In this analysis however, we have not taken advantage of
the fact that these sources are pulsating. By analysing the
polarization of the pulsed flux, rather than the total flux,
it should be possible to include all points up to 100 keV
and thereby increase the sensitivity. On the other hand the
polarization direction will probably change over the pulsa-
tion period which will have the opposite effect of reducing
the sensitivity. How important this effect is depends on the
precise source geometry, radiation beaming and our view-
ing angle.
As PoGOLite’s field-of-view is rather large (∼ 5 deg2),
the pointing errors with respect to the axis of rotation must
be small to avoid introducing systematic errors in the po-
larization measurements. The attitude control system used
for PoGOLite will assure accurate pointing to within a few
arcminutes, keeping the systematic error below 1% [10]. Al-
though this figure refers to the whole energy band, we do
not expect any such effect to change the results presented
here. A comprehensive study of systematic effects is beyond
the scope of this paper, but will be crucial once PoGOLite
is in operation.
The performance of the PoGOLite instrument has been
extensively evaluated, both with laboratory-based tests
[57], accelerator-based tests [28], and simulations [32].
These tests show that it will be able to detect low (∼ 10%)
degrees of polarization even for 100 mCrab sources. What
has not previously been tested is its sensitivity to a polar-
ization degree that varies with energy. Despite the rela-
tively modest inherent energy resolution, our results show
that PoGOLite has the capacity to detect changes in po-
larization degree with energy. The simulations show that
significant results can be obtained in a 35h observation,
attainable in the long duration flights already planned for
PoGOLite. We stress that the design is not optimized for
such detections, and future instruments will in all likeli-
hood develop this technique further.
6. Conclusions
The Compton technique applied to an array of plastic
scintillators is an effective method to measure broad en-
ergy band polarization, which is demonstrated with the
proposed PoGOLite mission, using Geant4 simulations. In
particular, energy dependence can be detected. However, in
our model of polarization from X-ray binaries, we require
the reflection to contribute a large fraction of the observed
flux and/or have high degree of polarization for energy de-
pendence to be detected. Similarly, for accreting magnetic
neutron stars, sharp energy variations in the polarization
are needed for a clear detection. This is made easier if the
lower energies contain most of the polarization.
Acknowledgments
The authors gratefully acknowledge support from the
Knut and Alice Wallenberg Foundation, the Swedish Na-
tional Space Board, the Swedish Research Council, the
Kavli Institute for Particle Astrophysics and Cosmology
(KIPAC) at Stanford University through an Enterprise
Fund, and the Ministry of Education, Science, Sports and
Culture (Japan) Grant-in-Aid in Science No.18340052.
J.K. and N.K. acknowledge support by JSPS.KAKENHI
(16340055).J.K. was also supported by a grant for the in-
ternational mission research, which was provided by the In-
stitute for Space and Astronautical Science (ISAS/JAXA).
T.M. acknowledges support by Grants-in-Aid for Young
Scientists (B) from Japan Society for the Promotion of
Science (No. 18740154).
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http://www.particle.kth.se/pogolite
http://heasarc.gsfc.nasa.gov/docs/astroe/prop_tools/suzaku_td/
http://www.particle.kth.se/pogolite
Introduction
Measurement of polarization
Expected objects of interest
The Compton technique
Basic principle
PoGOLite
Simulations
Geant4
Simulation setup
Source Models
Background
Analysis and Results
Data Processing
A 2 measure
Cygnus X-1
Magnetic NS
Discussion
Conclusions
References
|
0704.1604 | Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector
Mesons in Holographic QCD | EFI-07-08
Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector Mesons
in Holographic QCD
Sophia K. Domokos and Jeffrey A. Harvey
Enrico Fermi Institute and Department of Physics,
University of Chicago, Chicago Illinois 60637, USA
(Dated: April 2007)
We show that holographic models of QCD predict the presence of a Chern-Simons coupling
between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the
coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the
five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the
coupling mixes transverse ρ and a1 polarization states. At sufficiently large baryon number densities,
it produces an instability, which causes the ρ and a1 mesons to condense in a state breaking both
rotational and translational invariance.
INTRODUCTION
Models which use the gravity/gauge correspondence to
treat strongly-coupled QCD as a five-dimensional theory
of gravity have progressed dramatically in recent years
[1, 2, 3]. Particularly at high energies, these theories dif-
fer significantly from QCD – yet those models which in-
corporate light quarks [4] and chiral symmetry-breaking
of the form observed in QCD [5] do capture much of the
important low-energy structure of the theory, and give
rise to a spectrum of mesons whose masses, decay con-
stants, and couplings match those of QCD to within 20%.
The gravity/gauge approach includes both top-down
models of QCD arising from D-brane constructions in
string theory [5], and bottom-up phenomenological mod-
els, which attempt to capture the essential dynamics us-
ing a simple choice of five-dimensional metric (AdS5)
and a minimal field content consisting of a scalar X
and gauge fields AaLµ, and A
Rµ [6, 7]. These fields are
holographically dual to the quark bilinear q̄αqβ , and to
the SU(Nf )L × SU(Nf)R flavor currents q̄LγµtaqL and
µtaqR of QCD, respectively.
These holographic models can be used to study QCD
at finite baryon density [8, 9]. In this paper we focus
on a novel effect, in which a Chern-Simons term leads to
mixing between vector and axial-vector mesons. We will
use the model introduced in [6, 7] and for the most part
follow the conventions and notation of [6].
THE MODEL
We work in a slice of AdS5 with metric
ds2 =
−dz2 + dxµdxµ
, 0 < z ≤ zm . (1)
The fifth coordinate, z, is dual to the energy scale of
QCD. We generate confinement by imposing an IR cutoff
zm, and specifying the IR boundary conditions on the
fields. The UV behavior, meanwhile, is governed by z →
In AdS/CFT calculations, boundary contributions to
the action must be treated with care. In the full AdS
space, the only boundary is in the UV (at z = 0). UV-
divergent contributions to the action and to other quanti-
ties are canceled by counterterms. For details see [10, 11].
In the model at hand, the IR boundary at z = zm may
contribute to the action. We follow the approach of [6, 7]
by (1) dropping IR boundary terms, and (2) taking pa-
rameters normally fixed by IR boundary conditions on
the classical solution as input parameters of the model.
We generalize the gauge symmetry to U(Nf )L ×
U(Nf )R and add a Chern-Simons term which gives the
correct holographic description of the QCD flavor anoma-
lies [3]. The Chern-Simons term does not depend on the
metric and on general grounds will be present in any holo-
graphic dual description of QCD. The U(1) axial symme-
try in QCD is anomalous, but in the spirit of the large Nc
approximation we treat it as an exact symmetry of QCD
with massless quarks. Including the anomaly would not
affect our conclusions.
The Lagrangian is thus
d4xdz
|DX |2 + 3|X |2 − 1
(F 2L + F
+SCS .
The Chern-Simons term is given by
SCS =
[ω5(AL)− ω5(AR)] (3)
where dω5 = TrF
3, Nc is the number of colors, and
AL,R = ÂL,Rt̂ + A
a where ta are the generators
of SU(Nf )L,R normalized so that Tr t
atb = δab/2 and
t̂ = 1/
2Nf is the generator of the U(1) subalgebra
of U(Nf). In what follows, we take Nf = 2 so that
a = 1, 2, 3. We will often work with the vector and axial-
vector fields V = (AL +AR)/2 and A = (AL −AR)/2.
CLASSICAL BACKGROUND
We expand around a nontrivial solution to the classical
equations of motion for the scalar X . Following [6, 7] we
http://arxiv.org/abs/0704.1604v1
find the scalar background
X0(z) =
≡ v(z)
1 (4)
where the coefficient M of the non-normalizable term is
proportional to the quark mass matrix, and Σ is the q̄q
expectation value. We take bothM and Σ to be diagonal:
M ≡ mq1 and Σ ≡ σ1. As shown in [6, 7], we can
fix the five-dimensional coupling g5 by comparison with
the vector current two-point function in QCD at large
Euclidean momentum. This leads to the identification
g25 =
. (5)
The model is thus defined by three parameters: zm, mq
and σ. Note that including the U(1) gauge fields and
Chern-Simons coupling does not mandate the addition
of any new parameters. We use zm = 1/(346 MeV),
mq = 2.3 MeV and σ = (308 MeV)
3, which correspond
to values found through a global fit to seven observables
(Model B) in [6].
A background with a static, constant quark density is
described by a classical solution to the equation of motion
for the time component of the U(1) vector gauge field V̂µ,
which is dual to the quark number current. Solving the
V̂0 equation of motion at zero four-momentum yields
V̂0(z) = A+
Bz2 . (6)
By the general philosophy of AdS/CFT, the coefficient
of the non-normalizable term, A, is proportional to the
coefficient with which the operator dual to V̂0 enters the
gauge theory Lagrangian. Since V̂µ is dual to the quark
number current, A must be proportional to the quark
chemical potential. Meanwhile, the coefficient of the nor-
malizable term, B, is proportional to the expectation
value of the operator dual to V̂0: the quark number den-
sity. We now obtain the normalizations of A and B. The
action evaluated for the background Eq. (6) is given by
a boundary term:
V̂0∂zV̂0|z=0 =
d4x . (7)
At finite temperature and baryon number, the Euclidean
action is equal to the grand canonical potential. Using
Eq. (5), this implies that
nqµq (8)
with nq the quark number density and µq the quark
chemical potential. To fix A we separate U(Nf )L,R into
U(1)L,R and SU(Nf )L,R components and note that the
Chern-Simons term contains the coupling
d4xdzǫMNPQ(ÂL0 TrF
PQ−ÂR0 TrFRMNFRPQ)
where the indices M,N,P,Q run over 1, 2, 3, z and the
trace is over SU(Nf ). Defining the SU(Nf)L,R instanton
numbers by
nL,R =
d3xdzǫMNPQ TrF
PQ (10)
and taking Â
constant, this reduces to the coupling
nL − ÂR0 nR
. (11)
Using the connection between instantons and Skyrmion
configurations of the pion field carrying non-zero baryon
number [12, 13, 14, 15, 16], we can interpret an instanton
with nL = −nR = Nb as a state with baryon number Nb.
Eq. (11) then fixes A = µb/Nc = µq with µq the quark
chemical potential; Eq. (8) fixes B = 24π2nq/Nc.
QUADRATIC ACTION
In vacuum, the spectrum of the theory consists of
towers of scalar, vector, pseudoscalar, and axial-vector
mesons given by mode-expanding the five-dimensional
fields along the holographic (z) direction, and integrat-
ing over z. In this section, we identify the spectrum
of excitations and their dispersion relations at non-zero
baryon density by expanding the action to quadratic or-
der around the background given by Eqns. (4),(6).
We focus on the π mesons and the isospin triplet vector
ρ and axial-vector a1 mesons, ignoring contributions from
heavier mesons, and from the scalar σ which arises from
fluctuations in the magnitude of X . Couplings similar to
those for the ρ− a1 mesons exist for the isoscalar ω and
f1 mesons. For simplicity, we omit these as well.
Pions arise as Nambu-Goldstone modes associated
with the breaking of U(Nf )L × U(Nf )R to U(Nf )V .
We write X(x, z) = X0(z) exp(i2π
ata) and expand
to quadratic order in πa. The four-dimensional pion
field is obtained by writing πa(x, z) = πa(x)ψπ(z).
Similarly, the ρa and a1 mesons appear by writing
V aµ (x, z) = g5ρ
µ(x)ψρ(z), A
µ(x, z) = g5a
µ(x)ψa(z). The
wave functions ψπ(z), ψρ(z), and ψa(z) are solutions of
the quadratic equations of motion for fields with four-
momentum q2 = m2 and with boundary conditions
ψ(0) = ∂zψ(zm) = 0. For details see [6, 7].
Making the above substitutions and expanding to
quadratic order yields the four-dimensional action
a∂µπa − 1
aπa − 1
(ρaµν)
(aaµν)
+µǫijk (ρai ∂ja
k + a
i ∂jρ
, (12)
with ρµν , aµν the field strengths for ρµ, aµ. The Chern-
Simons term with coefficient µ mixes the ρ and a1
mesons. It arises from reduction of a term of the form
dV̂ TrAdV in the expansion of Eq. (3).
As usual, to obtain Eq. (12) one must remove the mix-
ing between aaµ and ∂µπ
a by performing the transforma-
tion aaµ → aaµ+ξ∂µπa and then rescaling the pion field to
obtain a canonical kinetic energy term [17]. This leads to
a pion contribution to the Chern-Simons term. A total
spatial derivative, it does not contribute to the equations
of motion and may be dropped.
Since the ρ has JPC = 1−− and the a1 has J
1++, the Chern-Simons coupling is even under P and odd
under C. This is indeed consistent with a background
having non-zero baryon number, which preserves P and
violates C: the coupling is rotationally invariant, but not
Lorentz invariant due to the preferred rest frame of the
baryons.
We can deduce the existence of the Chern-Simons cou-
pling in four-dimensional terms as follows. The reduction
of the five-dimensional Chern-Simons term [5, 22] gives
rise to the usual gauged WZW action [18, 19, 20], as well
as a set of couplings which arise from inexact bulk terms.
These include a ρ − a1 − ω coupling which, in the pres-
ence of a coherent ω field in nuclear matter, gives rise to
a coupling of the form given in Eq. (12). The ρ− a1 − ω
coupling has been considered previously in a general dis-
cussion of chiral effective Lagrangians [25], and is implicit
in the formulae of [23]. Related terms appear in [21, 22].
In AdS/QCD, different forms of the gauged WZW action
can be obtained by the addition of UV counterterms [24],
but these will not cancel the Chern-Simons coupling and
lead to explicit breaking of chiral symmetry beyond that
given by the quark mass term in Eq. (4).
The mass of the ρ meson is given by mρ = 2.405/zm,
while ma1 must be determined from a numerical solution
of the equation of motion. Model B of [6] finds mρ =
832 MeV, ma1 = 1200 MeV which should be compared
to the experimental values mρ = 775.8 ± 0.5 MeV and
ma1 = 1230 ± 40 MeV [26]. The parameter µ in the
Chern-Simons coupling is given by
µ = 18π2nbz
mI (13)
where I is the dimensionless overlap integral
dzzψρ(z)ψa1(z) . (14)
Numerical evaluation of the integral gives I = 0.54.
A typical baryon density in nuclear matter, n0b ≃
0.16/(fermi)3 , gives
µ ≃ 1.05 GeV
. (15)
PHENOMENOLOGICAL APPLICATIONS
We now outline two potentially observable conse-
quences of the Chern-Simons coupling between the ρ and
a1. Details will appear elsewhere.
Mixing of transverse ρ and a1 states
We consider plane-wave solutions to the equations of
motion resulting from Eq. (12), dropping the pion fields
and focusing on the ρ and a1 dispersion relation and po-
larization vectors. Without loss of generality, we consider
propagation along x3:
ρµ(x) = ǫ
µ(q)e
−iq·x, aµ(x) = ǫ
µ(q)e
−iq·x (16)
with q = (q0, 0, 0, q3). For convenience, we suppress the
SU(2) indices in the following. The components ρ0, ρ3,
a0, and a3 have standard dispersion relations, unaffected
by the Chern-Simons coupling. The transverse compo-
nents ρ1, ρ2, a1, and a2 mix through a derivative cou-
pling. The equations of motion yield the dispersion rela-
tion for the transverse polarizations
(m2ρ +m
(m2a1 −m2ρ)2 + 16µ2q
The lower sign in Eq. (17) gives a state which is pure ρ
as q3 → 0. At non-zero q3, it is a mixture of transverse
ρ and a1 states with orthogonal polarization vectors:
iM2(q3)
= − iM
2(q3)
where we have defined ∆2 = m2a1 − m
ρ and M2(q3) =
∆4 + 16µ2q2
− ∆2)/2. The upper sign in Eq. (17)
gives a pure a1 state for q3 = 0, while for non-zero q3,
= − iM
2(q3)
iM2(q3)
. (19)
For µ greater than some momentum-dependent critical
value, the dispersion relation Eq. (17) leads to tachyonic
modes (modes having dq0/dq3 > 1). For very large mo-
menta, this critical value becomes
µcrit =
(m2ρ +m
)/2 ≃ 1.09 GeV . (20)
For a range of µ below µcrit the dispersion relation with
the lower sign in Eq. (17) exhibits interesting anomalous
behavior, the analysis of which is beyond the scope of
this letter.
It would be interesting to explore signatures of these
mixed polarization states in the quark-gluon plasma and
in nuclear matter.
Vector Meson Condensation
To identify the tachyonic instability which occurs for
µ > µcrit we start with the energy density corresponding
to Eq. (12) for the diagonal component of the ρ and a
fields, aa = aδa3, ρa = ρδa3. Completing the square and
dropping the terms involving the electric components of
the field strengths, which play no role in the instability,
we find
H = 1
(m2a − µ2)~a · ~a+
(m2ρ − µ2)~ρ · ~ρ
( ~Ba − µ~ρ)2 +
( ~Bρ − µ~a)2 (21)
where ~Bρ = ~∇× ~ρ , ~Ba = ~∇× ~a.
Applying the ansatz
~a = v cos(µx3)x̂2, ~ρ = v sin(µx3)x̂1 , (22)
the last two terms in Eq. (21) vanish, while the average
of the first two terms over x3 is negative for µ
2 > µ2
leading to an instability to v 6= 0. Understanding the sta-
bilization of the configuration Eq. (22) requires general-
izing H to include higher order terms. Note that Eq. (22)
breaks both rotational and translational symmetry, ex-
hibiting a structure similar to the smectic phase of liquid
crystals which includes an interesting set of topological
defects.
The critical value Eq. (20) is remarkably close to the
estimate Eq. (15) for µ at ordinary nuclear densities. If
this model is accurate then there should be a conden-
sate of vector and axial-vector mesons in nuclear matter
with baryon densities at or slightly above n0b . In ordi-
nary nuclei, there are finite size effects as well as other
corrections to the ρ and a1 interactions which will have
to be included to determine whether this condensate oc-
curs. Neutron stars are more likely to produce such a
condensate, as they are thought to contain matter at a
density somewhat greater than n0b . The interplay be-
tween this condensate and other conjectured effects in
nuclear matter, such as pion condensation and color su-
perconductivity, deserves further study.
ACKNOWLEDGEMENTS
We thank O. Lunin and J. Rosner for helpful conversa-
tions. JH thanks the Galileo Galilei Institute in Arcetri,
Florence for hospitality while this work was being com-
pleted. The work of SD and JH was supported in part by
NSF Grant No. PHY-00506630 and NSF Grant 0529954.
Any opinions, findings, and conclusions or recommenda-
tions expressed in this material are those of the authors
and do not necessarily reflect the views of the National
Science Foundation.
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|
0704.1605 | Total Quantum Zeno Effect beyond Zeno Time | Total Quantum Zeno Effect beyond Zeno Time
D. Mundarain1, M. Orszag2 and J. Stephany1
Departmento de F́ısica, Universidad Simón Boĺıvar,
Apartado Postal 89000, Caracas 1080A, Venezuela
Facultad de F́ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile
In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems
for times larger than the correlation time of the bath. The effect is observed for some particular
systems in which one can chose appropriate observables which frequent measurements freeze the
system into the initial state. For a two level system in a squeezed bath one can show that there are
two bath dependent observables displaying Total Zeno Effect when the system is initialized in some
particular states. We show also that these states are intelligent states of two conjugate observables
associated to the electromagnetic fluctuations of the bath.
I. INTRODUCTION
An interesting consequence of the fact that frequent
measurements can modify the dynamics of quantum sys-
tems is known as Quantum Zeno Effect (QZE) [1, 2, 3,
4, 5] . In general QZE is related to suppression of in-
duced transitions in interacting systems or reduction of
the decay rate in unstable systems. Also, the opposite
effect of enhancing the decay process by frequent mea-
surements has been predicted and is known as Anti-Zeno
Effect (AZE). The experimental observation of QZE in
the early days was restricted to oscillating quantum sys-
tems [6] but recently, both QZE and AZE were success-
fully observed in irreversible decaying processes.[7, 8, 9].
Quantum theory of measurements predicts reduction
of the decay rate in unstable systems when the time be-
tween successive measurements is smaller than the Zeno
Time which is known to be smaller than the correlation
time of the bath. This effect is universal in the sense that
it does not depend on the measured observable whenever
the time between measurements is very small. This ob-
servation does not preclude the manifestation of Zeno
Effect for times larger than the correlation time for some
well selected observables in a particular bath. In this
work we show that is possible for a two-level system inter-
acting with a squeezed bath to select a couple observables
whose measurements beyond the correlation time for ad-
equately prepared systems lead to the total suppression
of transitions, i.e Total Zeno Effect.
This work is organized as follows: In section (II) we
discuss some general facts and review some results ob-
tained in reference [10] which are needed for our discus-
sion. In Section (III) we define the system we deal with
and identify the observables and the corresponding initial
states which are shown to display Total Zeno Effect. In
section (V) we show that the initial states which show To-
tal Zeno Effect are intelligent spin states, i.e states that
saturate the Heisenberg Uncertainty Relation for two fic-
titious spin operators. Finally, we discuss the results in
Section (VI).
II. TOTAL ZENO EFFECT IN UNSTABLE
SYSTEMS
Consider a closed system with Hamiltonian H and an
observable A with discrete spectrum. If the initial state
of the system is the eigenstate |an〉 of A with eigenvalue
an, the probability of survival in a sequence of S measure-
ments, that is the probability that in all measurements
one gets the same result an, is
Pn(∆t, S) =
where
∆2nH = 〈an|H 2|an〉 − 〈an|H |an〉2 (2)
and ∆t is the time between consecutive measurements.
In the limit of continuous monitoring ( S → ∞,∆t → 0
and S∆t → t ), Pn → 1 and the system is freezed in the
initial state.
In an unstable system and for times larger than the
correlation time of the bath, the irreversible evolution
of the system can be described in terms of the Liouville
operator L{ρ} by using the master equation;
= L{ρ} . (3)
In this case the survival probability in a sequence of S
measurements is:
Pn(∆t, S) = (1 + ∆t 〈an|L{|an〉〈an|}|an〉)S (4)
Then, the survival probability in the limit of contin-
uous monitoring is time dependent and is easy to show
that it is given by
Pn(t) = exp {〈an|L{|an〉〈an|}|an〉t} . (5)
In fact for non zero bath correlation time (τD 6= 0) one
cannot take the continous monitoring limit and the equa-
tion (5) is an aproximation since ∆t cannot be strictly
zero and at the same time be larger than τD. In that case
this expression is valid only when the time between con-
secutive measurements is small enough but greater than
http://arxiv.org/abs/0704.1605v1
the correlation time of the bath. For mathematical sim-
plicity in what follows we consider the zero correlation
time limit and then one is allowed to take the limit of
continuous monitoring.
¿From equation (5) one observes that the Total Zeno
Effect is possible when
〈an|L{|an〉〈an|}|an〉 = 0 . (6)
Then, for times larger than the correlation time, the pos-
sibility of having Total Zeno Effect depends on the dy-
namics of the system ( determined by the interaction
with the baths), on the observable to be measured and
on the particular eigenstate of the observable chosen as
the initial state of the system.
If equation (6) is satisfied, then equation (5) must be
corrected, taking the next non-zero contribution in the
expansion of ρ(∆t). In that case the eq. (4) becomes:
Pn(∆t, s) =
1 + 〈an|L{L{|an〉〈an|}}|an〉∆t2/2
Then the survival probility for continous monitoring is
Pn(t) = exp{
〈an|L{L{|an〉〈an|}}|an〉∆t
t} (8)
In general L is proportional to γ, the decay constant for
vacuum. Then as one can see a decay rate proportional to
γ2∆t appears. and the decay time is ∝ 1
, which is in
general a number much larger than the typical evolution
time of the system since ∆t ≪ γ. This observation is
particularly important for system in which one cannot
take the zero limit in ∆t, i.e when one has a bath with
a non zero correlation time. Notice that as the spectrum
of the bath gets broader, τD becomes smaller, and one is
able to choose a smaller ∆t, approaching in this way the
ideal situation and the Total Zeno Effect.
III. TOTAL ZENO OBSERVABLES
In the interaction picture the Liouville operator for a
two level system in a broadband squeezed vacuum has
the following structure [11],
L{ρ} = 1
γ (N + 1)
2σρσ† − σ†σρ− ρσ†σ
2σ†ρσ − σσ†ρ− ρσσ†
−γMeiφσ†ρσ† − γMe−iφσρσ (9)
where γ is the vacuum decay constant and N,M =
N(N + 1) and ψ are the parameters of the squeezed
bath. Here σ and σ† are the ladder operators for a two
level system,
(σx − iσy) σ† =
(σx + iσy) (10)
with σx, σy and σz the Pauli matrices.
Let us introduce the Bloch representation of the two
level density matrix
(1 + ~ρ · ~σ) (11)
Using this representation and the master equation one
can obtain the following set of differential equation for
the components of the Bloch vector (ρx, ρy, ρz):
ρ̇x = −γ (N + 1/2 +M cos(ψ)) ρx + γM sin(ψ)ρy
ρ̇y = −γ (N + 1/2−M cos(ψ)) ρy + γM sin(ψ)ρx
ρ̇z = −γ (2N + 1)ρz − γ (12)
which has the following solutions:
ρx(t) =
ρx(0) sin
2(ψ/2) + ρy(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2−M) t
ρx(0) cos
2(ψ/2)− ρy(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2+M) t (13)
ρy(t) =
ρy(0) cos
2(ψ/2) + ρx(0) sin(ψ/2) cos(ψ/2)
e−γ(N+1/2−M) t
ρy(0) sin
2(ψ/2)− ρx(0) sin(φ/2) cos(ψ/2)
e−γ(N+1/2+M) t (14)
ρz(t) = ρz(0)e
−γ(2N+1)t +
2N + 1
e−γ(2N+1)t − 1
These equations describe the behavior of the system
when there are no measurements.
Consider now the hermitian operator σµ associated to
the fictitious spin component in the direction of the uni-
tary vector µ̂ = (cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)) de-
fined by the angles θ and φ,
σµ = ~σ · µ̂ = σx cos(φ) sin(θ)+σy sin(φ) sin(θ)+σz cos(θ)
The eigenstates of σµ are,
|+〉µ = cos(θ/2) |+〉+ sin(θ/2) exp (iφ) |−〉 (17)
|−〉µ = − sin(θ/2) |+〉+ cos(θ/2) exp (iφ) |−〉 (18)
If the system is initialized in the state |+〉µ the survival
probability at time t is
P+µ (t) = exp {F (θ, φ) t } (19)
where
F (θ, φ) = µ〈+| L { |+〉µ µ〈+| } |+〉µ . (20)
In this case the function F (θ, φ) has the structure
F (θ, φ) = −1
γ (N + 1)
ρz(0) + ρ
z(0) +
ρ2x(0) +
ρ2y(0)
(ρz(0)− ρ2z(0)−
ρ2x(0)−
ρ2y(0)
γMρx(0)(cos(ψ)ρx(0)− sin(ψ)ρy(0))
γMρy(0)(sin(ψ)ρx(0) + cos(ψ)ρy(0)) (21)
where now ~ρ(0) = µ̂ is a function of the angles..
In figure (1) we show F (φ, θ) for N = 1 and ψ = 0 as
function of φ and θ. The maxima correspond to F (φ, θ) =
0. For arbitrary values of N and ψ there are two maxima
corresponding to the following angles:
φM1 =
π − ψ
and cos(θM ) = − 1
2 (N +M + 1/2)
φM2 =
π − ψ
+ π and cos(θM ) = − 1
2 (N +M + 1/2)
Theta
FIG. 1: F (φ, θ) for N = 1 and ψ = 0
These preferential directions given by the vectors µ̂1 =
(cos(φM1 ) sin(θ
M ), sin(φM1 ) sin(θ
M ), cos(θM )) and µ̂2 =
(cos(φM2 ) sin(θ
M ), sin(φM2 ) sin(θ
M ), cos(θM ))) define the
operators σµ1 and σµ2 which show Total Zeno Effect if
the initial state of the system is the eigenstate |+〉µ1 or
respectively |+〉µ2 , then each preferential observable has
only one eigenstate displaying Total Zeno Effect. These
eigenstates are:
|+〉µ1 =
|+〉+ i
exp{−iψ
}|−〉 (24)
|+〉µ2 =
|+〉 − i
exp{−iψ
}|−〉 (25)
The other eigenstates of the observables do not dis-
play Total Zeno Effect. As final remark is important to
observe that in the previous calculations we have ever
chosen the state |+〉µ in order to optimize the function
F (φ, θ). In fact one can select the state |−〉µ but the
final observables displaying Total Zeno Effect will be in
the same preferential directions indicated above.
IV. MASTER EQUATION AND
MEASUREMENTS
Besides of the Total Zeno effect obtained in the cases
specified previously it is also very interesting to discuss
the effect of measurements for other choices of the initial
state, the states which do not display Total Zeno Effect.
To be specific let us consider measurements of the ob-
servable σµ = ~σ · µ̂. The modified master equation with
the measurement of σµ is given by [10]:
= Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ) (26)
where
Pµ = |+〉µ µ〈+| (27)
and L{ρ} is given by (9). This equation can be solved
using the Bloch representation of the density matrix. In
this case we can write the density operator in terms of
a second set of rotated Pauli matrices that includes the
Pauli observable which we are measuring :
(1 + ρµσµ + ρασα + ρβσβ) (28)
where σα and σβ are two Pauli matrices projected in two
orthogonal direction to the vector µ̂. During the process
of measurement one obtains always eigenvectors of σµ
observable, these eigenvectors have the property of being
zero valued for the other two observables. Then during
the measurement process the quantities ρα and ρβ are
equal to zero because these quantities coprrespond to the
mean values of the respectives observables. Then in this
case the density matrix can be written in term of one
parameter which corresponds to the mean value of the
observable that is bein measured;
(1 + ρµσµ) (29)
ρµ = 〈σµ〉 = Tr {ρσµ} (30)
Then the master equation is reduced to the following
differential equation:
ρ̇µ = Tr {ρ̇σµ}
= Tr {(Pµ L {ρ} Pµ + (1− Pµ) L {ρ} (1− Pµ)) σµ}
= Tr {L {ρ}σµ} (31)
This equation could induced to think that the evolution
with and withouth measurements are equal, but we must
remember that the density matrix in the right hand side
of (??) is the density with measurements. Substituting
the form of the density matrix during the measuring pro-
cess one can obtain a real differential equation for ρµ:
ρ̇µ = α+ βρµ (32)
where
Tr {L {1}σµ} (33)
Tr {L {σµ}σµ} (34)
In our case and measuring σµ1 one obtains
α = 2 γ (N −M + 1/2) (35)
β = −α = −2 γ (N −M + 1/2) (36)
The solution to the differential equation is
ρµ(t) = 1 + (ρµ(o) − 1) e−αt (37)
one can observe the Total Zeno Effect when ρµ(o) = 1
which correspond to having as initial state |+〉µ1 .
In figure (2) we show the evolution of 〈σµ1〉, that is the
mean value of observable σµ1 , when the system is initial-
ized in the state |+〉µ1 . without measurements (master
equation (9)) and with frequent monitoring of σµ1 (mas-
ter equation (26)). Consistently with our discussion of
frequent measurements, the system is freezed in the state
|+〉µ1 (Total Zeno Effect).
In figure (3) we show the time evolution of 〈σµ1〉 when
the initial state is |−〉µ1 without measurements and with
measurements of the same observable as in previous case.
One observes that with measurements the system evolves
from |−〉µ1 to |+〉µ1 . In general for any initial state the
system under frequent measurements evolves to |+〉µ1
which is the stationary state of Eq. ( 26) whenever we
do measurements in σµ1 . Analogous effects are observed
if one measures σµ2 . In contrast, for measurements in
other directions different from those defined by µ̂1 or µ̂2
, the system evolves to states which are not eigenstates
of the measured observables.
< σ 1 > (t)
FIG. 2: 〈σµ1(t)〉 for N = 1 and ψ = 0. Solid circles: no mea-
surements. Empty circles: with measurements. One measures
σµ1 and the initial state is |+〉µ1
.5 0
.5 1
< σ 1 > (t)
.5 0
.5 1
< σ 1 > (t)
FIG. 3: 〈σµ1(t)〉 for N = 1 y ψ = 0. Solid circles: no mea-
surements. Empty circles: with measurements. One measures
σµ1 and the initial state is |−〉µ1
V. INTELLIGENT STATES
Aragone et al [12] considered well defined angular mo-
mentum states that satisfy the equality (∆Jx∆Jy)
| 〈Jz〉 |2 in the uncertainty relation. They are called
Intelligent States in the literature. The difference with
the coherent or squeezed states, associated to harmonic
oscillators, is that these Intelligent States are not Mini-
mum Uncertainty States (MUS), since the uncertainty is
a function of the state itself.
In this section we show that the states |+〉µ1 and |+〉µ2
are intelligent states of two observables associated to the
bath fluctuations. The master equation (9) can be writ-
ten in an explicit Lindblad form
2SρS† − ρS†S − S†Sρ
using only one Lindblad operator S,
N + 1σ −
N exp {iψ}σ† (39)
S = cosh(r)σ − sinh(r) exp {iψ}σ† (40)
Obviously any eigenstate of S satisfies the condi-
tion (6). It is very easy to show that the S opera-
tor has two eigenvectors |λ±〉 with eigenvalues λ± =
M exp{iψ/2}. It is also easy to observe that these
two states are exactly the same states founded in the
previous section, |λ+〉 = |+〉µ1 and |λ−〉 = |+〉µ2 .
Consider now the standard fictitious angular momen-
tum operators for the two level system are {Jx =
σx/2, Jy = σy/2, Jz = σz/2} and also two rotated op-
erators J1 and J2 which are consistent with the electro-
magnetic bath fluctuations in phase space (see fig. 2 in
ref [10]) and which satisfy the same Heisenberg uncer-
tainty relation that Jx and Jy . They are,
J1 = exp{iψ/2Jz}Jx exp{−iψ/2Jz}
= cos(ψ/2)Jx − sin(ψ/2)Jy (41)
J2 = exp{iψ/2Jz}Jy exp{−iψ/2Jz}
= sin(ψ/2)Jx + cos(ψ/2)Jy (42)
These two operators are associated respectively with the
major and minor axes of the ellipse which represents the
fluctuations of bath.
In terms of J1 y J2 we have
J− = σ = (Jx − iJy) = exp{iψ/2}(J1 − iJ2) , (43)
J+ = σ
† = (Jx + iJy) = exp{−iψ/2}(J1 + iJ2) . (44)
Then S can be written in the following form:
S = exp{iψ/2} (cosh(r) − sinh(r)) (J1 − iαJ2) (45)
cosh(r) + sinh(r)
cosh(r) − sinh(r)
= exp{2r} (46)
Following Rashid et al ( [13]) we define a non hermitian
operator J−(α)
J−(α) =
(J1 − iαJ2)
(1 − α2)1/2
so that
S = exp{iψ/2} (cosh(r) − sinh(r)) (1− α2)1/2 J−(α)
After some algebra one obtains that
S = 2λ+ J−(α) (49)
¿From this equation one can observe that the eigen-
states of S are also eigenstates of J−(α) with eigenvalues
±1/2. It is known that the eigenstates of J−(α) are in-
telligent states of J1 and J2, i.e they satisfy the equality
condition in the Heisenberg uncertainty relation for these
observables:
∆2J1∆
2J2 =
|〈Jz〉|2
VI. DISCUSSION
We have shown that Total Zeno Effect is obtained for
two particular observables σµ1 or σµ2 , for which the az-
imuthal phases in the fictitious spin representation de-
pend on the phase of the squeezing parameter of the bath
and the polar phases depend on the squeeze amplitude.
In this sense, the parameters of the squeezed bath specify
some definite atomic directions.
When performing frequent measurements on σµ1 ,
starting from the initial state |+〉µ1 , the system freezes
at the initial state as opposed to the usual decay when
no measurements are done. On the other hand, if the
system is initially prepared in the state |−〉µ1 , the fre-
quent measurements on σµ1 will makes it evolve from
the state |−〉µ1 to |+〉µ1 . More generally, when perform-
ing the measurements on σµ1 , any initial state evolves
to the same state |+〉µ1 which is the steady state of the
master equation (26) in this situation.
The above discussion could appear at a first sight sur-
prising. However, taking a more familiar case of a two-
level atom in contact with a thermal bath at zero temper-
ature, if one starts from any initial state, the atom will
necessarily decay to the ground state. This is because
the time evolution of 〈σz〉 is the same with or without
measurements of σz . In both cases the system goes to
the ground state, which is an eigenstate of the measured
observable σz. In the limit N,M → 0, σµ1 → −σz , and
the state |+〉µ1 → |−〉z, which agrees with the known
results.
Finally, we found that the two eigentates of the two
preferential observables displaying QZE are also eigen-
states of S operator and consequently intelligent states
of J1, J2 which are rotated versions of Jx, Jy obsevables.
A. Acknowledgements
Two of the authors(D.M. and J.S.) were supported
by Did-Usb Grant Gid-30 and by Fonacit Grant No G-
2001000712.
M.O was supported by Fondecyt # 1051062 and Nu-
cleo Milenio ICM(P02-049)
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|
0704.1606 | Asteroseismic Signatures of Stellar Magnetic Activity Cycles | Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 1 November 2018 (MN LATEX style file v2.2)
Asteroseismic Signatures of Stellar Magnetic Activity Cycles
T. S. Metcalfe1,2, W. A. Dziembowski3,4, P. G. Judge1, M. Snow5
1 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307-3000 USA
2 Scientific Computing Division, National Center for Atmospheric Research, Boulder, CO 80307-3000 USA
3 Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland
4 Copernicus Astronomical Centre, Bartycka 18, 00-716 Warsaw, Poland
5 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80309-0392 USA
1 November 2018
ABSTRACT
Observations of stellar activity cycles provide an opportunity to study magnetic dynamos
under many different physical conditions. Space-based asteroseismology missions will soon
yield useful constraints on the interior conditions that nurture such magnetic cycles, and will
be sensitive enough to detect shifts in the oscillation frequencies due to the magnetic varia-
tions. We derive a method for predicting these shifts from changes in the Mg II activity index
by scaling from solar data. We demonstrate this technique on the solar-type subgiant β Hyi,
using archival International Ultraviolet Explorer spectra and two epochs of ground-based as-
teroseismic observations. We find qualitative evidence of the expected frequency shifts and
predict the optimal timing for future asteroseismic observations of this star.
Key words: stars: activity – stars: individual (β Hyi) – stars: interiors – stars: oscillations
1 INTRODUCTION
Astronomers have been making telescopic observations of sunspots
since the time of Galileo, gradually building an historical record
showing a periodic rise and fall in the number of sunspots every
∼11 years. We now know that sunspots are regions with an en-
hanced local magnetic field, so this 11-year cycle actually traces
a variation in surface magnetism. Attempts to understand this be-
havior theoretically often invoke a combination of differential rota-
tion, convection, and meridional flow to modulate the field through
a magnetic dynamo (e.g., see Rempel 2006; Dikpati & Gilman
2006).
Although we can rarely observe spots on other solar-type stars
directly, these areas of concentrated magnetic field produce strong
emission in the Ca II H and K resonance lines in the optical, and
the Mg II h and k lines in the ultraviolet. Wilson (1978) was the
first to demonstrate that many solar-type stars exhibit long-term
cyclic variations in their Ca II H and K emission, analogous to those
seen in full-disc solar observations through the magnetic activity
cycle. Early analysis of these data revealed an empirical correlation
between the mean level of magnetic activity and the rotation pe-
riod normalized by the convective timescale (Noyes et al. 1984a),
as well as a relation between the rotation rate and the period of
the observed activity cycle (Noyes et al. 1984b), which generally
supports a dynamo interpretation.
Significant progress in dynamo modeling unfolded after he-
lioseismology provided detailed constraints on the Sun’s interior
structure and dynamics. These observations also established that
variations in the mean strength of the solar magnetic field lead to
significant shifts (∼0.5 µHz) in the frequencies of even the lowest-
degree p-modes (Libbrecht & Woodard 1990; Salabert et al. 2004).
These shifts can provide independent constraints on the physical
mechanisms that drive the solar dynamo, through their influence
on the outer boundary condition for the pulsation modes. They are
thought to arise either from changes in the near-surface propagation
speed due to a direct magnetic perturbation (Goldreich et al. 1991),
or from a slight decrease in the radial component of the turbulent
velocity in the outer layers and the associated changes in tempera-
ture (Dziembowski & Goode 2004, 2005).
Space-based asteroseismology missions, such as MOST
(Walker et al. 2003), CoRoT (Baglin et al. 2006), and Kepler
(Christensen-Dalsgaard et al. 2007) will soon allow additional tests
of dynamo models using other solar-type stars (see Chaplin et al.
2007). High precision time-series photometry from MOST has al-
ready revealed latitudinal differential rotation in two solar-type
stars (Croll et al. 2006; Walker et al. 2007), and the long-term mon-
itoring from future missions is expected to produce asteroseismic
measurements of stellar convection zone depths (Monteiro et al.
2000; Verner et al. 2006). By combining such observations with
the stellar magnetic activity cycles documented from long-term sur-
veys of the Ca II or Mg II lines, we can extend the calibration of
dynamo models from the solar case to dozens of independent sets
of physical conditions.
The G2 subgiant β Hyi is the only solar-type star that presently
has both a known magnetic activity cycle (Dravins et al. 1993)
and multiple epochs of asteroseismic observations (Bedding et al.
2001, 2007). In this paper we reanalyze archival International Ul-
traviolet Explorer (IUE) spectra for an improved characterization
of the magnetic cycle in this star, and we use it to predict the activity
related shifts in the observed radial p-mode oscillations. We com-
pare these predictions with recently published asteroseismic data,
c© 0000 RAS
http://arxiv.org/abs/0704.1606v1
2 Metcalfe et al.
and we suggest the optimal timing of future observations to maxi-
mize the amplitude of the expected p-mode frequency shifts.
2 ARCHIVAL IUE SPECTRA
The activity cycle of β Hyi was studied by Dravins et al. (1993),
who used high resolution IUE data of the Mg II resonance lines over
11 years, from June 1978 to the end of October 1989. They consid-
ered these data to be consistent with a cycle period between 15 and
18 years. Since the work of Dravins et al., a significant number of
additional IUE spectra were obtained by E. Guinan from early 1992
to the end of 1995. Our analysis of all of these spectra reveals the
beginning of a new cycle in 1993-1994.
In 1997, the IUE project reprocessed the entire database using
improved and uniform reduction procedures (“NEWSIPS”). Using
the NEWSIPS merged high resolution extracted spectra, we have
reanalyzed the entire IUE dataset containing useful echelle data
of the Mg II lines. Data were excluded when the NEWSIPS soft-
ware mis-registered the spectral orders, when continuum data near
279.67 nm were saturated, or when continuum data were more than
1σ below the mean (to reject additional poorly registered spectral
orders) or more than 1.5σ above the mean.
The classic definition of the Mg II index (Heath & Schlesinger
1986) uses wing irradiances at 276 and 283 nm. The wings in their
formulation had to be so far away from the cores due to the 1.1 nm
spectral bandpass of their instrument. In the IUE spectra, pixels
at those wavelengths are saturated, so the photospheric reference
levels need to be measured much closer to the emission cores.
Snow & McClintock (2005) have shown that at moderate resolu-
tion the variability of the inner wings of the Mg II absorption fea-
ture is very similar to the variability of the classic wing irradiances.
Therefore, we can construct a modified Mg II index using only the
unsaturated IUE data that still captures the full chromospheric vari-
ability.
The chromospheric line cores (0.14 and 0.12 nm wide band-
passes centered at 279.65 and 280.365 nm, in vacuo) and two bands
in the photospheric wings of the lines (0.4 nm wide bands, edge-
smoothed with cosine functions, centered at 279.20 and 280.70 nm)
were integrated, and the ratio of total core to total wing fluxes was
determined. Figure 1 shows the core to wing indices determined
from each usable spectrum from 1978 to the end of 1995. The post-
1992 data permit us to revise the cycle period estimate downwards
to ∼12.0 years, with more confidence than was previously possi-
ble. This period was derived by fitting a simple sinusoid to the data
using the genetic algorithm PIKAIA (Charbonneau 1995). The op-
timal fit yields minima at 1980.9 and 1992.8, and a maximum at
1986.9. The fit suggests that the next maximum occurred in 1998.8,
a minimum in 2004.8, and a future maximum predicted for 2010.8.
The reduced χ2 of the fit was calculated using flux uncertainties
for individual IUE observations of 7%, estimated from the varia-
tion in the ratios of the two wing fluxes, which vary far less than
this in the SOLSTICE solar data. This reduced χ2 has a minimum
value of 0.77. The probability of such a value occurring at random
is 24%, whereas a χ2 of 1.1 has a random probability of 76%. The
2 = 1.1 hypersurface contours suggests that the uncertainties are
roughly ±1 yr for the phase and +3.0−1.7 yr for the period, making our
new period estimate marginally consistent with the range quoted
by Dravins et al. (1993). The χ2 contours are ovals because these
uncertainties are correlated, allowing us to set the following formal
limits on the epochs of maximum: 1986.1–1988.0, 1997.5–2002.4,
and 2007.8–2017.8. These large uncertainties reinforce the need for
Figure 1. Core to wing ratios of the summed Mg II h and k lines determined
from IUE high dispersion observations of β Hyi (small points) and 3-month
seasonal averages with the uncertainties used in the evaluation of χ2 shown
as error bars (large points). The curve is an optimized simple sinusoid fit
obtained using a genetic algorithm applied to the seasonally averaged data,
intended only to estimate the period and phase of the stellar activity cycle,
which is listed in the legend. The IUE index is shown on the left, while the
corresponding NOAA index is shown on the right.
an activity cycle monitoring program specifically for the southern
hemisphere.
To compare the IUE Mg II index measurements to the National
Oceanic & Atmospheric Administration (NOAA) composite data
of solar activity1, we must determine the appropriate scaling fac-
tor. The SOLar-STellar Irradiance Comparison Experiment (SOL-
STICE) on the SOlar Radiation and Climate Experiment (SORCE;
McClintock et al. 2005) measures the solar irradiance every day,
and has a resolution of 0.1 nm in this region. We convolved the
IUE spectra with the SOLSTICE instrument function and then mea-
sured the wings and cores of both solar and stellar data in exactly
the same way. In particular, we used 279.15-279.35 nm as the blue
wing, and 280.65-280.85 nm for the red wing. The emission cores
were defined as 279.47-279.65 nm and 280.21-280.35 nm. We de-
termined the relation between the SOLSTICE modified Mg II index
and the NOAA long-term record using a standard linear regression
method (see Snow et al. 2005; Viereck et al. 2004). Since the mod-
ified IUE data has the same bandpass as the SOLSTICE data, the
scaling factors derived from SOLSTICE solar data will also ap-
ply to the stellar IUE data2. The Mg II index for β Hyi scaled
to the NOAA composite data is shown on the right axis of Fig-
ure 1. For the analysis in Section 3, we adopt a full amplitude of
∆iMgII = 0.015 in the NOAA index.
3 SCALING P-MODE SHIFTS FROM SOLAR DATA
In general, we can evaluate activity related frequency shifts from
the variational expression,
∆ν j =
d3xK jS
2I jν j
, (1)
where
1 http://www.sec.noaa.gov/ftpdir/sbuv/NOAAMgII.dat
2 To transform between indices: NOAA = 0.211 + 0.0708 SOLSTICE;
SOLSTICE = 0.297 + 1.11 IUE; NOAA = 0.232 + 0.079 IUE
c© 0000 RAS, MNRAS 000, 000–000
Asteroseismic Signatures of Activity Cycles 3
I j =
d3xρ|ξ|3 ≡ R5ρ̄Ĩ (2)
is the mode inertia, j ≡ (n,ℓ,m), and we need to know both the
source S(x) and the corresponding kernels K(x). The source must
include the direct influence of the growing mean magnetic field,
as well as its indirect effect on the convective velocities and tem-
perature distribution. Separate kernels for these effects were cal-
culated by Dziembowski & Goode (2004), but there is no theory
available to calculate the combined source. Moreover, it is unclear
whether the model of small-scale magnetic fields, adopted from
Goldreich et al. (1991), is adequate. Therefore, we will attempt to
formulate an extrapolation of the solar p-mode frequency shifts
based on changes in the Mg II activity index measured for the Sun
and for β Hyi in Section 2.
For p-modes, the dominant terms in all of the kernels are pro-
portional to |divξ|2. Thus, we write
K j(x) = |divξ j|
2 = q j(D)Y
ℓ , (3)
where D is the depth beneath the photosphere. A model-dependent
coefficient will be absorbed into the source, which we write in the
S(x) =
Sk(D)P2k(cosθ). (4)
Solar data imply that S is strongly concentrated near the photo-
sphere. Therefore data from all p-modes, regardless of their ℓ value,
may be used to constrain S . We might also expect that the source
normalization is correlated with the Mg II index.
If we want to calculate ∆ν j according to Eq. (1), we need
all terms of S up to k = ℓ. To assess the solar S , we have mea-
surements of the centroid shifts and the even-a coefficients (see
Dziembowski & Goode 2004). For the ℓ = 0 modes we only need
to know the k = 0 term, and for this the centroid data are sufficient.
Let us begin with this simple case.
3.1 Radial modes
Theoretical arguments and the observed pattern of solar frequency
changes suggest that the dominant source must be localized near
the photosphere. Therefore, it seems reasonable to try to fit the mea-
sured p-mode frequency shifts by adopting
S0(D) = 1.5× 10
−11A0δ(D − Dc) µHz
, (5)
with adjustable parameters A0 and Dc. The numerical coefficient is
arbitrary, and was chosen for future convenience. With Eqs. (3-5),
we get from Eq. (1)
∆ν j = A0
Q j(Dc), (6)
where R and M (as well as L below) are expressed in solar units,
frequencies are expressed in µHz, and
Q j = 1.5× 10
−11 q j
ν j Ĩ j
. (7)
The solar values of A0 and Dc can be determined by fitting the
centroid frequency shifts ∆ν j from SOHO MDI data for p-modes
with various spherical degrees, ℓ. Since at n = 1 the approximation
inherent in Eq. (3) is questionable, we use data only for the higher
orders.
For the Sun, we have
A0,⊙(Dc) =
w j∆ν j
, (8)
Figure 2. Determinations of A0,⊙ from SOHO MDI data with Dc fixed at
0.3 Mm (top panel), and using the optimal value of Dc for each set (middle
panel), with the corresponding changes in the Mg II index from the NOAA
composite data (bottom panel).
where ∆ν j are the measured shifts and w j are the relative weights.
The values of Q j are calculated from a solar model. The best value
of Dc is that which minimizes the dispersion,
σ(Dc) =
A0,⊙ −
. (9)
It is also reasonable to assume that A0 should be proportional to
the change in the Mg II activity index, ∆iMgII, and that Dc is pro-
portional to the pressure scale height at the photosphere. Thus, we
A0 = A0,⊙
∆iMgII
∆iMgII,⊙
Dc ∝ Hp = Dc,⊙L
0.25 R
. (11)
To determine A0,⊙, we used SOHO MDI frequencies for all
pn modes with n > 1, ℓ from 0 to ∼181, and ν between 2.5 and
4.2 mHz. The data were combined into 38 sets, typically cover-
ing 0.2 years. We averaged the frequencies from the first 5 sets,
corresponding to solar minimum (1996.3-1997.3), and subtracted
them from the frequencies in subsequent sets to evaluate A0,⊙ using
Eq. (8). The results are shown in Figure 2, where the points in the
top panel were obtained at fixed Dc = 0.3 Mm, which is represen-
tative of the highest activity period. In the middle panel, the value
of Dc was determined separately for each set and the error bars
represent the dispersion. In the bottom panel, we show the corre-
sponding changes in the solar Mg II index calculated from NOAA
composite data. A tight correlation between A0,⊙ and ∆iMgII,⊙ is
clearly visible.
Although the optimum value of Dc is weakly correlated with
the activity level, the dispersion changes very little between Dc =
0.2 and 0.4 Mm, so we fixed the value of Dc,⊙ to 0.3 Mm. In Fig-
c© 0000 RAS, MNRAS 000, 000–000
4 Metcalfe et al.
Figure 3. The observed p-mode frequency shifts averaged from four sets
of data obtained near the solar maximum in 2002.0 (top panel), and the
same shifts normalized by Q j (bottom panel) showing that most of the
frequency and ℓ-dependence is included in our parametrization. Different
symbols show the roughly equal number of modes with ℓ ≤ 30 (circles),
31 ≤ ℓ≤ 75 (squares), and 76 ≤ ℓ≤ 181 (triangles).
ure 3, we show the quality of the fit to the observed frequency
shifts for selected p-modes using Eq. (6) with the adopted value
of Dc. The upper panel shows the frequency shifts averaged from
four sets of data near the activity maximum in 2002.0, while the
lower panel shows ∆ν j/Q j . Note that most of the frequency and
ℓ-dependence appears to have been fit by our parametrization. The
slight rise at frequencies below 3 mHz could be eliminated by al-
lowing a spread of the kernel toward lower depths. However, since
the signal is more significant at higher frequencies, we believe that
adding a finite radial extent would be an unnecessary complication.
There were two activity maxima during solar cycle 23. The
first was centered near 2000.6 and the second at 2002.0. The av-
erage values of (A0,⊙,∆iMgII,⊙) are (0.3116, 0.0135) for five data
sets around the first maximum and (0.3669, 0.0178) for four data
sets around the second maximum. For future applications, we adopt
A0,⊙/∆iMgII,⊙ = 22.
With this specification, we get from Eqs. (6) and (10)
∆ν j = 3.3× 10
∆iMgII
q j(Dc)
ν j Ĩ j
, (12)
where
Dc = 0.3L
0.25 R
mM (13)
and again frequencies are expressed in µHz, while R, M, and L
are in solar units. This is our expression for predicting the radial
p-mode frequency shifts on the basis of changes in the NOAA
composite Mg II index. In Table 1, we list the frequency shifts
(∆ν j) calculated from Eq. (12) for the radial modes of β Hyi
observed by Bedding et al. (2007), adopting ∆iMgII = 0.015. The
mode parameters q j and Ĩ j were calculated from a model of β Hyi
generated using the Aarhus STellar Evolution Code (ASTEC;
Christensen-Dalsgaard 1982).
Table 1. Predicted radial p-mode frequency shifts between activity max-
imum and minimum for β Hyi, calculated with Eq. (12) and adopting
∆iMgII = 0.015. Frequencies are from Table 1 of Bedding et al. (2007).
n Frequency (µHz) ∆ν j (µHz)
13 833.72 0.061
14 889.87 0.091
15 945.64 0.116
16 1004.21 0.139
17 1062.06 0.168
18 1118.93 0.199
19 1176.48 0.234
3.2 Non-radial modes
Now from Eqs. (1) and (4). we have
∆νnlm =
Skκk,lm
2Inlνnl
, (14)
where,
κk,lm =
dθdφ|Y mℓ |
2P2k(cosθ) sinθ. (15)
As in Eq. (5), we can adopt
Sk(D) = 1.5× 10
−11Akδ(D − Dc,k) µHz
. (16)
For k > 0, the solar amplitudes Ak and effective depths, Dc,k
can be determined by fitting measurements of shifts in the a2k co-
efficients. The relation is
∆a2k,ℓm = AkZk,ℓQk,nℓ(Dc,k), (17)
where
Zk,ℓ = (−1)
k (2k − 1)!!
(2ℓ+ 1)!!
(2ℓ+ 2k + 1)!!
(ℓ− 1)!
(ℓ− k)!
(cf. Dziembowski & Goode 2004, their Eq. 2), and
Qk,nℓ = 1.5× 10
−11 qnℓ(Dc,k)
ν j Ĩnℓ
(compare to our Eq. 7).
The prediction of frequency shifts for non-radial modes re-
quires an additional assumption of the same scaling for all required
Ak amplitudes, which amounts to assuming the same Butterfly di-
agram as observed on the Sun. Moreover, since the shifts depend
on |m| and multiplets are not expected to be resolved, we need to
adopt the inclination angle (i) to correctly weight the contributions
from all of the components. Since we do not know i for β Hyi, we
restrict our numerical predictions to the radial modes.
4 ASTEROSEISMIC OBSERVATIONS
The detection of solar-like oscillations in β Hyi was first reported
by Bedding et al. (2001), and later confirmed by Carrier et al.
(2001). These two detections of excess power were based on data
obtained during a dual-site campaign organized in June 2000 using
the 3.9-m Anglo-Australian Telescope (AAT) at Siding Spring Ob-
servatory and the 1.2-m Swiss telescope at the European Southern
Observatory (ESO) in Chile. Both sets of observations measured
a large frequency separation between 56-58 µHz, but neither was
sufficient for unambiguous identification of individual oscillation
modes.
c© 0000 RAS, MNRAS 000, 000–000
Asteroseismic Signatures of Activity Cycles 5
Nearly 30 individual modes in β Hyi with ℓ=0-2 were detected
during a second dual-site campaign organized in September 2005,
and reported by Bedding et al. (2007). The authors also reanalyzed
the combined 2000 observations using an improved extraction al-
gorithm for the AAT data, allowing them to identify some of the
same oscillation modes at this earlier epoch. Motivated by the first
tentative detection of a systematic frequency offset between two
asteroseismic data sets for α Cen A (0.6±0.3 µHz; Fletcher et al.
2006), they compared the two epochs of observation for β Hyi and
found the 2005 frequencies to be systematically lower than those in
2000 by 0.1±0.4 µHz, consistent with zero but also with the mean
value in Table 1.
A comparison of the individual modes from these two data
sets (T. Bedding, private communication) allows a further test of
our predictions. Of the 14 modes that were detected with S/N> 4
in both 2000 and 2005, only one was known to be a radial (ℓ = 0)
mode, while four had ℓ = 1, three had ℓ = 2, two were mixed modes,
and four had no certain identification. Without a known inclination,
we can only calculate the shifts for radial modes, but the magnitude
of the shift is largest at high frequencies (see Table 1). Fortunately,
the radial mode that is common to both data sets (ℓ = 0, n = 18) has a
frequency above the peak in the envelope of power, improving our
chances of measuring a shift. The best estimate of the mode fre-
quency from each data set comes from the noise-optimized power
spectrum, since this maximizes the S/N of the observed peaks. The
noise-optimized frequency for the ℓ = 0, n = 18 mode was 1119.06
and 1118.89 µHz in the 2000 and 2005 data sets, respectively.
Considering the quoted uncertainty for this mode from Table 1 of
Bedding et al. (2007), the frequency was 0.17±0.62 µHz lower in
2005 than in 2000, again consistent with zero but similar to the
predicted shift for this mode in Table 1.
5 DISCUSSION
Our reanalysis of archival IUE spectra for β Hyi allows us to test
our predictions of the relationship between the stellar activity cy-
cle and the systematic frequency shift measured from multi-epoch
asteroseismic observations. The optimal period and phase of the ac-
tivity cycle from Section 2 suggest that β Hyi was near magnetic
minimum (2004.8) during the 2005 observations (2005.7), while it
was descending from magnetic maximum (1998.8) during the 2000
campaign (2000.5). The systematic frequency shift of 0.1±0.4 µHz
reported by Bedding et al. (2007) between these two epochs, and
the observed shift of 0.17±0.62 µHz in the only radial mode (ℓ = 0,
n = 18) common to both data sets are not statistically significant.
They are both nominally in the direction predicted by our analysis
of the activity cycle (lower frequencies during magnetic minimum)
and they have approximately the expected magnitude (cf. Table 1),
but the formal uncertainties on the period and phase of the activity
cycle do not permit a definitive test.
Future asteroseismic observations of β Hyi would sample the
largest possible frequency shift relative to the 2005 data if timed
to coincide with the magnetic maximum predicted for 2010.8+7−3 .
Long-term monitoring of the stellar activity cycles of this and other
southern asteroseismic targets (e.g. α Cen A/B, µ Ara, ν Ind),
which are not included in the Mt. Wilson sample, would allow
further tests of our predictions. For asteroseismic targets that have
known activity cycles from long-term Ca II H and K measurements
(e.g. ǫ Eri, Procyon), it would be straightforward to calibrate our
predictions to this index from comparable solar observations.
While our current analysis involves a simple scaling from so-
lar data, future observations may allow us to refine magnetic dy-
namo models by looking for deviations from this scaling relation
and attempting to rectify the discrepancies. By requiring the models
to reproduce the observed activity cycle periods and amplitudes—
along with the resulting p-mode shifts and their frequency depen-
dence for a variety of solar-type stars at various stages in their
evolution—we can gradually provide a broader context for our un-
derstanding of the dynamo operating in our own Sun.
ACKNOWLEDGMENTS
We would like to thank D. Salabert for inspiring this work with an
HAO colloquium on low-degree solar p-mode shifts in May 2005,
Keith MacGregor and Margarida Cunha for thoughtful discussions,
and the Copernicus Astronomical Centre for fostering this collab-
oration during a sponsored visit in September 2006. We also thank
the SOHO/MDI team, and especially Jesper Schou for easy access
to the solar frequency data, Tim Bedding for providing frequency
data for β Hyi, and Jørgen Christensen-Dalsgaard for the use of his
stellar evolution code. This work was supported in part by an NSF
Astronomy & Astrophysics Fellowship under award AST-0401441,
by Polish MNiI grant No. 1 P03D 021 28, and by NASA contract
NAS5-97045 at the University of Colorado. The National Center
for Atmospheric Research is a federally funded research and devel-
opment center sponsored by the U.S. National Science Foundation.
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Introduction
Archival IUE spectra
Scaling p-mode shifts from solar data
Radial modes
Non-radial modes
Asteroseismic Observations
Discussion
|
0704.1608 | Diffractive parton distributions from the analysis with higher twist | Diffractive parton distributions from the analysis
with higher twist
Krzysztof Golec-Biernat(a,b)∗ and Agnieszka Luszczak(b)†
aInstitute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland
bInstitute of Physics, University of Rzeszów, Rzeszów, Poland
(October 29, 2018)
Abstract
Diffractive parton distributions of the proton are determined from fits to diffractive data
from HERA. In addition to the twist–2 contribution, the twist–4 contribution from longi-
tudinally polarised virtual photons is considered, which is important in the region of small
diffractive masses. A new prediction for the longitudinal diffractive structure function is
presented which differs significantly from that obtained in the pure twist–2 analyses.
e-mail: [email protected]
e-mail: [email protected]
http://arxiv.org/abs/0704.1608v3
1 Introduction
The diffractive deep inelastic scattering (DDIS) at HERA provide a very interesting example of
the interplay between hard and soft aspects of QCD interactions. On one side, the virtuality of
the photon probe is large (Q2 ≫ Λ2QCD), while on the other side, the scattered proton remains
almost intact, loosing only a small fraction of its initial momentum. Its transverse momentum
with respect to the photon-proton collision axis is also small. In addition to the scattered incident
particles, a diffractive system forms which is well separated in rapidity from the scattered proton.
The most important observation made at HERA is that diffractive processes in DIS are not rare,
quite the contrary, they constitute up to 15% of deep inelastic events. What’s more, the ratio of
the diffractive and inclusive cross sections is constant as a function of energy of the γ∗p system
or as a function of the photon virtuality. The latter fact reflects the logarithmic dependence on
Q2 of diffractive structure functions in the Bjorken limit.
In the t-channel picture, the diffractive interactions can be viewed as a vacuum quantum
number exchange between the diffractive system and the proton. In old days of Regge phe-
nomenology such a mechanism of interactions was termed a pomeron. With the advent of
quantum chromodynamic we gain a new way of understanding the pomeron by modelling it
with the help of gluon exchanges projected onto the color singlet state. In the lowest approxi-
mation, the pomeron is a two gluon exchange which is independent of energy. By considering
radiative corrections to this process in the high energy limit, the famous BFKL pomeron [1–4]
was constructed with a strong, power-like dependence on energy. This dependence ultimately
violates unitarity which means that exchanges with more gluons have to be considered. A sys-
tematic program to sum exchanges with gluon number changing vertices was formulated in [5,6]
and developed in [7–10]. Other, somewhat more intuitive formulation, called Color Glass Con-
densate [11–14], is based on the idea of parton saturation [15] in which deep inelastic scattering
occurs on a dense gluonic system in the proton. In these approaches unitarization is supposed
to change the asymptotic energy behaviour of the cross sections involving the pomeron from
power-like to logarithmic.
DDIS is particularly sensitive to the pomeron energy behaviour since diffractive scattering
amplitudes are squared in diffractive cross sections. Thus, unitarization effects play more im-
portant role than for the total cross section which is proportional to the imaginary part of the
scattering amplitude. This observation is a basis of a successful description of the first diffractive
data from HERA in which the diffractive system was formed by the quark-antiquark (qq) and
quark-antiquark-gluon (qqg) systems. They can be viewed in the space of Fourier transformed
transverse momenta as color dipoles [16,17]. In the approach we follow in the forthcoming pre-
sentation, the pomeron interaction is modelled by a two-gluon exchange which is subsequently
substituted by the effective dipole–proton cross section fitted to inclusive DIS data [18, 19]. In
this way unitary is achieved.
In an alternative approach to DDIS, the diffractive structure functions are defined in terms of
diffractive parton distributions (DPD). They are evolved in Q2 with the help of the Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [20–22]. Thus in the Bjorken limit, the
diffractive structure functions depend logarithmically on Q2, i.e. they provide the twist–2 de-
scription of DDIS. The theoretical justification of this approach is provided by the collinear fac-
torisation theorem which is valid for hard diffractive scattering in ep collisions [23–27]. However,
collinear factorisation fails in hadron–hadron scattering due to nonfactorizable soft interactions
between incident hadrons [28,29]. Thus, unlike inclusive parton distributions, the DPD are not
Figure 1: Kinematic variables relevant for diffractive DIS.
universal objects and in general can only be used for diffractive processes in the ep deep inelas-
tic scattering. Nevertheless, the scale of nonuniversality can be estimated by applying them to
hadronic reactions.
The relation between the color dipole approach with the qq and qqg diffractive components
and the DGLAP based description was studied in detail in [30]. In short, after extracting the
twist–2 part, the dipole approach provides Q2-independent quark and gluon DPD. In addition,
the qqg component, which was computed assuming strong ordering between transverse momenta
of the gluon and the qq pair, gives the first step in the Q2-evolution of the gluon distribution.
The twist–2 approach, which is based on the DGLAP evolution equations, extends the two
component dipole picture by taking into account more complicated diffractive final state. In
the performed up till now twist-2 analyses, the diffractive parton distributions are determined
through fits to diffractive data from HERA [31] We will follow this approach with an important
modification.
The dipole approach teaches us one important lesson concerning the seemingly subleading
twist–4 contribution, given by the qq pair from longitudinally polarised virtual photons (Lqq).
Formally, it is is suppressed by a power of 1/Q2 with respect to the leading twist–2 transverse
contribution. However, the perturbative QCD calculation shows that for small diffractive masses,
M2 ≪ Q2, the longitudinal contribution dominates over the twist–2 one which tends to zero
in this limit. The effect of the Lqq component is particularly important for the longitudinal
diffractive structure function FDL which is supposed to be determined from the high luminosity
run data at HERA. Thus, we claim that it is absolutely necessary to consider the twist–4
contribution in the determination of the diffractive parton distributions. The analysis which we
present confirms its relevance for the prediction for FDL , which differs significantly from that
based on the pure DGLAP analysis. This is the main result of our paper.
The paper is organised as follows. In Section 2 we provide basic formulae for the kinematical
variables and quantities measured in diffractive deep inelastic scattering. We also describe the
three contributions which we include in the description of the diffractive structure functions, i.e.
the twist–2, twist–4 and Regge contributions. In Section 3 we describe performed fits while in
Section 4 we present their impact on the determination of the diffractive parton distributions
and diffractive structure functions. We finish with conclusions and outlook.
2 Basic formulae
We consider diffractive deep inelastic scattering: ep → e′p′X, shown schematically in Fig. 1.
After averaging over the azimuthal angle of the scattered proton, the four-fold differential cross
section is given in terms of the diffractive structure functions FD2 and F
dβ dQ2 dxIP dt
2πα2em
1 + (1− y)2
FD2 −
1 + (1− y)2
where y = Q2/(xBs) and s is the ep centre-of-mass energy squared. The expression in the curly
bracket is called reduced cross section:
σDr = F
1 + (1− y)2
FDL . (2)
Both structure functions depend on four kinematic variables (β,Q2, xIP , t), defined as follows
xIP =
Q2 +M2 − t
Q2 +W 2
, β =
Q2 +M2 − t
, (3)
where −Q2 is virtuality of the photon, t = (p − p′)2 < 0 is the square of four-momentum
transferred into the diffractive system, M is invariant diffractive mass and W is invariant energy
of the γ∗p system. The Bjorken variable xB = xIP β. For most of the diffractive events |t| is
much smaller then other scales, thus it can be neglected in eqs. (3). The diffractive structure
functions are measured in a limited range of t, thus the integrated structure functions are defined
2,L (β,Q
2, xIP ) =
∫ tmax
dt FD2,L(β,Q
2, xIP , t) , (4)
The integrated reduced cross section σ
r is defined in a similar way.
2.1 Twist–2 contribution
In the QCD approach based on collinear factorisation, the diffractive structure functions are
decomposed into the leading and higher twist contributions
FD2,L = F
D(tw2)
2,L + F
D(tw4)
2,L + . . . . (5)
The twist–2 part is given in terms of the diffractive parton distributions through the standard
collinear factorisation formulae [23,32–34]. In the next-to-leading logarithmic approximation we
D(tw2)
2 (x,Q
2, xIP , t) = SD +
CS2 ⊗ SD +C
2 ⊗GD
D(tw2)
L (x,Q
2, xIP , t) =
CSL ⊗ SD + C
L ⊗GD
where αs is the strong coupling constant and C
2,L are coefficients functions known from inclusive
DIS [35,36]. The integral convolution is performed for the longitudinal momentum fraction
(C ⊗ F )(β) =
dz C (β/z)F (z) . (8)
Notice that in the leading order, when terms proportional to αs are neglected, the longitudinal
structure function F
D(tw2)
L = 0. The functions SD and GD are given by diffractive quark and
gluon distributions, q
D and gD:
e2f β
D(β,Q
2, xIP , t) + q
D(β,Q
2, xIP , t)
GD = βgD(β,Q
2, xIP , t) (10)
Note that β = x/xIP plays the role of the Bjorken variable in DDIS. In the infinite momentum
frame, the DPD are interpreted as conditional probabilities to find a parton with the momentum
fraction x = βxIP in a proton under the condition that the incoming proton stays intact losing
a small fraction xIP of its momentum. A formal definition of the diffractive parton distributions
based on the quark and gluon twist-2 operators is given in [23,25].
The DPD are evolved in Q2 by the DGLAP evolution equations [37] for which the variables
(xIP , t) play the role of external parameters. In this analysis we assume Regge factorisation for
these variables:
D(β,Q
2, xIP , t) = fIP (xIP , t) q
IP (β,Q
2) (11)
gD(β,Q
2, xIP , t) = fIP (xIP , t) gIP (β,Q
2) . (12)
For convenience, the functions q
IP (β,Q
2) and gIP (β,Q
2) are called pomeron parton distributions.
The motivation for such a factorisation is a model of diffractive interactions with a pomeron
exchange [38]. In this model fIP is the pomeron flux
fIP (xIP , t) =
F 2IP (t)
1−2αIP (t)
IP , (13)
where αIP (t) = αIP (0) + α
IP t is the pomeron Regge trajectory and the formfactor
F 2IP (t) = F
IP (0) e
−BD |t| (14)
describes the pomeron coupling to the proton. We set F 2IP (0) = 54.4 GeV
−2 [34], BD =
5.5 GeV−2 and α′IP = 0.06 GeV
−2 [31], while the pomeron intercept αIP (0) is fitted to data.
The pomeron quark distributions are flavour independent, thus they are given by one function,
a singlet quark distribution ΣIP :
IP (β,Q
2) = q
IP (β,Q
ΣIP (β,Q
2) (15)
where Nf is a number of active flavours. The question about Regge factorisation is an issue
which should be tested experimentally. In our approach, the pomeron is a model of diffractive
interactions which only provides energy dependence through the xIP -dependent pomeron flux.
Its normalisation is only a useful convention since the normalisations of the pomeron parton
distributions in eqs. (11) are fitted to data.
2.2 Twist-2 charm contribution
We describe the charm quark diffractive production using twist-2 formulae for the cc pair gener-
ation from a gluon. These are formulae analogous to the inclusive case in which the diffractive
Figure 2: The qq̄ and qq̄g components of the diffractive system in the dipole approach.
gluon distribution gD is substituted for the inclusive one [39]:
D(cc)
2,L (β,Q
2, xIP , t) = 2β e
gD(z, µ2c , xIP , t) , (16)
where a = 1 + 4m2c/Q
2 and the factorisation scale µ2c = 4m
c with the charm quark mass
mc = 1.4 GeV. The coefficient functions read
C2(z, r) =
z2 + (1− z)2 + 4z(1− 3z)r − 8z2r2
1 + α
α {−1 + 8z(1 − z)− 4z(1 − z)r} (17)
CL(z, r) = −4z
2r ln
1 + α
+ 2αz(1 − z) (18)
with α =
1− 4rz/(1 − z). The cc pair can only be produced if invariant mass of the diffractive
system M2 fulfils the following condition
M2 = Q2
> 4m2c . (19)
2.3 Twist–4 contribution
The computation of the twist–4 contribution, proportional to 1/Q2, is a nontrivial task and
one could be tempted to assume that this contribution is suppressed at large Q2 as in inclusive
DIS. However, by analysing diffractive final states in the dipole approach it was found that
for diffractive mass M2 ≪ Q2 (β → 1), the twist–4 contribution dominates over the vanishing
twist–2 one [19,40,41].
This observation is made on the basis of the perturbative QCD calculations in which the
diffractive state is formed by the qq and qqg systems interacting with a proton through a colorless
gluonic exchange which is a model of the pomeron interactions in QCD. In the simplest case, two
gluons projected onto the color singlet state are exchanged, see Fig. 2. The computed amplitudes
do not depend on energy in such a case which problem can be cured in a more sophisticated
approach by modelling the dipole-proton cross section which fulfils unitarity conditions [19].
Independ of the details of the pomeron description, the diffractive mass (or β) dependence
is a genuine prediction of pQCD calculations. It appears that the leading in Q2 behaviour
components, qq and qqg from transverse virtual photons, vanish for β → 1. This is not the case
for the qq production from longitudinal photons (Lqq) which is formally suppressed by 1/Q2
with respect to the leading components. Thus, this particular β-dependence makes the Lqq
contribution dominant for β → 1, see Fig. 3.
The presence of the Lqq component has important consequence for the longitudinal diffractive
structure function which is supposed to be determined from the HERA data. The formula given
below is an important element in the description of FDL in the region of large β:
FDLqq̄ =
16π4xIP
e−BD |t|
(1− β)4
2(1−β)
k2/Q2
φ20(k, xIP ) (20)
where the function φ0(k, xIP ) is given in terms of the dipole cross section σ̂(xIP , r) and the Bessel
functions K0 and J0:
φ0(k, xIP ) = k
dr r K0
J0(kr) σ̂(xIP , r) . (21)
Strictly speaking, eq. (20) contains all inverse powers of Q2 but the part proportional to 1/Q2
(called twist–4) dominates. The dipole-proton cross section describes the interaction of a color
dipole, formed by the qq or qqg systems, with a proton. Following [18] we choose
σ̂(xIP , r) = σ0 {1− exp (−r
2Q2s/4)} (22)
where Q2s = (xIP /x0)
−λ GeV2 is a saturation scale which provides the energy dependence of
the twist–4 contribution. The parameters σ0 = 29 mb, x0 = 4 · 10
−5 and λ = 0.28 are taken
from [18] (Fit 2 with charm). This form of the dipole cross section provides successful description
of the first HERA data on both inclusive and diffractive structure functions [18,19]. A different
parametrisation of σ̂, without the saturation scale, is also given in [42–44]. We checked that a
very similar description of FDLqq̄ was found in a recent analysis [45] based on the Color Glass
Condensate parameterisation of the dipole scattering amplitude [46].
The relation between the dipole approach with three diffractive components and the DGLAP
approach with diffractive parton distributions was analysed at length in [30]. Summarising this
relation, the twist–2 part of the qq component gives a diffractive quark distribution. The twist-2
part of the qqg component forms a first step of the DGLAP evolution which starts from a given
gluon distribution. Both diffractive parton distributions do not depend on Q2, thus they may
serve as initial conditions for the DGLAP equations at the scale which is not determined. From
this perspective, the DGLAP approach offers a description of more complicated diffractive state
with any number of partons ordered in transverse momenta. However, the pQCD calculations
tell us that the twist–2 analysis of diffractive data should include the twist–4 contribution since
it cannot be neglected at large β. This is the strategy which we follow in our analysis.
We also borrow from the dipole approach a general form in β of the initial quark distribution
which vanishes at the endpoints β = 0, 1 (see eq. (31) in which Aq and Cq are positive). A very
important aspect of Regge factorisation (11) can also be motivated by the dipole approach. It
is a consequence of geometric scaling of the dipole cross section (22) [30,47].
2.4 Reggeon contribution
The diffractive data from the H1 collaboration for higher values of xIP hints towards a contribu-
tion which decreases with energy. This effect can be described by reggeon exchanges in addition
to the rising with energy pomeron exchange. Following [48, 49], we consider the dominant
isoscalar (f2, ω) reggeon exchanges which lead to the following contribution to F
fR(xIP , t)FR(β,Q
2) . (23)
This contribution breaks Regge factorisation of the diffractive structure function, however, its
presence is necessary for xIP > 0.01 [50]. The reggeon flux fR is given by the formula analogous
to eq. (13)
fR(xIP , t) =
F 2R(0)
e−|t|/Λ
R |ηR(t)|
1−2αR(t)
IP , (24)
where αR(t) = 0.5475 + 1 · t is the reggeon trajectory. From the Regge phenomenology of
hadronic reactions ΛR = 0.65GeV and the reggeon–proton couplings are given by [49]: F
(0) =
194GeV−2 and F 2ω(0) = 52GeV
−2. The functions
|ηR(t)|
2 = 4cos2[παR(t)/2] , |ηR(t)|
2 = 4 sin2[παR(t)/2] (25)
are signature factors for even (f2) and odd (ω) reggeons, respectively. We could also consider
isovector reggeons (a2, ρ) but their couplings to the proton are much smaller and we neglect
them. Finally, the reggeon structure function FR is given by [49]
FR(β) = AR β
−0.08 (1− β)2 , (26)
where the normalisation AR is a fitted parameter. Thus, in the first approximation, we neglect
the Q2-dependence of the reggeon contribution.
3 Fit details
Collab. No. points Data |t|-range Q2-range β-range
H1 [50] 72 LP [0.08, 0.5] [2 , 50] [0.02 , 0.7]
ZEUS [51] 80 LP [0.075 , 0.35] [2 , 100] [0.007 , 0.48]
H1 [31] 461 MY < 1.6 [|tmin| , 1] [3.5 , 1600] [0.01 , 0.9]
ZEUS [52] 198 MY < 2.3 [|tmin| ,∞] [2.2 , 80] [0.003 , 0.975]
Table 1: Kinematic regions of diffractive data from HERA. LP means leading proton data and
MY is invariant mass of a dissociated proton. Dimensionfull quantities are in units of 1 GeV.
In our analysis we use diffractive data from the H1 [31,50] and ZEUS [51,52] collaborations.
In Table 1 we show their kinematic limits in which they have been measured. The minimal value
value of |t| is given by
|tmin| ≃
1− xIP
m2p , (27)
where mp is the proton mass. The leading proton data from H1, measured in the range given
in Table 1, were corrected by the H1 collaboration to the range |tmin| < |t| < 1 GeV
The ZEUS data are given for the diffractive structure function FD2 , thus we use in our
analysis the following formulae
FD2 = F
D(tw2)
2 + F
2 + F
Lqq̄ (28)
FDL = F
D(tw2)
L + F
Lqq̄ . (29)
No Data Fit αIP (0) AR Aq Bq Cq Ag Bg Cg χ
1 H1 (LP) tw-2 1.098 0.29 1.75 1.49 0.5∗ 2.09 0.67 0.80 0.48
2 ZEUS (LP) tw-2 1.145 1.05 2.13 1.51 0.5∗ 10.0* 1.03 2.26 0.40
3 H1 tw-2 1.117 0.49 1.33 1.63 0.34 0.17 -0.16 -1.10 1.04
4 tw-(2+4) 1.119 0.48 1.62 1.98 0.59 0.04 -0.56 -1.68 1.17
5 ZEUS tw-2 1.093 0.0∗ 1.68 1.01 0.5∗ 0.49 -0.03 -0.40 1.35
6 tw-(2+4) 1.092 0.0∗ 1.20 0.85 0.57 0.07 -0.52 -1.48 1.82
Table 2: The fit parameters to H1 nd ZEUS data. The presence of twist–4 in the fits is marked
by tw-(2+4). The parameters with an asterisk are fixed in the fits.
The longitudinal twist-4 contribution is present on the r.h.s. of eq. (28) since FD2 is the sum of
the contributions from the transverse and longitudinal polarised virtual photon. The H1 data,
however, are presented for the reduced cross section (2). Thus we substitute relations (28) and
(29) in there and use
σDr =
D(tw2)
2 + F
1 + (1− y)2
D(tw2)
2(1− y)
1 + (1− y)2
FDLqq . (30)
The expression in the curly brackets is the twist–2 contribution while the last term is the twist–4
one. Notice that the difference between FD2 and σ
r is most important for y → 1.
We fit the diffractive parton distributions at the initial scale Q20 = 1.5 GeV
2, assuming the
Regge factorised form (11) with the following pomeron parton distributions [31]:
βΣIP (β) = Aq β
Bq (1− β)Cq (31)
βgIP (β) = Ag β
Bg (1− β)Cg . (32)
The six indicated parameters are fitted to data. We additionally multiplied both distributions
by a factor exp{−a/(1 − β)} with a = 0.01 to secure their vanishing for β = 1. This factor
is only important when Cq or Cg becomes negative in the fits. For the evolution, we use the
next-to-leading order DGLAP equations with ΛQCD = 407 MeV for Nf = 3 flavours [53].
The pomeron flux in eq. (11) is integrated over t in the limits given in Table 1 which leads
to the form
fIP (xIP ) =
F 2IP (0)
e−B|tmin| − e−B|tmax|
1−2αIP (0)
IP . (33)
The shrinkage parameter B equals
B = BD + 2α
IP ln(1/xIP ) (34)
with BD = 5.5GeV
−2 and α′IP = 0.06GeV
−2 [50].
In summary, we have eight fit parameters altogether: the pomeron intercept αIP (0), reggeon
normalisation AR in eq. (26) and six parameters in eqs. (31,32)
4 Fit results
The data sets from Table 1 were obtained in different kinematical regions, using different methods
of their analysis. Thus, we decided to perform fits to each data set separately. The values of
the fit parameter are shown in Table 2. The difference between them can be attributed to the
scale of uncertainty of our analysis. In each case we preformed two fits: with and without the
twist–4 formula added to the twist–2 contribution.
4.1 Leading proton data
We started from fits to the leading proton data. The fit parameters in this case are displayed
in the first two rows of Table 2. We only show the twist–2 fit results since they are not changed
in fits with the twist–4 term. This happenes because the leading proton data comes from the
region of β values where the twist–4 contribution is small (β ≤ 0.7 for H1 and β < 0.5 for
ZEUS), see Fig. 3.
The data with a dissociated proton (DP) which are measured in the region of large β influence
most the value of the parameter Cg which controls the behaviour of the gluon distribution at
β → 1. For the LP data Cg is positive and the gluon distribution is suppressed near β ≈ 1, while
for the DP data Cg is negative and the gluon distribution is strongly enhanced. This shows that
the data with β > 0.7 are crucial for the proper analysis. Without this kinematic region we lose
important information about diffractive interactions. Thus, from now on we concentrate on the
analysis of the DP data.
4.2 H1 data
The fit parameters to the H1 data with a dissociate proton are given in the third and fourth rows
of Table 2. We see that the fit quality is practically the same for both fits, with and without the
twist–4 contribution. The presence of the reggeon term improves fit quality by 30 units of χ2
for 461 experimental points. A good quality of the fits is illustrated in Fig. 4 which also shows
that the reduced cross sections (30) from the twist–2 (solid lines) and twist–(2+4) fits (dashed
lines) are very close to each other.
In Fig. 5 we show our results on the reduced cross section for the largest measured value of
β = 0.9. In this region, the twist–4 contribution, shown by the dotted lines, cannot be neglected.
We see that the curves from both the twist–2 (solid) and twist–(2+4) (dashed) fits describe data
reasonable well. However, the curves with twist–4 have a steeper dependence on xIP (energy)
than in the pure twist–2 analysis. This observation is by far more pronounced in the analysis of
the ZEUS data performed for the structure function FD2 .
The diffractive parton distributions from our fits are shown in Fig. 6 in terms of the pomeron
parton distributions, βΣIP (β,Q
2) and βgIP (β,Q
2). Being independent of the pomeron flux, such
a presentation allows for a direct comparison of the results from fits to different data sets. We see
that the singlet quark distributions are quite similar while the gluon distributions are different.
In the fit with twist–4, the gluon distribution is stronger peaked near β ≈ 1. This somewhat
surprising result can be understood by looking at the logarithmic slope of FD2 for fixed values
of β. From the LO DGLAP equations we have schematically:
∂ lnQ2
∂ lnQ2
= Pqq ⊗ ΣIP + PqG ⊗GIP − ΣIP
Pqq (35)
where the negative term describes virtual corrections. For large β, the measured slope is negative
which means that the virtual emission term must dominate over the real emission ones. The
addition of the twist–4 contribution to FD2 , proportional to 1/Q
2, contributes a negative value
to the slope which has to be compensated by a larger gluon distribution in order to describe the
same data.
In Fig. 7 we present our most important results. On the left panel, the FD2 structure function
is shown from both fits, with and without the twist–4 contribution (shown by the dotted lines).
We see no significant difference between these two results. However, the longitudinal structure
function FDL differs significantly for the two fits (right panel) due to the twist–4 contribution.
Let us emphasise that both sets of curves were found in the fits which well describe the existing
data on σDr , including the large β region. Thus, an independent measurement of F
L in this
region would be an important test of the QCD mechanism of diffraction.
4.3 ZEUS data
The results of same fits performed for the ZEUS data are shown in the last two rows of Table 2.
This time the Regge contribution (26) is not necessary since fits give the reggeon normalisation
AR ≈ 0. In general, the fit quality is worse than for the H1 data.
As shown in Fig. 8, the biggest difference between the twist–2 and twist–(2+4) results occurs
at large β values. This is analysed in detail in Fig. 9. We see that the presence of the twist–4
term in the fit (dashed lines) improves the agreement with the data in this region. In particular,
a steep dependence of FD2 on xIP is better reproduced by the twist–(2+4) fit then by the twist–2
one (solid lines). This dependence is to large extend driven by the twist–4 contribution (dotted
lines).
The behaviour of the diffractive parton distributions and structure functions, shown in
Figs. 10 and 11, respectively, is very similar to that found for the H1 data. The gluon dis-
tribution from the fit with twist–4 is stronger peaked near β ≈ 1 and the longitudinal structure
functions in the large β region is dominated by the twist–4 contribution.
We summarise the effect of the twist–4 contribution in Fig. 12 showing the predictions for
the longitudinal diffractive structure function FDL . Ignoring this contribution, we find the two
solid curves coming from the pure twist–2 analysis of the H1 (upper) and ZEUS (lower) data.
With twist–4, the dashed curves are found, the upper one from the H1 data and the lower one
from the ZEUS data. There is a significant difference between these two predictions in the
region of large β. We believe that the effect of the twist–4 contribution will be confirmed by the
forthcoming analysis of the HERA data.
5 Conclusions
We performed fits of the diffractive parton distributions to new diffractive data from the H1 and
ZEUS collaborations at HERA. In addition to the standard twist–2 formulae, we also considered
the twist–4 contribution which dominates in the region of large β. This contribution comes from
the diffractive production of the qq pair by the longitudinally polarised virtual photons. The
effect of the twist–4 contribution on the diffractive parton distributions and structure functions
was carefully examined. The twist–4 contribution leads to the gluon distribution which is peaked
stronger at β ≈ 1 than in the case without twist–4.
The main result of our analysis is a new prediction for the longitudinal diffractive struc-
ture function FDL . The twist–4 term significantly enhances F
L in the region of large β. A
measurement of this function at HERA in the region of large β should confirm the presented
expectations which are based on the perturbative QCD calculations. The obtained diffractive
parton distributions can also be used in the analysis of diffractive processes at the LHC, in
particular, to the estimation of the background to the diffractive Higgs production, see [54] for
a recent discussion.
Acknowledgements
This research has been partly supported by MEiN research grant 1 P03B 028 28 (2005-08)
and by EU grant HEPTOOLS, MRTN-CT-2006-035505. The hospitality of the Galileo Galilei
Institute for Theoretical Physics in Florence and partial support of INFN during the completion
of this work is gratefully acknowledged.
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Diffractive structure function
Q2=8 GeV2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 3: Three contributions to FD2 from: qq and qqg from transverse (T) and longitudinal (L)
photons [19] for xIP = 0.003. The twist–4 contribution Lqq̄ is indicated by the yellow band. Old
ZEUS data points are shown.
H1 DATA
β=0.01
β=0.01 β=0.04β=0.04 β=0.1β=0.1 β=0.2β=0.2 β=0.4β=0.4 β=0.65β=0.65 β=0.9 Q
β=0.9 Q
6.56.5
8.58.5
200200
Figure 4: Reduced cross section σ
r for H1 data as a function of xIP . Solid lines: twist–2 fit,
dashed lines: twist–(2+4) fit.
H1 DATA (β=0.9)
Q2= 3.5 GeV2
dashed: tw-(2+4) fit
solid: tw-2 fit
Q2= 5 GeV2
dotted: tw-4 contribution
Q2= 6.5 GeV2
Q2= 8.5 GeV2
Figure 5: Reduced cross section σ
r for H1 data at β = 0.9 for four values of Q
2 against fit
curves.
DPD (H1)
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 6: Pomeron parton distributions: singlet βΣIP (β,Q
2) (left) and gluon βgIP (β,Q
2) (right)
from H1 data.
DSF (H1)
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 7: Diffractive structure functions F
2 (left) and F
L (right) from fits to H1 data for
xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
ZEUS DATA
β=0.652
β=0.652 β=0.231β=0.231 β=0.07β=0.07 β=0.022β=0.022 β=0.007β=0.007
β=0.003
β=0.003
β=0.735β=0.735 β=0.308β=0.308 β=0.1β=0.1 β=0.032β=0.032 β=0.010β=0.010
β=0.004
β=0.004
β=0.807β=0.807 β=0.4β=0.4 β=0.143β=0.143 β=0.047β=0.047 β=0.015β=0.015
β=0.007
β=0.007
β=0.848β=0.848 β=0.471β=0.471 β=0.182β=0.182 β=0.062β=0.062 β=0.020β=0.020
β=0.009
β=0.009
β=0.907β=0.907 β=0.61β=0.61 β=0.28β=0.28 β=0.104β=0.104 β=0.034β=0.034
β=0.015
β=0.015
β=0.949β=0.949 β=0.75β=0.75 β=0.43β=0.43 β=0.182β=0.182 β=0.063β=0.063
β=0.029
β=0.029
β=0.975β=0.975 β=0.86β=0.86 β=0.604β=0.604 β=0.313β=0.313 β=0.121β=0.121
Figure 8: Diffractive structure function F
2 as a function xIP for ZEUS data. Solid lines:
twist–2 fit, dashed lines: twist–(2+4) fit.
ZEUS DATA
dotted: tw-4 contribution
β = 0.86 Q2= 55 GeV2
solid: tw-2 fit
dashed: tw-(2+4) fit
β = 0.91 Q2= 14 GeV2
β = 0.95 Q2= 27 GeV2
β = 0.975 Q2= 55 GeV2
Figure 9: Diffractive structure function F
2 for ZEUS data at large values of β against fit
curves.
DPD (ZEUS)
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 10: Pomeron parton distributions βΣIP (β,Q
2) (left) and βgIP (β,Q
2) (right) from fits to
ZEUS data.
DSF (ZEUS)
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
solid: tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 11: Diffractive structure functions F
2 (left) and F
L (right) from fits to ZEUS data
for xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
Diffractive FL
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
0.005
0.015
0.025
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Predictions for F
L for xIP = 10
−3 and Q2 = 10 GeV2 from the twist–(2+4) fits to
the H1 (upper dashed line) and ZEUS (lower dashed line) data. The solid lines show predictions
from pure twist–2 fits to the H1 (upper) and ZEUS (lower) data.
Introduction
Basic formulae
Twist–2 contribution
Twist-2 charm contribution
Twist–4 contribution
Reggeon contribution
Fit details
Fit results
Leading proton data
H1 data
ZEUS data
Conclusions
|
0704.1609 | GRB 061121: Broadband spectral evolution through the prompt and
afterglow phases of a bright burst | Draft version November 21, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
GRB 061121: BROADBAND SPECTRAL EVOLUTION THROUGH THE PROMPT AND AFTERGLOW
PHASES OF A BRIGHT BURST.
K.L. Page1, R. Willingale1, J.P. Osborne1, B. Zhang2, O. Godet1, F.E. Marshall3, A. Melandri4, J.P. Norris5,6,
P.T. O’Brien1, V. Pal’shin7, E. Rol1, P. Romano8,9, R.L.C. Starling1, P. Schady10, S.A. Yost11, S.D. Barthelmy3,
A.P. Beardmore1, G. Cusumano12, D.N. Burrows13, M. De Pasquale10, M. Ehle14, P.A. Evans1, N. Gehrels3, M.R.
Goad1, S. Golenetskii7, C. Guidorzi8,9, C. Mundell4, M.J. Page10, G. Ricker15, T. Sakamoto3, B.E. Schaefer16,
M. Stamatikos3, E. Troja1,12, M.Ulanov7, F. Yuan11 & H. Ziaeepour9
Draft version November 21, 2018
ABSTRACT
Swift triggered on a precursor to the main burst of GRB 061121 (z = 1.314), allowing observations to be made
from the optical to gamma-ray bands. Many other telescopes, including Konus-Wind, XMM-Newton, ROTSE and
the Faulkes Telescope North, also observed the burst. The gamma-ray, X-ray and UV/optical emission all showed a
peak ∼ 75 s after the trigger, although the optical and X-ray afterglow components also appear early on – before,
or during, the main peak. Spectral evolution was seen throughout the burst, with the prompt emission showing a
clear positive correlation between brightness and hardness. The Spectral Energy Distribution (SED) of the prompt
emission, stretching from 1 eV up to 1 MeV, is very flat, with a peak in the flux density at ∼ 1 keV. The optical-
to-X-ray spectra at this time are better fitted by a broken, rather than single, power-law, similar to previous results
for X-ray flares. The SED shows spectral hardening as the afterglow evolves with time. This behaviour might be
a symptom of self-Comptonisation, although circumstellar densities similar to those found in the cores of molecular
clouds would be required. The afterglow also decays too slowly to be accounted for by the standard models. Although
the precursor and main emission show different spectral lags, both are consistent with the lag-luminosity correlation
for long bursts. GRB 061121 is the instantaneously brightest long burst yet detected by Swift. Using a combination of
Swift and Konus-Wind data, we estimate an isotropic energy of 2.8 × 1053 erg over 1 keV – 10 MeV in the GRB rest
frame. A probable jet break is detected at ∼ 2 × 105 s, leading to an estimate of ∼ 1051 erg for the beaming-corrected
gamma-ray energy.
Subject headings: gamma-rays: bursts — X-rays: individual (GRB 061121)
1. INTRODUCTION
Electronic address: [email protected]
1 Department of Physics and Astronomy, University of Leicester,
Leicester, LE1 7RH, UK
2 Department of Physics & Astronomy, University of Nevada,
Las Vegas, NV 89154-4002, USA
3 NASA/Goddard Space Flight Center, Greenbelt, MD 20771,
4 Astrophysics Research Institute, Liverpool John Moores Uni-
versity, Twelve Quays House, Egerton Wharf, Birkenhead, CH41
5 Denver Research Institute, University of Denver, Denver, CO
80208, USA
6 Visiting Scholar, Stanford University
7 Ioffe Physico-Technical Institute, Laboratory for Experimen-
tal Astrophysics, 26 Polytekhnicheskaya, Saint Petersburg 194021,
Russian Federation
8 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46,
I-23807, Merate (LC), Italy
9 Dipartimento di Fisica, Universitá di Milano-Bicocca, Piazza
delle Scienze 3, I-20126, Milano, Italy
10 Mullard Space Science Laboratory, University College Lon-
don, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK
11 University of Michigan, 2477 Randall Laboratory, 450 Church
St., Ann Arbor, MI 48104, USA
12 INAF-IASF, Sezione di Palermo, via Ugo La Malfa 153, 90146,
Palermo, Italy
13 Department of Astronomy and Astrophysics, Pennsylvania
State University, 525 Davey Lab, University Park, PA 16802, USA
14 XMM-Newton Science Operations Centre, European Space
Agency, Villafranca del Castillo, Apartado 50727, E-28080 Madrid,
Spain
15 Center for Space Research, Massachusetts Institute of Tech-
nology, 70 Vassar Street, Cambridge, MA 02139, USA
16 Department of Physics and Astronomy, Louisiana State Uni-
versity, Baton Rouge, LA 70803, USA
Gamma-Ray Bursts (GRBs) are intrinsically extremely
luminous objects, approaching values of 1054 erg s−1 if
the radiation is isotropic (e.g., Frail et al. 2001; Bloom
et al. 2003). This energy is emitted over all bands in the
electromagnetic spectrum; to understand GRBs as fully
as possible, panchromatic observations are required over
all time frames of the burst.
The Swift multi-wavelength observatory (Gehrels et
al. 2004) is designed to detect and follow-up GRBs.
With its rapid slewing ability, Swift is able to follow
bursts and their afterglows from less than a minute af-
ter the initial trigger, and can often still detect them
weeks, and sometimes months, later. On rare occa-
sions, such as when Swift triggers on a precursor to the
main burst, the prompt emission, as well as the after-
glow, can be observed at X-ray and UV/optical wave-
lengths. GRB 061121, the subject of this paper, is only
the third GRB Swift has detected in this manner (af-
ter GRB 050117 – Hill et al. 2006 and GRB 060124 –
Romano et al. 2006), out of the almost 200 bursts trig-
gered on in the first two years of the mission.17 Of
these, GRB 061121 is the second well-sampled event
(GRB 060124 was the first), and the first for which the
UV/Optical Telescope (UVOT) was in event mode.
In addition to the small number of precursor triggers,
around 10% of Swift bursts show detectable emission over
17 GRB 050820A would possibly have also been in this category,
but Swift entered the South Atlantic Anomaly (SAA) just as a
dramatic increase in count rate began (Cenko et al. 2006; Page et
al. 2005a; Cummings et al. 2005; Page et al. 2005b; Chester et al.
2005); Swift does not actively collect data during SAA passages.
http://arxiv.org/abs/0704.1609v1
mailto:[email protected]
2 K.L. Page et al.
the BAT bandpass by the time the narrow field instru-
ments (NFIs) are on target.
Besides the Swift observations of prompt emission,
there have been a small number of prompt optical mea-
surements of GRBs, thanks to the increasing number
of robotic telescopes around the world. A variety of
behaviours has been found, with some optical (and in-
frared) light-curves tracking the gamma-ray emission
(e.g., GRB 041219A – Vestrand et al. 2005; Blake
et al. 2005), while others appear uncorrelated (e.g.,
GRB 990123 – Akerlof et al. 1999, Panaitescu & Kumar
2007, though see also Tang & Zhang 2006; GRB 050904
– Boër et al. 2006; GRB 060111B – Klotz et al. 2006;
GRB 060124 – Romano et al. 2006). GRB 050820A (Ves-
trand et al. 2006) showed a mixture of both correlated
and uncorrelated optical flux.
Where correlations exist between different energy
bands, it is likely that there is a common origin for
the components. In the uncorrelated cases, the opti-
cal emission may be due to an external reverse shock
(e.g., Sari & Piran 1999; Mészáros & Rees 1999), while
the prompt gamma-rays are caused by internal shocks.
Cenko et al. (2006) suggest that the early optical data
for GRB 050820A are produced by the forward shock
passing through the band. In the case of GRB 990123,
Panaitescu & Kumar (2007) have suggested that the
gamma-rays arose from inverse Comptonisation, while
the optical emission was due to synchrotron processes;
they do not assume a specific mechanism for the energy
dissipation, allowing for the possibility of either internal
or reverse-external shocks.
It is unclear whether precursors are ubiquitous fea-
tures of GRBs, often remaining undetected because of
a low signal-to-noise ratio or being outside the energy
bandpass of the detector, or whether only some bursts
exhibit them. A detailed discussion of the precursor phe-
nomenon is beyond the scope of this paper and will be
addressed in a future publication.
In this paper, we report on the multi-wavelength ob-
servations of both the prompt and afterglow emission of
GRB 061121. §2 details the observations made by Swift,
Konus-Wind, XMM-Newton, ROTSE18 and the Faulkes
Telescope North (FTN), with multi-band comparisons
being made. In §3, we discuss the precursor, prompt and
afterglow emission, with a summary given in §4.
Throughout the paper, the main burst (∼ 60–200 s
after the trigger) will be referred to as the prompt emis-
sion, and the emission seen over −5 to +10 s as the
precursor, where the BAT trigger time T0 = 0 s. Er-
rors are given at 90% confidence (e.g., ∆χ2 = 2.7 for
one interesting parameter) unless otherwise stated, and
the convention Fν,t ∝ ν
−βt−α (with the photon spec-
tral index, Γ = β + 1 where dN/dE ∝ E−Γ) has been
followed. We have assumed a flat Universe, with Hubble
constant, H0 = 70 km s
−1 Mpc−1, cosmological constant,
ΩΛ = 0.73 and Ωmatter = 1−ΩΛ.
2. OBSERVATIONS AND ANALYSES
Two years and one day after launch, the Burst Alert
Telescope (BAT; Barthelmy et al. 2005) triggered on a
precursor to GRB 061121 at 15:22:29 UT on 21st Novem-
ber, 2006. Swift slewed immediately, resulting in the
18 Robotic Optical Transient Search Experiment
NFIs being on target and beginning to collect data 55 s
(X-ray Telescope: XRT; Burrows et al. 2005a) and 62 s
(UVOT; Roming et al. 2005) later. This enabled broad-
band observations of the main burst event, which peaked
∼ 75 s after the trigger, leading to spectacular multi-
wavelength coverage of the prompt emission. The most
accurate Swift position for this burst was that determined
by the UVOT: RA = 09h 48m 54.s55, decl = −13◦ 11′ 42.′′4
(J2000.0; 90% confidence radius of 0.′′6; Marshall et al.
2006); the refined XRT position is only 0.′′1 from these
coordinates (Page et al. 2006b).
GRB 061121 was declared a ‘burst of interest’ by the
Swift team (Gehrels et al. 2006a), to encourage an inten-
sive ground- and space-based follow-up programme. In
addition to the Swift observations, the prompt emission
of GRB 061121 was detected by RHESSI19 (Bellm et
al. 2006), Konus-Wind and Konus-A (Golenetskii et al.
2006). Later afterglow observations were obtained in the
X-ray (XMM-Newton – Schartel 2006) and radio (VLA20
– Chandra & Frail 2006) bands. ATCA21 and WSRT22
also observed in the radio band between ∼5.2 day and
∼6.2 day after the burst, but did not detect the after-
glow (van der Horst et al. 2006a,b), implying it had faded
since the VLA observation.
Likewise, extensive optical follow-up observations were
performed: ROTSE-IIIa (Yost et al. 2006), FTN (Me-
landri et al. 2006), Kanata 1.5-m telescope (Uemura et
al. 2006), the University of Miyazaki 30-cm telescope
(Sonoda et al. 2006), MDM23 (Halpern et al. 2006a,b;
Halpern & Armstrong 2006a,b), P6024 (Cenko 2006),
ART25 (Torii 2006), the CrAO26 2.6-m telescope (Efimov
et al. 2006a,b) and SMARTS/ANDICAM27 (at infrared
wavelengths, too; Cobb 2006) all detected the optical
afterglow. Spectroscopic observations were performed
at the Keck telescope about 12 minute after the trig-
ger, finding a redshift of z = 1.314 for the optical af-
terglow, based on absorption features (Perley & Bloom
2006; Bloom et al. 2006).
GRB 061121 has the highest instantaneous peak flux
of all the long bursts detected by Swift to date (e.g.,
Angelini et al. in prep).
2.1. Gamma-ray Data
2.1.1. BAT
Temporal Analysis— After the initial precursor, the BAT
count rate returned to close to the instrumental back-
ground level, until T0+60 s, at which point the much
brighter main burst began. This is characterised by a
series of overlapping peaks, each brighter than the previ-
ous one, after which the gamma-ray flux decayed (from
∼T0+75 s to ∼T0+140 s). Event data were collected un-
til almost 1 ks after the trigger, thus covering the entire
emission period.
19 Reuven Ramaty High Energy Solar Spectroscopic Imager
20 Very Large Array
21 Australia Telescope Compact Array
22 Westerbork Synthesis Radio Telescope
23 Michigan-Dartmouth-MIT Observatory
24 Palomar 60 inch
25 Automated Response Telescope
26 Crimean Astrophysical Observatory
27 Small and Moderate Aperture Research Telescope System/A
Novel Double-Imaging CAMera
GRB 061121: Broadband observations 3
T90, over 15-150 keV, and incorporating both the pre-
cursor and main emission, is 81 ± 5 s, measured from
8.8–89.8 s after the trigger28. Figure 1 shows the mask-
weighted BAT light-curve in the four standard energy
bands [15–25, 25–50, 50–100, 100-150 keV; 64 ms bin-
ning between 50-80 s after the trigger, with 1 s bins
at all other times; units of count s−1 (fully illuminated
detector)−1], with light-curves from other instruments:
the precursor and the pulses of the main burst are de-
tected over all gamma-ray bands, although the precursor
is only marginal over the 100-150 keV BAT band. There
is also a soft tail (detected below ∼ 50 keV, when suffi-
ciently coarse time bins are used) visible until about 140 s
after the trigger (see bottom panel of Figure 1), corre-
sponding to a similar feature in the X-ray light-curves.
Spectral analysis— For the precursor,
T90,pre = 7.7 ± 0.5 s (15–150 keV). A spectrum ex-
tracted over this interval can be well modelled by a single
power-law, with Γ = 1.68 ± 0.09 (χ2/dof = 26.2/23); no
significant improvement was found by using the Band
function (Band et al. 1993) or a cut-off power-law and a
thermal model led to a slightly (χ2 ∼ 8) worse fit. The
15–150 keV fluence for this time interval is 4 × 10−7
erg cm−2.
Considering only the main event,
T90,main = 18.2 ± 1.1 s (measured from 61.8–80.0 s
post-trigger). Fitting a power-law to the mean
spectrum during this time also results in a good fit
(Γ = 1.40 ± 0.01; fluence = 1.31 × 10−5 erg cm−2 over
15–150 keV; χ2/dof =51.6/56 ); again, neither the Band
function nor a cut-off power-law improves upon this.
There is significant spectral evolution during the T90
period, as shown in Figure 2: at times when the count
rate is higher, the spectrum is harder. This behaviour
was also common in earlier bursts, as well as previous
Swift detections (e.g. Golenetskii et al. 1983; Ford et al.
1995; Borgonovo & Ryde 2001; Goad et al. 2007). The
precursor shows a similar dependence of hardness ratio
on count rate, suggesting that the emission processes
in the precursor and the main burst are the same or
similar.
2.1.2. Konus-Wind
Temporal Analysis— Konus-Wind (Aptekar et al. 1995)
triggered on the main episode of GRB 061121, while
Konus-A triggered on the precursor (Golenetskii et
al. 2006). Because of the spatial separation of
Swift and Wind, the light travel-time between the
spacecraft is 1.562 s: the Konus-Wind trigger time,
T0,K−W = T0,BAT + 61.876 s. All Konus light-curves
have been plotted with respect to the BAT trigger,
corrected for the light travel-time. Figure 1 shows
the Konus-Wind data plotted over the standard energy
bands, with 64 ms binning; the bottom panel plots the
coarser time resolution (2.944 s) ‘waiting mode’ data,
showing that Konus-Wind did see slightly enhanced
emission at the time of the precursor. The background
levels (which have been subtracted in each case) were
1005, 370 and 193.4 count s−1 for bands 21–83, 83–360
and 360–1360 keV, respectively.
28 Errors on the BAT T90 are estimated to be typically 5–10%,
depending on the shape of the light-curve.
Spectral analysis— Table 1 gives the spectral fits to
the Konus-Wind data in three separate time intervals
shown by vertical lines in Figure 1 (Konus-Wind spec-
tral intervals are automatically selected onboard): up
to the end of the ‘bump’ around 70 s (the ‘start’
of the burst), the burst maximum and, finally, until
most of the emission has died away (the burst tail).
The data were fitted with a cut-off power-law, where
dN/dE ∼ E−Γ × e[−(2−Γ)E/Epeak], leading to the photon
indices and peak energies given in the table. The Band
function was used to estimate upper limits for the pho-
ton index above the peak; the values for the peak energy
and Γ obtained from the Band function were the same
as when fitting the cut-off power-law. Little variation in
the spectral slope for energies below the peak is seen over
these intervals, though the peak itself may have moved to
somewhat higher energies during the burst emission. Ex-
tracting BAT spectra over the same time intervals, and
fitting with the same model (fixing Epeak at the value
determined from the Konus-Wind data) results in con-
sistent spectral indices.
2.2. X-ray Data
2.2.1. XRT
Temporal Analysis— The XRT identified and centroided
on an uncatalogued X-ray source in a 2.5 s Image Mode
(IM) frame, as soon as the instrument was on target.
This was quickly followed by a pseudo Piled-up Photo
Diode (PuPD) mode frame. Following damage from a
micrometeoroid impact in May 2005 (Abbey et al. 2005),
the Photo Diode mode (Low Rate and Piled-up) has been
disabled [see Hill et al. (2004) for details on the different
XRT modes]; however, the XRT team are currently work-
ing on a method to re-implement these science modes and
to update the ground software to process the files. The
pseudo PuPD point presented here is the first use of such
data.
Data were then collected in Windowed Timing (WT)
mode starting at a count rate of ∼ 1280 count s−1 (pile-
up corrected – see below); the rate rapidly increased to
a maximum of ∼ 2500 count s−1 at T0 + 75 s, mak-
ing GRB 061121 the brightest burst yet detected by the
XRT. Following this peak, the count-rate decreased, with
a number of small flares superimposed on the underlying
decay (see Figure 1). Photon Counting (PC) mode was
automatically selected when the count rate was below
about 10 count s−1. Around 1.5 ks, the XRT switched
back into WT mode briefly, due to an enhanced back-
ground linked to the sunlit Earth and a relatively high
CCD temperature.
Because of the high count rate, the early WT data were
heavily piled-up; see Romano et al. (2006) for informa-
tion about pile-up in this mode. To account for this, an
extraction region was used which excluded the central
20 pixels (diameter; 1 pixel = 2.′′36) and extended out
to a total width of 60 pixels. Likewise, the first three
orbits of PC data were piled-up, and the data were thus
extracted using annular regions (inner exclusion diame-
ter decreasing from 12 to 6 to 4 pixels as the afterglow
faded; outer diameter 60 pixels). The count rate was
then corrected for the excluded photons by a comparison
of the Ancillary Response Files (ARFs) generated with
and without a correction for the Point Spread Function
4 K.L. Page et al.
White UVOT
0.3−2 keV XRT
2−10 keV
15−25 keV BAT
25−50 keV
50−100 keV
100−150 keV
21−83 keV Konus
83−360 keV
360−1360 keV
0 50 100 150
time since BAT trigger (s)
21−1360 keV
(waiting mode)
50 100 150
time since BAT trigger (s)
15−50 keV
Fig. 1.— Top panels: Swift UVOT, XRT, BAT and Konus-Wind light-curves of GRB 061121; 1σ error bars are shown for the UVOT
and XRT data. Each instrument detected the peak of the main burst, with the precursor being detected over all gamma-ray energies. The
vertical lines in the 360–1360 keV panel indicate the start and stop times for the spectra given in Table 1. Bottom panel: The 15-50 keV
BAT light-curve, with 10-s bins, showing a tail out to ∼140 s.
GRB 061121: Broadband observations 5
start time (s) stop time (s) Γ Epeak (keV) ΓBand χ
2/dof
61.876 70.324 1.40
+0.08
−0.09
<2.1 72/75
70.324 75.188 1.23
+0.05
−0.06
<2.9 88/75
75.188 83.380 1.30
+0.11
−0.13
<2.3 81/75
61.876 83.380 1.32
+0.04
−0.05
<2.7 95/75
TABLE 1
Konus-Wind cut-off power-law spectral fit results. Times are given with respect to the BAT trigger. ΓBand is the upper
limit obtained for the spectral index above Epeak when fitting with the Band function.
6 K.L. Page et al.
65 70 75 80
time since trigger (s)
0 1 2 3
BAT count rate (count s−1 detector−1)
Fig. 2.— Top panels: Light-curves, hardness ratios (HR) and the
variation in Γ using a single power-law fit during the main emission.
The BAT light-curve (top panel) is in units of count s−1 (fully il-
luminated detector)−1 , and the corresponding hardness ratio plots
(50–150 keV)/(15–50 keV) using 1-s binning. The XRT light-curve
shows counts over 0.3–10 keV, while the hardness ratio compares
(1–10 keV)/(0.3–1 keV) over 1-s bins. Bottom panel: BAT hard-
ness ratio versus count rate, showing that the emission is harder
when brighter. Data from the precursor are shown as grey circles,
with the main burst in black. The grey line shows a fit to the data,
of the form HR = 0.14 CR + 0.39.
(PSF); the ratio of these files provides an estimate of the
correction factor. Nousek et al. (2006) give more details
on this method. Occasionally, the afterglow was partially
positioned over the CCD columns disabled by microm-
eteoroid damage mentioned above. In these cases, the
data were corrected using an exposure map.
From T0 + 3 × 10
5 s onwards, the afterglow had faded
sufficiently for a nearby (41.′′5 away), constant (count
rate ∼ 0.003 count s−1) source to contaminate the GRB
region; this source is coincident with a faint object in
the Digitized Sky Survey and is marginally detected in
the UVOT V filter. Thus, beyond this time, the ex-
traction region was decreased to a diameter of 30 pix-
els, and the count rates corrected for the loss in PSF (a
factor of ∼ 1.08). The spectrum of this nearby source
can be modelled with a single power-law of Γ = 1.5+0.2
−0.1,
100 1000 104 105 1061
time since trigger (s)
Fig. 3.— Swift-XRT light-curve of GRB 061121. The star and
triangle show the initial Image Mode and pseudo PuPD point (see
text for details), followed by WT mode data (black) during the
main burst (and at the end of the first orbit) and PC mode data
(in grey).
10050 200 500
time since trigger (s)
Fig. 4.— Swift flux light-curve of GRB 061121, showing the early
X-ray data (star, triangle and crosses) and the BAT data (grey
histogram) extrapolated into the 0.3–10 keV band pass in units
of erg cm−2 s−1, together with the UVOT flux density light-curve
(light grey circles – V -band; dark grey circles – White filter) in units
of erg cm−2 s−1 Å−1, scaled to match the XRT flux observed at
the start of the ‘plateau’ phase.
with NH = (1.8
−1.2) × 10
21 cm−2, in comparison with
the Galactic value in this direction of 5.09 × 1020 cm−2
(Dickey & Lockman 1990).
Figure 3 shows the XRT light-curve, starting with the
IM point (see Hill et al. 2006 for details on how IM data
are converted to a count rate) and followed by the pseudo
PuPDmode data. The importance of these early pre-WT
data is clear, confirming that the XRT caught the rise of
the main burst.
After the bright burst, the afterglow began to follow
the ‘canonical’ decay, seen in many Swift bursts (Nousek
et al. 2006; Zhang et al. 2006a). Such a decay can
be parameterised by a series of power-law segments; in
this case, fitting the data beyond 200 s after the trig-
ger (= 125 s after the main peak), two breaks in the
light-curve were identified, with the decay starting off
very flat (α = 0.38 ± 0.08) and eventually steepening
GRB 061121: Broadband observations 7
α1 0.38 ± 0.08 Plateau phase
Tbreak,1 2258
α2 1.07
+0.04
−0.06
Shallow phase
Tbreak,2 (3.2
) × 104 s
α3 1.53
+0.09
−0.04
Steep phase
TABLE 2
XRT power-law light-curve fits from 200 s after the
trigger onwards; times are referenced to the BAT
trigger. The names used in the text for the different
epochs of the light-curve are listed in the last column.
to α = 1.07+0.04
−0.06 at ∼ 2.3 ks and then α =1.53
+0.09
−0.04 at
∼ 32 ks (Table 2). The addition of the second break
vastly improved the fit by ∆χ2 = 112.4 for two degrees
of freedom. However, we note that O’Brien et al. (2006)
and Willingale et al. (2007) advocate a different descrip-
tion of the temporal decline; we return to this in §3.
Fitting the decay of the main peak (75–200 s, keeping
T0 as the trigger time) with a power-law, the slope is
very steep, with α0 = 5.1 ± 0.2. However, both Zhang
et al. (2006a) and Liang et al. (2006) have shown that
the appropriate time origin is the start of the last pulse.
Thus, a model of the form f(t) ∝ (t−t0)
−α0 was used,
finding t0 = 58 ± 1 s and a slope of α0 = 2.2
−0.3; this
is a statistically significant improvement on the power-
law fit using the precursor T0 (∆χ
2 = 32 for one extra
parameter).
Figure 4 plots the Swift data in terms of flux (the BAT
data have been extrapolated into the 0.3–10 keV band,
using the joint fits with the XRT described in §2.4.1) and
flux density for UVOT. The BAT and XRT data are fully
consistent with each other at all overlapping times.
Spectral Analysis— The XRT data also show that strong
spectral evolution was present throughout the period of
the prompt emission; this is discussed in conjunction
with the BAT data in §2.4.1. Considering the X-ray data
alone, there is some indication that the spectra may be
better modelled with a broken, rather than single, power-
law, although the break energies cannot always be well
constrained (see Figure 5). For each spectrum [covering
periods of 2 s during the main pulse, followed by two
spectra of 5 s (80–85 s and 85–90 s) where the emission
is fainter], the low-energy slopes were tied together for
each spectrum (i.e., the slope measured is that averaged
over all of the spectra), as were the high-energy indices,
and the rest-frame column density, NH,z, was fixed at
(9.2 ± 1.2) × 1021 cm−2 from the best fit to the data
from later times (see below); only the break energy and
the normalisation were allowed to vary. When simul-
taneously fitting all 11 spectra, χ2/dof decreased from
142/134 to 127/132. Individually, the spectral fits were
typically improved by χ2 of between 2–5.
The X-ray data during the GRB 051117A flares (Goad
et al. 2007) were found to be better modelled with bro-
ken power-laws, with the break energy moving to harder
energies during each flare rise, and then softening again
as the flux decayed. Likewise, Guetta et al. (2006) found
breaks in the X-ray spectra obtained during the flares in
GRB 050713A. The same pattern may be occurring here,
and there is certainly an indication of spectral curvature.
The observed flux calculated from the spectrum corre-
sponding to the peak of the emission (74–76 s) was mea-
70 80 90
time since trigger (s)
70 80 90
time since trigger (s)
70 80 90
time since trigger (s)
Fig. 5.— Fitting the X-ray data over 0.3–10 keV with a broken
power-law (Γ1 =0.69
+0.13
−0.07
and Γ2 =1.61
+0.14
−0.13
for all spectra), the
break energy seems to move through the band, towards higher
energies when the emission is brighter. Arrows indicate upper or
lower 90% limits.
sured to be 1.66 × 10−7 erg cm−2 s−1 (over 0.3–10 keV);
the unabsorbed value was 1.77 × 10−7 erg cm−2 s−1.
The PC spectra were also extracted for the various
phases of the light-curve (‘plateau’, ‘shallow’ and ‘steep’
– defined in Table 2); the results of the fitting are pre-
sented in Table 3. In each phase, the spectrum could
be well modelled by a single power-law (no break re-
quired), with excess absorption in the rest-frame of the
GRB (modelled using ztbabs and the ‘Wilms’ abun-
dance in xspec; Wilms et al. 2000). Together with the
WT spectrum from ∼ 200–590 s after the trigger (in the
plateau stage), the first two PC spectra (plateau and
shallow) are fully consistent with a constant photon in-
dex of Γ = 2.07 ± 0.06 and NH,z = (9.2 ± 1.2) × 10
cm−2.
Following the second apparent break in the light-curve,
around 3.2 × 104 s, the spectrum hardened slightly, to a
photon index of Γ = 1.83 ± 0.11 (or 1.87 ± 0.08 using
NH,z = 9.2 × 10
21 cm−2).
2.2.2. XMM-Newton
XMM-Newton (Jansen et al. 2001) performed a Target
of Opportunity observation of GRB 061121 (Observation
ID 0311792101) less than 6.5 hr after the trigger (Schartel
2006) and collected data for ∼ 38 ks (MOS1, MOS2;
Turner et al. 2001) and ∼ 35 ks (PN; Strüder et al. 2001).
This observation is mainly during the ‘shallow’ phase,
though also covers a short timespan after the break at
around 32 ks.
Figure 6 plots the PN flux light-curve and hardness
ratio during the XMM-Newton observation, showing the
lack of spectral evolution during this time frame; a hard-
ness ratio calculated for the Swift data was in agreement
with this finding. The decay slope over this time (MOS1,
MOS2, PN and joint) is consistent with the Swift results
(α ∼ 1.3; note this crosses the time of the second break
in the decay).
The XMM-Newton EPIC29 spectra show clear evidence
for excess NH, in agreement with the Swift data. In addi-
tion, fitting with excess NH in the rest-frame of the GRB
29 European Photon Imaging Camera
8 K.L. Page et al.
Epoch time since Γ NH,z χ
2/ν corresponding
trigger (s) (1021 cm−2) α
Plateau 590–1560 2.14 ± 0.12 10.8
62.5/52 0.38 ± 0.08
Shallow 4900–22245 2.04 ± 0.10 8.9
67.5/70 1.07
+0.04
−0.06
Steep 34550–1152750 1.83 ± 0.11 8.0
48.0/55 1.53
+0.09
−0.04
Plateau 590–1560 2.09 ± 0.08 9.2 ± 1.2 (tied) 63.5/53 0.38 ± 0.08
Shallow 4900–22245 2.05 ± 0.06 9.2 (tied) 67.6/71 1.07
+0.04
−0.06
Steep 34550–1152750 1.87 ± 0.08 9.2 (tied) 48.7/56 1.53
+0.09
−0.04
TABLE 3
XRT PC spectral fits - rest-frame NH free and then tied between all three spectra. The temporal decay slopes, α,
corresponding to each stage are also given. The Galactic absorbing column of NH = 5.09 × 10
20 cm−2 was always
included in the model.
3×104 3.5×104 4×104 4.5×104 5×104 5.5×104 6×104
time since BAT trigger (s)
Fig. 6.— XMM-Newton EPIC-PN light-curve and hardness ratio
of GRB 061121. The horizontal line shows the hardness ratio is
consistent with a constant value of ∼ 1.46, indicating there is no
spectral evolution during this time.
gives a significantly better fit than at z = 0, as shown in
Figure 7. When fitting in the observer’s frame there is a
noticeable bump in the residuals around 0.6 keV; fitting
with NH at z = 1.314 removes this feature. The data
are of sufficiently high signal-to-noise that the redshift of
the absorber can be estimated from the spectrum. Limits
can be placed on the redshift and absorbing column, re-
spectively, of z > 1.2 and NH,z > 4.6 × 10
21 cm−2 at 99%
confidence, in agreement with the spectroscopic redshift
from Bloom et al. (2006) within the statistial uncertain-
ties. At their value of z = 1.314, the excess NH,z from
the EPIC-PN spectrum is (5.3 ± 0.2)× 1021 cm−2, lower
than the best fit to the Swift data from the simultaneous
‘shallow’ decay section, but more similar to the values ob-
tained from fitting the optical-to-X-ray Spectral Energy
Distributions (SEDs) in §2.4.2. In agreement with the
simultaneous XRT PC mode data, there is no evidence
for a break in the EPIC spectrum over this time period.
Spectra from neither the Reflection Grating Spectrom-
eter (den Herder et al. 2001) nor EPIC show obvious
absorption or emission lines.
2.2.3. Chandra
Chandra performed a 33 ks Target of Opportunity ob-
servation at ∼ 61 day after the trigger. No source was
detected at the position of the X-ray afterglow, with a
3σ upper limit of 2.5 × 10−15 erg cm−2 s−1.
2.3. Optical/UV Data
zNH = 5.3x10
21 cm−2
10.5 2 5
channel energy (keV)
NH = 1.3x10
21 cm−2
Fig. 7.— EPIC-PN spectrum of the late-time afterglow of
GRB 061121, with an excess absorbing column both in the rest-
frame of the GRB and the observer’s frame. The spectrum is much
better modelled with an excess column at z = 1.314.
2.3.1. UVOT
The UVOT detected an optical counterpart in the ini-
tial White filter30 observation, starting 62 s after the
trigger, and subsequently in all other filters (optical and
UV). The UVOT followed the typical sequence for GRB
observations, with the early data being collected in event
mode, which has a frame time of 8.3 ms during this ob-
30 The White filter covers a broad bandpass of λ ∼ 1600−6500 Å.
GRB 061121: Broadband observations 9
servation.31 Photometric measurements were obtained
from the UVOT data using a circular source extraction
region with a 5− 6′′ radius. uvotmaghist was used to
convert count rates to magnitudes and flux; no normali-
sation between the different filters was applied.
As in the gamma-ray and X-ray bands, the main burst
was detected, with an increase in count rate seen between
∼ 50–75 s after the trigger (Figures 1 and 4). However,
although an increase in count rate is seen for the UVOT
data, it is by a smaller factor than observed for the XRT.
After ∼ 110 s, the UVOT emission stops decaying and re-
brightens slightly, until 140 s after the trigger, at which
time it flattens off and then starts to fade again (Fig-
ure 4). The slower decay between ∼ 100–200 s may be
indicative of the contribution of an additional (afterglow)
component beginning to dominate.
A single UV/optical light curve was created from all
the UVOT filters in order to get the best measurement of
the optical temporal decay. This was done by fitting each
filter dataset individually (between 200 and 1 × 105 s)
and finding the normalisation, which was then modified
to correspond to that of the V -band light-curve. The
decay across all the filters beyond 200 s after the trigger
can be fitted with a single slope of αUVOT = 0.68 ± 0.02;
the individual U , B and V decay rates are consistent
with one another. No break in the light-curve is seen out
to ∼ 100 ks.
2.3.2. ROTSE
ROTSE-IIIa, at the Siding Spring Observatory in Aus-
tralia, first imaged GRB 061121 21.6 s after the trigger
time under poor (windy) seeing conditions. A variable
source was immediately identified, at a position coinci-
dent with that determined by the UVOT (Yost et al.
2006).
The ROTSE data (unfiltered, but calibrated to the R-
band) have been included in Figure 10 (discussed later).
It is noticeable that the peak around 75 s seen in the Swift
data is not readily apparent in the ROTSE measure-
ments. The bandpass of the UVOT White filter is more
sensitive to photons with wavelengths of λ < 4500Å32,
while the ROTSE bandpass is redder. This, together
with poor seeing conditions during the observation, may
explain why the ROTSE light-curve does not clearly show
the main emission.
2.3.3. Faulkes Telescope North
The FTN, at Haleakala on Maui, Hawaii, began ob-
servations of GRB 061121 225 s after the burst trigger,
performing a BV Ri′ multi-colour sequence (Melandri
et al. 2006). R-band photometry was performed rela-
tive to the USNO-B 1.0 ‘R2’ magnitudes. Magnitudes
were then corrected for Galactic extinction using the
dust-extinction maps by Schlegel et al. (1998), and con-
verted to fluxes using the absolute flux calibration from
Fukugita et al. (1995). The photometric R-band points
have been included in Figure 10.
31 The data have been adjusted to take into account an incor-
rect onboard setting (between 2006-11-10 and 2006-11-22), which
resulted in the wrong frame times being stored in the headers of
the UVOT files (Marshall 2006a).
32 See http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/
2.4. Broadband Modelling
2.4.1. Gamma-rays – X-rays
Spectral Analysis— Because the BAT was in event
mode throughout the observation of the main burst of
GRB 061121, detailed spectroscopy could be performed.
Unfortunately this was not the case during the prompt
observation of GRB 060124 (Romano et al. 2006).
Figure 2 demonstrates the spectral evolution seen in
both the BAT and XRT during the prompt emission.
Spectra were extracted over 2 s intervals, in an attempt
to obtain sufficient signal to noise while not binning over
too much of the rapid variability. The BAT data are
hardest around 68 s and 75 s (the second of these times
corresponding to the peak of the main emission); the
XRT hardness peaks about 70 s, which could be a further
indication of the softer data lagging the harder. The joint
spectrum (Γjoint comes from a simple absorbed power-
law fit to the simultaneous BAT and XRT data) is at its
hardest during the brightest part of the emission. The
joint fit also hardens around 68–70 s, between the times
when the BAT and XRT data respectively are at their
hardest. The onboard spectral time-bin selection pre-
vents the Konus-Wind data from being sliced into corre-
sponding times, so constraints have not been placed on
the high energy cut-off, Epeak. Breaks in the XRT-BAT
power-laws can only be poorly constrained.
In Figure 4, the BAT and XRT data were converted to
0.3–10 keV fluxes using the time-sliced power-law fits to
the simultaneous BAT and XRT spectra. Without the
use of such varying conversion factors, the derived BAT
and XRT fluxes would have been inconsistent with each
other.
A broadband spectrum, covering 0.3 keV to 4 MeV in
the observer’s frame (XRT, BAT and Konus-Wind) for
∼ 70–75 s post trigger was fitted by the absorbed cut-off
power-law model described in §2.1.2. A constant factor
of up to 10% was included between the BAT and Konus-
Wind data, to allow for calibration uncertainties. The
best fit (χ2/dof = 301/167) gives Γ = 1.19 ± 0.01, with
Epeak = 670
−47 keV. NH,z was fixed at 9.2 × 10
21 cm−2
(from the X-ray fits in §2.2.1). Allowing Γ to vary be-
tween the three spectra hints at further spectral curva-
ture, although the differences are marginal, significant at
only the 2σ level.
The isotropic equivalent energy (calculated using
the time-integrated flux over the full T90 period) is
2.8 × 1053 erg in the 1 keV – 10 MeV band (GRB rest
frame), meaning that GRB 061121 is consistent with the
Amati relationship (Amati et al. 2002). See §3.3.2 for a
beaming-corrected gamma-ray energy limit.
Lag Analysis— A lag analysis (e.g., Norris et al. 1996)
between the BAT bands leads to interesting results.
Comparing bands 50–100 keV and 15–25 keV, the precur-
sor emission yields a spectral lag of 600 ± 100 ms, while
the main emission has a much smaller lag of 1 ± 6 ms.
Note that the calculation was performed using 64 ms
binning for the precursor and 4 ms binning for the main
burst; see Norris (2002) and Norris & Bonnell (2006) for
more details on the procedure. This lag for the main
emission is rather small for a typical long burst, however
both lags are consistent with the long-burst luminosity-
lag relationship generally seen (Norris et al. 2000). The
http://swiftsc.gsfc.nasa.gov/docs/heasarc/caldb/swift/
10 K.L. Page et al.
0 2 4 6
time delay (s)
1−4 keV
4−10 keV
15−25 keV
25−50 keV
50−100 keV
100−150 keV
Fig. 8.— Autocorrelation function of the BAT and XRT data
during the prompt emission of GRB 061121, showing that the main
burst peak is broader at softer energies.
short spectral lag for the main emission, and the longer
value for the precursor are also found when comparing
the 100–350 keV and 25–50 keV bands.
Similarly, comparison of the hard and soft (2–10 keV
and 0.3–2 keV) XRT bands reveals a lag of approximately
2.5 s, as the emission softens through the main burst.
The X-ray data also lag behind the gamma-ray data,
and the optical behind the X-ray.
Link et al. (1993) and Fenimore et al. (1995) used a
sample of BATSE33 (Paciesas et al. 1999) bursts to in-
vestigate the relationship between the duration of bursts
and the energy band considered. They found that the
bursts, and smaller structures within the main emission,
generally become shorter with increasing energy (see also
Cheng et al. 1995; Norris et al. 1996; in’t Zand & Fen-
imore 1996; Piro et al. 1998). Figure 8 plots the auto-
correlation function over various X-ray and gamma-ray
bands, to reinforce the point that the peak is narrower
the harder the band – over X-ray as well as gamma-ray
energies. Comparison of the light-curves over the differ-
ent energy bands in Figure 1 demonstrates this as well. A
similar behaviour was also found for GRB 060124, where
Romano et al. (2006) compared the T90 values obtained
for the main burst over the X-ray and gamma-ray bands.
Fenimore et al. (1995) found that the width of the auto-
correlation function, W ∝ E−0.4, where E is the energy
at which the function was determined; the six measure-
ments from GRB 061121 are consistent with this finding.
2.4.2. Optical – X-rays
Using the Swift X-ray and UV/optical data, R and
i′ band data from the Faulkes Telescope and Rc data
from the Kanata telescope (Uemera et al. 2006), SEDs
were produced at epochs corresponding to the peak of
the emission (72–75 s post BAT trigger), the plateau
stage and during the shallow decay. Fitting at the differ-
ent epochs gives an estimation of the broadband spectral
variation.
For each of the UVOT lenticular filters, the tool
uvot2pha was used to produce spectral files compatible
with xspec, and for the latter two epochs the count rate
33 Burst And Transient Source Experiment
in each band was set to that determined from a power-law
fit to the individual filter light curves over the time inter-
val in question, using α = 0.68. To determine the Faulkes
Telescope R and i′ band flux during the plateau stage, a
power law was fitted to the complete data set (220–1229 s
post BAT trigger for R and 467–1401 s for i′) with the
decay index left as a free parameter. The R magnitude
at the mid-time of the shallow stage (6058 s) was deter-
mined from the Kanata R-band magnitude reported at
6797 s (Uemera et al. 2006), assuming the same decay
index as observed in the UVOT data. An uncertainty of
0.2 mag was assumed as the systematic uncertainty for
the photometric calibration of the ground based data.
At a redshift of z = 1.314, the beginning of the Lyman-
α forest is redshifted to an observer-frame wavelength of
∼ 2812 Å which falls within the UVW1 filter bandpass,
the reddest of the UV filters. A correction was applied to
the three UV filter fluxes to account for this absorption,
based on parameters from Madau (1995) and Madau et
al. (1996); see also Curran et al. (in prep).
The methods used for simultaneous fitting of the SED
components are described in detail in Schady et al.
(2007a). The SEDs were fitted with a power-law, or
a broken power-law, as expected from the synchrotron
emission, and two dust and gas components, to model
the Galactic and host galaxy photoelectric absorption
and dust extinction. The column density and reddening
in the first absorption system were fixed at the Galactic
values. [The Galactic extinction along this line of sight
is E(B − V ) = 0.046 (Schlegel et al. 1998).]
The second photoelectric absorption system was set to
the redshift of the GRB, and the neutral hydrogen col-
umn density in the host galaxy was determined assuming
Solar abundances. The dependence of dust extinction on
wavelength in the GRB host galaxy was modelled us-
ing three extinction laws, taken from observations of the
Milky Way (MW), the Large Magellanic Cloud (LMC)
and the Small Magellanic Cloud (SMC) and parame-
terised by Pei (1992) and Cardelli et al. (1989). The
greatest differences observed in these extinction laws are
the amount of far UV extinction (which is greatest in
the SMC and least in the MW) and the strength of the
2175 Å absorption feature (which is most prominent in
the MW and negligible in the SMC).
Fitting these data together, a measurement of the spec-
tral slope and optical and X-ray intrinsic extinctions (for
the second two epochs) were obtained (Table 4); the AV
values given in the table are in addition to the AV = 0.151
associated with the Milky Way itself. The slope above
the break energy (which lies towards the low energy end
of the X-ray bandpass for each phase) was assumed to
be exactly 0.5 steeper than the spectral slope below the
break (the condition required for a cooling break), since
allowing all of the parameters to vary leads to uncon-
strained fits. Figure 9 shows, as an example, the fit to
the data in the plateau stage.
A Milky Way dust extinction law provides the best
overall fit to the data, using a broken power-law model,
although the LMC model is equally acceptable.
During the plateau phase, and adopting the bro-
ken power-law model parameters given in Table 4, we
find gas-to-dust ratios of (1.6 ± 0.7), (2.6 ± 0.7) and
(3.0 ± 0.7) ×1022 cm−2 mag−1 for MW, LMC and SMC
GRB 061121: Broadband observations 11
1015 1016 1017 1018
Frequency (Hz)
Fig. 9.— Broken power-law fit to the UVOT, XRT and ground-
based R and i′ spectral energy distribution of GRB 061121 between
∼ 596–1566 s after the trigger (plateau phase) plotted in the ob-
server’s frame. The arrows indicate the beginning of the Lyman-α
forest (1215Å in the rest-frame) and the absorption feature in the
MW dust extinction law (2175Å), which is shown by a dotted line.
The solid line corresponds to the LMC extinction, and the dashed
one to the SMC extinction.
fits respectively. We can compare these estimates to the
measured values for the MW of (4.93 ± 0.45) × 1021
cm−2 mag−1 (Diplas & Savage 1994) and the LMC and
SMC of (2.0 ± 0.8) and (4.4 ± 1.1) ×1022 cm−2 mag−1 ,
respectively (Koornneef 1982; Bouchet et al. 1985). The
MW fit to GRB 061121, which is found to be marginally
the best model, is consistent with the LMC gas-to-dust
ratio only, at the 90% confidence level. The ratios derived
from the LMC and SMC fits are consistent with both the
LMC and SMC gas-to-dust ratios. We note that all fits
are inconsistent with the MW ratio at this confidence
level, following the trend seen in pre-Swift bursts (e.g.,
Starling et al. 2007 and references therein), and that if
a metallicity below Solar were adopted, the gas-to-dust
ratio of GRB 061121 would increase, moving it further
towards the SMC value.
3. DISCUSSION
Swift triggered on a precursor to GRB 061121 leading
to comprehensive broadband observations of the prompt
emission, as well as the later afterglow. We discuss these
here, together with possible mechanisms involved.
3.1. Precursor
Lazzati (2005) found that about 20% of BATSE bursts
showed evidence for gamma-ray emission above the back-
ground between 10 to ∼200 s before the main burst, typi-
cally with non-thermal spectra which tended to be softer
than the main burst. GRB 060124 (Romano et al. 2006)
and GRB 061121 show the same behaviour.
Precursor models have been proposed for emission well-
separated from, or just prior to, the main burst. Early
emission occurring only a few seconds before the main
burst has been explained by the fireball interacting with
the massive progenitor star – though the spectrum of
such emission is expected to be thermal (Ramirez-Ruiz
et al. 2002a). Lazzati et al. (2007) investigated shocks in
a cocoon around the main burst; their model predicts a
non-thermal precursor as the jet breaks out of the sur-
face of the star. A high-pressure cocoon is formed as the
sub-relativistic jet head forces its way out of the star. As
the head of the jet breaks through the surface, the energy
of this cocoon is released through a nozzle and can give
rise to a precursor (Ramirez-Ruiz et al. 2002a,b). Within
the framework of this model, observers located at view-
ing angles of 5◦ < θ < 11◦ are expected to see first a
relatively bright precursor, then a dark phase with lit-
tle emission, followed, when the jet enters the unshocked
phase, by a bright GRB; this is very similar to the light-
curve observed for GRB 061121. Waxman & Mészáros
(2003) demonstrate that both a series of thermal X-ray
precursors (becoming progressively shorter and harder)
and nonthermal emission can be produced by an emerg-
ing shocked jet, although the nonthermal component is
expected to be in the MeV range. There could also be
an accompanying inverse Compton component, formed
by the thermal X-rays being upscattered by the jet.
The same type of smooth, wide-pulse, low intensity
emission as seen in some precursors, but occurring af-
ter the main emission is also occasionally seen (e.g.,
Hakkila & Giblin 2004; Nakamura 2000). Hakkila & Gib-
lin (2004) discuss two examples where postcursor emis-
sion is found to have a longer lag than expected from the
lag-luminosity relation, smoother shape and to be softer.
In the case of the GRB 061121 precursor, the spectrum is,
indeed, softer than the main event, and shows a compar-
atively smooth profile. The emission does have a longer
lag than the main emission, but it is still consistent with
the lag-luminosity relation.
There are two expected effects which could lead to such
a difference in lags for separate parts of a single burst: the
much lower luminosity for the precursor (resulting from
a much smaller Lorentz factor; the measured fluence of
the precursor is about a factor of 30 smaller than the
fluence of the main emission) is a natural explanation,
while the precursor being emitted at a greater off-axis
angle could also have an effect. In this second case, ejecta
are considered to emerge at different angles with respect
to the jet axis; not all of the solid angle of the jet will be
‘filled’ uniformly.
Such late postcursor emission is unlikely to be linked to
the jet breakout from the stellar surface, and it may not
be sensible to attribute apparently similar phenomena
(in the form of pre- and postcursors) to entirely different
processes.
Pre/postcursor emission could be due to the decelera-
tion of a faster front shell, resulting in slower shells catch-
ing up and colliding with it (Fenimore & Ramirez-Ruiz
1999; Umeda et al. 2005; note, however, that a faster
shell would be inconsistent with the precursor having a
smaller Lorentz factor as suggested to explain the lag
discrepancy), or late activity of the central engine. The
presence of flares in about 50% of Swift bursts is gener-
ally attributed to continuing activity of the central en-
gine (Burrows et al. 2005b; Zhang et al. 2006a) and the
appearance of broken power-laws in the X-ray spectra of
12 K.L. Page et al.
X-ray Model Extinction NH,z Γ1 Ebreak Γ
χ2/dof
Epoch (1021 cm−2) (keV)
Peak PL SMC 1.6
0.99 ± 0.01 · · · · · · 0.64 25/27
LMC 1.9
1.06 ± 0.01 · · · · · · 0.98 23/27
MW 2.4
1.16 ± 0.01 · · · · · · 1.51 22/27
BKN PL SMC 2.7+9.5
0.72+0.08
−0.15
0.17+0.79
−0.15
1.22 0.51 22/26
LMC 3.0+7.7
0.77+0.08
−0.20
0.18+0.53
−0.17
1.27 0.72 22/26
MW 3.0+7.7
0.77+0.10
−0.21
0.09+0.30
−0.09
1.27 1.03 22/26
Plateau PL SMC 1.42± 0.51 1.58± 0.02 · · · · · · 0.62± 0.05 167/59
LMC 1.98± 0.54 1.64± 0.03 · · · · · · 0.94± 0.08 152/59
MW 2.71± 0.69 1.71± 0.03 · · · · · · 1.39± 0.10 136/59
BKN PL SMC 3.89
+0.72
−1.01
+0.03
−0.02
+0.36
−0.12
1.96 0.52± 0.04 84/58
LMC 4.40
+0.77
−1.30
+0.04
−0.02
+0.16
−0.14
2.01 0.74± 0.06 80/58
MW 3.91
+0.77
−0.75
+0.04
−0.03
+0.25
−0.20
2.08 1.03
+0.09
−0.08
79/58
Shallow PL SMC 2.72± 0.49 1.69± 0.02 · · · · · · 0.65± 0.04 162/77
LMC 3.37
+0.53
−0.49
1.75± 0.03 · · · · · · 0.98
+0.07
−0.06
146/77
MW 4.60
+0.65
−0.60
1.87± 0.04 · · · · · · 1.63
+0.12
−0.11
127/77
BKN PL SMC 4.02+0.62
−0.67
1.58+0.02
−0.03
1.30+0.19
−0.11
2.08 0.50± 0.04 101/76
LMC 4.41+0.69
−0.63
1.62± 0.03 1.30+0.16
−0.14
2.12 0.72± 0.06 99/76
MW 4.78+0.75
−0.65
1.67± 0.04 1.35+0.16
−0.17
2.17 1.02+0.11
−0.10
102/76
a Γ2 is set to be equal to Γ1 + 0.5 in each broken power-law fit, as would be expected if the change in index were due to a cooling
break.
b In the fit to the peak epoch, AV is fixed to the average best-fit value found in the same model fits to plateau and shallow stage
data. The AV values are given for the observer’s frame of reference.
TABLE 4
Power-law (PL) and broken power-law (BKN PL) fits to the simultaneous UVOT and XRT spectra of GRB 061121, for
three different dust extinction models: Small and Large Magellanic Clouds (SMC and LMC) and the Milky Way (MW).
Γ1 and Γ2 are the photon indices below and above the spectral break for the BKN PL models. The data points have not
been corrected for reddening.
both flares and the prompt emission (Guetta et al. 2006;
Goad et al. 2007) hints of a common mechanism.
3.2. Prompt Emission
The prompt emission mechanism for GRBs is still de-
bated and the origin of Epeak is not fully understood
(Mészáros et al. 1994; Pilla & Loeb 1998; Lloyd & Pet-
rosian 2000; Zhang & Meszaros 2002; Rees & Mészáros
2005; Pe’er et al. 2005). The standard synchrotron
model predicts fast cooling (Ghisellini et al. 2000) with
a photon index, Γ, of 3/2 and (p/2)+1 below and above
the peak energy, respectively (e.g., Zhang & Mészáros
2004). The Konus-Wind spectral index below Epeak is
shallower than 3/2, which may suggest a slow cooling
spectrum with p < 2 [Epeak being the cooling frequency
and Γ =(p+1)/2] or additional heating. A slow-cooling
spectrum can be retained by assuming that the magnetic
fields behind the shock decay significantly in 104–105 cm,
so that synchrotron emission happens in small scale mag-
netic fields (Pe’er & Zhang 2006).
The SED at the peak time (SED 2 in Figure 11,
discussed below) has a peak flux density of around
1 keV, below which the optical to X-ray spectral slope
is 0.11 ± 0.09. This slope is harder than expected from
the standard synchrotron model (which predicts an in-
dex of 1/3). There should, however, be spectral cur-
vature around the break, which could flatten the index
(Lloyd & Petrosian 2000), so the data could still be con-
sistent with the synchrotron model. An alternative to
synchrotron emission, in the form of ‘jitter’ radiation is
discussed by Medvedev (2000), though that model pre-
dicts an even steeper index of 1 below the jitter break
frequency.
Figures 4 and 10 show that all three instruments on-
board Swift saw the prompt emission around 75 s after
the BAT trigger. However, it is noticeable that most of
the emission is in the gamma-ray and X-ray bands, with
the optical showing a relatively small increase in bright-
ness in comparison. Assuming the observed process is
synchrotron, then the prompt emission which is detected
by the UVOT will be the low-frequency extension of this
in the internal shock. No reverse shock is apparent.
3.3. Afterglow Emission
3.3.1. Broken Power-law Decline Models
The afterglow of GRB 061121 was observed over an
even broader energy range (from radio to X-rays) than
the prompt emission, with multi-colour data being ob-
tained from ∼ 100–105 s after the trigger. The X-ray
light-curve shows evidence for substantial curvature at
later times (see Figure 3), as has been found for other
Swift GRBs (e.g., GRBs 050315 – Vaughan et al. 2006;
060614 – Gehrels et al. 2006b). The standard practice
has been to fit such a decay using a series of power-
law segments as a function of time. An alternative
exponential-to-power-law description of the light-curve
is given in §3.3.2.
Nousek et al. (2006) and Zhang et al. (2006a) have
both discussed the canonical shape that many Swift af-
terglows seem to follow: steep to plateau to shallow,
with some light-curves showing a further steepening. In
these previous works, the extrapolation of the BAT data
into the XRT band was incorporated into the derivation
of the steep decay at the start of the canonical light-
curve shape. In the case of GRB 061121, the full curve
GRB 061121: Broadband observations 13
can be seen entirely in X-rays, suggesting that the pre-
vious extrapolations are reliable. For the afterglow of
GRB 061121, only data after the end of the main burst
have been modelled with power-laws. The early steep
decline, which might be attributable to the curvature ef-
fect (Kumar & Panaitescu 2000; Dermer 2004; Fan &
Wei 2005), is not considered here.
According to the model proposed in Nousek et al.
(2006) and Zhang et al. (2006a), the plateau phase of
the light-curve is due to energy injection in the fireball.
The plateau phase of GRB 061121 is consistent with an
injection of energy since the luminosity index, q, is nega-
tive, which is the requirement for injection to modify the
afterglow (Zhang et al. 2006a); the later two stages both
have q > 1. However, as will be discussed in §3.3.2, the
plateau and final transition to the power-law decay are
only visible in the X-ray data for GRB 061121; the start
of the final decay is much earlier in the V and R-bands
(see Figure 10). One might expect that energy injection
would affect all the energy bands simultaneously, rather
than just the X-rays.
From the standard afterglowmodel computations (e.g.,
Zhang & Mészáros 2004), we find that none of the closure
relations fit the entire dataset completely: although the
shallow phase (after the end of energy injection, between
T + 2.3 ks and T + 32 ks) could be consistent with the
evolution of a blast-wave which had already entered the
slow cooling regime when deceleration started [i.e., ν >
max(νm, νc) where νc is the cooling frequency and νm
is the synchrotron injection frequency; Sari et al. 1998;
Chevalier & Li 2000], the steeper part of the decay curve
(T> 32 ks) is not consistent with any of the models. This
lack of consistency suggests that a different approach is
required.
The change in decay slope between the shallow to
steep phases (∼ 32 ks) cannot be easily identified with
a jet-break. It certainly seems unlikely that the simplest
side-spreading jet model could be applicable, since the
post-break decay index (α ∼ 1.5) is not steep enough
(a post-jet decay has α = p, where p is the electron in-
dex). There is also some indication that the X-ray spec-
tral slope hardens after the break, whereas no change
in spectral signature is expected over a jet-break. In
the case of a non-laterally expanding jet (Panaitescu
& Mészáros 1999), α = (3β/2) + 0.25 [for a homoge-
neous circumstellar medium (CSM); Panaitescu et al.
(2006)], which does, indeed, fit the data after this break:
[1.5 × (0.9 ± 0.08)] + 0.25 = 1.6 ± 0.1; the measured
α is 1.53. Such a confined jet has been suggested as an
explanation for the observed decay in a number of pre-
vious bursts (e.g., GRB 990123 – Kulkarni et al. 1999;
GRB 050525A – Blustin et al. 2006; GRB 061007 –
Schady et al. 2007b). The UVOT data obtained around
this time show little evidence for a break, whereas jet
breaks should occur across all energy bands simultane-
ously. However non-simultaneity could be explained by
a multi-component outflow, where the X-ray emission is
produced within a narrow jet, while the optical compo-
nent comes from a wider jet with lower Lorentz factor
(Panaitescu & Kumar 2004; Oates et al. 2007). There
remains the issue, however, that α should steepen by
0.75 over a jet break (Mészáros & Rees 1999), whereas
the maximum observed change (within the 90% errors)
is only ∆α < 0.61, excluding ∆α = 0.75 at almost 3σ;
also, again there should be no spectral evolution across
the break. There is, however, a probable jet break at
later times, which will be covered in the next Section.
Other multi-component models [see, e.g., Oates et al.
(2007) and references therein] also fail to explain the
data, because of the lack of observed energy injection
(plateau phase) in the optical data.
Panaitescu et al. (2006a) discuss chromatic breaks in
Swift light-curves, and postulate that these could be due
to a change in microphysical parameters within a wind
environment. However, this model requires the cooling
frequency to lie between the X-ray and optical bands
and, as will be discussed in §3.3.2, this does not seem to
be the case here.
3.3.2. Exponential-to-power-law Decline Model
As first described by O’Brien et al. (2006), and further
expanded by Willingale et al. (2007), GRB light-curves
can be well modelled by one or two components com-
prised of an early exponential rise followed by a power-
law decay phase. Of these components, the first repre-
sents the prompt gamma-ray emission and early X-ray
decay. The second, when detected, dominates at later
times, forming what we see as the afterglow. These re-
sults show that fitting an intrinsically curved decay with
multiple power-law segments runs the risk of incorrectly
identifying temporal breaks (see also Sakamoto et al. in
prep). In this Section the models of O’Brien et al. (2006)
and Willingale et al. (2007) are applied to the multi-band
afterglow data of GRB 061121.
Figure 10 brings together the BAT, XRT, UVOT, FTN
and ROTSE data, along with further optical and ra-
dio points taken from the GCN Circulars (Halpern et
al. 2006a,b; Halpern & Armstrong 2006a,b; Chandra &
Frail 2006; van der Horst et al. 2006a,b) and the upper
limit from Chandra, to form a multi-energy decay plot.
The data have been plotted as ‘time since trigger + 4 s’
in order to include the precursor on a log time-scale. The
optical points have all been corrected for extinction using
AV = 1.2 (a combination of the Galactic value of 0.151
and an estimate of AV ∼ 1 for the GRB host galaxy –
see §2.4.2).
The contribution from the host galaxy reported by
Malesani et al. (2006) and Cobb (2006) has been sub-
tracted from the V - and R-band flux values. The magni-
tude of the host in the V -band is 22.4, which only changes
the last two or three V -band points by a small amount.
For the R-band we have no direct measurement, but the
last group of MDM exposures gave an R magnitude of
22.7, corresponding to a flux level of 2.8 µJy, and the flux
level is still declining at that epoch (∼ 3.3 × 105 s), so
an R-band flux level of 2.5 µJy was adopted for the host.
The error bars shown on the last few points reflect the
large uncertainty in the galaxy contribution subtracted.
The curved dotted lines in Figure 10 are the fits to
the data using the exponential-to-power-law model, fol-
lowed by a break to a steeper decay around 105 s. These
models are parameterised by the power-law decay, α,
and Ta, the time at which this decay is established.
For the X-ray data, Ta,X is found to be 5250
−460 s
and αa,X = 1.32 ± 0.03. Fits were also performed to
the V - and R-band data, yielding: αa,V = 0.66 ± 0.04
(with Ta,V = 70
−70 s) and αa,R = 0.84 ± 0.03
14 K.L. Page et al.
Fig. 10.— Flux density light-curves for the gamma-ray, X-ray, optical and radio data obtained for GRB 061121. The vertical dotted lines
indicate the times used for the SED plots shown in Figure 11, while the curved dotted lines show the fit to the X-ray and optical data,
including a late-time break, as described in the text.
Fig. 11.— SEDs for the four time intervals indicated in Figure 10.
SED 2 (the peak of the burst emission) includes the Konus-Wind
data, although these have not been included in Figure 10. The
solid lines represent the power-law fits to the BAT, XRT and Konus
data, while the dashed lines join the radio, optical and 1 keV points.
Spectral evolution over time is clearly seen.
(Ta,R = 230
−230 s).
The non-detection by Chandra almost two months af-
ter the burst shows there must have been a further steep-
ening in the X-ray regime, and the optical data are not
inconsistent with this finding. Constraining the tempo-
ral index after the late break to be α = 2 (a typical slope
for a post-jet-break decay), break times of ∼ 2.5 × 105,
∼ 2.5 × 104 and ∼ 105 s are estimated for the X-ray, V -
and R-band respectively; note that the UVOT V -band
value is particularly uncertain, given the small number
of data points at late times. Within the uncertainties,
these times are likely to be consistent, so the turnover
could be achromatic, as required for a jet break. From
Willingale et al. (2007), a jet break might be expected at
∼ 100 × Ta,X – i.e., 5.5 × 10
5 s, which is in agreement
with these fits.
As can be seen from these numbers and the mod-
els plotted in Figure 10, the X-ray data clearly show
the transition from the plateau to the power-law decay,
whereas the start of the final decay is much earlier in the
V - and R-bands. The V -band decay is also significantly
flatter (by α ∼ 0.2) than that estimated for the R-band.
As previously stated, the V , B and U light-curves are
all consistent with this slow decay. There have been few
multi-colour optical decay curves obtained for GRB af-
terglows, and, of these, the different filters [in the case of
GRB 061007 (Schady et al. 2007; Mundell et al. 2007)
X-ray and gamma-ray data as well as the optical] tend
to track each other (e.g., Guidorzi et al. 2005; Blustin
et al. 2006; de Ugarte Postigo et al. 2007). In the case
of GRB 061121, we find that the R-band data are fading
more rapidly than the V . GRB 060218, which was as-
sociated with a supernova (e.g., Campana et al. 2006a),
shows changes throughout the optical spectra, because of
a combination of shock break-out and radioactive heat-
ing of the supernova ejecta. There is a large difference
between the decays of the blue (V , U , B) and red (R)
data for GRB 061121, which cannot be easily explained
by a synchrotron spectrum. Although no supernova has
been detected in this case, we speculate that some form
of pre-supernova thermal emission could possibly be af-
fecting the optical data, adding energy into the blue end
of the spectrum, thus slowing its decline.
After the break in the decays around 105 s, the light-
curves across all bands become more consistent with one
GRB 061121: Broadband observations 15
another, although there are only limited data at such a
late time.
The vertical dotted lines in Figure 10 show the times
of the SEDs plotted in Figure 11; again, all points were
corrected for an extinction of AV = 1.2, so that they
represent the true SEDs (with the frequency in the ob-
server’s frame). The solid lines represent actual fits to
the X-ray and gamma-ray data, while the dashed lines
just join the separate radio, optical and 1 keV points.
The times of these SEDs, which clearly show spectral
evolution, correspond to (1) before the main BAT peak,
56 s after trigger; (2) at the BAT peak, 76 s after trig-
ger; (3) just after the start of the plateau, 300 s after the
trigger; (4) in the main decay at 65 ks (chosen because
radio measurements were taken at this time). SEDs 3
and 4 do not contain any BAT or Konus data, since the
gamma-ray flux had decayed by this point; the highest
energy point in these corresponds to the maximum en-
ergy (10 keV) of the X-ray fits.
Table 4 demonstrates that the optical and X-ray spec-
tra during the peak emission are best fitted with a broken
power-law model, with the break energy at the very low
energy end of the X-ray bandpass. SED 2 in Figure 11
shows that this spectral break corresponds to the peak
frequency in a flux density plot (β1 is less than zero in
this case). Only during SED 2 is the optical flux density
lower than that of the higher energy data. Figure 4 also
shows that the optical emission is less strong than the
X-ray and gamma-ray data during the main burst.
Table 5 shows the values of α for the X-ray and opti-
cal decays (i.e., before and after the break) in SED 4, at
65 ks, with their corresponding spectral indices. For the
initial stages of the power-law decay (Ta < t < 65000 s)
the evolution of the afterglow SED and the coupling be-
tween the temporal and spectral indices are not com-
pletely consistent with the standard model: although the
R-band decay, with αa,R = 0.84 ± 0.03, is in good agree-
ment with the homogeneous CSM model below the cool-
ing break, the X-ray and V -band flux decays are slower
than expected from the measured spectral indices; they
are in best agreement with the same constant density
model below νc, however.
The point at which the power-law decay dominates
the exponential in the optical bands is noticeably earlier
than in the X-ray (< few hundred seconds, rather than
∼ 5000 s) and, as mentioned above, the decay indices
are significantly different for all three (X-ray, V and R)
bands (see Figure 10). At the time of SED 3, the X-ray
data are not decaying (i.e., this is during the plateau),
yet both the V and R-band data have already entered
the power-law decline phase. The R-band is decaying
faster than the V -band, so the spectral index through
the optical range is becoming harder. The X-ray spec-
tral index shows a similar hardening trend (see Table 3),
so the SED measured from optical to 10 keV is gradually
getting harder. Such spectral hardening from the plateau
to the final decay is a feature of many X-ray afterglows
(Willingale et al. 2007).
This slow hardening of the broadband spectrum with
time could be a signature of synchrotron self-Compton
emission (Sari & Esin 2001; Panaitescu & Kumar 2000).
The strength of the self-Compton component in the af-
terglow depends on the flux of low energy photons (radio-
optical) and the electron density in the shock. Using the
formulation in Sari & Esin (2001) the density required is
given by
n1 = 3× 10
f ICmax
(E52tday)
−1/3cm−3 (1)
where f ICmax/f
max is the ratio of the peak flux of the
seed synchrotron spectrum (i.e., the source of low energy
photons) and the peak flux of the self-Compton emis-
sion; E52 is the isotropic burst energy in units of 10
52 erg;
tday is the time in days after the burst (which determines
the distance through the CSM swept up by the external
shock). From Figure 11 (SEDs 1, 3 and 4) we see that
f ICmax/f
max ∼ 0.001 if the X-ray flux has a significant con-
tribution from a self-Compton component at tday = 0.75.
A value of E52 = 30 gives n1 ≈ 10
5 cm−3. Even as-
suming the emission at 0.75 days is not dominated by
the self-Comptonisation, and so taking the f ICmax/f
ratio to be a factor of ten smaller, the density would be
∼ 5× 103 cm−3, which is still high. It seems unlikely that
self-Compton emission is the cause of the spectral hard-
ening of the SED unless the CSM density encountered
by the external shock is extremely large. However, there
have been suggestions that GRBs may form in molecular
clouds (Galama & Wijers 2001; Campana et al. 2006b,c),
which have densities of 104 or more particles per cubic
centimetre in the cores (Miyazaki & Tsuboi 1999; Wil-
son et al. 1999). Typically one might expect greater red-
dening than is found here (Table 4), though Waxman &
Draine (2000) discuss the possibility of dust destruction.
The spectrum will be redshifted as the jet slows down,
so the optical and X-ray spectral indices should, if any-
thing, become softer – the opposite of what is seen here.
Although spectral hardening with time is suggested from
the data, it is not be easily explained by current models.
Whether or not there is a Comptonised component,
the later SEDs clearly indicate that there is a break in
the spectrum somewhere between the optical and the X-
ray; this is also shown by the fits in Table 4, where the
UVOT–XRT spectra are better fitted with broken power-
laws, with Ebreak towards the low energy end of the X-ray
bandpass. Since both the optical and X-ray bands ap-
pear to be below the cooling frequency, from the closure
relations given in Table 5, this change in slope cannot
be identified with a cooling break; its origin remains un-
clear.
The redshift of z = 1.314 and the isotropic energy of
Eiso ∼ 3 × 10
53 erg (§2.4.1) can be used to place con-
straints on the jet opening angle. From Sari et al. (1999),
and assuming that the jet break occurs at T0+2× 10
we have θj ∼ 4
)1/8 ( n
where n and ηγ are
the density of the CSM and the efficiency of the fireball
in converting the energy in the ejecta into gamma-rays.
Taking ηγ = 0.2 and n = 3 cm
−3 (following Ghirlanda et
al. 2004), this gives Eγ ∼ 1.7 × 10
51 erg for the beaming-
corrected gamma-ray energy released, which is within the
range previously determined (e.g., Frail et al. 2001) and
consistent with the Ghirlanda relationship (Ghirlanda et
al. 2004).
4. SUMMARY AND CONCLUSIONS
Swift triggered on a precursor to GRB 061121, leading
16 K.L. Page et al.
GRB models α(β) α(βa,X)
α(βopt)
a αbopt
V -band R-band
CSM SCc (νm < ν < νc)
1.49 ± 0.10 1.32 ± 0.03 0.80 ± 0.09 0.66 ± 0.04 0.84 ± 0.03
Wind SCc (νm < ν < νc)
1.99 ± 0.10 1.30 ± 0.09
CSM or Wind SCc & FCd
0.99 ± 0.10 0.30 ± 0.09
(ν > max(νc, νm))
a Decay calculated from the measured spectral index
b Observed power-law decay index.
c Slow cooling.
d Fast cooling.
TABLE 5
Closure relations for exponential-plus-power-law model fits to the X-ray data (βa,X = 0.99± 0.07) and the
optical-to-X-ray band (βopt = 0.53 ± 0.06) from the time of SED 4 (65 ks after the burst).
to unprecedented coverage of the prompt emission by all
three instruments onboard, with the gamma-ray, X-ray
and optical/UV bands all tracking the main peak of the
burst. GRB 061121 is the instantaneously brightest long
Swift burst detected thus far, both in gamma-ray and X-
rays. The precursor and main burst show spectral lags of
different lengths, though both are consistent with the lag-
luminosity relation for long GRBs (Gehrels et al. 2006b).
The SED of the prompt emission, stretching from 1 eV
to 1 MeV shows a peak flux density at around 1 keV and
is harder than the standard model predicts. There is def-
inite curvature in the spectra, with the prompt optical-
to-X-ray spectrum being better fitted by a broken power-
law, similar to results found for fitting X-ray flares (e.g.,
Guetta et al. 2006; Goad et al. 2007).
The afterglow component, in both the optical and
X-ray, starts early on – before, or during, the main
burst peak (see also O’Brien et al. 2006; Willingale
et al. 2007; Zhang et al. 2006b). The broadband
SEDs reveal gradual spectral hardening as the afterglow
evolves, both within the X-ray regime (Γ flattening from
∼ 2.05 to ∼ 1.87) and between the V - and R-band op-
tical data (αV ∼ 0.66 compared with αR ∼ 0.84). Self-
Comptonisation could explain the hardening, although a
molecular-cloud-core density would be required. A prob-
able jet-break occurs around T0 + 2 × 10
5 s, shown by a
late-time non-detection by Chandra. Before this break,
the X-ray and V -band decays are too slow to be readily
explained by the standard models.
This extremely well-sampled burst shows clearly that
there remains much work to be done in the field of GRB
models. A single, unified model for all GRB emission
observed should be the ultimate goal.
5. ACKNOWLEDGMENTS
The authors gratefully acknowledge support for this
work at the University of Leicester by PPARC, at PSU
by NASA and in Italy by funding from ASI. This work is
partly based on observations with the Konus-Wind ex-
periment (supported by the Russian Space Agency con-
tract and RFBR grant 06-02-16070) and on data ob-
tained with XMM-Newton, an ESA science mission with
instruments and contributions directly funded by ESA
Member States and NASA. We thank the Liverpool GRB
group at ARI, Liverpool John Moores University, in par-
ticular C.J. Mottram, D. Carter, R.J. Smith and A.
Gomboc for their assistance with the FTN data acqui-
sition and interpretation. The Faulkes Telescopes are
operated by the Las Cumbres Observatory Global Tele-
scope Network. We also thank J.E. Hill and A.F. Abbey
for discussions and help with the PuPD data, P. Cur-
ran for assistance with the UVOT-XRT SED creation,
D. Grupe for the Chandra upper limit and B. Cobb and
D. Malesani for information regarding the magnitude of
the host galaxy. Thanks as well to C. Akerlof, E. Rykoff,
A. Phillips and M.C.B. Ashley from the ROTSE team.
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|
0704.1610 | High-altitude signatures of ionospheric density depletions caused by
field-aligned currents | Microsoft Word - Karlsson et al 2007.doc
High-altitude signatures of ionospheric density depletions caused by
field-aligned currents
T. Karlsson1, N. Brenning1, O. Marghitu2,3, G. Marklund1, S. Buchert4
1 Space and Plasma Physics, Royal Institute of Technology (KTH), Stockholm, Sweden.
2 Institute for Space Sciences, Bucharest, Romania.
3 Also at Max-Planck-Institut für extraterrestrische Physik, Garching, Germany.
4 Swedish Institute of Space Physics, Uppsala, Sweden.
Abstract. We present Cluster measurements of large electric fields correlated with intense downward
field-aligned currents, and show that the data can be reproduced by a simple model of ionospheric
plasma depletion caused by the currents. This type of magnetosphere-ionosphere interaction may be
important when considering the mapping between these two regions of space.
1. Introduction
A system of magnetic field-aligned current sheets closing via Pedersen currents in the
ionosphere will set up an ionospheric electric field. For constant conductivity, and for
sheets extending to infinity along the field-line and one of the perpendicular
directions, we get:
1 1 1P
P P P P
E j d d Bτν τν νμ ν μ
= = = =
Σ Σ Σ ∂ Σ∫ ∫ (1)
where ν is the direction perpendicular to the sheet, τ the tangential direction, Eν is the
normal electric field, JP and ΣP, the height integrated Pedersen current and
conductivity, Bτ the tangential magnetic field, j// the field-aligned current (positive for
downward currents) and μ0 the magnetic permeability of vacuum. This kind of
correlation between Eν and Bτ can be seen rather often in the dayside auroral oval (e.g.
Ishii et al., 1992). When the conductivity is not constant, the above correlation breaks
down; in this paper we will present data from the Cluster spacecrafts, where this
correlation is replaced with a correlation between Eν and j//, i.e. the derivative of Bτ .
2. Cluster data
We present electric and magnetic field data from the EFW (Gustafsson et al., 1997)
and FGM (Balogh et al., 2001) instruments on the Cluster satellites, which have an
apogee of 19.8 RE and a perigee of 4.0 RE, in radial distance. We first present data
from a northern hemisphere auroral oval crossing, on Feb 18, 2004, from 08:58:20 to
09:10:00 UT. The Cluster radial distance during this time period was about 4.2 RE,
and the satellite separations between approximately 350 and 1100 km. In Figure 1 we
show the residual magnetic field vectors along the satellite tracks projected onto a
plane perpendicular to the geomagnetic field. The two perpendicular directions in the
figure roughly correspond to geomagnetic North, and East. The diamonds at the
bottom end of the tracks indicate the satellite positions at 08:58:20 UT. (The data is
color coded: black – S/C 1, red – S/C 2, green – S/C 3, blue – S/C 4.) The satellites
move relatively close to a pearls-on-a-string configuration. The main feature of the
data is the crossing of three sheets of field-aligned current, from bottom to top a
relatively smooth sheet of upward current approximately 800 km wide, a thinner sheet
of downward current (≈250 km), and finally a wider sheet of predominantly upward
currents (~1000 km wide). (The meridional mapping factor to ionospheric altitude is
11.6.) This current system remains essentially stationary in space for the whole 200 s
period between the crossings of the central current sheet by S/C 1 and S/C 4, which is
the reason we have chosen to present this event. We have applied minimum variance
analysis on the magnetic field data from all four S/C, and have used the average
resulting angle of 5.8° to establish the sheet-aligned coordinate system. We have then
used the infinite current sheet approximation to calculate the field-aligned current j//
from the tangential component of the residual magnetic field Bτ . In Figure 2 we
present j// and the normal electric field Eν measured by Cluster. All values are mapped
to ionospheric altitude. Also presented is the result of a model calculation described in
Section 3. The correlation between Eν and j// is clear for all S/C in the downward
current region. This type of correlation is rather uncommon, but a manual inspection
of around 300 auroral zone crossings resulted in identification of 23 similar events,
i.e. in about 8% of the crossings, all for downward currents. 17 of the 23 events where
encountered during winter conditions and 15 on the night side.
3. Comparison data – model
The close relation between the electric field and the local downward field-aligned
current (DFAC) suggests that there is a relation between the DFAC and the
conductivity, since an infinitesimally thin current sheet gives a negligible contribution
to the ionospheric closure current across the sheet, Jν. However, with a coupling to a
local decrease in the conductivity it can produce a local increase in Eν (Figure 3).
Such decreases in the conductivity coupled to DFACs have been modeled by Doe at
al. (1995), Blixt and Brekke (1996), Karlsson and Marklund (1998, 2005), and
Streltsov and Marklund (2006). A few radar observations of ionospheric density
cavities which may be related to this mechanism have been reported by Doe et al.
(1993), Aikio et al. (2002), and Nilsson et al. (2005). The reason that a cavity is
formed in DFAC regions is that the parallel current is mainly carried by electrons,
whereas the Pedersen current is carried by ions. In regions where the downward
parallel and perpendicular currents couple there will then be a net outflow of current
carriers.
Here we model this interaction in a heuristic way by prescribing the conductances by
, // //
, for downward
0, for ward
down s
P P 0
k j j
Σ = Σ − ⎨
Σ = Σ
(2)
where ΣP,0 and kdown,s (>0) are constants, with s = 1-4, for the four spacecraft
crossings. We ignore any effects on the conductance from the upward currents, since
we will concentrate on the electric field behavior in the downward current region. We
also set a minimum value for the Pedersen conductivity of 0.2 S, which represents the
background conductivity due to galactic cosmic rays, which are always present.
Current continuity and the assumption of an infinite current sheet yields
( ) ( ) ( ) ( ) ( ) ( )
0 0 0 0P HJ j d E E
ν ν τν
ν ν ν ν ν ν ν′ ′= + Σ − Σ∫ (3)
where Jν is the height-integrated ionospheric current normal to the sheet. Eτ is
constant if we use the electrostatic approximation ( 0∇× ≡E ). (2) and (3) then give
( ) ( ) ( )
( ) ( )
1P H H
P P P
E E E j d
ν ν τ ν
ν ν ν ν
ν ν ν
Σ Σ − Σ
′ ′= + +
Σ Σ Σ ∫ (4)
Using the observed values for j// along each of the satellite tracks, we can calculate Eν,
as a function of ΣP,0 = ΣP(ν0), Eν (ν0), Eτ, and kdown,s. Before t ≈ 320 s the electric field
is small and rather constant and we can assume that it can be mapped to the
ionosphere and be taken as the background field of our model. However, there is an
offset in the electric field component aligned with the direction towards the sun, due
to a photo electron sheet. Using data from the electron drift instrument (EDI) on S/C 1
we correct for this and then take the average electric field for 60 s prior to the crossing
of the large DFAC, which we use as our background ionospheric electric field: Eν (ν0)
= 0, Eτ = -6 mV/m (values mapped to the ionosphere).
In principle the conductance could be calculated from the electron data, but this is a
very uncertain procedure in the absence of energetic precipitating electrons, and
outside the scope of this paper. Instead we assume a reasonable background
conductance. The results are rather robust with respect to the chosen value of ΣP,0, but
the numerical value of kdown will of course vary within a factor of 2-3 depending on
the choice of conductance. By trial and errorr we then find that the following
parameters reproduce the electric field behavior in the DFAC region well:, ΣP,0 = 5 S
and kdown,1 = 0.33 Sm2/μA, kdown,2 = 0.43 Sm2/μA, kdown,3 = 0.44 Sm2/μA, kdown,4 = 0.68
Sm2/μA, where the subscript on the k’s indicate S/C number. Eν thus calculated is
plotted in green in Figure 2. Thus the same set of parameters, except for kdown,
reproduces the DFAC electric field quite well. It is interesting that kdown has an
increasing trend with time; in Figure 4 we plot the values of kdown as a function of time
from the first crossing of the current sheet. The crossing time is defined as the time
when the current maximum is encountered, and the error bars in the t-direction
indicate when the current is half the maximum value. A linear fit is reasonable which
means that we can write 0( )downk t tκ= − , with κ = 1.4·10
-3 Sm2/μAs, and t0 ≈ -200 s,
consistent with a gradual deepening of the density cavity, beginning about 200 s
before the first satellite crossing.
Revisiting the data from the simulations by Karlsson et al. [1998] we can calculate
κ. In the simulations, the development of kdown settles down to a reasonably linear
dependence on time after the first tens of seconds, from which we can estimate κ. The
value of κ depends on various initial conditions of the simulations but for some
realistic situations varied from around 1·10-5 to 2·10-3 Sm2/μAs, which is in agreement
with the above measurement.
For this event the horizontal ionospheric current Jν ,//, resulting from the feeding
field-aligned currents was comparable to the current associated with the background
electric field: | Jν ,//| ≈ 20 mA/m, |Eτ ΣH(ν0)| ≈ 30 mA/m. Below we show two cases
where one of these current contributions dominates over the other one.
First (Figure 5a) we show data from a northern hemisphere auroral oval pass on Apr
27, 2002, with MLT ≈ 22, ILat ≈ 66º, and the geocentric distance 4.9 RE. We show
data only from S/C 4, but similar signatures can be seen on S/C 2 and 3. Using the
same method as above we calculate j//, ΣP, and Eν. In the figure the modeled Eν and ΣP
is plotted in red. j// is not shown, but has a maximum (downward) value of 34 μA/m2.
For this event upward accelerated electrons are observed from t ≈ 70950 s, which
complicates the mapping of the background ionospheric electric field. We instead
here consider it as a free parameter. The fact that the constant background current
(driven by the background electric field) dominates over Jν ,// (440 mA/m vs. 90
mA/m) means that the electric field traces out the form of the conductivity, which in
turn traces out the DFAC. We thus get a very detailed correlation between the electric
field and the DFAC, and a unipolar Eν field signature at the density cavity.
In Figure 5b we present data from an auroral crossing on Jan 11, 2005. MLT ≈ 22,
ILat ≈ 66º, geocentric distance 4.3 RE, max(j//,down) = 32 μA/m2. Here the background
ionospheric current is dominated by Jν ,//. This means that the ionospheric current is
not constant across the low-conductivity region, and we should not expect such a
detailed correlation between Eν and j// as in the above case. In fact, what we see is that
the electric field is large inside the low-conductivity region of the DFAC, but since
the ionospheric current changes sign inside this region, the electric field also does, and
produces a bipolar electric field signature. A small westward background electric field
shifts the zero crossing of the total current Jν slightly from that of Jν ,//.
4. Discussion and conclusions
The correlation between large electric fields and DFACs presented here is consistent
with them being associated with ionospheric low-conductivity regions. A correlation
between the electric field and the derivative of the magnetic field could also be the
result of a partially reflected Alfvén wave, but this would not explain why we only
observe this correlation for downward currents, or the preference for night-
/wintertime conditions. The correlation is also not consistent with the signatures of a
U-shaped potential structure. There, the largest current is associated with the centre of
the structure, where the perpendicular electric field has its minimum. In fact, in order
for the electric field correlation with the DFAC to map all the way out to Cluster
altitudes, we must assume that there is no field-aligned potential drop along the
magnetic field line. In that case the correlation represents the naked high-altitude
signature of the ionospheric density depletion. In many cases we would expect large
DFACs to be associated with such a parallel potential drop [e.g. Elphic et al, 1998];
this may be one of the reasons why events of the type we have presented here are
relatively rare; we will only see them before such a potential drop has developed.
Another reason could be that generally rather low background conductivities will be
required.
Reversing the argument, observations of large perpendicular electric fields at
magnetospheric altitudes is generally taken as proof that there is a parallel potential
drop above the ionosphere. Our results show that this is not necessarily true, but that
at least part of this potential drop may be situated deep in the ionosphere, in the E and
lower F regions, where the currents partially close through the developing density
cavity [Karlsson and Marklund, 1998]. This should be taken into account when
interpreting high-altitude electric field data.
For the first event, the current system is stable for around 200 s. The close to linear
evolution of kdown, can be seen as a first observational comparison with modeling of
the temporal evolution of ionospheric density cavities. The 200 s time scale is,
according to the modeling work quoted above, a typical time scale for creating a deep
ionospheric plasma depletion. We would expect to see this type of events for
conditions of some moderate geomagnetic activity (to create large DFACs), but not
during e.g. the substorm expansion phase, where the current systems would probably
move around too much on time scales faster than the depletion time. We have checked
this by inspecting the Auroral Electrojet index for the 23 events. Only four of the
events where encountered during the expansion phase, whereas the rest were observed
during periods that had a medium level of activity; growth or recovery phase or steady
magnetospheric convection events. This is further support for the model presented
above.
Acknowledgements
The authors are grateful to G. Haerendel for some suggestions and comments.
References
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D. Fontaine, A. Vaivads, and A. Fazakerley (2004), Temporal evolution of two
auroral arcs as measured by the Cluster satellite and coordinated ground-based
instruments, Ann. Geophys., 22, 4089–4101.
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Fornacon, K.-H., Georgescu, E., Glassmeier, K.-H., Harris, J.P., Musmann, G.,
Oddy, T.M. and Schwingenschuh, K (2001), The Cluster magnetic field
investigation: Overview of in-flight performance and initial results, Ann. Geophys.,
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McFadden, C. W. Carlson, W. Peria, C. A. Cattell, D. Klumpar, E. Shelley, W.
Peterson. E. Moebius, L. Kistler, R. Pfaff (1998), The auroral current circuit and
field-aligned currents observed by FAST, Geophys. Res. Lett., 25, 2033-2036.
Gustafsson, G., R. Bostrom, B. Holback, G. Holmgren, A. Lundgren, K. Stasiewicz,
L. Åhlen, F. S. Mozer, D. Pankow, P. Harvey, P. Berg, R. Ulrich., A. Pedersen, R.
Schmidt, A. Butler, A. W. C. Fransen, D. Klinge, M. Thomsen, C.-G. Fälthammar,
P.-A. Lindqvist, S. Christenson, J. Holtet, B. Lybekk, T. A. Sten , P. Tanskanen, K.
Lappalainen, and J. Wygant (1997), The electric field and wave experiment for the
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Ishii, M., M. Sugiura, T. Iyemori, J. A. Slavin (1992), Correlation between magnetic
and electric field perturbations in the field-aligned current regions deduced from
DE-2 observations, J. Geophys. Res., 97, 13877-13887.
Karlsson, T. and G. T. Marklund (1998), Simulations of Small-Scale Auroral Current
Closure in the Return Current Region, in Physics of Space Plasmas, No. 15, eds. T.
Chang and J. R. Jasperse, MIT Center for Theoretical Geo/Cosmo Plasma Physics,
Cambridge, Massachusetts, 401-406.
Karlsson, T., G. Marklund, N. Brenning, I. Axnäs (2005), On Enhanced Aurora and
Low-Altitude Parallel Electric Fields, Phys. Scr., 72, 419-422.
Nilsson, H., A. Kozlovsky, T. Sergienko, A. Kotikov (2005), Radar observations in
the vicinity of pree-noon auroral arcs, Ann. Geophys., 23, 1785-1796.
Streltsov, A. V., and G. T. Marklund (2006), Divergent electric fields in downward
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Figure 1. Magnetic field perpendicular to the geomagnetic field for the time period
2004-02-18 08:58:20-09:10:00 UT.
Figure 2. Measured (blue) and modeled (green) normal electric field, and measured
field-aligned current (red) for the same time period as Figure 1. Note that in this
figure downward current is plotted as negative current.
Figure 3. Schematic of the connection between DFAC, current closure and iono-
spheric cavity formation.
Figure 4. Temporal evolution of kdown.
Figure 5. Measured (black) and modeled (red) quantities. Time is in seconds from
00:00:00 UT.
|
0704.1611 | Burgers Turbulence | BurgersTurbulence
Jérémie Bec
Laboratoire Cassiopée UMR6202, CNRS, OCA; BP4229, 06304 Nice Cedex 4, France.
Konstantin Khanin
Department of Mathematics, University of Toronto, M5S 3G3 Toronto, Ontario, Canada.
Abstract
The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original
one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the
problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random
initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to
statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a
nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and
their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative
anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence.
The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.
Key words: Burgers equation, turbulence, Lagrangian systems.
Contents
1 From interface dynamics to cosmology 2
1.1 The Burgers equation in statistical mechanics 2
1.2 The adhesion model in cosmology 3
1.3 A benchmark for hydrodynamical turbulence 3
2 Basic tools 4
2.1 Inviscid limit and variational principle 4
2.2 Variational principle for the viscous case 5
2.3 Singularities of Burgers turbulence 5
2.4 Remarks on numerical methods 7
3 Decaying Burgers turbulence 8
3.1 Geometrical constructions of the solution 8
Email addresses: [email protected] (Jérémie
Bec), [email protected] (Konstantin Khanin).
3.2 Kida’s law for energy decay 10
3.3 Brownian initial velocities 12
4 Transport of mass in the Burgers/adhesion
model 14
4.1 Mass density and singularities 15
4.2 Evolution of matter inside shocks 16
4.3 Connections with convex optimization
problems 19
5 Forced Burgers turbulence 21
5.1 Stationary régime and global minimizer 21
5.2 Topological shocks 22
5.3 Hyperbolicity of the global minimizer 24
5.4 The case of extended systems 26
6 Time-periodic forcing 29
Preprint submitted to Physics Reports 1 February 2008
http://arXiv.org/abs/0704.1611v1
6.1 Kicked Burgers turbulence 29
6.2 Connections with Aubry–Mather theory 33
7 Velocity statistics in randomly forced
Burgers turbulence 37
7.1 Shocks and bifractality – a replica variational
approach 38
7.2 Dissipative anomaly and operator product
expansion 39
7.3 Tails of the velocity gradient PDF 41
7.4 Self-similar forcing and multiscaling 43
8 Concluding remarks and open questions 45
Acknowledgements 46
References 46
1 From interface dynamics to cosmology
At the end of the thirties, the Dutch scientist J.M. Burg-
ers [26] introduced a one-dimensional model for pressure-
less gas dynamics. He was hoping that the use of a simple
model having much in common with the Navier–Stokes
equation would significantly contribute to the study of
fluid turbulence. This model now known as the Burgers
equation
∂tv + v ∂xv = ν ∂
xv, (1.1)
has not only the same type of hydrodynamical (or advec-
tive) quadratic nonlinearity as the Navier–Stokes equa-
tion that is balanced by a diffusive term, but it also has
similar invariances and conservation laws (invariance un-
der translations in space and time, parity invariance,
conservation of energy and momentum in one dimension
for ν = 0).
Such hopes appeared to be shattered when in the fifties,
Hopf [67] and Cole [33] showed that the Burgers equation
can be integrated explicitly. This model thus lacks one
of the essential properties of Navier–Stokes turbulence:
sensitivity to small perturbations in the initial data and
thus the spontaneous arise of randomness by chaotic dy-
namics. Unable to cope with such a fundamental aspect,
the Burgers equation then lost its interest in “explain-
ing” fluid turbulence.
In spite of this, the Burgers equation reappeared in the
eighties as the asymptotic form of various nonlinear dis-
sipative systems. Physicists and astrophysicists then de-
voted important effort to the understanding of its multi-
dimensional form and to the study of its random solu-
tions arising from random initial conditions or a random
forcing. The goal of this paper is to review selected works
that exemplify this strong renewal of interest in Burgers
turbulence.
The rest of this section is dedicated to the description of
several physical situations where the Burgers equation
arises. We will then see in section 2 that in any dimen-
sion and in the limit of vanishing viscosity, the solutions
to the Burgers equation can be expressed in an explicit
manner in the decaying case or in an implicit manner in
the forced case, in terms of a variational principle that
permits a systematic classification of its various singu-
larities (shocks and others) and of their local structure
(normal form). Section 3 is dedicated to the study of the
decay of the solutions to the one-dimensional unforced
Burgers equation with random initial data. The multi-
dimensional decaying problem is discussed in section 4.
The motivation comes from cosmology where large-scale
structures can be described in terms of mass transport
by solutions to the Burgers equation. The basic prin-
ciples of the forced Burgers turbulence are discussed
in section 5 where the notions of global minimizer and
topological shocks are introduced. Section 6 is dedicated
to the study of the solutions to the periodically kicked
Burgers equation and their relation with Aubry–Mather
theory for commensurate-incommensurate phase transi-
tions. Section 7 reviews various studies of the stochas-
tically forced Burgers equation in one dimension with a
particular emphasize on questions that are arising from
the statistical study of turbulent flows. Finally, section 8
encompasses concluding remarks and a non-exhaustive
list of open questions on the problem of the Burgers tur-
bulence.
1.1 The Burgers equation in statistical mechanics
The Burgers equation appears in condensed matter, in
statistical physics, and also beyond physics in vehicle
traffic models (see [32], for a review on this topic). When
a random forcing term is added - usually a white noise
in time - it is used to describe various problems of in-
terface deposition and growth (see, for instance, [5]). An
instance frequently studied is the Kardar–Parisi–Zhang
(KPZ) model [74]. This continuous version of ballistic
deposition models accounts for the lateral growth of the
interface. Let us indeed consider an interface where par-
ticles deposit with a random flux F that depends both
on time t and on the horizontal position ~x. The growth
of the local height h happens in the direction normal to
the interface and its time evolution is given by
|∇h|2 = ν∇2h+ F, (1.2)
where the first term of the right-hand side represents the
relaxation due to a surface tension ν. The gradient of
(1.2) gives the multidimensional Burgers equation
∂t~v + ~v · ∇~v = ν∇2~v −∇F, ~v = −∇h, (1.3)
forced by the random potential F . As we will see later,
shocks generically appear in the solution to the Burgers
equation in the inviscid limit ν → 0. They correspond
to discontinuities of the derivative of the height h. The
KPZ model is hence frequently used to understand the
appearence of roughness in various interface problems, as
for instance front propagation in randomly distributed
forests (see, e.g., [101]).
The Hopf–Cole transformation Z = exp(h/2ν) allows
rewriting (1.2) as a linear problem with random coeffi-
cients.
∂tZ = ν∇2Z +
F Z, (1.4)
This equation appears in many complex systems, as for
instance directed polymers in random media [75,22].
Indeed the solution Z(~x, t) is exactly the partition
function of an elastic string in the random potential
(1/2ν)F (~x, t), subject to the constraint that its bound-
ary is fixed at (~x, t). Note that here, the time variable t
is actually a space variable in the main direction of the
polymer.
1.2 The adhesion model in cosmology
The multidimensional Burgers equation has important
applications in cosmology where it is closely linked to
what is usually referred to as the Zel’dovich approxi-
mation [112]. In the limit of vanishing viscosity ν → 0
the Burgers equation is known as the adhesion model
[62]. Right after the decoupling between baryons and
photons, the primitive Universe is a rarefied medium
without pressure composed mainly of non-collisional
dust interacting through Newtonian gravity. The initial
density of this dark matter fluctuates around a mean
value ρ̄. These fluctuations are responsible for the for-
mation of the large-scale structures in which both the
dark non-baryonic matter and the luminous baryonic
matter concentrate. A hydrodynamical formulation of
the cosmological problem leads to a description where
matter evolves with a velocity ~v, solution of the Euler–
Poisson equation (see, e.g., [98], for further details).
In the linear theory of the gravitational instability, that
is for infinitesimally small initial fluctuations of the den-
sity field, an instability is obtained with potential dom-
inant modes (i.e. ~v = −∇Ψ) and, in the suitable coordi-
nates, the gravitational interactions can be neglected. In
1970, Zel’dovich proposed to extend these two proper-
ties to the nonlinear régimes where density fluctuations
become important. In this approximation, he also pos-
tulates that the acceleration is a Lagrangian invariant,
leading to the formation of caustics. N-body simulations
however show that the large-scale structures of the Uni-
verse are much simpler than caustics: they resemble sort
of thin layers in which the particles tend to be trapped
(see figure 1(a)).
Fig. 1. (a) Projection of the matter distribution in a slice
obtained from an N-body simulation by the Virgo consor-
tium [71]. (b) Composite picture showing the superposition
of the results of an N-body simulation with the skeleton of
the results obtained from the adhesion model (from [78]).
It was shown by Gurbatov and Saichev [62] that these
structures are very well approximated by those obtained
when constraining the particles not to cross each other
but to stick together. Even if this mechanism is not col-
lisional but rather gravitational (probably due to insta-
bilities at small spatial scales), its effect can be modeled
by a small viscous diffusive term in the Euler–Poisson
equation and thus amounts to considering the Burgers
equation in the limit of vanishing viscosity.
1.3 A benchmark for hydrodynamical turbulence
As a nonlinear conservation law, and since its solution
can be easily known explicitly, the one-dimensional
Burgers equation frequently serves as a testing ground
for numerical schemes, and especially for those dedi-
cated to compressible hydrodynamics. For instance, it
is a central example for the validation of finite-volumes
schemes.
The Burgers equation was also used for testing statisti-
cal theories of turbulence. For instance, field theoretical
methods have frequently been applied to turbulence (see
[96,102]). These approaches had very little impact until
recently when they led to significant advances in the un-
derstanding of intermittency in passive scalar advection
(see, e.g., [46] for a review). In the past such attempts
were mostly based on a formal expansion of the non-
linearity using, for instance, Feynman graphs. Since the
Burgers equation has the same type of quadratic nonlin-
earity as the Navier–Stokes equation, such methods are
applicable in both instances. From this point of view,
it is important to know answers for Burgers turbulence
to questions that are generally asked for Navier–Stokes
turbulence. For instance, Burgers turbulence with a ran-
dom forcing is the counterpart of the hydrodynamical
turbulence model where a steady state is maintained by
an external forcing. The Burgers equation has frequently
been used as a model where the dissipation of kinetic
energy remains finite in the limit of vanishing viscosity
(dissipative anomaly). This allows singling out artifacts
arising from manipulation that ignore shock waves (see,
for instance, [51,40]).
Beyond statistical theory, Burgers turbulence gives a
simple hydrodynamical training ground for developing
mathematical tools to study not only Navier–Stokes tur-
bulence but also various hydrodynamical or Lagrangian
problems. The forced Burgers equation has recently been
at the center of studies that allowed unifying different
branches of mathematics. Mainly used in the past as a
simple illustration of the notion of entropy (or viscos-
ity) solution for conservation laws [83,95,85], the Burg-
ers equation was related in the eighties to the theory
of Hamiltonian systems developed by Kolmogorov [80],
Arnold [2] and Moser [93] (KAM), through the introduc-
tion of the weak KAM theory [43,47,48]. More recently,
the study of the solutions to the Burgers equation with a
random forcing was at the center of a “random” Aubry–
Mather theory related to random Lagrangian systems
[38,69]. A particular emphasis on these aspects of Burg-
ers turbulence is given throughout the present review.
For the application of the Burgers equation to the prop-
agation of random nonlinear waves in nondispersive me-
dia, we refer the reader to the book written by Gurba-
tov, Malakhov, and Saichev [61]. For a complete state of
the art on most mathematical apsects of Burgers turbu-
lence, we refer the reader to the lecture notes by Woy-
czyński [110].
2 Basic tools
In this section we introduce various analytical, geomet-
rical and numerical tools that are useful for constructing
solutions to the Burgers equation, with and without forc-
ing, in the limit of vanishing viscosity. All these tools are
derived from a variational principle that allows writing
in an implicit way the solution at any time. This varia-
tional principle leads to a straightforward classification
of the various singularities that are generically present
in the solution to the Burgers equation.
2.1 Inviscid limit and variational principle
We consider here the multidimensional viscous Burgers
equation with forcing
∂t~v + (~v · ∇)~v = ν∇2~v −∇F (~x, t), (2.1)
where ~x lives on a prescribed configuration space Ω of
dimension d. For a potential initial condition, ~v(~x, t0) =
−∇Ψ0(~x), the velocity field remains potential by con-
struction at any later time, ~v = −∇Ψ, where the poten-
tial Ψ satisfies the equation
∂tΨ −
|∇Ψ|2 = ν∇2Ψ + F. (2.2)
Note that if one sets abruptly ν = 0 in (2.2), then −Ψ
solves the Hamilton–Jacobi equation associated to the
Hamiltonian H(~q, ~p) = |~p|2 + F (~q, t). In the unforced
case, −Ψ is a solution of the Hamilton–Jacobi equation
associated to the dynamics of free particles. The Hopf–
Cole transformation [67,33] uses a change of unknown
Ψ(~x, t) = 2ν ln Θ(~x, t). The new unknown scalar field Θ
is solution of the (imaginary-time) Schrödinger equation
∂tΘ = ν∇2Θ +
F Θ, (2.3)
with the initial condition Θ(~x, t0) = exp(Ψ0(~x)/(2ν)).
The solution can be expressed through the Feynman-
Kac formula
Θ(~x, t)=
Ψ0( ~Wt0)−
F ( ~Ws, s) ds
, (2.4)
where the brackets 〈·〉 denote the ensemble average with
respect to the realizations of the d-dimensional Brown-
ian motion ~Ws with variance 2ν defined on the configu-
ration space Ω and which starts at ~x at time t. The limit
ν → 0 is obtained by a classical saddle-point argument.
The main contribution will come from the trajectories
~W minimizing the argument of the exponential; the ve-
locity potential can then be expressed as a solution of
the variational principle
Ψ(~x, t) = − inf
~γ(·)
[A(~γ, t0, t) − Ψ0(~γ(t0))] , (2.5)
where the infimum is taken over all trajectories ~γ that
are absolutely continuous (e.g. piece-wise differentiable)
with respect to the time variable and that satisfy ~γ(t) =
~x. The action A(~γ, t0, t) associated to the trajectory ~γ
is defined by
A(~γ, t0, t) =
|~̇γ(s)|2 − F (~γ(s), s)
ds, (2.6)
where the dot stands for time derivative. The kinetic
energy term |~̇γ|2/2 comes from the propagator of the d-
dimensional Brownian motion ~W . This variational for-
mulation of the solution to the Burgers equation was
obtained first by Hopf [67], Lax [83] and Oleinik [95]
for scalar conservation laws. Its generalization to mul-
tidimensional Hamilton–Jacobi equations was done by
Kruzhkov [82] (see also [85]). In the case of a random
forcing potential F , it was shown by E, Khanin, Mazel
and Sinai [38] that this formulation is still valid after re-
placing the action by a stochastic integral. It is also im-
portant to notice that the variational formulation (2.5)
in the limit of vanishing viscosity is valid irrespective of
the configuration space Ω on which the solution is de-
fined.
The minimizing trajectories ~γ(·) necessarily satisfy at
times s < t the Newton (or Euler–Lagrange) equation
~̈γ = −∇F (~γ(s), s), (2.7)
with the boundary conditions (at the final time t)
~γ(t) = ~x and ~̇γ(t) = ~v(~x, t). (2.8)
Note that these equations are only valid backward in
time. Extending them to times larger than t requires
knowing that the Lagrangian particle will neither cross
the trajectory of another particle, nor be absorbed by
a mature shock. This requires global knowledge of the
solution that satisfies the variational principle (2.5).
When the forcing term is absent from (2.1), it is easily
checked that the variational principle reduces to
Ψ(~x, t) = max
Ψ0(~x0) −
|~x− ~x0|2
, (2.9)
where the maximum is taken over all initial positions
~x0 in the configuration space Ω. The Euler–Lagrange
equation takes then the particularly simple form
~̈γ = 0, i.e. ~x = ~x0 + t ~v0(~x0), (2.10)
which simply means that the initial velocity is conserved
along characteristics.
Typically there exist Eulerian locations ~xwhere the min-
imum in (2.5) – or the maximum in (2.9) in the unforced
case – is reached for several different trajectories ~γ. Such
locations correspond to singularities in the solution to
the Burgers equation. After their appearance, the veloc-
ity potential Ψ contains angular points corresponding to
discontinuities of the velocity field ~v = −∇Ψ.
2.2 Variational principle for the viscous case
The derivation of the variational principle (2.5) makes
use of the Hopf–Cole transformation and of the
Feynman–Kac formula. There is in fact another ap-
proach which also yields a variational formulation of
the solution to the viscous Hamilton–Jacobi equation
(2.2). Indeed it turns out that the solution to (2.2) can
be obtained in the following way. Consider solutions to
the stochastic differential equation
d~γ~u = ~u(~γ~u, s) ds+
2ν d ~Ws , (2.11)
where ~u is a stochastic control, that is an arbitrary time-
dependent velocity field which depends (progressively
measurably) on the noise ~W . Limiting ourselves to so-
lutions satisfying the final condition ~γ~u(t) = ~x, we can
write
Ψ(~x, t) = − inf
〈A~u(~γ~u, t0, t) − Ψ0(~γ~u(t0))〉 , (2.12)
where the brackets 〈·〉 now denote average with respect
to ~Ws and the action is given by
A~u(~γ~u, t0, t) =
|~u(s)|2 − F (~γ~u(s), s)
ds. (2.13)
It is obvious that this variational principle gives (2.5) in
the inviscid limit ν → 0. Note that this approach has the
advantage to be applicable not only to Burgers dynamics
but to any convex Lagrangian (see [50,58]).
2.3 Singularities of Burgers turbulence
The singularities appearing in the course of time play
an essential role in understanding various aspects of the
statistical properties in the inviscid limit. The shocks –
discontinuities of the velocity field – and other singular-
ities, such as preshocks, generally not associated to dis-
continuities, are often responsible for non-trivial univer-
sal behaviors. In order to understand the contribution
of each kind of singularities, it is first important to know
in a detailed manner their genericity and their type.
As we have seen in the previous section, the potential
solutions to the multidimensional Burgers equation can
be expressed in the inviscid limit in terms of the vari-
ational principle (2.5) (that reduces to (2.9) in the un-
forced case). There typically exist Eulerian locations ~x
where the minimum is either degenerate or attained for
several trajectories. A co-dimension can be associated
to such points by counting the number of relations that
are necessary to determine them. The singular locations
of co-dimension c form manifolds of the Eulerian space-
time with dimension (d− c). The singularities with the
lower co-dimension are the shocks corresponding to the
Eulerian positions where two different trajectories min-
imize (2.5); they form Eulerian manifolds of dimension
(d− 1): in one dimension the shocks are isolated points,
in two dimensions they are lines, in three dimensions
surfaces, etc. There also exist Eulerian manifolds with
three different minimizing trajectories. In one dimen-
sion, they are isolated space-time events corresponding
to the merger of two shocks. In two dimensions, they are
triple points where three shock lines meet. In three di-
mensions they are filaments corresponding to the inter-
section of three shock surfaces. There also exist Eulerian
locations where the minimum in (2.5) is reached for four
different trajectories, etc.
points
termination
shock lines
points
triple
Fig. 2. Typical aspect of the singularities present at a fixed
time in the solution for (a) d = 2 and (b) d = 3.
The generic form of such singularities and their typi-
cal metamorphoses occurring in the course of time were
studied in details and classified for d = 2 and d = 3 by
Arnold, Baryshnikov and Bogaevsky in the Appendix
of [62] and in a more detailed paper by Bogaevsky [17].
This classification is based on two criteria: (i) the num-
ber of trajectories minimizing (2.5) and (ii) the multi-
plicity of each of these minima. The shocks correspond-
ing to locations with two distinct minimizers are hence
denoted by A21. At a fixed time, the A
1 singularities are
discrete points in one dimension. In two dimensions (see
figure 2(a)) they form curve segments with extremities
that can be either triple points A31 or isolated termina-
tion points of the type A3 corresponding to a degenerate
minimum. In three dimensions (see figure 2(b)) the sin-
gular manifold is formed by shock surfaces of A21 points.
The boundaries of these surfaces are either made of de-
generate A3 points or of triple lines made of A
1 points.
The triple lines intersect at isolated A41 points or inter-
sect shock boundaries at particular singularities called
A1A3 where the minimum is attained in two points, one
of which is degenerate.
It is important to remark here that degenerate singu-
larities (of the type A3 or of higher orders A5, A7, etc.)
introduce in the solution points where the velocity gra-
dients becomes arbitrarily large. This is not the case of
the An1 singularities which correspond to discontinuities
of the velocity but are associated to bounded values of
its gradients. As we will see in sections 4 and 7, these
degenerate singularities are responsible for an algebraic
behavior of the probability density function of velocity
gradients, velocity increments and of the mass density.
Fig. 3. Illustration of the similarities between the singular
manifold in space time for d = 1 and at fixed time for d = 2
(b). The two manifolds contain the same type of singularities
with the same co-dimensions. The restrictions on the possible
metamorphoses in dimension d = 1 are the following: a point
of the type A3 can only exist at the bottom extremity of
a shock trajectory; the A31 points necessarily correspond to
the merger of two shocks; shock trajectories cannot have a
horizontal tangent.
The singularities with co-dimensions (d+ 1) generically
appear in the solution at isolated times. They corre-
spond to instantaneous changes in the topological struc-
ture of the singular manifold, called metamorphoses and
can be also classified (see [17]). In one dimension, there
are two generic metamorphoses: shock formations (the
preshocks) corresponding to a specific space-time loca-
tion where the minimum is degenerate (A3 singularities)
and shock mergers associated to space-time positions
where the minimum is attained for three different tra-
jectories (A31 singularities). We see that some of the sin-
gularities generically present in two dimensions appear
at isolated times in three dimensions. Actually, all the
singularities generically present in dimension (d+1) ap-
pear in dimension d on a discrete set of space time, that
is at isolated positions and instants of time. However, it
has been shown in [17] that the irreversible dynamics of
the Burgers equation restricts the set of possible meta-
morphoses. The admissible metamorphoses are charac-
terized by the following property: after the bifurcation,
the singular manifold must remain locally contractible
(homotopic to a point in the neighborhood of the Eu-
lerian location of the metamorphosis). This topological
restriction is illustrated for the one-dimensional case in
figure 3. Note that this constraint actually holds for all
solutions to the Hamilton–Jacobi equation in the limit
of vanishing viscosity, as long as the Hamiltonian is a
convex function.
In order to determine precisely how all these singularities
contribute to the statistical properties of the solution, it
is important to know the local structure of the velocity
(or potential) field in their vicinity. Variousnormal forms
can be obtained from the multiplicity of the minimum
in the variational formulation of the solution (2.5). In
the case without forcing, they can be obtained from a
Taylor expansion of the initial velocity potential. This
will be used in next section to determine the tail of the
probability distribution of a mass density field advected
by a velocity solution to the Burgers equation.
2.4 Remarks on numerical methods
All the traditional methods used to solve equations of
fluid dynamics, or more generally any partial differen-
tial equations, can be used to obtain the solutions to the
Burgers equation. However, as we have seen above, the
solution typically has singularities (discontinuities of the
velocity) in the limit of vanishing viscosity. Hence meth-
ods which rely on the smoothness of the solution require
a non-vanishing viscosity, which is introduced either in
an explicit way to ensure stability (as, e.g., for pseudo-
spectral methods) or in an implicit way through the
discretization procedure (as for finite-differences meth-
ods). In both cases the value of the viscosity is deter-
mined from the mesh size and, even in one dimension,
their uses might be very disadvantageous. We will now
demonstrate various numerical methods that allow ap-
proximating the solutions to the Burgers equation di-
rectly in the limit of vanishing viscosity ν → 0.
2.4.1 Finite volumes
The one-dimensional Burgers equation with no forcing is
a scalar conservation law. Its entropic solutions (or vis-
cosity solutions) can thus be approximated numerically
by finite-volume methods. Instead of constructing a dis-
crete approximation of the solution on a grid, such meth-
ods consist in considering an approximation of its mean
value on a discrete partitioning of the system into finite
volumes. One then needs to evaluate for each of these
volumes the fluxes exchanged with each of its neighbors.
Various approximations of these fluxes were introduced
by Godunov, Roe, and Lax and Wendroff (see, e.g., [35],
Vol. 3, for a review). These methods require to dicretize
both space and time. The time step being then related
to the spatial mesh size by a Courant–Friedrichs–Lewy
type condition. Thus to integrate the equation during
times comparable to one eddy turnover time, they re-
quire a computational time O(N2) where N is the res-
olution. As we now show there actually exist numerical
schemes that allow constructing the solution to the de-
caying Burgers equation for arbitrary times without any
need to compute the solution at intermediate times.
2.4.2 Fast Legendre transform
As we have seen in section 2.1, the solution to the un-
forced Burgers equation is given explicitly by the varia-
tional principle (2.9). A method based on the idea of us-
ing this formulation together with a monotonicity prop-
erty of the Lagrangian map ~x0 → ~x = ~X(~x0, t) was
given in [94]. It is called the fast Legendre transform
whose principles were already sketched in [23]. Both Eu-
lerian and Lagrangian positions are discretized on reg-
ular grids. Then, for a fixed Eulerian location ~x(i) on
the grid, one has to find the corresponding Lagrangian
coordinate ~x
0 maximizing (2.9). A naive implementa-
tion would require O(NdEN
L) operations if the Eulerian
and the Lagrangian grids contain NdE and N
L points re-
spectively. Actually the number of operations can be re-
duced to O((NE lnNL)
d) by using the monotonicity of
the Lagrangian map, that is the fact that for any pair
of Lagrangian positions ~x
0 and ~x
0 , one has at any
time [ ~X(~x
0 , t) − ~X(~x
0 , t)] · (~x
0 − ~x
0 ) ≥ 0. In the
case of orthogonal grids, this property allows perform-
ing the maximization by exploring along a binary tree
the various possibilities; thus the number of operations
is reduced to lnNL for each of the NE positions on the
Eulerian grid. Such algorithms give access to the solu-
tion not only directly in the limit of vanishing viscosity
but also by jumping directly from the initial time to an
arbitrary time.
This method can also be used for the forced Burgers
equation, approximating the forcing by a sum of im-
pulses at discrete times and letting the solution decay
between two such kicks. This gives an efficient algorithm
for the forced Burgers equation directly applicable in the
limit of vanishing viscosity.
2.4.3 Particle tracking methods
In one dimension, Lagrangian methods can be imple-
mented in a straightforward manner after noticing that
particles cannot cross each other and that it is advis-
able to track not only fluid particles but also shocks (see,
e.g., [6]). Lagrangian methods can in principle be used
to solve the Burgers equation in any dimension. How-
ever the shock dynamics is meaningful only for poten-
tial solutions. Outside the potential framework almost
nothing is known about the construction of the solution
beyond the first crossing of trajectories. In the potential
case, a particle method can be formulated by choosing
to represent the solution in the position-potential (~x,Ψ)
space instead of the position-velocity (~x,~v) space. An
idea in two dimensions, which was not yet implemented,
consists in considering a meshing of the hyper-surface
defined by the velocity potential. If such a mesh contains
only triple points, such singularities are preserved by the
dynamics and can be tracked using the results discussed
below in section 4.2 and by checking at all time steps in
an exhaustive manner at all the metamorphoses encoun-
tered by triple points.
3 Decaying Burgers turbulence
We focus in this section on the solutions to the d-
dimensional unforced potential Burgers equation
∂t~v + ~v ·∇~v = ν∇2~v, ~v(~x, 0)=~v0(~x)=−∇Ψ0(~x). (3.1)
As showed in section 2.1, the solution can be expressed
in the limit of vanishing viscosity ν → 0 in terms of a
variational principle that relates the velocity potential
at time t to its initial value:
Ψ(~x, t) = max
Ψ0(~x0) −
|~x− ~x0|2
. (3.2)
The next subsection describes several geometrical con-
structions of the solution that are helpful to determine
various statistical properties of the decaying prob-
lem (3.1). This is illustrated in subsections 3.2 and 3.3
which are devoted to the study of the decay of smooth
homogeneous and of Brownian initial data, respectively.
The study of the solutions to the Burgers equation trans-
porting a density field is of particular interest in the ap-
plication of the Burgers equation in cosmology within
the framework of the adhesion model. This question will
be discussed in section 4.
3.1 Geometrical constructions of the solution
3.1.1 The potential Lagrangian manifold
The variational formulation of the solution (3.2) has
a simple geometrical interpretation in the position-
potential space (~x,Ψ) of dimension d + 1. Indeed, con-
sider the d-dimensional manifold parameterized by the
Lagrangian coordinate ~x0 and defined by
~x = ~x0 − t∇Ψ0(~x0)
Ψ = Ψ0(~x0) −
|∇Ψ0(~x0)|2.
(3.3)
The first line corresponds to the position where the gra-
dient of the argument of the maximum function in (3.2)
vanishes while the second line is just its argument eval-
uated at the maximum. For a sufficiently regular initial
potential Ψ0 (at least twice differentiable) and for suf-
ficiently small times, equation (3.3) unambiguously de-
fines a single-valued function Ψ(~x, t). However, there ex-
ists generically a time t⋆ at which the manifold is folding.
Figure 4(a) (upper) shows in one space dimension the
typical shape of the Lagrangianmanifold defined by (3.3)
after the critical time t⋆. For some Eulerian positions ~x,
there is more than one branch and cusps are present at
Eulerian locations where the number of branches change.
The situation is very similar in higher dimensions as il-
lustrated for d = 2 in figure 4(b). Clearly from the varia-
tional principle (2.9), the correct solution to the inviscid
Burgers equation is obtained by taking the maximum,
that is the highest branch. The velocity potential is by
construction always continuous but it contains angular
points corresponding to discontinuities of the velocity
~v = −∇Ψ. Such singularities are located at Eulerian lo-
cations where the maximum in (2.9) is degenerate and
attained for different ~x0. As already discussed in sec-
tion 2.3 the different singularities appearing in the solu-
tions can be classified in any dimension.
Below we describe other geometrical constructions of the
solutions to the decaying Burgers equation in the limit
of vanishing viscosity that are based on the variational
principle (2.9).
3.1.2 The velocity Lagrangian manifold
In one dimension, when the velocity field is always po-
tential, the method based on the study of the poten-
tial manifold in the (x,Ψ) space described above can
be straightforwardly extended to the position-velocity
phase space. Consider the Lagrangian manifold defined
x = x0 − t v0(x0)
v = v0(x0).
(3.4)
Maxwell
rule
Fig. 4. (a) Lagrangian manifold for d = 1 in the (x,Ψ)
plane (upper) and in the (x, v) plane (lower); the heavy lines
correspond to the correct Eulerian solutions. (b) Lagrangian
manifold in the (~x,Ψ) space for d = 2.
The regular parts of the graph of the solution are nec-
essarily contained in this manifold. However, for times
larger than t⋆, folding appears and the naive solution
would be multi-valued. To construct the true solution
one should find a way to choose among the different
branches. In one dimension, there is a simple relation
between the potential Lagrangian manifold in the (x,Ψ)
plane and those of the (x, v) plane defined by (3.4):
the potential manifold is obtained by taking the “multi-
valued integral” that can be defined by transforming the
spatial integral into an integral with respect to the arc
length. The maximum representation (2.9) implies that
the velocity potential is continuous. Hence a shock cor-
responds to an Eulerian position x where two points be-
longing to different branches define equal areas in the
(x, v) plane. In the case of a single loop of the manifold,
this is equivalent to applying the Maxwell rule to deter-
mine the shock position (see figure 4(a) - lower). This
construction of the solution can become rather involved
as soon as the number of shocks becomes large or that
several mergers have taken place. For the moment there
xshock
interval
regular
points
Φ, Φc
Fig. 5. Convex hull construction in terms of the Lagrangian
potential (a) for d = 1 and (b) for d = 2.
is no generalization to dimension higher than one of this
Maxwell rule construction of the solution. For such an
extension, one needs to develop a geometrical framework
to describe the Lagrangian manifold in the (~x,~v) space.
Such approaches would certainly shed some light on the
problem of constructing non-potential solutions to the
Burgers equation in the limit of vanishing viscosity.
3.1.3 The convex hull of the Lagrangian potential
Another geometrical construction of the solution, which
is valid in any dimension makes use of the Lagrangian
potential
Φ(~x0, t) = tΨ0(~x0) −
|~x0|2
. (3.5)
Clearly, the negative gradient of the Lagrangian poten-
tial gives the naive Lagrangian map
~X(~x0, t) = −∇~x0Φ(~x0, t) = ~x0 + t~v0(~x0), (3.6)
that is satisfied by Lagrangian trajectories as long as
they do not enter shocks. The maximum formulation of
the solution (2.9) can be rewritten as
tΨ(~x, t) +
|~x|2
= max
(Φ(~x0) + ~x0 · ~x), (3.7)
which represents the potential as, basically, a Legendre
transform of the Lagrangian potential. An important
property of the Legendre transform is that the right-
hand side. of (3.7) is unchanged if the Lagrangian po-
tential Φ is replaced by its convex hull, that is the in-
tersection of all the half planes containing its graph. In
other terms, the convex hull Φc of the Lagrangian po-
tential Φ is defined as Φc(~x0, t) = inf g(~x0), where the
infimum is taken over all convex functions g satisfying
g(·) ≥ Φ(·, t). This is illustrated in one dimension in fig-
ure 5(a) which shows both regular points (Lagrangian
points which have not fallen into a shock) and one shock
interval, situated below the segment which is a part of
the convex hull. In two dimensions, as illustrated in fig-
ure 5(b), the convex hull is typically formed by regular
points, by ruled surfaces, and by triangles which corre-
spond, to the regular part of the velocity field, the shock
lines, and the shock nodes, respectively.
Note that in one dimension, there exists an equivalent
construction which is directly based on the Lagrangian
map x0 7→ X(x0, t) defined by (3.6). Working with the
convex hull is equivalent to the Maxwell rule applied
to the non-invertible regions of the Lagrangian map. A
shock corresponds to a whole Lagrangian interval having
a single point as an Eulerian image. One then talks about
a Lagrangian shock interval.
3.1.4 The paraboloid construction
Finally, the maximum representation (3.7) leads in a
straightforwardway to another geometrical construction
of the solution. As illustrated in figure 6 in both one and
two dimensions, a paraboloid with apex at ~x and ra-
dius of curvature proportional to t is moved down in the
(~x0,Ψ0) space until it touches the surface defined by the
initial velocity potential Ψ0 at the Lagrangian location
associated to ~x. The location ~x0 where the paraboloid
touches the graph of the potential is exactly the pre-
image of ~x. If it touches simultaneously at several loca-
tions, a shock is located at the Eulerian position ~x. One
constructs in this way the inverse Lagrangian map.
3.2 Kida’s law for energy decay
An important issue in turbulence is that of the law of
decay at long times when the viscosity is very small.
Before turning to the Burgers equation it is useful to re-
call some of the features of decay for the incompressible
Navier–Stokes case. It is generally believed that high-
Reynolds number turbulence has universal and non-
trivial small-scale properties. In contrast, large scales,
important for practical applications such as transport of
heat or pollutants, are believed to be non-universal. This
is however so only for the toy model of turbulence main-
tained by prescribed large-scale random forces. Very
high-Reynolds number turbulence, decaying away from
its production source, and far from boundaries can relax
Fig. 6. Paraboloid construction of solution for (a) d = 1 and
(b) d = 2.
under its internal nonlinear dynamics to a (self-similarly
evolving) state with universal and non-trivial statistical
properties at all scales. Von Kármán and Howarth [109],
investigating the decay for the case of high-Reynolds
number homogeneous isotropic three-dimensional tur-
bulence, proposed a self-preservation (self-similarity)
ansatz for the spatial correlation function of the ve-
locity: the functional shape of the correlation function
remains fixed, while the integral scale L(t) grows in
time and the mean kinetic energy E(t) = V 2(t) decays,
both following power laws; there are two exponents
which can be related by the condition that the energy
dissipation per unit mass |Ė(t)| should be proportional
to V 3/L. But an additional relation is needed to actu-
ally determine the exponents. The invariance in time of
the energy spectrum at low wavenumbers, known as the
“permanence of large eddies” [53,84,63] can be used to
derive the law of self-similar decay when the initial spec-
trum E0(k) ∝ kn at small wavenumbers k with n below
a critical value equal to 3 or 4, the actual value being
slope 1/t
Random
position
Fig. 7. Snapshot of solution of decaying Burgers turbulence
at long times when spatial periodicity is assumed.
disputed because of the “Gurbatov phenomenon” (see
the end of this section). One then obtains a law of decay
E(t) ∝ t−2(n+1)/(3+n). (Kolmogorov [79] proposed a law
of energy decay V 2(t) ∝ t−10/7, which corresponds to
n = 4 and used in its derivation the so-called “Loitsyan-
sky invariant”, a quantity actually not conserved, as
shown by Proudman and Reid [100].) When the initial
energy spectrum at low wavenumbers goes to zero too
quickly, the permanence of large eddies cannot be used,
because the energy gets backscattered to low wavenum-
bers by nonlinear interactions. For Navier–Stokes tur-
bulence the true law of decay is then known only within
the framework of closure theories (see, e.g., [84]).
For one-dimensional Burgers turbulence, many of the
above issues are completely settled. First, we observe
that the problem of decay is quite simple if spatial peri-
odicity is assumed. Indeed, all the shocks appearing in
the solution will eventually merge into a single shock per
period, as shown in figure 7. The position of this shock
is random and the two ramps have slope 1/t, as is easily
shown using the parabola construction of subsection 3.1.
Hence, the law of decay is simply E(t) ∝ t−2. Nontrivial
laws of decay are obtained if the Burgers turbulence
is homogeneous in an unbounded domain and has the
“mixing” property (which means, roughly, that correla-
tions are vanishing when the separation goes to infinity).
The number of shocks is then typically infinite but their
density per unit length decreases in time because shocks
are constantly merging. The E(t) ∝ t−2(n+1)/(3+n) law
mentioned above can be derived for Burgers turbulence
from the permanence of large eddies when n ≤ 1 [63].
For n = 0, this t−2/3 law was actually derived by Burg-
ers himself [27].
The hardest problem is again when permanence of
large eddies does not determine the outcome, namely
for n > 1. This problem was solved by Kida [77] (see
also [51,61,63]).
We now give some key ideas regarding the derivation of
Kida’s law of energy decay. We assume Gaussian, homo-
geneous smooth initial conditions, such that the poten-
tial is homogeneous. Note that a homogeneous function
is not, in general, the derivative of another homogeneous
function. Here this is guaranteed by assuming that the
Fig. 8. An initial potential which is everywhere below the
parabola x20/(2t) + Ψ. The probability of such events gives
the cumulative probability to have a potential at time t less
than Ψ.
initial spectrum of the kinetic energy is of the form
E0(k) ∝ kn for k → 0 with n > 1 . (3.8)
This condition implies that the mean square initial po-
tential
k−2E0(k) dk has no infrared (small-k) diver-
gence (the absence of an ultraviolet divergence is guar-
anteed by the assumed smoothness).
A very useful property of decaying Burgers turbulence,
which has no counterpart for Navier–Stokes turbulence,
is the relation
E(t) =
〈Ψ〉 , (3.9)
which follows by taking the mean of the Hamilton–
Jacobi equation for the potential (2.2) in the absence of
viscosity and of a driving force. Hence, the law of energy
decay can be obtained from the law for the mean po-
tential. The latter can be derived from the cumulative
probability of the potential which, by homogeneity, does
not depend on the position. By (2.9), its expression at
x = 0 is
Prob{Pot.<Ψ}=Prob
∀x0, Ψ0(x0)<
. (3.10)
Expressed in words, we want to find the probability
that the initial potential does not cross the parabola
x20/(2t) + Ψ (see figure 8). Since, at large times t, the
relevant Ψ is going to be large, the problem becomes
that of not crossing a parabola with small curvature and
very high apex. The crossings, more precisely the up-
crossings, are spatially quite rare and, for large t, form
a Poisson process [92] for which
Prob. no crossing ≃ e−〈N(t)〉, (3.11)
where 〈N(t)〉 is the mean number of up-crossings. By
the Rice formula (a consequence of the identity δ(λx) =
(1/|λ|)δ(x)),
〈N(t)〉=
〈∫ +∞
dx0 δ(m(x0)−Ψ)
, (3.12)
where H is the Heaviside function and
m(x0) ≡ Ψ0(x0) −
. (3.13)
Since Ψ0(x0) is Gaussian, the right-hand side of (3.12)
can be easily expressed in terms of integrals over the
probability densities of Ψ0(x0) and of dΨ0(x0)/dx0 (as
a consequence of homogeneity these variables are uncor-
related and, hence, independent). The resulting integral
can then be expanded by Laplace’s method for large t,
yielding
〈N(t)〉 ∼ t1/2Ψ−1/2e−Ψ
, t→ ∞. (3.14)
When this expression is used in (3.11) and the result is
differentiated with respect to Ψ to obtain the probability
density function (PDF) of p(Ψ), the latter is found to
be concentrated around Ψ⋆ = (ln t)
1/2. It then follows
that, at large times, we obtain Kida’s log-corrected 1/t
law for the energy decay
〈Ψ〉 ∼ (ln t)1/2, E(t) ∼
t(ln t)1/2
, L(t) ∼
. (3.15)
The Eulerian solution, at large times, has the ramp
Fig. 9. The Eulerian solution at large times t. The ramps
have slope 1/t. In time-independent scales, the figure would
be stretched horizontally and squeezed vertically by a factor
proportional to t.
structure shown in figure 9 with shocks of typical
strength V (t) = E1/2(t), separated by a distance L(t).
The fact that Kida’s law is valid for any n > 1, and not
just for n ≥ 2 as thought originally, gives rise to an inter-
esting phenomenon now known as the “Gurbatov effect”:
if 1 < n < 2 the large-time evolution of the energy spec-
trum cannot be globally self-similar. Indeed, the perma-
nence of large eddies, which is valid for any n < 2 dic-
tates that the spectrum should preserve exactly its initial
n behavior at small wavenumbers k, with a constant-
in-time Cn. Global self-similarity would then imply a
t−2(n+1)/(3+n) law for the energy decay, which would
contradict Kida’s law. Actually, as shown in [63], there
are two characteristic wavenumbers with different time
dependences, the integral wavenumber kL(t) ∼ (L(t))−1
and a switching wavenumber ks(t) below which holds
the permanence of large eddies. It was shown that the
same phenomenon is present also in the decay of a pas-
sive scalar [45]. Whether or not a similar phenomenon is
present in three-dimensional Navier–Stokes incompress-
ible turbulence, or even in closure models, is a contro-
versial matter [44,97].
For decaying Burgers turbulence, if we leave aside the
Gurbatov phenomenon which does not affect energy-
carrying scales, the following may be shown. If we rescale
distances by a factor L(t) and velocity amplitudes by a
factor V (t) = E1/2(t) and then let t → ∞, the spatial
(single-time) statistical properties of the whole random
velocity field become time-independent. In other words,
there is a self-similar evolution at large times. Hence,
dimensionless ratios such as the velocity flatness
F (t) ≡
v4(t)
[〈v2(t)〉]2
(3.16)
have a finite limit as t → ∞. A similar property holds
for the decay of passive scalars [28]. We do not know if
this property holds also for Navier–Stokes incompress-
ible turbulence or if, say, the velocity flatness grows with-
out bounds at large times.
3.3 Brownian initial velocities
Initial conditions in the Burgers equation that are Gaus-
sian with a power-law spectrum ∝ k−α have been fre-
quently studied because they belong in cosmology to the
class of scale-free initial conditions (see [98,34]). We con-
sider here the one-dimensional case with Brownian mo-
tion as initial velocity, corresponding to α = 2.
Brownian motion is continuous but not differentiable;
thus, shocks appear after arbitrarily short times and are
actually dense (see figure 10(a)). Numerically supported
conjectures made in [104] have led to a proof by Sinai
[105] of the following result: in Lagrangian coordinates,
the regular points, that is fluid particles which have not
yet fallen into shocks, form a fractal set of Hausdorff
dimension 1/2. This implies that the Lagrangian map
forms a Devil’s staircase of dimension 1/2 (see figure 11).
Note that when the initial velocity is Brownian, the La-
grangian potential has a second space derivative delta-
correlated in space; this can be approximately pictured
as a situation where the Lagrangian potential has very
wild oscillations in curvature. Hence, it is not surprising
that very few points of its graph can belong to its convex
hull (see figure 10(b)).
We will now highlight some aspects of Sinai’s proof of
this result. The idea is to use the construction of the
solution in terms of the convex hull of the Lagrangian
potential (see section 3.1), so that regular points are ex-
actly points where the graph of the Lagrangian poten-
tial coincides with its convex hull. For this problem, the
Hausdorff dimension of the regular points is also equal
to its box-counting dimension, which is easier to deter-
mine. One obtains it by finding the probability that a
−5 −2.5 0 2.5 5
t = 0
t = 1
Fig. 10. Snapshot of the solution resulting from Brownian
initial data in one dimension. (a) Velocity profile at initial
time t = 0 and at time t = 1; notice the dense proliferation
of shocks. (b) Lagrangian potential together with its convex
hull.
Fig. 11. The Lagrangian map looks like a Devil’s staircase:
it is constant almost everywhere, except on a fractal Can-
tor-like set (from [107]).
small Lagrangian interval of length ℓ contains at least
one regular point which belongs simultaneously to the
graph of the Lagrangian potential Φ and to its convex
hull. In other words, one looks for points, such asR, with
the property that the graph of Φ lies below its tangent
at R (see figure 12). Following Sinai, this can be equiva-
lently formulated by the box construction with the fol-
lowing constraints on the graph:
Left: graph of the potential below the half line Γ−,
Right: graph of the potential below the half line Γ+,
1 : enter (AF ) with a slope larger
than that of Γ− by O(ℓ
2 : exit (CB) with a slope less than
that of Γ+ by O(ℓ
3 : cross (FC) and stay below (ED).
It is obvious that such conditions ensure the existence of
at least one regular point, as seen by moving (ED) down
parallel to itself until it touches the graph. Note that A
and the slope of (AB) are prescribed. Hence, one is cal-
culating conditional probabilities; but it may be shown
that the conditioning is not affecting the scaling depen-
dence on ℓ.
Φ( )
Fig. 12. The box construction used to find a regular point R
within a Lagrangian interval of length ℓ (from [105,107]).
As the Brownian motion v0(x0) is a Markov process, the
constraints Left, Box and Right are independent and
hence,
P reg. (ℓ) ≡ Prob{regular point in interval of length ℓ}
= Prob{Left}×Prob{Box}×Prob{Right} . (3.17)
The sizes of the box were chosen so that Prob{Box} is
independent of ℓ:
Prob {Box} ∼ ℓ0. (3.18)
Indeed, Brownian motion and its integral have scaling
exponent 1/2 and 3/2, respectively, and the problem
with ℓ≪ 1 can be rescaled into that with ℓ = 1 without
changing probabilities.
It is clear by symmetry that Prob{Left} and Prob{Right}
have the same scaling in ℓ. Let us concentrate on
Prob{Right}. We can write the equation for the half
line Γ+ in the form
Γ+: x0 7→Φ(x′′0 )+δℓ3/2
(x′′0 )+γℓ
(x0−x′′0), (3.19)
where γ and δ are positive O(1) quantities. Hence, intro-
ducing α ≡ x0 − x′′0 , the condition Right can be written
to the leading order as
v0(x0) + γℓ
dx0 + δℓ
3/2 +
> 0, (3.20)
for all α > 0. By the change of variable α = βℓ and use of
the fact that the Brownian motion has scaling exponent
1/2, one can write the condition Right as
(v0(x0) + γ) dx0 > −δ, for all α ∈ [0, ℓ−1]. (3.21)
Without affecting the leading order, one can replace the
Brownian motion by a stepwise constant random walk
with jumps of ±1 at integer x0’s. The integral in (3.21)
has a simple geometric interpretation, as highlighted in
figure 13, which shows a random walk starting from
the ordinate γ and the arches determined by succes-
sive zero-passings. The areas of these arches are denoted
S⋆, S1, ...Sn, S⋆⋆.
Fig. 13. The arches construction which uses the zero-passings
of a random walk to estimate the integral of Brownian motion
(from [105,107]).
It is easily seen that
Prob{Right} ∼ Prob{S1 > 0, S1 + S2 > 0, . . .
S1 + · · · + Sn > 0 }, (3.22)
where n = O(ℓ−1/2) is the number of zero-passings of
the random walk in the interval [0, ℓ−1]. The probability
(3.22) can be evaluated by random walk methods (see,
e.g.,[49], Chap. 12, section 7), yielding
Prob{Right} ∼Prob{n first sums>0}
∝ n−1/2 ∝ ℓ1/4. (3.23)
By (3.17), (3.18) and (3.23), the probability to have a
regular point in a small interval of length ℓ behaves as
ℓ1/2 when ℓ → 0. Thus, the regular points have a box-
counting dimension 1/2.
This rigorous result on the fractal dimension of regular
points served as a basis in [4] for a proof of the bifrac-
tality of the inverse Lagrangian map when the initial
velocity is Brownian. Namely, the moments Mq(ℓ) =
〈(x0(x+ ℓ) − x0(x))〉 behave as ℓτq at small separation
ℓ and the exponents τq experience the phase transition
τq = 2q for q ≤ 1/2 (3.24)
τq = 1 for q ≥ 1/2 (3.25)
At the moment, this is the only rigorous result on the
bifractal nature of the solutions to the Burgers equation
in the case of non-differentiable initial velocity. In par-
ticular, the case of fractional Brownian motion is still
opened.
4 Transport of mass in the Burgers/adhesion
model
In the cosmological application of the Burgers equation,
i.e. for the adhesion model, it is of particular interest
to analyze the behavior of the density of matter, since
the large-scale structures are characterized as regions
where mass is concentrated. This is done by associat-
ing to the velocity field ~v solution to the d-dimensional
decaying Burgers equation (3.1), a continuity equation
for the transport of a mass density field ρ. In Eulerian
coordinates, the mass density ρ satisfies
∂tρ+ ∇ · (ρ~v) = 0 , ρ(~x, 0) = ρ0(~x) . (4.1)
A straightforward consequence of (4.1) and of the for-
mulation of Burgers dynamics in terms of characteristics
~X(~x0, t) is that, at the Eulerian locations where the La-
grangian map is invertible, the mass density field ρ can
be expressed as
ρ(~x, t)=
ρ0(~x0)
J(~x0, t)
, where ~X(~x0, t)=~x,
and J(~x0, t)=det
(∂X i)/(∂x
. (4.2)
Large but finite values of the density will be reached
at locations where the Jacobian J of the Lagrangian
map becomes very small. As we will see in section 4.1,
they contribute a power-law behavior in the tail of the
probability density function of ρ.
The expression (4.2) is no more valid when the Jacobian
vanishes (inside shocks). Then the density field becomes
infinite and mass accumulates on the shock. We will see
in section 4.2 that the evolution of the mass inside the
singularities of the solution can be obtained as the ν →
0 limit of the well-posed viscous problem. Finally, we
will discuss in section 4.3 some of the applications of
the Burgers equation to cosmology, and in particular
how, assuming the dynamics of the adhesion model, the
question of reconstruction of the early Universe from
its present state can be interpreted as a convex optimal
mass transportation problem.
4.1 Mass density and singularities
We give here the proof reported in [54] that in any dimen-
sion large densities are localized near “kurtoparabolic”
singularities residing on space-time manifolds of co-
dimension two. In any dimension, such singularities con-
tribute universal power-law tails with exponent −7/2
to the mass density probability density function (PDF)
p(ρ), provided that the initial conditions are smooth.
In one dimension, the mass density at regular points can
be written as
ρ(X(x0, t), t) =
ρ0(x0)
1 − t[(d2Ψ0)/(dx20)]
. (4.3)
We suppose here that the initial density ρ0 is strictly
positive and that both ρ0 and Ψ0 are sufficiently regu-
lar statistically homogeneous random fields. Large val-
ues of ρ(x, t) are obtained in the neighborhood of La-
grangian positions with a vanishing Jacobian, i.e. where
d2Ψ0(x0)/dx
0 = 1/t. Once mature shocks have formed,
the Lagrangian points with vanishing Jacobian are in-
side shock intervals and thus not regular. The only points
with a vanishing Jacobian that are at the boundary of
the regular points are obtained at the preshocks, that is
when a new shock is just born at some time t⋆. Such
points, that we denote by x⋆0, are local minima of the ini-
tial velocity gradient which have to be negative, so that
the following relations are satisfied:
(x⋆0) =
(x⋆0) = 0,
(x⋆0) < 0 . (4.4)
There is of course an extra global regularity condition
that the preshock Lagrangian location x⋆0 has not been
captured by a mature shock at a time previous to t⋆.
This global condition affects only constants but not the
scaling behavior of p(ρ) at large ρ.
We now Taylor-expand the initial density and the initial
velocity potential in the vicinity of x⋆0. By adding a suit-
able constant to the initial potential, shifting x⋆0 to the
origin and making a Galilean transformation canceling
the initial velocity at x⋆0, we obtain the following “nor-
mal forms” for the Lagrangian potential (3.5) and for
the density
Φ(x0, t)≃
τx20+ζx
0, ρ(X(x0, t), t)≃
τ+12ζx20
, (4.5)
where
t− t⋆
and ζ =
< 0 . (4.6)
The Lagrangian potential bifurcates from a situation
where it has a single maximum at τ < 0 through a de-
generate maximum with quartic behavior at τ = 0, to
a situation where convexity is lost and where it has two
maxima at x±0 = ±
−τ/(4ζ) for τ > 0. As a result of
our choice of coordinates, the symmetry implies that the
convex hull contains a horizontal segment joining these
two maxima (see. figure 14(a)).
, t)
τ < 0
τ = 0
τ > 0
Fig. 14. Normal form of the Lagrangian potential. (a) in
one dimension, in the time-neighborhood of a preshock; at
the time of the preshock (τ = 0), the Lagrangian potential
changes from a single extremum to three extrema and devel-
ops a non-trivial convex hull (shown as a dashed line). (b)
in two dimension, the space neighborhood of a shock ending
point has a structure similar to the spatio-temporal normal
form of a preshock in one dimension when replacing the x0,2
variable by the time τ ; the continuous line is the separatrix
between the regular part and the ruled surface of the convex
hull; the dotted line corresponds to the locations where the
Jacobian of the Lagrangian map vanishes.
We see from (4.5) that the Eulerian density ρ is propor-
tional to x20 in Lagrangian coordinates at t = t⋆. Since
X = −∂x0Φ, the relation between Lagrangian and Eu-
lerian coordinates is cubic, so that at τ = 0, the den-
sity has a singularity ∝ x−2/3 in Eulerian coordinates.
At any time t 6= t⋆, the density remains bounded ex-
cept at the shock position. Before the preshock (τ < 0),
it is clear that ρ < −ρ0/τ , while after (τ > 0), exclu-
sion of the Lagrangian shock interval [x−0 , x
0 ] implies
that ρ < ρ0/(2τ). Clearly, large densities are obtained
only in the immediate space-time neighborhood of the
preshock. More precisely, it follows from (4.5) that hav-
ing ρ(x, t) > µ requires simultaneously
|τ | < ρ0
and |x| < (−12ζ)−1/2
. (4.7)
The tail of the cumulative probability of the density can
be determined from the fraction of Eulerian space-time
where ρ exceeds a given value. This leads to
P>(µ; x, t) = Prob{ρ(x, t)>µ} ≃ C(x, t)µ−5/2, (4.8)
where the constant C can be expressed as
C(x, t) = At
|ζ|−1/2p3(x, t, ζ) dζ, (4.9)
A is a positive numerical constant and p3 designates the
joint probability distribution of the preshock space-time
position and of its “strength” coefficient ζ (see [54] for
details). This algebraic law for the cumulative probabil-
ity implies that the PDF of the mass density has a power-
law tail with exponent −7/2 at large values. Actually
this law was first proposed in [37] for the large-negative
tail of velocity gradients in one-dimensional forced Burg-
ers turbulence, a subject to which we shall come back in
section 7.
In higher dimensions it was shown in [54] that the main
contribution to the probability distribution tail of the
mass density does not stem from preshocks but from
“kurtoparabolic” points. Such singularities (called A3
according to the classification of [62], which is summa-
rized in section 2.3) correspond to locations which be-
long to the regular part of the convex hull of the La-
grangian potential Φ(~x0, t) and where its Hessian van-
ishes. The name kurtoparabolic comes from the Greek
“kurtos” meaning “convex”. These points are located
on the spatial boundaries of shocks and generically form
space-time manifolds of co-dimension 2 (persisting iso-
lated points for d = 2, lines for d = 3, etc.). As in
one dimension, the normal form of such singularities is
obtained by Taylor-expanding in a suitable coordinate
frame the Lagrangian potential to the relevant order
Φ(~x0, t) ≃ ζx40,1+
2≤j≤d
x20,j +βjx
0,1x0,j
, (4.10)
where the different coefficients satisfy inequalities that
ensure that the surface is below its tangent plane at
~x0 = 0. The typical shape of the Lagrangian potential in
two dimensions is shown in figure 14(b). The positions
where the Jacobian of the Lagrangian map vanishes can
be easily determined from this normal form. The convex
hull of Φ and the area where the mass density exceeds the
value µ can also be constructed explicitly. An important
observation is that, in any dimension, the scalar product
of the vector ~y0 = (x0,2, . . . , x0,d) with the vector ~β =
(β2, . . . , βd) plays locally the same role as time does in
the analysis of one-dimensional preshocks.
When µ → ∞, the cumulative probability can be esti-
mated as
P>(µ; x, t) ∝ µ−3/2
︸ ︷︷ ︸
from x0,1
× µ−1
from ~β·~y0
× 1 × · · · × 1
︸ ︷︷ ︸
from other components of ~y0
from time
. (4.11)
The only non-trivial contributions come from x0,1 and
from the component of ~y0 along the direction of ~β, all
the other components and time contributing order-unity
factors. Hence, the cumulative probability P>(µ) is pro-
portional to µ−5/2 in any dimension, so that the PDF of
mass density has a power-law behavior with the univer-
sal exponent −7/2.
As we have seen, the theory is not very different in one
and higher dimension even if kurtoparabolic points are
persistent only in the latter case. This is due to the pres-
ence of a time-like direction in the case d ≥ 2.
4.2 Evolution of matter inside shocks
As we have seen in the previous subsection, the mass
density becomes very large in the neighborhood of kur-
toparabolic points (A3 singularities) corresponding to
the space-time boundaries of shocks. Such singularities
dominate the tail of the mass density probability dis-
tribution and contribute a power-law behavior with ex-
ponent −7/2. However the mass distribution depends
strongly on what happens inside the shocks where the
density is infinite. Indeed, after the formation of the first
singularity a finite fraction of the initial mass gets con-
centrated inside these low-dimensional structures. De-
scribing the mass distribution requires understanding
how matter evolves once concentrated in the shocks. But
before it will be useful to explain briefly the time evolu-
tion of the shock manifold.
4.2.1 Dynamics of singularities
Suppose that ~X(t) denotes the position of a shock at
time t. We suppose this singularity to be of type An1
(see section 2.3), so that at this position, the veloc-
ity field is discontinuous; we denote by ~v1, . . . , ~vn the n
different limiting values it takes at that point. At any
time we generically have n ≤ d + 1 and occasionally
n = d+2 at the space-time positions of shock metamor-
phoses corresponding to instants when two Ad1 singular-
ities merge. We first restrict ourselves to persistent sin-
gularities, meaning that n ≤ d+ 1. In the neighborhood
of ~X(t), it is easily checked that the velocity potential
can be written as
Ψ(~x, t) = Ψ( ~X(t), t) + max
j=1..n
~vj · ( ~X(t) − ~x)
+o(‖~x− ~X(t)‖) . (4.12)
This expansion divides locally the physical space in n
subdomains Ωj where ~vj · ( ~X(t) − ~x) is maximum, i.e.
~y ∈ Ωj ⇔ (~vi − ~vj) · (~y− ~X(t)) ≥ 0, 1 ≤ i ≤ n . (4.13)
Writing the expansion (4.12) amounts to approximating
the velocity potential by a continuous function which is
piecewise linear on the subdomains Ωj . The boundaries
between the Ωj ’s define the local shock manifold. The
maximum in (4.12) ensures that we are focusing on en-
tropic solutions to the Burgers equation (solutions ob-
tained in the limit of vanishing viscosity) and results in
the convexity of the local approximation of the poten-
tial. Note also that the position ~x = ~X(t) of the reference
singular point corresponds by construction to the unique
intersection of all subdomains Ωj . Remember that we
have assumed that locally, the solution does not have
higher-order singularity.
The approximation (4.12) fully describes the local struc-
ture of the singularity. If n = 2, corresponding to ~X(t)
being the position of a simple shock, it is easily checked
from (4.12) that there will actually exist a whole shock
hyper-plane given by the set of positions ~y satisfying
(~v1 − ~v2) · ( ~X(t) − ~y) = 0 . (4.14)
If n > 2, meaning that ~X(t) is an intersection be-
tween different shocks, all the singular manifolds of
co-dimension m ≤ n are present in the expansion and
are given by the set of positions ~y satisfying
~vi1 · ( ~X(t) − ~y) = · · · = ~vim · ( ~X(t) − ~y) , (4.15)
with 1 ≤ i1 < · · · < im ≤ n.
We next apply the variational principle (3.2) in order to
solve the decaying problem between times t and t + δt
with the initial condition given by (4.12). This yields an
approximation of the potential at time t+ δt:
Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t)
+ max
j=1..n
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
. (4.16)
Note that here, δt and ‖~x− ~X(t)‖ are chosen sufficiently
small in a suitable way to ensure that (i) any singularity
of higher co-dimension does not interfere with the posi-
tion of ~X(t) between times t and t+ δt and that (ii) the
subleading terms are always dominated by the kinetic
energy contribution ‖~x− ~y‖2/(2δt).
The two maxima in ~y and in j of (4.16) can be inter-
changed, under the condition that the maximum in ~y is
restricted to the domain Ωj defined in (4.13). The po-
tential at time t+ δt can thus be written as
Ψ(~x, t+ δt) ≃ Ψ( ~X(t), t)
+ max
j=1..n
~y∈Ωj
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
. (4.17)
We next remark that for all ~x, j and ~y, one has
~vj · ( ~X(t) − ~y) −
‖~x− ~y‖2
≤ ~vj · ( ~X(t) − ~x+ δt~vj) −
‖~vj‖2 ,(4.18)
which gives an upper-bound to the maximum over ~y ∈
Ωj in (4.17). Suppose now that the maximum over the
index j is achieved for j = j0. This means that for all
1 ≤ i ≤ n and ~y ∈ Ωi
~vi · ( ~X(t) − ~y) −
‖~x− ~y‖2
≤ max
~z∈Ωj0
~vj0 · ( ~X(t) − ~z) −
‖~x− ~z‖2
≤ ~vj0 · ( ~X(t) − ~x+ δt~vj0) −
‖~vj‖2 . (4.19)
Let Ωi0 be the domain containing the vector (~x− δt~vj0).
Then, (4.19) applied to i = i0 and ~y = ~x− δt~vj0 trivially
implies that
(~vi0 − ~vj0 ) · (~x− δt~vj0 − ~X(t)) ≥ 0 , (4.20)
which together with the definition (4.13) for Ωi0 leads
to i0 = j0. Hence, to summarize, if the first maximum
is reached for j = j0 then the second maximum is nec-
essarily reached for ~y = ~x− δt~vj0 .
It is clear that the approximation (4.16) of the velocity
potential at time t + δt preserves the local structure of
the singular manifold. Indeed, for m ≤ n, the positions
~y satisfying
~v1 · ( ~X(t) − ~y) +
‖~v1‖2 = · · ·
· · ·= ~vm · ( ~X(t) − ~y) +
‖~vm‖2 (4.21)
form a (d−m)-dimensional shock manifold. The trajec-
tory ~X(t) of the reference singular point satisfies
~v1 ·
‖~v1‖2 = · · · = ~vn ·
‖~vn‖2 , (4.22)
which can be rewritten as
‖d ~X/dt− ~v1‖ = · · · = ‖d ~X/dt− ~vn‖ . (4.23)
This gives n relations for the d components of the vector
d ~X/dt. These relations allow determining the normal
velocity of the singular manifold. The tangent velocity
remains undetermined. The velocity of the singularity
located at ~X(t) is completely determined only if n = d,
i.e. for point singularities. For instance when d = 1, the
dynamics of shocks is given by
(u1 + u2) , (4.24)
meaning that they move with a velocity equal to the half
sum of their right and left velocities. For d = 2, only the
positions of triple points (singularities of type A31 corre-
sponding to the intersection of three shock lines) are well
determined. It is easily checked that the two-dimensional
velocity vector d ~X/dt is the circumcenter of the trian-
gle formed by the three limiting values (~v1, ~v2, ~v3) that
are achieved by the velocity field at this position (see
figure 15).
v 3v node
v mass
Fig. 15. Determination of the velocity of a triple point and
of that of the mass inside it when the three limiting values
of the velocity ~v1, ~v2, and ~v3 form an obtuse triangle. The
dash-dotted circle is the circumcircle whose center gives the
velocity of the singularity and the dashed circle is the small-
est circle containing the triangle whose center gives the ve-
locity of mass.
4.2.2 Dynamics of the mass inside the singular mani-
One of the central themes of this review article is a con-
nection between Lagrangian particle dynamics and the
inviscid Burgers equation. In the unforced case the ve-
locity is conserved along particle trajectories minimizing
the Lagrangian action (see section 2). At a given mo-
ment of time, all particles whose trajectories are not min-
imizers have been absorbed by the shocks. In the one-
dimensional case when shocks are isolated points, par-
ticles absorbed by shocks just follow the dynamics of a
shock point. However, in the multi-dimensional case the
geometry of the singular shock manifold can be rather
complicated. This results in a non-trivial particle dy-
namics inside the singular manifold. In other words, the
particle absorbed by shocks have a rich afterlife and
the main problem is to describe their dynamical proper-
ties inside the singular manifold. This problem was ad-
dressed by I. Bogaevsky in [18].
The basic idea is to consider first particle transport by
the velocity field given by smooth solutions to the viscous
Burgers equation. Indeed, let ~vν(x, t) be a solution to
the viscous Burgers equation
ν + (~vν · ∇)~vν = ν∇2~vν −∇F (~x, t).
Then the dynamics of a Lagrangian particle labeled by
its position ~x0 at time t = 0 is described by the system
of ordinary differential equations
~̇Xν(~x0, t) = ~v
ν( ~Xν(~x0, t), t), ~X
ν(~x0, 0) = ~x0, (4.25)
where the dots stand for time derivatives. It is possible to
show that in the inviscid limit ν → 0 solutions to (4.25)
converge to limiting trajectories { ~X(~x0, t)}. These lim-
iting trajectories are not disjoint anymore. In fact, two
trajectories corresponding to different initial positions
~x10 and ~x
0 can merge:
~X(~x10, t
∗) = ~X(~x20, t
∗). This corre-
sponds to absorption of particles by the shock manifold.
Of course, two trajectories coincide after they merge:
~X(~x10, t) =
~X(~x20, t) for t ≥ t∗. Particles which until time
t never merged with any other particles correspond to
minimizers. Such trajectories obviously satisfy the lim-
iting differential equation:
~̇X(~x0, t) = ~v( ~X(~x0, t), t), ~X(~x0, 0) = ~x0, (4.26)
where ~v(x, t) is the entropic solution of the inviscid Burg-
ers equation which is well defined outside of the shock
manifold. However, we are mostly interested in the dy-
namics of particles which have merged with other par-
ticles and thus were absorbed by shocks. One can prove
that for such trajectories one-sided time derivatives exist
~X(t) = lim
∆t→0+
~X(t+ ∆t) − ~X(t)
(4.27)
and satisfy a “one-sided” differential equation:
~X(t) = ~v(s)( ~X(t), t). (4.28)
Here ~v(s)(·, t) is the velocity field on the shock manifold
(index s stands for shocks). It turns out that ~v(s)(~x, t)
and the corresponding shock trajectories satisfy a vari-
ational principle, described hereafter. Denote by Ψ(~x, t)
a potential of the viscous velocity field ~v(~x, t): ~v(~x, t) =
−∇Ψ(~x, t). As we have pointed out many times be-
fore,−Ψ(~x, t) corresponds to a minimum Lagrangian ac-
tion among all the Lagrangian trajectories which pass
through point ~x at time t. Shocks correspond to a situa-
tion where the minimum is attained for several different
trajectories. Correspondingly, one has several smooth
branches such that Ψ(~x, t) = Ψi(~x, t), 1 ≤ i ≤ k. Sup-
pose a particle moves from a point of shock (~x, t) with
a velocity ~v. Then at infinitesimally close time t+ δt its
position will be ~x+~vδt. In linear approximation (see pre-
vious subsection) the Lagrangian action of this infinites-
imal piece of trajectory is equal to [|~v|2/2 − F (~x, t)]δt.
Of course, the action minimizing trajectory at the point
(~x + ~vδt, t + δt) does not pass through a shock point
(~x, t). Hence, the minimum action −Ψ(~x + ~vδt, t + δt)
is smaller than −Ψ(~x, t) + [‖~v‖2/2 − F (~x, t)]δt for any
velocity ~v. However, we can put a variational condition
on ~v which requires the difference between −Ψ(~x, t) +
[‖~v‖2/2 − F (~x, t)]δt and −Ψ(~x + ~vδt, t + δt) to be as
small as possible. This is exactly the variational principle
which determines the velocity ~v = ~v(s)(~x, t) at a shock
point. It is easy to see that in linear approximation
Ψ(~x+~vδt, t+δt) = max
1≤i≤k
[Ψi(~x+~vδt, t+δt)]
= Ψ(~x, t) − min
1≤i≤k
[−∇Ψi(~x, t) · ~v − ∂tΨi(~x, t)] δt. (4.29)
Let us denote by ~vi the limiting velocities −∇Ψi(~x, t)
at the shock point (~x, t). Then, using Hamilton–Jacobi
equation for the velocity potential
∂tΨi(~x, t) =
‖∇Ψi(~x, t)‖2 + F (~x, t)
‖~vi‖2 + F (~x, t) (4.30)
we have
Ψ(~x+ ~vδt, t+ δt) = Ψ(~x, t) −
− min
1≤i≤k
~vi ·~v −
‖~vi‖2
δt−F (~x, t)δt. (4.31)
Hence, the difference of actions can be written as
∆A=−Ψ(~x, t)+ 1
‖~v‖2δt+ Ψ(~x+~vδt, t+δt)
‖~v‖2δt− min
1≤i≤k
~vi · ~v −
‖~vi‖2
1≤i≤k
‖~v − ~vi‖2δt. (4.32)
Obviously minimization of ∆A over ~v corresponds to a
center of a minimum ball covering ~vi. It implies that
such a center gives the velocity ~v(s)(~x(t), t) of particles
concentrated at a shock point (~x, t). It is interesting that
this variational principle implies that a particle absorbed
by a shock cannot leave the singular shock manifold in
the future.
Let us now consider the first nontrivial generic exam-
ple of a shock point, namely a triple point in two di-
mensions d = 2. The point ( ~X(t), t) is thus the intersec-
tion of three shock lines. In this case there are exactly
three smooth branches Ψi(·, t) with limiting velocities
~vi = −∇Ψi, 1 ≤ i ≤ 3. As we have seen in previous sec-
tion the motion of the triple point is determined by con-
tinuity of the velocity potential at ( ~X, t). The “geomet-
rical velocity” d ~X/dt of the triple point is then the cir-
cumcenter of the triangle formed by the three velocities
~v1, ~v2, ~v3. It is easy to see that d ~X/dt = ~v
(s) only in the
case when the vectors ~v1, ~v2, and ~v3 form an acute trian-
gle. If so, a cluster of particles follows the triple point. In
the opposite case when the triangle is obtuse, the parti-
cles leave the node. Such a situation is presented in fig-
ure 15, where the mass leaves the node along the shock
line delimiting the values ~v1 and ~v3 of the velocity.
The analysis presented above was carried out for the
Burgers equation jointly with A. Sobolevskĭı as a part of
ongoing work on a similar theory for the case of a general
Hamilton–Jacobi equation, with a Hamiltonian that is
convex in the momentum variable. The formal extension
of this analysis to the latter case is straightforward and
can be left to the interested reader; however at present a
rigorous justification of it, employing methods of [18], is
known only for the case of H(x, ẋ, t) = a(x, t)|ẋ|2, with
a(x, t) > 0.
4.3 Connections with convex optimization problems
As discussed in section 1.2, Burgers dynamics is known
in cosmology as the adhesion model and frequently used
to understand the formation of the large-scale structures
in the Universe. Recently, this model was used as a basis
for developing new techniques for one of the most chal-
lenging questions in modern cosmology, namely recon-
struction. This problem aims at reconstructing the dy-
namical history of the Universe through the mass den-
sity initial fluctuations that evolved into the distribution
of matter and galaxies which is nowadays observed (see,
e.g., [98]). The main difficulty encountered is that the
velocities of galaxies (the peculiar velocities) are usually
unknown, so that most approaches lead to non-unique
solutions to this ill-posed problem. The reconstruction
technique we present here, which was proposed in [55,25],
is based on the observation that, to the leading order,
the mass is initially uniformly distributed in space (see,
e.g., [98]). This observation, together with the Zeldovich
approximation, leads to a reformulation of the prob-
lem as a well-posed instance of an optimal mass trans-
portation problem between the initial (uniform) and the
present (observed) distributions of mass. More precisely
it amounts to a convex optimization problem related
to the Monge–Ampère equation and dually, as found
by Kantorovich [73], to a linear programming problem.
This is the reason why the name MAK (Monge–Ampère–
Kantorovich) has been proposed for this method in [55].
Namely, one has to find the transformation from initial
to current positions (the Lagrangian map) that maps
the initial density ρ(~x0, 0) = ρ0 to the field ρ(~x, t) which
is nowadays observed. One then use a well-known fact
in cosmology: because of the expansion of the Universe,
the initial velocity field of the self-gravitating matter is
slaved to the initial gravitational field (see, e.g., [25]).
This observation implies that the initial velocity field is
potential and allows one to deduce from it the sublead-
ing fluctuations of the mass density.
The MAK reconstruction technique is based on two cru-
cial assumptions. First the Lagrangian map ~x0 7→ ~x =
~X(~x0, t) is assumed to be potential, i.e. ~X = ∇x0Φ(~x0).
Second, the Lagrangian potential Φ(~x0) is assumed to
be a convex function. As explained in [25] these two hy-
potheses are motivated by the adhesion model (and thus
inviscid Burgers dynamics) where they are trivially sat-
isfied. As we will see later the reverse is actually true: the
potentiality of the Lagrangian map and the convexity of
the potential is equivalent to assuming that the latent ve-
locity field is a solution to the Burgers equation. We will
now see how, under these hypotheses, the reconstruction
problem relates to Monge–Ampère equation. Conserva-
tion of mass trivially implies that ρ(~x, t)d3x = ρ0d
which can be rewritten in terms of the Jacobian matrix
(∂X i)/(∂x
0) as
ρ( ~X(~x0, t), t)
. (4.33)
Potentiality of the Lagrangian map leads to
∂xi0∂x
ρ(∇x0Φ, t)
. (4.34)
The problem with this formulation is that the unknown
potential Φ enters the right-hand side of the equation
in a non-trivial way. Convexity of the Lagrangian po-
tential Φ is next used to reformulate the problem in
term of the inverse Lagrangian map. Indeed, if Φ is con-
vex, the inverse Lagrangian map is also potential, i.e.
~x0 = ~X0(x, t) = ∇xΘ(~x) with the potential Θ itself con-
vex. The two potentials Φ and Θ are moreover related
by Legendre transforms:
Θ(~x) = max
[~x · ~x0 − Φ(~x0)], (4.35)
Φ(~x0) = max
[~x · ~x0 − Θ(~x)]. (4.36)
In terms of the inverse Lagrangian potential Θ the con-
servation of mass (4.34) reads
∂xi∂xj
= ρ(~x, t), (4.37)
which is exactly the elliptic Monge-Ampère equation.
This time, the difficulty expressed above has disappeared
since the unknown potential Θ does not enter the right-
hand side of the equation. Note that we have implic-
itly assumed here that the present distribution of mass
has no singularity. The case of a singular distribution
could actually be treated using a weak formulation of
the Monge-Ampère equation, which amounts to apply-
ing conservation of mass on any subdomain but requires
allowing the inverse Lagrangian map to be multival-
ued. The next step in the design of the MAK method is
to reformulate (4.37) as an optimal transport problem
with quadratic cost. Indeed, as shown in [24], the map
~X(~x0, t) (and its inverse ~X0(~x, t)) minimizing the cost
‖ ~X(~x0, t) − ~x0‖2ρ0 d3x0
‖~x− ~X0(~x, t)‖2ρ(~x, t) d3x, (4.38)
is a potential map whose potential is convex and is the
solution to the Monge–Ampère equation (4.37). This can
be understood using a variational approach as proposed
in [55]. Suppose we perform a small displacement δ ~X0(~x)
of the inverse Lagrangian map ~X0(~x, t) solution of the
optimal transport problem. On the one hand the only ad-
missible displacement are those satisfying the constraint
to map the initial density field ρ0 to the final one ρ(~x, t).
It is shown in [25] that this is equivalent to require that
∇x · [ρ(~x, t)δ ~X0(~x)] = 0. On the other hand one easily
see that the variation of the cost function corresponding
to the variation δx reads
δI = −2
[~x− ~X0(~x, t)] · [ρ(~x, t)δ ~X0(~x)] d3x. (4.39)
This integral can be interpreted as the scalar product
(in the L2 sense) between ~x− ~X0(~x0, t) and ρ(x)δ ~X(~x0).
Hence the optimal solution, which should satisfy δI = 0
for all δ ~X0, is such that the displacement ~x − ~X0(~x0, t)
(or equivalently ~X(~x0) − ~x0) is orthogonal to all
divergence-free vector fields. This means that it is nec-
essarily the gradient of a potential, from which it follows
that ~X(~x0, t) = ∇x0Φ(~x0). Convexity follows from the
observation that the Lagrangian map ~x0 7→ ~X has to
satisfy
(~x0 − ~x′0) · [ ~X(~x0) − ~X(~x′0)] ≥ 0. (4.40)
Indeed, if that was not the case, one can easily check that
any map where the Lagrangian pre-image of a neighbor-
hood of ~x0 and of one of ~x0
are inverted would lead to a
smaller cost. Formulated in terms of potential maps, the
relation (4.40) straightforwardly implies convexity of Φ.
This finishes the proof of equivalence between Monge–
Ampère equation and the optimal transport problem
with quadratic cost.
The goal of reformulating reconstruction as an optimiza-
tion problem is mostly algorithmic. Once discretized,
the problem of finding the optimal map between initial
and final positions amounts is equivalent to solving a so-
called assignment problem. An efficient method to deal
numerically with such problems is based on the auction
algorithm [15] and was used in [25] with data stemming
from N -body cosmological simulations. As summarized
0.2 0.3 0.7 0.80.4 0.5 0.6
simulation coordinate
1 2 3 4
distances
Fig. 16. Test of the MAK reconstruction for a sample of
N ′ = 17, 178 points from a N-body simulation (from [25]).
The scatter diagram plots reconstructed versus true initial
positions. The histogram inset gives the distribution (in per-
centages) of distances between true and reconstructed initial
positions; the horizontal unit is the distance between two
sampled points. The width of the first bin is less than unity to
ensure that only exactly reconstructed points fall in it. More
than sixty percent of the points are exactly reconstructed.
in figure 16, the MAK reconstruction method leads to
very promising results. More than 60% of the discrete
points are assigned to their actual Lagrangianpre-image.
Such a number has to be compared with other recon-
struction methods for which the success rate barely ex-
ceed 40% for the same data set.
Even if the mapping from initial to final positions is
unique, the peculiar velocities are not well defined ex-
cept if we have some extra knowledge of what is happen-
ing at intermediate times 0 ≤ t′ ≤ t. Of course the den-
sity field ρ(~x′, t′) is unknown. However, there are triv-
ial physical requirements. First the two mass transport
problems between 0 and t′ and between t′ and t have
both to be optimal. This means that one looks for two
Lagrangian maps, ~X1 from 0 to t
′ and ~X2 from t
′ to t
which are minimizing the respective costs
‖ ~X1(~x0) − ~x0‖2ρ0 d3x0,
‖~x− ~X−12 (~x)‖
2ρ(~x, t) d3x. (4.41)
The second physical requirement is that the composition
of these two optimal maps have to give the Lagrangian
map between times 0 and t, namely ~X(~x0, t) = ~X2( ~X1).
Under these two conditions there is equivalence between
the optimal transport with a quadratic cost and the
Burgers dynamics supplemented by the transport of a
density field (see [13] for details).
5 Forced Burgers turbulence
5.1 Stationary régime and global minimizer
We consider in this section solutions to the forced Burg-
ers equation. As we have seen in section 2, the solution in
the limit of vanishing viscosity can be expressed at any
time t in terms of the initial condition at time t0 through
a variational principle which consists in minimizing an
action along particle trajectories. The statistically sta-
tionary régime toward which the solution converges at
large time can be studied assuming that the by reject-
ing the initial time t0 is at minus infinity. The solution
is then given by the variational principle
Ψ(~x, t)=−inf
~γ(·)
‖~̇γ(s)‖2−F (~γ(s), s)
, (5.1)
where the infimum is taken over all (absolutely continu-
ous) curves ~γ : (−∞, t] → Ω such that ~γ(t) = ~x. In this
setting, the action is computed over the whole half line
(−∞, t] and the argument of the infimum does not de-
pend anymore on the initial condition. Of course, (5.1)
defines Ψ up to an additive constant. This means that
only the differences Ψ(~x, t)−Ψ(0, t) can actually be de-
fined. A trajectory ~γ minimizing (5.1) is called a one-
sided minimizer. It is easily seen from (5.1) that all the
minimizers are solutions of the Euler–Lagrange equation
~̈γ(s) = −∇F ( ~γ(s), s) , (5.2)
where the dots denote time derivatives. This equation
defines a 2d-dimensional (possibly random) dynamical
system in the position-velocity phase space (~γ, ~̇γ). The
Lagrangian one-sided minimizers ~γ defined over the half-
infinite interval (−∞, t] play a crucial role in the con-
struction of the global solution and of the stationary
régime. Namely, a global solution to the randomly forced
inviscid Burgers equation is given by~v(~x, t) = ~̇γ(t) where
~γ(t) = ~x. To prove that such half-infinite minimizers ex-
ist, one has to take the limit t0 → −∞ for minimizers
defined on the finite time interval [t0, t]. The existence of
this limit follows from a uniform bound on the absolute
value of the velocity |~̇γ| (see, e.g., [38]). Obtaining such
a bound becomes the central problem for the theory, as
we shall now see.
When the configuration space Ω where the solutions live
is compact (bounded), one can expect the velocity of a
minimizer to be uniformly bounded. Indeed, in this case
the displacement of a minimizer for any time interval is
then bounded by the diameter of the domain Ω, so that
action minimizing trajectories cannot have large veloc-
ities. For forcing potential that are delta-correlated in
time, it has been shown by E et al. [38] in one dimension
and by Iturriaga and Khanin [68,69] in higher dimen-
sions that the minimizing problem (5.1) has a unique
solution Ψ with the following properties:
• Ψ is the unique statistically stationary solution to the
Hamilton–Jacobi equation (2.2) in the inviscid limit
ν → 0;
• Ψ is almost everywhere differentiable with respect to
the space variable ~x;
• −∇Ψ uniquely defines a statistically stationary solu-
tion to the Burgers equation in the inviscid limit;
• there exists a unique one-sided minimizer at those Eu-
lerian positions ~x where the potential Ψ is differen-
tiable; the locations where Ψ is not differentiable cor-
respond to shocks.
• There exists a unique minimizer ~γ(g) that minimizes
the action calculated from −∞ to any time t. It is
called the global minimizer (or two-sided minimizer)
and corresponds to the trajectory of a fluid particle
that is never absorbed by shocks. Moreover, all one-
sided minimizers are asymptotic to it as s→ −∞.
All the properties above follow from the variational ap-
proach. In fact, the variational principle (2.12) imply
similar statements in the viscous case. Of course, when
viscosity is positive the unique statistically stationary
solution is smooth. However, one can show that the sta-
tionary distribution corresponding to such solutions con-
verges to inviscid stationary distribution in the limit
ν → 0 [58]. Although the variational proofs are concep-
tual, general and simple, they are based on the fluctua-
tion mechanism and therefore do not give a good control
of the rate of convergence to the statistically stationary
regime. Exponential convergence would follow from the
hyperbolicity of the global minimizer. Although one ex-
pects hyperbolicity holds in any dimension, mathemat-
ically it is an open problem. At present a rigorous proof
of hyperbolicity is only available in dimension one [38].
The assumption of compactness of the configuration
space Ω is essential in the construction of the stationary
régime. As we will see in subsection 5.4, the situation
is much more complex in the non-compact case when
for instance the solution is defined on the whole space
Ω = Rd.
5.2 Topological shocks
To introduce the notion of topological shock we first fo-
cus on the one-dimensional case in a periodic domain,
i.e. in Ω = T = R/Z. If we “unwrap” at a given time t
the configuration space to its universal cover R (see fig-
ure 17(a)), we then obtain an infinite number of global
minimizer γ
k , which at all time s ≤ t satisfy γ
k+1(s) =
k (s) + 1. All the one-sided minimizers converge back-
ward in time to one of these global minimizers. The topo-
logical shock (or main shock) is defined as the set of x
positions giving rise to several minimizers approaching
two successive replicas of the global minimizer. This par-
ticular shock is also the only shock that has existed for
all times.
This construction can easily be extended to higher
dimensions (see [10]). For this we unwrap the d-
dimensional torus Td to its universal cover, the full
space Rd (see figure 17(b) for d = 2). Then, the different
replicas of the periodic domain define a lattice of global
minimizers ~γ
parameterized by integer vectors ~k. The
backward-in-time convergence on the torus of the one-
sided minimizers to the global minimizer implies that
a minimizer associated to a location ~x in Rd at time t
will be asymptotic to one of the global minimizer ~γ
the lattice. Hence, every position ~x which has a unique
one-sided minimizer is associated to an integer vector
~k(~x). This defines a tiling of space at time t. The tiles
O~k are the sets of points whose associated one-sided
minimizers are asymptotic to the ~k-th global minimizer.
The boundaries of the O~k’s are the topological shocks.
They are the locations from which at least two one-
sided minimizers approach different global minimizers
on the lattice. Indeed, a point where two tiles O~k1 and
O~k2 meet, has at least two one-sided minimizers, one
of which is asymptotic to ~γ
and another to ~γ
course, there are also points on the boundaries where
three or more tiles meet and thus where more than two
one-sided minimizers are asymptotic to different global
minimizers. For d = 2 such locations are generically iso-
lated points corresponding to the intersections of three
or more topological shock lines, while for d = 3, they
form edges and vertices where shock surfaces meet. Note
Fig. 17. Space-time sketch of the unwraping of the periodic
domain Td to the whole space Rd for d = 1 (a) and d = 2 (b).
that, generically, there exist other points inside O~k with
several minimizers. They correspond to shocks of “lo-
cal” nature because at these locations, all the one-sided
minimizers are asymptotic to the same global minimizer
and hence, to each other. In terms of Lagrangian
dynamics, the topological shocks play a role dual to that
of the global minimizer. Indeed, all the fluid particles
are converging backward-in-time to the global mini-
mizer and are absorbed forward-in-time by the topo-
logical shocks. For the transportation of mass when we
assume that the Burgers equation is supplemented by
a continuity equation for the mass density, all the mass
concentrate at large times in the topological shocks.
The global structure of the topological shocks is related
to the various singularities generically present in the so-
lution to the Burgers equation that were detailed in sec-
tion 2.3. Generically there are no locations associated to
more than (d+1) minimizers. As one expects to see only
generic behavior in a random situation, the probability
to have points with more than (d + 1) one-sided mini-
mizers is zero. It follows that there are no points where
(d+ 2) tiles O~k meet, which is an important restriction
on the structure of the tiling. For d = 2 it implies that
the tiling is constituted of curvilinear hexagons. Indeed,
suppose each tile O~k is a curvilinear polygon with s ver-
Fig. 18. (a) Position of the topological shock on the torus;
the two triple points are represented as dots. (b) Snapshot
of the velocity potential ψ(x, y, t) for d = 2 in the statisti-
cal steady state, obtained numerically with 2562 grid points.
Shock lines, corresponding to locations where ψ is not dif-
ferentiable, are represented as black lines on the bottom of
the picture; the four gray areas are different tiles separated
by the topological shocks; the other lines are local shocks.
tices corresponding to triple points. For a large piece
of the tiling that consists of N tiles, the total num-
ber of vertices is nv ∼ sN/3 and the total number of
edges is ne ∼ sN/2. The Euler formula implies that
1 = nv − ne +N ∼ (6− s)N/6, and we necessarily have
s = 6, corresponding to an hexagonal tiling. As shown
in figure 18(a), this structure corresponds on the peri-
odicity torus T2, to two triple points connected by three
shock lines that are the curvilinear edges of the hexagon
O~0. The connection between the steady-state potential
and the topological shocks is illustrated numerically on
figure 18(b). The different tiles covering the periodic do-
main were obtained by tracking backward in time fluid
particle trajectories and by determining to which peri-
odic image of the global minimizer they converge.
In dimensions higher than two, the structure of topolog-
ical shocks is more complicated. For instance it is not
possible to determine in a unique manner the shape of
the polyhedra forming the tiling. However, it has been
shown by Matveev [87] that for d = 3 the minimum poly-
hedra forming such tiling has 24 vertices and 36 edges
and is composed of 8 hexagons and 6 rectangles (see fig-
ure 19). It is of interest to note that the structure of
topological shocks is in direct relation with the notions
of complexity and minimum spines of manifolds intro-
duced by Matveev from a purely topological viewpoint.
Fig. 19. Sketch of the simplest configuration of the topolog-
ical shock in dimension d = 3.
Algebraic characterization of the topological shock
In two dimensions, when periodic boundary conditions
are considered, very strong constraints are imposed on
the structure of the solution. In particular, the topol-
ogy of the torus T2 imply that the topological shocks
generically form a periodic tiling of R2 with curvilinear
hexagons. However, this tiling can be of various alge-
braic types. Consider the tile O~0 surrounded by its six
immediate neighbors O~ki , where the integer vectors
are labeled in anti-clockwise order, ~k1 having the small-
est polar angle (see figure 20). It is easily seen that the
periodicity of the tiling implies
~k3 = ~k2 − ~k1, ~k4 = −~k1, ~k5 = −~k2
and ~k6 = ~k1 − ~k2, (5.3)
so that the whole information on the algebraic struc-
ture of the tiling is contained in the vectors ~k1 and ~k2
which form a matrix S from the group SL(2,Z) of 2× 2
integer matrices with unit determinant. The matrix S
gives information on the number of times each shock line
turns around the torus before reconnecting to another
triple point. Figure 18(a) corresponds to the simplest
case when S is the identity matrix. When the forcing is
stochastic, the matrix S is random and stationary solu-
tions to the two-dimensional Burgers equation define a
stationary distribution on SL(2,Z).
(i, j)
(i−1, j+1)
(i+1, j)
(i+1, j−1)
(i, j+1)
(i−1, j)
(i, j−1)
Fig. 20. The algebraic structure of the topological shock in
dimension d = 2 is determined by the indexes corresponding
to immediate neighbors of the tiling considered.
Certainly, topological shocks evolve in time and may
change their algebraic structure. This happens through
bifurcations (or metamorphoses) described in section
2.3. In two dimensions, the generic mechanism which
transforms the algebraic structure of topological shocks
is the merger of two triple points. This metamorphosis
is called the flipping bifurcation and corresponds to the
appearance at time t⋆ of an A
1 singularity in the solu-
tion associated to a position with four minimizers. The
mechanism transforming the algebraic structure of the
topological shock is illustrated in figure 21. Issues such
as the minimum number of flips needed to transform the
matrix S1 associated to the algebraic structure of the
topological shock to another matrix S2 are discussed in
in [1].
Fig. 21. Sketch of the tiling before, at the flipping time t∗ and
after it. This example corresponds to a bifurcation from the
matrix S1 = [
1] to S2 = [
2]. The dashed boxes represent
the periodicity domain [0, 1]2.
5.3 Hyperbolicity of the global minimizer
The nature of the convergence to a statistical steady
state is determined by the local properties of the global
minimizer. The hyperbolicity of this action-minimizing
trajectory implies an exponential convergence, so that
the global picture of the solution is reached very rapidly,
after just a few turnover times.
Since the trajectory of the global minimizer is unique
and can be extended to arbitrary large times, it corre-
sponds to an ergodic invariant measure for the stochastic
flow defined by the Euler–Lagrange equation (5.2). Con-
ditioned by the random force, this measure is simply the
delta measure sitting at the location (~γ(g)(0), ~̇γ(g)(0)).
By the Oseledets ergodic theorem (see, e.g. [98]), 2d non-
random Lyapunov exponents can be associated to the
global minimizer trajectory. Since the flow is symplectic
these non-random exponents come in pairs with oppo-
site signs. That is
λ1 ≥ · · · ≥ λd ≥ 0 ≥ −λd ≥ · · · ≥ −λ1 . (5.4)
Hyperbolicity is defined as the non-vanishing of all these
exponents. Thus, the issue of hyperbolicity can be ad-
dressed in terms of the backward-in-time convergence of
the one-sided minimizers to the global one or, better, in
terms of forward-in-time dynamics. In the latter case,
this amounts to looking how fast Lagrangian fluid par-
ticles are absorbed by shocks. For this we consider the
set Ωreg(T ) of locations ~x such that the fluid particle
emanating from ~x at time t = 0 survives, i.e. is not ab-
sorbed by any shock, until the time t = T . The long-time
shrinking of Ωreg as a function of time is asymptotically
governed by the Lyapunov exponents. To ensure the ab-
sence of vanishing Lyapunov exponents, it is sufficient to
show that the diameter of Ωreg(T ) decays exponentially
as T → +∞.
In one dimension, it has been shown in [38] that this is
indeed the case, and particularly that there exists posi-
tive constants α, β, A and B such that
diamΩreg(T ) ≥ Ae−αT
≤ Be−βT . (5.5)
Unfortunately this proof of hyperbolicity is purely one-
dimensional and at present time there is no extension of
this result to higher dimensions.
In two dimensions, the behavior of diamΩreg(T ) at large
times was studied numerically in [10] by using the fast
Legendre transform described in section 2.4 and a forc-
ing that is a sum of independent random impulses con-
centrated at discrete times. The ideas of this numerical
method are related to the Lagrangian structure of the
flow. This easily permits to track numerically the set Ωreg
of regular Lagrangian locations. As seen from figure 22,
the diameter of this set decays exponentially fast in time
for three different types of forcing, providing good ev-
idence of the hyperbolicity of the global minimizer for
d = 2.
0 0.25 0.5 0.75 1
time T
Algebraic [∝ k−3]
Gaussian [∝ exp (−k2)]
Step [= Const. if k ≤3]
Fig. 22. Time evolution of the diameter of the Lagrangian
set Ω(T ) (points corresponding to the regular region) for
three different types of forcing spectrum; average over 100
realizations and with 2562 grid points (from [10]).
Hyperbolicity of the global minimizer implies existence
at any time t of two d-dimensional smooth manifolds
u x,t( )
( )tΓ(u)
global
minimizer
shock
a preshock
Fig. 23. Sketch of the unstable manifold for d = 1 in the
(x, v) plane. Shock locations (A21 singularities) are obtained
by applying Maxwell rules to the loops. A preshock (A3
singularity) is represented; it corresponds to the formation
of a loop in the manifold. The velocity profile which is the
actual solution to the Burgers equation is represented as a
bold line.
in phase space (~γ, ~̇γ) that are invariant by the Euler–
Lagrange dynamics (5.2): a stable (attracting) manifold
Γ(s)(t) and an unstable (repelling) manifold Γ(u)(t), de-
fined as the instantaneous location of trajectories con-
verging to the global minimizer forward in time and
backward in time, respectively. Since all the minimizers
converge backward in time to the global minimizer, the
graph in the position-velocity phase space (~x,~v) of the
solution in the statistical steady state is made of pieces of
the unstable manifold Γ(u)(t) with discontinuities along
the shocks lines or surfaces. In other words, shocks ap-
pear as jumps between two different folds of the unstable
manifold. The smoothness of the unstable manifold is an
important property; for instance, it implies that when
d = 2, the topological shock lines are smooth curves.
In one dimension, where hyperbolicity is ensured, the
main shock corresponds to a jump between the right
branch and the left branch of the unstable manifold. Its
position can be obtained geometrically after observing
that the area b covered by the unstable manifold, once
the latter is cut by the main shock, should be equal to
the first integral of motion which is conserved, i.e.
v(x, t) dx =
v0(x) dx . (5.6)
The other shocks (or secondary shocks that have existed
only for a finite time) cut through the double-fold loops
of the unstable manifold (see figure 23). Their locations
can be obtained by a Maxwell rule applied to those loops.
Indeed, the difference of the two areas defined by cutting
such a loop at some position x is equal to the difference
of actions of the two trajectories emanating from the up-
per and lower locations and, thus, vanishes at the shock
location. We will see in section 6 that this construction
of the solution is also valid when the forcing is periodic
in time, problem which can be related to Aubry–Mather
theory relative to commensurate-incommensurate phase
transitions.
The above geometrical construction of the solution has
much in common with that appearing in the unforced
problem. Indeed, as we have seen in section 3.1, when
F = 0 the solution can be obtained geometrically by
considering in the (~x,~v) space, the Lagrangian manifold
defined by the position and the velocity of the fluid par-
ticles at a given time. This analogy gives good ground
predicting that some universal properties associated to
the unforced problem will still hold in the forced case, as
we will indeed see in section 7. Another instance concerns
transport of mass in higher dimension. We have seen in
section 4.1 that, for the unforced case, large but finite
mass densities are localized near boundaries of shocks
(“kurtoparabolic” singularities) contributing power-law
tails with the exponent −7/2 to the probability density
function of the mass density. When a force is applied the
smoothness of the unstable manifold associated to the
global minimizer should lead to the same universal law.
5.4 The case of extended systems
So far, we have discussed the global structure of the
solution to the forced Burgers equation with periodic
boundary conditions. Is is however of physical interest
to understand instances when the size of the domain is
much larger than the typical length scale of the forcing.
In this section, we will focus on describing, in the one-
dimensional case, the singular structure of the solution
in unbounded domains. Based on the formalism of [11],
we achieve this goal by considering a spatially periodic
forcing with a characteristic scale Lf much smaller than
the system size L. More precisely, for a fixed size L we
consider the stationary régime corresponding to the limit
t → ∞ and then study the limit L → ∞ by keeping
constant the energy injection rate (i.e. the L2 norm of
the forcing grows like L).
In order to get an idea of the behavior of the solution,
the limit of infinite aspect ration L/Lf was investigated
numerically in [11]. As seen from figure 24(a) numerical
observations suggest that at any time in the statistical
steady state, the shape of the velocity profile is simi-
lar to the order-unity aspect ratio problem, duplicated
over independent intervals of sizeLf . In particular, when
tracking backward in time the trajectories of fluid par-
ticles the minimizers converge to each other in a very
non-uniform way. Figure 24(b) shows that the minimiz-
ers form different branches, which are converging to each
other backward in time; in space time a tree structure is
obtained. As shown in figure 25(a) a similar behavior is
observed for shocks.
The velocity field at a given time t, consists of smooth
pieces separated by shocks. Let us denote by {Ωj} the
set of intervals in [0, L), on which the solution u(·, t)
is smooth. The boundaries of the Ωj ’s are the shocks
positions. Each of these shocks is associated to a root-
like structure formed by the trajectories of the various
Fig. 24. (a) Upper: snapshot of the velocity field for
L = 256Lf . Lower: zoom of the field in a interval of
length 10Lf . (b) Minimizing trajectories in space time for
L = 256Lf and over a time interval of length T = 100
shocks that have merged at times less than t to form the
shock under consideration (see figure 25(a)). This root-
like structure contains the whole history of the shock
and in particular its age (i.e. the length of the deeper
branch of the root structure). Indeed, if the root has a
finite depth, the shock considered has only existed for a
finite time. A T -global shock is defined as a shock whose
associated root is deeper than −T . They can alterna-
tively be defined geometrically by considering the left-
most and the rightmost minimizer associated to it. After
tracing them backward for a sufficiently long time, these
two minimizers are getting close and eventually converge
to each other exponentially fast (see figure 24(b)). For
a T -global shock, the time when the two minimizers are
getting within a distance smaller than the forcing cor-
relation length Lf is larger than T . As we have seen in
section 5.2, the existence in one dimension of a main
X (t)1 X (t)2
space
Fig. 25. (a) Shock trajectories for aspect ratio L/Lf = 32
and with T = 10. The different gray areas correspond to
the space-time domains associated to the different smooth
pieces Ωj of the velocity field at time t = 0. (b) Sketch of
the space-time evolution of a given smooth piece Ωj located
between two shock trajectories X1(t) and X2(t) that merge
at time Tj .
shock in the spatially periodic situation follows from a
simple topological argument. The main shock can also
be defined as the only shock that has existed forever in
the past. It is hence infinitely old, contrary to all other
shocks, all of them being created at a finite time and
having a finite age. When the periodicity condition is
dropped, the main shock disappears and it is useful to
consider the T -global shocks that mimic the behavior of
a main shock over time scales larger than T .
One can dually define T -global minimizers. All the
smoothness intervals Ωj defined above, except that
which contains the global minimizer, will be entirely
absorbed by shocks after a sufficient time (see figure
25(b)). For each of these pieces, one can define a life-time
Tj as the time when the last fluid particle contained in
this piece at time t enters a shock. It corresponds to the
first time for which the shock located on the left of this
smooth interval at time t merges with the shock located
on the right. When the life-time of such an interval is
greater than T , the trajectory of the last surviving fluid
particle is here called a T -global minimizer. Note that,
when T → ∞, the number of T -main shocks and of
T -global minimizers is one, recovering respectively the
notions of main-shock and of two-sided minimizer.
time T
= 64
= 128
= 256
slope = −2/3
Fig. 26. Density of T -main shocks as a function of T for three
different system sizes L/Lf = 64, 128 and 256; average over
100 realizations. Lower inset: local scaling exponent.
Hence, at a given instant t, and for any timelag T ,
the spatial domain [0, L) contains a certain number of
T -objects. We define their spatial density as being the
number of such objects, averaged with respect to the
forcing realizations, divided by the size of the domain L.
The density ρ(T ) of T -main shocks was investigated nu-
merically in [11] for the kicked case by using a two-step
method: first, the simulation was run until a large time t
for which the statistically stationary régime is reached;
secondly, each shock present at time t was tracked
backward-in-time down to the instant of its creation,
giving an easy way to characterize the density ρ(T ). It
is seen in figure 26 that, for three different aspect ratios
L/Lf , the density ρ(T ) displays a power-law behavior
ρ(T ) ∝ T−2/3 for the intermediate time asymptotics
Lf/urms ≪ T ≪ L/urms.
We now present a simple phenomenological theory aim-
ing to explain the scaling exponent 2/3. We consider the
solution at a fixed time (t = 0, for instance). Denote by
ℓ(T ) the typical spatial separation scale for two nearest
T -global shocks. Obviously, ℓ(T ) ∼ 1/ρ(T ). The mean
velocity of the spatial segment of length ℓ is given by
[y, y+ℓ]
u(x, 0) dx (5.7)
Since the expected value 〈u(x, 0)〉 = 0, and that the
integral in (5.7) is over an interval of size much larger
than the forcing correlation length, it is equivalent to
a sum of independent centered random variables and
scales as the Brownian motion. Hence, for large ℓ one
has the following asymptotics
[y, y+ℓ]
u(x, 0) dx ∼
ℓ, (5.8)
which gives bℓ ∼ ℓ−1/2 for mean velocity fluctuations.
Consider now the rightmost minimizer corresponding to
the left T -global shock and the leftmost minimizer re-
lated to the right one. Since there are no T -global shocks
in between, it follows that the two minimizers we se-
lected get close to each other backward-in-time around
times of the order of −T . This means that the backward-
in-time displacement of a spatial segment of length O(ℓ)
is itself O(ℓ) for time intervals of the order of T . The
corresponding displacement is given as the sum of two
competing behaviors: the first, which can be understood
as a drift induced by the local mean velocity bℓ, is due
to the mean velocity fluctuations and is responsible for
a displacement ∝ bℓT ; the second contribution is due to
a standard diffusive scale ∝ T 1/2 expressing the diffu-
sive behavior of the minimizing trajectories. Taking into
account both terms we obtain
ℓ ∼ B1T ℓ−1/2 +B2T 1/2, (5.9)
where B1 and B2 are numerical constants. It is easy to
see that the dominant contribution comes from the first
term. Indeed, if the second term were to dominate, then
ℓ would be much larger than T , which contradicts (5.9).
Hence, one has ℓ ∼ B1T ℓ−1/2, leading to the scaling
behavior
ℓ(T ) ∝ T 2/3, ρ(T ) ∝ T−2/3. (5.10)
As we have already discussed, T -global shocks are shocks
older than T . Denote by p(A) the probability density
function (PDF) for the age of shocks. More precisely,
p(A) is a density in the stationary régime of a probability
distribution of the age A(t) of a shock, say the nearest
to the origin. It follows from (5.10) that the probability
of shocks whose age is larger than A decays like A−2/3;
this implies the following asymptotics for the PDF p(A):
p(A) ∝ A−5/3. (5.11)
Actually, the power-law behavior of the density ρ(T ) of
T -global shocks can be interpreted in term of an inverse
cascade in the spectrum of the solution (although there
is no conserved energy-like quantity). Indeed, the fluctu-
ations (5.8) of the mean velocity suggest that, for large-
enough separations ℓ, the velocity potential increment
scales like
|ψ(x+ ℓ, t) − ψ(x, t)| ∝ ℓ1/2. (5.12)
This behavior is responsible for the presence of an inter-
mediate power-law range with exponent −2 in the spec-
trum of the velocity potential at wavenumbers smaller
than the forcing scale (see figure 27). In order to observe
Fig. 27. Spectrum 〈ψ̂2(k)〉 of the velocity potential in the
stationary régime for the aspect ratio L/Lf = 128. This
spectrum contains two power-law ranges: at wavenumbers
k ≫ L/Lf , the traditional ∝ k
−4 inertial range connected
to the presence of shocks in the solution and, for k ≪ L/Lf ,
an “inverse cascade” ∝ k−2 associated to the large-scale
fluctuations of ψ
the k−2 range at small wavenumbers, the spectrum of
the forcing potential must decay faster than k−2; other-
wise the leading behavior is non-universal but depends
on the functional form of the forcing correlation.
The one-dimensional randomly forced Burgers equation
in an unbounded domain has been studied in [66] with
a different type of forcing: it was assumed that the forc-
ing potential has at any time its global maximum and
its global minimum in a prescribed compact region of
space. It was proven that with these particular settings
the statistically stationary régime exists and is very sim-
ilar to that arising in compact domains. In particular,
there exists a unique global minimizer located in a finite
spatial interval for all times and all other minimizers are
asymptotic to it in the limit t → −∞. The main idea
behind considering such type of forcing potential is to
ensure that the potential energy plays a dominant role
in comparison with the kinetic (elastic) term in the ac-
tion. This leads to effective compactification and allows
estimates on the velocities of fluid particles. As we al-
ready mentioned in section 5.1, these estimates are very
important and pave the way to the construction of the
whole theory of the statistically stationary régime.
Note finally that it was shown in [76] that for special
cases of forcing potentials F (x, t), the velocity of a min-
imizers can be arbitrarily large. More specifically, one
can construct pathological forcing potentials such that
minimizers are accelerated and reach infinite velocities.
Randomness is of course expected to prevent such a type
of non-generic blow-up.
6 Time-periodic forcing
This section is devoted to the study of the solutions
to the one-dimensional Burgers equation with time-
periodic forcing. In this case many of the objects we
have discussed above can be constructed almost explic-
itly: the global minimizer, the main shock etc. Also,
a mathematical analysis is then much simpler. For in-
stance, hyperbolicity of the global minimizer follows
immediately from first principles. Finally, the case of
time-periodic forcing is directly related to the Aubry-
Mather theory as we explain below.
6.1 Kicked Burgers turbulence
We shall be concerned here with the initial-value prob-
lem for the one-dimensional Burgers equation when the
force is concentrated in Dirac delta functions at discrete
times:
f(x, t) =
fj(x) δ(t − tj), (6.1)
where both the “impulses” fj(x) and the “kicking times”
tj are prescribed (deterministic or random). The kicking
times are ordered and form a finite or infinite sequence.
The impulses fj(x) are always taken spatially smooth,
i.e. acting only at large scales. The general scheme we
are presenting below holds for any sequence of impulses
fj(x) and kicking time. Later on we shall assume that
they define a time-periodic forcing. The precise meaning
we ascribe to the Burgers equation with such forcing is
that at time tj , the solution u(x, t) changes discontinu-
ously by the amount fj(x)
u(x, tj+) = u(x, tj−) + fj(x), (6.2)
while, between tj+ and t(j+1)− the solution evolves ac-
cording to the unforced Burgers equation.
We shall also make use of the formulation in terms of the
velocity potential ψ(x, t) and the force potentials Fj(x)
u(x, t) = −∂xψ(x, t), fj(x) = −
Fj(x). (6.3)
The velocity potential satisfies
∂tψ =
(∂xψ)
2 + ν∂xxψ +
Fj(x) δ(t − tj), (6.4)
ψ(x, t0) = ψ0(x), (6.5)
where ψ0(x) is the initial potential.
Using the variational principle we obtain the following
“minimum representation” for the potential in the limit
of vanishing viscosity which relates the solutions at any
two times t > t′ between which no force is applied:
ψ(x, t) = −min
(x− y)2
2(t− t′)
− ψ(y, t′)
. (6.6)
As before, when t′ is the initial time, the position y which
minimizes (6.6) is the Lagrangian coordinate associated
to the Eulerian coordinate x. The map y 7→ x is called
the Lagrangian map. By expanding the quadratic term it
is easily shown that the calculation of ψ(·, t) from ψ(·, t′)
is equivalent to a Legendre transformation. For details,
see [104,107].
We now turn to the forced case with impulses applied at
the kicking times tj . Let tJ(t) be the last such time before
t. Using (6.6) iteratively between kicks and changing
the potential ψ(y, tj+1) discontinuously by the amount
Fj+1(y) at times tj+1, we obtain
ψ(x, t) = − min
{yj}j0≤j≤J
[A({yj};x, t; j0)) − ψ0(yj0)] , (6.7)
A({yj};x, t; j0) ≡
(x − yJ)2
2(t− tJ)
(yj+1 − yj)2
2(tj+1 − tj)
−Fj+1(yj+1)
, (6.8)
where A(j0;x, t; {yj}) is called the action. We shall as-
sume that the force potential and the initial condition
are periodic in the space variable and the period is taken
to be unity. This assumption is very important for the
discussion below.
For a given initial condition at tj0 we next define a “min-
imizing sequence” associated to (x, t) as a sequence of
yj’s (j = j0, j0 + 1, . . . , J(t)) at which the right-hand
side of (6.7) achieves its minimum. Differentiating the
action (6.8) with respect to the yj ’s one gets necessary
conditions for such a sequence, which can be written as
a sequence of (Euler–Lagrange) maps
vj+1 = vj + fj(yj), (6.9)
yj+1 = yj + vj+1(tj+1 − tj)
= yj + (vj + fj(yj))(tj+1 − tj), (6.10)
where
yj − yj−1
tj − tj−1
. (6.11)
These equations must be supplemented by the initial
and final conditions:
vj0 = u0(yj0), (6.12)
x= yJ + vJ+1(t− tJ). (6.13)
It is easily seen that u(x, t) = vJ+1 = (x− yJ)/(t− tJ ).
Observe that the “particle velocity” vj is the velocity of
the fluid particle which arrives at yj at time tj and which,
of course, has remained unchanged since the last kick (in
Lagrangian coordinates). Equation (6.9) just expresses
that the particle velocity changes by fj(yj) at the the
kicking time tj .
Note that (6.9)-(6.10) define an area-preserving and (ex-
plicitly) invertible map.
As in the case of continuous-in-time forcing we can for-
mulate the Burgers equation in the half-infinite time in-
terval (−∞, t] without fully specifying the initial con-
dition u0(x) but only its (spatial) mean value 〈u〉 ≡
u0(x)dx.
The construction of the solution in a half-infinite time
interval is done by extending the concept of minimizing
sequence to the case of dynamics starting at t0 = −∞.
For a half-infinite sequence {yj} (j ≤ J), let us define the
action A({yj};x, t;−∞) by (6.8) with j0 = −∞. Such
a half-infinite sequence will be called a “minimizer” (or
“one-sided minimizer”) if it minimizes this action with
respect to any modification of a finite number of yj’s.
Specifically, for any other sequence {ŷj} which coincides
with {yj} except for finitely many j’s (i.e. ŷj = yj , j ≤
J − k, k ≥ 0), we require
A({ŷj};x, t; J − k) ≥ A({yj};x, t; J − k). (6.14)
Of course, the Euler–Lagrange relations (6.9)-(6.10) still
apply to such minimizers. Hence, if for a given x and t we
know u(x, t), we can recursively construct the minimizer
{yj} backwards in time by using the inverse of (6.9)-
(6.10) for all j < J and the final condition – now an
initial condition – (6.13) with vJ+1 = u(x, t). This is
well defined except where u(x, t) has a shock and thus
more than one value.
One way to construct minimizers is to take a sequence
of initial conditions at different times t0 → −∞. At each
such time some initial condition u0(x) is given with the
only constraint that it have the same prescribed value
for 〈u〉. Then, (finite) minimizing sequences extending
from t0 to t are constructed for these different initial
conditions. This sequence of minimizing sequences has
limiting points (sequences themselves) which are pre-
cisely minimizers (E et al. 1998). The uniqueness of such
minimizers, which would then imply the uniqueness of
a solution to the Burgers equation in the time interval
]−∞, t], can only be shown by using additional assump-
tions, for example for the case of random forcing or when
the forcing is time-periodic.
If 〈u〉 = 0, the sequence {yj} minimizes the action
A({yj};x, t;−∞) in a stronger sense. Consider any
sequence {ŷj} such that, for some integer P we have
ŷj = yj + P , j ≤ J − k, k ≥ 0 and which differs ar-
bitrarily from {yj} for j > J − k. (In other words, in
a sufficiently remote past the hatted sequence is just
shifted by some integer multiple of the spatial period.)
We then have
A({ŷj};x, t;−∞) ≥ A({yj}, x, t;−∞). (6.15)
Indeed, for 〈u〉 = 0, the velocity potential for any initial
condition is itself periodic. In this case a particle can be
considered as moving on the circle S1 and its trajectory
is a curve on the space-time cylinder. The yj ’s are now
defined modulo 1 and can be coded on a representative
0 ≤ yj < 1. The Euler–Lagrange map (6.9)-(6.10) is still
valid provided (6.10) is defined modulo 1.
The condition of minimality implies now that yj and
yj+1 are connected by the shortest possible straight seg-
ment. It follows that |vj+1| = ρ(yj , yj+1)/(tj+1 − tj),
where ρ is the distance on the circle between the points
yj, yj+1, namely ρ(a, b) ≡ min{|a−b|, 1−|a−b|}. Hence,
the actionA can be rewritten in terms of cyclic variables:
A({yj};x, t;−∞) =
ρ2(x, yJ )
2(t− tJ)
ρ2(yj+1, yj)
2(tj+1 − tj)
− Fj+1(yj+1)
. (6.16)
The concept of “global minimizers” can be defined in
a usual way. Namely, global minimizers correspond to
one-sided minimizers that can be continued to a bilat-
eral sequence {yj ,−∞ < j < +∞} while keeping the
minimizing property. Such global minimizers correspond
to trajectories of fluid particles that, from t = −∞ to
t = +∞, have never been absorbed in a shock. As before
we define a “main shock” as a shock which has always
existed in the past.
From now on we shall consider exclusively the case where
the kicking is periodic in both space and time. Specifi-
cally, we assume that the force in the Burgers equation
is given by
f(x, t) = g(x)
δ(t− jT ), (6.17)
g(x) ≡− d
G(x), (6.18)
where G(x), the kicking potential, is a deterministic
function of x which is periodic and sufficiently smooth
(e.g. analytic) and where T is the kicking period. The
initial potential ψinit(x) is also assumed smooth and pe-
riodic. This implies that the initial velocity integrates to
zero over the period. The case where this assumption is
relaxed will be considered later in connection with the
Aubry–Mather theory.
The numerical experiments of [9] reported here have
Fig. 28. Snapshots of the velocity for the unique time-peri-
odic solution corresponding to the kicking force g(x) shown
in the upper inset; the various graphs correspond to six out-
put times equally spaced during one period. The origin of
time is taken at a kick. Notice that during each period, two
new shocks are born and two mergers occur. (From [9].)
been made with the kicking potential
G(x) =
sin 3x+ cosx, (6.19)
and a kicking period T =1. Other experiments were done
with (i) G(x) = − cosx and (ii) G(x) = (1/2) cos(2x) −
cosx. The former potential produces a single shock and
no preshock. As a consequence it displays no −7/2 law
in the PDF of gradients. The latter potential gives es-
sentially the same results as reported hereafter but has
an additional symmetry. To avoid non-generic behaviors
that could result from this symmetry, it was chosen to
focus on the forcing potential given by (6.19).
The number of collocation points chosen for such simu-
lations was mostly Nx = 2
17 ≈ 1.31 × 105, with a few
simulations done at Nx = 2
20 (for the study of the re-
laxation to the periodic régime presented below). Since
the numerical method allows going directly to the de-
sired output time (from the nearest kicking time) there
is no need to specify a numerical time step. However, in
order to perform temporal averages, e.g. when calculat-
ing PDF’s or structure functions, without missing the
most relevant events (which can be sharply localized in
time) sufficiently frequent temporal sampling is needed.
The total number of output times Nt ≈ 1000, is thus
chosen such that the increment between successive out-
put times is roughly the two-thirds power of the mesh
(this is related to the cubic structure of preshocks, see
section 2.3).
Figure 28 shows snapshots of the time-periodic solution
at various instants. It is seen that shocks are always
present (at least two) and that at each period two new
shocks are born at t⋆1 ≈ 0.39 and t⋆2 ≈ 0.67. There is one
main shock which remains near x = π and which collides
space x
Main Shock
Fig. 29. Evolution of shock positions during one period. The
beginnings of lines correspond to births of shocks (preshocks)
at times t⋆1 and t⋆2; shock mergers take place at times tc1
and tc2. The “main shock”, which survives for all time, is
shown with a thicker line.
0 2 4 6 8 10 12 14
number of kicks
sin(x)
sin(2x)
sin(3x)
slope −0.74
Fig. 30. Exponential relaxation to a time-periodic solution
for three different initial velocity data as labeled. The hori-
zontal axis gives the time elapsed since t = 0. (From [9].)
with the newborn shocks at tc1 ≈ 0.44 and tc2 ≈ 0.86.
Figure 29 shows the evolution of the positions of shocks
during one period.
It was found that, for all initial conditions u0(x) used, the
solution u(x, t) relaxes exponentially in time to a unique
function u∞(x, t) of period 1 in time. Figure 30 shows the
variation of
|u(x, n−)−u∞(x, 1−)| dx/(2π) for three
different initial conditions as a function of the discrete
time n.
The phenomenon of exponential convergence to a unique
space- and time-periodic solution is something quite gen-
eral: whenever the kicking potentialG(x) is periodic and
analytic and the initial velocity potential is periodic (so
that the mean velocity 〈u〉 =0 at all times), there is ex-
ponential convergence to a unique piecewise analytic so-
lution. This can be proved rigorously (see Appendix to
[9]) in the case when the functions G(x) have a unique
point of maximum with a non-vanishing second deriva-
tive (Morse generic functions). Here, we just explain the
main ideas of the proof and give some additional prop-
erties of the unique solution.
One very elementary property of solutions is that, for
any initial condition of zero mean value, the solution
after at least one kick satisfies
|u(x, t)| ≤ (1/2) + max
|dG(x)/dx|. (6.20)
Indeed, at a time t = n− just before any kick we have
x = y+u(x, n−) where y is the position just after the pre-
vious kick of the fluid particle which goes to x at time n−.
It follows from the spatial periodicity of the velocity po-
tential that the location y which minimizes the action is
within less than half a period from x. Thus, |u(x, n−)| ≤
1/2. The additional maxx |dG(x)/dx| term comes from
the maximum change in velocity from one kick. Hence
the solution is bounded. Note that if the spatial and tem-
poral periods are L and T , respectively, the bound on
the velocity becomes L/(2T ) + maxx |dG(x)/dx|.
The convergence at large times to a unique solution can
be understood in terms of the two-dimensional conserva-
tive (area-preserving) dynamical system defined by the
Euler–Lagrange map (6.9)-(6.10). By construction, we
have u(x, 1+) = û(x) − dG(x)/dx, where û(x) is the so-
lution of the unforced Burgers equation at time t = 1−
from the initial condition u(x) at time t = 0+. The map
u 7→ û(x) + g(x), where g(x) ≡ −dG(x)/dx, here de-
notedBg, solves the kicked Burgers equation over a time
interval one. The problem is to show that the iterates
Bng u0 converge as n→ ∞ to a unique solution.
If it were not for the shocks it would suffice to consider
the two-dimensional Euler–Lagrange map. Note that,
for the case of periodic kicking, this map has an obvi-
ous fixed point P , namely (x = xc, v = 0), where xc is
the unique point maximizing the kicking potential. It is
easily checked that this fixed point is an unstable (hy-
perbolic) saddle point of the Euler–Lagrange map with
two eigenvalues λ = 1 + c +
c2 + 2c and 1/λ, where
c = −∂2xxG(xc)/2.
Like for any two-dimensional map with a hyperbolic
fixed point, there are two curves globally invariant by
the map which intersect at the fixed point. The first is
the stable manifold Γ(s), i.e. the set of points which con-
verge to the fixed point under indefinite iteration of the
map; the second is the unstable manifold Γ(u), i.e. the
set of points which converge to the fixed point under in-
definite iteration of the inverse map, as illustrated in fig-
ure 31(a). Any curve which intersects the stable manifold
transversally (at the intersection point, the two curves
xx xl rc
Fig. 31. (a) Sketch of a hyperbolic fixed point P with stable
(Γ(s)) and unstable (Γ(u)) manifolds. The dashed line gives
the orbit of successive iterates of a point near the stable
manifold. (b) Unstable manifold Γ(u) on the (x, v)-cylinder
(the x-coordinate is defined modulo 1) which passes through
the fixed point P = (xc, 0). The bold line is the graph of
u∞(x, 1−). The main shock is located at xl = xr. Another
shock at x1 corresponds to a local zig-zag of Γ
(u) between A
and B.
are not tangent to each other) will, after repeated appli-
cations of the map, be pushed exponentially against the
unstable manifold at a rate determined by the eigenvalue
1/λ. In the language of Burgers dynamics, the curve in
the (x, v) plane defined by an initial condition u0(x) will
be mapped after time n into a curve very close to the
unstable manifold. In fact, for the case studied numeri-
cally, 1/λ ≈ 0.18 is within one percent of the value mea-
sured from the exponential part of the graph shown in
figure 30. Note that if the initial condition u0(x) contains
the fixed point, the convergence rate becomes (1/λ)
(even higher powers of 1/λ are possible if the initial con-
dition is tangent to the unstable manifold).
The fixed point P is actually a very simple global min-
imizer: (yj = xc, vj = 0) for all positive and negative
j’s. It follows indeed by inspection of (6.16) that any
deviation from this minimizer can only increase the ac-
tion; actually, this trajectory minimizes both the kinetic
and the potential part of the action. Note that the cor-
z (0)l rz (0)
����������������������������������������������
����������������������������������������������
����������������������������������������������
����������������������������������������������
����������������������������������������������
z (0)1z c
Fig. 32. Minimizers (trajectories of fluid particles) on the
(x, t)-cylinder. Time starts at −∞. Shock locations at t = 0−
are characterized by the presence of two minimizers (an in-
stance is at x1). The main shock is at xl = xr. The fat line
x = xc is the global minimizer.
responding fluid particle is at rest forever and will never
be captured by a shock (it is actually the only particle
with this property). It is easy to see that any minimizer
is attracted exponentially to such a global minimizer as
t→ −∞. Thus, any point (yj , vj) on a minimizer belongs
to the unstable manifold Γ(u) and, hence, any regular
part of the graph of the limiting solution u∞(x) belongs
to the unstable manifold Γ(u). This unstable manifold is
analytic but can be quite complicated. It can have sev-
eral branches for a given x (see figure 31(b)) and does
not by itself define a single-valued function u∞(x). The
solution has shocks and is only piecewise analytic. Con-
sideration of the minimizers is required to find the po-
sition of the shocks in the limiting solution: two points
with the same x corresponding to a shock, such as A and
B on figure 31(b) should have the same action.
Finally, we give the geometric construction of the main
shock, the only shock which exists for an infinite time.
Since the eigenvalue λ is positive, locally, minimizers
which start to the right of xc approach the global min-
imizer from the right, and those which start to the left
approach it from the left. Take the rightmost and left-
most points xr and xl on the periodicity interval such
that the corresponding minimizers approach the global
minimizer from the right and left respectively (see fig-
ure 32). These points are actually identical since there
cannot be any gap between them that would have min-
imizers approaching the global minimizer neither from
the right nor the left. The solution u∞(x) has then its
main shock at xl = xr.
6.2 Connections with Aubry–Mather theory
In the previous subsection, the study of the solutions
to the periodically kicked Burgers equation was limited
to initial conditions with a vanishing spatial average
b. With a non-vanishing mean velocity b, which in the
forced case cannot be eliminated by a Galilean invari-
ance, many of the properties of the solutions described
above are still valid. However the action now depends on
b. Global minimizers {y(g)j , j ∈ Z} exist in this case as
well. However generically they are not unique and do not
correspond to fixed points of the Euler–Lagrange map
(6.9)-(6.10). A global minimizer now minimizes the ac-
A∞({yk}) =A({yk}; +∞;−∞)
(yk+1−yk−b)2−G(yk+1)
. (6.21)
This action is exactly the potential energy associated
to an infinite chain of atoms linked by elastic springs
and embedded in a periodic potential, problem known
as the Frenkel–Kontorova model [52]. The parameter b
represents the equilibrium length l of the springs and
the spatial period L of the external potential (see fig-
ure 33) is equal to 1. A global minimizer of (6.21) rep-
Fig. 33. Sketch of the Frenkel–Kontorova model for the equi-
librium states of an atom chain in a periodic potential.
resents an equilibrium configuration of this system. The
properties of this equilibrium, or ground states are de-
termined by the competition between two tendencies: on
the one hand the atoms tend to stabilize at those loca-
tions where the potential is minimum; on the other hand,
the springs tend to maintain them at a fixed distance of
each other. When b = 0 this competition disappears and
the equilibrium is given by yk = xc, where xc is the loca-
tion at which G attains its global minimum. For b 6= 0,
the situation is more delicate and the structure of the
ground states involves, as we shall now see, a problem
of commensurate-incommensurate transition. The prop-
erties of ground states were studied in great details by
Aubry [3] and Mather [86].
The relations between the Burgers equation with a
time-periodic forcing and Aubry–Mather theory were
discussed for the first time in [70] and in [38]. The the-
ory was further developed in [36,106]. For integer values
of b, the global minimizer is trivially associated to the
fixed point (x, v) = (xc, b) of the Euler–Lagrange map
(6.9)-(6.10), which corresponds to a fluid trajectory lo-
cated at integer times at x = xc and which moves on
distance of b spatial periods during one temporal period.
A similar argument implies that it is enough to study
values of b in the interval [0, 1).To each global minimizer
{y(g)j , j ∈ Z} is associated a rotation number defined as
ρ ≡ lim
j+1 − y
, (6.22)
which represents the time-average velocity of the mini-
mizer. For a fixed value of the spatial average b of the
velocity, all global minimizers associated to the solution
of the Burgers equation have the same rotation number
ρ. Indeed, as the dynamics is restricted to a compact do-
main of the configuration space (in our case T), two min-
imizers with different rotation numbers necessarily cross
each other; this is an obvious obstruction to the action
minimization property. In the case of rational rotation
numbers the global minimizers correspond to periodic
orbits of the dynamical system defined by the Euler–
Lagrange map. An important feature is that for rational
ρ, the rotation number does not change when varying
b over a certain closed interval [bmin, bmax], called the
mode-locking interval. On the contrary, irrational ρ cor-
respond to a unique value of the parameter b. Such “ir-
rational” values of b form a Cantor set of zero Lebesgue
measure. In particular, the graph of ρ as a function of
the parameter b is a “Devil staircase” (see figure 34).
0 0.2 0.4 0.6 0.8 1
Fig. 34. Rotation number ρ as a function of the spatial mean
of the velocity b for the standard map.
When ρ is rational (ρ = p/q in irreducible form), global
minimizers correspond to a periodic orbit of period q.
It is easy to see that such an orbit generates q different
but closely related global minimizers. Of course each of
these global minimizer is the image of another one by the
Euler–Lagrange map and is mapped back to itself after
q iterations. This procedure generates a periodic orbit,
which turns out to be hyperbolic one. Hence, each of
the q global minimizers has a one-dimensional unstable
manifold associated to it. The solution to the Burgers
equation is formed by branches of these various mani-
folds with jumps between them defining q global shocks.
The picture is very different for values of b corresponding
to irrational rotation numbers. Consider velocities and
positions of all global minimizers at a fixed moment of
time, say t = 0. They form a subset G of the phase space
T × R. Then two cases have to be distinguished:
• The set G forms a closed invariant curve for the Euler–
Lagrange map. This invariant curve has a one-to-one
projection onto the base T and dynamics on the curve
is conjugated to a rigid rotation by angle ρ. The lim-
iting solution of the Burgers equation is given by the
invariant curve and does not contain any shocks.
• The set G forms an invariant Cantor set and the limit-
ing solution of the Burgers equation contains an infi-
nite number of shocks, none of which is a main shock.
The Kolmogorov [80], Arnold [2] and Moser [93] the-
ory (frequently referred to as KAM) describes invari-
ant curves (or invariant tori) for small analytic per-
turbations of integrable Hamiltonian systems, and thus
the various types of dynamical trajectories. The KAM
theory ensures that for sufficiently small perturbations,
most of the invariant curves associated to Diophantine
irrational rotation numbers are stable with respect to
small analytic perturbations of the system. Diophantine
irrational numbers possess fast converging approxima-
tions by rational numbers (in a suitable technical sense).
However, these invariant curves may disappear from the
perturbed system when an interaction corresponding to
a non-integrable perturbation gets sufficiently strong.
Aubry–Mather theory provides another variational de-
scription for the KAM invariant curves. But even more
importantly, it describes the invariant Cantor sets that
appear instead of invariant curves in the case of strong
nonlinear interactions. We have mentioned already that
these Cantor sets correspond to global minimizers. Thus
Aubry–Mather theory provides information about the
global minimizers and, hence, allows one to study in such
a situation the properties of limiting entropic solutions
and, in particular, the structure of shocks.
A numerical study of the Burgers equation in the inviscid
limit, with periodic forcing and a non-vanishing spatial
average of the velocity, reveals the appearance of shock
accumulations. Such events occur for the values of the
mean velocity b near the end-points of the mode-locking
intervals, corresponding to rational rotation numbers.
The shock accumulation phenomenon is due to the fact
that the end-points bmin, bmax of the mode-locking in-
tervals can be approximated by convergent sequences of
“irrational” values of the parameter b. This implies ac-
cumulation of shocks, since for irrational rotation num-
bers the number of shocks is infinite.
The limiting solution u∞(x, t) is completely determined
by the function û(x) defined in the previous subsection.
The function û(x) corresponds to a stroboscopic section
of u∞ right after each impulse. The regular parts of û
are made of single-valued functions related to the unsta-
ble manifolds. The shocks correspond to jumps, either
between different branches of the same manifold (sec-
ondary shocks), or between the manifolds associated to
different global minimizers (main shocks).
When the rotation number is rational (ρ = p/q), there
are q global minimizers. The positions of the q main
shocks of û are determined by a requirement that the
area defined by the graph of the solution is equal to the
conserved quantity b. The latter constraint shows that
the values of b compatible with the rotation number p/q
belong to an interval [bmin, bmax] bounded by the mini-
mum and maximum areas defined by the unstable man-
ifolds, as illustrated in figure 35. The detailed shape of
Fig. 35. Sketch of the unstable manifolds of the two global
minimizers associated to the rotation number ρ = 1/2. The
values bmin and bmax given by this configurations are repre-
sented as grey areas.
the manifolds can actually not be sketched on a figure.
Generically the unstable manifold of a global minimizer
corresponding to a particular point of the basic periodic
orbit of period q intersects transversally with the stable
manifold of another minimizer corresponding to another
point of the periodic orbit. Such an intersection leads to
formation of a heteroclinic tangle, a notion which can be
traced back to the work of Poincaré. The heteroclinic in-
tersection results in the formation of an infinite number
of zig-zags of the unstable manifolds. These zig-zags are
accumulating along the stable manifold and come arbi-
trary close to the corresponding point of the periodic or-
bit. The zig-zags contract exponentially in one direction
(along the stable manifold) and are stretched exponen-
tially in the other direction. It is easy to see that the
accumulation of zig-zags generates an infinite number of
“potential” shocks of smaller and smaller size which also
accumulate near the periodic orbit. When the param-
eter b is located well inside the mode-locking interval,
the position of the main shock cuts off the accumulated
shocks of small size so that the total number of shocks
is of the order of unity. However, when b gets closer and
closer to bmax or bmin, the main shocks move closer to
the periodic points and a larger number of the small ac-
cumulating shocks appears in the solution. This mecha-
nism leads to an infinite number of shocks in the solution
when b is equal to bmin or bmax (see figure 36(a)). Both
Fig. 36. (a) Accumulations of shocks occurring for b = bmin or
b = bmax, due to the presence of an infinite number of loops
of the unstable manifold in the homocline or heterocline
tangle. (b) Shock accumulation at the fixed point (0, 0) of
the standard map. Here, λ = 0.1 and b = 0.15915. The latter
value is close to the upper bound of the interval associated
to ρ = 0. The upper inset is a zoom near (0, 0), illustrating
the accumulation of shocks.
the distances between two consecutive shocks and the
sizes of the shocks decrease exponentially fast with the
number of shocks; the rate is given by the stable eigen-
value associated to the hyperbolic periodic orbit. It is
interesting to mention that when b = bmin or b = bmax
the main shocks merge with the periodic orbit associ-
ated to the global minimizers. Hence, for the end-points
of the mode-locking interval the main shocks disappear.
To illustrate numerically the change in behavior of the
solution to the Burgers equation when the mean velocity
b changes, we focus here on the simple periodic kicking
potential G(x) = (λ/2π) cos(2πx) where λ is a free pa-
rameter. The associated Euler–Lagrangemap then reads
T : (y, v) 7→(y+v+λ sin(2πy), v+λ sin(2πy)). (6.23)
This transformation is usually called the standard map
(or Chirikov–Taylormap). It is one of the simplest model
for studying the presence of chaos in Hamiltonian dy-
Fig. 37. General aspect in position-velocity phase space of
the dynamical system defined by the standard map (6.23)
for two different values of the parameter (a) λ = 0.1 and
(b) λ = 0.3. The corresponding time-periodic solutions to
the kicked Burgers equation are represented as bold lines in
both cases. The results are presented for the spatial mean
velocities b = 0, b = 0.3 and b = 0.5.
namical systems and in particular particularly to study
the KAM theory.
Figure 36(b) illustrates the accumulation of shocks due
to the homoclinic or heteroclinic tangling for the first
transition (starting from b = 0). This transition cor-
responds to a rotation number of the global minimizer
changing value from ρ = 0 to ρ > 0. When b is increased
and gets close to the critical value, shocks accumulate
on the left-hand side of the global minimizer located at
(y, v) = (0, 0).
Other numerical experiments were performed in order
to observe the destruction of invariant curves and the
accumulation of shocks on Cantor sets for irrational ro-
tation numbers. It is of course impossible numerically to
set the rotation number to an irrational value. Indeed,
the values of b for which ρ is irrational are in a Cantor
set. It is however possible to be very close to irrational
rotation numbers. Figure 37 illustrates the changes in
the behavior of the solutions to the periodically kicked
Burgers equation when varying the parameter λ. The
time-asymptotic solutions associated to various values
of the mean velocity b are shown for λ = 0.1 and λ = 0.3.
For the latter value, all KAM invariant curves have al-
ready disappeared. For b = 0 and for all values of λ the
global minimizer trivially corresponds to the fixed point
(0, 0) with a vanishing rotation number. For b = 0.5
there are two global minimizers associated to the ratio-
nal rotation number ρ = 1/2. For λ = 0.1 and b = 0.3
the rotation number is much closer to an irrational than
in previous cases. The solution is then very close to the
invariant curve associated to this value. Note that the
main shock is actually located close to x ≈ 0.85. It is so
small that it can hardly be seen. When λ = 0.3 the value
b = 0.3 of the mean velocity no more corresponds to a
rotation number close to an irrational value; it is now
in the mode-locking interval associated to ρ = 1/3. This
change in the rotation number reflects the dependence of
the mode-locking intervals [bmin, bmax] on the parameter
λ. The interval of values of b associated to ρ = 0 is rep-
resented as a function of λ in figure 38. Such a structure
is frequently called an Arnold tongue (see, e.g., [72]).
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 38. Evolution as a function of the parameter λ of the
mode-locking interval [bmin, bmax] associated to the rotation
number ρ = 0. Such a graph is frequently referred to as an
Arnold tongue.
Finally, we discuss the structure of shocks in the case
when the global minimizers form a Cantor set. There are
then infinitely many gaps with no global minimizers. It
is known in this case that all the gaps can be split into
the finite number of images of the main gaps. For the
standard map there is only one main gap. Its end-points
(x1, v1) and (x2, v2) belong to the Cantor set associated
to the global minimizers. All other gaps can be obtained
by iterating this main gap with the Euler-Lagrange map
(Standard map) for both positive and negative times:
(x1i , v
i ) = T i(x1, v1), (x2i , v2i ) = T i(x2, v2), i ∈ Z. One
can show that the length of the ith gap tends to zero as
i→ ±∞. Since global minimizers are hyperbolic trajec-
tories one can connect the end-points of the main gap
by two smooth curves: the stable manifold Γ(s) and the
unstable manifold Γ(u). As i→ ∞ the iterates of the sta-
ble manifold T iΓ(s) tend to a straight segment connect-
ing the i-th gap with end-points at (x1i , v
i ) and (x
i , v
The same is true for iterates of the unstable manifold
T iΓ(u) in the limit i → −∞. On the contrary, negative
iterates of the stable manifold and positive of the unsta-
ble one form exponentially long curves connecting corre-
sponding gaps. As usual we are interested in the iterates
of the unstable manifold since they appear in the time-
periodic solution of the Burgers equation. Such a solu-
tion is formed by the iterates of the unstable manifold
connecting all the gaps. Note that in the case of large
negative i, the unstable manifold is close to a straight
segment; hence there are no shocks located inside the
corresponding gaps. Conversely, for large positive i, the
unstable manifold is exponentially long and possesses
large zig-zags. Hence, the solution to the Burgers equa-
tion has one or several shocks inside such gaps. Since
there are no shocks for gaps with large enough negative
i, it follows that all the shocks have a finite age. In other
words, the time-periodic solution has no main shocks.
At the moment it was not possible to study numeri-
cally the strange behavior of the solutions to the Burgers
equation corresponding to global minimizers living on
Cantor-like sets. Looking for such cases requires a very
high spatial resolution in order to minimize the numeri-
cal error in the approximation of the solution. Moreover,
a large number of values for the parameters b and λ has to
be investigated in order to observe such a phenomenon.
This would require heavy computer ressources. How-
ever, many other aspects of the Aubry–Mather theory
for Hamiltonian systems can be studied numerically us-
ing the Burgers equation with periodic kicks. For in-
stance it could be very useful for analyzing the higher
dimensional versions.
7 Velocity statistics in randomly forced Burgers
turbulence
The universality of small-scale properties in fully devel-
oped Navier–Stokes turbulence has frequently been in-
vestigated, assuming that a steady state is maintained
by an external large-scale forcing. It is generally conjec-
tured that the velocity increments have universal sta-
tistical properties with respect to such a force. Under-
standing this issue in simpler models of turbulence has
motivated much work for over ten years. A toy model
which has been extensively studied is the passive trans-
port of a scalar field by random flows (see, e.g., [46]).
Tools borrowed from statistical physics and field theory
were used to describe and explain the anomalous scaling
laws observed in the scalar spatial distribution. It was
shown that the scale invariance symmetry is broken by
geometrical constraints on tracer configurations that are
statistically conserved by the dynamics. Universality of
the intermittent scaling exponents with respect to the
forcing was proven for the case where energy is injected
at large scales [31,57,103,14].
Issues of universality for the nonlinear Burgers turbu-
lence model has also been very much on the focus. The
possibility to solve exactly a hydrodynamical problem
displaying the same kind of quadratic nonlinearity as
Navier–Stokes turbulence constitutes of course the cen-
tral motivation. Three independent approaches were
published almost simultaneously in 1995 and were at
the origin of the growing interest in Burgers turbulence.
First, an analogy was made in [22] between forced Burg-
ers turbulence and the problem of a directed polymer in
a random medium. This analogy was used to show that
the shocks appearing in the solution lead to anomalous
scaling laws for the structure functions. The strong
intermittency could be related to the replica-symmetry-
breaking nature of the disordered system associated
to Burgers turbulence. This approach is discussed in
subsection 7.1. Second, ideas using operator product
expansions borrowed from quantum field theory were
proposed in [99]. The goal was to close in the inertial
range the equations governing the correlations of the ve-
locity field in one dimension. This treatment of the dis-
sipative anomaly is described in subsection 7.2. It yields
a prediction for the probability density function (PDF)
of velocity increments and gradients and in particular
to a power-law behavior for the PDF of ∂xv at large
negative values [99]. However, the value of the exponent
of this algebraic tail has been a matter of controversy.
An overview of the various works related to this issue is
given in subsection 7.3. Finally, the turbulent model of
the one-dimensional Burgers equation with a self-similar
forcing was proposed in [30] as one of the simplest non-
linear hydrodynamical problem displaying multiscaling
of the velocity structure function. As stressed in subsec-
tion 7.4 this problem is easily tractable numerically and
some of the numerical observations can be confirmed by
a one-loop renormalization group expansion.
In what follows we consider the solutions to the Burg-
ers equation with a homogeneous Gaussian random forc-
ing that is delta-correlated in time. Namely, the spatio-
temporal correlation of the forcing potential is taken to
〈F (~x, t)F (~x′, t′)〉 = B(~x − ~x′) δ(t− t′) . (7.1)
The function B contains information on the spatial
structure of the forcing. It can be either smooth (i.e.
concentrated at large spatial scales) or asymptoti-
cally self-similar (i.e. behaving as a power law at small
separations). In the former case the solution reaches
exponentially fast a statistically stationary régime in
any space dimension. The construction of the solution
in this régime in terms of global minimizer and main
shock is described in detail in section 5. When B does
not decrease sufficiently fast at small separations (e.g.
B(r) ∼ r2h with h < 1 as r → 0 in one dimension), there
is no rigorous proof of the existence of a statistically sta-
tionary régime. However we assume in the sequel that
such a stationary régime exists in order to perform a
statistical analysis of the solutions to Burgers equation.
7.1 Shocks and bifractality – a replica variational ap-
proach
The replica solution for Burgers turbulence proposed
in [22] is based on its analogy with the problem of a di-
rected polymer in a random medium. As already stated
in the Introduction, the viscous Burgers equation forced
by the potential F is equivalent to finding the partition
function Z of an elastic string in the quenched spatio-
temporal disorder V (~x, t) = F (~x, t)/2ν (remember that
t has to be interpreted as the space direction in which
the polymer is oriented). This relation is obtained by ap-
plying to the velocity potential Ψ the Hopf–Cole trans-
formation Z(~x, t)=exp(Ψ(~x, t)/2ν). The solution of the
problem can be written in terms of the path integral
Z(~x, t) =
~γ(t)=~x
exp(−H(~γ)) d[~γ(·)] ,
with H(~γ) = 1
∥~̇γ(s)
+ F (~γ(s), s)
ds. (7.2)
In the analogy between Burgers turbulence and directed
polymers, the polymer temperature is assumed to be
unity and its elastic modulus is 1/(2ν). The strength
of the potential fluctuations applied to the polymer de-
pends on the viscosity and is ∝ ε1/2Lf/(2ν) (where ε is
the energy injection rate and Lf is the spatial scale of
forcing). In order to calculate the various moments of the
velocity field ~v = −∇Ψ, one needs to average the loga-
rithm of the partition function Z, a celebrated problem
in disordered systems.
Bouchaud, Mézard and Parisi proposed in [22] the use
of a replica trick in order to estimate the average free
energy 〈lnZ〉. The first step is to write the zero-replica
limit lnZ = limn→0 (Zn − 1)/n. Then, the moments
〈Zn〉 are used to generate an effective attraction between
replicas: they are written as the partition functions of
the disorder-averaged Hamiltonian Hn(~γ1, . . . , ~γn) asso-
ciated to n replicas of the same system [90]
∥~̇γi(s)
B(~γi(s)−~γj(s))
,(7.3)
where B denotes the spatial part of the forcing potential
correlation. The next step is to study this problem by a
variational approach. The Hamiltonian Hn is replaced
by an effective Gaussian quadratic Hamiltonian that can
be written as
Heff =
~γi(τ)Gij(τ−τ ′)~γj(τ ′)dτdτ ′. (7.4)
The kernel Gij is then chosen in such a way that it
minimizes the free energy. It is shown in [22] that the
optimal Gaussian Hamiltonian is the solution of a sys-
tem of equations that can be solved following the ansatz
proposed in [89]. When d > 2 this approach singles
out two régimes depending on the Reynolds number
Re = ε1/3L
f /ν. These régimes are separated by the
critical value Rec = [2(1−2/d)1−d/2]1/3. When Re < Rec
the optimal solution is of the form Gij = G0 δij +G1 and
obeys the replica symmetry. In finite-size systems it cor-
responds to a linear velocity profile. When Re > Rec
the correct solution is given by the one-step replica-
symmetry-breaking scheme (see [89]). The off-diagonal
elements of Gij are then parameterized with two func-
tions depending on whether the indices i and j belong to
the same block or to different blocks. Qualitatively, the
one-step replica-symmetry-breaking approach amounts
to the assumption that the instantaneous velocity po-
tential can be written as a weighted sum of Gaussians,
leading to an approximation of the velocity field as
~v(~x, t) ≃
α(~x− ~rα) e
−Re (fα+‖~x−~rα‖
2/2L2
−Re (fα+‖~x−~rα‖2/2L2f )
, (7.5)
where the fα’s are independent variables with a Poisson
distribution of density exp(−f). The ~rα are uniformly
and independently distributed in space. In (7.5) the sum
over α is running from 1 to a large-enough integer M .
The typical shape of the approximation of the veloc-
ity field given by (7.5) is represented in figure 39(a) in
the two-dimensional case. In the limit of large Reynolds
numbers the random velocity field given by (7.5) typi-
cally contains cells of width ∝ Lf . The width of a shock
separating two cells is of the order of Lf/Re.
The replica approximation (7.5) leads to an estimate of
the PDF p(∆v, r) of the longitudinal velocity increment
∆v = (~v(~x+ r ~e, t) − ~v(~x, t)) · ~e, where ~e is an arbitrary
unitary vector. When Re ≫ 1 and r ≪ Lf this approxi-
mation takes the particularly simple asymptotic form
p(∆v, r) ≈ δ
∆v − Uf
, (7.6)
where Uf = Re ν/Lf is the typical velocity associated
to the scale Lf and g is a scaling function that is deter-
mined explicitly in [22]. This approximation is in agree-
ment with the following qualitative picture. With a prob-
ability almost equal to one, the two points ~x and ~x+ r ~e
lie in the same cell; the velocity increment is then given
by the typical velocity gradient which, according to the
Fig. 39. (a) Typical shape of the velocity field given by the
replica approximation in dimension d = 2 obtained from
(7.5) for Re = 103. The contour lines represent the velocity
modulus. Note the cell structure of the domain. (b) Scaling
exponents of the pth order structure function.
approximation (7.5), is order Uf/Lf . With a probabil-
ity r/Lf the two points are sitting on different sides of
a shock separating two such cells and the associated ve-
locity difference is of the order of Uf .
The structure functions of the velocity field given by
the various moments of ∆v can be straightforwardly
estimated from the approximation (7.6). Their scaling
behavior 〈∆vp〉 ∼ rζp at small separations r display
a bifractal behavior as sketched in figure 39(b). When
p < 1, the first term on the right-hand side of (7.6) domi-
nates and 〈∆vp〉 ∝ Upf (r/Lf )p. For p > 1 the shock con-
tribution is dominating the small-r behavior and thus
〈∆vp〉 ∝ Upf (r/Lf ).
This approach, which makes use of replica tricks, is as we
have seen able to catch the leading scaling behavior of
velocity structure functions in any dimension. It is based
on approximations of the velocity field by the superposi-
tion (7.5) of Gaussian velocity potentials. A first advan-
tage of this method is that it catches the generic aspect
of the solution including the hierarchy of high-order sin-
gularities appearing in the solution when Re → ∞ which
was examined in section 2.3. This method also gives
predictions regarding the dependence on Re of the sta-
tistical properties of the solution. However, as stressed
in [22], the validity of this approximation is expected to
hold only in the limit of infinite space dimension d. In
particular, it is known that for d ≤ 2 a full continuous
replica-symmetry-breaking scheme is needed [89]. Nev-
ertheless, as we have seen, there is enough evidence that
this approach describes very well the qualitative aspects
of the solution.
7.2 Dissipative anomaly and operator product expan-
The replica-trick approach described in the previous
subsection cannot reproduce one of the main statis-
tical features of the solution, namely the tails of the
velocity increments PDF p(∆v, r). Indeed the predic-
tion (7.6) based on a variational approximation of the
velocity field implies that p identically vanishes when
∆v > Uf (r/Lf ). In order to study the quantitative be-
havior of the PDF p(∆v, r) in the inviscid limit ν → 0
(or equivalently Re → ∞), Polyakov [99] proposed to
use an operator product expansion. This approach leads
to an explicit expression for p(∆v, r) and predicts a
super-exponential tail at large positive values and a
power-law behavior for negative ones. Such predictions
have immediate implications for the asymptotics of the
PDF p(ξ) of the velocity gradient ξ = ∂xv. The work
of Polyakov was the starting point of a controversy on
the value of the exponent of the left tail of p(ξ). Before
returning to this issue in the next subsection, we give
in the sequel a quick overview of the original work by
Polyakov.
We henceforth focus on the one-dimensional solutions
to the Burgers equation with Gaussian forcing whose
autocorrelation is given by (7.1). Following [99] (see also
[19,20]) we introduce the characteristic function of the
n-point velocity distribution
Zn(λj , xj ; t) ≡
e λ1 v(x1,t)+···+λn v(xn,t)
. (7.7)
For a finite value of the viscosity ν, it is easily seen that
this quantity is a solution to a Fokker–Planck (master)
equation obtained by differentiating Zn with respect to
t and using the Burgers equation and the fact that the
forcing is Gaussian and δ-correlated in time. This leads
b(xi − xj)λi λj Zn + D(n)ν , (7.8)
where b ≡ (d2B)/(dr2) denotes the spatial part of the
correlation of the forcing applied to the velocity field.
D(n)ν denotes the contribution of the dissipative term and
reads
D(n)ν ≡ ν
v(xj , t)
λj v(xj ,t)
. (7.9)
This term does not vanish in the limit ν → 0 since the
solutions to the Burgers equation develop singularities
with a finite dissipation. It has been proposed in [99] to
use an analogy with the anomalies appearing in quantum
field theory in order to tackle this term in the inviscid
limit. The important assumption is then made that the
singular term in the operator product expansion relates
linearly to the characteristic function Zn. Since this ex-
pansion should preserve the statistical symmetries of the
Burgers equation, it leads to the replacement in all av-
erages of the singular limit limν→0 ν λ (∂
xv) e
λ v by the
asymptotic expression
b − 1
eλ v , (7.10)
where the coefficients a, b and c are parameters that can
be determined only indirectly. However their possible
values can be restricted by requiring that Zn is the char-
acteristic function of a probability distribution which is
non-negative, finite, normalizable, and that the dissipa-
tive term D(n)ν acts as a positive operator. Finding these
coefficients is similar to an eigenvalue problem in quan-
tum mechanics.
We now come to a crucial point in Polyakov’s approach.
Important restrictions on the form of the different
anomalous terms in (7.10) result from the fact that the
solutions to the Burgers equation obey a certain form of
Galilean invariance. A notion of “strong Galilean prin-
ciple” is introduced for invariance of the n-point distri-
bution of velocity under the transformation v 7→ v + v0
with v0 an arbitrary constant. As a consequence, the n-
point characteristic function Zn has to be proportional
to δ(λ1 + · · ·+λn). The operators appearing in the limit
ν → 0 have to be consistent with such an invariance.
In [99] it is argued that this symmetry is automatically
broken by the forcing that introduces a typical velocity
〈v2〉1/2 ∝ b1/3(0)L1/3. However Polyakov assumes this
“strong Galilean principle” to be asymptotically recov-
ered in the limit L → ∞ of infinite-size systems. In the
case of finite-size systems, when L is of the order of the
correlation length Lf of the forcing, the strong Galilean
symmetry is broken because of the conservation of the
spatial average of v which introduces a characteristic
velocity v0 = (1/L)
v(x, t) dx. However, the Galilean
symmetry should be recovered when averaging the cor-
relation functions with respect to the mean velocity v0.
This symmetry restoration was introduced in [20] where
it is referred to as the “weak Galilean principle”. The
n-point characteristic function associated to an aver-
age velocity v0 relates to that associated to a vanishing
mean velocity by
Zn(λj , xj ; t; v0) = e
Zn(λj , xj ; t; 0) .
After averaging with respect to v0, one obtains
Zn(λj , xj ; t) = 2π δ
Zn(λj , xj ; t; 0) . (7.11)
One can easily check that (7.8), together with the dis-
sipative term given by (7.10), are compatible with this
expression for the n-point characteristic function Zn.
Moreover, any higher-order term in the expansion (7.10)
of the dissipative anomaly would violate Galilean invari-
ance.
To obtain the statistical properties of the solution, one
needs to further restrict the values of the three free pa-
rameters a, b, and c appearing in the expansion (7.10).
Following [99] this can be done by considering the case
n = 2 that corresponds to the equation for the PDF of
velocity differences. Performing the change of variables
λ1,2 = Λ ± µ and x1,2 = X ± y/2, and assuming that
λ ≪ µ and y ≪ Lf (so that the spatial part of the
forcing correlation is to leading order b(y) ≃ b0 − b1y2),
the stationary and space-homogeneous solutions to the
master equation (7.8)) satisfy
− (2b0Λ2 + b1µ2y2)Z2 =
= aZ2 +
. (7.12)
It is next assumed in [99] (see also [20]) that the velocity
difference v(x1, t) − v(x2, t) is statistically independent
of the mean velocity (v(x1, t) + v(x2, t))/2. This implies
that the two-point characteristic function factorizes as
Z2 = Z
2 (Λ)Z
2 (µ, y), where the two functions Z
2 and
Z−2 satisfy the closed equations
− 2b0Λ2Z+2 = cΛ
, (7.13)
∂2Z−2
− b1µ2y2Z−2 = aZ
. (7.14)
The solution to the first equation corresponds to a Gaus-
sian distribution which is normalizable only if c < 0. As
shown numerically in [20] this distribution is representa-
tive of the bulk of the one-point velocity PDF. Informa-
tion on the solutions to the second equation can be ob-
tained assuming the scaling property Z−2 (µ, y) = Φ(µy),
which amounts to considering only those contributions
to the distribution of velocity differences stemming from
velocity gradients ξ = ∂xv. This yields a prediction the
negative and positive tails of the PDF of velocity gradi-
ents:
p(ξ) ∝ |ξ|−α when ξ → −∞ , (7.15)
p(ξ) ∝ ξβ exp(−C ξ3) when ξ → +∞ , (7.16)
where C is a constant, which depends only on the
strength of the forcing. The two exponents α and β are
related to the coefficient b of the anomaly by
α = 2b + 1 and β = 2b − 1 . (7.17)
The value of b remains undetermined but is prescribed
to belong to a certain range. This approach was first
designed in [99] for infinite-size systems where strong
Galilean invariance holds. In that case consistency with
such an invariance leads to dropping the third term in
the operator product expansion (i.e. c = 0). Positivity
and normalizability of the two-point velocity PDF and
non-positivity of the anomalous dissipation operator im-
ply that the two other coefficients form a one-parameter
family with 3/4 ≤ b ≤ 1. In particular, this implies
that the left tail of the velocity gradient PDF with ex-
ponent α should be shallower than ξ−3. As we will see in
the next section, strong evidence has been obtained that
p(ξ) ∝ ξ−7/2 for ξ → −∞. This seems to contradict the
approach based on operator product expansion. How-
ever, as argued in [20], the breaking of strong Galilean in-
variance occurring in finite-size systems and resulting in
the presence of the c anomaly broadens the range of ad-
missible values for b. In particular it allows for the value
b = 5/4 which corresponds to the exponent α = 7/2.
7.3 Tails of the velocity gradient PDF
After the numerical work of Chekhlov and Yakhot [29],
the asymptotic behavior at large positive and negative
values of the PDF of velocity derivatives ξ = ∂xv for
the one-dimensional randomly forced Burgers equation
attracted much attention. A broad consensus emerged
around the prediction of Polyakov [99] that p(ξ) dis-
plays tails of the form (7.16) and (7.15), but the values
of the exponents α and β were at the center of a contro-
versy. Note that the presence of a super-exponential tail
∝ exp(−C ξ3) at large positive arguments has been con-
firmed by the use of instanton techniques [60] and that
the only remaining uncertainty concerns the exponent
of the algebraic prefactor. A standard approach to de-
termine the exponents α and β appearing in (7.15) and
(7.16) makes use of the stationary solutions to the in-
viscid limit of the Fokker–Planck equation for the PDF,
namely
−ξp+ν∂ξ
∂3xv | ∂xv=ξ
= b̃∂2ξp . (7.18)
Here the brackets 〈·|·〉 denote conditional averages and
the right-hand side expresses the diffusion of probability
due to the delta-correlation in time of the forcing. The
main difficulty in studying the solutions of (7.18) stems
from the treatment of the dissipative term Dν(ξ) =
∂3xv|∂xv=ξ
in the limit ν → 0. The value α = 3
is obtained if a piecewise linear approximation is made
for the solutions of the Burgers equation [21]. Gotoh and
Kraichnan [59] argued that the dissipative term is to
leading order negligible and presented analytical and nu-
merical arguments in favor of α = 3 and β = 1. However,
the inviscid limit of (7.18) contains anomalies due to the
singular behavior of Dν(ξ) in the limit ν → 0. As we
have seen in previous section, the approach based on the
use of an operator product expansion [99] leads to a rela-
tion involving unknown coefficients which must be deter-
mined, e.g., from numerical simulations [111,19,20], and
restricts the possible values to 5/2 ≤ α ≤ 3 [6]. Anoma-
lies cannot be studied without a complete description of
the singularities of the solutions, such as shocks, and a
thorough understanding of their statistical properties.
E, Khanin, Mazel and Sinai made a crucial observa-
tion in [37] that large negative gradients stem mainly
from preshocks, that is the cubic-root singularities in
the velocity preceding the formation of shocks (see sec-
tion 2.3). They then used a simple argument for de-
termining the fraction of space-time where the veloc-
ity gradient is less than some large negative value. This
leads to α = 7/2, provided preshocks do not cluster.
Later on, this approach has been refined by E and Van-
den Eijnden who proposed to determine the dissipative
anomaly of (7.18) using formal matched asymptotics [39]
or bounded variation calculus [42]. As we shall see be-
low, with the assumption that shocks are born with a
zero amplitude, that their strengths add up during colli-
sions, and that there ar no accumulations of preshocks,
the value α = 7/2 was confirmed [42]. Other attempts to
derive this value using also isolated preshocks have been
made [81,6]. Note that there are simpler instances, in-
cluding time-periodic forcing [9] (see section 6) and de-
caying Burgers turbulence with smooth random initial
conditions [8,42] (see section 4.1), which fall in the uni-
versality class α = 7/2, as can be shown by systematic
asymptotic expansions using a Lagrangian approach.
We give here the flavor of the approach used in [39]
in order to estimate the dissipative anomaly D0(ξ) =
limν→0D
ν(ξ). One first notices that for |ξ| ≫ b̃1/3, the
forcing term in the right-hand side of (7.18) becomes
negligible, so that stationary solutions to the Fokker–
Planck equation satisfy
p(ξ) ≈ |ξ|−3
dξ′ ξ′Dν(ξ′). (7.19)
A straightforward consequence of this asymptotic ex-
pression is that, if the integral in the right-hand side de-
creases as ξ → −∞ (i.e. if ξDν(ξ) is integrable), then
p(ξ) decreases faster than |ξ|−3, and thus α > 3.
To get some insight into the behavior ofDν as ν → 0, one
next observes that the solutions to the one-dimensional
Burgers equation contain smooth regions where viscos-
ity is negligible, which are separated by thin shock layers
where dissipation takes place. The basic idea consists in
splitting the velocity field v into the sum of an outer so-
lution away from shocks and of an inner solution near
them for which boundary layer theory applies. Matched
asymptotics are then used to construct a uniform ap-
proximation of v. To construct the inner solution near
a shock centered at x = x⋆, one performs the change of
variable x 7→ x̃ = (x−x⋆)/ν and looks for an expression
of ṽ(x̃, t) = v(x⋆+νx̃, t) in the form of a Taylor expan-
sion in powers of ν: ṽ = ṽ0+νṽ1+o(ν). At leading order,
the inner solution satisfies
[ṽ0 − v⋆] ∂x̃ṽ0 = ∂2x̃ṽ0, (7.20)
where v⋆ = (dx⋆)/(dt). This expression leads to the well-
known hyperbolic tangent velocity profile
ṽ0 = v⋆ −
. (7.21)
Here, s = v(x⋆+, t)−v(x⋆−, t) denotes here the velocity
jump across the shock and is given by matching condi-
tions to the outer solution. The term of order ν is then
a solution of
∂tṽ0 + [ṽ0 − v⋆] ∂x̃ṽ1 = ∂2x̃ṽ1 + f(x, t). (7.22)
In order to evaluate the dissipative anomaly, it is con-
venient to assume spatial ergodicity so that the viscous
term in (7.18) can be written as
Dν(ξ) = ν∂ξ lim
dx ∂3xv δ(∂xv−ξ). (7.23)
In the limit ν → 0 the only remaining contribution stems
from shocks and is thus given by the inner solution. Using
the expansion of the solution up to the first order in ν,
this leads to writing the dissipative term in the limit of
vanishing viscosity as (see Appendix of [41] for details)
D0(ξ) =
ds s [p+(s, ξ) + p−(s, ξ)] , (7.24)
where ρ is the density of shocks and p+ (respectively p−)
is the joint probability of the shock jump and of the value
of the velocity gradient at the right (respectively left) of
the shock. This expression guarantees the finiteness of
the dissipative anomaly, and in particular the fact that
the integral in the right-hand side of (7.19) is finite in
the limit ν → 0 and converges to 0. As a consequence,
this gives a proof that the exponent α of the left tail of
the velocity gradient PDF is larger than 3.
To proceed further, E and Vanden Eijnden proposed to
estimate the probability densities p+ and p− by deriving
master equations for the joint probability of the shock
strength s, its velocity v⋆ and the values ξ
± of the veloc-
ity gradient at its left and at its right. This is done in [42]
using a formulation of Burgers dynamics stemming from
bounded variation calculus. More precisely, it is shown
in [108] that the Burgers equation is equivalent to con-
sidering the solutions to the partial differential equation
∂tv + v̄∂xv = f , (7.25)
where v̄(x, t) = (v(x+, t) + v(x−, t))/2. Basically this
means that Burgers dynamics can be formulated in
terms of the transport of the velocity field by its average
v̄. This formulation straightforwardly yields a master
equation for v(x±, t) and ∂xv(x±, t) which is then used
to estimate p± and the dissipative anomaly (7.24). Al-
though the treatment of the master equation does not
involve any closure hypothesis, it is not fully rigorous:
in particular it requires the assumption that shocks are
created with zero amplitude and that shock amplitudes
add up during collision. However such an approaches
strongly suggests that α = 7/2 and β = 1.
Obtaining numerically clean scaling for the PDF of gra-
dients is not easy with standard schemes. Let us ob-
serve that any method involving a small viscosity, either
introduced explicitly (e.g. in a spectral calculation) or
stemming from discretization (e.g. in a finite difference
calculation), may lead to the presence of a power-law
range with exponent −1 at very large negative gradi-
ents [59]. This behavior makes the inviscid |ξ|−α range
appear shallower than it actually is, unless extremely
high spatial resolution is used. In contrast, methods that
directly capture the inviscid limit with the appropriate
shock conditions, such as the fast Legendre transform
method [94], lead to delicate interpolation problems.
They have been overcome in the case of time-periodic
forcing [9] but with white-noise-in-time forcing, it is dif-
ficult to prevent spurious accumulations of preshocks
leading to α = 3.
To avoid such pitfalls, a Lagrangian particle and shock
tracking method was developed in [6]. This method is
able to separate shocks and smooth parts of the solution
and is particularly effective for identifying preshocks.
The main idea is to consider the evolution of a set of
N massless point particles accelerated by a discrete-in-
time approximation of the forcing with a uniform time
step. When two of these particles intersect, they merge
and create a new type of particle, a shock, characterized
by its velocity (half sum of the right and left velocities
of merging particles) and its amplitude. The particle-
like shocks then evolve as ordinary particles, capture
further intersecting particles and may merge with other
shocks. In order not to run out of particles too quickly,
the initial small region where particles have the least
chance of being subsequently captured is determined by
localization of the global minimizer of the Lagrangian
action (see section 5.1). The calculation is then restarted
from t = 0 for the same realization of forcing but with
a vastly increased number of particles in that region.
This particle and shock tracking method gives complete
control over shocks and preshocks.
slope = −7/2
−3.75
−3.25
−7/2±1%
Fig. 40. PDF of the velocity gradient at negative values in
log-log coordinates obtained by averaging over 20 realiza-
tions and a time interval of 5 units of time (after relaxation
of transients). The simulation involves up to N = 105 parti-
cles and the forcing is applied at discrete times separated by
δt = 10−4. Upper inset: local scaling exponent (from [6]).
Figure 40 shows the PDF of the velocity gradients in
log-log coordinates at negative values, for a Gaussian
forcing restricted to the first three Fourier modes with
equal variances such that the large-scale turnover time is
order unity. Quantitative information about the value of
the exponent is obtained by measuring the “local scaling
exponent”, i.e. the logarithmic derivative of the PDF
calculated in this case using least-square fits on half-
decades. It is seen that over about five decades, the local
exponent is within less than 1% of the value α = 7/2
predicted by E et al. [37].
7.4 Self-similar forcing and multiscaling
As we have seen in section 7.1, the solutions to the Burg-
ers equation in a finite domain and with a large-scale
forcing have structure functions (moments of the ve-
locity increment) displaying a bifractal scaling behav-
ior. Such a property can be easily interpreted by the
presence of a finite number of shocks with a size order
unity in the finite system. Somehow this double scal-
ing and its relationship with singularities gives some in-
sight on the multiscaling properties that are expected
in the case of turbulent incompressible hydrodynamics
flows. There is a general consensus that the turbulent
solutions to the Navier–Stokes equations display a full
multifractal spectrum of singularities which are respon-
sible for a nonlinear p-dependence of the scaling expo-
nents ζp associated to the scaling behavior of the p-th
order structure function [53]. The construction of simple
tractable models which are able to reproduce such a be-
havior has motivated much work during the last decades.
Significant progress, both analytical and numerical, has
been made in confirming multiscaling in passive-scalar
and passive-vector problems (see, e.g., [46] for a review).
However, the linearity of the passive-scalar and passive-
vector equations is a crucial ingredient of these studies,
so it is not clear how they can be generalized to fluid
turbulence and the Navier–Stokes equation.
After the work of Chekhlov and Yakhot [30], it appeared
that the Burgers equation with self-similar forcing could
be the simplest nonlinear partial differential equation
which has the potential to display multiscaling of veloc-
ity structure functions. We report in this section various
works that tried to confirm or to weaken this statement.
Let us consider the solutions to the one-dimensional
Burgers equation with a forcing term f(x, t) which is ran-
dom, space-periodic, Gaussian and whose spatial Fourier
transform has correlation
〈f̂(k, t)f̂(k′, t′)〉 = 2D0 |k|β δ(t− t′) δ(k + k′). (7.26)
The exponent β determines the scaling properties of the
forcing. When β > 0 the force acts at small scales; for
instance β = 2 corresponds to thermal noise for the ve-
locity potential, and thus to the KPZ model for interface
growth [74]. It is well known in this case (see, e.g., [5])
that the solution displays simple scaling (usually known
as KPZ scaling), such that ζq = q for all q. More gener-
ally, the case β > 0 can be exactly solved using a one-
loop renormalization group approach [88].
As stressed in [64], renormalization group techniques fail
when β < 0 and the forcing acts mostly at large scales
and non-linear terms play a crucial role. When β < −3,
the forcing is differentiable in the space variable, the so-
lution is piecewise smooth and contains a finite number
of shocks with sizes order unity. The scaling exponents
are then ζp = min (1, p). In the case of non-differentiable
forcing (−3 < β < 0), the presence of order-unity shocks
and dimensional arguments suggest that the scaling ex-
ponents are ζp = min (1, −pβ/3). However, very little is
known regarding the distribution of shocks with inter-
mediate sizes. In particular, there is no clear evidence
whether or not they form a self-similar structure at small
scales. We summarize here some studies which were done
on Burgers turbulence with self-similar forcing to show
how difficult it might be to measure scaling laws of struc-
ture functions and in particular how logarithmic correc-
tions can masquerade anomalous scaling.
For this we focus on the case β = −1 which has attracted
much attention; indeed, dimensional analysis suggests
that ζp = p/3 when p ≤ 3, leading to a K41-type −5/3
energy spectrum. Early studies [29,30] seemed to confirm
this prediction using pseudo-spectral viscous numerical
Fig. 41. Representative snapshots of the velocity v (jagged
line) in the statistically stationary régime, and of the inte-
gral of the force f over a time step (rescaled for plotting
purposes).
simulations at rather low resolutions (around ten thou-
sands gridpoints). It was moreover argued in [64,65] that
a self-similar forcing with −1 < β < 0, could lead to gen-
uine multifractality. The lack of accuracy in the determi-
nation of the scaling exponents left open the question of
a weak anomalous deviation from the dimensional pre-
diction. This question was recently revisited in [91] with
high-resolution inviscid numerical simulations using the
fast Legendre transform algorithm (see section 2.4.2).
A typical snapshot of the forcing and of the solution in
the stationary régime are represented in figure 41. It is
clear that because of shocks the velocity develops small-
scale fluctuations much stronger than those present in
the force. However one notices that shock dynamics and
spatial finiteness of the system lead, as predicted, to the
presence of few shocks with order-unity sizes.
0 1 2 3 4 5
ξ p 0 1 2 3 4 5
Fig. 42. Scaling exponents ζp versus order p for
N = 216(⋄), 218(∗), and 220(◦) grid points. Error bars (see
text) are shown for the case N = 220. The deviation of ζp
from the exponents for bifractal scaling (full lines), shown
as an inset, naively suggests multiscaling (from [91])
Structure functions were measured with high accuracy.
They typically exhibit a power-law behavior over nearly
three decades in length scale; this is more than two
decades better than in [30]. In principle one expects to
be able to measure the scaling exponents with enough
accuracy to decide between bifractality and multiscal-
ing. Surprisingly the naive analysis summarized in fig-
ure 42 does suggest multiscaling: the exponents ζp de-
viate significantly from the bifractal-scaling prediction
(full lines). Since the goal here is to have a precise han-
dle on the scaling properties of velocity increments, it is
important to carefully define how the scaling exponents
are measured. They are estimated from the average log-
arithmic derivative of Sabsp (r) = 〈|v(x+r)−v(x)|p〉 over
almost two decades in the separation r. The error bars
shown are given by the maximum and minimum devia-
tions from this mean value in the fitting range. Note also
that the observed multiscaling is supported by the fact
that there is no substantial change in the value of the
exponents when changing the number N of grid points
in the simulation from 216 to 220: any dependence of
ζp upon N is much less than the error bars determined
through the procedure described above.
−5 −4 −3 −2 −1
−6 −4 −2
Fig. 43. Log-log plots of Sabs3 (r) (dashed line), S3(r)
(crosses), and 〈(δ+v)3〉 (squares) versus r. The continuous
line is a least-square fit to the range of points limited by two
vertical dashed lines in the plot. Inset: An explicit check of
the von Kármán–Howarth relation (7.27) from the simula-
tions with N = 220 reported in [91]. The dashed curve is the
integral of the spatial part of the forcing correlation and the
circles represent the numerical computation of the left-hand
side.
As found in [91], the observed deviations of the scaling
exponents from bifractality are actually due to the con-
tamination by subleading terms in Sabsp (r). To quantify
this effect, let us focus on the third-order structure func-
tion (p = 3) for which one measures ζ3 ≈ 0.85±0.02 over
nearly four decades (see figure 43). To estimate sublead-
ing terms we first notice that the third-order structure
function S3(r) ≡ 〈(v(x + r) − v(x))3〉, which is defined,
this time, without the absolute value, obeys an analog of
the von Kármán–Howarth relation in fluid turbulence,
namely
S3(r) =
b(r′)dr′, (7.27)
where b(·) denotes the spatial part of the force corre-
lation function, i.e. 〈f(x + r, t′)f(x, t)〉 = b(r)δ(t − t′).
This relation, together with the correlation (7.26) and
β = −1, implies the behavior S3(r) ∼ r ln r at small
separations r. As seen in figure 43, the graph of S3(r)
in log-log coordinates indeed displays a significant cur-
vature which is a signature of logarithmic corrections.
The next step consists in decomposing the velocity in-
crements δrv = v(x + r, t) − v(x, t) into their positive
δ+r v and negative δ
r v parts. It is clear that
Sabs3 (r) = −
(δ−r v)
(δ+r v)
S3(r) =
(δ−r v)
(δ+r v)
, (7.28)
so that
(δ+r v)
= (Sabs3 (r) + S3(r))/2. As seen in fig-
ure 43 the log-log plot of
(δ+r v)
as a function of r
is nearly a straight line with slope ≈ 1.07 very close to
unity. This observation is confirmed in [91] by indepen-
dently measuring the PDFs of positive and negative ve-
locity increments. Assuming that
(δ+r v)
∼ B r, one
obtains the following prediction for the small-r behav-
iors of the third-order structure functions
Sabs3 (r) ∼ −Ar ln r +B r,
S3(r) ∼ Ar ln r +B r. (7.29)
This suggests that the only difference in the small-
separation behaviors of Sabs3 (r) and S3(r) is the sign in
the balance between the leading term ∝ r ln r and the
subleading term ∝ r. In a log-log plot this difference
amounts to shifting the graph away from where it is
most curved and thus makes it straighter, albeit with a
(local) slope which is not unity. This explains why signif-
icant deviations from 1 are observed for ζ3. Note that a
similar approach can be used for higher-order structure
functions. It leads for instance to S4(r) ≈ Cr −Dr4/3,
where C and D are two positive constants. The nega-
tive sign before the sub-leading term (r4/3) is crucial.
It implies that, for any finite r, a naive power-law fit
to S4 can yield a scaling exponent less than unity. The
presence of sub-leading, power-law terms with oppo-
site signs also explains the small apparent “anomalous”
scaling behavior observed for other values of p in the
simulations. Note that similar artifacts involving two
competing power-laws have been described in [16,7].
The work reported in this section indicates that a naive
interpretation of numerical measurements might result
in predicting artificial anomalous scaling laws. In the
case of Burgers turbulence for which high-resolution nu-
merics are available and statistical convergence of the
averages can be guaranteed, we have seen that it is not
too difficult to identify the numerical artifacts which
are responsible for such a masquerading. However this
is not always the case. For instance, it seems reason-
able enough to claim that attacking the problem of mul-
tiscaling in spatially extended nonlinear systems, such
as Navier–Stokes turbulence, requires considerable the-
oretical insight that must supplement sophisticated and
heavy numerical simulations and experiments. Note fi-
nally that, up to now, the question of the presence or not
of anomalous scaling laws in the Burgers equation with
a self-similar forcing with exponent −1 < β < 0 remains
largely open.
8 Concluding remarks and open questions
This review summarizes recent work connected with the
Burgers equation. Originally this model was introduced
as a simplification of the Navier–Stokes equation with
the hope of shedding some light on issues such as turbu-
lence. This hope did not materialize. Nevertheless many
of the interesting questions that have been addressed
for Burgers turbulence are eventually transpositions of
similar questions for Navier–Stokes turbulence. One par-
ticularly important instance is the issue of universality
with respect to the form of the forcing and of the initial
condition. For Burgers turbulence most of the universal
features, such as scaling exponents or functional forms
of PDF tails are dominated by the presence of shocks
and other singularities in the solution. This applies both
to the case of decaying turbulence driven by random ini-
tial conditions and randomly forced turbulence. In the
latter case one is mostly interested in analysis of station-
ary properties of solutions, for example stationary dis-
tribution for velocity increments or gradients. Another
set of questions is motivated by more mathematical con-
siderations. It mainly concerns the construction of a sta-
tionary invariant measure when Burgers dynamics in a
finite-size domain is supplemented by an external ran-
dom source of energy. Again it has been shown that the
presence of shocks, and in particular of global shocks,
plays a crucial role in the construction of the statisti-
cally stationary solution. Both physical and mathemat-
ical questions lead to a similar answer: one first needs to
describe and control shocks. The main message to retain
for hydrodynamical turbulence is hence a strong confir-
mation of the common wisdom that it cannot be fully
understood without a detailed description of singulari-
ties. Moreover, the behavior depends not only on the lo-
cal structure of singularities, but also on their distribu-
tion at larger scales. Here a word of caution: for incom-
pressible fully developped Navier–Stokes turbulence, we
have no evidence that the universal scaling properties
observed in experiments and simulations stem from real
singularities. Indeed the issue of a finite-time blow-up of
the three-dimensional Euler equation is still open (see,
e.g. [56]). Another important observation that can be
drawn from the study of Burgers turbulence is that both
the tools used and the answers obtained strongly de-
pend on the kind of setting one considers: decay versus
forced turbulence, finite-size versus infinite-size systems,
smooth versus self-similar forcing, etc.
Besides turbulence, the random Burgers equation has
various applications in cosmology, in non-equilibrium
statistical physics and in disordered media. Among
them, the connection to the problem of directed poly-
mers has attracted much attention. As already noted in
the Introduction, there is a mathematical equivalence
between the zero-viscosity limit of the forced Burgers
equation and the zero-temperature limit for directed
polymers. We have seen in section 5.4 that the so-called
KPZ scaling, which usually is derived for a finite tem-
perature, can be established can be established also in
the zero-temperatur limit, using the action minimizer
representation. Such an observation leads to two related
questions: to what extent can the limit of zero temper-
ature give an insight into finite-temperature polymer
dynamics and how can the global minimizer formalism
be extended to tackle the finite-temperature setting?
It looks plausible that in polymer dynamics, or more
generally in the study of random walks in a random
potential, the trajectories carrying most of the Gibbs
probability weight are defining corridors in space time.
These objects can concentrate near the trajectories of
global minimizers but, at the moment, there is no for-
malism to describe them, nor attempts to quantify their
contribution to the Gibbs statistics.
Another important open question concerns the multi-
dimensional extensions of the Burgers equation. As we
have seen, when the forcing is potential, the potential
character of the velocity field is conserved by the dynam-
ics. This leads to the construction of stationary solutions
which carry many similarities with the one-dimensional
case. Up to now there is only limited understanding of
what happens when the potentiality assumption of the
flow is dropped. This problem has of course concrete
applications in gas dynamics and for disperse inelastic
granular media (see, e.g., [12]). An interesting question
concerns the construction of the limit of vanishing vis-
cosity, given that the Hopf–Cole transformation is in-
applicable in the non-potential case. Understanding ex-
tensions of the viscous limiting procedure to the non-
potential case might give new insight into the problem of
the large Reynolds number limit in incompressible tur-
bulence. Another question related to non-potential flow
concerns the interactions between vorticity and shocks.
For instance, in two dimensions the vorticity is trans-
ported by the flow. This results in its growth in the highly
compressible regions of the flow. The various singulari-
ties of the velocity field should hence be strongly affected
by the flow rotation and, in particular, the shocks are
expected to have a spiraling structure.
We finish with few remarks on open mathematical prob-
lems. As we have seen in the one-dimensional case, one
can rigorously prove hyperbolicity of the global mini-
mizer. In the multi-dimensional case it is also possible
to establish the existence and, in many cases, unique-
ness of the global minimizer. However, the very impor-
tant question of its hyperbolicity is still an open prob-
lem. If proven, hyperbolicity would allow for rigorous
analysis of the regularity properties of the stationary so-
lutions and of the topological shocks. There are many
interesting problems – even basic issues of existence and
uniqueness – in the non-compact case where at present a
mathematical theory is basically absent. Finally, a very
challenging open problem concerns the extension of the
results on the evolution of matter inside shocks to the
case of general Hamilton-Jacobi equations.
Acknowledgements
Over the years of our work on Burgers turbulence, we
profited a lot from numerous discussions with Uriel
Frisch whose influence on our work is warmly acknowl-
edged. We also want to express our sincere gratitude
to all of our collaborators: W. E, U. Frisch, D. Gomes,
V.H. Hoang, R. Iturriaga, D. Khmelev, A. Mazel, D. Mi-
tra, P. Padilla, R. Pandit, Ya. Sinai, A. Sobolevskĭı, and
B. Villone.
While writing this article, we benefited from discus-
sions with M. Blank, I. Bogaevsky, K. Domelevo, V.
Epstein, and A. Sobolevskĭı. Finally, our thanks go to
Itamar Procaccia whose encouragements and patience
are greatly appreciated.
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From interface dynamics to cosmology
The Burgers equation in statistical mechanics
The adhesion model in cosmology
A benchmark for hydrodynamical turbulence
Basic tools
Inviscid limit and variational principle
Variational principle for the viscous case
Singularities of Burgers turbulence
Remarks on numerical methods
Decaying Burgers turbulence
Geometrical constructions of the solution
Kida's law for energy decay
Brownian initial velocities
Transport of mass in the Burgers/adhesion model
Mass density and singularities
Evolution of matter inside shocks
Connections with convex optimization problems
Forced Burgers turbulence
Stationary régime and global minimizer
Topological shocks
Hyperbolicity of the global minimizer
The case of extended systems
Time-periodic forcing
Kicked Burgers turbulence
Connections with Aubry--Mather theory
Velocity statistics in randomly forced Burgers turbulence
Shocks and bifractality -- a replica variational approach
Dissipative anomaly and operator product expansion
Tails of the velocity gradient PDF
Self-similar forcing and multiscaling
Concluding remarks and open questions
Acknowledgements
References
|
0704.1612 | Analytical evaluation of the X-ray scattering contribution to imaging
degradation in grazing-incidence X-ray telescopes | Astronomy & Astrophysics manuscript no. 7228 c© ESO 2022
March 1, 2022
Analytical evaluation of the X-ray scattering contribution
to imaging degradation in grazing-incidence X-ray telescopes
D. Spiga
INAF/Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC) - Italy
Received 02 February 2007 / Accepted 27 March 2007
ABSTRACT
Aims. The focusing performance of X-ray optics (conveniently expressed in terms of HEW, Half Energy Width) strongly depend on
both mirrors deformations and photon scattering caused by the microroughness of reflecting surfaces. In particular, the contribution
of X-ray Scattering (XRS) to the HEW of the optic is usually an increasing function H(E) of the photon energy E. Therefore, in
future hard X-ray imaging telescopes of the future (SIMBOL-X, NeXT, Constellation-X, XEUS), the X-ray scattering could be the
dominant problem since they will operate also in the hard X-ray band (i.e. beyond 10 keV). In order to ensure the imaging quality at
all energies, clear requirements have to be established in terms of reflecting surfaces microroughness.
Methods. Several methods were proposed in the past years to estimate the scattering contribution to the HEW, dealing with the surface
microroughness expressed in terms of its Power Spectral Density (PSD), on the basis of the well-established theory of X-ray scattering
from rough surfaces. We faced that problem on the basis on the same theory, but we tried a new approach: the direct, analytical
translation of a given surface roughness PSD into a H(E) trend, and – vice versa – the direct translation of a H(E) requirement into a
surface PSD. This PSD represents the maximum tolerable microroughness level in order to meet the H(E) requirement in the energy
band of a given X-ray telescope.
Results. We have thereby found a new, analytical and widely applicable formalism to compute the XRS contribution to the HEW
from the surface PSD, provided that the PSD had been measured in a wide range of spatial frequencies. The inverse problem was also
solved, allowing the immediate evaluation of the mirror surface PSD from a measured function H(E). The same formalism allows
establishing the maximum allowed PSD of the mirror in order to fulfill a given H(E) requirement. Practical equations are firstly
developed for the case of a single-reflection optic with a single-layer reflective coating, and then extended to an optical system with
N identical reflections. The results are approximately valid also for multilayer-coated mirrors to be adopted in hard X-rays. These
results will be extremely useful in order to establish the surface finishing requirements for the optics of future X-ray telescopes.
Key words. Telescopes – Methods: analytical – Instrumentation: high angular resolution
1. Introduction
The adoption of grazing-incidence optics in X-ray telescopes in
the late 70s allowed a great leap forward in X-ray astronomy
because they endowed the X-ray instrumentation with imag-
ing capabilities in the soft X-ray band (E < 10 keV). The
excellent performances of the soft X-ray telescopes ROSAT
(Aschenbach 1988), Chandra (Weisskopf 2003) and Newton-
XMM (Gondoin et al. 1998) are well known.
To date, the utilized technique to focus soft X-rays consists
in systems of double-reflection mirrors with a single layer coat-
ing (Au, Ir) in total external reflection at shallow grazing inci-
dence angles. In this case, the incidence angle θi (as measured
from the mirror surface) cannot exceed the critical angle for total
reflection, otherwise the mirror reflectivity would be very low.
The critical angle is inversely proportional to E, the energy of
the photons to be focused. Using Au coatings, for instance, the
incidence angle cannot exceed ∼ 0.4 deg for photon energies
E ≈ 10 keV.
An extension of this technique to the hard X-ray energy band
(E > 10 keV) can be pursued by combining long focal lengths
(> 10 m), very small incidence angles (0.1 ÷ 0.25 deg), and
wideband multilayer coatings to enhance the reflectance of the
mirrors at high energies (Joensen et al. 1995; Tawara et al. 1998).
A very long focal length is hardly managed using a single
Send offprint requests to: [email protected]
spacecraft, therefore the optics and the focal plane instruments
should be carried by two separate spacecrafts in formation-
flight configuration. This is the baseline for the future X-ray
telescopes SIMBOL-X (Pareschi & Ferrando 2006) and XEUS
(Parmar et al. 2004). Other hard X-ray imaging telescopes of
the future are NeXT (Ogasaka et al. 2006) and Constellation-X
(Petre et al. 2006).
The focusing and reflection efficiency of X-ray optics can be
tested and calibrated on ground by means of full-illumination
X-ray facilities like PANTER (Bräuninger et al. 2004;
Freyberg et al. 2006), successfully utilized in the last years to
calibrate the optics of a number of soft X-ray telescopes. The
PANTER X-ray facility now allows testing in soft (0.2÷ 10 keV)
and hard (15 ÷ 50 keV) X-rays multilayer-coated optics pro-
totypes for future X-ray telescopes (Pareschi et al. 2005;
Romaine et al. 2005). The source distance finiteness causes
some departures of the optic performances, with respect to
the case with the source at astronomical distance: effective
area loss, different incidence angles on paraboloid and hyper-
boloid, focal length displacement, a slight focal spot blurring
(Van Speybroeck & Chase 1972). However, there effects can
be quantified and subtracted from experimental data. After this
treatment, the focusing-concentration performances of the optic
can be experimentally characterized as a function of the inci-
dent photon energy, in terms of Half-Energy Width (HEW) and
Effective Area (EA).
http://arxiv.org/abs/0704.1612v2
2 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
The focusing performance, in particular, is altered by mirror
deformations that may arise in the manufacturing, handling, in-
tegration, positioning processes. The consequent imaging degra-
dation can be calculated from the measured departures of the
mirrors from the nominal profile, by means of a ray-tracing pro-
gram. As long as the geometrical optics approximation can be
applied, the effect is independent of the photon energy. The fig-
ure errors contribution to the HEW can be also directly measured
using a highly collimated beam of visible/UV light in a precision
optical bench. In this case, however, the light diffraction has to
be carefully estimated and subtracted.
Another drawback is the X-ray scattering (XRS) caused by
the microroughness of reflecting surfaces (Church et al. 1979;
Stearns et al. 1998; Stover 1995; and many others). The XRS
spreads a variable fraction of the reflected beam intensity in the
surrounding directions: the result is the effective area loss in the
specular direction (i.e. in the focus) and a degradation of the
imaging quality. The XRS is an increasing function of the pho-
ton energy; due to the impact that the XRS can have on astro-
nomical X-ray images quality, the height fluctuations rms of the
mirror surface should not exceed few angströms. Loss of effec-
tive area is also caused by interdiffusion of layers in multilayer
coatings, which enhances the X-ray transmission and absorption
throughout the stack. On the other hand, an uniform interdiffu-
sion does not cause X-ray scattering (Spiller 1994), hence it does
not contribute to the focusing degradation.
The microroughness of an X-ray mirror can be measured on
selected samples using several metrological instruments, each of
them sensitive to a definite interval of spatial scales l̂: Long Trace
Profilometers (10 cm > l̂ > 0.5 mm: Takács et al. 1999), opti-
cal interference profilometers (5 mm > l̂ > 10 µm) and Atomic
Force Microscopes (100 µm > l̂ > 5 nm) can be suitable in-
struments to provide a detailed profile characterization of X-ray
mirrors surface. It is convenient to present the deviation of sur-
face from the ideality in terms of Power Spectral Density (PSD),
because its values do not depend on the measurement technique
in use (see ISO 10110 Standard). In addition, the XRS diagram,
and consequently the HEW, can be immediately computed from
the PSD at any photon energy (Church et al. 1979).
In the past years, several approaches were elaborated to
relate a mirror PSF (Point Spread Function) to the PSD
of its surface. Among a wealth of works, we can cite
(De Korte et al. 1981) the assumption of a Lorentzian model
for the PSD to fit the mirror PSFs at some photon energies,
allowing the derivation of two parameters (roughness rms and
correlation length) of the model PSD. Christensen et al. (1988)
perform a fit of experimental high-resolution XRS data dealing
with the surface correlation function. Harvey et al. (1988) re-
late the PSF of Wolter-I optics to the parameters of an exponen-
tial self-correlation function along with a transfer function-based
approach. Willingale (1988) derived the surface PSD of a mirror
from the wings of a few PSFs, measured at PANTER at some
soft X-ray photon energies. O’Dell et al. (1993) interpret the PSF
of a focusing mirror on the basis of surface roughness and partic-
ulate contamination. Zhao & Van Speybroeck (2003) construct
from the PSD of a focusing mirror a model surface and compute
the X-ray scattering PSF from the Fraunhofer diffraction theory.
In the present work that problem is faced in a new and differ-
ent way, looking for a general and simple link between measured
roughness and mirror HEW. More precisely, we considered the
following question: for an X-ray grazing-incidence optic, what
is the maximum acceptable PSD of the surface that fulfills the
angular resolution (HEW) requirements of the telescope, in all
the energy band of sensitivity?
In this work we shall give a definite answer to this question.
In the sect. 2 we shall summarize the causes of imaging degra-
dation. In the sect. 3 we show how to evaluate H(E), the XRS
contribution to the HEW of a focusing mirror at the photon en-
ergy E, from any surface microroughness PSD, measured over a
very wide range of spatial frequencies. We shall see in the sect. 4
that for the special class of fractal surfaces we can even relate
the power-law indexes of PSD and HEW, and in the sect. 5 we
see how to treat the other cases. Then we prove in the sect. 6 that
the formalism can be reversed, providing thereby an independent
evaluation of the surface PSD from an analytical calculation over
H(E), and in the sect. 7 we extend the results to focusing mirrors
with more than one reflection. Finally, an example of computa-
tion is provided in the sect. 8.
2. Contributions to the imaging degradation
We shall henceforth indicate with λ the wavelength of photons
impinging on the mirror, and we shall consider the HEW as a
function of λ instead of the photon energy E. For isotropical re-
flecting surfaces in grazing incidence, the X-ray scattering dis-
tribution lies essentially in the incidence plane, so we denote the
incidence angle on the mirror as θi and the scattering angle as θs,
both measured from the surface plane (a schematic of the scatter-
ing geometry is drawn in fig. 1). If we do not consider the optic
roundness errors, the longitudinal deviations from the nominal
profile of a focusing mirror can be classified on the basis of their
typical length l̂. According to De Korte et al. (1981), they are:
1. Power errors: errors with l̂ equal to the mirror length L. They
consist in a single-concavity deformation of the profile with
respect to the nominal one.
2. Regularity errors: errors in the spatial range from 0.1 L <
l̂ < 0.5 L.
3. Surface roughness: surface defects with l̂ < 0.1 L.
However, other criteria were also formulated to separate fig-
ure errors from roughness. Consider a single Fourier component
of the surface profile with wavelength l̂ and root mean square σ.
That Fourier component is dominated by figure error if it fulfills
the condition (Aschenbach 2005)
4π sin θiσ > λ. (1)
Otherwise, it is dominated by microroughness. In other words,
surface defects within the smooth-surface approximation can be
mainly considered as microroughness. To understand the impor-
tance of this approximation, we write the optical path difference
∆s of X-rays reflected by two points of the surface with a hori-
zontal spacing l̂ and vertical spacing σ̂ = 2
2σ (for optically-
polished surfaces, σ is a increasing function of l̂, and usually
σ≪ 10−3l̂) as
∆s = l̂(cos θs − cos θi) + σ̂(sin θi + sin θs) (2)
that, for small incidence angles, becomes
∆s = l̂ sin θi(θs − θi) + σ̂(θi + θs). (3)
If that component is responsible for X-ray scattering, it has
to be ∆s ≈ λ, to cause the diffraction from surface features with
a l̂ spacing and σ̂ height. Conversely, the ”figure errors”, which
are treated with the methods of the geometrical optics, should
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 3
Fig. 1. The geometry of X-ray scattering: the strictly speaking
”reflected” rays (i.e. in the focus direction) are characterized by
the equality θs = θi, the others are scattered apart. The rough sur-
face is a simulated one, assuming a PSD with power-law index
n = 2.4 (see sect. 4).
be characterized by the inequality ∆s ≫ λ. Note that this condi-
tion becomes similar to the eq. 1 in the limit |θi − θs | → 0. The
application of this criterion and of the subsequent X-ray scatter-
ing theory requires the incident radiation to be spatially coher-
ent over the spatial scale l̂, so that the properties of the reflected
wavefront are determined only by the coherence properties of
the mirror surface. This in turn requires the angular diameter of
the source φS to fulfill the inequality (Holý et al. 1999)
l̂ sin θi
. (4)
This equation sets a maximum to the values of l̂ that can be used
in the application of the results presented in this work. The limi-
tation can affect X-ray sources at finite distance, like those used
for X-ray optics calibrations in full-illumination setup. For very
distant astronomical X-ray sources, the condition 4 is met even
for larger l̂, up to l̂ ≈ L.
It is worth pointing out that, for a given reflecting surface,
the separation of figure errors from microroughness is strongly
affected by the incidence/scattering angles. In fact, even for large
l̂, ∆s can become comparable with λ, if θi and θs are sufficiently
small: thus, the spatial wavelength window of interest for X-ray
scattering can shift to the large l̂ domain (or, equivalently, to the
range of low spatial frequencies f = 1/l̂), provided that the con-
dition 4 is fulfilled.
Let us now consider how to separate the figure and scattering
terms in HEW data. In absence of XRS, the mirror PSF would
be independent of the energy and due only to figure errors (i.e.
in the approximation of the geometrical optics). The resulting
HEW would be also constant. Instead, due to the XRS, the figure
PSF is convolved with the X-ray scattering PSF to return the
PSF(λ) being measured (Willingale 1988; Stearns et al. 1998;
and many others),
PS F(λ) = PS Ffig ⊗ PS FXRS(λ). (5)
The resulting HEW will depend on the photon wavelength,
as it does the PSF. In order to isolate the scattering term from the
total PSF a deconvolution should be carried out, provided that
the PSFfig is known. However, if we assume that the XRS and
the mirror deformations are statistically independent, the total
HEW can be approximately calculated as the squared sum of the
two contributions:
HEW2(λ) ≈ HEW2fig + H
2(λ). (6)
An estimation of HEWfig can be obtained:
1. from the application of a ray-tracing code to several mea-
surements of the mirror profile,
2. from reliable extrapolation of the HEW(λ) curve to E → 0,
in absence of low-energy diffraction effects like dust contam-
ination, studied in detail by O’Dell et al. (1993)
3. from a direct measurement of the HEW in visible/UV light,
provided that the diffraction at the mirror edges can be reli-
ably calculated and subtracted.
Once known the measured HEW(λ) experimental trend and
the HEWfig term, the eq. 6 can be used to isolate the scattering
contribution from the experimental HEW trend: we shall prove
in the next section that the H(λ) function is immediately related
to the reflecting surface 1D Power Spectral Density (PSD) P( f )
P( f ) =
z(x)e−2πi f dx
where z(x) is a height profile (of length L) of the mirror, mea-
sured in any direction (Stover 1995): the surface is assumed to
be isotropic, and the spectral properties of the profile to be rep-
resentative of the whole surface. The PSD is often measured in
nm3 units, and for optically-polished surfaces it is usually a de-
creasing function of the frequency f .
PSD measurements have always a finite extent [ fmin, fmax],
determined by the length and the spatial resolution of the mea-
sured profile. As well known, the surface rms σ is simply com-
puted from the PSD by integration over the spatial frequencies f :
∫ fmax
P( f ) d f (8)
note that the integration range should always be specified.
3. Estimation of H(λ) for single-reflection focusing
mirrors
3.1. Single-layer coatings
Firstly, we suppose the mirror to be plane and single-layer
coated. For a surface with roughness rms σ, the specular beam
intensity obeys the well-known Debye-Waller formula
R = RF exp
16π2σ2 sin2 θi
, (9)
here RF is the reflectivity at the grazing incidence angle θi, as
calculated from Fresnel’s equations (zero roughness). However,
it should be noted in the eq. 9 that neither the spatial frequen-
cies range where the PSD should be integrated is specified, nor
the separation between reflected and scattered ray is clearly in-
dicated: these ambiguities can be solved as follows.
Let us derive the total scattered intensity Is from the con-
servation of the energy: for smooth surfaces, i.e. fulfilling the
inequality 2σ sin θi ≪ λ, we can approximate
Is = I0RF
1 − exp
16π2σ2 sin2 θi
≈ I0RF
16π2σ2 sin2 θi
.(10)
4 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
In grazing incidence, X-ray scattering lies mainly in the inci-
dence plane. Moreover, the normalized scattered intensity per ra-
dian at the scattering angle θs (either θs > θi or θs < θi) is related
to the PSD along with the well-known formula at first-order ap-
proximation (Church et al. 1979; Church & Takács 1986), valid
for smooth, isotropic surfaces and for scattering directions close
to the specular ray (i. e. |θs − θi| ≪ θi),
sin3 θiRFP( f ) (11)
where P( f ) is the Power Spectral Density of the surface (eq. 7)
and I0 is the flux intensity of the incident X-rays. If the scattered
intensity is evaluated at the scattering angle θs, the PSD can be
immediately evaluated as a function of the spatial frequency f :
f = l̂−1 =
cos θi − cos θs
sin θi(θs − θi)
. (12)
In the eq. 12 the approximation was justified by the assumption
|θs − θi| ≪ θi and the negative frequencies are conventionally
assumed to scatter at θs < θi: the assumed approximations make
the XRS diagram symmetric, because the PSD is an even func-
tion.
For a single-reflection mirror shell, the extension of the for-
mulae above-mentioned is straightforward by regarding |θs − θi|
as the angular distance at which the PSF is evaluated. The fo-
cal image is the superposition of many identical XRS diagrams
on the image plane, generated by every meridional section of the
mirror shell: since a π angle rotation of every meridional plane of
the shell sweeps the whole image plane, the scattered intensity is
spread over a π angle. The integration on circular coronae used
to compute the mirror PSF (at positive angles) compensates this
factor multiplying the XRS diagram by 2π (De Korte et al. 1981).
The remaining 2-fold factor accounts for the negative frequen-
cies in the surface PSD. We shall henceforth suppose that the fac-
tor 2 is embedded in the PSD definition. Therefore, the eqs. 11
and 12 can be used to describe the XRS contribution to the PSF.
We are now interested in the scattered power at angles larger
than a definite angle αmeasured from the focus. Due to the steep
fall of scattering intensity for increasing angles, the integral has
a finite value
I [|θs − θi| > α] =
∫ π−θi
dθs. (13)
Combining eqs. 11 and 13, one obtains:
I [|θs − θi| > α] = I0RF
16π2 sin3 θi
∫ π−θi
P( f ) dθs (14)
with respect to the definition used in the eqs. 7 and 11, a factor
2 was included in the PSD. The upper integration limit corre-
sponds to a photon back-scattering: at first glance, this seems
to violate our small-scattering angle assumption (eqs. 11 and
12), but it should be remembered that only the angles close to θi
contribute significantly to the integral in eq. 13: hence its value
should not be significantly affected by a particular choice of the
upper integration limit. After a variable change from θs to f
(eq. 12), the eq. 14 becomes (approximating cos θi ≈ 1 in the
upper integration limit):
I [|θs − θi| > α] = I0RF
16π2 sin2 θi
P( f ) d f (15)
where f0 = α sin θi/λ is the spatial frequency corresponding to
the scattering at the angle α. As expected, this equation equals
the integrated scattering according to the eq. 10, provided that
we identify I [|θs − θi| > α] with Is, and the squared roughness
rms with
P( f ) d f . (16)
The eq. 16 is in agreement with the eq. 8, but it states clearly the
window of spatial frequencies involved in the XRS. Therefore,
for a definite angular limit α the ”reflected beam” intensity can
be simply calculated by using the Debye-Waller formula, pro-
vided that σ2 is computed from the PSD integration beyond the
frequency f0, which corresponds to an X-ray scattering at α. The
upper integration limit is a very high frequency (close to 1/Å):
hence, the atomic structure of the surface is not important in the
integral of the eq. 16. Moreover, considering that the PSD trend
for optically-polished surfaces decreases steeply for increasing
f , the largest contribution to the integral should be given by the
frequencies close to f0.
Now we can evaluate H(λ), the scattering term of the HEW.
For simplicity, in the following we will suppose that the HEW
is obtained from the collection of all the reflected/scattered pho-
tons: this allows us to avoid problems related to the finite size of
the detector, and to extend the surface roughness PSD up to very
large spatial frequencies. By definition, H(λ) is twice the angu-
lar distance from focus at which the integrated scattered power
halves the total reflected intensity:
I [|θs − θi| > α] =
I0RF (17)
we immediately derive, from the eq. 9,
16π2σ2 sin2 θi
, (18)
where σ2 has now the meaning as per the eq. 16. Solving the
eq. 18 for σ2 and equating to the integral of the PSD,
P( f ) d f =
λ2 ln 2
16π2 sin2 θi
, (19)
once known the PSD from topography measurements over a
wide range of spatial frequencies, the PSD numerical integra-
tion in the eq. 19 allows to recover f0. In turn, f0 is related to
H(λ) through the eq. 12, that we write in the following form
H(λ) =
2λ f0
sin θi
, (20)
where H is measured in radians. Note that the condition
H(λ) ≪ θi is very important, for the eq. 20 to hold. Small
scattering angles and grazing incidence are also very important
for the considerations that follow.
3.2. Multilayer coatings
The obtained result (eq. 19) can be extended to mirrors with
multilayer coatings, used to enhance the grazing incidence re-
flectivity of mirrors in hard X-rays (E > 10 keV). In general,
the multilayer cannot be characterized by means of a single
PSD, due to the evolution of the roughness throughout the stack
(Spiller et al. 1993; Stearns et al. 1998). Moreover, due to the
interference of scattered waves at each multilayer interface, the
final scattering pattern is more structured than eq. 11, with peaks
whose height depends on the phase coherence of the interfaces
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 5
Fig. 2. Dependence of the spectral exponents for different in-
dexes n of a power-law PSD, for a single-reflection focusing
mirror. In the forbidden region (n > 3) γ would be negative.
(Kozhevnikov 2003). The HEW term can be computed numeri-
cally from the XRS diagram.
In order to extend the eq. 19 to mirrors coated with a graded
multilayer, we have to assume the additional requirements:
1. the PSD is constant and completely coherent throughout
the multilayer stack: i.e., the deposition process does not
cause additional roughness and replicates simply the profile
of the substrate. Therefore, all the PSDs and all the cross-
correlation between interface profiles equal the PSD mea-
sured at the multilayer surface. This is often observed in
the l̂ > 10 µm regime (Canestrari et al. 2006), where most
of frequencies f0 fall when the incidence angle is less than
0.5 deg. Most of microroughness growth, indeed, takes place
for 10 µm> l̂ > 0.1 µm.
2. the multilayer reflectivity Rλ(θi) at the photon wavelength λ
changes gradually over angular scales of H(λ). Ideally, this
condition should be fulfilled by wideband multilayer coat-
ings for astronomical X-ray mirrors.
Under these hypotheses, a quite tedious calculation reported
in appendix A shows that the eq. 19 can be approximately ap-
plied also with multilayer coatings. The following developments
also apply in that case.
4. H(λ) for a fractal surface
We apply now the equations 19 and 20 to the typical (monodi-
mensional) PSD model for optically-polished surfaces, a power-
law (Church 1988)
P( f ) =
, (21)
where the power-law index n is a real number in the interval
1 < n < 3 and Kn is a normalization factor. A power-law PSD is
typical of a fractal surface, and it represents the high-frequency
regime of a K-correlation model PSD (Stover 1995). This model
exhibits a saturation for f → 0 that avoids the PSD divergence.
In practice, the fractal behavior dominates in almost all spatial
frequencies of interest for X-ray optics.
There are interesting reasons for which n can take val-
ues on the interval (1:3). In fact, for a surface in the 3D
space, n is related to its Hausdorff-Besicovitch dimension D
(Barabási & Stanley 1995) along with the equation n = 7 − 2D
(see Church 1988; Gouyet 1996). The restriction 1 < n < 3 for
a fractal surface is therefore necessary to have 3 > D > 2.
A power-law PSD is particularly interesting because the in-
tegral on left-hand side of the eq. 19 can be explicitly calculated:
f 1−n0 −
n − 1
λ2 ln 2
16π2 sin2 θi
. (22)
As 1 − n < 0, in grazing incidence the (2/λ)1−n term can be
neglected with respect to f 1−n0 . By isolating the frequency f0 and
using the eq. 20 to derive H(λ), we obtain after some algebra,
for the scattering term of the HEW,
H(λ) = 2
16π2Kn
(n − 1) ln 2
sin θi
. (23)
This equation states that:
1. The H(λ) function for a power-law PSD has a power-law
dependence on the photon energy E ∝ 1/λ, i.e., H(E) ∝ Eγ.
The power-law index γ is related to the PSD power-law index
n through the simple equation:
3 − n
n − 1
. (24)
As 1 < n < 3, γ is positive, i.e. H is an increasing function
of the photon energy. For a fixed value of Kn, the HEW di-
verges quickly for n ≈ 1 but very slowly for n ≈ 3: a PSD
power-law index close to 2-3 would hence be preferable in
order to reduce the degradation of focusing performances for
increasing energies.
2. H(λ) depends on the sine of the incidence angle at the γth
power. In other words, the HEW depends only on the ratio
sin θi/λ: this scaling relation shows that for a given power-
law PSD (with n < 3) at a given photon wavelength λ we can
reduce the HEW by decreasing the incidence angle.
3. H(λ) increases with the PSD normalization Kn, as expected:
the dependence is also a power law with spectral index
n − 1
. (25)
As for γ(n), the closeness of n to the maximum allowed value
for fractal surfaces makes less severe the roughness effect on
imaging degradation.
The functions β and γ are plotted in fig. 2. For instance, if
n = 2, γ = β = 1, and H(E) increases linearly with both pho-
ton energy and Kn coefficient. The divergence of indexes β, γ for
n ≈ 1 makes apparent the importance of obtaining steep PSDs
in the optical polishing of X-ray mirrors. Finally, it is worth not-
ing that for n > 3 there is the theoretical possibility of a slight
decrease of H(E) for increasing energy because γ(n) becomes
negative.
To clarify the dependence of the HEW on the power-law in-
dex n and the incidence angle, we depict in fig. 3 and 4 some ex-
amples of H(E) simulations (single reflection) for some power-
law PSDs in the photon energy range 0.1-50 keV. The H(E)
curves were computed using the eq. 23. In fig. 3 the incidence
angle θi is fixed at 0.5 deg and the index n is variable; a constant
n = 1.8 and a variable θi is instead assumed in the simulations of
fig. 4. Note in fig. 3 the slower H(E) increase for larger n and the
common intersection point, determined by the particular choice
of the incidence angle and the σ = 4 Å value in the window of
spatial wavelengths [100 ÷ 0.01 µm].
6 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Fig. 3. H(E) simulations assuming power-law PSDs with con-
stant σ = 4 Å in the spatial wavelengths range [100 ÷ 0.01µm],
but variable power-law index n. The incidence angle is fixed at
θi = 0.5 deg.
Fig. 4. H(E) simulations assuming a power-law PSD with
power-law index n = 1.8 and with σ = 4 Å in the spatial wave-
lengths range [100 ÷ 0.01 µm], but variable incidence angle θi.
5. Numerical integration of the PSD
A power-law PSD is a modelization that can be used for
optically-polished surfaces. If the polishing process is not op-
timized or a reflecting layer is grown onto a optically polished
substrate, several deviations from a power-law trend can be ob-
served. A typical ”bump”, for instance, can be present in the PSD
of multilayer coatings, often in the range of spatial wavelengths
[10 ÷ 0.1 µm], as a result of the replication of the substrate to-
pography and of fluctuations intrinsically related to the random
deposition process (Spiller et al. 1993; Stearns et al. 1998). If
the PSD deviates significantly from a power-law, the eq. 23 can-
not be used. However, if the surface PSD has been extensively
measured over a wide range of spatial frequencies [ fm, fM] (wide
enough to have fm < f0(λ) for all λ), the HEW scattering term
H(λ) can be computed by numerical integration (eqs. 19 and 20),
on condition that the following approximation is valid:
P( f ) d f ≈
P( f ) d f . (26)
The condition above is usually satisfied when f0 ≪ fM i.e. when
the following inequality holds:
H(λ)≪
2λ fM
sin θi
. (27)
As we are also interested in computing H(λ) in hard X-rays
(small λ), there is the possibility that the two integrals in the
eq. 26 differ by a significant factor. In this case the integral can
be corrected by adding the remaining term
P( f ) d f =
P( f ) d f +
P( f ) d f , (28)
that can be evaluated, in principle, by measuring the mirror re-
flectivity within an angular acceptance corresponding to the spa-
tial frequency fM, and using the Debye-Waller formula to derive
σ2; then, the importance of measuring the PSD in a very wide
frequencies interval becomes apparent. The value of f0 depends
strongly on both incidence angle and photon energy: for soft X-
rays (< 10 keV) and very small angles (< 0.2 deg) the character-
istic spatial wavelength l̂ = 1/ f0 often falls in the millimeter or
centimeter range.
It should be noted that, if the detector is small, a fraction of
the scattered photons can be lost; to account for the finite an-
gular radius of the detector d (as seen from the optic principal
plane), one should integrate the PSD over the smaller interval
[ f0, d sin θi/λ] to recover the measured H(λ) trend. As an alter-
native method, one can compare the theoretical predictions of
eqs. 19 and 20 with the experimental H(λ) values, as calculated
from the Encircled Energy normalized to the photon count fore-
seen by the Fresnel equations (i.e. with zero roughness), rather
than to the maximum of the measured Encircled Energy func-
tion.
6. Computation of the PSD from the H(λ) trend
If the approach described above can be used to simulate the
HEW trend from a measured surface PSD, the reverse problem,
i.e. the derivation of surface PSD from the measured HEW trend
is also possible. This requires that the figure error contribution
had been reliably measured, in order to isolate the scattering
term function H(λ) using the eq. 6.
This problem is interesting for three reasons at least:
1. it is a quick, non-destructive surface characterization method
in terms of its PSD.
2. The measurement is extended to a large portion of the illu-
minated optic, hence local surface features are averaged and
ruled out from the PSD.
3. For a given HEW(λ) requirement in the telescope sensitiv-
ity energy band, it allows establishing the maximum allowed
In order to find an analytical expression for the PSD, we note
that the spatial frequency f0 that scatters at an angular distance
H/2 from the specular beam is a function only of λ, along with
the eq. 20. Solving for f0, we have
f0(λ) ≈ H(λ)
sin θi
. (29)
We suppose that all scattered photons are collected, so we can
assume the eq. 19 as valid. By deriving both sides of eq. 19 with
respect to λ, we have
P( f ) d f
8π2 sin2 θi
λ, (30)
that is,
P( f0) =
8π2 sin2 θi
λ, (31)
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 7
and, using the eq. 29 to compute the derivative of f0:
+ f0P( f0)
sin θi
dH(λ)
P( f0) =
8π2 sin2 θi
λ. (32)
Now remember that, in grazing incidence, f0 ≪ 2λ−1 by several
orders of magnitude. Even if P( f ) is not a power-law, it is always
a steeply decreasing function of f . Moreover, it should have over
[ f0, 2λ
−1] an average PSD index ñ > 1, for the reasons explained
in the sect. 4. This means that
P( f0)
P(2λ−1)
, (33)
therefore, in practical cases the 2λ−1P(2λ−1) term in the eq. 32 is
negligible with respect to f0P( f0). Consequently, we can neglect
the first term of eq. 31: then we have
P( f0) ≈
8π2 sin2 θi
λ. (34)
Combining this with the eq. 29 and collecting the constants, we
obtain the final result
P( f0)
4π2 sin3 θi
≈ 0. (35)
The eq. 35 enables the computation of the PSD (at the spatial
frequency given by the eq. 29) along with the derivative of the
ratio H(λ)/λ with respect to λ.
The obtained equation shows that P( f ) is inversely propor-
tional to the derivative of H(λ)/λ. This result seems strange at
first glance, because by decreasing H(λ) one would obtain a
larger P( f ) (a rougher surface). One should remember, indeed,
that by reducing H(λ) we increase P( f0), but f0 is shifted to-
wards the low frequencies domain, where P( f0) is expected to
be higher. In fact, the ”rough” or ”smooth” feature of the surface
depends on whether f0 or P( f0) varies more rapidly, i.e. on the
overall H(λ) trend.
We can also check the correctness of the eq. 35 by computing
the PSD for the particular case of the H(λ) derived from the inte-
gration of a power-law PSD (the eq. 23, derived under the same
approximation, the eq. 33). If the results are correct, the substi-
tution of the HEW trend of the eq. 23 in the eq. 35 should return
the original PSD (eq. 21). The straightforward, but lengthy cal-
culation (carried out in appendix B) shows that the substitution
returns
P( f0) =
as expected.
The eq. 35 should be approximately valid also for graded
multilayers with a slowly-decreasing reflectivity (see sect. 3.2),
however, due to the approximations needed to extend the eq. 19
to the multilayers, the resulting PSD should be considered a
”first guess” in this case. Then, the matching of the PSD to the
required HEW trend should be checked by means of a detailed
computation of the XRS PSF(λ).
7. Extension to X-ray mirrors with multiple
reflections
The formalism exposed in the previous sections can be extended
to a double-reflection optic (like a Wolter-I one). In this op-
tical configuration, photons are firstly reflected by a parabolic
surface and subsequently by a hyperbolic one. If the smooth-
surface condition is satisfied, multiple scattering is often negli-
gible (Willingale 1988) and the scattering diagrams of the two
reflecting surfaces can be simply summed (De Korte et al. 1981;
Stearns et al. 1998). The source is assumed to be at infinite dis-
tance, then X-rays impinge on the two surfaces at the same an-
gle θi. If the surface PSDs are the same for both reflections, the
scattering diagram will be simply doubled. Thus, the integrated
scattered intensity is also doubled:
Is = 2I0R
1 − exp
16π2σ2 sin2 θi
. (37)
The RF factor is squared in the eq. 37 because each ray is re-
flected twice: in absence of scattering the reflected power would
be I0R
F, so the half-power scattering angle condition reads
|θs − θi| >
F, (38)
and, combining the eqs. 37 and 38, we obtain
16π2σ2 sin2 θi
. (39)
Solving for σ2, and using the eq. 16,
P( f ) d f =
λ2 ln(4/3)
16π2 sin2 θi
, (40)
that differs from the eq. 19 only in the factor ln(4/3) instead of
ln 2 on right-hand side. Consequently, the corresponding differ-
ential equation is
P( f0)
ln(4/3)
4π2 sin3 θi
≈ 0. (41)
Similar equations can be derived for an optical system with
an arbitrary number of reflections N: to compute the H(λ) from
the PSD,
P( f ) d f =
16π2 sin2 θi
2N − 1
. (42)
If the PSD is a power-law P( f ) = Kn/ f
n we can generalize the
eq. 23:
H(λ) = 2
2N − 1
16π2Kn
(n − 1)
sin θi
, (43)
note the divergence of the logaritmic factor for increasing N,
due to the negative exponent 1/(1 − n). This indicates that H(λ)
increases rapidly with the number of reflections, as expected.
Finally, we can also generalize the differential eq. 35 to an
arbitrary number of reflections,
P( f0)
4π2 sin3 θi
≈ 0. (44)
In the eqs. 42 and 44, f0 is always related to H(λ) by the eq. 29.
8 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Fig. 5. an hypothetical PSD with reasonable values and a PSD
break around a 100 µm spatial wavelength (dashed line). This
PSD is adopted to compute the corresponding HEW trends for
1,2,3 reflections at a 0.3 deg grazing incidence angle (fig. 6). The
achieved HEW trends were used to re-calculate the respective
PSDs (solid line). For clarity, we do not plot the single PSDs,
but just their overlap.
8. An example
As an application of the equations reported above, we shall
make use of a simulated surface PSD with reasonable values,
that is not a power-law. The PSD (see fig. 5, dashed line) is ex-
tended from 105 µm down to a 0.01 µm spatial wavelength, with
a break around 100 µm: at the lowest frequencies the PSD is
steep (n ≈ 2.3), whereas at the largest frequencies it is smoother
(n ≈ 1.3). From the discussion in sect. 4 concerning the relation
between the exponents of the PSD and the HEW (eq. 24), we
should expect that the PSD break causes a slope change in the
function H(λ): however, as the actual PSD is not a power-law,
the H(λ) function should be computed by means of the eqs. 20
and 42. Before carrying out the integration, we can remark qual-
itatively that, as we increase the photon energy, the highest fre-
quencies in the PSD (where the PSD index becomes smaller)
become important; hence, we can expect a steeper increase of
the HEW at the highest energies.
The analysis is made quantitative in fig. 6, where we show
the calculated HEW trends from the PSD in fig. 5 (the dashed
line) by means of the eqs. 20 and 42, assuming 1,2,3 reflections
at the same grazing incidence angle (0.3 deg). The approxima-
tion of eq. 26 was adopted. In addition to the scattering term, 15
arcsec of HEW due to figure errors were added in quadrature.
The HEW increases slowly (concave downwards) at low ener-
gies, corresponding to a frequency f0 in the steeper part of the
PSD. Then it increases more steeply (concave upwards) when
the energy becomes large enough to set f0 in the portion of the
spectrum with n ≈ 1.3. By increasing the number of reflections,
the HEW values also increase, and the ”turning point” where the
HEW starts to diverge (arrows in fig. 6) shifts at lower X-ray
energies. All the calculation is based on the assumption that the
contribution of the PSD over the maximum measured frequency
fM = 0.01 µm is negligible. Otherwise, the computed HEW val-
ues will be underestimated (see sect. 5).
In addition to the general trend of the HEW, there are oscil-
lations due to small irregularities in the adopted PSD: the cal-
culation is, in fact, very sensitive to small variations of the PSD
values. Notice that for a definite energy all the frequencies larger
than f0 contribute to the HEW value, even if the largest contri-
Fig. 6. the HEW trend computed from the PSD for 1,2,3 reflec-
tions, plus 15 arcsec of HEW due to figure errors. The HEW
trends were used to compute back the PSD (the solid line in
fig. 5) to verify the reversibility of the calculation. The energy
at which the concavity change takes place is also indicated (ar-
rows).
bution comes from frequencies near f0: this is a consequence of
the steeply decreasing trend of the PSD.
We checked the reversibility of the result by computing the
PSD from the HEW trends (after subtracting in quadrature 15
arcsec figure error) by means of the eq. 44 with the respec-
tive value of N. The resulting PSDs (the solid line in fig. 5)
were overplotted to the initial PSD, with a perfect superposition.
Each obtained PSD has, indeed, an extent of spatial frequencies
smaller than the initial one: the overall PSD ranges from 104
to 11 µm (vs. the initial 105 ÷ 0.01 µm), and the smaller wave-
lengths could be computed from the HEW trend with N = 3. The
limitation in spatial frequency ranges occurs for two reasons:
1. small f − large l̂: all the power scattered by the lowest fre-
quencies is found at angles less than 1/2 HEW even for the
lowest energies being considered: therefore, that part of the
spectrum is not necessary to compute the HEW in the energy
range of interest;
2. large f − small l̂: the PSD is computed from a derivative,
therefore the information concerning the absolute magnitude
of the HEW is substantially lost. This information was in-
cluded in the integral of the PSD (eq. 42) for the maximum
considered energy.
Therefore, from the integral in the eq. 42 we cannot re-
cover the PSD over the minimum computed spatial wavelength
(11 µm, using the HEW trend with N = 3), but we can at
least calculate the value of σ at spatial wavelengths smaller than
11 µm. Substituting the incidence angle and the minimum pho-
ton wavelength being considered (λ = 0.24 Å) in the eq. 42
with N = 3 and with the approximation of the eq. 26, we ob-
tain σ = 1.6 Å, in perfect agreement with the value computed
from the original PSD.
Summing up, for a given incidence angle the H(λ) function
in a definite photon energy range is equivalent to the PSD in
a corresponding range of spatial frequencies f (or equivalently,
spatial wavelengths l̂), plus the integral of the PSD beyond the
maximum frequency being computed. Therefore, requirements
of a definite HEW(λ) function in designing an X-ray optical sys-
tem can be translated in terms of PSD in a frequencies range
[ fmin, fmax] plus the surface rms at frequencies beyond fmax. The
usefulness of such a relationship is apparent.
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 9
9. Conclusions
In the previous pages we have developed useful equations to
compute the contribution of the X-ray scattering to the HEW of
a grazing incidence X-ray optic, by means of a simple integra-
tion. The formalism has been inverted in order to derive the PSD
of the surface from the function H(λ), and it can be extended
to an arbitrary number of reflections at the same incidence an-
gle. The equations are valid for a single-layer coating mirror, but
they can be approximately applied to a multilayer-coated mir-
ror. This approach is particularly useful in order to establish the
surface finishing level needed to keep the X-ray scattering HEW
of X-ray optics within the limits fixed by the X-ray telescope
requirements.
It should be remarked that the reasoning was developed for
the Half-Energy Width, but it can be extended to any angular
diameter including a fraction η of the energy spread around the
focal point. To do this, it is sufficient to substitute the logarithmic
factors in equations 42, 43, 44,
2N − 1
N − 1 + η
, (45)
and for instance, to compute the 90%-energy diameter for a dou-
ble reflection mirror, simply substitute η = 0.9 and N = 2. The
proof is straightforward: however, one should always keep in
mind that the energy diameters computed with this method can
be considered valid only if they are much smaller than the inci-
dence angle θi.
Notice that in the development of the exposed formalism we
have supposed, in addition to the smooth-surface condition, two
additional hypotheses:
1. the source is at infinite distance from the mirror
2. the X-ray detector is large enough to collect all the scattered
photons.
In order to apply the mentioned equations to experimental
calibrations of X-ray optics at existing facilities (like MPE-
PANTER), where the source is at a finite distance and the de-
tector has a finite size, some corrections should be taken into
account. We will deal with their quantification in a subsequent
paper.
Appendix A: Extension to multilayer coatings
Here we provide with a plausibility argument to extend the for-
malism of sect. 3.1 to mirror shells with multilayer coatings (see
sect. 3.2). The intensity of a scattered wave at each interface is
proportional to its PSD as per the eq. 11, and the overall scatter-
ing diagram will be their coherent interference. To simplify the
notation, we neglect the X-ray refraction and we suppose that
the incidence angle is beyond the critical angles of the multilayer
components. The electric field scattered by the kth interface can
be written as
Ek = E0TkrkXk( f ) exp(−iφk), (A.1)
where rk is the single-boundary amplitude reflectivity, E0 the
incident electric field amplitude, the weights Tk are the rel-
ative amplitudes of the electric field in the stack (in scalar,
single-scattering approximation), and account for the extinction
of the incident X-rays due to gradual reflection and absorption.
Xk( f ) is the single-boundary scattering power (proportional to
the PS D( f ) amplitude), and φk is the phase of the scattered wave
at θs by the k
th interface
φk = 2π
sin θi + sin θs
zk, (A.2)
where zk is the depth of the k
th interface with respect to the outer
surface of the multilayer. Now, the measured intensity is
|Escatt|2 =
= |E0|2|Xk( f )|2
rkTk exp(−iφk)
. (A.3)
Now, |E0|2 = I0, the incident X-ray flux intensity, and |Xk( f )|2 is
proportional to the interfacial PSD P( f ), which is independent
of k by hypothesis. Assuming the proportionality factor of eq. 11
for |Xk( f )|2, we obtain for the scattering diagram
sin3 θiP( f )
rkTk exp(−iφk)
(A.4)
and if we set
Kλ(θi, θs) =
rkTk exp(−iφk)
, (A.5)
the eq. A.4 becomes analogous to the eq. 11, with Kλ(θi, θs)
playing the role of RF. Note that Kλ(θi, θi) = Rλ(θi), the mul-
tilayer reflectivity in single reflection approximation. As before,
we write the scattering diagram for a mirror with axial symmetry
as a function of the angular distance from the focus α = |θi − θs|
averaging the contributions of negative and positive frequencies
sin3 θiP( f )[Kλ(θi, θi − α) + Kλ(θi, θi + α)]. (A.6)
For a single reflection optic, we can calculate the scattered power
over H/2, where H is the scattering term of optic Half-Energy
Width:
Is[α > H/2] =
I0Rλ(θi). (A.7)
Now, the steps 18 and 19 can be repeated:
Kλ(θi, θi + α) + Kλ(θi, θi − α)
Rλ(θi)
P( f ) d f =
λ2 ln 2
8π2 sin2 θi
(A.8)
where f0 is still defined by the eq. 20. For small scattering angles
(α ≪ θi), since we assumed a slow variation of Rλ over angular
scales of H/2 (and the same occurs for Kλ), we can approximate
Kλ(θi, θi ± α) ≈ Rλ(θi) ± α
∂Kλ(θi, θs)
θs=θi
. (A.9)
Substituting in the eq. A.8, we obtain
P( f ) d f ≈
λ2 ln 2
16π2 sin2 θi
(A.10)
because the two derivatives have opposite sign and cancel out.
This is the same equation found for the case of a single-layer
coating (eq. 19).
10 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Appendix B: Derivation of the PSD from the HEW
for a fractal surface (single reflection)
We recall here the H(λ) trend for a power-law PSD (eq. 23):
H(λ) = 2
16π2Kn
(n − 1) ln 2
sin θi
(B.1)
we verify that it returns a power-law PSD if substituted in the
differential eq. 35:
P( f0)
4π2 sin3 θi
= 0. (B.2)
To simplify the notation, we write simply H instead of H(λ): by
carrying out the derivation,
n − 1
16π2Kn
(n − 1) ln 2
(sin θi)
n−1 λ
n−1−3. (B.3)
Using again the eq. B.1:
n − 1
−3 (B.4)
hence, the related PSD is
P( f0) = −
4π2 sin3 θi
λ3 ln 2
4π2H sin3 θi
n − 1
. (B.5)
Now, we can derive (n − 1)/2 from the eq. B.1,
n − 1
4π2HKn
)−n ( sin θi
(B.6)
and combining the eqs. B.5-B.6, one obtains
P( f0) = Kn
H sin θi
, (B.7)
that is, by recalling the eq. 29,
P( f0) =
(B.8)
i.e., the expected power-law PSD.
Acknowledgements. Many thanks to G. Pareschi, O. Citterio, R. Canestrari, S.
Basso, F. Mazzoleni, P. Conconi, V. Cotroneo (INAF/OAB) for support and
useful discussions. The author is indebted to MIUR (the Italian Ministry for
Universities) for the COFIN grant awarded to the development of multilayer
coatings for X-ray telescopes.
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Introduction
Contributions to the imaging degradation
Estimation of H() for single-reflection focusing mirrors
Single-layer coatings
Multilayer coatings
H() for a fractal surface
Numerical integration of the PSD
Computation of the PSD from the H() trend
Extension to X-ray mirrors with multiple reflections
An example
Conclusions
Extension to multilayer coatings
Derivation of the PSD from the HEW for a fractal surface (single reflection)
|
0704.1613 | Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory
with quantum mechanics''' | Reply to “Comment on ‘On the inconsistency of the
Bohm-Gadella theory with quantum mechanics’ ”
Rafael de la Madrid
Department of Physics, University of California at San Diego, La Jolla, CA 92093
E-mail: [email protected]
Abstract. In this reply, we show that when we apply standard distribution theory to
the Lippmann-Schwinger equation, the resulting spaces of test functions would comply
with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov,
and of the theory of ultradistributions were wrong. As well, we point out several
differences between the “standard method” of constructing rigged Hilbert spaces in
quantum mechanics and the method used in Time Asymmetric Quantum Theory.
PACS numbers: 03.65.-w, 02.30.Hq
1. Introduction
The authors of [1] allege to have shown that the conclusions of [2] regarding the
inconsistency of Time Asymmetric Quantum Theory (TAQT) with quantum mechanics
are false. In this reply, we will show that the arguments of [1] are missing essential
aspects of [2], and that therefore the conclusions of [2] still stand.
The most important claims of [1] are the following:
1. There are many examples of TAQT, and the present author has inadvertently
constructed another one.
2. The flaws of the Quantum Arrow of Time (QAT) pointed out in [2] are actually not
flaws, because the original derivation of the QAT was misquoted from its source [3].
3. The crucial argument of [2] regarding the exponential blowup of the test functions
ϕ̂±(z) does not prevent ϕ̂±(z) from being of Hardy class.
As we shall see, all these claims do not stand close scrutiny. In order to show why,
in Sec. 2 we will outline the method to construct rigged Hilbert spaces in quantum
mechanics based on the theory of distributions [4]. We shall refer to this method as
the “standard method” and show that the resulting rigged Hilbert spaces are not of
Hardy class. We shall also explain the meaning of the exponential blowup of ϕ̂±(z) and
why it implies that the spaces of test functions are not of Hardy class. In Sec. 3, we
briefly outline the method to introduce rigged Hilbert spaces of Hardy class in TAQT
and compare such method with the “standard method.” It will then be apparent that
using the method of TAQT, one can introduce any arbitrary rigged Hilbert space for the
http://arxiv.org/abs/0704.1613v1
Gamow states. In order to address claim 2, we show (again) in Sec. 4 that no matter
how one introduces it, the Quantum Arrow of Time has little to do with the actual time
evolution of a quantum system. To address claim 3, in Sec. 5 we use classic results of
Paley and Wiener and of Gelfand and Shilov to show that the “standard method” of
dealing with the Lippmann-Schwinger equation leads to rigged Hilbert spaces that are
not of Hardy class. Section 7 concludes that the arguments of [2] still stand.
2. The “standard method”
In this section, we illustrate the main features of the “standard method” to construct
rigged Hilbert spaces in quantum mechanics [5]. Such “standard method” is based on
the theory of distributions [4]. For the sake of clarity, we shall use the spherical shell
potential of height V0,
V (~x) = V (r) =
0 0 < r < a
V0 a < r < b
0 b < r < ∞ .
(2.1)
For l = 0, the Hamiltonian acts as (we take ~2/2m = 1)
H = − d
+ V (r) . (2.2)
The regular solution is
χ(r;E) =
E r) 0 < r < a
J1(E)ei
E−V0 r + J2(E)e−i
E−V0 r a < r < b
J3(E)ei
E r + J4(E)e−i
E r b < r < ∞ .
(2.3)
The Jost functions and the S matrix are given by
J+(E) = −2iJ4(E) , J−(E) = 2iJ3(E) , (2.4)
S(E) =
J−(E)
J+(E)
. (2.5)
The solutions of the Lippmann-Schwinger equation can be written as
〈r|E±〉 ≡ χ±(r;E) =
χ(r;E)
J±(E)
. (2.6)
When V tends to zero, these eigensolutions tend to the “free” eigensolution:
〈r|E〉 ≡ χ0(r;E) =
E r) . (2.7)
These eigenfunctions are delta-normalized and therefore their associated unitary
operators,
(U±f)(E) =
dr χ±(r;E) f(r) ≡ f̂±(E) , E ≥ 0 , (2.8)
(U0f)(E) =
dr χ0(r;E) f(r) ≡ f̂0(E) , E ≥ 0 , (2.9)
transform from L2([0,∞), dr) onto L2([0,∞), dE).
The Lippmann-Schwinger and the “free” eigenfunctions can be analytically
continued from the scattering spectrum into the whole complex plane. We shall denote
such analytically continued eigenfunctions by χ±(r; z) and χ0(r; z). Whenever they
exist, the analytic continuations of (2.8) and (2.9) are denoted by
f̂±(z) =
dr χ±(r; z) f(r) , (2.10)
f̂0(z) =
dr χ0(r; z) f(r) , (2.11)
where here and in the following z belongs to a two-sheeted Riemann surface.
The resonant energies are given by the poles zn of the S matrix, and their associated
Gamow states are
u(r; zn) = Nn
J3(zn) sin(
zn r) 0 < r < a
J1(zn)
J3(zn)e
zn−V0 r +
J2(zn)
J3(zn)e
zn−V0 r a < r < b
zn r b < r < ∞ ,
(2.12)
where Nn is a normalization factor.
The theory of distributions [4] says that a test function ϕ(r) on which a distribution
d(r) acts is such that the following integral is finite:‡
〈ϕ|d〉 ≡
dr ϕ(r)d(r) < ∞ , (2.13)
where 〈ϕ|d〉 represents the action of the functional |d〉 on the test function ϕ. With some
variations, this is the “standard method” followed by [7–15] to introduce spaces of test
functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the
method followed by the present author runs (somewhat) parallel to [8], not to TAQT.
In order to use (2.13) to construct the rigged Hilbert spaces for the analytically
continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to
obtain the growth of χ±(r; z), χ0(r; z) and u(r; zn). Because the regular solution blows
up exponentially [16],
|χ(r; z)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r , (2.14)
the growth of the eigenfunctions (2.6), (2.7) and (2.12) blows up exponentially:
|χ±(r; z)| ≤ C 1
J±(z)
|z|1/4 r
1 + |z|1/2 r
z |r , (2.15)
‡ In quantum mechanics, we need to impose a few more requirements, but we will not need to go into
such details here.
|χ0(r; z)| ≤ C
|z|1/4 r
1 + |z|1/2 r
z |r , (2.16)
|u(r; zn)| ≤ Cn
|zn|1/2 r
1 + |zn|1/2 r
zn |r . (2.17)
When we plug this exponential blowup into the basic requirement (2.13) of the “standard
method,” we see that the test functions on which those distributions act must fall off at
least exponentially.
By using the Gelfand-Shilov theory of M an Ω functions [4], it was shown in [15]
that when a and b are positive real numbers satisfying
= 1 , (2.18)
and when ϕ+(r) is an infinitely differentiable function whose tails fall off like e−r
a/a, then
ϕ+(z) grows like e|Im(
z)|b/b in the infinite arc of the lower half-plane of the Riemann
surface:
If |ϕ+(r)| < Ce−
a as r → ∞, then |ϕ̂+(z)| ≤ Ce
b as |z| → ∞ . (2.19)
It was shown in [2] that when ϕ+(r) ∈ C∞0 , ϕ̂+(z) blows up exponentially in the infinite
arc of the lower half-plane of the Riemann surface:
If |ϕ+(r)| = 0 when r > A, then |ϕ̂+(z)| ≤ CeA|Im
z | as |z| → ∞ . (2.20)
From the above estimates, we concluded in [2] that the ϕ+’s obtained from the “standard
method” cannot be Hardy functions, since ϕ̂+(z) does not tend to zero as |z| tends to
infinity.
The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞
from estimates such as (2.19) or (2.20), and therefore they conclude that nothing
prevents ϕ̂+(z) from tending to zero and therefore from being Hardy functions. Their
conclusion is not true, because their argument does not take the nature of (2.19) and
(2.20) into account. After we explain the meaning of those estimates, it will be clear
why they prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface.
In order to understand what (2.19) and (2.20) mean, we start with the simple sine
function sin(
E r). When E ≥ 0, the sine function oscillates between +1 and −1:
| sin(
E r)| ≤ 1 , E ≥ 0 . (2.21)
As E tends to infinity, such oscillatory behavior remains, and in such limit the sine
function does not tend to zero. When we analytically continue the sine function,
z r) , (2.22)
the oscillations are bounded by
| sin(
z r)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r . (2.23)
Thus, as |z| tends to infinity, sin(
z r) oscillates wildly, and the magnitude of its
oscillation is tightly bounded by the exponential function. It is certain that as |z|
tends to infinity, sin(
z r) does not tend to zero, even though the function vanishes
z r = ±nπ, n = 0, 1, . . .
It just happens that the solutions of the Lippmann-Schwinger equation follow the
same pattern. When E is positive, the eigensolutions are oscillatory and bounded by
|χ±(r;E)| ≤ C 1
J±(E)
|E|1/4 r
1 + |E|1/2 r
. (2.24)
When the energy is complex, their oscillations get wild and are bounded by Eq. (2.15).§
Thus, the analytic continuations of the Lippmann-Schwinger eigenfunctions oscillate
wildly, and the magnitude of their oscillation is tightly bounded by an exponential
function (multiplied by factors that don’t cancel the exponential blowup when |z| → ∞).
Because in Eqs. (2.10) and (2.11) we are integrating over r, the exponentially-
bounded oscillations of χ±(r; z) get transmitted into ϕ̂±(z). The estimates (2.19) and
(2.20) bound the oscillation of the test functions of the “standard method,” except
for factors that don’t cancel the exponential blowup. It is the exponentially-bounded
oscillations of ϕ̂±(z) what prevent ϕ̂±(z) from tending to zero in any infinite arc of the
Riemann surface and therefore from being of Hardy class.
A somewhat simpler way to understand the above estimates is by looking at the
“free” incoming and outgoing wave functions ϕin and ϕout. Because in the energy
representation such wave functions are the same as the “in” and “out” wave functions,
ϕ̂in(E) = 〈E|ϕin〉 = 〈+E|ϕ+〉 = ϕ̂+(E) , (2.25)
ϕ̂out(E) = 〈E|ϕout〉 = 〈−E|ϕ−〉 = ϕ̂−(E) , (2.26)
in TAQT the analytic continuation of ϕ̂in(E) and ϕ̂out(E) are also of Hardy class. Since
ϕ̂in,out(z) =
z r)ϕin,out(r) , (2.27)
it is evident that the exponential blowup (2.23) of sin(
z r) will prevent ϕ̂in,out(z) from
tending to zero as |z| → ∞ in any half-plane of the Riemann surface. Thus, ϕ̂in,out(z)
are not of Hardy class, contrary to TAQT.
Strictly speaking, the bounds (2.19) and (2.20) are not the tightest ones. We should
include polynomial corrections, see Eq. (B.15) in [15], and the effect of
|z|1/4r
1+|z|1/2r
and 1J±(z)
to obtain the tightest bounds. We shall not obtain those corrections here, because they
do not cancel the exponential blowup at infinity, and because in this reply we shall use
instead other classic bounds, see Sec. 5.
Let us summarize this section. In standard quantum mechanics, once the
Lippmann-Schwinger equation is solved, the properties of ϕ̂±(z) are already determined
by Eqs. (2.10) and (2.11), and there is no room for any extra assumption on their
properties. This means, in particular, that the Hardy axiom cannot be simply assumed.
Rather, the Hardy axiom must be proved using Eqs. (2.10) and (2.11).‖ It simply
§ The points at which J±(z) = 0 do not affect the essence of the argument.
‖ This is what in [2] it was meant by the assertion that the Hardy axiom is not a matter of assumption
but a matter of proof.
happens that the “standard method” yields ϕ̂±(z) and ϕ̂in,out(z) that oscillate wildly.
Because these oscillations are bounded by exponential functions, ϕ̂±(z) and ϕ̂in,out(z) do
not tend to zero as |z| tends to infinity in any half-plane of the Riemann surface—hence
they are not of Hardy class.
3. TAQT vs. the “standard method”
In TAQT, one doesn’t solve the Lippmann-Schwinger equation in order to afterward
obtain the properties of ϕ̂±(z) using Eq. (2.10). Instead, one transforms into the energy
representation (using U± in our example) and then imposes the Hardy axiom. If H2±
denotes the spaces of Hardy functions from above (+) and below (−), S denotes the
Schwartz space, and Φ̃± denote their intersection restricted to the positive real line,
Φ̃± = H2± ∩ S|R+ , (3.1)
then the Hardy axiom states that the functions ϕ̂±(z) belong to Φ̃∓:
ϕ̂±(z) ∈ Φ̃∓ . (3.2)
This means that in the position representation, the Gamow states and the analytic
continuation of the Lippmann-Schwinger eigenfunctions act on the following spaces:
ΦBG∓ = U
± Φ̃∓ . (3.3)
It is obvious that the choices (3.2)-(3.3) are arbitrary. One may as well choose another
dense subset of L2([0,∞), dE) with different properties and obtain a different space of
test functions for the Gamow states. What is more, ΦBG± are different from the spaces
of test functions obtained through the “standard method,” because the functions ϕ̂±(z)
of the “standard method” are not of Hardy class.
The authors of [1] claim that the present author has inadvertently constructed
an example of TAQT. That such is not the case can be seen not only from the
differences between the “standard method” and the method used in TAQT to introduce
rigged Hilbert spaces, but also from the outcomes. For example, whereas in the
position representation the “standard method” calls for just one rigged Hilbert
space for the Gamow states and for the analytically continued Lippmann-Schwinger
eigenfunctions [15], TAQT uses two rigged Hilbert spaces
ΦBG± ⊂ L2([0,∞), dr) ⊂ Φ×BG± . (3.4)
One of the rigged Hilbert spaces is used for the “in” solutions and for the anti-resonant
states, whereas the other one is used for the “out” solutions and for the resonant states.
Another difference is that in TAQT, the solutions of the Lippmann-Schwinger equation
for scattering energies have a time asymmetric evolution [17], whereas the “standard
method” yields that such time evolution runs from t = −∞ to t = +∞, see [14].
Incidentally, this is an instance where TAQT differs not only mathematically but also
physically from standard quantum mechanics, because in standard scattering theory,
the time evolution of a scattering process goes from the asymptotically remote past
(t → −∞) to the asymptotically far future (t → +∞). This is not so in TAQT [17].
It seems hardly necessary to clarify what the present author means by “standard
quantum mechanics.” Standard quantum mechanics means the Schrödinger equation,
and standard scattering theory means the Lippmann-Schwinger equation. In standard
quantum mechanics, one assumes that these equations describe the physics and then
solves them. Because of the scattering and resonant spectra, their solutions lie within
rigged Hilbert spaces. The construction of such rigged Hilbert spaces follows by
application of the “standard method.” By contrast, TAQT simply assumes that the
solutions of the Schrödinger and the Lippmann-Schwinger equations comply with the
Hardy axiom, without ever showing that the actual solutions of those equations comply
with such axiom.
It was claimed in [2] that there is no example of TAQT. The authors of [1] dispute
such claim and assert that there are many examples. The present author disagrees
with their assertion, because assuming that for a large class of potentials the solutions
of the Lippmann-Schwinger equation comply with the Hardy axiom is not the same
as having an example where it is shown that the actual solutions of the Lippmann-
Schwinger equation comply with the Hardy axiom. In fact, to the best of the present
author’s knowledge, no advocate of TAQT has ever used Eq. (2.10) to discuss the
analytic properties of ϕ̂±(E) = 〈±E|ϕ±〉 in terms of the actual solutions χ±(r;E) of
the Lippmann-Schwinger equation.
The authors of [1] inadvertently acknowledge that there is no example of TAQT
when they say that they still need “to identify the form and properties” of the functions
of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging
that they don’t know whether the standard Gamow states defined in the position
representation are well defined as functionals acting on ΦBG±. If TAQT had an example,
it would be known.
4. The Quantum Arrow of Time (QAT)
Advocates of TAQT argue that their choice (3.3) is not arbitrary but rather is rooted
on a causality principle. Such causality principle is the “preparation-registration arrow
of time,” sometimes referred to as the “Quantum Arrow of Time” (QAT). For the “in”
states ϕ+, the causal statement of the QAT is written as
ϕ̃+(t) ≡
dE e−iEtϕ̂+(E) = 0 , for t > 0 . (4.1)
By one of the Paley-Wiener theorems, Eq. (4.1) is equivalent to assuming that ϕ̂+(E)
is of Hardy class from below. The corresponding causal statement for the “out” wave
functions ϕ− implies that ϕ− is of Hardy class from above. Hence, in TAQT, the
choice (3.3) is not arbitrary but a consequence of causality.
It was pointed out in [2] that the QAT is flawed. The argument was twofold. First,
it was pointed out that the original derivation [3] of Eq. (4.1) made use of the following
flawed assumption:
0 = 〈E|ϕin(t)〉 = 〈+E|ϕ+(t)〉 = e−iEtϕ̂+(E) , for all energies, (4.2)
which can happen only when ϕ+ and ϕin are identically 0. It was then pointed out that
even though one may simply assume the causal statement (4.1) and forget about how
it was derived, such causal statement says little about the actual time evolution of a
quantum system, because the quantum mechanical time evolution of ϕ+ is not given by
Eq. (4.1):
ϕ+(t) = e−iHtϕ+ 6= ϕ̃+(t) . (4.3)
To counter this argument, the authors of [1] claim that the derivation of the QAT
was misquoted from the original source [3], and that the flawed assumption (4.2) was
never used to derive the QAT (4.1). It seems therefore necessary to quote the original
derivation (see [3], page 2597):¶
“We are now in the position to give a mathematical formulation of the QAT: we
choose t = 0 to be the time before which all preparations of φin(t) are completed and
after which the registration of ψout(t) begins. This means that for t > 0 the energy
distribution of the preparation apparatus must vanish: 〈E, η|φin(t)〉 = 0 for all values
of the quantum numbers E and η (η are the additional quantum numbers which we
usually suppress). As the mathematical statement for ‘no preparations for t > 0’ we
therefore write (the slightly weaker condition)
dE 〈E|φin(t)〉 =
dE 〈+E|φ+(t)〉 =
dE 〈+E|e−iHt|φ+〉 (4.4)
dE 〈+E|φ+〉e−iEt ≡ F(t) for t > 0 . (4.5)
The readers can decide whether or not the flawed hypothesis (4.2) was used to derive
the QAT (4.5).
Nevertheless, it is actually not very relevant whether the authors of [3] used (4.2)
to derive (4.1). As pointed out in [2], and as mentioned above, even though one can
forget (4.2) and simply assume (4.1) as the causal condition to be satisfied by ϕ+, such
causal condition has little to do with the time evolution of a quantum system, see again
Eq. (4.3). In particular, as even the author of [6] has asserted, the t that appears in
Eq. (4.1) is not the same as the parametric time t that labels the evolution of a quantum
system.+ Thus, as far as standard quantum mechanics is concerned, the causal content
of the QAT is physically vacuous, and therefore, regardless of how one motivates it,
there is no physical justification for the choice (3.3).
5. TAQT vs. the “classic results”
In this section, we are going to compare the Hardy axiom of TAQT with some
classic results of Paley and Wiener, of Gelfand and Shilov and of the theory of
ultradistributions, which we shall collectively refer to as the “classic results.” More
¶ In this quote, φin, φ+, F(t) and Eq. (4.5) correspond, respectively, to ϕin, ϕ+, ϕ̃+(t) and Eq. (4.1).
+ All this shows that the new term TAQT is a misnomer. A better name is Bohm-Gadella theory,
because it was these two authors who proposed the theory and summarized it in [18].
precisely, we will see that the spaces of test functions ϕ̂± obtained by the “standard
method” would be of Hardy class only if the “classic results” were wrong.
The direct comparison with the “classic results” is more easily done in one
dimension, and therefore we shall use the example of the one-dimensional rectangular
barrier potential:
V (x) =
0 −∞ < x < a
V0 a < x < b
0 b < x < ∞ .
(5.1)
For this potential, the “in” and “out” eigensolutions are well known and can be found
for example in [12]. We shall denote them by χ±l,r(x;E), where the labels l,r denote
left and right incidence. When we analytically continue these eigenfunctions, or when
we consider the Gamow states for this potential, the “standard method” calls for test
functions ϕ±l,r(x) for which the following integrals are finite:
ϕ̂±l,r(z) =
dxχ±l,r(x; z)ϕ(x) . (5.2)
Just as in the example discussed in Sec. 2, the test functions ϕ(x) must at least fall off
faster than exponentials.
To further simplify the discussion, we need to recall that, because of Eqs. (2.25)
and (2.26), the Hardy axiom assumes that the “free” wave functions ϕ̂inl,r(E) and ϕ̂
l,r (E)
are also of Hardy class. These “free” functions are given by (hereafter, we just consider
ϕinl,r, since the analysis for ϕ
l,r is the same)
ϕ̂inl (E) =
dx e−ikx ϕin(x) , (5.3)
ϕ̂inr (E) =
dx eikx ϕin(x) , (5.4)
where k =
E is the wave number. The total wave function is given by the sum of left
and right components:
ϕ̂in(E) = ϕ̂inl (E) + ϕ̂
r (E) . (5.5)
It is simpler to work with k rather than with E and define
ϕ̂inl,r(k) ≡
2k ϕ̂inl,r(E) ; (5.6)
that is,
ϕ̂inl (k) =
dx e−ikx ϕin(x) , k ≥ 0 , (5.7)
ϕ̂inr (k) =
dx eikx ϕin(x) , k ≥ 0 . (5.8)
The “total” wave function in the wave-number representation, ϕ̂in(k) = ϕ̂inl (k)+ ϕ̂
r (k),
is thus the Fourier transform of ϕ(x),
ϕ̂in(k) =
dx e−ikx ϕin(x) , k ∈ R . (5.9)
Its analytic continuation will be denoted as
ϕ̂in(q) =
dx e−iqx ϕin(x) , q ∈ C . (5.10)
At this point, we are ready to introduce two classic theorems. The first one is due
to Paley and Wiener (see Theorem IX.11 in [19]):
Theorem 1 (Paley-Wiener). An entire analytic function ϕ̂(q) is the Fourier
transform of a C∞0 (R) function ϕ(x) with support in the segment {x | |x| < A} if,
and only if, for each N there is a CN so that
|ϕ̂(q)| ≤ CN e
A|Im(q)|
(1 + |q|)N
(5.11)
for all q ∈ C.
This theorem says that the Fourier transform of a C∞0 function is an analytic
function that grows exponentially, and that such exponential growth is mildly corrected
(but not canceled) by a polynomial falloff.
The second theorem we shall use is due to Gelfand and Shilov [4]. Before stating it,
we need some definitions. Let a and b denote two positive real numbers satisfying (2.18).
Let us define Φa,b as the set of all differentiable functions ϕ(x) (−∞ < x < ∞) satisfying
the inequalities
dnϕ(x)
∣∣∣∣ ≤ Cne
−α |x|
a (5.12)
with constants Cn and α > 0 which may depend on the function ϕ. Let us define the
space Φ̂a,b as the set of entire analytic functions ϕ̂(q), q = Re(q) + i Im(q), which satisfy
the inequalities
|qnϕ̂(q)| ≤ Cne+β
|Im(q)|b
b , (5.13)
where the constants Cn and β > 0 depend on the function ϕ. It is obvious that
the elements of Φa,b are functions that, together with their derivatives, decrease at
infinity faster than e−
a , whereas the elements of Φ̂a,b are analytic functions that grow
exponentially at infinity as e+
|Im(q)|b
b , except for a polynomial correction that doesn’t
cancel the exponential blowup.
Theorem 2 (Gelfand-Shilov). The space Φ̂a,b is the Fourier transform of Φa,b.
This theorem means that the smooth functions that fall off at infinity faster than
e−|x|
a/a are, in Fourier space, analytic functions that grow exponentially like e+|Im(q)|
The bounds (5.11) and (5.13) are to be understood in the same way as the
bounds (2.19) and (2.20). That is, the bounds (5.11) and (5.13) mean that ϕ̂(q) is
an oscillatory function that grows exponentially in the infinite arc of the q-plane, the
oscillation being tightly bounded by Eqs. (5.11) and (5.13) when ϕ(x) belongs to C∞0
and Φa,b, respectively. Note that after the addition of the corresponding polynomial
corrections, the bounds (2.19) and (2.20) are entirely analogous to the bounds (5.11)
and (5.13)—the operators U± are after all Fourier-like transforms [12].
Let us now apply the above theorems to the functions ϕin(x) obtained by the
“standard method.” In order for Eq. (5.10) to make sense, ϕin(x) must fall off faster
than exponentials. If we choose ϕin(x) to fall off like e−|x|
a/a, then the Gelfand-Shilov
theorem tells us that ϕ̂in(q) grows like e+|Im(q)|
b/b. Even when we impose that ϕin(x)
is C∞0 , which is already a very strict requirement, the Paley-Wiener theorem says that
ϕ̂in(q) grows exponentially. This means, in particular, that the ϕ̂in(q) do in general not
tend to zero in the infinite arc of the q-plane, because if they did, the Paley-Wiener
and the Gelfand-Shilov theorems would be wrong. Because of Eq. (5.6), ϕ̂in(z) does in
general not tend to zero as |z| tends to infinity in the lower half-plane of the second
sheet. Hence the space of ϕ̂in’s is not of Hardy class from below.
The space of ϕ̂+’s cannot be of Hardy class from below either, because if it were,
|z|→∞
ϕ̂+(z) = 0 , (5.14)
where the limit is taken in the lower half plane of the second sheet. By Eq. (2.25),
this implies that also the space of ϕ̂in’s would be of Hardy class and comply with this
limit, which we know is not possible due to the “classic results.” Thus, the “standard
method” yields spaces of test functions that do not comply with the Hardy axiom. This
is precisely what it was meant in [2] by the assertion that TAQT is inconsistent with
standard quantum mechanics.
To finish this section, we note that if we chose the test functions as in [8], then
we would be dealing with ultradistributions. In Fourier space, the test functions for
ultradistributions grow faster than any exponential as we follow the imaginary axis,
see [8] and references therein. Thus, if the “standard method” yielded spaces of Hardy
functions, that property of ultradistributions would be false.
6. Further remarks
The authors of [1] claim that it is inaccurate to state that the proponents of TAQT
dispense with asymptotic completeness. This statement should be compared with the
first quote in section 6 of [2].
The authors of [1] also claim that TAQT obtains the resonant states by solving
the Schrödinger equation subject to purely outgoing boundary conditions. This claim
should be compared with the second quote in section 6 of [2].
The authors of [1] also dispute the assertion of [2] that TAQT sometimes uses the
whole real line as though it coincided with the scattering spectrum of the Hamiltonian.
A glance at, for example, the QAT (4.1) seems to support such assertion.
7. Conclusions
In standard scattering theory, one assumes that the physics is described by the
Lippmann-Schwinger equation. When one solves such equation, one finds that its
solutions must be accommodated by a rigged Hilbert space, and that its time evolution
runs from t = −∞ till t = +∞ [14]. When one analytically continues the solutions
of the Lippmann-Schwinger equation, one finds that they must be accommodated by
one rigged Hilbert space, which also accommodates the resonant (Gamow) states. The
construction of such rigged Hilbert space is determined by standard distribution theory.
By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger
equations belong to two rigged Hilbert spaces of Hardy class. In TAQT, one never
explicitly solves the Lippmann-Schwinger equation for specific potentials in the position
representation. Instead, one assumes that its solutions satisfy the Hardy axiom. Unlike
in standard scattering theory, in TAQT the time evolution of the solutions of the
Lippmann-Schwinger equation does not run from t = −∞ till t = +∞.
By comparing the properties of the actual solutions of the Lippmann-Schwinger
equation with the Hardy axiom, we have seen that such actual solutions would comply
with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov,
and of the theory of ultradistributions were wrong. We have (again) stressed the fact
that the Quantum Arrow of Time, which is the justification for using the rigged Hilbert
spaces of Hardy class, has little to do with the time evolution of a quantum system. We
have stressed that using the method of TAQT to introduce rigged Hilbert spaces, we
could accommodate the Gamow states in a landscape of arbitrary rigged Hilbert spaces,
see also [5].
Our claim of inconsistency should not be taken as a claim that TAQT is
mathematically inconsistent or that TAQT doesn’t have a beautiful mathematical
structure. What the present author claims is that TAQT is not applicable in quantum
mechanics and is in fact a different theory.
To finish, we would like to mention that the “classic theorems” are not in conflict
with using Hardy functions in quantum mechanics. They are in conflict only with the
Hardy axiom. Thus, our results do not apply to other works that use Hardy functions
in a different way [20].
Acknowledgment
This research was supported by MEC and DOE.
References
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[14] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3949 (2006); quant-ph/0603176.
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[16] J.R. Taylor, Scattering theory, John Wiley & Sons, Inc., New York (1972).
[17] A. Bohm, P. Kielanowski, S. Wickramasekara, “Complex energies and beginnings of time suggest
a theory of scattering and decay,” quant-ph/0510060.
[18] A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes
in Physics 348, Berlin (1989).
[19] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York
(1975).
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http://arxiv.org/abs/quant-ph/0109154
http://arxiv.org/abs/quant-ph/0110165
http://arxiv.org/abs/quant-ph/0210167
http://arxiv.org/abs/quant-ph/0407195
http://arxiv.org/abs/quant-ph/0502053
http://arxiv.org/abs/quant-ph/0603176
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http://arxiv.org/abs/quant-ph/0612027
Introduction
The ``standard method''
TAQT vs. the ``standard method''
The Quantum Arrow of Time (QAT)
TAQT vs. the ``classic results''
Further remarks
Conclusions
|
0704.1614 | Modelling the Galactic bar using OGLE-II Red Clump Giant Stars | Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 6 September 2021 (MN LATEX style file v2.2)
Modelling the Galactic bar using OGLE-II Red Clump Giant Stars
Nicholas J. Rattenbury1, Shude Mao1, Takahiro Sumi2, Martin C. Smith3 ⋆
1 University of Manchester, Jodrell Bank Observatory, Macclesfield, SK11 9DL, UK
2 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
3 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
Accepted ........ Received .......; in original form ......
ABSTRACT
Red clump giant stars can be used as distance indicators to trace the mass distribution of
the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing
collaboration database to constrain analytic tri-axial models for the Galactic bar. We find the
bar major axis is oriented at an angle of 24◦ – 27◦ to the Sun-Galactic centre line-of-sight.
The ratio of semi-major and semi-minor bar axis scale lengths in the Galactic plane x0, y0,
and vertical bar scale length z0, is x0 : y0 : z0 = 10 : 3.5 : 2.6, suggesting a slightly more
prolate bar structure than the working model of Gerhard (2002) which gives the scale length
ratios as x0 : y0 : z0 = 10 : 4 : 3.
Key words: Galaxy: bulge - Galaxy: centre - Galaxy: structure
1 INTRODUCTION
It is now generally accepted that the Galactic bulge is a tri-axial,
bar-like structure. Observational evidence for a bar has arisen
from several sources, such as the study of gas kinematics (e.g.
Binney et al. 1991), surface brightness (e.g. Blitz & Spergel 1991),
star counts (e.g. Nakada et al. 1991; Stanek et al. 1994) and mi-
crolensing (e.g. Udalski et al. 1994); see Gerhard (2002) for a re-
view.
Observational data have been used to constrain dynami-
cal models of the Galaxy. Dwek et al. (1995) used the COBE-
DIRBE multi-wavelength observations of the Galactic centre
(Weiland et al. 1994) to constrain several analytic bar models.
Stanek et al. (1997) used optical observations of red clump gi-
ant (RCG) stars to constrain theoretical bar models. Similarly,
Babusiaux & Gilmore (2005) and Nishiyama et al. (2005) traced
the bulge RCG population in the infrared. This work uses a sam-
ple of stars 30 times larger than that of Stanek et al. (1997), with
a greater number of fields distributed across a larger area of the
Galactic bulge, thus allowing finer constraints to be placed on the
bar parameters than those determined by Stanek et al. (1997).
Our current understanding of the Galactic bar is that it is ori-
entated at about 15− 40◦ to the Sun–Galactic centre line-of-sight,
with the near end in the first Galactic longitude quadrant. The
bar length is around 3.1 – 3.5 kpc with axis ratio approximately
10 : 4 : 3 (Gerhard 2002). The above bar parameters are gener-
ally accepted as a working model, however they are not well de-
termined. Our understanding of the complete structure of the inner
Galactic regions is similarly incomplete. For example, recent infra-
red star counts collected by the Spitzer Space Telescope for Galac-
⋆ e-mail: (njr, smao)@jb.man.ac.uk; [email protected];
[email protected]
tic longitudes l = 10◦ – 30◦ are best explained assuming a long thin
bar oriented at an angle of ∼ 44◦ to the Sun–Galactic centre line
(Benjamin et al. 2005) while most previous studies (performed at
|l| . 12◦) prefer a short bar with an opening angle of ∼ 20◦. Re-
cently, Cabrera-Lavers et al. (2007) report that NIR observations of
RCGs support the hypothesis that a long thin bar oriented at ∼ 45◦
co-exists with a distinct short tri-axial bulge structure oriented at
∼ 13◦. In addition, there may be some fine features, such as a ring
in the Galactic bulge (Babusiaux & Gilmore 2005), or a secondary
bar (Nishiyama et al. 2005), that are not yet firmly established. It is
therefore crucial to obtain as many constraints as possible in order
to better understand the structure of the inner Galaxy.
In this paper we present an analysis of RCG stars observed
in the Galactic bulge fields during the second phase of the OGLE
microlensing project (Udalski et al. 2000). These stars are bright
and they are approximately standard candles, hence their magni-
tudes can be taken as an approximate measure of their distances.
Number counts in 34 central bulge fields with −4◦ 6 l 6 6◦
and −6◦ 6 b 6 3◦ are used to constrain analytic tri-axial bar
models, and thereby obtain estimates on bar parameters. We repeat
the analysis with 44 fields with −6.8◦ 6 l 6 10.6◦. We find the
fitted bar parameters support the general orientation and shape of
the bar reported by other groups. This paper is organised as fol-
lows: in Section 2 we describe the OGLE microlensing experiment
and photometry catalogue and we illustrate how RCG stars can be
used as approximate distance indicators; in Section 3 we detail how
RCGs in the OGLE-II proper motion catalogue are selected; in Sec-
tion 4 we compute the distance modulus to the red clump in 45
OGLE-II fields and thereby trace the central mass density of the
Galaxy; in Section 5 we describe how RCG star count histograms
for each field can be used to constrain analytic bar models of the
inner Galaxy; our results and their comparison to previous works is
c© 2005 RAS
http://arxiv.org/abs/0704.1614v1
2 Rattenbury et al.
given in Section 6 and in Section 7 we discuss the implications and
limitations of these results.
2 DATA
The OGLE (Udalski et al. 2000) and MOA (Bond et al. 2001;
Sumi et al. 2003) microlensing collaborations currently make rou-
tine observations of crowded stellar fields towards the Galactic
bulge, and issue alerts when a microlensing event is detected. A
result of this intense monitoring is the creation of massive photom-
etry databases for stars in the Galactic bulge fields. Such databases
are extremely useful for kinematic and population studies of the
central regions of the Galaxy.
Sumi et al. (2004) obtained the proper motions for millions of
stars in the OGLE-II database for a large area of the sky. Fig. 1
shows the OGLE-II fields towards the Galactic bulge. In this paper
we focus on the population of red clump giant stars at the Galac-
tic centre. Red clump giants are metal-rich horizontal branch stars
(Stanek et al. 2000 and references therein). Theoretically, one ex-
pects their magnitudes to have (small) variations with metallicity,
age and initial stellar mass (Girardi & Salaris 2001). Empirically
they appear to be reasonable standard candles in the I-band with
little dependence on metallicities (Udalski 2000; Zhao et al. 2001).
3 METHODS
Stanek et al. (1997) used RCG stars in 12 fields (see Fig. 1)
observed during the first phase of the OGLE microlensing ex-
periment, OGLE-I, to constrain several analytic models of the
Galactic bar density distribution. Babusiaux & Gilmore (2005),
Nishiyama et al. (2005) and Cabrera-Lavers et al. (2007) similarly
used IR observations of RCGs to trace the bulge stellar density. We
follow similar procedures to extract RCG stars from the OGLE-II
Galactic bulge fields and to constrain analytic models.
3.1 Sample selection
We compute the reddening-independent magnitude IV−I for all
stars in each of the 45 OGLE-II fields:
IV−I = I − AI/(AV − AI) (V − I)
where AI and AV are the extinctions in the I and V bands deter-
mined by Sumi (2004). We select stars which have I < 4(V −
I)+k, where k is a constant chosen for each field that excludes the
main-sequence dwarf stars, and IV−I < 14.66, which corresponds
to the magnitude of RCG stars closer than 15 kpc1. Fig. 2 shows the
sample of stars selected from the IV−I, (V −I) CMD for OGLE-II
field 1.
The selected stars are then collected in IV−I magnitude bins,
see Fig. 3. A function comprised of quadratic and Gaussian com-
ponents is used to model this number count histogram in each of
the OGLE-II fields:
N(x ≡ IV−I) = a+bx+cx2+
2πσRC
− (IV−I,RC − x)
2σRC2
1 We assume the fiducial RCG star at 15 kpc has an absolute magnitude and
colour of I0 = −0.26 and (V − I)0 = 1.0 respectively, with AI/(AV −
AI) = 0.96.
0.5 1 1.5 2 2.5 3
17PSfrag replacements
V − I
Figure 2. Reddening-independent magnitude vs colour diagram for OGLE-
II field 1. The red clump is clearly visible. Stars are selected (black dots)
using the criteria I < 4(V − I) + k, where k is a constant chosen for each
field, and IV−I < 14.66 (solid lines, see text).
12 12.5 13 13.5 14 14.5 15
PSfrag replacements
Figure 3. Number count histogram for selected stars in OGLE-II field 1.
The heavy solid line is the best-fitting model of the form given by equa-
tion (1). The last histogram bin is generally not included in the fitting due
to incompleteness effects near the limiting magnitude of IV−I < 14.66.
where σRC is the spread of the red clump giant magnitudes,
IV −I,RC is the mean apparent magnitude of the red clump, NRC
and a, b and c are coefficients for the Gaussian and quadratic com-
ponents respectively. These six parameters are allowed to vary for
each of the OGLE-II fields and the best-fitting values obtained by
minimising χ2 =
[(N − Nobs,i)/σi]2 where the sum is
taken over all i = 1 . . . 26 histogram bins which cover the range
12 6 IV−I 6 14.6. The error on histogram number counts is as-
sumed to be σi =
Nobs,i , i.e. Poissonian.
The errors on the mean magnitude and distribution width of
the red clump stars, ξIV−I,RC and ξσRC respectively, are deter-
mined using a maximum-likelihood analysis (see e.g. Lupton et al.
1987).
We determine the distance modulus to the red clump in each
field as:
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 3
−8−6−4−20246810
replacem
Galactic longitude (◦)
Figure 1. The position of the 45 OGLE-II fields used in this analysis. The solid black regions indicate the location of fields used in Stanek et al. (1997).
µ = 5 log(d)− 5 = I − I0,RC
= IV −I,RC +R (V − I)0,RC − I0,RC (2)
where d is the distance to the red clump measured in parsecs,
IV −I,RC is the fitted peak reddening-independent magnitude of the
red clump, R is the mean value of AI/(AV − AI) for each field.
I0,RC = −0.26 ± 0.03 (Alves et al. 2002) and (V − I)0,RC =
1.0± 0.08 (Paczynski & Stanek 1998) are the mean absolute mag-
nitude and colour of the population of local red clump giant stars.
We assume that the properties of the local population of red clump
giant stars are the same as the Galactic bulge population, however
it is likely that population effects are significant. We discuss the
effects of red clump population effects in Section 4.1.
4 DISTANCE MODULI OF RCGS
Table 1 lists the fitted parameters IV −I,RC ± ξIV−I,RC and σRC ±
ξσRC , along with the mean value of R for each field. The dis-
tance modulus, µ, computed from equation (2) showed that there
was a significant offset: the fitted mean red clump giant magni-
tudes were uniformly too faint compared to that expected of typ-
ical local RCG stars at 8 kpc, resulting in overly large distance
moduli. Sumi (2004) found that the OGLE-II RCGs are approxi-
mately 0.3 mag fainter than expected when assuming that the pop-
ulation of RCGs in the bulge is the same as local. For this reason,
we apply an offset of 0.3 mag to the distance moduli computed
above. The shifted distance moduli µ′ = µ − 0.3 are given in Ta-
ble 1. The true line-of-sight dispersion, σlos, can be approximated
by σlos = (σ
RC − σ2RC,0 − σ2e )1/2 where σRC is the Gaussian dis-
persion fitted using equation (1), σRC,0 is the intrinsic dispersion
of the RCG luminosity function and σe is an estimate of the pho-
tometric errors (Babusiaux & Gilmore 2005). We use σRC,0 = 0.2
(see Section 2) and σe = 0.02, along with the tabulated values of
σRC to determine σlos. Fig. 4 shows the mean distance to the red
clump stars in each of the 45 OGLE-II fields listed in Table 1. The
fields with Galactic longitude −4◦ 6 l 6 6◦ show clear evidence
of a bar, with a major axis oriented at ≃ 25◦ to the Sun-Galactic
centre line-of-sight. For fields with l > 6◦ and l < −4◦ the mean
position of the red clump stars do not continue to trace the major
axis of a linear structure. Babusiaux & Gilmore (2005) find similar
evidence that the position of the red clump stars is not predicted by
a bar for l = −9.7◦, suggesting that this is a detection of the end
of the bar, or the beginning of a ring-like structure. We investigate
these possibilities in Section 6.1.
The uncertainties on the mean position of the red clump are
large; the largest term in the error expression for ξIV−I,RC arises
from the relatively large uncertainty in the intrinsic colour of the
RCGs. The true line-of-sight dispersions, σlos, are consistent with
a wide range of bar position angles, but the mean position of the
red clump in each direction strongly suggest a bar oriented along
a direction consistent with that determined by the previous work
referred to in Section 1.
c© 2005 RAS, MNRAS 000, 000–000
4 Rattenbury et al.
Table 1. Fitted values of the red clump mean magnitude, IV −I,RC, and Gaussian dispersion, σRC for RCG stars selected from 45 OGLE-II fields. The
mean selective extinction R is also given. The shifted distance modulus µ′ = µ − 0.3 for each field is computed via equation (2), see text. σlos is the true
line-of-sight distance dispersion of the red clump giant stars. N is the total number of stars selected from each CMD, see Section 3.1
Field l b IV −I,RC σRC R µ
′ σlos N
1 1.08 -3.62 13.616 ±0.005 0.2936 ±0.0043 0.964 ±0.02 14.55 ±0.09 0.21 31002
2 2.23 -3.46 13.536 ±0.005 0.3130 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.24 33813
3 0.11 -1.93 13.664 ±0.003 0.2461 ±0.0026 0.964 ±0.04 14.60 ±0.09 0.14 66123
4 0.43 -2.01 13.655 ±0.003 0.2517 ±0.0026 0.964 ±0.04 14.59 ±0.09 0.15 65748
5 -0.23 -1.33 13.630 ±0.003 0.2809 ±0.0030 0.964 ±0.06 14.56 ±0.10 0.20 43493
6 -0.25 -5.70 13.606 ±0.011 0.4197 ±0.0088 0.964 ±0.03 14.54 ±0.09 0.37 12085
7 -0.14 -5.91 13.589 ±0.012 0.4255 ±0.0100 0.964 ±0.03 14.52 ±0.09 0.38 11328
8 10.48 -3.78 13.366 ±0.002 1.4277 ±0.0013 0.964 ±0.03 14.30 ±0.09 1.41 10248
9 10.59 -3.98 13.383 ±0.012 0.5855 ±0.0110 0.964 ±0.03 14.32 ±0.09 0.55 9971
10 9.64 -3.44 13.407 ±0.002 1.4178 ±0.0013 0.964 ±0.03 14.34 ±0.09 1.40 12068
11 9.74 -3.64 13.438 ±0.015 0.3902 ±0.0125 0.964 ±0.04 14.37 ±0.09 0.33 11345
12 7.80 -3.37 13.381 ±0.008 0.4692 ±0.0072 0.964 ±0.04 14.31 ±0.09 0.42 15936
13 7.91 -3.58 13.389 ±0.009 0.4234 ±0.0081 0.964 ±0.03 14.32 ±0.09 0.37 15698
14 5.23 2.81 13.550 ±0.006 0.3131 ±0.0053 0.964 ±0.04 14.48 ±0.09 0.24 27822
15 5.38 2.63 13.564 ±0.007 0.3027 ±0.0059 0.964 ±0.04 14.50 ±0.09 0.23 24473
16 5.10 -3.29 13.487 ±0.007 0.3200 ±0.0057 0.964 ±0.03 14.42 ±0.09 0.25 22055
17 5.28 -3.45 13.474 ±0.007 0.3270 ±0.0058 0.964 ±0.03 14.41 ±0.09 0.26 23132
18 3.97 -3.14 13.471 ±0.005 0.2983 ±0.0044 0.964 ±0.02 14.40 ±0.09 0.22 32457
19 4.08 -3.35 13.491 ±0.006 0.3026 ±0.0048 0.964 ±0.03 14.42 ±0.09 0.23 30410
20 1.68 -2.47 13.583 ±0.004 0.2726 ±0.0031 0.964 ±0.03 14.52 ±0.09 0.18 49900
21 1.80 -2.66 13.596 ±0.004 0.2886 ±0.0034 0.964 ±0.02 14.53 ±0.09 0.21 45578
22 -0.26 -2.95 13.741 ±0.004 0.2669 ±0.0034 0.964 ±0.04 14.67 ±0.09 0.18 42914
23 -0.50 -3.36 13.724 ±0.004 0.2778 ±0.0037 0.964 ±0.04 14.66 ±0.09 0.19 36030
24 -2.44 -3.36 13.817 ±0.004 0.2638 ±0.0037 0.964 ±0.04 14.75 ±0.09 0.17 35351
25 -2.32 -3.56 13.810 ±0.004 0.2695 ±0.0039 0.964 ±0.03 14.74 ±0.09 0.18 31801
26 -4.90 -3.37 13.815 ±0.005 0.2897 ±0.0046 0.964 ±0.02 14.75 ±0.09 0.21 26940
27 -4.92 -3.65 13.794 ±0.006 0.2782 ±0.0050 0.964 ±0.02 14.73 ±0.09 0.19 24603
28 -6.76 -4.42 13.785 ±0.010 0.3081 ±0.0082 0.964 ±0.02 14.72 ±0.09 0.23 13702
29 -6.64 -4.62 13.762 ±0.009 0.2792 ±0.0083 0.964 ±0.02 14.70 ±0.09 0.19 12893
30 1.94 -2.84 13.570 ±0.004 0.2746 ±0.0034 0.964 ±0.03 14.50 ±0.09 0.19 41748
31 2.23 -2.94 13.535 ±0.004 0.2892 ±0.0036 0.964 ±0.02 14.47 ±0.09 0.21 40623
32 2.34 -3.14 13.528 ±0.004 0.3001 ±0.0038 0.964 ±0.02 14.46 ±0.09 0.22 35954
33 2.35 -3.66 13.559 ±0.005 0.3206 ±0.0045 0.964 ±0.02 14.49 ±0.09 0.25 30882
34 1.35 -2.40 13.608 ±0.003 0.2711 ±0.0031 0.964 ±0.03 14.54 ±0.09 0.18 52216
35 3.05 -3.00 13.533 ±0.005 0.3049 ±0.0040 0.964 ±0.02 14.47 ±0.09 0.23 36796
36 3.16 -3.20 13.508 ±0.005 0.3104 ±0.0042 0.964 ±0.02 14.44 ±0.09 0.24 34437
37 0.00 -1.74 13.636 ±0.003 0.2488 ±0.0025 0.964 ±0.05 14.57 ±0.10 0.15 72098
38 0.97 -3.42 13.637 ±0.005 0.2911 ±0.0038 0.964 ±0.02 14.57 ±0.09 0.21 34675
39 0.53 -2.21 13.687 ±0.003 0.2524 ±0.0027 0.964 ±0.04 14.62 ±0.09 0.15 60217
40 -2.99 -3.14 13.854 ±0.004 0.2459 ±0.0036 0.964 ±0.04 14.79 ±0.09 0.14 35426
41 -2.78 -3.27 13.857 ±0.004 0.2543 ±0.0035 0.964 ±0.04 14.79 ±0.09 0.16 34118
42 4.48 -3.38 13.494 ±0.006 0.3354 ±0.0051 0.964 ±0.03 14.43 ±0.09 0.27 27377
43 0.37 2.95 13.839 ±0.004 0.2654 ±0.0033 0.964 ±0.05 14.77 ±0.10 0.17 40730
45 0.98 -3.94 13.595 ±0.006 0.3302 ±0.0047 0.964 ±0.03 14.53 ±0.09 0.26 29009
46 1.09 -4.14 13.616 ±0.006 0.3189 ±0.0047 0.964 ±0.03 14.55 ±0.09 0.25 26027
4.1 RCG population effects
Sumi (2004) found that the extinction-corrected magnitudes of
RCG stars in the OGLE-II bulge fields were 0.3 mag higher than
that of a RCG with an intrinsic magnitude equal to those of lo-
cal RCGs, placed at a distance of 8kpc. Sumi (2004) notes that
the cause of this offset is uncertain but may be resolved when de-
tailed V-band OGLE-II photometry of RR Lyrae stars will allow
improved extinction zero-point estimations for all fields. There is
evidence that the properties of bulge RCGs are different to local
RCG stars, in particular age and metallicity, resulting in a differ-
ent average absolute magnitude (see e.g. Percival & Salaris (2003);
Salaris et al. (2003)). This could in part explain the observed off-
set of 0.3 mag between local and bulge RCG stars. The distance
modulus plotted in Fig. 4 is as for equation (2):
µ = IV −I,RC +R (V − I)0,RC − I0,RC + κ
where κ = −0.3 mag is the offset between the observed bulge
RCG population and a fiducial local RCG placed at 8 kpc. Includ-
ing the difference in intrinsic magnitude between local and bulge
RCG populations we have:
µ = IV −I,RC +R (V − I)0,RC − I0,RC +∆IRC + ν
where ∆IRC is the intrinsic RCG magnitude difference between lo-
cal and bulge populations and ν is a magnitude offset arising from
effects other than population differences. The theoretical popula-
tion models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 5
−2−1012
PSfrag replacements
x (kpc)
l = −10◦l = −5◦l = +5◦l = +10◦
Figure 4. Structure of the inner region of the Milky Way as traced by red
clump giant stars extracted from the OGLE-II microlensing survey data.
The mean position of the red clump for 45 OGLE-II fields is indicated by
black dots with errors shown by black lines. The grey lines along each line
of sight indicate the 1-σ spread in distances obtained from fitting equa-
tion (1) to the red clump data in each field and correcting for intrinsic red
clump luminosity dispersion and photometric errors; see text. A position
angle of 25◦ is shown by the thin solid line. Lines-of-sight at l = ±5◦ and
l = ±10◦ are indicated along the top axis.
The remaining magnitude offset ν = −0.2 can be accounted for
by decreasing the assumed value of the Galactocentric distance R0
from 8 kpc to 7.3 kpc. This assumption of R0 is in agreement
with the value of 7.52±0.10 (stat) ±0.32 (sys) kpc determined by
Nishiyama et al. (2006), and that of 7.63±0.32 kpc determined by
Eisenhauer et al. (2005). We note that as long as the stellar popula-
tion is uniform in the Galactic bulge (an assumption largely consis-
tent with the data), then this unknown offset only affects the zero
point (and hence the distance to the Galactic centre), and the abso-
lute bar scale lengths. However the ratios between bar scale lengths
are very robust, which we demonstrate in Section 6.2 where we
determine the bar parameters for several values of R0.
A metallicity gradient across the Galactic bulge would have an
effect on the intrinsic colour and absolute magnitude of bulge RCG
stars as a function of Galactic longitude. Pérez et al. (2006) find
that there is a connection between metallicity gradient and struc-
tural features in a sample of 6 barred galaxies. Minniti et al. (1995)
measured metallicities for K giants in two fields at 1.5 kpc and 1.7
kpc from the Galactic centre. The average metallicity was [Fe/H]
= −0.6, lower than that of K giants in Baade’s window. However,
no such metallicity gradient was reported for Galactocentric dis-
tance range 500 pc – 3.5 kpc by Ibata & Gilmore (1995). Similarly,
Santiago et al. (2006) determine that there is no metallicity gradient
within angles 2.2◦ to 6.0◦ of the Galactic centre (corresponding to
300 – 800 pc from the GC assuming a Sun-GC distance of 8 kpc).
5 MODELLING THE BAR
We continue to investigate the structure of the Galactic bar by fit-
ting analytic models of the stellar density to the observed RCG data
in the OGLE-II fields. Following Stanek et al. (1997) we use the an-
alytic models of Dwek et al. (1995) to fit the observed data. Three
model families (Gaussian, exponential and power-law) are used:
ρG1 = ρ0 exp(−r2/2) (3)
ρG2 = ρ0 exp(−r2s /2) (4)
ρG3 = ρ0r
exp(−r3) (5)
ρE1 = ρ0 exp(−re) (6)
ρE2 = ρ0 exp(−r) (7)
ρE3 = ρ0K0(rs) (8)
ρP1 = ρ0
1 + r
ρP2 = ρ0
r(1 + r)3
ρP3 = ρ0
1 + r2
where K0 is the modified Bessel function of the second kind and
The co-ordinate system has the origin at the Galactic centre,
with the xy plane defining the mid-plane of the Galaxy and the
z direction parallel to the direction of the Galactic poles. The x
direction defines the semi-major axis of the bar. The functions are
rotated by an angle α around the z-axis. An angle of α = ±π
corresponds to the major axis of the bar pointing towards the Sun.
The functions can also be rotated by an angle β around the y axis,
corresponding to the Sun’s position away from the mid-plane of the
Galaxy.
We aim to fit the observed number count histograms for each
field as a function of magnitude. Given a magnitude IV−I, the num-
ber of stars with this magnitude is (Stanek et al. 1997):
N(IV−I) = c1
Z smax
ρ(s)s
Φ(L)Lds (12)
where the integration is taken over distance smin = 3kpc < s <
smax = 13 kpc. We perform the integration over this range of R0±
5 kpc as we do not expect the tri-axial bulge density structure to
exceed these limits. The constant c1 is dependent on the solid angle
subtended around each line of sight. The luminosity L is given by
L = c2s
−0.4IV−I
c© 2005 RAS, MNRAS 000, 000–000
6 Rattenbury et al.
and c2 is a constant. The luminosity function Φ(L) is
Φ(L) = N0
2π σRC
− (L− LRC)
2σ2RC
where LRC is the luminosity of the red clump and σRC is the in-
trinsic spread in red clump giant luminosity and is held constant.
There are ten parameters to be determined in the above equa-
tions: the three bar scale lengths x0, y0, z0; the bar orientation and
tilt angles α and β; the luminosity function parameters N0, NRC ,
γ and LRC and the density function parameter ρ0.
There is evidence that the centroid of the bar is offset from
the centre of mass of the Galaxy, a feature commonly observed
in external galaxies (Stanek et al. 1997; Nishiyama et al. 2006 and
references therein). We include an additional parameter in the mod-
elling process, δl, which allows for this possible centroid offset.
Theoretical number counts are computed for the i = 1 . . .M fields
at longitudes li as Ni(li + δl) where the offset parameter δl is de-
termined over all fields for a given density model.
We apply an exponential cut-off to the density functions simi-
lar to that of Dwek et al. (1995):
f(r) =
1.0 r =
x2 + y2
6 rmax
− (r−rmax)
r > rmax
where r is in kpc, r0 = 0.5 kpc and rmax is a cut-off radius. The
theoretical constraint on the maximum radius of stable stellar or-
bits is the co-rotation radius, rmax. We adopt the co-rotation value
of rmax = 2.4 kpc determined by Binney et al. (1991) in equa-
tion (13). We later repeat the modelling process without this theo-
retical cut-off, see Section 6.2.
Stanek et al. (1997) included a further fitting parameter to ac-
count for a possible metallicity gradient across the bulge, but found
that this did not significantly affect the bar parameters. The discus-
sion in Section 4.1 suggests that there is no appreciable metallic-
ity gradient over the bulge region investigated here. We therefore
do not include an additional model parameter corresponding to a
metallicity gradient in the model fitting analysis.
The model fitting was performed using a standard non-linear
Neadler-Mead minimisation algorithm. For each of the nine density
profile models we minimise χ2:
(Nik(IV−I)− bNik(IV−I))2
where the summations are taken over each of the 26 IV−I histogram
bins in M OGLE-II fields, Nk(IV−I) is the observed histogram
data for field k and bNk(IV−I) is the model number count histogram
from equation (12). The error in Nik(IV−I) was taken to be σik =
Nik(IV−I).
6 RESULTS
The 11 parameters x0, y0, z0, α, β, A, N0, NRC, γ, LRC and
δl were fitted for each of the nine density profiles given in equa-
tions. (3 – 11) for the 34 OGLE-II fields2 which have −4◦ 6 l 6
6◦. A naive interpretation of Fig. 4 is that fitting all fields including
those with l > 6◦ and l < −4◦ with a single bar-like structure
2 Field 5 was excluded due to poor understanding of the dust extinction in
this field.
would not be successful and indeed initial modelling using data
from all fields resulted in bar angles of ≃ 45◦. As seen in Fig. 4
this result is consistent with the mean position angle of all fields,
but clearly does not describe the bar correctly. We therefore begin
modelling the bar using data from the central 34 fields which have
−4◦ 6 l 6 6◦. In Section 6.1 we proceed by including the data
from fields with l > 6◦ and l < −4◦ in the modelling process and
in Section 6.2 we consider the effect of changing R0.
The range of field latitudes is small and therefore the infor-
mation available in the latitude direction from the total dataset un-
likely to be able to constrain β strongly. Initial modelling runs held
β constant at 0◦. Once a χ2 minimum was determined for each
model holding β = 0◦, this parameter was allowed to vary with the
ten other parameters.
Figure 5 shows the number count histograms of observed red
clump giants in the 34 OGLE-II fields with −4◦ 6 l 6 6◦, along
with the best-fitting model (equation (12)) using the E2 density
profile given in equation (7) as an example3. Each set of axes is
arranged roughly in order of decreasing Galactic longitude. The
magnitude of the red clump peak increases with decreasing longi-
tude, consistent with a bar potential.
The observed number count histograms are reasonably well-
fitted by all models in most fields. The most obvious exceptions
are fields 6, 7, 14, 15 and 43. These fields are at the most extreme
latitudes represented in the data. In the case of fields 14 and 15
the algorithm fails to fit satisfactorily both observed histograms for
a given model. Upon closer inspection of the observed histogram
data for these two fields, it is noted that the total number of stars in
the field 15 histogram is significantly less than that for field 14. The
models preferred by all fields clearly cannot reproduce the decline
in total star count between fields 14 and 15. Upon inspection of the
CCD pixel position of stars for field 15 it was found that there are
no stars recorded in a region along one edge of the field. The lack
of stars in this strip is due to missing V -band data for these stars.
Similarly, there is a lack of stars in field 15 in the lower right corner,
due to heavy dust extinction in this area.
All models predict the peak of the red clump to be at a lower
magnitude than that observed for field 43. A similar offset is seen
in the other high latitude fields 14 and 15. These fields are all at
latitude b ≃ 3◦ and the observed offset in all three fields is likely
to be related to this common position in latitude. The cause of this
effect is currently unknown. The observed offset between the maxi-
mum density position predicted by the tri-axial bar models and that
observed for some fields may imply some asymmetry of the bar in
the latitude direction, as suggested by Nishiyama et al. (2006) and
references therein.
Fields 6 and 7, both at b ≃ −6◦, are also poorly fitted by
all models. The histograms of observed red clump stars in these
fields show a curious double peak. Clearly the density functions
equations (3 – 11) are inadequate for describing such a feature.
This double-peaked structure may be the result of another popu-
lation of stars lying along the line of sight, at a distance differ-
ent to the main bulge population. The best-fitting model curves
typically have a peak at magnitudes corresponding to the highest
peak in the observed histogram, thereby suggesting that the sec-
ond population, if real, and composed of a significant fraction of
RCGs, exists between us and the bulge population. A population
of asymptotic giant branch (AGB) stars (Alves & Sarajedini 1999;
3 Similar figures showing the best-fitting models to the data using all den-
sity profiles equations (3 – 11) appear in the on-line supplementary material.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 7
15 17 14 16 42
19 18 36 35 33
32 31 2 30 21
4000 20 L 34 L 46 R 1 R
200045 R
38 L 39 R 4 R 3 R
43 L 22 L 23 L 7 R
13 14 15
10006 R
12 13 14
13 14
13 14
13 14 15
PSfrag replacements
Figure 5. Red clump number count histograms and best-fitting profiles using density model E2 for 34 OGLE-II fields with −4◦ 6 l 6 6◦ . Fields are arranged
roughly in order of descending Galactic longitude. Field numbers are given in each set of axes. A ‘L’ or ‘R’ in the axes of rows 4 – 6 indicates whether the
vertical scale corresponds to left or right vertical axis of the row.
c© 2005 RAS, MNRAS 000, 000–000
8 Rattenbury et al.
Faria et al. 2007) may also be the cause of the secondary peak in
the number count histograms for these and similar fields (e.g. field
10, Fig. 9). Closer investigation of the stellar characteristics and
kinematics of stars in these fields may help in describing the origin
of the second number count peak.
The best-fitting parameter values for all the nine density mod-
els are presented in Table 2. Two sets of parameter values are given,
one where the tilt angle, β, was held at 0.0◦, and the other where β
was allowed to vary. The values of χ2 are well in excess of the num-
ber of degrees of freedom: Ndof = 26 × 34 − 11 = 873, and can
therefore not be used to set reliable errors on the parameters values
in the usual way. It is likely that the errors on the histogram data
are underestimated, resulting in extreme values of χ2. We assumed
that the errors on the number count data would be Poissonian and
we have not added any error in quadrature that might arise from any
systematic effects. Furthermore, as mentioned above, the extreme
latitude fields 6, 7, 14, 15 and 43 are poorly fit by every model. The
cause for this is unknown, however the effect is to add a relatively
high contribution to the total value of χ2 compared to other fields.
The values of χ2 can only be used to differentiate between
the various models in a qualitative manner. In terms of relative per-
formance, model E3 best reproduces the observed number count
histograms for the fields tested here, and model G2 provides the
worst fit to the data.
The position angle of the bar semi-major axis with respect to
the Sun-Galactic centre line-of-sight is α′ and is related to the ro-
tation angle α by α′ = α− sgn(α)π/2. We find α′ = 20◦ – 26◦.
The addition of β as another variable parameter does not result in a
significant improvement in χ2. The lack of information in the lati-
tude direction means that the data have little power in constraining
parameters such as β.
The absolute values of the scale lengths in Table 2 cannot
be directly compared between models. The axis ratios x0/y0 and
x0/z0 can be compared however. Fig. 6 shows the axis ratios x0/y0
and x0/z0 for the nine models tested. The ratio of the major bar axis
scale length to the minor bar axis in the plane of the Galaxy, x0/y0,
has values 3.2 – 3.6, with the exception of that for model E1, for
which x0/y0 = 4.0. Stanek et al. (1997) found x0/y0 = 2.0 –
2.4, with the exception of model E1 for which x0/y0 = 2.9. It is
interesting to note that our values of x0/y0 have a similar range
to that of Stanek et al. (1997), when we exclude the outlying result
from the same model (E1). Our results do suggest a slimmer bar,
i.e. higher values of x0/y0 compared to Stanek et al. (1997). Both
these results and the results of Stanek et al. (1997) are consistent
with the value of x0/y0 ≃ 3± 1 reported by Dwek et al. (1995).
The ratio of the major bar axis scale length to the verti-
cal axis scale length x0/z0 was found to be 3.4 – 4.2, with the
same exception of model E1, which has an outlying value of
x0/z0 = 6.6. Again, the range of ratio values is comparable to
that of Stanek et al. (1997) who found x0/z0 = 2.8 – 3.8, with the
exception of model E1 again, which had x0/z0 = 5.6.
The mean axis ratios are x0 : y0 : z0 are 10 : 2.9 : 2.5.
Excluding the outlying results from the E1 model, the ratios are
10 : 3.0 : 2.6. These results suggest a bar more prolate than the
general working model with 10 : 4 : 3 (Gerhard 2002).
6.1 Evidence of non-bar structure? Including wide longitude
fields
Fig. 4 shows the mean position of the bulge red clump stars. At
longitudes −4◦ 6 l 6 6◦ the bulge red clump stars follow the main
axis of the bar. At greater angular distances, the mean positions of
G1 G2 G3 E1 E2 E3 P1 P2 P3
PSfrag replacements
Model
x0/y0, 34 fields
x0/z0, 34 fields
x0/y0, 44 fields
x0/z0, 44 fields
Figure 6. Scale length ratios x0/y0 (squares) and x0/z0 (triangles) for all
models. Solid and open symbols show the ratios for the best-fitting models
using the 34 fields with −4◦ 6 l 6 6◦, and all 44 fields (see Section 6.1)
respectively.
the red clump stars are clearly separated from the main bar axis.
Babusiaux & Gilmore (2005) find similar evidence and suggest that
this could indicate either the end of the bar or the existence of a
ring-like structure. We used the OGLE-II data to investigate these
possibilities.
The OGLE-II data from fields with longitude l < −4◦ and
l > 6◦ were excluded from the original bar modelling based on
the findings illustrated by Fig. 4, and because initial modelling ef-
forts using all data from all fields simultaneously failed to produce
satisfactory results. Using the best-fitting model (E3) determined
using the central OGLE-II fields, Fig. 7 shows the predicted num-
ber count densities for fields with l < −4◦ and l > 6◦, overlaid on
the observed number count histograms.
It is clear from the left hand column of Fig. 7 that the mean
position of the red clump in fields with l > 6◦ are significantly
removed from that predicted by the E3 linear bar model. However,
the right hand column of Fig. 7 shows that the observed position of
the red clump is in rough agreement with that predicted from the
model for fields with l < −4◦. We test this further, by taking lines
of sight through the best-fitting E3 model and computing the num-
ber count profile using equation (12). The mean magnitude of the
red clump is converted to a distance for each line of sight. Fig. 8
shows the density contours of the analytic E3 model overlaid with
the mean position of the red clump determined in this way. The fea-
tures of Fig. 4 at longitudes |l| & 5◦ are qualitatively reproduced in
Fig. 8. This suggests that the observed data are likely to be consis-
tent with a single bar-like structure, rather than requiring additional
structure elements. We note however, that the abrupt departure of
the location of the density peak in Fig. 4 at longitudes l . 5◦ is
best reproduced in Fig. 8 by the analytical density model when the
exponential cut-off of equation (13) is imposed.
Fields with l < −4◦ and l > 6◦ should be included in the
modelling of the bar, as they are likely to provide further constraints
on the final model. Initial attempts to model the bar using all fields
failed because the fitting algorithm found a model with a bar angle
of 45◦, consistent with the observed data, but not consistent with
the current understanding of the bar. Fig. 8 shows that the mor-
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 9
13 14 15
13 14 15
PSfrag replacements
Figure 7. Predicted number count profiles using the best-fitting E3 model (equation (12)) with the observed number count histograms for OGLE-II fields with
l > 6◦ (left column) and l < −4◦ (right column). The predicted number count profiles are significantly different for fields with l > 6◦, however the predicted
magnitude peak of the red clump roughly coincides with those observed for fields with l < −4◦ .
c© 2005 RAS, MNRAS 000, 000–000
10 Rattenbury et al.
Table 2. Best fitting parameter values for all density models (equations 3 – 11) fitted to the number count histograms from the 34 OGLE-II fields with
−4◦ 6 l 6 6◦. Two sets of parameter values are given, one where the tilt angle, β, was held at 0.0◦, and the other where β was allowed to vary. α′ is the
position angle of the bar semi-major axis with respect to the Sun-Galactic centre line-of-sight.
Bar orientation (◦) Scale lengths (pc) Axis ratios
Model β α′ x0 y0 z0 χ2 x0 : y0 : z0
G1 0.00 21.82 1469.00 449.28 391.80 15302.31
-0.99 21.82 1469.41 448.78 391.62 15282.57 10.0 : 3.1 : 2.7
G2 0.00 19.76 1206.92 375.94 353.81 16558.28
-0.28 19.79 1203.37 377.22 354.24 16547.35 10.0 : 3.1 : 2.9
G3 0.00 25.57 5289.91 1512.32 1277.52 14306.20
-0.30 25.55 5288.64 1511.72 1277.98 14306.06 10.0 : 2.9 : 2.4
E1 0.00 21.63 2143.51 540.08 325.49 15766.77
0.06 21.62 2135.11 539.86 325.48 15753.98 10.0 : 2.5 : 1.5
E2 0.00 23.68 1034.30 306.39 261.43 11930.61
-0.01 23.68 1034.24 306.38 261.47 11930.58 10.0 : 3.0 : 2.5
E3 0.00 21.92 1039.86 323.80 299.08 10722.69
0.47 21.97 1039.62 323.75 298.75 10711.13 10.0 : 3.1 : 2.9
P1 0.00 24.66 1988.33 562.49 478.93 13952.47
0.08 24.66 1988.10 562.39 478.84 13952.09 10.0 : 2.8 : 2.4
P2 0.00 25.07 3906.98 1088.90 927.00 15183.47
0.76 25.03 3907.42 1089.36 927.61 15180.89 10.0 : 2.8 : 2.4
P3 0.00 23.81 1992.91 580.32 491.30 12123.07
0.10 23.82 1993.85 580.33 491.09 12122.82 10.0 : 2.9 : 2.5
−3 −2 −1 0 1 2 3
PSfrag replacements
x (kpc)
Figure 8. Mean position of the red clump determined using the best-fitting
analytic bar model (E3). Gray lines indicate 11 lines-of-sight through the
E3 model, with l = −10◦,−9.8◦, . . . , 10◦ and b = 0.0◦ and contours in-
dicating the density profile of the model. The dashed line shows the orienta-
tion of the bar major axis, with the Sun positioned at (0,−8) kpc. Number
count profiles are generated using equation (12). The mean red clump mag-
nitude for each line-of-sight was converted to a distance, and plotted as solid
circles. Black circles indicate the mean position of the red clump when the
exponential cut-off, equation (13) is applied. Grey circles indicate the mean
RCG position when this cut-off is not imposed. The observed features in
Fig. 4 at longitudes |l| & 5◦ are more clearly reproduced when the cut-off
is applied to the analytic model.
phology indicated by the observed data in Fig. 4 can be explained
using a bar structure oriented at ∼ 20◦ to the Sun-GC line-of-sight.
Cabrera-Lavers et al. (2007) also note that the line-of-sight density
for tri-axial bulges reaches a maximum where the line-of-sight is
tangential to the ellipsoidal density contours. These authors also
quantify the difference between the positions of maximum density
and the intersection of the line-of-sight with the major axis of the
The fitting procedure for all nine tri-axial models was re-
peated, including now the 10 OGLE-II fields with l < −4◦ or
l > 6◦. Figure 9 shows the best-fitting number count profiles for the
ten non-central fields for the ‘G’ type of analytic tri-axial model4.
The best-fitting number count profiles for fields with−4◦ 6 l 6 6◦
are not significantly different to those shown in Figure 5. The main
features of the observed number count profiles are reproduced by
the best-fitting analytic models. There are however instances where
details of the observed number counts are not reproduced. Magni-
tude bins with IV−I & 13 are poorly fitted by the G-type models in
positive longitude fields. Two of the E-type models (E2, E3) sim-
ilarly fail to trace these data. The number count profile for model
E1 shows a flattened peak, resulting from the pronounced box-like
nature of the density profile. All the P-type models appear to repro-
duce the data at these magnitudes with similar profiles. All models
predict a RCG peak at magnitudes greater than that observed in
fields 28 and 29. This offset between predicted and observed RCG
peak location is also seen in fields 26 and 27 but to a lesser degree.
The best-fitting parameters are shown in Table 3. The model
which results in the lowest value of χ2 is still E3. Model G2 also
remains the worst-fitting model to the data.
We now consider the change to the best-fitting model param-
eters when data from all fields are used in the analysis. The po-
sition angle of the bar semi-major axis with respect to the Sun-
Galactic centre line-of-sight is now α′ = 24◦ – 27◦. The bar
4 Similar figures showing the best-fitting models using the ‘E’ and ‘P’ type
density profiles appear in the on-line supplementary material.
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 11
111111
101010
131313
13 14 15
1000 121212
2000262626
272727
292929
13 14 15
282828
PSfrag replacements
Figure 9. Best-fitting number count profiles using the Gaussian ‘G’ type tri-axial models with the observed number count histograms for OGLE-II fields with
l > 6◦ (left column) and l < −4◦ (right column). Solid, dashed and dot-dashed lines correspond to model subtype 1, 2 and 3 respectively.
c© 2005 RAS, MNRAS 000, 000–000
12 Rattenbury et al.
Table 3. Best fitting parameter values for all density models (equations 3-11) fitted to the number count histograms from all 44 OGLE-II fields.
Bar orientation (◦) Scale lengths (pc) Axis ratios
Model β α′ x0 y0 z0 χ2 x0 : y0 : z0
G1 0.75 27.06 1505.20 568.49 392.19 23808.97 10.0 : 3.8 : 2.6
G2 -7.68 24.49 1276.56 473.28 359.91 26571.42 10.0 : 3.7 : 2.8
G3 -0.72 26.56 4787.17 1680.64 1279.94 17397.76 10.0 : 3.5 : 2.7
E1 1.95 23.82 1901.43 626.89 324.93 19170.60 10.0 : 3.3 : 1.7
E2 0.97 26.43 986.60 356.65 260.88 15674.39 10.0 : 3.6 : 2.6
E3 2.50 25.54 1045.67 378.20 294.89 15110.75 10.0 : 3.6 : 2.8
P1 2.19 25.30 1810.98 609.40 478.93 16720.85 10.0 : 3.4 : 2.6
P2 3.67 25.16 3513.34 1160.72 927.54 18069.53 10.0 : 3.3 : 2.6
P3 -0.84 26.06 1876.75 658.13 487.76 15228.11 10.0 : 3.5 : 2.6
G1 G2 G3 E1 E2 E3 P1 P2 P3
PSfrag replacements
Model
x0,M=44/x0,M=34
y0,M=44/y0,M=34
z0,M=44/z0,M=34
Figure 10. Ratio of bar scale lengths from best-fitting models using data
from all 44 OGLE-II fields to those using data from the central 34 OGLE-II
fields.
scale lengths x0, y0 and z0 also changed; Fig. 10 shows the ra-
tio of the best-fitting scale lengths determined using all 44 fields
to those found using only the central 34 fields. On average, upon
including the data from wide longitude fields, the semi-major axis
scale length, x0, decreased by ∼ 5%; the semi-minor axis scale
length, y0, increased by ∼ 16% and the vertical scale length, z0,
remained essentially unchanged. The relatively large change in the
semi-minor axis scale length is intuitively understandable, as a bar
position angle of ∼ 25◦ with respect to the Sun-GC line-of-sight
mean the semi-minor axis direction has a large vector component
in the Galactic longitude direction. The additional constraints of
the data from fields at extended Galactic longitudes can therefore
have a pronounced effect on the fitted values of y0. Similarly, we
expect some weak correlation between the semi-major and semi-
minor axis scale lengths, evidenced by the slight decrease on av-
erage for the x0 values upon adding the data from wide longitude
fields. It is unsurprising that the values of z0 are unchanged, as
adding data from the wide longitude fields has little power in fur-
ther constraining model elements in the direction perpendicular to
the Galactic midplane.
As above, we consider the ratio of the axis scale lengths for
each model (see Fig. 6). The ratio x0/y0 for all nine models using
all field data has values 2.7 – 3.0. This range is now completely
consistent with the range 2.5 – 3.3 reported by Bissantz & Gerhard
(2002). The ratio x0/z0 now lies in the range 3.6 – 3.9, with the
exception of that for model G1 which has x0/z0 = 5.9. The range
x0/z0 = 3.6 – 3.9 is narrower than that found previously, 3.4 – 4.2,
yet still higher on average than 2.8 – 3.8 reported by Stanek et al.
(1997). The scale length ratios are now x0 : y0 : z0 = 10 : 3.5 :
2.6, which compared to those determined using only the central
OGLE-II fields (x0 : y0 : z0 = 10 : 3.0 : 2.6) are closer to the
working model proposed by Gerhard (2002) which has x0 : y0 :
z0 = 10 : 4 : 3.
There are features in the number count histograms that are not
faithfully reproduced by the analytic tri-axial bar models for wide
longitude fields. Specifically, the predicted number count disper-
sions around the RCG peak magnitudes for fields with l > 6◦ are
too small compared to that observed, see Fig. 9. While the location
of the observed maximum line-of-sight density can be reproduced
using just a bar, it is possible that the reason why the analytic bar
models underestimate the observed number count dispersions for
fields with l > 6◦ is because these models do not include elements
which correspond to extended aggregations of stars at or near the
ends of the bar. Clumps of stars at the ends of the bar would in-
crease the line-of-sight density dispersion at wide longitudes. Such
extra aggregations of stars have been observed in galaxies with
boxy or peanut-shaped bulges (Bureau et al. 2006). Their origin
may be due to the superposition of members of the x1 family of
orbits (Patsis et al. 2002; Binney & Tremaine 1987) many of which
have loops near the end of the bar. Alternatively, the aggregations
of stars near the ends of the bar may be the edge-on projection
of an inner ring (Bureau et al. 2006 and references therein). The
presence of another density structure such as a long thin bar ori-
ented at ∼ 45◦ (Cabrera-Lavers et al. 2007) might also contribute
to the relatively large line-of-sight density dispersion. The effects
of such a structure would be most pronounced at wide longitudes.
The presence of a spiral arm might similarly contribute to the large
line-of-sight density dispersion for these fields.
6.2 Varying R0
We consider in this section the effect on the fitted model parameters
of changing the Galactocentric distance R0. It was noted in Sec-
tion 4.1 that there is an offset between the peak magnitude of RCGs
observed in the OGLE-II data and that expected for a RCG, with ab-
solute magnitude similar to that of local RCGs, placed at a distance
of 8.0 kpc. The observed RCGs are systematically 0.3 mag fainter
than the fiducial local RCG at 8.0 kpc. There are two possible rea-
sons for this magnitude offset. Firstly, the adopted Galactocentric
distance of R0 = 8.0 kpc may be incorrect; secondly, there may be
population variations between local and bulge RCGs, resulting in
different intrinsic magnitudes in the two populations. In Section 4.1
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 13
we noted that the offset predicted from the theoretical population
models of Girardi & Salaris (2001) estimate ∆IRC ≃ −0.1. We
postulated that if the remaining magnitude offset is completely ac-
counted for by a change in the Galactocentric distance, this would
mean that R0 = 7.3 kpc. Conversely, if we adopted the value
of R0 = 7.6 ± 0.32 kpc determined by Eisenhauer et al. (2005),
the magnitude offset would become −0.18 mag, where including
the population effect predicted by Girardi & Salaris (2001) would
leave an unaccounted-for offset of −0.08 mag. In this section we
present the results of further modelling where we apply magnitude
offsets which correspond to different values of R0.
In Section 6, we note that for the OGLE-II fields 6, 7, 14, 15
and 43, which have latitudes removed from the majority of OGLE-
II fields, there are systematic offsets between the RCG peak in ob-
served number count histograms and the predictions of all nine tri-
axial models tested. For this reason, we limited the fields used to
those which have Galactic latitude −5◦ 6 b 6 −2◦. We also noted
in Section 6 that due to the lack of field coverage in the latitude
direction the data are not effective at constraining the bar tilt an-
gle β. For this reason, β was held constant at 0.0◦ in the following
modelling analysis. During the previous modelling efforts of Sec-
tion 6 it was also found that applying the exponential cut-off of
equation (13) had little effect on determining the best-fitting bar
parameters. This cut-off was not applied in the following analysis.
The best-fitting bar parameters were determined for the nine
tri-axial bar models for the OGLE-II fields occupying the latitude
strip−5◦ 6 b 6 −2◦ using magnitude offsets corresponding to R0
values of 8.0 kpc, 7.6 kpc and 7.3 kpc. The resulting best-fitting
parameters are listed in Table 4.
From the results in Table 4 we see that the fitted bar opening
angle α increases for all models as R0 decreases. The fitted scale
lengths x0, y0 and z0 all decrease linearly as R0 decreases. The val-
ues of χ2 in Table 4 indicate that a smaller value of R0 is favoured
by all models. The ratio between the scale heights remains remark-
ably constant with varying R0, indicating that while the orientation
of the bar changes slightly with varying R0, the overall shape of
the bar does not change.
The mean values of the scale length ratios for all models, and
all values of R0 are x0 : y0 : z0 = 10 : 3.6 : 2.7. This result
is very close to that determined using data from all 44 OGLE-II
fields, see Section 6 above.
7 DISCUSSION
Red clump giant stars in the OGLE-II microlensing survey cata-
logue can be used as tracers of the bulge density over a large re-
gion towards the Galactic centre. Nine analytic tri-axial bar mod-
els were initially fitted to the number count histograms of red
clump stars observed in 34 OGLE-II fields which have −4◦ 6
l 6 6◦. The models all have the major axis of the bar oriented
at 20◦ – 26◦ to the Sun-Galactic centre line-of-sight. This orien-
tation is in agreement with the results of Stanek et al. (1997) and
Nikolaev & Weinberg (1997) which give a bar angle of 20◦ – 30◦,
and is marginally in agreement with the value of 12 ± 6◦ from
López-Corredoira et al. (2000). Bissantz & Gerhard (2002) obtain
best-fitting non-parametric models to the COBE/DIRBE L band
map of the inner Galaxy with bar angles of 20◦ – 25◦.
We find the ratio of the bar major axis scale length to minor
axis scale length in the plane of the Galaxy to be x0/y0 = 3.2
– 3.6, higher than the value of 2.0 – 2.4 reported by Stanek et al.
(1997), but consistent with the upper end of the range 2.5 – 3.3
found by Bissantz & Gerhard (2002). The ratio of bar major axis
scale length to vertical axis scale length was found to be x0/z0 =
3.4 – 4.2, again higher on average than that reported by Stanek et al.
(1997) who found x0/z0 = 2.8 – 3.8, and higher than the value
of ≃ 3.3 generally adopted (Gerhard 2002). The working model
proposed by Gerhard (2002) gives the scale length ratios as x0 :
y0 : z0 = 10 : 4 : 3. Our results suggest a more prolate model with
x0 : y0 : z0 = 10 : 3.0 : 2.6.
The observed separation of the mean position of red clump
giants from the bar major axis at |l| & 5◦ was shown to be a ge-
ometric effect, rather than evidence of a more complicated struc-
ture such as a ring. The observed data from these fields were used
to further constrain the bar models. The resulting bar position an-
gles was found to be 24◦ – 27◦. This narrower range is consis-
tent with the several values of the bar position angle found by pre-
vious studies. The bar scale length ratios were determined to be
x0 : y0 : z0 = 10 : 3.5 : 2.6, which are closer to the working
model proposed by Gerhard (2002) than those values found using
data from only the central 34 OGLE-II fields.
Reasons for the difference between the bar axis ratios deter-
mined here and the general working model of Gerhard (2002) may
include RCG population effects noted above in Section 4.1. The in-
trinsic luminosity of bulge RCG stars was assumed to be the same
as the local population, however it was found that an offset of −0.3
mag had to be applied to the computed distance moduli in order
to obtain results consistent with standard bar models. Sumi (2004)
found the mean magnitude of observed bulge clump stars is found
to be 0.3 mag fainter to that expected assuming (i) the properties
of bulge RCGs are the same as the local population and (ii) the
distance to the Galactic centre is 8 kpc. A possible implication is
that the adopted distance to the Galactic centre of 8 kpc may be in-
correct. The bar modelling procedure was repeated for three values
of the Sun-Galactic centre distance R0: 7.3 kpc, 7.6 kpc and 8.0
kpc, using data from OGLE-II fields which have −5◦ 6 b 6 −2◦.
Some fields with latitudes outside this range were found to have
number count histograms which were not reproducible by any lin-
ear tri-axial model of the bar tested in this work, and were there-
fore excluded. In addition, most of the OGLE-II fields latitudes
−4◦ < b < −2◦ were excluded. The low amount of informa-
tion in the latitude direction means that the data have little leverage
in determining some of the model parameters. Without more data
in these regions, model fitting algorithms may not be able to refine
some model parameters such as the tilt angle β. The three values
of R0 used are the ‘default’ value of 8.0 kpc; 7.6 kpc, as deter-
mined by Eisenhauer et al. (2005); and 7.3 kpc, which corresponds
to the value of R0 consistent with the observed mean RCG magni-
tudes in the OGLE-II bulge fields assuming the population effects
of Girardi & Salaris (2001). The fitted scale lengths x0, y0 and z0
were found to increase linearly with increasing values of R0, while
the bar opening angle α decreased slightly with increasing R0. The
shape of the bar, as quantified by the ratio of axis scale lengths,
was found to be insensitive to different values of R0. The mean ra-
tio of scale lengths over all models and values of R0 was found to
be x0 : y0 : z0 = 10 : 3.6 : 2.7, which is slightly closer to the
working model of Gerhard (2002) for the Galactic bar than that de-
termined using all 44 OGLE-II fields. The goodness-of-fit measure
χ2 decreased for all models as the value of R0 was lowered.
Improved modelling for the Galactic bar may be possible
through the addition of further elements to the methods presented
here. The possibility of a metallicity gradient across the bulge was
not accounted for in this work. Including spiral terms (see e.g.
Evans & Belokurov 2002) in the analytic density profiles may sim-
c© 2005 RAS, MNRAS 000, 000–000
14 Rattenbury et al.
Table 4. Best-fitting parameters for all density models (equations 3-11) fitted to the number count histograms for OGLE-II fields with Galactic latitude
−5◦ 6 b 6−2◦ . Three values of Galactocentric distance R0 were used. The bar tilt angle, β, was held at 0.0◦.
Scale lengths (pc) Axis ratios
Model R0 (kpc) α (◦) x0 y0 z0 χ2 x0 : y0 : z0
8.0 26.74 1525.19 569.35 382.73 14622.23 10.0 : 3.7 : 2.5
G1 7.6 27.67 1430.76 528.21 363.45 14384.62 10.0 : 3.7 : 2.5
7.3 28.63 1357.09 494.16 349.04 14165.15 10.0 : 3.6 : 2.6
8.0 25.91 1313.82 467.27 337.77 15676.58 10.0 : 3.6 : 2.6
G2 7.6 26.65 1235.00 435.18 320.48 15303.12 10.0 : 3.5 : 2.6
7.3 27.42 1173.14 408.84 307.58 14993.34 10.0 : 3.5 : 2.6
8.0 24.16 4587.68 1658.73 1330.48 9712.92 10.0 : 3.6 : 2.9
G3 7.6 25.32 4266.68 1528.53 1272.76 9502.48 10.0 : 3.6 : 3.0
7.3 26.52 4011.04 1418.58 1226.61 9353.45 10.0 : 3.5 : 3.1
8.0 21.41 1710.52 626.99 343.73 11530.12 10.0 : 3.7 : 2.0
E1 7.6 22.48 1595.52 575.62 327.43 11028.20 10.0 : 3.6 : 2.1
7.3 23.70 1509.95 532.02 316.25 10707.93 10.0 : 3.5 : 2.1
8.0 24.56 974.73 351.07 264.40 9135.87 10.0 : 3.6 : 2.7
E2 7.6 25.63 907.63 323.75 250.86 9085.08 10.0 : 3.6 : 2.8
7.3 26.75 855.05 301.02 240.66 9043.39 10.0 : 3.5 : 2.8
8.0 23.87 1023.09 365.74 297.10 9341.10 10.0 : 3.6 : 2.9
E3 7.6 24.75 954.56 338.68 281.02 9254.02 10.0 : 3.5 : 2.9
7.3 25.69 900.61 316.34 268.93 9180.48 10.0 : 3.5 : 3.0
8.0 22.05 1698.22 609.30 485.75 9418.38 10.0 : 3.6 : 2.9
P1 7.6 23.34 1550.61 554.41 456.64 9288.41 10.0 : 3.6 : 2.9
7.3 24.68 1435.21 509.34 434.79 9191.40 10.0 : 3.5 : 3.0
8.0 21.69 3192.12 1144.07 919.70 10148.09 10.0 : 3.6 : 2.9
P2 7.6 23.03 2894.57 1035.28 861.54 9957.52 10.0 : 3.6 : 3.0
7.3 24.43 2663.48 946.72 817.88 9813.78 10.0 : 3.6 : 3.1
8.0 23.61 1827.34 651.79 487.70 8606.41 10.0 : 3.6 : 2.7
P3 7.6 24.77 1686.69 599.37 460.45 8566.67 10.0 : 3.6 : 2.7
7.3 25.88 1584.40 555.15 444.66 8479.57 10.0 : 3.5 : 2.8
ilarly result in a closer reproduction of the observed number count
profiles.
The observed maximum line-of-sight density can be repro-
duced using just a bar, without requiring additional structure ele-
ments. However, the finer details of the number count histograms,
especially at wide longitudes, were not reproduced by the tri-axial
bar models used here. The predicted number count dispersions
around the peak RCG magnitude were underestimated by the ana-
lytical models, particularly for fields with l > 6◦. It is possible that
these finer features can be reproduced using models which include
extra stellar agglomerations at the ends of the bar. The additional
stellar densities could arise from specific stellar orbits aligned with
the bar, or due to the projection effect of an inner ring. A long thin
bar as postulated by Cabrera-Lavers et al. (2007) might similarly
increase the line-of-sight density dispersion, particularly for wide
longitude fields, as might the presence of a spiral arm. Modelling
the bar using non-parametric methods (see e.g. Efstathiou et al.
1988) may provide valuable insight into the existence and nature
of such additional features. Some preliminary work applying these
methods has begun.
Data from current and future surveys of the Galactic bulge
region will be useful for refining the constraints on the bar param-
eters. The third evolution of the OGLE microlensing experiment,
OGLE-III, is currently in progress, covering a larger region of the
central Galactic region than OGLE-II.
Infra-red observations of the bulge have the advantage of
lower extinction effects due to dust, compared to optical observa-
tions. Current catalogues which would be suitable for investigat-
ing the structure of the Galactic bar include the point source cata-
logue from the 2MASS All Sky data release (Skrutskie et al. 2006),
the Galactic Plane Survey from the UKIRT Infra-Red Deep Sky
Survey (Lawrence et al. 2006; Dye et al. 2006) and data from the
ISOGAL (Omont et al. 2003), Spitzer/GLIMPSE (Benjamin et al.
2005) and AKARI (Ishihara & Onaka 2006) space telescope sur-
veys. Proposed infra-red surveys towards the Galactic centre in-
clude the Galactic Bar Infra-red Time-domain (GABARIT) survey
which aims to monitor the Galactic bar region in the K band for mi-
crolensing events (Kerins 2006, private communication). The anal-
ysis of red clump giant star counts in these surveys may result in
tighter constraints on the properties of the bar, and can be combined
with constrains on stellar kinematics from proper motion surveys
c© 2005 RAS, MNRAS 000, 000–000
Modelling the Galactic bar 15
(Rattenbury et al. 2006) in order to develop dynamical models of
the inner Galaxy.
ACKNOWLEDGEMENTS
We thank D. Faria, Ł. Wyrzykowski, R. James and W. Evans
for helpful discussions. NJR acknowledges financial support by a
PPARC PDRA fellowship and the LKBF. NJR thanks the Kapteyn
Astronomical Institute, RuG, for visitor support. This work was
partially supported by the European Community’s Sixth Frame-
work Marie Curie Research Training Network Programme, Con-
tract No. MRTN-CT-2004-505183 ‘ANGLES’.
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Introduction
Data
Methods
Sample selection
Distance Moduli of RCGs
RCG population effects
Modelling the bar
Results
Evidence of non-bar structure? Including wide longitude fields
Varying R0
Discussion
|
0704.1615 | Dynamical Coupled-Channel Model of $\pi N$ Scattering in the W $\leq$ 2
GeV Nucleon Resonance Region | Dynamical Coupled-Channel Model of πN Scattering
in the W ≤ 2 GeV Nucleon Resonance Region∗
(From EBAC, Thomas Jefferson National Accelerator Facility)
B. Juliá-Dı́az,1, 2 T.-S. H. Lee,1, 3 A. Matsuyama,1, 4 and T. Sato1, 5
1 Excited Baryon Analysis Center (EBAC),
Thomas Jefferson National Accelerator Facility, Newport News, VA 22901, USA
2Departament d’Estructura i Constituents de la Matèria,
Universitat de Barcelona, E–08028 Barcelona, Spain
3Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
4Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan
5Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Abstract
As a first step to analyze the electromagnetic meson production reactions in the nucleon reso-
nance region, the parameters of the hadronic interactions of a dynamical coupled-channel model,
developed in Physics Reports 439, 193 (2007), are determined by fitting the πN scattering data.
The channels included in the calculations are πN , ηN and ππN which has π∆, ρN , and σN reso-
nant components. The non-resonant meson-baryon interactions of the model are derived from a set
of Lagrangians by using a unitary transformation method. One or two bare excited nucleon states
in each of S, P , D, and F partial waves are included to generate the resonant amplitudes in the
fits. The parameters of the model are first determined by fitting as much as possible the empirical
πN elastic scattering amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting
parameters by directly comparing the predicted differential cross section and target polarization
asymmetry with the original data of the elastic π±p → π±p and charge-exchange π−p → π0n pro-
cesses. The predicted total cross sections of πN reactions and πN → ηN reactions are also in good
agreement with the data. Applications of the constructed model in analyzing the electromagnetic
meson production data as well as the future developments are discussed.
PACS numbers: 13.75.Gx, 13.60.Le, 14.20.Gk
∗ Notice: Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-
06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to
publish or reproduce this manuscript for U.S. Government purposes.
http://arxiv.org/abs/0704.1615v2
I. INTRODUCTION
It is now well recognized that a coupled-channel approach is needed to extract the nucleon
resonance (N∗) parameters from the data of πN and electromagnetic meson production re-
actions. With the recent experimental developments [1, 2], such a theoretical effort is needed
to analyze the very extensive data from Jefferson Laboratory (JLab), Mainz, Bonn, GRAAL,
and Spring-8. To cope with this challenge, a dynamical coupled-channel model (MSL) for
meson-baryon reactions in the nucleon resonance region has been developed recently [3]. In
this paper we report a first-stage determination of the parameters of this model by fitting
the πN scattering data up to invariant mass W = 2 GeV.
The details of the MSL model are given in Ref. [3]. Here we will only briefly recall its
essential features. Similar to the earlier works on meson-exchange models [4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] of pion-nucleon scattering, the
starting point of the MSL model is a set of Lagrangians describing the interactions between
mesons (M =γ, π, η , ρ, ω, σ, . . .) and baryons (B = N,∆, N∗, . . .). By applying a unitary
transformation method [13, 27], an effective Hamiltonian is then derived from the considered
Lagrangian. It can be cast into the following more transparent form
Heff = H0 + ΓV + v22 + hππN , (1)
where H0 =
m2α + ~p
α with mα denoting the mass of particle α, and
ΓV = {
ΓN∗→MB) +
hM∗→ππ}+ {c.c.} , (2)
v22 =
MB,M ′B′
vMB,M ′B′ + vππ , (3)
hππN =
ΓN∗→ππN +
[(vMB,ππN) + (c.c.)] + vππN,ππN . (4)
Here c.c. denotes the complex conjugate of the terms on its left-hand-side. In the above
equations, MB = γN, πN , ηN, π∆, ρN, σN , represent the considered meson-baryon states.
The resonance associated with the bare baryon state N∗ is induced by the vertex interactions
ΓN∗→MB and ΓN∗→ππN . Similarly, the bare meson states M
∗ = ρ, σ can develop into reso-
nances through the vertex interaction hM∗→ππ. Note that the masses M
N∗ and m
M∗ of the
bare states N∗ and M∗ are the parameters of the model which must be determined by fit-
ting the πN and ππ scattering data. They differ from the empirically determined resonance
positions by mass shifts which are due to the coupling of the bare states to the scattering
states. The term v22 contains the non-resonant meson-baryon interaction vMB,M ′B′ and ππ
interaction vππ. The non-resonant interactions involving ππN states are in hππN . All of
these interactions are energy independent, an important feature of the MSL formulation.
We note here that the Hamiltonian defined above does not have a πN ↔ N vertex. By
applying the unitary transformation method, this un-physical process as well as any vertex
interaction A ↔ B + C with a mass relation mA < mB + mC are eliminated from the
considered Hilbert space and their effects are absorbed in the effective interactions v22 and
hππN . This procedure defines the Hamiltonian in terms of physical nucleons and greatly
simplifies the formulation of a unitary reaction model. In particular, the complications
due to the nucleon mass and wavefunction renormalizations do not appear in the resulting
scattering equations. This makes the numerical calculations involving the ππN channel
much more tractable in practice. The details of this approach are discussed in Refs. [13, 27]
as well as in the earlier works on πNN interactions [28].
Starting from the above Hamiltonian, the coupled-channel equations for πN and γN
reactions are then derived by using the standard projection operator technique [29], as given
explicitly in Ref. [3]. The obtained scattering equations satisfy the two-body (πN, ηN , γN)
and three-body (ππN) unitarity conditions. The π∆, ρN and σN resonant components
of the ππN continuum are generated dynamically by the vertex interaction ΓV of Eq. (2).
Accordingly, the ππN cuts are treated more rigorously than the commonly used quasi-
particle formulation within which these resonant channels are treated as simple two-particle
states with a phenomenological parametrization of their widths. The importance of such a
dynamical treatment of unstable particle channels was well known in earlier studies of πN
scattering [4, 30] and πNN reactions [31].
A complete determination of the parameters of the model Hamiltonian defined by Eqs.(1)-
(4) requires good fits to all of the data of πN and γN reactions up to invariant mass W ≤
about 2 GeV. Obviously, this is a very complex task and can only be accomplished step
by step. Our strategy is as follows. We need to first determine the parameters associated
with the hadronic interaction parts of the Hamiltonian. With the fits to ππ phase shifts in
Ref. [32], the ππ interactions hρ,ππ and hσ,ππ and the corresponding bare masses for ρ and
σ have been determined in an isobar model with vππ = 0. We next proceed in two stages.
The first-stage is to determine the ranges of the parameters of the interactions ΓN∗→MB and
vMB,M ′B′ . This will be achieved by fitting the πN scattering data from performing coupled-
channel calculations which neglect the more complex three-body interaction term hππN . This
simplification greatly reduces the numerical complexity and the number of parameters to be
determined in the fits. This first-stage fit will provide the starting parameters to fit both
the data of πN scattering and πN → ππN reactions. In this second-stage, the parameters
associated with ΓN∗→MB and vMB,M ′B′ will be refined and the parameters of hππN are then
determined. The dynamical coupled-channel calculations for such more extensive fits are
numerically more complex, as explained in Ref. [3].
In this work we report on the results from our first-stage determination of the parameters
of ΓN∗→MB and vMB,M ′B′ of Eqs.(2)-(3) withMB,M
′B′ = πN, ηN, π∆, ρN, σN . We proceed
in two steps. We first locate the range of the model parameters by fitting as much as possible
the empirical πN elastic scattering amplitudes up to W = 2 GeV of SAID [33]. We then
refine and confirm the resulting parameters by directly comparing our predictions with
the original πN scattering data. Our procedures are similar to what have been used in
determining the nucleon-nucleon (NN) potentials [34] from fitting NN scattering data.
The constructed model can describe well almost all of the empirical πN amplitudes in
S, P , D, and F partial waves of SAID [33]. We then show that the predicted differential
cross sections and target polarization asymmetry are in good agreement with the original
data of elastic π±p → π±p and charge-exchange π−p → π0n processes. Furthermore the
predicted total cross sections of the πN reactions and πN → ηN reactions agree well with
the data. Thus the constructed model is at least comparable to, if not better than, all of the
recent πN models [11, 12, 13, 19, 20, 22, 23, 24, 26]. It can be used to perform a first-stage
extraction of the γN → N∗ parameters by analyzing the photo- and electro-production of
single π meson. It has also provided us with a starting point for performing the second-stage
determination of the model parameters by also fitting the data of πN → ππN reactions.
Our efforts in these directions are in progress and will be reported elsewhere.
In Section II, we recall the coupled-channel equations presented in Ref. [3]. The calcula-
MB,M’B’
t MB,M’B’ t MB,M’B’
t MB,M’B’
vMB,M’B’
FIG. 1: Graphical representation of Eqs.(5)-(21).
tions performed in this work are described in Section III. The fitting procedure is described
in Section IV and the results are presented in Section V. In Section VI we give a summary
and discuss future developments.
II. DYNAMICAL COUPLED-CHANNEL EQUATIONS
With the simplification that ππN interaction hππN of Eq. (4) is set to zero, the meson-
baryon (MB) scattering equations derived in Ref. [3] are illustrated in Fig. 1. Explicitly,
they are defined by the following equations
TMB,M ′B′(E) = tMB,M ′B′(E) + t
MB,M ′B′(E) , (5)
where MB = πN, ηN, π∆, ρN, σN . The full amplitudes TπN,πN(E) can be directly used to
calculate πN scattering observables. The non-resonant amplitude tMB,M ′B′(E) in Eq. (5) is
defined by the coupled-channel equations,
tMB,M ′B′(E) = VMB,M ′B′(E) +
M ′′B′′
VMB,M ′′B′′(E) GM ′′B′′(E) tM ′′B′′,M ′B′(E) (6)
VMB,M ′B′(E) = vMB,M ′B′ + Z
MB,M ′B′(E) . (7)
Here the interactions vMB,M ′B′ are derived from the tree-diagrams illustrated in Fig. 2 by
using a unitary transformation method [13, 27]. It is energy independent and free of singu-
larity. On the other hand, Z
MB,M ′B′(E) is induced by the decays of the unstable particles
v v v vs u t c
FIG. 2: Mechanisms for vMB,M ′B′ of Eq. (7): v
s direct s-channel, vu crossed u-channel, vt one-
particle-exchange t-channel, vc contact interactions.
MB,M’B’
FIG. 3: One-particle-exchange interactions Z
π∆,π∆(E), Z
ρN,π∆ and Z
σN,π∆ of Eq. (7).
(∆, ρ, σ) and thus contains moving singularities due to the ππN cuts, as illustrated in
Fig.3. Here we note that if the ππN interaction term hππN of Eq.(4) is included, the driving
term Eq. (7) will have an additional term Z
MB,M ′B′(E) which involves a 3-3 ππN amplitude
tππN,ππN , as given in Ref. [3], and hence is much more difficult to calculate. As explained in
Section I, we neglect this term in this first-stage fit to the πN scattering data.
The second term in the right-hand-side of Eq. (5) is the resonant term defined by
tRMB,M ′B′(E) =
Γ̄MB→N∗
(E)[D(E)]i,jΓ̄N∗
→M ′B′(E) , (8)
[D−1(E)]i,j = (E −M0N∗
)δi,j − Σ̄i,j(E) , (9)
where M0N∗ is the bare mass of the resonant state N
∗, and the self-energies are
Σ̄i,j(E) =
→MBGMB(E)Γ̄MB→N∗
(E) . (10)
The dressed vertex interactions in Eq. (8) and Eq. (10) are (defining ΓMB→N∗ = Γ
N∗→MB)
Γ̄MB→N∗(E) = ΓMB→N∗ +
M ′B′
tMB,M ′B′(E)GM ′B′(E)ΓM ′B′→N∗ , (11)
Γ̄N∗→MB(E) = ΓN∗→MB +
M ′B′
ΓN∗→M ′B′GM ′B′(E)tM ′B′,MB(E) . (12)
It is useful to mention here that if there is only one N∗ in the considered partial wave, the
resonant amplitude (Eq. (8)) can be written as
tRMB,M ′B′(E) =
Γ̄MB→N∗
(E)Γ̄N∗
→M ′B′(E)
E − ER(E) + iΓR(E)2
ER(E) = M
N∗ + Re[Σ̄(E)] , (14)
ΓR(E) = −2 Im[Σ̄(E)] , (15)
where,
Σ̄(E) =
ΓN∗→MBGMB(E){
M ′B′
[δMB,M ′B′ + tMB,M ′B′(E)GM ′B′(E)]}ΓM ′B′→N∗(E) .
The form Eq. (13) is similar to the commonly used Breit-Wigner form, but the resonance
position ER(E) and width ΓR(E) are determined by the N
∗ → MB vertex and the non-
resonant amplitude tMB,M ′B′ . This is the consequence of the unitarity condition and is
an important and well known feature of a dynamical approach. Namely, the resonance
amplitude necessarily includes the non-resonant mechanisms. This feature is consistent
with the well developed formal reaction theory [29]. Eq. (16) indicates that it is essential
to understand the non-resonant mechanisms in extracting the bare vertex functions ΓN∗,MB
which contain the information for exploring the N∗ structure. The parameterization used
for ΓN∗,MB will be explained in Section III. We also note here that the energy dependence of
ER(E) and ΓR(E), defined by Eqs (14)-(15), is essential in determining the resonance poles
in the complex E-plane.
The meson-baryon propagators GMB in the above equations are
GMB(k, E) =
E −EM(k)−EB(k) + iǫ
for the stable particle channels MB = πN, ηN , and
GMB(k, E) =
E − EM(k)− EB(k)− ΣMB(k, E)
for the unstable particle channels MB = π∆, ρN, σN . The self-energies [36] in Eq. (18) are
Σπ∆(k, E) =
E∆(k)
MπN(q)
[M2πN(q) + k
2]1/2
|f∆,πN(q)|2
E − Eπ(k)− [(EN(q) + Eπ(q))2 + k2]1/2 + iǫ
ΣρN (k, E) =
Eρ(k)
Mππ(q)
[M2ππ(q) + k
2]1/2
|fρ,ππ(q)|2
E − EN(k)− [(2Eπ(q))2 + k2]1/2 + iǫ
, (20)
ΣσN (k, E) =
Eσ(k)
Mππ(q)
[M2ππ(q) + k
2]1/2
|fσ,ππ(q)|2
E −EN (k)− [(2Eπ(q))2 + k2]1/2 + iǫ
, (21)
where MπN(q) = Eπ(q) + EN (q) and Mππ(q) = 2Eπ(q). The vertex function f∆,πN(q) is
taken from Ref. [13], fρ,ππ(q) and fσ,ππ(q) are from the isobar fits [32] to the ππ phase shifts.
They are also given explicitly in [3].
Here we note that the driving term Z
MB,M ′B′ of Eq. (7) is also determined by the same
vertex functions f∆,πN(q), fρ,ππ(q) and fρ,ππ(q) of Eqs. (19)-(21). This consistency is essential
for the solutions of Eq. (6) to satisfy the unitarity condition.
III. CALCULATIONS
We solve the coupled-channel equations defined by Eqs.(5)-(21) in the partial-wave repre-
sentation. The input of these equations are the partial-wave matrix elements of ΓN∗→MB and
vMB,M ′B′ of Eqs.(2)-(3), with MB,M
′B′ = πN, ηN , π∆, ρN, σN , and Z
MB,M ′B′ of Eq. (7)
with MB,M ′B′ = π∆, ρN, σN . The calculations of these matrix elements have been given
explicitly in the appendices of Ref. [3]. Here we only mention a few points which are needed
for later discussions.
In deriving the non-resonant interactions vMB,M ′B′ of Eq. (7) we consider the tree-
diagrams (Fig. 2) generated from a set of Lagrangians with π, η, σ, ρ, ω, N , and ∆ fields.
The higher mass mesons, such as a0, a1 included in other meson-exchange πN models,
such as the Jülich model [19], are not considered. The employed Lagrangians are ( in the
convention of Bjorken and Drell [37])
LπNN = −
ψ̄Nγµγ5~τψN · ∂µ ~φπ , (22)
LπN∆ = −
~TψN · ∂µ ~φπ , (23)
Lπ∆∆ =
ψ̄∆µγ
νγ5 ~T∆ψ
∆ · ∂ν~φπ , (24)
LηNN = −
ψ̄Nγµγ5ψN∂
µφη . (25)
LρNN = gρNN ψ̄N [γµ −
ν ] ~ρµ ·
ψN , (26)
LρN∆ = −i
νγ5 ~T · [∂µ ~ρν − ∂ν ~ρµ]ψN + [h.c.] , (27)
Lρ∆∆ = gρ∆∆ψ̄∆α[γ
µ − κρ∆∆
σµν∂ν ] ~ρµ · ~T∆ψα∆ , (28)
Lρππ = gρππ[ ~φπ × ∂µ ~φπ] · ~ρµ , (29)
LNNρπ =
gρNN ψ̄Nγµγ5~τψN · ~ρµ × ~φπ , (30)
LNNρρ = −
µν~τψN · ~ρµ × ~ρν . (31)
LωNN = gωNN ψ̄N [γµ −
ν ]ωµψN , (32)
Lωπρ = −
ǫµαλν∂
α ~ρµ∂λ ~φπω
ν , (33)
LσNN = gσNN ψ̄NψNφσ (34)
Lσππ = −
∂µ~φπ∂µ~φπφσ . (35)
To solve the coupled-channel equations, Eq. (6), we need to regularize the matrix elements
of vMB,M ′B′ , illustrated in Fig. 2. Here we follow Ref. [13] in order to use the parameters
determined in the ∆ (1232) region as the starting parameters in our fits. For the vs and
vu terms of Fig. 2, we include at each meson-baryon-baryon vertex a form factor of the
following form
F (~k,Λ) = [~k2/[(~k2 + Λ2)]2 (36)
with ~k being the meson momentum. For the meson-meson-meson vertex of vt of Fig. 2, the
form Eq. (36) is also used with ~k being the momentum of the exchanged meson. For the
contact term vc, we regularize it by F (~k,Λ)F (~k′,Λ′).
With the non-resonant amplitudes generated from solving Eq. (6), the resonant ampli-
tude tRMB,M ′B′ Eq. (8) then depends on the bare mass M
N∗ and the bare N
∗ → MB vertex
functions. As discussed in Ref. [3], these bare N∗ parameters can perhaps be taken from
a hadron structure calculation which does not include coupling with meson-baryon contin-
uum states or meson-exchange quark interactions. Unfortunately, such information is not
available to us. We thus use the following parameterization
ΓN∗,MB(LS)(k) =
(2π)3/2
CN∗,MB(LS)
Λ2N∗,MB(LS)
Λ2N∗,MB(LS) + (k − kR)2
(2+L/2) [
.(37)
where L and S are the orbital angular momentum and the total spin of the MB system,
respectively. The above parameterization accounts for the threshold kL dependence and the
right power (2 + L/2) such that the integration for calculating the dressed vertex Eq. (11)-
(12) is finite. Nevertheless as we will discuss in Section V this parameterization could be
too naive.
The partial-wave quantum numbers for the considered channels are listed in Table I. The
numerical methods for handling the moving singularities due to the ππN cuts in Z
MB,M ′B′
(Fig. 3) in solving Eq. (6) are explained in detail in Ref [3]. To get the πN elastic scattering
amplitudes, we can use either the method of contour rotation by solving the equations on
the complex momentum axis k = ke−iθ with θ > 0 or the Spline-function method developed
in Refs. [38, 39] and explained in detail in Ref. [3]. We perform the calculations using these
two very different methods and they agree within less than 1%. When Z
MB,M ′B′ is neglected,
Eq. (6) can be solved by the standard subtraction method since the resonant propagators,
Eqs. (18), for unstable particle channels π∆, ρN , and σN are free of singularity on the real
momentum axis. A code for this simplified case has also been developed to confirm the
results from using the other two methods.
The method of contour rotation becomes difficult at high W since the required rotation
angle θ is very small. The Spline function method has no such limitation and we can perform
calculations at W > 1.9 GeV without any difficulty. Typically, 24 and 32 mesh points
are needed to get convergent solutions of the coupled-channel integral equation (6). Such
mesh points are also needed to get stable integrations in evaluating the dressed resonance
quantities Eqs. (10)-(12).
(LS) of the considered partial waves
πN ηN π∆ σN ρN
S11 (0,
) (0, 1
) (2, 3
) (1, 1
) (0, 1
), (2, 3
S31 (0,
) − (2, 3
) − (0, 1
), (2, 3
P11 (1,
) (1, 1
) (1, 3
) (0, 1
) (1, 1
), (1, 3
P13 (1,
) (1, 1
) (1, 3
),(3, 3
) (2, 1
) (1, 1
),(1, 3
), (3, 3
P31 (1,
) − (1, 3
) − (1, 1
), (1, 3
P33 (1,
) − (1, 3
),(3, 3
) − (1, 1
),(1, 3
), (3, 3
D13 (2,
) (2, 1
) (0, 3
),(2, 3
) (1, 1
) (2, 1
), (0, 3
), (4, 3
D15 (2,
) (2, 1
) (2, 3
) , (4, 3
) (3, 1
) (2, 1
), (2, 3
), (4, 3
D33 (2,
) − (0, 3
),(2, 3
) − (2, 1
), (0, 3
), (2, 3
D35 (2,
) − (2, 3
), (4, 3
) − (2, 1
), (2, 3
), (4, 3
F15 (3,
) (3, 1
) (1, 3
),(3, 3
) (2, 1
) (3, 1
), (1, 3
), (3, 3
F17 (3,
) (3, 1
) (3, 3
),(5, 3
) (4, 1
) (3, 1
), (3, 3
), (5, 1
F35 (3,
) − (1, 3
),(3, 3
) − (3, 1
), (1, 3
), (3, 3
F37 (3,
) − (3, 3
),(5, 3
) - (3, 1
), (3, 3
), (5, 3
TABLE I: The orbital angular momentum (L) and total spin (S)of the partial waves included in
solving the coupled channel Equation (6).
IV. FITTING PROCEDURE
With the specifications given in Section III, the parameters associated with Z
MB,M ′B′ of
Eq. (7) are completely determined from fitting the ππ phase shifts in Refs. [13] and [32]. Thus
the considered model has the following parameters: (a) the coupling constants associated
with the Lagrangians listed in Eqs. (22)-(35), (b) the cutoff Λ for each vertex of vMB,M ′B′
(Fig. 2), (c) the coupling strength CN∗,MB(LS) and range kR and ΛN∗,MB(LS) of the bare
N∗ → MB vertex Eq. (37), and (d) the bare mass M0N∗ of each N∗ state. We determine
these by fitting the πN scattering data.
Our fitting procedure is as follows. We first perform fits to the πN scattering data up to
about 1.4 GeV and including only one bare state, the ∆ (1232) resonance. In these fits, the
starting coupling constant parameters of vMB,M ′B′ are taken from the previous studies of
πN and NN scattering, which are also given in Ref. [3]. Except the πNN coupling constant
fπNN all coupling constants and the cutoff parameters are allowed to vary in the χ
2-fit to
the πN data. The coupled-channel effects can shift the coupling constants greatly from their
starting values. We try to minimize these shifts by allowing the cutoff parameters to vary
in a very wide range 500 MeV < Λ < 2000 MeV. Some signs of coupling constants, which
could not be fixed by the previous works [40], are also allowed to change. We then use the
parameters from these fits at low energies as the starting ones to fit the amplitudes up to
2 GeV by also adjusting the resonance parameters, M0N∗ , CN∗,MB(LS), kR and ΛN∗,MB(LS).
Here we need to specify the number of bare N∗ states in each partial wave. The simplest
approach is to assume that each of 3-star and 4-star resonances listed by the Particle Data
Group [35] is generated from a bare N∗ state of the model Hamiltonian Eq. (1). However,
this choice is perhaps not well justified since the situation of the higher mass N∗’s is not so
clear.
We thus start the fits including only the bare states which generate the lowest and well-
established N∗ resonance in each partial wave. The second higher mass bare state is then
included when a good fit can not be achieved. We also impose the condition that if the
resulting M0N∗ is too high > 2.5 GeV, we remove such a bare state in the fit. This is due
to the consideration that the interactions due to such a heavy bare N∗ state could be just
the separable representation of some non-resonant mechanisms which should be included
in vMB,M ′B′ . In some partial waves the quality of the fits is not very sensitive to the N
couplings to π∆, ρN , and σN . But the freedom of varying these coupling parameters is
needed to achieve good fits.
It is rather difficult to fit all partial waves simultaneously because the number of resonance
parameters to be determined is very large. We proceed as follows. We first fit only 3 or
4 partial waves which have well established resonant states, and whose amplitudes have an
involved energy dependence. These are the S11, P11, S31 and P33 partial waves. These fits are
aimed at identifying the possible ranges of the parameters associated with vMB,M ′B′ . This
step is most difficult and time consuming. We then gradually extend the fits to include more
partial waves. For some cases, the fits can be reached easily by simply adjusting the bare
N∗ parameters. But it often requires some adjustments of the non-resonance parameters to
obtain new fits. This procedure has to be repeated many times to explore the parameter
space as much as we can. We carry out this very involved numerical task by using the fitting
code MINUIT and the parallel computation facilities at NERSC in US and the Barcelona
Supercomputing Center in Spain.
The most uncertain part of the fitting is to handle the large number of parameters asso-
ciated with the bare N∗ states. Here the use of the empirical partial-wave amplitudes from
SAID is an essential step in the fit. It allows us to locate the ranges of the N∗ parameters
partial-wave by partial-wave for a given set of the parameters for the non-resonant vMB,M ′B,.
Even with this, the information is far from complete for pinning down the N∗ parameters.
Perhaps the N∗ parameters associated with the πN state are reasonably well determined in
this fit to the πN scattering data. The parameters associated with ηN , π∆, ρN and σN
can only be better determined by also fitting to the data of πN → ηN and πN → ππN
reactions. This will be pursued in our second-stage calculations, as discussed in section I.
It is useful to note here that the leading-order effect due to Z(E) of the meson-baryon
interaction Eq. (7) on πN elastic scattering is
δvπN,πN =
MB,M ′B′=π∆,ρN,σN
vπN,MBGMB(E)Z
MB,M ′B′GM ′B′(E)vM ′B′,πN . (38)
We have found by explicit numerical calculations that δvπN,πN is much weaker than vπN,πN
and hence the coupled channel effects due to Z
MB,M ′B′ on πN elastic scattering amplitude
are weak. One example obtained from our model is shown in Table II. Thus we first
perform the fits without including Z(E) term to speed up the computation. We then refine
the parameters by including this term in the fits.
V. RESULTS
As mentioned in section I, we first locate the range of the parameters by fitting the
empirical πN scattering amplitude of SAID [33]. We then check and refine the resulting
parameters by directly comparing our predictions with the original πN scattering data.
Re[tπN,πN ] Re[tπN,πN (Z
(E) = 0)] Im[tπN,πN ] Im[tπN,πN(Z
(E) = 0)]
S11 −0.00481 −0.00557 0.0841 0.0827
P11 0.0937 0.103 0.636 0.640
P13 0.169 0.181 0.275 0.275
D13 0.202 0.194 0.299 0.309
D15 0.117 0.116 0.0179 0.0179
F15 0.290 0.291 0.157 0.155
F17 0.0360 0.0359 0.00293 0.00289
S31 −0.433 −0.437 0.496 0.504
P31 −0.253 −0.230 0.434 0.448
P33 0.0506 0.0306 0.510 0.457
D33 −0.00504 −0.0135 0.106 0.104
D35 0.0551 0.0551 0.0540 0.0537
F35 −0.0214 −0.0229 0.0259 0.0283
F37 0.0625 0.0626 0.00502 0.00512
TABLE II: The effect of Z
MB,M ′B′ on the πN scattering amplitudes tπN,πN from solving Eq. (6)
at W = 1.7 GeV. The normalization is tπN,πN = (e
2iδπN − 1)/(2i), where δπN is the πN scattering
phase shift which could be complex at energies above the π production threshold.
Our fits to the empirical amplitudes of SAID [33] are given in Figs. 4-5 and Figs. 6-7 for
the T = 1/2 and T = 3/2 partial waves, respectively. The resulting parameters are presented
in Appendix I. The parameters associated with the non-resonant interactions, vMB,M ′B′ with
MB,M ′B′ = πN, ηN , π∆, ρN, σN , are given in Table III for the coupling constants of the
starting Lagrangian Eqs.(22)-(35) and Table IV for the cutoffs of the form factors defined
by Eq. (36). The resulting bare N∗ parameters are listed in Tables V-VII
From Figs. 4-7, one can see that the empirical πN amplitudes can be fitted very well.
The most significant discrepancies are in the imaginary part of S31 in Fig.7. The agreement
is also poor for the F17 in Fig.4-5 and D35 in Figs.6-7, but there are rather large errors in
the data. Our parameters are therefore checked by directly comparing our predictions with
the data of differential cross sections dσ/dΩ and target polarization asymmetry P of elastic
π±p → π±p and charge-exchange π−p → π0n processes. Our results (solid red curves) are
shown in Figs.8-12. Clearly, our model is rather consistent with the available data, and are
close to the results (dashed blue curves) calculated from the SAID’s amplitudes. Thus our
model is justified despite the differences with the SAID’s amplitudes seen in Fig.4-7.
It will be important to further refine our parameters by fitting the data of other πN
scattering observables, such as the recoil polarization and double polarization. Hopefully,
such data can be obtained from the new hadron facilities at JPARC in Japan.
Our model is further checked by examining our predictions of the total cross sections σtot
which can be calculated from the forward elastic scattering amplitudes by using the optical
theorem. The total elastic scattering cross sections σel can be calculated from the predicted
partial wave amplitudes. With the normalization < ~k|~k′ >= δ(~k − ~k′) used in Ref. [3], we
σel(W ) =
T=1/2,3/2
σelT (W ) (39)
1200 1600 2000
1200 1600 2000
1200 1600 2000
-0.15
-0.05
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
-0.04
-0.02
11 P13
FIG. 4: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2 are
compared with the energy independent solutions of Ref. [33].
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
11 P13
FIG. 5: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 1/2
are compared with the energy independent solutions of Ref. [33].
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
-0.15
-0.05
1200 1600 2000
W (MeV)
-0.08
-0.06
-0.04
-0.02
1200 1600 2000
W (MeV)
-0.15
-0.05
1200 1600 2000
W (MeV)
31 P33
FIG. 6: Real parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2 are
compared with the energy independent solutions of Ref. [33].
σelT (W ) =
(4π)2
ρπN (W )
(2J + 1)
|T TJπN(LS),πN(LS)(k, k,W )|2 , (40)
where ρπN(W ) = πkEπ(k)EN (k)/W with k determined by W = Eπ(k) + EN(k) and the
amplitude T TJL′S′(πN),LS(πN)(k, k;W ) is the partial-wave solution of Eq. (5). Similarly, the
total πN → ηN cross sections can be calculated from
σtotπN→ηN =
(4π)2
πN(W )ρ
ηN (W )
(2J + 1)
|T T=1/2,JηN(LS),πN(LS)(k
′, k,W )|2 (41)
where ρηN (W ) = πk
′Eη(k
′)EN(k
′)/W with k′ determined by W = Eη(k
′) + EN (k
′). We
can also calculate the contribution from each of the unstable channels, π∆, ρN , and σN ,
to the total πN → ππN cross sections. For example, we have for the πN → π∆ → ππN
contribution in the center of mass frame
σrecπ∆(W ) =
∫ W−mπ
mN+mπ
E∆(k)
Γπ∆(k, E)/(2π)
|W − Eπ(k)− E∆(k)− Σπ∆(k, E)|2
σπN→π∆(k,W ) (42)
where k is defined by MπN = Eπ(k) +EN(k), EπN(k) = [M
πN + k
2]1/2, Σπ∆(k, E) is defined
in Eq.(19), Γπ∆(k, E) = −2Im(Σπ∆(k, E)), and
σπN→π∆(k,W ) = 4πρπN (k0)ρπ∆(k)
L′S′,LS,J
2J + 1
(2SN + 1)(2Sπ + 1)
|T Jπ∆(L′S′),πN(LS)(k, k0;W )|2
where k0 is defined by W = Eπ(k0) + EN (k0) and ρab(k) = πkEa(k)Eb(k)/W . The am-
plitude T JL′S′(π∆),LS(πN)(k, k0;W ) is the partial-wave solution of Eq.(5). The corresponding
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
1200 1600 2000
W (MeV)
31 P33
FIG. 7: Imaginary parts of the calculated πN partial wave amplitudes (Eq. (5)) of isospin T = 3/2
are compared with the energy independent solutions of Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p π
p π
W=1440 MeV W=1440 MeV
W=1650 MeV W=1650 MeV
W=1800 MeV W=1800 MeV
FIG. 8: Differential cross section for several different center of mass energies. Solid red curve
corresponds to our model while blue dashed lines correspond to the SP06 solution of SAID [33].
All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p π
p π
W=1235 MeV W=1235 MeV
W=1535 MeV W=1535 MeV
W=1680 MeV W=1680 MeV
FIG. 9: Differential cross section for several different center of mass energies. Similar description
as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p π
p π
W=1230 MeV W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV W=1800 MeV
FIG. 10: Target polarization asymmetry, P , for several different center of mass energies. Similar
description as Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p π
p π
W=1230 MeV W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV W=1800 MeV
FIG. 11: Target polarization asymmetry, P , for several different center of mass energies. Descrip-
tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
expressions for the unstable channels ρN and σN can be obtained from Eqs. (42)-(43) by
changing the channel labels.
The predicted σtot (solid curves) along with the resulting total elastic scattering cross
sections σel compared with the data of π+p reaction are shown in Fig. 13. Clearly, the
model can account for the data very well within the experimental errors. Here only the
T = 3/2 partial waves are relevant. Equally good agreement with the data for π−p reaction
are shown in the left side of Fig. 14. In the right side, we show how the contributions from
each channel add up to get the total cross sections. The comparison of the contribution
from ηN channel with the data is shown in Fig. 15. It is possible to improve the fit to this
data by adjusting N∗ → ηN parameters. But this can be done correctly only when the
differential cross section data of πN → ηN are included in the fit. This is beyond the scope
of this work and will be pursed in our second-stage calculations.
The contributions from π∆, ρN and σN intermediate states to the π−p → ππN total
cross sections calculated from our model can be seen in the right side of Fig. 14. These
predictions remain to be verified by the future experiments. The existing πN → ππN data
are not sufficient for extracting model independently the contributions from each unstable
channel.
The results shown in Figs. 13-15 indicate that our parameters are consistent with the
total cross section data.
We now discuss the parameters presented in Appendix A. It is rather difficult to compare
the resulting non-resonant coupling constants listed in Table III with the values from other
works, since the coupling strengths are also determined by the cutoff parameters listed in
0 40 80 120 160
θ (cm)
0 40 80 120 160
θ (cm)
p π
p π
W=1230 MeV
W=1640 MeV
W=1440 MeV W=1680 MeV
W=1540 MeV
W=1800 MeV
FIG. 12: Target polarization asymmetry, P , for several different center of mass energies. Descrip-
tion as in Fig. 8. All data have been obtained through the SAID online applications. Ref. [33].
Table IV. Perhaps it is possible to narrow their differences by using a different parameter-
ization of the form factors. However, the fit is a rather time consuming process and hence
no attempt is made in this work to try other forms of form factors.
In Table V, we see that all of the bare masses are higher than the PDG’s resonance
positions. This can be understood from the expression Eq. (14) for the partial waves with
only one N∗ since one finds in general that Re[Σ̄(E)] < 0. For the S11, P11, P33 and D13
partial waves, two bare N∗ states are mixed by their interactions, as can be seen in Eq. (10).
Thus the relation between their bare masses and the resonance positions identified by PDG
is much more complex.
As we mentioned above, the fit to πN elastic scattering is can not determine well the
bare N∗ → π∆, ρN, σN parameters. Thus the results for these unstable particle channels
listed in Tables III-VII must be refined by fitting the πN → ππN data.
VI. SUMMARY AND FUTURE DEVELOPMENTS
Within the formulation developed in Ref. [3], we have constructed a dynamical coupled-
channel model of πN scattering by fitting the πN scattering data. The parameters of the
model are first determined by fitting as much as possible the empirical πN elastic scattering
amplitudes of SAID up to 2 GeV. We then refine and confirm the resulting parameters by
directly comparing the predicted differential cross section and target polarization asymmetry
with the original data of the elastic π±p→ π±p and charge-exchange π−p→ π0n processes.
1200 1400 1600 1800 2000
W (MeV)
FIG. 13: The predicted total cross sections of the π+p → X (solid curve)and π+p → π+p (dashed
curve) reactions are compared with the data. Squares and triangles are the corresponding data
from Ref. [35].
The predicted total cross sections of πN reactions and are also in good agreement with the
data. The model thus can be used as a starting point for analyzing the very extensive data
of electromagnetic π production reactions.
The predicted total cross sections of πN → ηN reactions are also in fair agreement with
the data. However, the parameters associated with the ηN channel need to be refined to
also fit the differential cross section data of πN → ηN before the model can be used to
analyze the data of electromagnetic η production reactions.
The main shortcoming of this work is that the ππN interaction term hππN of Eq.(4) is
not included in the calculations. As derived in Ref. [3], the effects due to this interaction
can be included by adding a term Z
MB,M ′B′(E), which contains the ππN → ππN scattering
amplitude, to the driving term VMB,M ′B′(E) of Eq.(6). Our effort in this direction is in
progress along with the development of a more complete determination of the parameters
of the model by fitting both the data of πN elastic scattering and πN → ππN reactions.
This is also essential to pin down the parameters of the interactions associated with the
π∆, ρN and σN states. Only when this second-stage is completed, we then can perform
dynamical coupled-channel analysis of the very extensive and complex data of photo- and
electro-production of two pions. This is an essential step to probe the W > about 1.7 GeV
resonance region where the information on N∗ is very limited and uncertain.
Finally, a necessary next step is to extract the resonance poles and the associated residues
from the predicted πN amplitudes. This is being pursued and will be published else-
where [46].
Acknowledgments
We would like to thank M. Paris for his assistance in using the parallel processors at
NERSC and A. Parreño for her help and encouragement to use the BSC. This work is sup-
ported by the U.S. Department of Energy, Office of Nuclear Physics Division, under contract
1200 1400 1600 1800
W (MeV)
1200 1400 1600 1800 2000
W (MeV)
πN, ηN
πN, ηN, π∆
πN, ηN, pD, σN
πN, ηN, π∆, σN, ρN
FIG. 14: Left: The predicted total cross sections of the π−p → X (solid curve) and π−p →
π−p + π0n (dashed curve) reactions are compared with the data. Open squares are the data on
π−p → X from Ref. [35], open triangles are obtained by adding the π−p → π−p and π−p → π0n
data obtained from Ref. [35] and SAID database [41] respectively. Right: Show how the predicted
contributions from each channel are added up to the predicted total cross sections of the π−p → X.
1400 1600 1800 2000
W (MeV)
FIG. 15: The predicted total cross sections of πp → ηp reaction are compared with the data [42, 43].
No. DE-AC02-06CH11357, and Contract No. DE-AC05-060R23177 under which Jefferson
Science Associates operates Jefferson Lab, and by the Japan Society for the Promotion of Sci-
ence, Grant-in-Aid for Scientific Research(c) 15540275. This work is also partially supported
by Grant No. FIS2005-03142 from MEC (Spain) and FEDER and European Hadron Physics
Project RII3-CT-2004-506078. The computations were performed at NERSC (LBNL) and
Barcelona Supercomputing Center (BSC/CNS) (Spain). The authors thankfully acknowl-
edges the computer resources, technical expertise and assistance provided by the Barcelona
Supercomputing Center - Centro Nacional de Supercomputacion (Spain).
APPENDIX A: PARAMETERS FROM THE FITS
Parameter SL Model
f2πNN/(4π) 0.08 0.08
mσ (MeV) 500.1 −
fπN∆ 2.2061 2.0490
fηNN 3.8892 −
gρNN 8.7214 6.1994
κρ 2.654 1.8250
gωNN 8.0997 10.5
κω 1.0200 0.0
gσNN 6.8147 −
gρππ 4. 6.1994
fπ∆∆ 1.0000 −
fρN∆ 7.516 −
gσππ 2.353 −
gωπρ 6.955 −
gρ∆∆ 3.3016 −
kρ∆∆ 2.0000 −
TABLE III: The parameters associated with the Lagrangians Eqs.(22)-(35). The results are from
fitting the empirical πN partial-wave amplitudes [33] of a given total isospin T = 1/2 or 3/2. The
parameters from the SL model of Ref. [13] are also listed.
Parameter (MeV) SL model (MeV)
ΛπNN 809.05 642.18
ΛπN∆ 829.17 648.18
ΛρNN 1086.7 1229.1
Λρππ 1093.2 1229.1
ΛωNN 1523.18 −
ΛηNN 623.56 −
ΛσNN 781.16 −
ΛρN∆ 1200.0 −
Λπ∆∆ 600.00 −
Λσππ 1200.0 −
Λωπρ 600.00 −
Λρ∆∆ 600.00 −
TABLE IV: Cut-offs of the form factors, Eq. (36), of the non-resonant interaction vMB,M ′B′ . The
results are from fitting the empirical πN partial-wave amplitudes [33] of a given total isospin
T = 1/2 or 3/2. The parameters from the SL model of Ref. [13] are also listed.
LTJ PDG’s Mass( MeV) M1 (MeV) M2 (MeV)
S11 1535; 1655 1800. 1880.
S31 1630 1850.
P11 1440; 1710 1763 2037
P13 1720 1711
P31 1910 1900.3
P33 1232; 1600 1391 1602.
D13 1520; 1700 1899.1 1988.
D15 1675 1898
D33 1700 1976
D35 1960 −
F15 1685 2187
F35 1890 2162
F37 1930 2137.8
TABLE V: The masses of the nucleon excited states included in the fits. (second and third
columns). The first column contains the masses of the nucleon resonances given by PDG [35].
πN ηN π∆ σN ρN
S11 (1) 7.0488 9.1000 −1.8526 −2.7945 2.0280 .02736
S11 (2) 9.8244 .60000 .04470 1.1394 −9.5179 −3.0144
S31 5.275002 − −6.17463 − −4.2989 5.63817
P11 (1) 3.91172 2.62103 −9.90545 −7.1617 −5.1570 3.45590
P11 (2) 9.9978 3.6611 −6.9517 8.62949 −2.9550 −0.9448
P13 3.2702 −.99924 −9.9888 −5.0384 1.0147 −.00343 1.9999 −.08142
P31 6.80277 − 2.11764 − 9.91459 0.15340
P33 (1) 1.31883 − 2.03713 9.53769 − −.3175 1.0358 0.76619
P33 (2) 1.3125 − 1.0783 1.52438 − 2.0118 −1.2490 0.37930
D13 (1) .44527 −.0174 −1.9505 .97755 −.481855 1.1325 −.31396 .17900
D13 (2) .46477 .35700 9.9191 3.8752 −5.4994 .28916 9.6284 −.14089
D15 .31191 −.09594 4.7920 .01988 −.45517 −.17888 1.248 −.10105
D33 .9446 − 3.9993 3.9965 − .16237 3.948 −.85580
F15 .06223 0.0000 1.0395 .00454 1.5269 −1.0353 1.6065 −.0258
F35 .173934 − −2.96090 −1.09339 − −.07581 8.0339 −.06114
F37 0.25378 − −0.3156 −0.0226 − .100 .100 .100
TABLE VI: The coupling constants CN∗,JTLS;MB of Eq. (37) with MB = πN, ηN, π∆, σN, ρN
for each of the resonances. When there are more than one value for π∆ and ρN channels, they
correspond to the possible quantum numbers (LS) listed in Table 2.
πN ηN π∆ σN ρN
S11 (1) 1676.4 598.97 554.04 801.03 1999.8 1893.6
S11 (2) 533.48 500.02 1999.1 1849.5 796.83 500.00
S31 2000.00 − 500.00 − 500.031 500.00
P11 (1) 1203.62 1654.85 729.0 1793.0 621.998 1698.90
P11 (2) 646.86 897.84 501.26 1161.20 500.06 922.280
P13 1374.0 500.23 500.00 500.770 640.50 500.00 500.10 1645.2
P31 828.765 − 1999.9 − 1998.8 2000.6
P33 (1) 880.715 − 507.29 501.73 − 606.78 1043.4 528.37
P33 (2) 746.205 − 846.37 780.96 − 584.98 500.240 1369.7
D13 (1) 1658. 1918.2 976.36 1034.5 1315.8 599.79 1615.1 1499.50
D13 (2) 1094.0 678.41 1960.0 660.02 1317.0 550.14 597.57 1408.7
D15 1584.7 1554.0 500.77 820.17 507.07 735.40 749.41 937.53
D33 806.005 − 1359.38 608.090 − 1514.98 1998.99 956.61
F15 1641.6 655.87 1899.5 522.68 500.93 500.76 500.0 1060.9
F35 1035.28 − 1227.999 586.79 − 1514.3 593.84 1506.0
F37 1049.04 − 1180.2 1031.81 − 600.02 600.00 600.02
TABLE VII: The range parameter ΛN∗,JTLS;MB (in unit of (MeV/c)) of Eq. (37) with MB =
πN, ηN, π∆, σN, ρN for each of the resonances. When there are more than one value for π∆ and
ρN channels, they correspond to the possible quantum numbers (LS) listed in Table 2.
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Introduction
Dynamical Coupled-channel equations
Calculations
Fitting Procedure
Results
Summary and Future Developments
Acknowledgments
Parameters from the fits
References
|
0704.1616 | Reply to Comment on "Chiral suppression of scalar glueball decay" | arXiv:0704.1616v1 [hep-ph] 12 Apr 2007
Published in Phys. Rev. Lett. 98, 149104 (2007)
Reply to “Comment on ‘Chiral suppression of scalar glueball decay’ ”
Michael Chanowitz1
Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
In [1] I observed that the amplitude for spin zero glue-
ball decay is proportional to the quark mass, M(G0 →
qq) ∝ mq, to all orders in perturbation theory, so that
the ratio Γ(G0 → uu+ dd)/Γ(G0 → ss) is calculable and
small, even though the individual rates are not pertur-
batively calculable because of soft t and u channel quark
exchanges. I noted that if hadronization of G0 → qq is an
important mechanism for G0 → ππ and G0 → KK, then
Γ(G0 → ππ) is much smaller than Γ(G0 → KK), ex-
plaining a previous LQCD result[2] and supporting iden-
tification of f0(1710) with G0. A more robust conse-
quence, emphasized in [3], is that mixing of G0 with
uu + dd (and perhaps also ss) mesons is suppressed, so
that the scalar (and pseudoscalar) may be the purest
glueballs. In both [1] and [3] I emphasized the neces-
sity to verify the existence and consequences of chiral
suppression by a reliable nonperturbative method, which
today can only be LQCD.
Chao et al. agree that G0 → qq is chirally suppressed
but propose that G0 → qqqq, which is not chirally sup-
pressed, is the dominant mechanism for G0 → ππ. In
the preceding Comment[4] and in a previous paper[5]
they exhibit an O(αS) amplitude for the exclusive process
G0 → ππ using light cone wave functions. Since pQCD
for exclusive processes converges much more slowly than
inclusive pQCD[6], the estimate is not quantitatively re-
liable at the experimentally interesting scale, mG = 1.7
GeV, where even the applicability of ordinary inclusive
pQCD is marginal. While the qqqq mechanism might in-
deed dilute or remove chiral suppression of G0 → ππ, it
is not possible to decide, since the magnitude of neither
the qq nor qqqq contributions are reliably calculable.
Comparing the amplitudes for M(G0 → qq) and
M(G0 → qqqq → ππ) in [1] and [4, 5] it appears that
both begin at first order in αS , but this impression is
misleading. It is easy to see that M(G0 → qqqq → ππ)
vanishes in the chiral limit at O(αS) for on-shell con-
stituent gluons. The qqqq mechanism requires the quark
from one gluon to combine with the antiquark from the
other gluon to form a color singlet pion. But G0 cms
(center of mass) kinematics then requires both quarks
to have the same energy fraction, x = 2Eq/mG and both
antiquarks to have fraction 1−x, with m2π = x(1−x)m
One of the q or q constituents of each pion is then mov-
ing in the opposite direction to the pion in the G0 cms.
Boosting to an infinite momentum frame, one constituent
is then at x = 1 and the other at x = 0, where the wave
function vanishes. In the chiral limit, mπ = 0, this is al-
ready apparent in the G0 cms. Since confining dynamics
may put the gluons off-shell of order ΛQCD, the ampli-
tude does not actually vanish but is suppressed of order
O(ΛQCD/mG).
In the revised Comment the authors have responded
to this observation with the added stipulation that the
G0 constituent gluons are maximally off-shell, of order
mG. Although this requirement was not imposed in [5],
the result is apparently unchanged. Certainly one conse-
quence is that fg, the effective G0gg coupling, cannot be
identified with the corresponding coupling f0 in [1] as is
claimed in [4, 5], but reflects the off-shell tail of the G0
wave function or implicitly contains a factor αS at the
hard scale mG reflecting hard gg → g
∗g∗ scattering to
push the gluons maximally off-shell. Alternatively, hard
scattering of qqqq can align the quarks suitably with the
final state pions, with the amplitude then explicitly of
order O(α2S).
The relative magnitude of the qq and qqqq mechanisms
for G0 → ππ is not obvious. For the qq mechanism we do
not know the magnitude of M(G0 → qq) because both
αS(Q) and the running mass mq(Q) are evaluated at a
soft scale, O(ΛQCD), and thus are not under perturbative
control. In addition we do not know the hadronization
rate from qq to ππ and KK compared to multi-meson fi-
nal states. On the other hand, Γ(G0 → ππ) via the qqqq
mechanism cannot be reliably estimated and is addition-
ally suppressed by the square of the coupling, αS(Q)
evaluated at the largest scale in the problem, Q = mG.
It is then important to stress the agreement, expressed
in both [1, 3] and [4], on the most important point: re-
liable nonperturbative methods are needed to determine
whether G0 → ππ is chirally suppressed. We eagerly
await LQCD “data” and data from BES II to clarify the
issue.
Acknowledgments: This work was supported by the
Director, Office of Science, Office of High Energy Physics
of the U.S. Department of Energy under contract DE-
AC02-05CH11231.
[1] M. Chanowitz, Phys. Rev. Lett. 95 (2005), hep-
ph/0506125
[2] J.Sexton, A.Vaccarino, D.Weingarten, Phys.Rev.Lett.75:
4563,1995, hep-lat/9510022.
[3] M. Chanowitz, Talk given at Charm 2006, Beijing, China,
5-7 Jun 2006. Published in Int.J.Mod.Phys.A21, 5535
(2006), hep-ph/0609217.
http://arxiv.org/abs/0704.1616v1
[4] K.T. Chao, X-G He, J.P. Ma, preceding Comment.
[5] K.T. Chao, X-G He, J.P. Ma, hep-ph/0512327.
[6] N. Isgur and C.H. Llewellyn-Smith, Phys. Rev. Lett. 52:
1080 (1984).
|
0704.1617 | High-resolution study of a star-forming cluster in the Cep-A HW2 region | Astronomy & Astrophysics manuscript no. cepa2007˙paper c© ESO 2021
September 15, 2021
High-resolution study of a star-forming cluster
in the Cep-A HW2 region
C. Comito1, P. Schilke1, U. Endesfelder1 , I. Jiménez-Serra2, and J. Martı́n-Pintado2
1 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
e-mail: [email protected]
2 Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı́ficas, Departamento de Astrofı́sica Molecular e Infrarroja,
C/Serrano 121, E-28006 Madrid, Spain
Received; accepted
ABSTRACT
Context. Due to its relatively small distance (725 pc), the Cepheus A East star-forming region is an ideal laboratory to study massive star
formation processes.
Aims. Based on its morphology, it has been suggested that the flattened molecular gas distribution around the YSO HW2 may be a
350-AU-radius massive protostellar disk. Goal of our work is to ascertain the nature of this structure.
Methods. We have employed the Plateau de Bure Interferometer⋆ to acquire (sub-)arcsecond-resolution imaging of high-density and shock
tracers, such as methyl cyanide (CH3CN) and silicon monoxide (SiO), towards the HW2 position.
Results. On the 1′′ (∼ 725 AU) scale, the flattened distribution of molecular gas around HW2 appears to be due to the projected superposition,
on the plane of the sky, of at least three protostellar objects, of which at least one is powering a molecular outflow at a small angle with respect
to the line of sight. The presence of a protostellar disk around HW2 is not ruled out, but such structure is likely to be detected on a smaller
spatial scale, or using different molecular tracers.
1. Introduction
Several theories are being considered to explain the forma-
tion of massive (M ≥ 8 M⊙) stars, which can be roughly
grouped into accretion-driven and coalescence-driven models
(cf. Stahler et al. 2000). In the latter case, high-mass stars
would form by merging of two or more lower-mass objects,
making the presence of stable massive accretion disks around
the protostar very unlikely. However, only models based on
disk-protostar interactions are capable of explaining the exis-
tence of jets and outflows: hence, the high incidence, in large
samples of massive YSOs, of highly collimated outflows (cf.
Beuther et al. 2002) has been interpreted as indirect evidence
for the existence of high-mass disks.
It is undoubted that the direct detection of accretion onto
massive protostars through rotating disks constitutes an im-
portant tile in the massive-star-formation-theory mosaic. From
an observational point of view, this task is made very difficult
by two factors: i) massive star-forming regions typically are
far away, a few kpc on average, making the direct observa-
tion of small-scale structure such as disks virtually impossible
with current instruments; and ii), massive stars form in clus-
Send offprint requests to: C. Comito
ters, making the surrounding region extremely complex, both
spatially and kinematically.
Located only ∼ 725 pc from the Sun (Johnson 1957),
Cepheus A is considered a very promising candidate for the
detection of a massive disk. Its well-studied bipolar outflow
(cf. Gómez et al. 1999, hereafter G99, and references therein)
is thought to be powered by the radio-continuum source HW2
(∼ 104 L⊙, Rodrı́guez et al. 1994). Curiel et al. (2006) report
the presence of very large tangential velocities in the HW2 ra-
dio jet, consistent with HW2 being a massive Young Stellar
Object (YSO). The distribution of H2O masers (Torrelles et
al. 1996) and of the SiO emission (G99) around HW2, both
oriented perpendicularly with respect to the direction of the
flow, have been interpreted as strongly supporting the existence
of accretion shocks onto a rotating and contracting molecu-
lar disk of ∼ 700-AU diameter, centered on HW2, with the
northeast-southwest outflow being triggered by the interaction
between such disk and HW2 itself. Similar conclusions have
been reached by Patel et al. (2005), based on SMA observa-
tions of CH3CN and dust emission. However, the fact that the
HW2 vicinities are crowded with YSOs (at least three within
an area of 0.′′6 × 0.′′6, Curiel et al. 2002), together with the re-
cent detection of an internally heated hot core within 0.′′4 from
the center of the outflow (Martı́n-Pintado et al. 2005, hereafter
http://arxiv.org/abs/0704.1617v1
2 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region
MP05) cast some doubts on this interpretation. Based on our
PdBI observations, we conclude that, on the 1′′ scale, the elon-
gated molecular structure around HW2 can be explained with
the superposition, on the plane of the sky, of at least three dif-
ferent hot-core-type sources, at least one of them being the ex-
citing source for a second molecular outflow.
2. Observations
In 2003 and 2004, with the Plateau de Bure Interferometer, we
have carried out observations of several high-density and shock
tracers (also cf. Schilke et al., in prep.), among which silicon
monoxide (SiO) and methyl cyanide (CH3CN), towards the
HW2 position (αJ2000 = 22
h56m17.9s, δJ2000 = +62
◦01′49.6′′).
A combination of high-spectral-resolution correlator units were
employed to achieve a channel width ∆v of up to ∼ 0.3 km s−1.
The five antennas in AB (extended) configuration provided a
HPBW of 2′′ × 1.′′6 for SiO(2-1) at 86 GHz, and of 0.′′9 × 0.′′7
for CH3CN(12 − 11) at 220 GHz. The data cubes were pro-
duced with natural weighting. All maps have been CLEANed.
Analysis of all molecular spectra has been performed after sub-
traction of the continuum emission.
3. Results
Fig. 1 (left panel) shows the Cep-A star-forming region within
a 1100-AU radius from HW2. The peak of the 241-GHz dust
emission (grey scale) roughly coincides with the HW2 posi-
tion and with the center of the large-scale outflow. The inte-
grated CH3CN emission is also centered on HW2 (contours),
and somewhat elongated almost perpendicularly to the direc-
tion of the large-scale outflow. Like other molecular tracers
(cf. Brogan et al. 2007), CH3CN displays two different velocity
components, centered around −5 and −10 km s−1 respectively.
The solid contours in Fig. 1, center panel, show the emission
of the CH3CN(123−113) transition, integrated between −7 and
−3 km s−1, whereas the emission in the range between −11.5
and −7.5 km s−1 is represented by the dashed contours (see
§ 3.2).
The center of SiO emission, instead, is at ∼ −10 km s−1.
Silicon monoxide peaks about 0.′′4 eastwards of HW2, at a po-
sition that coincides with the HC source of MP05 (triangle in
Fig. 1, see § 3.1), close to the −10-km s−1 CH3CN component.
In what follows, we will discuss in more detail the SiO and
CH3CN data.
3.1. SiO
Our dataset confirms that the spatial distribution of this shock
tracer is mainly concentrated in the HW2 region (its presence
in the large-scale outflow is limited to a few bullets at large dis-
tances from the center), although not centered on the HW2 po-
sition. This does indeed suggest that shock processes are taking
place in the (projected) immediate vicinities of HW2. However,
if the SiO emission were arising from accretion shocks onto a
rotating disk (as proposed by G99), we would expect to ob-
serve a similar velocity structure to that observed for the other
molecular tracers peaking around HW2. Instead, SiO seems to
be tracing a completely different kinematic component: unlike
any other line in our dataset, the SiO(2-1) line has a velocity
spread of at least 35 km s−1at the zero-flux level (∼ 15 km s−1
FWHM). A mass of about 90 M⊙ would be required to pro-
duce such large line width in a gravitationally bound enviro-
ment (assuming virial equilibrium, and that the emission arises
in a region of ∼ 350-AU radius). This value is about one order
of magnitude larger than the estimated mass of HW2, which
is expected to become a B0.5 star once in ZAMS (Rodriguez
et al. 1994). Fig. 2 shows a comparison between SiO(2-1) and
CH3CN(124 − 114).
We carried out a two-dimensional Gaussian fit of the SiO(2-
1) spatial distribution, for every spectral channel in the velocity
range −25 < vlsr −3.5 km s
−1. This corresponds to the spec-
tral interval in which the signal-to-noise (S/N) ratio of the SiO
transition is ≥ 9σ. For smaller S/N ratios, in fact, the error on
the fitted centroid position easily exceeds 50%. The result is a
distribution of the centroids of SiO emission as a function of
velocity. Fig. 1, right panel, shows that the centroid positions
are located in a well-defined two-lobed area, centered about
0.′′4 eastwards of HW2 and of the dust continuum emission
peak. Although the error on every single centroid position is
still relatively large (up to 30%), as a whole their distribution
describes a very clear velocity trend, with all the emission at
vlsr< −10 km s
−1 clustering in the left lobe, and all the emis-
sion at vlsr> −10 km s
−1 clustering in the right lobe. This re-
sult suggests that a second molecular outflow is being ejected
in the HW2 region. Our interpretation is supported by the re-
cent discovery of an intermediate-mass protostar, surrounded
by the hot molecular core HC (MP05) located in the region be-
tween the blue and red lobes of SiO emission (white triangle
in Fig. 1, right panel), hence a very likely candidate to be its
powering engine. With the current dataset it is not possible to
establish the exact inclination angle of the flow, but the large
velocity spread observed in the SiO(2-1) line, together with the
relatively concentrated spatial distribution of the SiO emission,
suggests that the inclination angle must be high, i.e., that the
SiO flow is being ejected at a small angle with respect to the
line of sight.
3.2. CH3CN
The dense molecular gas, as traced by CH3CN(123-113)
(Fig. 1, left panel) appears to be distributed around the HW2
position, and elongated in a direction roughly perpendicular to
the projected direction of the large-scale outflow on the plane
of the sky. From a morphological point of view, therefore, the
data are very suggestive of the presence of a ∼ 350-AU-radius
disk-like structure around HW2.
On the other hand, as already pointed out by Torrelles et
al. (1999) based on VLA observations of NH3, the kinemati-
cal picture describing the dense gas distribution in the region
is quite complex. A position-velocity cut along the major axis
of the elongated structure (indicated in Fig. 1, left and center
panels, with a dashed line) reveals a velocity spread of about
6 km s−1 (see Fig. 3), also observed by Patel et al. (2005).
However, the two intensity peaks along the axis share roughly
C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 3
Fig. 1. All panels: the grey levels represent the continuum emission at 241 GHz. Lowest level is 3.3 mJy/beam or 2σ, highest
is 22σ. The HW2 position is indicated with a white star. The solid, crossing lines show the opening angle of the large-scale
northeast-southwest outflow, inferred from HCN and 13CO observations obtained with PdBI together with the core-tracing data.
The contours trace the continuum-subtracted CH3CN emission at 220 GHz, and in the top left corner, the HPBW for the con-
tinuum (grey foreground) and CH3CN (black background ellipse) are shown. The position of the intermediate-mass protostar
HC (MP05) is indicated with a white triangle. Left panel: the integrated emission of the CH3CN(123-113) is shown in solid
contours. Center panel: integrated intensity of CH3CN(123-113) between −7 and −3 km s
−1 (solid contours) and between −11.5
and −7.5 km s−1 (dashed contours). The dotted square box indicates the area enlarged in the right panel. Right panel: The circles
show the centroid positions of the SiO(2-1) emission for every channel in the range −25 < vlsr −3.5 km s
−1, with a channel width
of 0.5 km s−1. The black circles stand for the blue-shifted (vlsr < −10 km s
−1), white circles for the red-shifted (vlsr > −10 km s
emission.
Fig. 2. Comparison between the line profiles of SiO(2-1) (grey
spectrum) and CH3CN(124 − 114) (transparent spectrum). At
vlsr=∼ −10 km s
−1, the center of the SiO emission coincides
with the position of source HC of MP05 (cf. § 3.2 and Fig. 1).
The SiO(2-1) spectrum arises from a region of about 1′′ ra-
dius around the HC position (50%-of-peak-emission level).
The CH3CN emission arises from the HC3 position (cf. § 3.2
and Fig. 4), and has been scaled by a factor 0.5 for a better
visual comparison. No recognizable counterpart to the ∼ −5-
km s−1 CH3CN component is observed for SiO.
the same systemic velocity (∼ −5 km s−1). The weaker, blue-
shifted component of emission (∼ −10 km s−1), appears to trace
rather the outskirt of a physically separated component than a
rotation-induced velocity gradient along the axis of the alleged
“disk”. The peak of the -10 km s−1 CH3CN emission is spa-
tially and kinematically close to the center of the small-scale
SiO outflow (see Fig. 1, right panel), it is likely associated to it
and/or to its exciting source.
The CH3CN integrated intensity is dominated by the two
−5-km s−1 peaks, which lie respectively about 0.′′6 to the north-
west, and 0.′′5 to the southeast of the HW2 position. The fact
that the two CH3CN peaks share roughly the same systemic
velocity, is not compatible with the “rotating disk” hypothesis.
In what follows, we will treat them as independent condensa-
tions, and to be consistent with the nomenclature introduced
by MP05, we will refer to them respectively as HC2 and HC3.
Fig. 4 compares the spectra observed towards the two positions.
It is clear that, in both cases, both the −5- and −10-km s−1 com-
ponents are present along the line of sight, although the contri-
bution from the latter is more substantial towards HC3, i.e.,
close to the peak of the −10-km s−1 SiO emission.
We have assumed the LTE approximation to fit the physical
parameters associated with the two different velocity compo-
nents. All transitions in the spectrum are fitted simultaneously,
in order to take line blending and optical depth effects properly
into account (a detailed description of the method can be found
in Comito et al. 2005). For the ∼ −5 km s−1component, our
fit reveals that the k= 0 through k= 4 transitions are optically
thick towards both positions. The data at this velocity can only
be reproduced by including a very compact, hot, dense object
in the model. The emission centered at ∼ −10-km s−1 can be
modeled with a cooler, more extended component. The results
of the fit, for the two positions, are summarized in Tab. 1.
Although the presence of secondary minima in the χ2 space
is unavoidable when so many parameters are varied to achieve
4 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region
Fig. 3. Position-velocity plot for CH3CN(123 − 113), along the
major axis of the elongated structure (dashed line in Fig. 1, left
and center panels). Levels range from 3σ to 27σ in 3-σ steps.
The bracketed velocity ranges show the intervals making up the
−5 (solid) and −10 (dashed) km s−1 components, whose spatial
distribution is shown in Fig. 1, center panel.
minimization, in this case the simultaneous fitting of intensity
ratios between optically thick and optically thin lines, between
ortho- and para-CH3CN transitions, and between
12C and 13C
isotopologues of methyl cyanide (see Fig. 4), places very strin-
Fig. 4. In grey, high-resolution (∆v = 0.3 km s−1) spectra of
the CH3CN(12-11) emission at 200 GHz, towards the HC2 and
HC3 positions (see Fig. 1, center panel). Overlayed in black
are the model spectra, resulting from the parameters listed in
Tab. 1, assuming LTE approximation and that [12CO]/[13CO]=
gent constraints on the viable parameter space, at least as far
as the compact (∼ −5 km s−1) component is concerned. Note
that the rotational temperatures derived in this fashion are sig-
nificantly higher than those derived by Patel et al. (2005). This
discrepancy can be explained with the optical depth correction
in our fit.
4. Discussion
The observed elongation of the molecular gas distribution
around HW2, over a radius of ∼ 0.′′5 (∼ 360 AU), appears to
be due to the projected superposition, on the plane of the sky,
of at least three protostellar objects, of which at least one is
triggering a molecular outflow at a small angle with respect to
the line of sight (§ 3.1, § 3.2). All lines in our dataset are con-
sistent with this interpretation. The distribution of molecular
gas around HW2 can, on a 1′′ scale, be interpreted as a clus-
ter of high- and intermediate-mass protostars in the Cepheus A
HW2 region. The analysis of the CH3CN spectra (§ 3.2) sug-
gests the presence of internally heated compact hot-core-type
objects like HC, likely hosting protostellar objects, although
the 1-mm continuum emission peaks on the HW2 position and
does not show any secondary clumps. This may be due to in-
sufficient dynamic range in our data, if the contribution, to the
241-GHz continuum, of free-free emission from the HW2 ther-
mal jet is large. Fig. 5 shows the variation of the measured
HW2 continuum flux density as a function of frequency, S ν,
between 1.5 and 327 GHz (data points from: Rodrı́guez et al.
1994; this work; Patel et al. 2005). A two-component least-
squares fit of the data yields S jet ∝ ν
(0.51±0.12) (consistent with
the value inferred by Rodrı́guez et al. 1994, and with the theo-
retical predictions for the radio continuum spectrum of a con-
fined thermal jet, Reynolds 1986) and S (sub)mm ∝ ν
(1.92±0.12)
(dashed and dashed-dotted lines respectively in Fig. 5), where
S jet + S (sub)mm = S ν (solid curve in Fig. 5). Based on this esti-
mate, the thermal jet (free-free) contribution at 241 GHz should
be ∼ 10% of the total flux.
However, the flux density variation in the (sub)mm-
wavelength portion of the spectrum increases basically on
a Rayleigh-Jeans slope, suggesting the presence of optically
thick emission from an unresolved (with our best spatial res-
olution, at 241 GHz with PdBI) continuum source, whose
size thus cannot be larger than ∼ 0.′′6. Although we cannot
determine the nature of this compact source, it makes sense
to hypothesize that it can be described by i), dust emission,
and/or ii), free-free emission from a Hii region (for example
associated to a photoevaporating disk). For case i), we adopt
Beckwith et al.’s (1990) values for the mass absorption coeffi-
cient, κν = 0.1(ν/10
12Hz)β, with β = 1, to estimate the lower
mass limit for an object with size θdust = 0.
′′5 (∼ 363 AU) to
produce optically thick emission at 87 GHz: Mτdust=1 ≥ 1 M⊙.
Based on the peak flux at 87, 241 (this work) and 327 GHz
(Patel et al. 2005), the above value of θdust yields a brightness
temperature, TB ≃ 80 K. Smaller source sizes would lead to
higher intrinsic temperatures and lower mass limits. In case ii),
we assume TB = 10
4 K (typical for Hii regions), which would
translate into a source size of θfree−free ≃ 0.
′′04 (∼ 30 AU). The
continuum source VLA-mm (Curiel et al. 2002), which is lo-
C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region 5
vlsr Source size N(CH3CN) Trot ∆v
(km s−1) (cm−2) (K) (km s−1)
HC2 -4.5 0.′′3 ∼ 3 × 1016 250 2.9
-8.9 1′′ ∼ 8 × 1014 150 4.0
HC3 -4.2 0.′′25 ∼ 3 × 1016 250 3.2
-10.0 0.′′45 ∼ 5 × 1015 150 4.5
Table 1. LTE model results for the CH3CN emission towards the HC2 and HC3 cores (Fig. 4). For a discussion on the error
estimate, cf. Comito et al. 2005.
cated ∼ 0.′′15 south of HW2 and whose size cannot be larger
than 30 AU, displays a much too weak emission at cm and mm
wavelengths, and we estimate its contribution to cover for at
most 5% of the observed (sub)mm flux.
In other words, we cannot discard any of the two hypothe-
ses for the observed optically thick continuum emission. At
higher frequencies, the spectral index would get flatter if the
optically thick emission were only due to free-free emission,
while it would remain the same for dust. Observations with a
resolution of < 0.5′′, which are within the reach of present day
interferometers, would be able to shed more light on the nature
of this object.
Fig. 5. Flux density as a function of frequency for Cep-A
HW2. For the Rodrı́guez et al. (1994, triangles) and Patel et
al. (2005, square) data points, the errorbars fall within the sym-
bols and are therefore not visible. The solid curve shows the
2-component least-squares fit of the data, as descrived in the
text. The single components resulting from the fit are also plot-
ted separately (dashed and dashed-dotted lines).
As these estimates show, our conclusions do not rule out at
all the existence, on a smaller scale, of an accretion disk around
HW2, which in fact is to be expected, based on the very pres-
ence of the HW2 jet. In fact, recent 7-mm VLA observations
of SO2 have led Jiménez-Serra et al. (2007) to claim the de-
tection of a disk-like structure with a size of 600 × 100 AU,
roughly centered on the HW2 position, part of which may be
photoevaporating. Although spatially almost coexistent on the
plane of the sky, this structure is characterized by a different
vlsr (−7.3 km s
−1, as opposed to ∼ −5 km s−1) and apparently a
different chemistry from the molecular gas traced by CH3CN.
Our above estimates on the nature of the black-body emission
in the (sub)mm regime are all consistent with Jiménez-Serra et
al.’s conclusions.
Overall, the Cepheus A HW2 allows, due to its proxim-
ity, a view into the heart of a massive star forming region. The
emerging picture is anything but simple: including the sources
detected by Curiel et al. (2002), and the hot cores HC, HC2
and HC3, at least 6 probable young stellar or protostellar ob-
jects are located within a radius of 1′′ or 725 AU. It remains an
open issue whether, under such circumstances we can expect to
observe a classical accretion disk feeding a single central star,
or rather some kind of circum-cluster disk or ring-like structure
(analogous perhaps to circumbinary rings like the one around
GG Tau, Guilloteau et al. 1999), and what such a structure may
look like, both from a morphological and from a kinematical
point of view. Higher spatial resolution is needed, but the chal-
lenge is to identify the right chemical tracer to investigate the
structures one is interested in. CH3CN, otherwise considered a
reasonably good disk tracer (e.g. for the disk in IRAS 20126,
Cesaroni et al. 1997), does not seem to trace the disk-like struc-
ture seen by Jiménez-Serra et al. (2007) at all.
Another issue is the physical location of the −10 km s−1
molecular component. Though it seems likely that the peak
of −10-km s−1 CH3CN emission is associated with the power-
ing source of the small-scale SiO outflow, its connection to the
somewhat more extended molecular emission at this systemic
velocity (cf. Brogan et al. 2007) remains to be confirmed.
Acknowledgements. The authors are grateful to the IRAM staff in
Grenoble, particularly to H. Wiesemeyer, J. M. Winters and R. Neri,
for their support in the data calibration process. An anonymous
referee has given a significant contribution to the improvement of
this paper. CC and PS have enjoyed many fruitful discussions with
Malcolm Walmsley. JMP and IJS acknowledge the support pro-
vided through projects number ESP2004-00665 and S-0505/ESP-
0277 (ASTROCAM).
6 C. Comito et al.: High-resolution study of a star-forming cluster in the Cep-A HW2 region
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Introduction
Observations
Results
CH3CN
Discussion
|
0704.1618 | A Renormalization group approach for highly anisotropic 2D Fermion
systems: application to coupled Hubbard chains | A Renormalization group approach for highly anisotropic 2D Fermion systems:
application to coupled Hubbard chains
S. Moukouri
Department of Physics and Michigan Center for Theoretical Physics
University of Michigan, 2477 Randall Laboratory, Ann Arbor MI 48109
I apply a two-step density-matrix renormalization group method to the anisotropic two-
dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the
exact one for the tight-binding model. I find a ground-state energy which agrees with the exact
value up to four digits for systems as large as 24 × 25. I then apply the method to the interact-
ing case. I find that for strong Hubbard interaction, the ground-state is dominated by magnetic
correlations. These correlations are robust even in the presence of strong frustration. Interchain
pair tunneling is negligible in the singlet and triplet channels and it is not enhanced by frustration.
For weak Hubbard couplings, interchain non-local singlet pair tunneling is enhanced and magnetic
correlations are strongly reduced. This suggests a possible superconductive ground state.
I. INTRODUCTION
Quasi-one dimensional organic1 and inorganic2 mate-
rials have been the object of an important theoretical in-
terest for the last three decades. The essential features of
their phase diagram may be captured by the anisotropic
Hubbard model (AHM),
H = −t‖
i,l,σ
i,l,σci+1,l,σ + h.c.) + U
ni,l,↑ni,l,↓
i,l,σ
ni,l,σni+1,l,σ − µ
i,l,σ
ni,l,σ
i,l,σ
i,l,σci,l+1,σ + h.c.). (1)
or a more general Hubbard-like model including longer
range Coulomb interactions. The indices i and l label
the sites and the chains respectively. For these highly
anisotropic materials, t⊥ ≪ t‖. Over the years, the
AHM has remained a formidable challenge to condensed-
matter theorists. Some important insights on this model
or its low energy version, the g-ology model, have been
obtained through the work of Bourbonnais and Caron3,4
and others. They used a perturbative renormalization
group approach to analyze the crossover from 1D to 2D
at low temperatures. More recently, Biermann et al.5 ap-
plied the chain dynamical mean-field approach to study
the crossover from Luttinger liquid to Fermi liquid in this
model. Despite this important progress, crucial informa-
tion such as the ground-state phase diagram, or most no-
tably, whether the AHM displays superconductivity, are
still unknown. So far it has remained beyond the reach
of numerical methods such as the exact diagonalization
(ED) or the quantum Monte Carlo (QMC) methods. ED
cannot exceed lattices of about 4 × 5. It is likely to re-
main so for many years unless there is a breakthrough
in quantum computations. The QMC method is plagued
by the minus sign problem and will not be helpful at low
temperatures. The small value of t⊥ implies that, in or-
der to see the 2D behavior, it will be necessary to reach
lower temperatures than those usually studied for the
isotropic 2D Hubbard model. Hence, even in the absence
of the minus sign problem, in order to work in this low
temperature regime, the QMC algorithm requires spe-
cial stabilization schemes which lead to prohibitive cpu
time.6
II. TWO-STEP DMRG
I have shown in Ref. 7 that this class of anisotropic
models may be studied using a two-step density-matrix
renormalization group (TSDMRG) method. The TS-
DMRG method is a perturbative approach in which the
standard 1D DMRG is applied twice. In the first step,
the usual 1D DMRG method9 is applied to find a set
of low lying eigenvalues ǫn and eigenfunctions |φn〉 of a
single chain. In the second step, the 2D Hamiltonian is
then projected onto the basis constructed from the tensor
product of the |φn〉’s. This projection yields an effective
one-dimensional Hamiltonian for the 2D lattice,
E‖[n]|Φ‖[n]〉〈Φ‖[n]| − t⊥
i,l,σ
i,l,σ c̃i,l+1,σ + h.c.)(2)
where E‖[n] is the sum of eigenvalues of the different
chains, E‖[n] =
l ǫnl ; |Φ‖[n]〉 are the corresponding
eigenstates, |Φ‖[n]〉 = |φn1〉|φn2〉...|φnL〉; c̃
i,l,σ, c̃i,l,σ, and
ñi,l,σ are the renormalized matrix elements in the single
chain basis. They are given by
i,l,σ)
nl,ml = (−1)ni〈φnl |c
i,l,σ|φml〉, (3)
(c̃i,l,σ)
nl,ml = (−1)ni〈φnl |ci,l,σ|φml〉, (4)
(ñi,l,σ)
nl,ml = 〈φnl |ni,l,σ|φml〉, (5)
where ni represents the total number of fermions from
sites 1 to i− 1. For each chain, operators for all the sites
are stored in a single matrix
http://arxiv.org/abs/0704.1618v1
l,σ = (c̃
1,l,σ, ..., c̃
L,l,σ), (6)
c̃l,σ = (c̃1,l,σ, ..., c̃L,l,σ), (7)
ñl,σ = (ñ1,l,σ, ..., ñL,l,σ). (8)
Since the in-chain degrees of freedom have been inte-
grated out, the interchain couplings are between the
block matrix operators in Eq.( 6, 7) which depend only
on the chain index l. In this matrix notation, the effec-
tive Hamiltonian is one-dimensional and it is also studied
by the DMRG method. The only difference compared to
a normal 1D situation is that the local operators are now
ms2 ×ms2 matrices, where ms2 is the number of states
kept during the second step.
The two-step method has previously been applied
to anisotropic two-dimensional Heisenberg models.7 In
Ref. 8, it was applied to the t− J model but due to the
absence of an exact result in certain limits, it was tested
against ED results on small ladders only. A systematic
analysis of its performance on a fermionic model on 2D
lattices of various size has not been done. In this paper,
as a prelude to the study of the AHM, I will apply the
TSDMRG to the anisotropic tight-binding model on a
2D lattice, i.e., model (1) with U = V = 0. I perform
a comparison with the exact result of the tight-binding
model. I was able to obtain agreement for the ground-
state energies on the order of 10−4 for lattices of up to
24 × 25. I then discuss how these calculations may be
extended to the interacting case, before presenting the
U 6= 0 results.
III. WARM UP: THE TIGHT-BINDING MODEL
The tight-binding Hamiltonian is diagonal in the mo-
mentum space, the single particle energies are,
ǫk = −2t‖coskx − 2t⊥cosky − µ, (9)
with k = (kx, ky), kx = nxπ/(Lx+1) and ky = nyπ/(Ly+
1) for open boundary conditions (OBC); Lx, Ly are re-
spectively the linear dimensions of the lattice in the par-
allel and transverse directions. The ground-state energy
of an N electron system is obtained by filling the low-
est states up to the Fermi level, E[0](N) =
However in real space, this problem is not trivial and it
constitutes, for any real space method such as the TS-
DMRG, a test having the same level of difficulty as the
case with U 6= 0. This is because the term involving U is
diagonal in real space and the challenge of diagonalizing
the AHM arises from the hopping term.
I will study the tight-binding model at quarter filling,
N/LxLy = 1/2, the nominal density of the organic con-
ductors known as the Bechgaard salts. Systems of up to
Lx×Ly = L×(L+1) = 24×25 will be studied. During the
first step, I keep enough states (ms1 is a few hundred) so
0 20 40 60 80 100
FIG. 1: Low-lying states of the 1D tight-binding model (full
line) and of the 1D Heisenberg spin chain (dotted line) for
L = 16 and ms2 = 96.
that the truncation error ρ1 is less than 10
−6. I target the
lowest state in each charge-spin sectorsNx±2, Nx±1, Nx
and Sz ± 1, Sz ± 2, Nx is the number of electrons within
the chain. It is fixed such that Nx/Lx = 1/2. There is a
total of 22 charge-spin states targeted at each iteration.
For the tight-binding model, the chains remain discon-
nected if t⊥ < ǫ0(Nx + 1) − ǫ0(Nx) or t⊥ < ǫ0(Nx) −
ǫ0(Nx − 1), where Nx is the number of electons on sin-
gle chain. In order to observe transverse motion, it is
necessary that at least t⊥ >∼ ǫ0(Nx + 1) − ǫ0(Nx) and
t⊥ >∼ ǫ0(Nx)− ǫ0(Nx − 1). These two conditions are sat-
isfied only if µ is appropietly chosen. The values listed in
Table (I) corresponds to µ = (ǫ0(Nx+1)−ǫ0(Nx−1))/2.
This treshold varies with L. I give in Table (I) the val-
ues of t⊥ chosen for different lattice sizes. In principle,
for the TSDMRG to be accurate, it is necessary that
∆ǫ = ǫnc − ǫ0, where ǫnc is the cut-off, be such that
∆ǫ/t⊥ ≫ 1. But in practice, I find that I can achieve
accuracy up to the fourth digit even if ∆ǫ/t⊥ ≈ 5 using
the finite system method. Five sweeps were necessary
to reach convergence. Note that this conclusion is some-
what different from my earlier estimate of ∆ǫ/t⊥ ≈ 10
for spin systems.8 This is because in Ref. 8, I used the
infinite system method during the second step.
The ultimate success of the TSDMRG depends on the
density of the low-lying states in the 1D model. For
fixed ms2 and L, it is, for instance, easier to reach
larger ∆ǫ/J⊥ in the anisotropic spin one-half Heisen-
berg model, studied in Ref. 7, than ∆ǫ/t⊥ for the tight-
binding model as shown in Fig.1. For L = 16, ms2 = 96,
and J⊥ = t⊥ = 0.15, I find that ∆ǫ/J⊥ ≈ 10, while
∆ǫ/t⊥ ≈ 5. Hence, the TSDMRG method will be more
accurate for a spin model than for the tight-binding
model. Using the infinite system method during the
second step on the anisotropic Heisenberg model with
J⊥ = 0.1, I can now reach an agreement of about 10
with the stochastic QMC method.
Two possible sources of error can contribute to reduce
the accuracy in the TSDMRG with respect to the conven-
8× 9 16× 17 24× 25
t⊥ 0.28 0.15 0.1
µ -1.2660 -1.3411 -1.3657
∆ǫ/t⊥ 6.42 5.40 5.78
TABLE I: Transverse hopping and chemical potential used in
the simulations for different lattice sizes
ms2 8× 3 16× 3 24× 3
64 -0.241524 -0.211929 0.204040
Exact -0.241524 -0.211931 0.204049
TABLE II: Ground-state energies of three-leg ladders.
tional DMRG. They are the truncation of the superblock
from 4×ms1 states to onlyms2 states and the use of three
blocks instead of four during the second step. In Table
(II) I analyze the impact of the reduction of the number of
states to ms2 for three-leg ladders. The choice of three-
leg ladders is motivated by the fact that at this point,
the TSDMRG is equivalent to the exact diagonalization
of three reduced superblocks. It can be seen that as far as
t⊥ >∼ ǫ0(Nx+1)−ǫ0(Nx) and t⊥ >∼ ǫ0(Nx)−ǫ0(Nx−1), the
TSDMRG at this point is as accurate as the 1D DMRG.
Note that the accuracy remains nearly the same irrespec-
tive of L as far as the ratio ∆ǫ/t⊥ remains nearly con-
stant. Since ∆ǫ decreases when L increases, t⊥ must be
decreased in order to keep the same level of accuracy for
fixed ms2. In principle, following this prescription, much
larger systems may be studied. ∆ǫ/t⊥ does not have to
be very large, in this case it is about 5, to obtain very
good agreement with the exact result.
The second source of error is related to the fact that
the effective single site during the second step is now a
chain having ms2 states, I am thus forced to use three
blocks instead of four to reduce the computational bur-
den. In Table (III), it can be seen that this results in a
reduction in accuracy of about two orders of magnitude
with respect to those of three leg-ladders. These results
are nevertheless very good given the relatively modest
computer power involved. All calculations were done on
a workstation.
The DMRG is less accurate when three blocks are used
instead of four. This can be understood by applying the
following view on the formation of the reduced density
matrix. The construction of the reduced density matrix
ms2 8× 9 16× 17 24× 25
64 -0.24761 -0.21401 0.20504
100 -0.24819 -0.21414 0.20509
120 -0.24832 -0.21419
Exact -0.24857 -0.21432 0.20519
TABLE III: Ground-state energies for different lattice sizes;
a single state was targeted in the second step.
may be regarded as a linear mapping uΨ : F
∗ → E,
where E is the system, F is the environment and, F∗
is the dual space of F. Using the decomposition of the
superblock wave function Ψ[0] =
i ⊗ φRi , with φLi ∈
E and ΦRi ∈ F, for any φ∗ ∈ F ∗,
〈φ∗|φRi 〉φLi . (10)
Let |k〉, k = 1, ...dimE and |l〉, l = 1, ...dimF be the
basis of E and F respectively. Then, |l〉 has a dual basis
〈l∗| such that 〈l∗|l〉 = δl,l∗ . The matrix elements of uΨ
in this basis are just the coordinates of the superblock
wave function Φ[0]k,l . The rank r of this mapping, which
is also the rank of the reduced density matrix is always
smaller or equal to the smallest dimension of E or F,
r < Min(dimE, dimF). Hence, if ms2 states are kept in
the two external blocks, the number of non-zero eigenval-
ues of ρ cannot be larger than ms2. Consequently, some
states which have non-zero eigenvalues in the normal four
block configuration will be missing. A possible cure to
this problem is to target additional low-lying states above
Ψ[0](N). The weight of these states in ρ must be small
so that their role is simply to add the missing states not
to be described accurately themselves. A larger weight
on these additional states would lead to the reduction
of the accuracy for a fixed ms2. In table (IV), I show
the improved energies when, besides the ground state,
I target the lowest states of the spin sectors Sz = −1
and Sz = +1 with N electrons. The weights were re-
spectively 0.995, 0.0025, and 0.0025 for the three states.
This lowers E[0](N) in all cases, but the gain does not
appear to be spectacular. But I do not know whether
this is due to my choice of perturbation of ρ or whether
even the algorithm with four blocks would not yield bet-
ter E[0](N). If the lowest sectors with N + 1 and N − 1
electrons which have Sz = ±0.5 are projected instead, I
find that the results are similar to those with Sz = ±1
sectors, there are possibly many ways to add the missing
states. A more systematic approach to this problem has
recently been suggested.10 It is based on using a local
perturbation to build a correction to the density matrix
from the site at the edge of the system. Here, such a
perturbation would be ∆ρ = αc
l ρcl, where α is a con-
stant, α ≈ 10−3 − 10−2, and c†l , cl are the creation and
annihilation operators of the chain at the edge of the sys-
tem. This type of perturbation resulted in an accuracy
gain of more than an order of magnitude in the case of a
spin chain.10 The three block method was found to be on
par with the four block method. It will be interesting to
see in a future study how this type of local perturbation
performs within the TSDMRG.
To conclude this section, as a first step to the investi-
gation of interacting electron models, I have shown that
the TSDMRG can successfuly be applied to the tight-
binding model. The agreement with the exact result is
very good and can be improved since the computational
power involved in this study was modest. The extension
ms2 8× 9 16× 17
64 -0.24803 -0.21401
100 -0.24828 -0.21417
Exact -0.24857 -0.21432
TABLE IV: Ground-state energies for different lattice sizes;
three states were targeted in the second step: the ground state
itself and the lowest states of Sz = 0 and Sz = 1 sectors.
0 1 2 3 4
FIG. 2: Width ∆ǫ for the low-lying states of the 1D Hubbard
chain as function of U for L = 16 and ms2 = 128.
to the AHM with U 6= 0 is straightforward. There is no
additional change in the algorithm since the term involv-
ing U is local and thus treated during the 1D part of the
TSDMRG. The role of U is to reduce ∆ǫ as shown in
Fig.2. For fixed L and ms2, ∆ǫ decreases linearly with
increasing U . For L = 16 and ms2 = 128, I anticipate
that for U <∼ 3 the interacting system results will be on
the same level or better than those of the non-interacting
case with ms2 = 100 for the same value of L.
IV. GROUND-STATE PROPERTIES OF
COUPLED HUBBARD CHAINS
I now proceed to the study of U 6= 0. One of the
main motivations for such a study is the possibility to
gain insight into the mechanism of superconductivity in
quasi 1D systems. The mechanism of superconductivity
in the quasi 1D organic materials Bechgaard and Fabre
salts, is still an open issue.13 Since these materials are 1D
above a crossover temperature Tx ≈ t⊥/π, it is broadly
accepted that the starting point for the the understand-
ing of their low T behavior should be pure 1D physics.
The occurence of the low T ordered phases is driven by
the interchain hopping t⊥. Two main hypotheses have
been suggested concerning superconductivity. The first
hypothesis (see a recent review in Ref. 13) relies on a
more conventional physics: t⊥ drives the system to a 2D
electron gas which is an anisotropic Fermi liquid which
becomes superconductive through a conventional BCS
0 2 4 6 8
-0.004
-0.003
-0.002
-0.001
FIG. 3: Transverse Green’s function G(y) for td = 0 (circles),
td = 0.1 (squares).
mechanism. However, it has been argued11 that given
the smallness of t⊥, the resulting electron-phonon cou-
pling would not be enough to account for the observed
Tc. The second hypothesis, which has gained strength
over the years given the absence of a clear phonon sig-
nature, is that the pairing mechanism originates from an
exchange of spin fluctuation.11
Interest in this issue was recently revived by the NMR
Knight shift experimental finding that the symmetry of
the Cooper pairs is triplet12 in (TMTSF )2(PF )6. No
shift was found in the magnetic susceptibility at the tran-
sition for measurement made under a magnetic field of
about 1.4 Tesla. A triplet pairing scenario was subse-
quently supported by the persistence of superconduc-
tivity under fields far exceeding the Pauli breaking-pair
limit19. However there is no simple explanation of this
scenario. Triplet pairing would be unfavorable in a BCS
like scenario for which a singlet s-wave is most likely.
Triplet pairing is also less likely in the spin fluctuation
mechanism for which a singlet d-wave is predicted by an-
lytical RG13 or by perturbative approaches17. It has be
argued that these difficulties in both mechanisms can be
circumvented. In the BCS case, the association of AFM
fluctuations with an open Fermi surface to the electron-
phonon mechanism may lead to a triplet pairing18. In the
spin fluctuation case, the addition of interchain Coulomb
interactions may favor a triplet f-wave in lieu of the sin-
glet d-wave13,17. The more exotic Fulde-Ferrel-Larkin-
Ovchinnikov phase can also been invoked to account for
the large paramagnetic limit. However, the Knight shift
result which was thought to bring a conclusion to this
long standing issue has only revived the old controversy.
The conclusion of this experiment itself has been recently
challenged. In Ref. 15, it was pointed out that the ob-
servation of triplet superconductivity claimed in Ref. 12
could be a spurious effect due to the lack of thermaliza-
tion of the samples. A recent Knight shift experimenent
performed at lower fields reveals a decrease in the spin
susceptibility. This is consistent with singlet pairing.16
0 2 4 6 8
-0.002
-0.001
0.001
FIG. 4: Transverse spin-spin correlation C(y) for td = 0 (cir-
cles), td = 0.1 (squares).
The 1D interacting electron gas is now fairly well
understood.3 There is no phase with long range order.
There are essentially four regions in the phase diagram,
characterized by the dominant correlations i.e., SDW,
charge density wave (CDW), singlet superconductivity
(SS) and triplet superconductivity (TS). The essential
question is whether the interchain hopping will simply
freeze the dominant 1D fluctuation into long-range order
(LRO) or create new 2D physics. The estimated values
of U and V for the Bechgaard salts suggest that they
are in the SDW region in their 1D regime. This suggests
that superconductivity in these materials is a 2D phe-
nomenon. Interchain pair tunneling was suggested soon
after the discovery of superconductivity in an organic
compound.14 Emery argued instead that a mechanism
similar to the Kohn-Luttinger mechanism might be re-
sponsible for superconductivity in the organic materials.
When t⊥ is turned on, pairing can arise from exchange of
short-range SDW fluctuations. The reason is that the os-
cillating SDW susceptibility atQ = (2kF , k⊥) would have
an attractive region if k⊥ 6= 0. In particular if k⊥ = π
as I found, then the interaction would be attractive be-
tween particles in neighboring chains. In this study, I
will restrain myself to the study of interchain pair tun-
neling. I was unable to compute correlation functions of
pairs in which each electron belongs to a different chain.
The reason is that in the DMRG method, for the correla-
tion functions to be accurate, at least two different blocks
should be involved. This means that for pair correlation
for which each electron of the pair is on a different chain,
at least four blocks are needed. However, the introduc-
tion of four blocks in the second step of the TSDMRG
leads to a prohibitive CPU time.
With the hope of frustrating an SDW ordering which
is usually expected, I will add an extra terms to model
(1) . These are the diagonal interchain hopping,
Hd = −td
i,l,σ
i,l,σci+1,l+1,σ +
0 2 4 6 8
-0.002
-0.001
0.000
0.001
0 1 2 3 4 5 6 7 8
-0.0001
0.0000
0.0001
FIG. 5: Transverse local singlet correlation SS(y) for td = 0
(circles), td = 0.1 (squares).
h.c) + (c
i+1,l,σci,l−1,σ + h.c), (11)
and the next-nearrest neighbor interchain hopping,
H ′⊥ = −t′⊥
i,l,σ
i,l,σci,l+2,σ + h.c).
I will also add the interchain Coulomb interaction,
HV = V⊥
i,l,σ
ni,l,σ.ni,l+1,σ (12)
I set t⊥ = 0.2, ms1 = 256, ms2 = 128, and (L × (L +
1) = 16×17. A second set of calculations with t⊥ = 0.15,
same values of ms1 and ms2, and (L× (L+1) = 24× 25
lead to the same conclusions. Therefore, they will not be
shown here. In order to analyze the physics induced by
the transverse couplings, I compute the following inter-
chain correlations: the transverse single-particle Green’s
function, shown in Fig.3,
G(y) = 〈cL/2,L/2+yc†L/2,L/2+1〉, (13)
the transverse spin-spin correlation function, shown in
Fig.4,
C(y) =
〈SL/2,L/2+ySL/2,L/2+1〉, (14)
the transverse local pairs singlet superconductive corre-
lation, shown in Fig.5,
SS(y) = 〈ΣL/2,L/2+yΣ†L/2,L/2+1〉, (15)
where
Σi,l = ci,l↑ci,l↓, (16)
the transverse triplet superconductive correlation, shown
in Fig.6,
ST (y) = 2〈ΘL/2,L/2+yΘ†L/2,L/2+1〉, (17)
where
Θi,l =
(ci,l↑ci+1,l↓ + ci,l↓ci+1,l↑), (18)
and the transverse non-local singlet pair superconductive
correlation function, shown in Fig.7,
SD(y) = 2〈∆L/2,L/2+y∆†L/2,L/2+1〉, (19)
where
∆i,l =
(ci,l↑ci+1,l↓ − ci,l↓ci+1,l↑). (20)
A. Strong-coupling regime
Let us first consider, the regime U >∼ 4, I choose
for instance U = 4, V = 0.85, µ = 0, and
td = t
⊥ = V⊥ = 0; besides single-particle hop-
ping, t⊥ also generates two-particle hopping both in
the particle-hole and particle-particle channels. These
two-particle correlation functions are roughly given
by the average values t2⊥〈c
i,lσci,l−σc
i,l+j−σci,l+jσ〉 and
t2⊥〈c
i,lσc
i,l−σci,l+jσci,l+j−σ〉 for an on-site pair created
at (i, l) and then destroyed at (i, l + j). It is expected
that the dominant two-particle correlation are SDW with
k⊥ = π. This is seen in Fig.(4-7). The transverse pair-
ing correlations are all found to be small with respect
to C(y). Among the pairing correlations, SS(y) decays
faster then ST (y) and SD(y). These results are consis-
tent with the view that the role of t⊥ is to freeze the
dominant 1D correlations into LRO.
When td 6= 0, it is expected that for a strong enough td,
the magnetic order will vanish because of the frustration
induced by td. A simple argument is that td induces an
AFM exchange between next-nearest neigbhors on chains
l and l + 1 which compete with the AFM exchange be-
tween nearest neigbhors. The hope is that there could be
a region of the phase diagram where superconductivity
could ultimately win either by pair tunneling between
the chains or by the Emery’s mechanism. However, in
Fig.(4-7) it can be seen that, while td slightly reduces
C(y), the dominant correlations are still SDW even for a
strong td/t⊥ = 0.5. SS(y), ST (y) and SD(y) are barely
affected by td. The fact that td does not strongly af-
fect the SDW order can be understood in the light of
recent study of coupled t− J chains8. It was shown that
the frustration strongly suppresses magnetic LRO only
0 1 2 3 4 5 6 7
-0.002
-0.001
0.000
0.001
0 1 2 3 4 5 6 7 8
-0.0002
-0.0001
0.0000
0.0001
FIG. 6: Transverse triplet superconductive correlation ST (y)
for td = 0 (circles), td = 0.1 (squares).
0 1 2 3 4 5 6 7
-0.002
-0.001
0.000
0.001
0 1 2 3 4 5 6 7 8
-0.0001
0.0000
0.0001
0.0002
FIG. 7: Transverse singlet non-local superconductive correla-
tion SD(y) for td = 0 (circles), td = 0.1 (squares).
close to half-filling. For large dopings, two neighboring
spins in a chain do not always points to opposite direc-
tion as the consequence, td does not necessarily frustrate
the magnetic order. This is illustrated in a simple sketch
in Fig.(8). td could even enhance it as seen in the study
of t− J chains. In Fig.3, it can be seen that td enhances
G(y). This enhancement, together with the decrease of
C(y), suggests a possible widening of an eventual Fermi
liquid region at finite T above the ordered phase. When
t⊥ 6= 0, I also found (not shown) that magnetic correla-
tion are not effectively suppressed even when t′⊥ = t⊥/2.
For this value, it would be expected that the ratio of the
effective exchange term generated by t′⊥ to that generated
by t⊥ is about one quarter. In the frustrated J1−J2 spin
chain, a spin gap opens around this ratio. This simple
picture does not seem to work here.
FIG. 8: sketch of the spin texture (arrows) in two consecutive
chains in an SDW. The bold horizontal lines represent the
chains. The full diagonal lines show bonds for which td tends
to increase the SDW order. The diagonal dotted lines show
bonds for which td frustrates the magnetic order.
B. Weak-coupling regime
I now turn in to the regime where U <∼ 4. I set U = 2,
V = 0, µ = −0.9271, t⊥ = 0.2, td = 0, and V⊥ = 0.4,
where V⊥ is the interchain Coulomb interaction between
nearest neighbors. It can be seen in Fig.9 that C(y) is
now strongly reduced with respect to its strong coupling
values. It is already within our numerical error for the
next-nearest neighbor in the transverse direction. This
is an indication that the ground state is probably not an
SDW. It is to be noted that this occurs even in the ab-
sence of td or t
⊥. This seems to be at variance with the
RG analysis which requires t′⊥ to destroy the magnetic
order. A possible explanation of this is that at half-filling
the perfect nesting occurs at the wave vector Q = (π, π)
for the spectrum of equation (9). Away from half-filling
the nesting is no longer perfect this leads to the reduc-
tion of magnetic correlations. The first correction to the
nesting is an effective frustration term which is roughly
t2⊥cos2k⊥. This expression is identical to a term that
could be generated by an explicit frustration t′⊥ = t
The discrepancy between the TSDMRG and the RG re-
sults could be that this nesting deviation is undereval-
uated in the RG analysis. This mechanism cannot be
invoked in the strong coupling regime where band effects
are small.
The suppression of magnetism is concommitant to a
strong enhancement of the singlet pairing correlations as
seen in Fig. 11. Triplet correlations, shown in Fig. 10,
remain very small. However, while it is clear from the
behavior of C(y) that the ground state is non magnetic.
This result strongly suggests that the ground state is a
superconductor in this regime. A finite size analysis is,
however, necessary to conclude whether this persists to
the thermodynamic limit. I cannot rule out the possibil-
ity of a Fermi liquid ground state, which is implied by
0 2 4 6 8
-0.002
-0.001
0.001
FIG. 9: Transverse spin-spin correlation C(y) for U = 4 (cir-
cles), U = 2 and V⊥ = 0.4 (squares).
0 1 2 3 4 5 6 7
-0.001
0.001
0.002
FIG. 10: Transverse triplet superconductive correlation
ST (y) for U = 4 (circles), U = 2 and V⊥ = 0.4 (squares).
strong single particle correlations.
V. CONCLUSION
In this paper, I have presented a TSDMRG study of
the competition between magnetism and superconductiv-
ity in an anisotropic Hubbard model. I have analyzed the
effect of the interchain hopping in the strong and weak
U regimes. In the strong-coupling regime, the results are
consistent with earlier predictions that the role of t⊥ is
to freeze the dominant 1D SDW correlations into a 2D
ordered state. But at variance with analytical predic-
tions, this is only true in the strong U regime. In this
regime, I find that even the introduction of frustration
does not disrupt the SDW order which remain robust
up to large values of the frustration. In the weak cou-
pling regime singlet pair correlations are dominant. The
ground state seems to be a superconductor. This behav-
ior is somewhat in agreement with experiments in the
Bechgaard or Fabre salts. The phase diagram is domi-
0 1 2 3 4 5 6 7
-0.001
0.001
0.002
FIG. 11: Transverse singlet non-local superconductive cor-
relation SD(y) for U = 4 (circles), U = 2 and V⊥ = 0.4
(squares).
nated by magnetism at low pressure (strong U) and by
superconductivity at high pressure (weak U). Because of
experimental relevance, I restricted myself to the com-
petion between magnetism and superconductivity. I did
not analyze CDW correlations. These are likely to be
important given that I applied open boundary conditions
which are known to generate Friedel oscillations20 that
very decay slowly from the boundaries. They may also
genuinely generated by V⊥, leading to a CDW ground
state instead of a superconductor.
Acknowledgments
I am very grateful to C. Bourbonnais for very helpful
exchanges. I wish to thank A.M.-S. Tremblay for helpful
discussions. This work was supported by the NSF Grant
No. DMR-0426775.
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Metals” Eds. P. Bernier, S. Lefrant and G. Bidan (Elsevier,
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1033 (1991).
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Clarendon Press Eds, P. 254-269 (2004).
5 S. Biermann, A. Georges, A. Lichtenstein, and T. Gia-
marchi, Phys. Rev. Lett. 87, 276405 (2001).
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bernatis, and R.T. Scalettar, Phys. Rev. B 40, 506 (1989).
7 S. Moukouri, Phys. Rev. B 70, 014403 (2004).
8 S. Moukouri, J. Stat. Mech. P02002 (2006)
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B 48, 10 345 (1993).
10 S.R. White, Phys. Rev. B 72, 180403 (2005).
11 V.J. Emery, Synthetic Metals, 13, 21 (1986).
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13 N. Dupuis, C. Bourbonnais and J.C. Nickel,
cond-mat/0510544.
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http://arxiv.org/abs/cond-mat/0510544
http://arxiv.org/abs/cond-mat/0701566
http://arxiv.org/abs/cond-mat/0001331
|
0704.1619 | Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge:
Observations and Model Comparisons | Mon. Not. R. Astron. Soc. 000, 000–000 (2005) Printed 2 November 2018 (MN LATEX style file v2.2)
Proper Motion Dispersions of Red Clump Giants in the Galactic
Bulge: Observations and Model Comparisons
Nicholas J. Rattenbury1, Shude Mao1, Victor P. Debattista2, Takahiro Sumi3
Ortwin Gerhard4, Flavio De Lorenzi4 ⋆
1 University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, UK
2 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195-1580, USA
3 Solar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
4 Max-Planck-Institut fuer extraterrestrische Physik, P.O. Box 1312,D-85741 Garching, Germany
Accepted ........ Received .......; in original form ......
ABSTRACT
Red clump giants in the Galactic bulge are approximate standard candles and hence they can
be used as distance indicators. We compute the proper motion dispersions of RCG stars in
the Galactic bulge using the proper motion catalogue from the second phase of the Optical
Gravitational Microlensing Experiment (OGLE-II, Sumi et al. 2004) for 45 fields. The proper
motion dispersions are measured to a few per cent accuracy due to the large number of stars
in the fields. The observational sample is comprised of 577736 stars. These observed data are
compared to a state-of-the-art particle simulation of the Galactic bulge region. The predictions
are in rough agreement with observations, but appear to be too anisotropic in the velocity
ellipsoid. We note that there is significant field-to-field variation in the observed proper motion
dispersions. This could either be a real feature, or due to some unknown systematic effect.
Key words: gravitational lensing - Galaxy: bulge - Galaxy: centre - Galaxy: kinematics and
dynamics - Galaxy: structure
1 INTRODUCTION
Many lines of evidence suggest the presence of a bar at the Galac-
tic centre, such as infrared maps (Dwek et al. 1995; Binney et al.
1997) and star counts (Stanek et al. 1997; Nikolaev & Weinberg
1997; Unavane & Gilmore 1998), see Gerhard (2002) for a review.
However, the bar parameters are not well determined. For example,
recent infra-red star counts collected by the Spitzer Space Tele-
scope are best explained assuming a bar at a ∼ 44◦ angle to the
Sun–Galactic centre line (Benjamin et al. 2005) while most previ-
ous studies prefer a bar at ∼ 20◦. In addition, there may be some
fine features, such as a ring in the Galactic bulge, that are not yet
firmly established (Babusiaux & Gilmore 2005). It is therefore cru-
cial to obtain as many constraints as possible in order to better un-
derstand the structure of the inner Galaxy.
Many microlensing groups monitor the Galactic bulge, in-
cluding the EROS (Aubourg et al. 1993), MACHO (Alcock et al.
2000), MOA (Bond et al. 2001; Sumi et al. 2003a) and OGLE
(Udalski et al. 2000) collaborations. In addition to discovering mi-
crolensing events, these groups have also accumulated a huge
amount of data about the stars in the Galactic bulge spanning sev-
eral years to a decade and a half.
⋆ e-mail: (njr, smao)@jb.man.ac.uk; [email protected];
[email protected]; [email protected];
[email protected]
Eyer & Woźniak (2001) first demonstrated that the data can
be used to infer the proper motions of stars, down to ∼ mas yr−1.
Sumi et al. (2004) obtained the proper motions for millions of stars
in the OGLE-II database for a large area of the sky. In this pa-
per, we focus on the red clump giants. These stars are bright and
they are approximately standard candles, hence their magnitudes
can be taken as a crude measure of their distances. As the OGLE-II
proper motions are relative, in this paper we compute the proper
motion dispersions of bulge stars for all field data presented by
Sumi et al. (2004), as they are independent of the unknown proper
motion zero-points. These results could aid theoretical modelling
efforts for the central regions of the Galaxy.
The structure of the paper is as follows. In section 2, we
describe the OGLE-II proper motion catalogue and compute the
proper motion dispersions for bulge stars in 45 OGLE-II fields. In
section 3 we describe the stellar-dynamical model of the Galaxy
used in this work and detail how the model was used to generate
proper motion dispersions. These model predictions are compared
to the observational results in section 4 and in section 5 we discuss
the implications of the results.
2 OBSERVED PROPER MOTION DISPERSIONS
The second phase of the OGLE experiment observed the Galactic
Centre in 49 fields using the 1.3m Warsaw telescope at the Las
c© 2005 RAS
http://arxiv.org/abs/0704.1619v1
2 Rattenbury et al.
−0.5 0 0.5 1 1.5 2
PSfrag replacements
(V − I)0
Figure 2. Extinction-corrected colour-magnitude diagram for stars in the
OGLE-II field 1. The ellipse defines the selection criteria for RCG stars
based on colour and magnitude, see text. Sample stars are also required to
have proper motion errors sl,b < 1 mas yr
−1 and total proper motion
µ < 10 mas yr−1.
Campanas Observatory, Chile. Data were collected over an interval
of almost four years, between 1997 and 2000. Each field is 0.24◦×
0.95◦ in size. Fig. 1 shows the position of the OGLE-II Galactic
Bulge fields which returned data used in this paper.
2.1 Red Clump Giants
The red clump giants are metal-rich horizontal branch stars
(Stanek et al. 2000 and references therein). Theoretically, one ex-
pects their magnitudes to have (small) variations with metallicity,
age and initial stellar mass (Girardi & Salaris 2001). Empirically
they appear to be reasonable standard candles in the I-band with
little dependence on metallicities (Udalski 2000; Zhao et al. 2001).
Below we describe the selection of RCG stars in more detail.
2.2 OGLE-II proper motion data
Bulge RCG stars are selected from the OGLE-II proper motion cat-
alogue by applying a cut in magnitude and colour to all stars in each
of the OGLE-II fields. We corrected for extinction and reddening
using the maps presented by Sumi (2004) for each field. Stars were
selected which are located in an ellipse with centre (V −I)0 = 1.0
, I0 = 14.6; and semi-major (magnitude) and semi-minor (colour)
axes of 0.9 and 0.4 respectively, see Fig. 2; a similar selection cri-
terion was used by Sumi (2004). Stars with errors in proper mo-
tion greater than 1 mas yr−1 in either the l or b directions were
excluded. Stars with total proper motion greater than 10 mas yr−1
where similarly excluded, as these are likely to be nearby disk stars,
see also section 3.2. Fields 44, 47-49 were not analysed due to the
low number of RCG stars appearing in these fields.
The proper motion dispersions for the longitude and latitude
directions (σl and σb) were computed for each field via a maxi-
mum likelihood analysis following Lupton et al. (1987). Assuming
a Gaussian distribution of proper motions with mean µ̄ and intrin-
sic proper motion dispersion σ, the probability of a single observed
proper motion µi with measurement error ξi is:
2π(σ2 + ξ2i )
(µi − µ̄)
2(σ2 + ξ2i )
Maximising the likelihood ln(L) = ln(
pi) for µ̄ and σ over all
observations we find:
∂ lnL
(µi − µ̄)
σ2 + ξ2i
= 0 (2)
⇒ µ̄ =
σ2 + ξ2i
−1 (3)
∂ lnL
σ2 + ξ2i
(µi − µ̄)
(σ2 + ξ2i )
= 0 (4)
which can be solved numerically to find σ2.
The values of µ̄ and σ obtained using the above maximum-
likelihood analysis are virtually identical to those obtained via the
equations in Spaenhauer et al. (1992). The errors on the observed
proper motion dispersion values were determined from a bootstrap
analysis using 500 samplings of the observed dataset.
2.3 Extinction
In order to ensure the correction for extinction and reddening above
does not affect the kinematic measurements, σl and σb were recom-
puted for each OGLE-II field using reddening-independent mag-
nitudes. Following Stanek et al. (1997) we define the reddening-
independent magnitude IV−I:
IV−I = I − AI/(AV − AI) (V − I) (5)
where AI and AV are the extinctions in the I and V bands deter-
mined by Sumi (2004). The position of the red clump in the IV−I,
(V − I) CMD varies from field to field. The red clump stars were
extracted by iteratively applying a selection ellipse computed from
the moments of the data (Rocha et al. 2002) rather than centred on a
fixed colour and magnitude. The selection ellipse was recomputed
iteratively for each sample until convergence. The proper motion
dispersions σl and σb computed using RCG stars selected in this
way are consistent with those determined using the original selec-
tion criteria on corrected magnitudes and colours.
2.4 Results
Table 1 lists the observed proper motion dispersions along with
errors for each of the 45 OGLE-II fields considered in this paper.
Figures 3 and 4 show the proper motion dispersions σl and σb
as a function of Galactic longitude and latitude. A typical value of
σl or σb of 3.0 mas yr
−1 corresponds to ∼ 110 kms−1, assum-
ing a distance to the Galactic centre of 8 kpc. The proper motion
dispersion profiles as a function of Galactic longitude shows some
slight asymmetry about the Galactic centre. This asymmetry may
be related to the tri-axial Galactic bar structure (Stanek et al. 1997;
Nishiyama et al. 2005; Babusiaux & Gilmore 2005). The most dis-
crepant points in Fig. 3 correspond to the low-latitude fields num-
bers 6 and 7 (see Fig. 1). The varying field latitude accounts for
some of the scatter in Fig. 3, however we note below in section 4.1
that there are significant variations in the observed proper motion
dispersion between some pairs of adjacent fields. Owing to the the
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 3
−8−6−4−20246810
replacem
Galactic longitude (◦)
Figure 1. The position of the 45 OGLE-II fields used in this analysis. The field used in Spaenhauer et al. (1992) is shown, located within OGLE-II field 45
with (l, b) = (1.0245◦,−3.9253◦).
lack of fields at positive Galactic latitude, any asymmetry about the
Galactic centre in the proper motion dispersions as a function of
Galactic latitude is not obvious, see Fig. 4. Field-to-field variations
in the proper motion dispersions similarly contribute to the scat-
ter seen in Fig. 4, along with the wide range of field longitudes,
especially for fields with −4◦ < b < −3◦.
Table 2 lists the proper motion dispersions and cross-
correlation term Clb in the OGLE-II Baade’s Window fields 45
and 46 along with those found by Kozłowski et al. (2006) using
HST data in four BW fields. The two sets of proper motion dis-
persions results are consistent at the ∼ 2σ level. It is important to
note that the errors on the proper motion dispersions in Table 1 do
not include systematic errors. We also note that the selection cri-
teria applied to stars in the HST data are very different to those
for the ground-based data, in particular the magnitude limits ap-
plied in each case. The bulge kinematics from the HST data of
Kozłowski et al. (2006) were determined for stars with magnitudes
18.0 < IF814W < 21.5. The approximate reddening-independent
magnitude range for the OGLE-II data was 12.5 . IV−I . 14.6.
The effects of blending are also very different in the two datasets. It
is therefore very reassuring that our results are in general agreement
with those obtained by Kozłowski et al. (2006) using higher reso-
lution data from the HST. For more comparisons between ground
and HST RCG proper motion dispersions, see section 4.
Figure 5 shows the cross-correlation term Clb as a function of
Galactic co-ordinate. There is a clear sinusoidal structure in the Clb
data as a function of Galactic longitude, with the degree of corre-
lation between σl and σb changing most rapidly near l ≃ 0
◦. The
Clb data as a function of Galactic latitude may also show some ev-
idence of structure. It is possible however, that this apparent struc-
ture is due to the different number of fields at each latitude, rather
than some real physical cause.
3 GALACTIC MODEL
The stellar-dynamical model used in this work was produced us-
ing the made-to-measure method (Syer & Tremaine 1996). The
model is constrained to reproduce the density distribution con-
structed from the dust-corrected L-band COBE/DIRBE map of
Spergel et al. (1996). An earlier dynamical model was built to
match the total column density of the disk (Bissantz & Gerhard
2002). This dynamical model matched the radial velocity and
proper motion data in two fields (including Baade’s window) quite
well. No kinematic constraints were imposed during the construc-
tion of the model. We refer the readers to Bissantz et al. (2004)
for more detailed descriptions. The model used here is constructed
as in that case with the further refinement that the vertical den-
sity distribution is also included. This is necessary as the vertical
kinematics (σb) will also be compared with observations in this
paper. However the density distribution near the mid-plane is con-
siderably more uncertain, in part because of the dust extinction cor-
rection. Thus the model used in this paper can only be considered
illustrative, not final. Further efforts to model the vertical density
distribution are currently under way and will be reported elsewhere
(Debattista et al. 2007, in preparation).
In Fig. 6, we present the mean motion of stars in the mid-plane
of the Galaxy from this model. A bar position angle of θ = 20◦ is
shown here, as this is the orientation favoured both by optical depth
measurements (Evans & Belokurov 2002) and by the red clump gi-
ant brightness distribution (Stanek et al. 1997) and was the angle
c© 2005 RAS, MNRAS 000, 000–000
4 Rattenbury et al.
Table 1. Observed proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II
fields. High precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of
stars selected from each field. Fields 44, 47-49 were not analysed due to the low number of RCG stars appearing in these fields.
Field Field centre PM Dispersions (mas yr−1) Clb N
l (◦) b (◦) Longitude σl Latitude σb
1 1.08 -3.62 3.10 ±0.02 2.83 ±0.02 -0.13 ±0.01 15434
2 2.23 -3.46 3.21 ±0.02 2.80 ±0.02 -0.14 ±0.01 16770
3 0.11 -1.93 3.40 ±0.01 3.30 ±0.02 -0.08 ±0.01 26763
4 0.43 -2.01 3.43 ±0.02 3.26 ±0.01 -0.11 ±0.01 26382
5 -0.23 -1.33 3.23 ±0.03 3.00 ±0.04 -0.04 ±0.02 3145
6 -0.25 -5.70 2.61 ±0.02 2.36 ±0.03 -0.06 ±0.01 7027
7 -0.14 -5.91 2.70 ±0.03 2.43 ±0.02 -0.05 ±0.01 6236
8 10.48 -3.78 2.80 ±0.03 2.29 ±0.02 -0.08 ±0.01 5136
9 10.59 -3.98 2.73 ±0.02 2.16 ±0.03 -0.06 ±0.01 5114
10 9.64 -3.44 2.77 ±0.02 2.27 ±0.02 -0.07 ±0.01 5568
11 9.74 -3.64 2.84 ±0.02 2.32 ±0.02 -0.10 ±0.01 5369
12 7.80 -3.37 2.66 ±0.03 2.31 ±0.03 -0.08 ±0.01 6035
13 7.91 -3.58 2.66 ±0.03 2.24 ±0.02 -0.07 ±0.01 5601
14 5.23 2.81 2.97 ±0.02 2.60 ±0.02 0.04 ±0.01 10427
15 5.38 2.63 3.02 ±0.02 2.64 ±0.03 -0.00 ±0.01 8989
16 5.10 -3.29 2.87 ±0.02 2.53 ±0.02 -0.12 ±0.01 9799
17 5.28 -3.45 2.81 ±0.02 2.42 ±0.01 -0.12 ±0.01 10268
18 3.97 -3.14 2.92 ±0.02 2.62 ±0.02 -0.13 ±0.01 14019
19 4.08 -3.35 2.90 ±0.02 2.60 ±0.02 -0.17 ±0.01 13256
20 1.68 -2.47 3.27 ±0.01 2.82 ±0.01 -0.12 ±0.01 17678
21 1.80 -2.66 3.31 ±0.02 2.90 ±0.02 -0.13 ±0.01 17577
22 -0.26 -2.95 3.17 ±0.02 2.84 ±0.02 -0.01 ±0.01 19787
23 -0.50 -3.36 3.15 ±0.01 2.84 ±0.02 -0.04 ±0.01 17996
24 -2.44 -3.36 2.96 ±0.01 2.48 ±0.01 0.02 ±0.01 16397
25 -2.32 -3.56 2.91 ±0.01 2.50 ±0.01 0.02 ±0.01 16386
26 -4.90 -3.37 2.68 ±0.02 2.17 ±0.01 0.02 ±0.01 13099
27 -4.92 -3.65 2.63 ±0.02 2.15 ±0.01 0.03 ±0.01 12728
28 -6.76 -4.42 2.63 ±0.03 2.12 ±0.02 -0.01 ±0.01 8367
29 -6.64 -4.62 2.66 ±0.03 2.09 ±0.02 -0.02 ±0.01 8108
30 1.94 -2.84 3.04 ±0.02 2.70 ±0.02 -0.12 ±0.01 17774
31 2.23 -2.94 3.11 ±0.02 2.74 ±0.01 -0.12 ±0.01 17273
32 2.34 -3.14 3.10 ±0.02 2.78 ±0.01 -0.13 ±0.01 15966
33 2.35 -3.66 3.08 ±0.02 2.77 ±0.02 -0.14 ±0.01 15450
34 1.35 -2.40 3.36 ±0.02 2.92 ±0.01 -0.11 ±0.01 16889
35 3.05 -3.00 3.09 ±0.02 2.72 ±0.02 -0.14 ±0.01 15973
36 3.16 -3.20 3.19 ±0.02 2.77 ±0.02 -0.16 ±0.01 14955
37 0.00 -1.74 3.29 ±0.02 3.04 ±0.01 -0.05 ±0.01 20233
38 0.97 -3.42 3.15 ±0.01 2.84 ±0.02 -0.12 ±0.01 15542
39 0.53 -2.21 3.21 ±0.01 3.00 ±0.01 -0.07 ±0.01 24820
40 -2.99 -3.14 2.84 ±0.01 2.47 ±0.02 0.05 ±0.01 13581
41 -2.78 -3.27 2.78 ±0.01 2.41 ±0.02 0.04 ±0.01 14070
42 4.48 -3.38 2.89 ±0.02 2.63 ±0.02 -0.15 ±0.01 10099
43 0.37 2.95 3.17 ±0.02 2.87 ±0.01 0.02 ±0.01 11467
45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2380
46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 1803
Table 2. Comparison between proper motion dispersions and cross-correlation term Clb in two of the OGLE-II fields (45 and 46) with proper motion
dispersions computed from four nearby HST fields (Kozłowski et al. 2006).
Field l (◦) b (◦) σl (mas yr
−1) σb (mas yr
−1) Clb Ref
119-A 1.32 -3.77 2.89 ±0.10 2.44 ±0.08 -0.14 ±0.04 1
119-C 0.85 -3.89 2.79 ±0.10 2.65 ±0.08 -0.14 ±0.04 1
OGLE-II 45 0.98 -3.94 2.97 ±0.04 2.61 ±0.04 -0.13 ±0.02 2
119-D 1.06 -4.12 2.75 ±0.10 2.56 ±0.09 -0.05 ±0.06 1
95-BLG-11 0.99 -4.21 2.82 ±0.09 2.62 ±0.09 -0.14 ±0.04 1
OGLE-II 46 1.09 -4.14 2.90 ±0.04 2.67 ±0.04 -0.16 ±0.03 2
1Kozłowski et al. (2006) 2This work.
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 5
−10−5051015
PSfrag replacements
Galactic longitude (◦)
Figure 3. Proper motion dispersion in the Galactic longitude (σl) and lati-
tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of
field Galactic longitude. Open circles correspond to fields 6, 7, 14, 15 and
43 which have relatively extreme galactic latitudes, see Fig. 1.
used in deriving the model. Clearly one can see that the mean mo-
tion follows elliptical paths around the Galactic bar. The analysis of
OGLE-II proper motions by Sumi et al. (2003b) is consistent with
this streaming motion.
3.1 Model stellar magnitudes
The model has a four-fold symmetry, obtained by a rotation of π
radians around the vertical axis and by positioning the Sun above
or below the mid-plane. The kinematics of model particles falling
within the solid angle of each OGLE-II field were combined to
those from the three other equivalent lines-of-sight. This procedure
allows an increase in the number of model particles used for the
predictions of stellar kinematics.
We assign magnitudes to stars in the Galactic model described
above which appear in the same fields as that observed by the
OGLE collaboration. Number counts as a function of I-band ap-
parent magnitude, I , were used to compute the fraction of RCG
stars in each of the OGLE-II fields. Figure 7 shows an example of
the fitted number count function Nk(I) for one of the k = 1 . . . 49
OGLE-II fields, where Nk(I) is of the form of a power-law and a
Gaussian (Sumi 2004):
Nk(I) = ak10
(bkI) + ck exp
−(I − Ip,k)
−6 −4 −2 0 2 4
PSfrag replacements
Galactic latitude (◦)
Figure 4. Proper motion dispersion in the Galactic longitude (σl) and lati-
tude (σb) directions for 45 OGLE-II Galactic bulge fields as a function of
field Galactic latitude. Open circles correspond to fields 6, 7, 14, 15 and 43
which have relatively extreme galactic latitudes, see Fig. 1.
where the constants ak, bk, ck, Ip,k, σk are determined for each of
the k OGLE-II fields, see Table 3. The fraction Rk of RCG stars
is evaluated as the ratio of the area under the Gaussian component
of equation (6) to the area under the full expression. The integrals
are taken over ±3σk around the RCG peak in Nk(I) for each of
the k OGLE-II fields. Fields 44 and 47-49 are not included as there
are insufficient RCGs in the OGLE-II fields to fit equation (6). Fig-
ure 7 shows that the model number count function fails to fit the
observed number counts well for magnitudes I ≃ 15.4. In order to
convert stellar density to a distribution of apparent magnitude, the
relevant quantity is ρr3 (Bissantz & Gerhard 2002). Depending on
the line-of-sight, this quantity can give asymmetric magnitude dis-
tributions through the bulge. Using the best-fitting analytic tri-axial
density models for the bulge (Rattenbury et al. 2007, in prepara-
tion), this asymmetry is observed and may explain the excess of
stars in the number count histograms, compared to the best-fitting
two-component fit of equation (6). The inability of equation (6) to
model completely all features in the observed number counts in
some cases leads to an additional uncertainty in the magnitude lo-
cation of the fitted Gaussian peak. Computing the apparent magni-
tude distribution as ∝ ρr3 also produces a small shift in the peak of
the magnitude distribution. This shift is ∼ +0.04 mag for l = 0◦,
b = 0◦. The proper motion dispersions computed here are unlikely
to be sensitive to these small offsets.
c© 2005 RAS, MNRAS 000, 000–000
6 Rattenbury et al.
Table 3. Values of fitted parameters in equation (6) for all 45 OGLE-II fields used in this analysis. R is the ratio of observed RCG stars to the total number
of stars in each field, evaluated over ±3σ around the RCG peak magnitude, Ip, where σ is the fitted Gaussian spread in equation (6). The magnitudes of
the model RCG stars are shifted by ∆m to correspond with the observed mean RCG magnitude in each field. The total number of model stars in each field
assigned RCG magnitudes and colours is nrcg and the total number of model stars in each field is nall. The corresponding total model weight values for each
field are given by wrcg and wall respectively. The large values of σ for fields 8-11 might be related to their position at large positive longitudes, and could
indicate a structure such as the end of the bar, a ring or spiral arm. An analysis of the bar morphology based on these results is underway (Rattenbury et al.
2007, in preparation).
Field a b c Ip σ R ∆m nrcg nall wrcg wall
1 0.11 0.27 1735.70 14.62 0.29 0.40 0.43 585 1773 277.2 842.4
2 0.15 0.26 1876.47 14.54 -0.29 0.43 0.41 621 1802 298.1 853.9
3 0.16 0.28 4692.78 14.66 0.25 0.44 0.54 1264 3626 668.5 1911.1
4 0.17 0.28 4438.63 14.65 0.24 0.44 0.52 1298 3653 670.8 1922.2
5 0.05 0.33 4581.59 14.70 0.28 0.33 0.55 1342 4668 755.7 2685.7
6 0.04 0.27 519.71 14.57 0.37 0.34 0.36 152 583 69.5 270.8
7 0.03 0.28 457.42 14.55 0.39 0.32 0.36 143 527 71.9 243.8
8 0.04 0.27 259.65 14.37 -0.51 0.22 0.35 96 561 41.7 236.2
9 0.04 0.27 270.90 14.34 0.51 0.25 -0.05 96 497 46.1 230.9
10 0.08 0.26 321.32 14.44 0.52 0.22 0.40 131 654 49.1 260.1
11 0.04 0.28 316.25 14.45 0.50 0.23 0.28 128 695 57.5 339.4
12 0.12 0.25 546.85 14.43 0.38 0.28 0.41 238 908 100.7 393.1
13 0.10 0.25 520.45 14.45 0.37 0.29 0.15 190 863 83.9 392.4
14 0.09 0.28 1309.28 14.55 0.32 0.35 0.34 458 1587 216.0 767.4
15 0.05 0.29 1154.52 14.57 0.33 0.31 0.55 421 1661 185.2 761.8
16 0.12 0.27 1042.72 14.50 0.35 0.33 0.50 397 1383 172.8 601.1
17 0.12 0.26 1069.07 14.48 0.34 0.35 0.25 406 1443 212.4 753.4
18 0.17 0.26 1569.83 14.49 0.31 0.40 0.35 527 1564 234.7 702.4
19 0.17 0.26 1429.23 14.51 0.32 0.40 0.44 434 1365 184.4 608.5
20 0.20 0.27 3012.09 14.58 0.26 0.42 0.53 939 2728 480.3 1398.3
21 0.15 0.27 2793.36 14.58 0.26 0.43 0.45 900 2554 443.5 1260.0
22 0.12 0.28 2574.77 14.74 0.28 0.42 0.51 830 2419 382.5 1113.3
23 0.09 0.28 2147.71 14.73 0.29 0.42 0.47 767 2126 384.2 1060.6
24 0.12 0.27 2130.41 14.82 0.28 0.42 0.50 595 1864 269.6 905.4
25 0.07 0.28 2002.91 14.82 0.28 0.42 0.51 581 1782 289.5 885.1
26 0.09 0.27 1452.89 14.83 0.31 0.38 0.55 375 1325 159.7 570.5
27 0.07 0.27 1319.67 14.81 0.32 0.39 0.40 387 1238 172.5 578.9
28 0.04 0.28 563.00 14.79 0.31 0.31 0.62 162 649 72.3 293.5
29 0.05 0.27 559.86 14.78 0.31 0.32 0.44 156 607 70.7 267.5
30 0.18 0.27 2533.75 14.57 0.27 0.42 0.41 754 2195 362.4 1026.7
31 0.17 0.27 2354.64 14.53 0.28 0.43 0.32 763 2229 361.9 1122.1
32 0.17 0.26 2062.96 14.53 0.28 0.42 0.41 638 1962 291.8 938.5
33 0.13 0.27 1614.83 14.56 0.31 0.41 0.34 559 1586 265.5 760.7
34 0.18 0.27 3210.56 14.60 0.27 0.43 0.42 990 2936 503.0 1473.9
35 0.16 0.26 1963.53 14.53 0.29 0.41 0.45 663 1925 307.7 913.7
36 0.16 0.26 1773.62 14.51 0.30 0.41 0.47 574 1902 301.1 943.5
37 0.18 0.28 4901.22 14.64 0.25 0.42 0.43 1439 4077 794.9 2218.5
38 0.12 0.27 2091.19 14.64 0.28 0.43 0.46 662 1945 319.2 948.1
39 0.18 0.28 3919.30 14.69 0.26 0.44 0.65 1217 3456 631.8 1804.2
40 0.09 0.28 2181.18 14.87 0.29 0.41 0.62 668 1936 315.1 933.3
41 0.10 0.28 2180.49 14.87 0.28 0.42 0.55 626 1905 318.2 965.4
42 0.13 0.26 1215.38 14.52 0.35 0.37 0.40 425 1389 190.2 637.7
43 0.10 0.28 2659.91 14.84 0.27 0.41 0.79 777 2290 345.8 1074.6
45 0.11 0.27 1541.36 14.59 0.31 0.40 0.38 485 1568 228.3 767.7
46 0.09 0.27 1428.63 14.60 0.30 0.41 0.38 454 1400 221.6 669.5
Each star in the galactic model is assigned a RCG magnitude
with probability Rk for each field. The apparent magnitude is com-
puted using the model distance. Stars which are not assigned a RCG
magnitude are assigned a magnitude using the power-law compo-
nent of equation (6), defined over the same limits used to compute
Rk. Here we implicitly assume that the RCG stars trace the overall
Galactic disk and bulge populations.
The RCG luminosity function is approximated by a Gaussian
distribution with mean magnitude −0.26 and σ = 0.2. These as-
sumptions are mostly consistent with observations (Stanek et al.
1997) and the fitted distribution from Udalski (2000), but there may
be small offsets between local and bulge red clump giants. It was
noted in Sumi (2004) that there is some as-yet unexplained off-
set (0.3 mag) in the extinction-corrected mean RCG magnitudes in
the OGLE fields. A possible explanation for this offset is that the
RCG population effects are large: so that the absolute magnitude
of RCG stars is significantly different for RCGs in the bulge com-
pared to local RCGs, as claimed by Percival & Salaris (2003) and
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 7
−10−5051015
−0.15
−0.05
−6 −4 −2 0 2 4
−0.15
−0.05
PSfrag replacements
Galactic longitude (◦)
Galactic latitude (◦)
Figure 5. Cross-correlation term Clb for 45 OGLE-II Galactic bulge fields
as a function of field Galactic longitude (top) and latitude (bottom). Open
circles in the top plot of Clb vs. l correspond to fields 6, 7, 14, 15 and 43
which have relatively extreme galactic latitudes, see Fig. 1.
Salaris et al. (2003). A different value of the distance to the Galac-
tic centre to that assumed here (8 kpc) would in part account for
the discrepancy, however would not remove it completely. Using a
value of 7.6 kpc (Eisenhauer et al. 2005; Nishiyama et al. 2006) as
the distance to the Galactic centre would change the zero-point by
0.12 mag, resulting in an offset value of 0.18 mag. It is also possi-
ble that reddening toward the Galactic centre is more complicated
than assumed in Sumi (2004). In order to compare the model proper
motion results with the observed data, it was necessary to shift the
mean model RCG magnitudes to correspond with that observed in
each of the OGLE fields. The model RCG magnitudes were fitted
with a Gaussian curve. The mean of the model RCG magnitudes
was then shifted by a value ∆m, see Table 3, to correspond with
the observed mean RCG magnitude in each of the OGLE fields. No-
tice that we concentrate on second-order moments (proper motion
dispersions) of the proper motion, so a small shift in the zero-point
has little effect on our results.
Every model particle has an associated weight, wi. The par-
ticle weight can take values 0 < wi . 20. In order to account
for this weighting, ⌈wi⌉ stars are generated for each particle with
the same kinematics but magnitudes determined as above. ⌈wi⌉ is
the nearest integer toward +∞. Each model star is then assigned
a weight, γi = wi/⌈wi⌉. Notice this procedure allows us to in-
crease the effective number of particles to better sample the lumi-
nosity function. The total number of stars and the number of stars
assigned RCG magnitudes in each field are listed in Table 3 as nall
Figure 6. Galactic kinematics from the model of Debattista et al (2007, in
preparation). Bulk stellar motion in the mid-plane of the Galaxy is shown
super-imposed on the stellar density. The Sun is located at the origin (not
shown). An example line of sight is shown. The model can be rotated to four
equivalent positions for each line of sight due to symmetry (see section 3.1).
12 12.5 13 13.5 14 14.5 15 15.5 16 16.5
PSfrag replacements
stars
non-RCG stars
Figure 7. Number count as function of apparent magnitude, I , for OGLE-II
field 1. The number count histogram is shown along with the fitted function
equation (6). The fraction of RCG stars, Rk , is evaluated over the magni-
tude range Ip ±3σ for each of the (k = 1 . . . 49) OGLE-II fields. The ratio
Rk is assumed to be the same at all stellar distances for each field.
and nrcg respectively. 81806 stars from the model were used to
compare model kinematics to observed values.
3.2 Model kinematics
Stars with apparent magnitudes within the limits mmin = 13.7 and
mmax = 15.5, were selected from the model data. This magnitude
range corresponds to the selection criteria imposed on the observed
data sample, see section 2.2. Model stars with total proper motions
greater than 10 mas yr−1 (corresponding to > 380 kms−1 at a
c© 2005 RAS, MNRAS 000, 000–000
8 Rattenbury et al.
distance of the Galactic centre) were excluded on the basis that
such stars would be similarly excluded from any observed sample.
The fraction of weight removed and number of stars removed in
this way only amounted to a few per cent of the total weight and
number of stars in each field. Bulge model stars were selected by
requiring a distance d > 6 kpc.
The mean proper motion and proper motion dispersions in the
latitude and longitude directions were computed along with their
errors for all model stars in each field which obey the above selec-
tion criteria. The weights on model stars, γi, were used to compute
these values.
We then tested whether the finite and discrete nature of the
model data gives rise to uncertainties in the measured proper mo-
tion dispersion values. We measured the intrinsic noise in the model
by comparing the proper motion dispersions computed for four
equivalent lines-of-sight through the model for each field. The
spread of the proper motion dispersions for each field was then used
as the estimate of the intrinsic noise in the model. The mean (me-
dian) value of these errors in the longitude and latitude directions
are 0.08 (0.06) and 0.12 (0.097) mas yr−1 respectively.
The statistical error for the proper motion dispersions in the
longitude and latitude directions for each field were combined in
quadrature with the error arising from the finite discrete nature of
the model data to give the total error on the proper motion disper-
sions computed from the model.
4 COMPARISON BETWEEN THEORETICAL MODEL
AND OBSERVED DATA
The observed and predicted proper motion dispersions for each of
the OGLE-II fields are shown in Table 4. Fig. 8 shows the observed
proper motion dispersions for each of the analysed OGLE-II fields
plotted against the predicted model proper motion dispersions.
Fig. 8 shows that the model predictions are in general agree-
ment with observed proper motion dispersions for the OGLE-
II fields. The model has been used previously to predict the
proper motion dispersions of 427 stars1 entries observed by
Spaenhauer et al. (1992) in a single 6′× 6′ field toward the bulge
(Bissantz et al. 2004). The model value of σl in this previous anal-
ysis was in agreement with the observed value, yet the model and
observed values of σb were significantly different. The 6
′× 6′ field
used by Spaenhauer et al. (1992) falls within the OGLE-II field
number 45. The model prediction of σl for stars in OGLE field 45
is completely consistent with the measured value. The model pre-
diction of σb shows a similar discrepancy to the previous analysis
of Bissantz et al. (2004).
Fig. 9 shows the ratio R = σl/σb and cross-correlation term
Clb = σlb/(σlσb) computed using the model and observed data.
Typically the model predicts more anisotropic motion with R > 1
than what is observed.
The model predictions for stellar kinematics in the latitude di-
rection may be problematic. This is not surprising as the model is
not well constrained toward the plane due to a lack of observational
data because of the heavy dust extinction. The problem is currently
under investigation. Similarly, the model predictions for σl degrade
as l increases. This is because the model performance has been op-
timised for regions close to the Galactic centre.
1 There are two repeated entries in Table 2 of Spaenhauer et al. (1992).
The significant difference between the observed proper mo-
tion dispersions of adjacent fields (e.g. fields 1 and 45) might hint at
some fine-scale population effect, whereby a group of stars surviv-
ing the selection criteria have a significant and discrepant kinematic
signature. Higher-accuracy observations using the HST support this
evidence of such population effects (Kozłowski et al. 2006).
No attempt has been made to account for the blending of flux
inherent in the OGLE-II crowded-field photometry. It is certain
that a fraction of stars in each OGLE-II field suffers from some
degree of blending (Kozłowski et al. 2006). To investigate this ef-
fect, we checked one field covering the lens MACHO-95-BLG-
37 (l = 2.54◦, b = 3.33◦, Thomas et al. 2005) from the HST
proper motion survey of Kozłowski et al. (2006), which falls inside
OGLE-II field number 2. HST images suffer much less blending,
but the field of view is small, and so it has only a dozen or so clump
giants. We derive a proper motion of σl = 3.13 ± 0.57 mas yr
and σb = 2.17±0.40 mas yr
−1. These values agree with our kine-
matics in field 2 within 0.2σ for σl and 1.6σ for σb. The errors in
our proper motion dispersions are very small (∼ km s−1 at a dis-
tance of the Galactic centre), but it is likely that we underestimate
the error bars on the observed data due to systematic effects such
as blending.
4.1 Understanding the differences
We now seek to understand the cause of the differences between
the model and the Milky Way, at least at a qualitative level. We
first consider the possibility that the difference can be explained by
some systematic effect. We compute the differences between ob-
served proper motion dispersions of nearest fields for fields with
separations less than 0.25 degrees. No pair of fields is used twice,
and the difference ∆ = σi − σj is always plotted such that
|bi| ≥ |bj |. ∆l,obs and ∆b,obs denote the difference in observed
proper motion dispersions between adjacent fields in the longitude
and latitude directions respectively. The equivalent quantities pre-
dicted from the model are denoted ∆l,mod and ∆b,mod. In Fig. 10
we see that the deviations ∆l,obs and ∆b,obs scatter about 0, but
have a quite broad distribution in both the l and b directions, with
several fields inconsistent with zero difference at 1σ (defined as the
sum in quadrature of the uncertainties of the corresponding quanti-
ties of the two fields under comparison). Several deviations are as
large as 0.2 mas yr−1 (corresponding to ≃ 8 kms−1 at the Galac-
tic centre) and many σ’s away from zero. In view of the fact that
these differences have mean close to zero, it is possible that these
deviations are due to some systematic effect rather than to intrinsic
substructure in the Milky Way. We return to this point briefly in the
discussion.
In the case of the model uncertainties, however, Fig. 11 shows
that in most cases the differences ∆l,mod and ∆b,mod are consistent
with zero at the 1σ level, indicating that these error estimates are
robust.
We now seek to explore the correlations of the residuals with
properties of the model. We plot residuals δl,b = (σmod − σobs),
where σmod and σobs are the model and observed proper motion
dispersions in the corresponding Galactic co-ordinate. The errorbar
length is (u2mod+u
1/2 where umod and uobs are the uncertain-
ties in the model and observed proper motion dispersions, respec-
tively. Plotting these quantities as a function of l, we note that there
is no significant correlation, but that the largest deviations in the
latitude proper motion dispersion occur close to l = 0, see Fig. 12.
In plotting δl,b as a function of b, the reason which becomes evi-
dent is that the fields closest to the mid-plane have the largest δb,
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 9
2.6 2.8 3 3.2 3.4
27 28
PSfrag replacements
Model σl (mas yr
Model σb (mas yr
Observed σb (mas yr
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
11 12
24 25
35 36
46PSfrag replacements
Model σl (mas yr
Observed σl (mas yr
Model σb (mas yr
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
Figure 8. Comparison between observed and predicted proper motion dispersions for stars in the OGLE-II proper motion catalogue of Sumi et al. (2004). Left:
Proper motion dispersions in the galactic longitude direction, σl. The OGLE-II field number is indicated adjacent to each point, see also Fig. 1. Fields with
galactic longitude |l| > 5◦ are shown in magenta; fields within Baade’s window are shown in red; all other fields in blue. Right: Proper motion dispersions in
the galactic latitude direction, σb, shown with the same colour scheme.
0.8 1 1.2 1.4 1.6 1.8
PSfrag replacements
Model σl/σb
Model Clb
Observed Clb
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
−0.1 0 0.1
PSfrag replacements
Model σl/σb
Observed σl/σb
Model Clb
BW fields
|l|>5◦
|l|<5◦
BW fields
|l|>5◦
|l|<5◦
Figure 9. Left: Ratio of proper motion dispersions R = σl/σb for the observed OGLE-II proper motion data and model predictions. The model generally
predicts more anisotropic motion, i.e. R > 1 than is observed in the data. Right: The cross-correlation term Clb = σlb/σlσb.
see Fig. 13. The density distribution in this region is uncertain due
to presence of dust and the large extinction corrections required.
This may explain why the residuals of σb seem to correlate more
with b than those of σl. We note that the σl residuals also seem to
have some dependence on b. A possible explanation is that there is
some additional effect due to dust which has not been accounted
5 DISCUSSION
Red clump giant stars in the dense fields observed by the OGLE-II
microlensing survey can be used as tracers of the bulge density and
motion over a large region toward the Galactic centre. The proper
motion dispersions of bulge RCG stars in the OGLE-II proper mo-
tion catalogue of Sumi et al. (2004) were calculated for 45 OGLE-
II fields. The kinematics derived from the ground-based OGLE-II
data were found to be in agreement with HST observations in two
fields from Kozłowski et al. (2006). It is reassuring that the results
presented here are consistent with those derived from the higher
resolution HST data, despite possible selection effects and blend-
The observed values of σl and σb were compared to predic-
tions from the made-to-measure stellar-dynamical model of De-
battista et al. (2007, in preparation). In general, the model gives
predictions qualitatively similar to observed values of σl and σb
for fields close to the Galactic centre. The model is in agree-
ment with observed OGLE-II data in the direction previously tested
by Bissantz et al. (2004). Using the definition of De Lorenzi et al.
(2007), the effective number of particles in the model used here
c© 2005 RAS, MNRAS 000, 000–000
10 Rattenbury et al.
Table 4. Proper motion dispersions in the longitude and latitude directions, σl, σb , and cross-correlation term Clb for bulge stars in 45 OGLE-II fields. High
precision proper motion data for bulge stars were extracted from the OGLE-II proper motion catalogue (Sumi et al. 2004). N is the number of stars selected
from each field. Field 44 was not used due to the low number of RCGs in this field.
PM Dispersions (mas yr−1) Clb
Field Field centre Longitude σl Latitude σb
l (◦) b (◦) Model Observed Model Observed Model Observed N
1 1.08 -3.62 3.02 ±0.08 3.10 ±0.02 2.46 ±0.08 2.83 ±0.02 0.01 ±0.08 -0.13 ±0.01 15434
2 2.23 -3.46 3.02 ±0.06 3.21 ±0.02 2.68 ±0.15 2.80 ±0.02 0.12 ±0.03 -0.14 ±0.01 16770
3 0.11 -1.93 3.19 ±0.08 3.40 ±0.01 2.64 ±0.02 3.30 ±0.02 0.02 ±0.01 -0.08 ±0.01 26763
4 0.43 -2.01 3.26 ±0.05 3.43 ±0.02 2.80 ±0.06 3.26 ±0.01 0.01 ±0.03 -0.11 ±0.01 26382
5 -0.23 -1.33 3.22 ±0.15 3.23 ±0.03 2.30 ±0.07 3.00 ±0.04 -0.01 ±0.03 -0.04 ±0.02 3145
6 -0.25 -5.70 3.26 ±0.16 2.61 ±0.02 2.42 ±0.23 2.36 ±0.03 -0.02 ±0.13 -0.06 ±0.01 7027
7 -0.14 -5.91 2.95 ±0.15 2.70 ±0.03 2.49 ±0.12 2.43 ±0.02 -0.15 ±0.16 -0.05 ±0.01 6236
8 10.48 -3.78 3.07 ±0.09 2.80 ±0.03 1.98 ±0.14 2.29 ±0.02 0.00 ±0.09 -0.08 ±0.01 5136
9 10.59 -3.98 3.28 ±0.21 2.73 ±0.02 2.03 ±0.07 2.16 ±0.03 -0.03 ±0.07 -0.06 ±0.01 5114
10 9.64 -3.44 3.30 ±0.32 2.77 ±0.02 2.89 ±0.62 2.27 ±0.02 0.09 ±0.08 -0.07 ±0.01 5568
11 9.74 -3.64 3.01 ±0.20 2.84 ±0.02 2.22 ±0.29 2.32 ±0.02 -0.08 ±0.09 -0.10 ±0.01 5369
12 7.80 -3.37 3.31 ±0.10 2.66 ±0.03 2.29 ±0.06 2.31 ±0.03 -0.09 ±0.06 -0.08 ±0.01 6035
13 7.91 -3.58 3.26 ±0.18 2.66 ±0.03 2.29 ±0.12 2.24 ±0.02 0.05 ±0.02 -0.07 ±0.01 5601
14 5.23 2.81 3.21 ±0.05 2.97 ±0.02 2.62 ±0.13 2.60 ±0.02 0.06 ±0.04 0.04 ±0.01 10427
15 5.38 2.63 3.31 ±0.12 3.02 ±0.02 2.46 ±0.07 2.64 ±0.03 0.04 ±0.04 -0.00 ±0.01 8989
16 5.10 -3.29 3.19 ±0.07 2.87 ±0.02 2.23 ±0.08 2.53 ±0.02 0.03 ±0.08 -0.12 ±0.01 9799
17 5.28 -3.45 3.09 ±0.09 2.81 ±0.02 2.50 ±0.07 2.42 ±0.01 -0.01 ±0.11 -0.12 ±0.01 10268
18 3.97 -3.14 3.20 ±0.09 2.92 ±0.02 2.48 ±0.08 2.62 ±0.02 0.02 ±0.03 -0.13 ±0.01 14019
19 4.08 -3.35 3.06 ±0.13 2.90 ±0.02 2.49 ±0.26 2.60 ±0.02 0.01 ±0.03 -0.17 ±0.01 13256
20 1.68 -2.47 3.12 ±0.06 3.27 ±0.01 2.66 ±0.05 2.82 ±0.01 0.07 ±0.03 -0.12 ±0.01 17678
21 1.80 -2.66 3.12 ±0.06 3.31 ±0.02 2.57 ±0.08 2.90 ±0.02 -0.02 ±0.03 -0.13 ±0.01 17577
22 -0.26 -2.95 3.17 ±0.04 3.17 ±0.02 2.46 ±0.12 2.84 ±0.02 0.01 ±0.03 -0.01 ±0.01 19787
23 -0.50 -3.36 3.13 ±0.17 3.15 ±0.01 2.62 ±0.10 2.84 ±0.02 -0.02 ±0.14 -0.04 ±0.01 17996
24 -2.44 -3.36 2.77 ±0.04 2.96 ±0.01 2.32 ±0.10 2.48 ±0.01 -0.04 ±0.04 0.02 ±0.01 16397
25 -2.32 -3.56 2.76 ±0.07 2.91 ±0.01 2.47 ±0.15 2.50 ±0.01 -0.04 ±0.03 0.02 ±0.01 16386
26 -4.90 -3.37 2.80 ±0.17 2.68 ±0.02 2.22 ±0.04 2.17 ±0.01 -0.00 ±0.03 0.02 ±0.01 13099
27 -4.92 -3.65 2.78 ±0.07 2.63 ±0.02 2.19 ±0.04 2.15 ±0.01 -0.06 ±0.02 0.03 ±0.01 12728
28 -6.76 -4.42 3.02 ±0.11 2.63 ±0.03 2.44 ±0.36 2.12 ±0.02 0.05 ±0.06 -0.01 ±0.01 8367
29 -6.64 -4.62 3.02 ±0.21 2.66 ±0.03 1.79 ±0.14 2.09 ±0.02 -0.00 ±0.11 -0.02 ±0.01 8108
30 1.94 -2.84 3.13 ±0.07 3.04 ±0.02 2.59 ±0.11 2.70 ±0.02 -0.04 ±0.08 -0.12 ±0.01 17774
31 2.23 -2.94 3.08 ±0.05 3.11 ±0.02 2.68 ±0.11 2.74 ±0.01 0.08 ±0.05 -0.12 ±0.01 17273
32 2.34 -3.14 3.10 ±0.09 3.10 ±0.02 2.56 ±0.04 2.78 ±0.01 0.11 ±0.02 -0.13 ±0.01 15966
33 2.35 -3.66 2.82 ±0.11 3.08 ±0.02 2.57 ±0.12 2.77 ±0.02 0.07 ±0.06 -0.14 ±0.01 15450
34 1.35 -2.40 3.18 ±0.06 3.36 ±0.02 2.62 ±0.03 2.92 ±0.01 0.04 ±0.02 -0.11 ±0.01 16889
35 3.05 -3.00 3.05 ±0.05 3.09 ±0.02 2.59 ±0.07 2.72 ±0.02 0.08 ±0.03 -0.14 ±0.01 15973
36 3.16 -3.20 3.00 ±0.06 3.19 ±0.02 2.95 ±0.40 2.77 ±0.02 -0.05 ±0.08 -0.16 ±0.01 14955
37 0.00 -1.74 3.29 ±0.04 3.29 ±0.02 2.70 ±0.04 3.04 ±0.01 -0.01 ±0.01 -0.05 ±0.01 20233
38 0.97 -3.42 3.01 ±0.07 3.15 ±0.01 2.60 ±0.14 2.84 ±0.02 0.07 ±0.07 -0.12 ±0.01 15542
39 0.53 -2.21 3.22 ±0.03 3.21 ±0.01 2.69 ±0.06 3.00 ±0.01 0.01 ±0.04 -0.07 ±0.01 24820
40 -2.99 -3.14 2.84 ±0.04 2.84 ±0.01 2.28 ±0.07 2.47 ±0.02 -0.09 ±0.07 0.05 ±0.01 13581
41 -2.78 -3.27 2.86 ±0.06 2.78 ±0.01 2.60 ±0.19 2.41 ±0.02 -0.16 ±0.07 0.04 ±0.01 14070
42 4.48 -3.38 3.07 ±0.05 2.89 ±0.02 2.44 ±0.15 2.63 ±0.02 0.02 ±0.02 -0.15 ±0.01 10099
43 0.37 2.95 3.13 ±0.06 3.17 ±0.02 2.72 ±0.10 2.87 ±0.01 0.04 ±0.07 0.02 ±0.01 11467
45 0.98 -3.94 3.02 ±0.05 2.97 ±0.04 2.42 ±0.14 2.61 ±0.04 0.06 ±0.11 -0.13 ±0.02 2380
46 1.09 -4.14 2.87 ±0.08 2.90 ±0.04 2.53 ±0.21 2.67 ±0.04 -0.03 ±0.06 -0.16 ±0.03 1803
is 3986. This relatively low number results in large errors on the
model proper motion dispersions and we therefore recommend re-
garding interpretations based on this model with some caution. An
improved model with a larger number of particles (the recent study
by De Lorenzi et al. (2007) has an effective particle number ∼ 106)
will decrease the errors on the model predictions and allow a more
useful comparison between model and observed proper motion dis-
persions.
The OGLE-II fields mostly extend over ∼ 17◦ in longi-
tude and about 5◦ in latitude across the Galactic bulge region
and can therefore provide a more powerful set of constraints on
stellar motions predicted by galactic models. The high-accuracy
proper motion data for the 45 fields and those obtained with HST
(Kozłowski et al. 2006) can be used as direct input in the made-to-
measure method to construct a better constrained dynamical model
of the Milky Way.
The statistical errors of our proper motion dispersions are
small (∼ km s−1), but systematic uncertainties (for example due
to incorrect dust extinction treatment) which were not included in
the analysis may be significant. Nevertheless, it is interesting to
note that there appears to be significant difference between the ob-
served proper motion dispersions of adjacent fields (e.g. fields 1
c© 2005 RAS, MNRAS 000, 000–000
Proper Motion Dispersions of Red Clump Giants in the Galactic Bulge 11
0.22 0.23 0.24 0.25
PSfrag replacements
Separation (◦)
Figure 10. Difference between observed proper motion dispersions for
pairs of fields with separations less than 0.25 degrees (corresponding to
≃ 40 pc at the Galactic centre).
and 45). This might hint at some fine-scale population effect,where
the kinematics of the bulge may be not in total equilibrium (e.g. due
to a small accretion event). Higher-accuracy observations using the
HST may provide further evidence of such population effects. We
note that Rich et al. (2006) suggest the possible existence of cold
structures using data from a radial velocity survey of Galactic bulge
M giant stars although their conclusion could be strengthened by a
larger sample of stars.
The OGLE-II proper motion catalogue (Sumi et al. 2004) for
millions of bulge stars is still somewhat under-explored. For exam-
ple, it will be interesting to explore the nature of the high proper
motion stars (µ > 10 mas yr−1) and search for wide binaries in
the catalogue. Some exploration along these lines is under way.
ACKNOWLEDGEMENTS
We thank Drs. Vasily Belokurov, Wyn Evans and Martin Smith for
helpful discussions, and the anonymous referee for their helpful
suggestions.
NJR acknowledges financial support by a PPARC PDRA
fellowship. This work was partially supported by the European
Community’s Sixth Framework Marie Curie Research Training
Network Programme, Contract No. MRTN-CT-2004-505183 ‘AN-
GLES’. VPD is supported by a Brooks Prize Fellowship at the Uni-
0.22 0.23 0.24 0.25
PSfrag replacements
Separation (◦)
Figure 11. Difference between model proper motion dispersions for pairs
of fields with separations less than 0.25 degrees.
versity of Washington and receives partial support from NSF ITR
grant PHY-0205413.
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Introduction
Observed Proper Motion Dispersions
Red Clump Giants
OGLE-II proper motion data
Extinction
Results
Galactic Model
Model stellar magnitudes
Model kinematics
Comparison between theoretical model and observed data
Understanding the differences
Discussion
|
0704.1620 | Intramolecular long-range correlations in polymer melts: The segmental
size distribution and its moments | Intramolecular long-range correlations in polymer melts:
The segmental size distribution and its moments
J.P. Wittmer,∗ P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel†
Institut Charles Sadron, CNRS, 23 rue du Loess, 67037 Strasbourg Cédex, France
(Dated: November 1, 2018)
Abstract
Presenting theoretical arguments and numerical results we demonstrate long-range intrachain
correlations in concentrated solutions and melts of long flexible polymers which cause a systematic
swelling of short chain segments. They can be traced back to the incompressibility of the melt
leading to an effective repulsion u(s) ≈ s/ρR3(s) ≈ ce/
s when connecting two segments together
where s denotes the curvilinear length of a segment, R(s) its typical size, ce ≈ 1/ρb3e the “swelling
coefficient”, be the effective bond length and ρ the monomer density. The relative deviation of
the segmental size distribution from the ideal Gaussian chain behavior is found to be proportional
to u(s). The analysis of different moments of this distribution allows for a precise determination
of the effective bond length be and the swelling coefficient ce of asymptotically long chains. At
striking variance to the short-range decay suggested by Flory’s ideality hypothesis the bond-bond
correlation function of two bonds separated by s monomers along the chain is found to decay
algebraically as 1/s3/2. Effects of finite chain length are considered briefly.
PACS numbers: 61.25.Hq,64.60.Ak,05.40.Fb
∗Electronic address: [email protected]
†URL: http://www-ics.u-strasbg.fr/~etsp/welcome.php
http://arxiv.org/abs/0704.1620v2
mailto:[email protected]
http://www-ics.u-strasbg.fr/~etsp/welcome.php
I. FLORY’S IDEALITY HYPOTHESIS REVISITED
A cornerstone of polymer physics. Polymer melts are dense disordered systems con-
sisting of macromolecular chains [1]. Theories that predict properties of chains in a melt
or concentrated solutions generally start from the “Flory ideality hypothesis” formulated
already in the 1940s by Flory [2, 3, 4]. This cornerstone of polymer physics states that
chain conformations correspond to “ideal” random walks on length scales much larger than
the monomer diameter [1, 4, 5, 6]. The commonly accepted justification of this mean-field
result is that intrachain and interchain excluded volume forces compensate each other if
many chains strongly overlap which is the case for three-dimensional melts [5]. Since these
systems are essentially incompressible, density fluctuations are known to be small. Hence,
all correlations are supposed to be short-ranged as has been systematically discussed first
by Edwards who developed the essential statistical mechanical tools [6, 7, 8, 9, 10] also used
in this paper.
One immediate consequence of Flory’s hypothesis is that the mean-squared size of chain
segments of curvilinear length s = m − n (with 1 ≤ n < m < N) should scale as R2e(s) ≡
〈r2〉 = b2es if the two monomers n and m on the same chain are sufficiently separated along
the chain backbone, and local correlations may be neglected (1 ≪ s). For the total chain
(s = N−1 ≫ 1) this implies obviously that R2e(N−1) = b2e(N−1) ≈ b2eN . Here, N denotes
the number of monomers per chain, r the end-to-end vector of the segment, r = ||r|| its
length and be the “effective bond length” of asymptotically long chains [6]. (See Fig. 1 for
an illustration of some notations used in this paper.) For the 2p-th moment (p = 0, 1, 2, . . .)
of the segmental size distribution G(r, s) in three dimensions one may write more generally
Kp(s) ≡ 1−
(2p+ 1)!
〈r2p〉
(b2es)
= 0 (1)
which is, obviously, consistent with a Gaussian segmental size distribution
G0(r, s) =
2πsb2e
. (2)
Both equations are expected to hold as long as the moment is not too high for a given
segment length and the finite-extensibility of the polymer strand remains irrelevant [6].
Deviations caused by the segmental correlation hole effect. Recently, Flory’s hypothe-
sis has been challenged both theoretically [11, 12, 13, 14, 15] and numerically for three-
dimensional solutions [16, 17, 18, 19, 20] and ultrathin films [21, 22]. These studies suggest
that intra- and interchain excluded volume forces do not fully compensate each other on
intermediate length scales, leading to long-range intrachain correlations. The general phys-
ical idea behind these correlations is related to the “segmental correlation hole” of a typical
chain segment [19]. As sketched in Fig. 2, this induces an effective repulsive interaction when
bringing two segments together, and swells (to some extent) the chains causing, hence, a
systematic violation of Eq. (1). Elaborating and clarifying various points already presented
briefly elsewhere [18, 19, 20], we focus here on melts of long and flexible polymers. Us-
ing two well-studied coarse-grained polymer models [23] various intrachain properties are
investigated numerically as functions of s and compared with predictions from first-order
perturbation theory. (For a discussion of intrachain correlations in reciprocal space see
Refs. [14, 15, 19].)
Central results tested in this study. The key claim verified here concerns the deviation
δG(r, s) = G(r, s) − G0(r, s) of the segmental size distribution G(r, s) from Gaussianity,
Eq. (2), for asymptotically long chains (N → ∞) in the “scale-free regime” (1 ≪ s ≪ N).
We show that the relative deviation divided by ce/
s scales as a function f(n) of n = r/be
δG(r, s)/G0(r, s)
= f(n) =
+ 9n−
. (3)
As we shall see, this scaling holds indeed for sufficiently large segment size r and curvilinear
length s. The indicated “swelling coefficient” ce has been predicted analytically,
24/π3
(ρ being the monomer number density), where we shall argue that the bond length of the
Gaussian reference chain of the perturbation calculation must be renormalized to the effective
bond length be. Accepting Eq. (3) the swelling of the segment size is readily obtained by
computing 〈r2p〉 = 4π
dr r2+2pG(r, s). For the 2p-th moment this yields
Kp(s) =
3(2pp!p)2
2(2p+ 1)!
. (5)
For instance, for the second moment (p = 1) this reduces to K1(s) = 1 − Re(s)2/b2es =
s ≈ ce/
s. We have replaced in Eq. (5) the theoretically expected swelling coefficient
ce by empirically determined coefficients cp. It will be shown, however, that cp/ce is close
to unity for all moments. Effectively, this reduces Eq. (5) to an efficient one-parameter
extrapolation formula for the effective bond length be of asymptotically long chains albeit
empirical and theoretical swelling coefficients may slightly differ. While we show how be may
be fitted, no attempt is made to predict it from the operational model parameters and other
measured properties such as the microscopic structure or the bulk compression modulus
[6, 9, 10].
Outline. We begin our discussion by sketching the central theoretical ideas in Sec. II.
There we will give a simple scaling argument and outline very briefly some elements of the
standard perturbation calculations we have performed to derive them (Sec. II B). Details of
the analytical treatment are relegated to Appendix A. The numerical models and algorithms
allowing the computation of dense melts containing the large chain lengths needed for a clear-
cut test are presented in Sec. III. Our computational results are given in Sec. IV. While
focusing on long chains in dense melts, we explain also briefly effects of finite chain size. The
general background of this work and possible consequences for other problems of polymer
science are discussed in the final Sec. V.
II. PHYSICAL IDEA AND SKETCH OF THE PERTURBATION CALCULATION
A. Scaling arguments
Incompressibility and correlation of composition fluctuations. Polymer melts are essen-
tially incompressible on length scales large compared to the monomer diameter, and the
density ρ of all monomers does not fluctuate. On the other hand, composition fluctuations
of labeled chains or subchains may certainly occur, however, subject to the total density
constraint. Composition fluctuations are therefore coupled and segments feel an entropic
penalty when their distance becomes comparable to their size [12, 19]. As sketched in
Fig. 2(a), we consider two independent test chains of length s in a melt of very long chains
(N → ∞). If s is sufficiently large, their typical size, R(s) ≈ be
s, is set by the effective
bond length be of the surrounding melt (taking apart finite chain-size effects). The test
chains interact with each other directly and through the density fluctuations of the sur-
rounding melt. The scaling of their effective interaction may be obtained from the potential
of mean force U(r, s) ≡ − ln(g(r, s)/g(∞, s)) where g(r) is the probability to find the second
chain at a distance r assuming the first segment at the origin (r = 0). Since the correlation
hole is shallow for large s, expansion leads to U(r, s) ≈ 1− g(r, s)/g(∞, s) ≈ c(r, s)/ρ with
c(r, s) being the density distribution of a test chain around its center of mass. This distri-
bution scales as c(r ≈ 0, s) ≈ s/R(s)d close to the center of mass (d being the dimension of
space) and decays rapidly at distances of order R(s) [5]. Hence, the interaction strength at
r/R(s) ≪ 1 is set by u(s) ≡ U(0, s) ≡ c(0, s)/ρ ≈ s/ρR(s)d ∼ s1−d/2 [12, 19]. Interestingly,
u(s) does not depend explicitly on the bulk compression modulus v. It is dimensionless and
independent of the definition of the monomer unit, i.e. it does not change if λ monomers
are regrouped to form an effective monomer (ρ → ρ/λ, s → s/λ) while keeping the segment
size R fixed.
Connectivity and swelling. To connect both test chains to form a chain of length 2s
the effective energy u(s) has to be paid and this repulsion will push the half-segments
apart. We consider next a segment of length s in the middle of a very long chain. All
interactions between the test segment and the rest of the chain are first switched off but we
keep all other interactions, especially within the segment and between the segment monomers
and monomers of surrounding chains. The typical size R(s) of the test segment remains
essentially unchanged from the size of an independent chain of same strand length. If we
now switch on the interactions between the segment and monomers on adjacent segments
of same length s, this corresponds to an effective interaction of order u(s) as before. (The
effect of switching on the interaction to all other monomers of the chain is inessential at
scaling level, since these other monomers are more distant.) Since this repels the respective
segments from each other, the corresponding subchain is swollen compared to a Gaussian
chain of non-interacting segments. It is this effect we want to characterize.
Perturbation approach in three dimensions. In the following we will exclusively consider
chain segments s which are much larger than the number of monomers g ≡ 1/vρ contained
in a blob [5], i.e. we will look on a scale where incompressibility matters. (The number g
is also sometimes called “dimensionless compressibility” [14].) Interestingly, when taken at
s = g the interaction strength takes the value
u(s = g) ≈
ρbdeg
(vρ)d/2−1
≈ Gz (6)
with Gz being the standard Ginzburg parameter used for the perturbation calculation
of strongly interacting polymers [6]. Hence, the segmental correlation hole potential
u(s) ≈ Gz(g/s)d/2−1 ≪ Gz for d > 2 and s ≫ g. Although for real polymer melts as
for computational systems large values of Gz ≈ 1 may sometimes be found, u(s) ∼ 1/
decreases rapidly with s in three dimensions, as illustrated in Fig. 2(b), and standard per-
turbation calculations can be successfully performed.
As sketched in the next paragraph these calculations yield quantities K[u] which are
defined such that they vanish (K[u = 0] = 0) if the perturbation potential u(s) is switched
off and are then shown to scale, to leading order, linearly with u. For instance, for the
quantity Kp(s), defined in Eq. (1), characterizing the deviation of the chain segment size
from Flory’s hypothesis one thus expects the scaling
Kp[u(s)] ≈ +u(s) ≈ +
ρRd(s)
. (7)
The +-sign indicated marks the fact that the prefactor has to be positive to be consis-
tent with the expected swelling of the chains. Consequently, the typical segment size,
R(s)/be
s ≈ 1 − u(s), must approach the asymptotic limit for large s from below. For
three dimensional solutions Eq. (7) implies that Kp(s) should vanish rapidly as 1/(ρb
(This is different in thin films where u(s) ≈ Gz decays only logarithmically [12] as may be
seen from Eq. (33) given below.) Taking apart the prefactors — which require a full calcu-
lation — this corresponds exactly to Eq. (5) with a swelling coefficient ce ≈ cp ≈ 1/ρb3e in
agreement with Eq. (4). Note also that the predicted deviations are inversely proportional
to b3e , i.e. the more flexible the chains, the more pronounced the effect. Similar relations
K[u] ∼ u may also be formulated for other quantities and will be tested numerically in
Sec. IV. There, we will also check that the linear order is sufficient.
B. Perturbation calculation
Generalities. Before delving more into our computational results we summarize here how
Eqs. (3-5) and related relations have been obtained using standard one-loop perturbation
calculation. The general task is to determine 〈A〉 ≈ 〈A〉0 (1+ 〈U〉0)−〈AU〉0 for measurable
quantities A such as the squared distance between two monomers n andm on the same chain,
A = r2nm. Here, 〈...〉0 denotes the average over the distribution function of the unperturbed
ideal chain of bond length b and U =
dl ṽ(rkl) the effective perturbation potential.
We discuss first the general results in the scale free regime (1 ≪ s ≪ N), argue then that b
should be renormalized to the effective bond length be and sketch finally the calculation of
finite chain-size effects.
The scale free regime. Following Edwards [6, 7, 8], the Gaussian (or “Random Phase”
[5]) approximation of the pair interaction potential in real space is
ṽ(r) = v
δ(r)− exp(−r/ξ)
4πrξ2
where v is a parameter which tunes the monomer interaction. (It is commonly associated
with the bare excluded volume of the monomers [6], but should more correctly be identified
with the bulk modulus effectively measured for the system. See the discussion of Eq. (15) of
Ref. ([13]).) The effective potential consists of a strongly repulsive part vδ(r) of very short
range, and a weak attractive part of range ξ where the correlation length of the density
fluctuations is given by ξ2 = b2g/12 with g = 1/ρv. In Fourier space Eq. (8) is equivalent to
ṽ(q) = v
q2 + ξ−2
with q being the wave vector. This is sufficient for calculating the scale free regime corre-
sponding to asymptotically long chains where chain end effects may be ignored. The different
graphs one has to compute are indicated in Fig. 1. For A = r2 (with 1 ≪ n < m ≪ N) this
yields, e.g.,
(m− n)−
24/π3
= b2es
. (10)
In the second line we have used the definition of the swelling coefficient ce indicated in
Eq. (4) and have set
b2e ≡ b2
1 + p
1 + p
with Gz ≡
vρ/b3ρ and p = 1. (The prefactor p has been added for convenience.) The
coefficient be of the leading Gaussian term in Eq. (10) — entirely due to the graph Ii
describing the interactions of monomers inside the segment between n and m — has been
predicted long ago by Edwards [6]. It describes how the effective bond length is increased
from b to be under the influence of a small excluded volume interaction. The second term in
Eq. (10) entails the 1/
s-swelling which is investigated numerically in this paper. It does
only depend on b and ρ but, more importantly, not on v — in agreement with the scaling
of u(s) discussed in Sec. IIA. The relative weights contributing to this term are indicated
in Fig. 1 in units of −
6/π3vξ2/b3
s. The diagrams I− and I+ are obviously identical in
the scale free limit. Note that the interactions described by the strongest graph Ii align the
bonds ln and lm while the others tend to reduce the effect.
For higher moments of the segment size distribution G(r, s) it is convenient to calculate
first the deviation of the Fourier-Laplace transformation of δG(r, s) and to obtain the mo-
ments from the coefficients of the expansion of this “generating function” in terms of the
squared wave vector q2. As explained in detail in the Appendix A this yields more generally
(2p+ 1)!
(b2es)
3(2pp!p)2
2(2p+ 1)!
)2p−3
where we have used Eq. (11) with general p. Obviously, Eq. (12) is consistent with our
previous finding Eq. (10) for p = 1. The corresponding segmental size distribution is
G(r, s) =
2πb2es
2πb2s
f(n) (13)
with n = r/b
s and f(n) being the same function as indicated in Eq. (3). The leading
Gaussian terms in Eqs. (12) and (13) depend on the effective bond length be, the second
only on the Kuhn length b of the reference chain. When comparing these result with Eqs. (3)
and (5) proposed in the Introduction, one sees that both equations are essentially identical —
taken apart, however, that they depend on b and be. Note the conspicuous factor (b/be)
in Eq. (12) which would strongly reduce the empirical swelling coefficients cp = ce(b/be)
for large p if b and be were different.
Interpretation of first-loop results in different contexts. The above perturbation results
may be used directly to describe the effect of a weak excluded volume v on a reference system
of perfectly ideal polymer melts with Kuhn segment length b where all interactions have been
switched off (v = 0). It is expected to give a good estimation for the effective bond length
be only for a small Ginzburg parameter: Gz ≪ 1. For the dense melts we want to describe
this does not hold (Sec. III) and one cannot hope to find a good quantitative agreement
with Eq. (11). Note also that large wave vectors contribute strongly to the leading Gaussian
term. The effective bond length be is, hence, strongly influenced by local and non-universal
effects and is very difficult to predict in general.
Our much more modest goal is to predict the coefficient of the 1/
s-perturbation and
to express it in terms of a suitable variational reference Hamiltonian characterized by a
conveniently chosen Kuhn segment b and the measured effective bond length be (instead of
Eq. (11)). Following Muthukumar and Edwards [10], we argue that for dense melts b should
be renormalized to be to take into account higher order graphs. No strict mathematical
proof can be given at present that the infinite number of possible graphs must add up in
this manner. Our hypothesis relies on three observations:
• The general scaling argument discussed in Sec. IIA states that we have only one
relevant length scale in this problem, the typical segment size R(s) ≈ be
s itself. The
incompressibility constraint cannot generate an additional scale. It is this size R(s)
which sets the strength of the effective interaction which then in turn feeds back to
the deviations of R(s) from Gaussianity. Having a bond length b in addition to the
effective bond length be associated with R(s) would imply incorrectly a second length
scale b
s varying independently with the bulk modulus v. (We will check explicitly
below in Fig. 13 that there is only one length scale.) This implies b/be = const v
• Thus, since by construction b/be = 1 for v → 0, it follows that both lengths should be
equal for all v.
• We know from Eq. (12) that the empirical coefficients cp = ce(b/be)2p−3 should depend
strongly on the moment considered if the ratio b/be is not close to unity. It will be
shown below (Fig. 6) that cp/ce ≈ 1 for all p. This implies b ≈ be.
Finite chain size effects. To describe properly finite chain size corrections Eq. (9) must
be replaced by the general linear response formula
ṽ(q)
+ ρF (q) (14)
with F (q) = NfD(x) being the form factor of the Gaussian reference chain given by Debye’s
function fD(x) = 2(e
−x − 1 + x)/x2 with x = (qb)2N/6 [6]. This approximation allows
in principle to compute, for instance, the (mean-squared) total chain end-to-end distance,
A = (rN −r1)2. One verifies readily (see [6], Eq. (5.III.9)) that the effect of the perturbation
may be expressed as
〈A〉0 〈U〉0 − 〈AU〉0 =
(2π)3
ṽ(q)
ds s2(N − s) exp
q2b2s
. (15)
We take now first the integral over s. In the remaining integral over q small q wave vectors
contribute to the
N -swelling while large q renormalize the effective bond length of the
dominant Gaussian behaviour linear in N (as discussed above). Since we wish to determine
the non-Gaussian corrections, we may focus on small wave vectors q ≪ 1/ξ. Since in this
limit 1/v = ρg ≪ ρF (q), one can neglect in Eq. (14) the 1/v contribution to the inverse
effective interaction potential. We thus continue the calculation using the much simpler
ṽ(N, x) = 1/(NρfD(x)). This allows us to express the swelling as
〈(rN − r1)2〉
I(xu). (16)
To simplify the notation we have set here finally b = be in agreement with the hypothesis
discussed above. The numerical integral I(xu) =
dx . . . over x is slowly convergent at
infinity. As a consequence the estimate I(∞) = 1.59 may be too large for moderate chain
lengths. In practice, convergence is not achieved for values xu(N) ≈ (b/ξ)2N corresponding
to the screening length ξ.
We remark finally that numerical integration can be avoided for various properties if the
Padé approximation of the form factor, F (q) = N/(1 + (qb)2N/12), is used. This allows
analytical calculations by means of the simplified effective interaction potential
ṽ(q) =
12ρb3
. (17)
This has been used for instance for the calculation of finite chain size effects for the bond-
bond correlation function discussed in Sec. IVC below [61].
III. COMPUTATIONAL MODELS AND TECHNICAL DETAILS
A. Bond fluctuation model
A widely-used lattice Monte Carlo scheme for coarse-grained polymers. The body of
our numerical data comes from the three dimensional bond fluctuation model (BFM) —
a lattice Monte Carlo (MC) algorithm where each monomer occupies eight sites of a unit
cell of a simple cubic lattice [24, 25, 26]. Our version of the BFM with 108 bond vectors
corresponds to flexible athermal chain configurations [23]. All length scales are given in units
of the lattice constant and time in units of Monte Carlo Steps (MCS). We use cubic periodic
simulation boxes of linear size L = 256 containing nmon = ρL
3 = 220 ≈ 106 monomers.
This monomer number corresponds to a monomer number density ρ = 0.5/8 where half of
the lattice sites are occupied (volume fraction 0.5). The large system sizes used allow us to
suppress finite box-size effects for systems with large chains. Using a mix of local, slithering
snake [27, 28, 29], and double-bridging [17, 23, 30, 31] MC moves we were able to equilibrate
dense systems with chain lengths up to N = 8192.
Equilibration and sampling of high-molecular BFM melts. Standard BFM implementa-
tions [26, 32, 33] use local MC jumps to the 6 closest lattice sites to prevent the crossing
of chains and conserve therefore the chain topology. These “L06” moves lead to very large
relaxation times, scaling at least as τe ∼ N3, as may be seen from Fig. 3 (stars). The relax-
ation time τe = R
e/6Ds indicated in this figure has been estimated from the self-diffusion
coefficient Ds obtained from the mean-square displacements of all monomers in the free
diffusion limit. (For the largest chain indicated for L06 dynamics only a lower bound for
τe is given.) Instead of this more realistic but very slow dynamical scheme we make jump
attempts to the 26 sites of the cube surrounding the current monomer position (called “L26”
moves). This allows the chains to cross each other which dramatically speeds up the dynam-
ics, especially for long chains (N > 512). If only local moves are considered, the dynamics
is perfectly consistent with the Rouse model [6]. As shown in Fig. 3, we find τe ≈ 530N2 for
L26 dynamics. This is, however, still prohibitive by large for sampling configurations with
the longest chain length N we aim to characterize [62].
Slithering snake moves. In addition to the local moves one slithering snake move per
chain is attempted on average per MCS corresponding to the displacement of N monomer
along the chain backbone. Note that in our units two spatial displacement attempts per
MCS are performed on average per monomer, one for a local move and one for a snake
move. (In practice, it is computationally more efficient for large N to take off a monomer
at one chain end and to paste it at the other leaving all other monomers unaltered. Before
dynamical measurements are performed the original order of beads must then be restored.)
Interestingly, a significantly larger slithering snake attempt frequency would not be useful
since the relaxation time of slithering snakes without or only few local moves increases
exponentially with mass [29, 34] due to the correlated motion of snakes [35]. In order to
obtain an efficient free snake diffusion (with a chain length independent curvilinear diffusion
coefficient Dc(N) ∼ N0 and τe ≈ N2/Dc(N) ∼ N2 [28, 29]) it is important to relax density
fluctuations rapidly by local dynamical pathways. As shown in Fig. 3 (squares), we find
a much reduced relaxation time τe ≈ 40N2 which is, however, still unconveniently large
for our longest chains. Note that most of the CPU time is still used by local moves. The
computational load per MCS remains therefore essentially chain length independent.
Advantages and pitfalls of double-bridging moves. Double-bridging (DB) moves are very
useful for high densities and help us to extend the accessible molecular masses close to
104. As for slithering snake moves we use all 108 bond vectors to switch chain segments
between two different chains. Only chain segments of equal length are swapped to conserve
monodispersity. Topolocial constraints are again systematically and deliberately violated.
Since more than one swap partner is possible for a selected first monomer, delicate detailed
balance questions arise. This is particularly important for short chains and is discussed
in detail in Ref. [23]. Technically, the simplest solution to this problem is to refuse all
moves with more than one swap partner (to be checked both for forward and back move).
The configurations are screened with a frequency fDB for possible DB moves where we
scan in random order over the monomers. The frequency should not be too large to avoid
(more or less) immediate back swaps and monomers should move at least out of the local
monomer cage and over a couple of lattice sites. We use fDB = 0.1 between DB updates for
the configurations reported here. (The influence of fDB on the performance has not been
explored systematically, but preliminary results suggest a slightly smaller DB frequency for
future studies.) The diffusion times over the end-to-end distance for this case are indicated
in Tab. I. As shown in Fig. 3, we find empirically τe(N) ≈ 13N1.62. For N = 8192 this
corresponds to 3 · 107 MCS. This allows us even for the largest chain lengths to observe
monomer diffusion over several Re within the 10
8 MCS which are feasible on our XEON-PC
processor cluster.
The efficiency of DB moves is commonly characterized in terms of the relaxation time
τee of the end-to-end vector correlation function [30, 31]. For normal chain dynamics this
would indeed characterize the longest relaxation time of the system, i.e. τe ≈ τee. For the
double-bridging this is, however, not sufficient since density fluctuations do not couple to the
bridging moves and can not be relaxed. We find therefore that configurations equilibrate
on time scales given by τe rather than by τee ≪ τe. This may be verified, for instance,
from the time needed for the distribution Re(s) (and especially its spatial components) to
equilibrate. The criterion given in the literature [30] is clearly not satisfactory and may lead
to insufficiently equilibrated configurations. In summary, equilibration with DB moves still
requires monomer diffusion over the typical chain size, however at a much reduced price.
Some properties of our configurations. The Tables I and II summarize some system
properties obtained for our reference density ρ = 0.5/8. Averages are performed over all
chains and 1000 configurations. These configurations may be considered to be independent
for N < 4096. Only a few independent configurations exist for the largest chain length
N = 8192 which has to be considered with some care. Taking apart this system, chains are
always much smaller than the box size. For asymptotically long chains, we obtain an average
bond length 〈|l|〉 ≈ 2.604, a root-mean-squared bond length l ≡ 〈l2〉1/2 ≈ 2.635 and an
effective bond length be ≈ 3.244 — as we will determine below in Sec. IVA. This corresponds
to a ratio C∞ ≡ b2e/l2 ≈ 1.52 and, hence, to a persistence length lp = l(C∞ + 1)/2 ≈ 3.32
[23]. Especially, we find from the zero wave vector limit of the total structure factor S(q)
a low (dimensionless) compressibility g = S(q → 0)/ρ ≈ 0.246 which compares well with
real experimental melts. From the measured bulk compression modulus v ≡ 1/g(ρ)ρ and
the effective bond length be one may estimate a Ginzburg parameter Gz =
vρ/b3eρ ≈ 0.96.
Following Ref. [13] the interaction parameter v is supposed here to be given by the full
inverse compressibility and not just by the second virial coefficient.
B. Bead spring model
Hamiltonian. Additionally, molecular dynamics simulations of a bead-spring model
(BSM) [36] were performed to dispel concerns that our results are influenced by the un-
derlying lattice structure of the BFM. The model is derived from a coarse-grained model
for polyvinylalcohol which has been employed to study polymer crystallization [37]. It is
characterized by two potentials: a non-bonded potential of Lennard-Jones (LJ) type and a
harmonic bond potential. While the often employed Kremer-Grest model [38] uses a 12− 6
LJ potential to describe the non-bonded interactions Unb(r), our non-bonded potential has
a softer repulsive part. It is given by
Unb(r) = 1.511
, (18)
which is truncated and shifted at the minimum at rmin ≈ 1.15. Note that all length scales
are given in units of σ0 and we use LJ units [39] for all BSM data (mass m = 1, Boltzmann
constant kB = 1). The parameters of the bond potential, Ub(r) = 1120(r− lb)2, are adjusted
so that the average bond length l(ρ = 0.84) ≈ lb = 0.97 is approximately the same as in the
standard Kremer-Grest model [38]. The average bond length and the root-mean-squared
bond length are almost identical for the BSM due to the very stiff bond potential. Since
rmin/l ≈ 1.16 bonded monomers penetrate each other significantly.
Equilibration and sampling. We perform standard molecular dynamics simulations in the
canonical ensemble with a Langevin thermostat (friction constant Γ = 0.5) at temperature
T = 1. The equations of motion are integrated by the velocity-Verlet algorithm [39]. To
improve the statistics for large chain length, we have implemented additional double-bridging
moves. Since only few of these MC moves are accepted per unit time, this does affect neither
the stability nor the accuracy of the molecular dynamics sweeps.
Some properties obtained. For clarity, we show only data for chain length N = 1024
and number density ρ = 0.84, the typical melt density of the Kremer-Grest model [38].
For the reported data we use periodic simulation boxes of linear size L ≈ 62 containing
nmon = 196608 monomers, but we have also sampled different boxes sizes (up to L = 77.5)
to check for finite box-size effects. For the reference density a dimensionless compressibility
g ≈ 0.08 is found which is about three times smaller than for our BFM melt. For the
effective bond length we obtain be ≈ 1.34, i.e. BSM chains (C∞ ≈ 1.91, lp ≈ 1.41) are
slightly stiffer than the corresponding BFM polymers. Fortunately, the product ρb3e ≈ 2 is
roughly similar in both models and one expects from Eq. (4) a similar swelling for large
s. Note finally that the Ginzburg parameter Gz ≈ 1.8 is much larger than for the BFM
systems. As we have emphasized in Sec. II, this should, however, not influence the validity
of the perturbation prediction of the expected 1/
s-swelling of the chains when expressed
in terms of the measured effective bond length.
IV. NUMERICAL RESULTS
As illustrated in Fig. 1, a chain segment of curvilinear length s > 0 is identified by two
monomers n and m = n + s on the same chain. We compute here various moments of
chain segment properties where we ensemble-average over all chains and all start points n.
The statistical accuracy must therefore always decrease for large s. We concentrate first on
the second moment (p = 1) of the segmental size distribution. Higher moments and the
segmental size distribution are discussed in Sec. IVE.
A. The swelling of chain segments
Scale free regime for 1 ≪ s ≪ N . The mean-squared segment size R2e(s) = 〈r2〉 is
presented in the Figs. 4, 5 and 6. The first plot shows clearly that chain segments are swollen,
i.e. R2e(s)/s increases systematically and this up to very large curvilinear distances s. Only
BFM data are shown for clarity. A similar plots exists for the BSM data. In agreement
with Eq. (5) for p = 1, the asymptotic Gaussian behavior (dashed line) is approached
from below and the deviation decays as u(s) ∝ 1/
s (bold line). The bold line indicated
corresponds to be = 3.244 and c1 ≈ ce ≈ 0.41 which fits nicely the data over several decades
in s — provided that chain end effects can be neglected (s ≪ N). Note that a systematic
underestimation of the true effective bond length would be obtained by taking simply the
largest R2e(s)/s ≈ 3.232 value available, say, for monodisperse chains of length N = 2048.
Finite chain-size effects. Interestingly, R2e(s)/s does not approach the asymptotic limit
monotonicly. Especially for short chains one finds a non-monotonic behavior for s → N .
This means that the total chain end-to-end distance Re(s = N − 1) must show even more
pronounced deviations from the asymptotic limit. This is confirmed by the dashed line
representing the b2e(N) ≡ R2e(N − 1)/(N − 1) data points given in Tab. I. We emphasize
that the non-monotonicity of R2e(s)/s becomes weaker with increasing N and that, as one
expects, the inner distances, as well as the total chain size, are characterized by the same
effective bond length be for large s or N . The non-monotonic behavior may be qualitatively
understood by the reduced self-interactions at chain ends which lessens the swelling on these
scales. These finite-N corrections have been calculated analytically using the full Debye
function for the effective interaction potential ṽ(q), Eq. (14). The prediction for the total
chain end-to-end vector given in Eq. (16) is indicated in Fig. 4 (dash-dotted line) where we
have replaced the weakly N -dependent integral I(xu) by its upper bound value for infinite
chains
R2e(N − 1)
b2e (N − 1)
1.59ce√
N − 1
. (19)
We have changed here the chain length N in the analytical formula (obtained for large chains
where N ≈ N − 1) to the curvilinear length N − 1. This is physically reasonable and allows
to take better into account the behavior of small chains. Note that Eq. (19) is similar to
Eq. (5) — apart from a slightly larger prefactor explaining the observed stronger deviations.
Theory compares well with the measured data for large N . It does less so for smaller N ,
as expected, where the chain length dependence of the numerical integral I(xu(N)) ≤ 1.59
must become visible. This explains why the data points are above the dash-dotted line.
Note also that additional non-universal finite-N effects not accounted for by the theory are
likely for small N . In contrast to this, Re(s) is well described by the theory even for rather
small s provided that N is large and chain end effects can be neglected. In summary, it is
clear that one should use the segment size Re(s) rather than the total chain size to obtain
in a computational study a reliable fit of the effective bond length be.
Extrapolation of the effective bond length of asymptotically long chains. The represen-
tation chosen in Fig. 4 is not the most convenient one for an accurate determination of be
and c1. How precise coefficients may be obtained according to Eq. (5) is addressed in the
Figs. 5 and 6. The fitting of the effective bond length be and its accuracy is illustrated in
Fig. 5 for BFM chains of length N = 2048. This may be first done approximately in linear
coordinates by plotting R2e(s)/s as a function of 1/
s (not shown). Since data for large s
are less visible in this representation, we recommend for the fine-tuning of be to switch then
to logarithmic coordinates with a vertical axis y = 1 − R2e(s)/b2es for different trial values
of be. The correct value of be is found by adjusting the vertical axis y such that the data
extrapolates linearly as a function of 1/
s to zero for large s. We assume for the fine-tuning
that higher order perturbation corrections may be neglected, i.e. we take Eq. (5) literally.
(We show below that higher order corrections must indeed be very small.) The plot shows
that this method is very sensitive, yielding a best value that agrees with the theory over
more than one order of magnitude without curvature. As expected, it is not possible to
rationalize the numerically obtained values be ≈ 3.244 for the BFM and be ≈ 1.34 for the
BSM using Eq. (11). According to Eq. (4) these fit values imply the theoretical swelling
coefficients ce = 0.41 for the BFM and ce = 0.44 for the BSM.
Empirical swelling coefficients. As a next step the horizontal axis is rescaled such that
all data sets collapse on the bisection line, i.e. using Eq. (5) we fit for the empirical swelling
coefficient c1 and compare it to the predicted value ce. This rescaling of the axes allows to
compare both models in Fig. 6. For clarity the BSM data have been shifted upwards. For
the BFM we find c1/ce ≈ 1.0, as expected, while our BSM simulations yield a slightly more
pronounced swelling with c1/ce ≈ 1.2.
Segmental radius of gyration. Also indicated in Fig. 6 is the segmental radius of gyration
Rg(s) (filled circles) computed as usual [6] as the variance of the positions of the segment
monomers around their center of mass. Being the sum over all s + 1 monomers, it has a
much better statistics compared to Re(s). The scaling used can be understood by expressing
the radius of gyration R2g(s) =
(s+1)2
l=n r
in terms of displacement vectors rkl
[6]. Using Eq. (5) and integrating twice this yields
6R2g(s)
b2e(s+ 1)
. (20)
Plotting the l.h.s. of this relation against the r.h.s. we obtain a perfect data collapse on the
bisection line where we have used the same parameters be and c1 as for the mean-squared
segment size. This is an important cross-check which we strongly recommend. Different
values indicate insufficient sample equilibration.
B. Chain connectivity and recursion relation
As was emphasized in Sec. IIA the observed swelling is due to an entropic repulsion
between chain segments induced by the incompressibility of the melt. To stress the role of
chain connectivity we repeat the general scaling argument given above in a form originally
proposed by Semenov and Johner for ultrathin films [12]. As shown in Fig. 7 we test the
relation
Kλ(s) ≡
R2e(λs)− λR2e(s)
λ)R2e(s)
≈ u(s) ≡ s
ρRe(s)d
with Kλ(s) being a direct measure of the non-Gaussianity (λ being a positive number)
comparing the size of a segment of length λs with the size of λ segments of length s joined
together. (The prefactor 1/(λ −
λ) in the definition of Kλ(s) has been introduced for
convenience.) Equivalently, this can be read as a measure for the swelling of a chain where
initially the interaction energy u between the segments has been switched off. Kλ is a
functional of u(s) with Kλ[u = 0] = 0. The analytic expansion of the functional must be
dominated by the linear term (as indicated by ≈ in the above relation) simply because u is
very small. Altogether, Eq. (21) yields a recursion relation relating Re(λs) with Re(s) for
any λ provided 1 ≪ s < λs ≪ N . It can be solved, leading (in lowest order) to Eq. (5) with
p = 1. This may be seen from the ansatz R2e(s) = b
es(1− ce/sω−1+ . . .) which readily yields
ω = 3/2 and ce ≈ 1/ρb3e .
Eq. (21) has been validated directly in Fig. 7 for λ = 2 (corresponding to two segments
of length s joined together) for the BFM and the BSM as indicated. In addition, for BFM
chains of length N = 2048 several values of λ have been given. As suggested by Eq. (4),
we have plotted Kλ(s) as a function of (c1/ce) u(s) with u(s) ≡
24/π3s/ρR3e(s) ≈ ce/
The prefactor of u(s) allows a convenient comparison with Fig. 5. Note the perfect data
collapse for all data sets. More importantly, the predicted linearity is well confirmed for
large segments (1 ≪ s) and this without any tunable parameter for the vertical axis, as was
needed in the previous Figs. 5 and 6.
C. Intrachain bond-bond correlations
Expectation from Flory’s hypothesis. An even more striking violation of Flory’s ideality
hypothesis may be obtained by computing the bond-bond correlation function, defined by
the first Legendre polynomial P (s) = 〈lm=n+s · ln〉 /l2 where the average is performed, as
before, over all possible pairs of monomers (n,m = n+ s) [63]. Here, li = ri+1 − ri denotes
the bond vector between two adjacent monomers i and i + 1 and l2 = 〈l2n〉n the mean-
squared bond length. The bond-bond correlation function is generally believed to decrease
exponentially [4]. This belief is based on the few simple single chain models which have
been solved rigorously [4, 40] and on the assumption that all long range interactions are
negligible on distances larger than the screening length ξ. Hence, only correlations along
the backbone of the chains are expected to matter and it is then straightforward to work
out that an exponential cut-off is inevitable due to the multiplicative loss of any information
transferred recursively along the chain [4].
Asymptotic behavior in the melt. That this reasoning must be incorrect follows imme-
diately from the relation
P (s) =
R2e(s) (22)
expressing the bond-bond correlation function as the curvature of the second moment of
the segment size distribution. It is obtained from the identity 〈ln · lm〉 = 〈∂nrn · ∂mrm〉 =
−∂n∂m 〈r2nm〉 /2. (Note that the velocity correlation function is similarly related to the
second derivative of the mean-square displacement with respect to time [41].) Hence, P (s)
allows us to probe directly the non-Gaussian corrections without any ideal contribution.
This relation together with Eq. (5) suggests an algebraical decay P (s) = cP/s
ω with
ω = 3/2 , cP = c1 (be/l)
2/8 ≈
ρl2be
of the bond-bond correlation function for dense solutions and melts, rather than the ex-
ponential cut-off expected from Flory’s hypothesis. This prediction (bold line) is perfectly
confirmed by the larger chains (N > 256) indicated in Fig. 8. In principle, the swelling
coefficient, c1 ∼ cP, may also be obtained from the power law amplitude of the bond-bond
correlation function, however, to lesser accuracy than by the previous method (Fig. 5). One
reason is that P (s) decays very rapidly and does not allow a precise fit beyond s ≈ 102. The
values of cP obtained from c1 are indicated in Tab. II. Data from the BSM have also been
included in the figure to demonstrate the universality of the result. The vertical axis has
been rescaled with cP which allows to collapse the data of both models.
Finite chain-size corrections. As can be seen for N = 16, exponentials are compatible
with the data of short chains. This might explain how the power law scaling has been
overlooked in previous numerical studies, since good statistics for large chains (N > 1000)
has only become available recently. However, it is clearly shown that P (s) approaches
systematically the scale free asymptote with increasing N . The departure from this limit is
fully accounted for by the theory if chain end effects are carefully considered (dashed lines).
Generalizing Eq. (23) and using the Padé approximation, Eq. (17), perturbation theory
yields
P (s) =
1 + 3x+ 5x2
1 + x
(1− x)2 (24)
where we have set x =
s/N . For x ≪ 1 this is consistent with Eq. (23). In the limit of
large s → N , the correlation functions vanish rigorously as P (s) ∝ (1 − x)2. Considering
that non-universal features cannot be neglected for short chain properties and that the
theory does not allow for any free fitting parameter, the agreement found in Fig. 8 is rather
satisfactory.
D. Higher moments and associated coefficients
Effective bond length and empirical swelling coefficients. The preceding discussion fo-
cused on the second moment of the segmental size distribution G(r, s). We have also com-
puted for both models higher moments 〈r2p〉 with p ≤ 5. If traced in log-linear coordinates
as y = (6pp! 〈r2p〉 /(2p + 1)!sp)1/p vs. x = s higher moments approach b2e from below —
just as the second moment presented in Fig. 4. The deviations from ideality are now more
pronounced and increase with p (not shown). The moments are compared in Fig. 6 with
Eq. (5) where they are rescaled as y = Kp(s) as defined in Eq. (1) and plotted as functions
of x =
3(2pp!p)2
2(2p+1)!
. The prediction is indicated by bold lines. It is important that the same
effective bond length be is obtained from the analysis of all functions Kp(s) as illustrated in
Fig. 6. Otherwise we would regard equilibration and statistics as insufficient.
The empirical swelling coefficients cp are obtained, as above in Sec. IVA, by shifting the
data horizontally. A good agreement with the expected cp/ce ≈ 1 is found for both models
and all moments as may be seen from Tab. II. This confirms the renormalization of the
Kuhn segment b → be of the Gaussian reference chain in agreement with our discussion in
Sec. II B. Otherwise we would have measured empirical coefficients decreasing strongly as
cp/ce ≈ (b/be)2p−3 with p. Since the effective bond length of non-interacting chains are known
for the BFM (b ≈ 2.688) and the BSM (b ≈ 0.97), one can simply check, say for p = 5, that
the non-renormalized values would correspond to the ratios c5/ce ≈ (2.688/3.244)7 ≈ 0.3 for
the BFM and c5/ce ≈ (0.97/1.34)7 ≈ 0.1 for the BSM. This is clearly not consistent with
our data.
It should be emphasized that both coefficients be and cp are more difficult to determine
for large p, since the linear regime for x ≪ 1 in the representation chosen in Fig. 6 becomes
reduced. For large x ≫ 1 one finds that y(x) → 1, i.e. 〈r2p〉 /b2pe sp → 0. This trivial
departure from both Gaussianity and the 1/
s-deviations we try to describe, is due to the
finite extensibility of chain segments of length s which becomes more marked for larger
moments probing larger segment sizes. The data collapse for both x-regimes is remarkable,
however. Incidentally, it should be noted that for the BSM the empirical swelling coefficients
are slightly larger than expected. At present we do not have a satisfactory explanation for
this altogether minor effect, but it might be attributed to the fact that neighbouring BSM
beads along the chain strongly interpenetrate — an effect not considered by the theory.
Non-Gaussian parameter αp. The failure of Flory’s hypothesis can also be demonstrated
by means of the standard non-Gaussian parameter
αp(s) ≡ 1−
(2p+ 1)!
〈r2pnm〉
〈r2nm〉
p (25)
comparing the 2p-th moment with the second moment (p = 1). In contrast to the closely
related parameter Kp(s) this has the advantage that here two measured properties are com-
pared without any tuneable parameter, such as be, which has to be fitted first. Fig. 9 presents
αp(s) vs. ce/
s for the three moments with p = 2, 3, 4. For each p we find perfect data col-
lapse for all chain lengths and both models and confirm the linear relationship αp(s) ≈ u(s)
expected. The lines indicate the theoretical prediction
αp(s) =
3 (2pp!p)2
2 (2p+ 1)!
which can be derived from Eq. (5) by expanding the second moment in the denominator. An
alternative derivation based on the coefficients of the expansion of the generating function
G(q, s) in q2 is indicated by Eq. (A2) in the Appendix. Having confirmed above that cp/ce ≈
1, we assume in Eq. (26) that cp = ce to simplify the notation. The prefactors 6/5, 111/35
and 604/105 for p = 2, 3 and 4 respectively are nicely confirmed. They increase strongly
with p, i.e. the non-Gaussianity becomes more pronounced for larger moments as already
mentioned. Note also the curvature of the data at small s due to the finite extensibility
of the segments which becomes more marked for higher moments. If one plots αp(s) as a
function of the r.h.s. of Eq. (26) all data points for all moments and even for too small s
collapse on one master curve (not shown) — just as we have seen before in Fig. (6).
Correlations of different directions. A similar correlation function is presented in Fig. 10
which measures the non-Gaussian correlations of different spatial directions. It is defined by
Kxy(s) ≡ 1−
〈x2 y2〉
〈x2〉 〈y2〉
for the two spatial components x and y of the vector r as illustrated by the sketch given
at the bottom of Fig. 10. Symmetry allows to average over the three pairs of directions
(x, y), (x, z) and (x, z). Following the general scaling argument given in Sec. II we expect
Kxy(s) ≈ u(s) ≈ ce/
s which is confirmed by the perturbation result
Kxy(s) =
= K2(s). (28)
This is nicely confirmed by the linear relationship found (bold line) on which all data from
both simulation models collapse perfectly. The different directions of chain segments are
therefore coupled. As explained in the Appendix (Eq. (A3)), Kxy(s) and α2(s) must be
identical if the Fourier transformed segmental size distribution G(q, s) can be expanded
in terms of q2 and this irrespective of the values the expansion coefficients take. Fig. 10
confirms, hence, that our computational systems are perfectly isotropic and tests the validity
of the general analytical expansion.
The correlation function Kxy is of particular interest since the zero-shear viscosity should
be proportional to
∼ 〈x2y2〉 = 〈x2〉 〈y2〉 (1 − Kxy(s)). We assume here following
Edwards [6] that only intrachain stresses contribute to the shear stress σxy. Hence, our
results suggest that the classical calculations [6] — assuming incorrectly Kxy = 0 — should
be revisited.
E. The segmental size distribution
We turn finally to the segmental size distribution G(r, s) itself which is presented in
Figs. 11, 12 and 13. From the theoretical point of view G(r, s) is the most fundamental
property from which all others can be derived. It is presented last since it is computationally
more demanding — at least if high accuracy is needed — and coefficients such as be may
be best determined directly from the moments. The normalized histograms G(r, s) are
computed by counting the number of segment vectors between r − dr/2 and r + dr/2 with
dr being the width of the bin and one divides then by the spherical bin volume. Since
the BFM model is a lattice model, this volume is not 4πr2dr but given by the number of
lattice sites the segment vector can actually point to for being allocated to the bin. Incorrect
histograms are obtained for small r if this is not taken into account. (Averages are taken
over all segments and chains, just as before.) Clearly, non-universal physics must show up
for small vector length r and small curvilinear distance s and we concentrate therefore on
values r ≫ σ and s ≥ 31.
When plotted in linear coordinates as in Fig. 11, G(r, s) compares roughly with the
Gaussian prediction G0(r, s) given by Eq. (2), but presents a distinct depletion for small
segment sizes with n ≡ r/be
s ≪ 1 and an enhanced regime for n ≈ 1. A second depletion
region for large n ≫ 1 — expected from the finite extensibility of the segments — can be best
seen in the log-log representation of the data (not shown). To analyse the data it is better
to consider instead of G(r, s) the relative deviation δG(r, s)/G0(r, s) = G(r, s)/G0(r, s)− 1
which should further be divided by the strength of the segmental correlation hole, ce/
As presented in Fig. 12 this yields a direct test of the key relation Eq. (3) announced in the
Introduction. The figure demonstrates nicely the scaling of the data for all s and for both
models. It shows further a good collapse of the data close to the universal function f(n)
predicted by theory (bold line). Note that the depletion scales as 1/n for small segment
sizes (dashed line). The agreement of simulation and theory is by all standards remarkable.
(Obviously, error bars increase strongly for n ≫ 1 where G0(r, s) decreases strongly. The
regime for very large n where the finite extensibility of segments matters has been omitted
for clarity.) We emphasize that this scaling plot depends very strongly on the value be which
is used to calculate the Gaussian reference distribution.
If a precise value is not available we recommend to use instead the scaling variable
m = r/Re(s) for the horizontal axis, i.e. to replace the scale be
s estimated from the
behaviour of asymptotically long chains by the measured (mean-squared) segment size
for the given s. The Gaussian reference distribution is then accordingly G0(m,Re(s)) =
(3/2πRe(s)
2) exp(−3
m2). The corresponding scaling plot is given in Fig. 13. It is simi-
lar and of comparable quality as the previous plot. Changing the scaling variable from
n = r/be
s to m = r/Re(s) ≈ (r/be
s)(1 + ce/2
s) changes somewhat the universal func-
tion. Expanding the previous result, Eq. (3), this adds even powers of m to the function
f(n) given in Eq. (3)
f(n) ⇒ f(m) =
+ 9m+
m2 − 9
. (29)
That the two additional terms in the function are correct can be seen by computing the
second moment 4π
drr4δG(r, s) which must vanish by construction. The rescaled relative
deviation is somewhat broader than in the previous plot due to the additional term scaling
as m2. As already stressed this scaling does not rely on the effective bond length be and
is therefore more robust. It has the nice feature that it underlines that there is only one
characteristic length scale relevant for the swelling induced by the segmental correlation
hole, the typical size of the chain segment itself.
V. CONCLUSION
Issues covered and central theoretical claims. We have revisited Flory’s famous ideality
hypothesis for long polymers in the melt by analyzing both analytically and numerically
the segmental size distribution G(r, s) and its moments for chain segments of curvilinear
length s. We have first identified the general mechanism that gives rise to deviations from
ideal chain behavior in dense polymer solutions and melts (Sec. II). This mechanism rests
upon the interplay of chain connectivity and the incompressibility of the system which
generates an effective repulsion between chain segments (Fig. 2). This repulsion scales like
u(s) ≈ ce/
s where the “swelling coefficient” ce ≈ 1/b3eρ sets the strength of the interaction.
It is strong for small segment length s, but becomes weak for s → N in the large-N limit.
The overall size of a long chain thus remains almost ‘ideal’, whereas subchains are swollen as
described by Eq. (5). Most notably, the relative deviation δG(r, s)/G0(r, s) of the segmental
size distribution from Gaussianity should be proportional to u(s). As a function of segment
size r, the repulsion manifests itself by a strong 1/r-depletion at short distances r ≪ be
and a subsequent shift of the histogram to larger distances (Eq. (3)).
Summary of computational results. Using Monte Carlo and molecular dynamics simu-
lation of two coarse-grained polymer models we have verified numerically the theoretical
predictions for long and flexible polymers in the bulk. We have explicitly checked (e.g.,
Figs. 7, 9, 13) that the relative deviations from Flory’s hypothesis scale indeed as 1/
Especially, the measurement of the bond-bond correlation function P (s), being the second
derivative of the second moment of G(r, s) with respect of s, allows a very precise verifi-
cation (Fig. 8) and shows that higher order corrections beyond the first-order perturbation
approximation must be small. The most central and highly non-trivial numerical verification
concerns the data collapse presented in Figs. 12 and 13 for the segmental size distribution
of both computational models. All other statements made in this paper can be derived and
understood from this key finding. It shows especially that the swelling coefficient ce must
be close to the predicted value, Eq. (4).
It is well known [10] that the effective bond length is difficult to predict at low com-
pressibility and no attempt has been done to do so in this paper. We show instead how the
systematic swelling of chain segments – once understood – may be used to extrapolate for
the effective bond length of asymptotically long chains. Figs. 5 and 6 indicate how this may
be done using Eq. (5). The high precision of our data is demonstrated in Fig. 12 by the
successful scaling of the segmental size distribution.
For several moments 〈r2p〉 we have also fitted empirical swelling coefficients cp using
Eq. (5). In contrast to the effective bond length be these coefficients are rather well pre-
dicted by one-loop perturbation theory if the bond length b of the reference Hamiltonian is
renormalized to the effective bond length be, as we have conjectured in Sec. II B. Since the
empirical swelling coefficients, cp ≈ ce(b/be)2p−3, would otherwise strongly depend on the
moment taken, as shown in Eq. (12), our numerical data (Tab. II) clearly imply b/be ≈ 1.
Minor deviations found for the BSM samples may be attributed to the fact that monomers
along the BSM chains do strongly overlap — an effect not taken into account by the theory.
To clarify ultimately this issue we are currently performing a numerical study where we
systematically vary both the compressibility and the bond length of the BSM.
General background and outlook. The most striking result presented in this work con-
cerns the power law decay found for the bond-bond correlation function, P (s) ∝ 1/s3/2
(Fig. 8). This result suggests an analogy with the well-known long-range velocity correla-
tions found in dense fluids by Alder and Wainwright nearly fourty years ago [41, 42]. In
both cases, the ideal uncorrelated object is a random walker which is weakly perturbed (for
d > 2) by the self-interactions generated by global constraints. Although these constraints
are different (momentum conservation for the fluid, incompressibility for polymer melts) the
weight with which these constraints increase the stiffness of the random walker is always
proportional to the return probability. It can be shown that the correspondence of both
problems is mathematically rigorous if the fluid dynamics is described on the level of the
linearized Navier-Stokes equations [43].
We point out that the physical mechanism which has been sketched above is rather
general and should not be altered by details such as a finite persistence length — at least
not as long as nematic ordering remains negligible and the polymer chains are sufficiently
long. (Similarly, velocity correlations in dense liquids must show an analytical decay for
sufficiently large times irrespective of the particle mass and the local static structure of
the solution.) While this paper focused exclusively on scales beyond the correlation length
of the density fluctuations, i.e. qξ ≪ 1 or s/g ≫ 1, where the polymer solution appears
incompressible, effects of finite density and compressibility can be readily described within
the same theoretical framework and will be presented elsewhere [43]. To test our predictions,
flexible chains should be studied preferentially, since the chain length required for a clear-
cut description increases strongly with persistence length. This is in fact confirmed by
preliminary and on-going simulations using the BSM algorithm.
In this work we have only discussed properties in real space as a function of the curvilin-
ear distance s. These quantities are straightforward to compute in a computer simulation
but are barely experimentally relevant. The non-Gaussian deviations induced by the seg-
mental correlation hole arise, however, also for an experimentally accessible property, the
intramolecular form factor (single chain scattering function) F (q). As explained at the
end of the Appendix, the form factor can be readily obtained by integrating the Fourier
transformed segmental size distribution given in Eq. (3). This yields
q2F (q) ≈ 12
in agreement with the result obtained in Refs. [14, 19] by direct calculation of the form factor
for very long equilibrium polymers. As a consequence of this, the Kratky plot (q2F (q) vs.
wave vector q) should not exhibit the plateau expected for Gaussian chains in the scale-free
regime, but rather noticeable non-monotonic deviations. See Fig. 3 of [19]. This result
suggests to revisit experimentally this old pivotal problem of polymer science.
Our work is part of a broader attempt to describe systematically the effects of correlated
density fluctuations in dense polymer systems, both for static [12, 13, 44, 45] and dynam-
ical [29, 35, 46] properties. An important unresolved question is for instance whether the
predicted long-range repulsive forces of van der Waals type (“Anti-Casimir effect”) [13, 45]
are observable, for instance in the oscillatory decay of the standard density pair-correlation
function of dense polymer solutions. Since the results presented here challenge an important
concept of polymer physics, they should hopefully be useful for a broad range of theoreti-
cal approaches which commonly assume the validity of the Gaussian chain model down to
molecular scales [47, 48, 49]. This study shows that a polymer in dense solutions should not
be viewed as one soft sphere (or ellipsoid) [50, 51, 52], but as a hierarchy of nested segmental
correlation holes of all sizes aligned and correlated along the chain backbone (Fig. 2 (b)). We
note that similar deviations from Flory’s hypothesis have been reported recently for linear
polymers [16, 17, 47] and polymer gels and networks [53, 54]. The repulsive interactions
should also influence the polymer dynamics, since strong deviations from Gaussianity are
expected on the scale where entanglements become important, hence, quantitative predic-
tions for the entanglement length Ne have to be regarded with more care. The demonstrated
swelling of chains should be included in the popular primitive path analysis for obtaining Ne
[55], especially if ‘short’ chains (N < 500) are considered. The effect could be responsible
for observed deviations from Rouse behavior [26, 56] as may be seen by considering the cor-
relation function Cpq ≡ 〈Xp ·Xq〉 of the Rouse modes Xp = 1N
dnrn cos(npπ/N) where
p, q = 0, . . . , N − 1 [6, 57]. Using (rn − rm)2 = r2n + r2m − 2rn · rm for the segment size, this
correlation function can be readily expressed as an integral over the second moment of the
segmental size distribution
Cpq = −
(rn − rm)2
cos(npπ/N) cos(mpπ/N) (31)
which can be solved using our result Eq. (5). This implies for instance for p = q that
Cpp =
2(πp)2
1− π√
. (32)
The bracket entails an important correction with respect to the classical description given by
the prefactor [6]. We are currently working out how static corrections, such as those for Cpp,
may influence the dynamics for polymer chains without topological constraints. (This may
be realized, e.g., within the BFM algorithm by using the L26 moves described in Sec. IIIA.)
Moreover, for thin polymer films of width H the repulsive interactions are known to be
stronger than in the bulk [12]. This provides a mechanism to rationalize the trend towards
swelling observed experimentally [58] and confirmed computationally [21]:
= log(s)/H. (33)
(Prefactors omitted for clarity.) Here Rx(s) and bx denote the components of the segment
size and the effective bond length parallel to the film. It also explains the (at first sight
surprising) systematic increase of the polymer dynamics with decreasing film thickness [22].
Specifically, the parallel component of the monomer mean-squared displacement gx(t) is
expected to scale as gx(t) ≈ R2x(s(t)) ∝ t1/4(1 + log(t)/H) for long reptating chains where
s(t) ∝ t1/4 [6]. (The corresponding effect for the three-dimensional bulk should be small,
however.) For the same reason (flexible) polymer chains close to container walls must be
more swollen and, hence, faster on intermediate time scales than their peers in the bulk.
Acknowledgments
We thank T. Kreer, S. Peter and A.N. Semenov (all ICS, Strasbourg, France),
S.P. Obukhov (Gainesville, Florida) and M. Müller (Göttingen, Germany) for helpful discus-
sions. A generous grant of computer time by the IDRIS (Orsay) is also gratefully acknowl-
edged. J.B. acknowledges financial support by the IUF and from the European Community’s
“Marie-Curie Actions” under contract MRTN-CT-2004-504052.
APPENDIX A: MOMENTS OF THE SEGMENTAL SIZE DISTRIBUTION AND
THEIR GENERATING FUNCTION
Higher moments of the segmental size distribution G(r, s) can be systematically obtained
from its Fourier transformation
G(q, s) =
d3r G(r, s) exp(iq · r),
which is in this context sometimes called the “generating function” [59]. For an ideal
Gaussian chain, the generating function is then G0(q, s) = exp(−sq2a2) where we have
used a2 = b2/6 instead of the bond length b2 to simplify the notation. Moments of the
size distribution are given by proper derivatives of G(q, s) taken at q = 0. For example,
〈r2p〉 = (−1)p∆pG(q, s)q=0 (with ∆ being the Laplace operator with respect to the wave vec-
tor q). A moment of order 2p is, hence, linked to only one coefficient A2p in the systematic
expansion, G(q, s) =
p=0A2pq
2p, of G(q, s) around q = 0. For our example this implies
= (−1)p(2p+ 1)! A2p (A1)
in general and more specifically for a Gaussian distribution 〈r2p〉0 =
(2p+1)!
spa2p. The non-
Gaussian parameters read, hence,
αp(s) ≡ 1−
(2p+ 1)!
〈r2p〉
〈r2〉p
= 1− p! A2p
, (A2)
which implies (by construction) αp = 0 for a Gaussian distribution. As various moments of
the same global order 2p are linked to the same A2p they differ by a multiplicative constant
independent of the details of the (isotropic) distribution G(q, s). For example, 〈r2〉 = 6|A2|,
〈r4〉 = 120A4, 〈x2〉 = 〈y2〉 = 2|A2|, 〈x2y2〉 = 8A4 with x and y denoting the spatial
components of the segment vector r. Using Eq. (A2) for p = 2 it follows that
Kxy(s) ≡ 1−
〈x2y2〉
〈x2〉〈y2〉
= 1− 2A4
= α2(s), (A3)
i.e. the properties α2(s) and Kxy(s) discussed in Figs. 9 and 10 must be identical in general
provided that G(q, s) is isotropic and can be expanded in q2.
We turn now to specific properties ofG(q, s) computed for formally infinite polymer chains
in the melt. In practice, these results are also relevant for small segments in large chains,
N ≫ s ≫ 1, and, especially, for segments located far from the chain ends. These chains are
nearly Gaussian and the generating function can be written as G(q, s) = G0(q, s) + δG(q, s)
where δG(q, s) = −〈UG〉0 + 〈U〉0〈G〉0 is a small perturbation under the effective interaction
potential ṽ(q) given by Eq. (9). To compute the different integrals it is more convenient to
work in Fourier-Laplace space (q, t) with t being the Laplace variable conjugate to s:
δG(q, t) =
ds δG(q, s)e−st.
As illustrated in Fig. 14, there a three contributions to this perturbation: one due to in-
teractions between two monomers inside the segment (left panel), one due to interactions
between an internal monomer and an external one (middle panel) and one due to interac-
tions between two external monomers located on opposite sides (right panel). In analogy to
the derivation of the form factor described in Ref. [14] this yields:
δG(q, t) = −
(q2a2 + t)2
q2a2 + t−
(q2a2 + t)2
4πqa2ξ2
Arctan
a/ξ +
a/ξ +
q2a2 + t
q2a2 + t
4πqa4
Arctan
a/ξ +
a/ξ +
q2a2 + t
4πqa6
Arctan
q2a2 + t
. (A4)
The graph given in the left panel of Fig. 14 corresponds to the first two lines, the middle
panel to the third line and the right panel to the last one. Seeking for the moments we
expand δG(q, t) around q = 0. Having in mind chain strands counting many monomers
(s ≫ 1), we need only to retain the most singular terms for t → 0. Defining the two
dimensionless constants d = vξ/3πa4 = 12vξ/πb4 and c = (3π3/2a3ρ)−1 =
24/π3/b3ρ this
expansion can be written as
δG(q, t) = −
d a2q2 +
Γ(3/2)
c a2q2 + . . . (A5)
d a4q4 − 1
Γ(5/2)
c a4q4 + . . .
d a6q6 +
Γ(7/2)
c a6q6 + . . .
+ . . .
where we have used Euler’s Gamma function Γ(α) [60]. The first leading term at each order
in q2 — being proportional to the coefficient d — ensures the renormalization of the effective
bond length. The next term scaling with the coefficient c corresponds to the leading finite
strand size correction. Performing the inverse Laplace transformation Γ(α)/tα → sα−1 and
adding the Gaussian reference distribution G0(q, s) this yields the A2p-coefficients for the
expansion of G(q, s) around q = 0:
A0 = 1
A2 = −a2s
1 + d− c√
1 + 2d−
A6 = −
1 + 3d− 216
A8 = . . . (A6)
More generally, one finds
A2p =
(−1)p
(sa2)p
1 + pd− 3(2
pp!p)2
2(2p+ 1)!
From this result and using Eq. (A1) one immediately verifies that the moments of the
distribution are given by the Eqs. (11) and (12). Using Eq. (A2) one justifies similarly
Eq. (26) for the non-Gaussian parameter αp.
These moments completely determine the segmental distribution G(r, s) which is indi-
cated in Eq. (13). While at least in principle this may be done directly by inverse Fourier-
Laplace transformation of the correction δG(q, t) to the generating function it is helpful to
simplify further Eq. (A4). We observe first that δG(q, t) does diverge for strictly incom-
pressible systems (v → ∞) and one must keep v finite in the effective potential whenever
necessary to ensure convergence (actually everywhere but in the diagram corresponding to
the interaction between two external monomers). Since we are not interested in the wave
vectors larger than 1/ξ we expand δG(q, t) for ξ → 0 which leads to the much simpler
expression
δG(q, t) ≈ − vξq
3πa2(a2q2 + t)2
t(3a2q2 + t)
(a2q2 + t)2
Arctan[ aq√
+ o(vξ3). (A8)
The first term diverges as
v for diverging v. It renormalizes the effective bond length in
the zero order term which is indicated in the first line of Eq. (13). The next two terms scale
both as v0. Subsequent terms must all vanish for diverging v and can be discarded. It is
then easy to perform an inverse Fourier-Laplace transformation of the two relevant v0 terms.
This yields
δG(x, s) = G0(x, s)
with x = r/a
6n. This is consistent with the expression given in the second line of
Eq. (13).
We note finally that the intramolecular form factor F (q) = 1
n,m=1 〈exp(iq · (rn − rm)〉
of asymptotically long chains can be readily obtained from Eq. (A8). Observing that
〈exp(iq · (rn − rm)〉 =
d3r exp(iq · r)G(r, s) = G(q, s) one finds
δF (q) = 2
ds δG(q, s) = 2 δG(q, t = 0) = −2 vξ
, (A10)
where we used the third term of Eq. (A8) in the last step. The first term in Eq. (A8) is
discarded as before, since it renormalizes the effective bond length in the reference form
factor: F0(q) = 12/b
2q2 ⇒ 12/b2eq2. It follows, hence, that within first-order perturbation
theory
F (q) = F0(q) + δF (q) ≈ F0(q)
(A11)
as indicated by Eq. (30) in the Conclusion. This is equivalent to the result 1/F (q)−1/F0(q) ≈
q3/32ρ discussed in Refs. [14, 19] for polymer melts and anticipated by Schäfer [11] by
renormalization group calculations of semidilute solutions.
N nch τe Re Rg be(N)
6bg(N)
16 216 1214 11.7 4.8 2.998 2.939
32 215 3485 17.1 7.0 3.066 3.030
64 214 1.1 · 104 24.8 10.1 3.116 3.094
128 8192 3.3 · 104 35.6 14.5 3.153 3.139
256 4096 1.0 · 105 50.8 20.7 3.179 3.171
512 2048 3.2 · 105 72.2 29.5 3.200 3.193
1024 1024 1.0 · 106 103 42.0 3.216 3.212
2048 512 3.2 · 106 146 59.5 3.227 3.223
4096 256 9.7 · 106 207 85.0 3.235 3.253
8192 128 2.9 · 107 294 120 3.249 3.248
TABLE I: Various static properties of dense BFM melts of number density ρ = 0.5/8: the chain
length N , the number of chains nch per box, the relaxation time τe characterized by the diffusion
of the monomers over the end-to-end distance and corresponding to the circles indicated in Fig. 3,
the root-mean-squared chain end-to-end distance Re and the radius of gyration Rg of the total
chain (s = N − 1). The last two columns give estimates for the effective bond length from the
end-to-end distance, be(N) ≡ Re/(N − 1)1/2, and the radius of gyration, bg(N) ≡ Rg/
N . The
dashed line in Fig. 4 indicates be(N)
2. Apparently, both estimates increase monotonicly with N
reaching be(N) ≈
6bg(N) ≈ 3.2 for the largest chains available. Note that
6bg(N) < be(N) for
smaller N .
Property BFM BSM
Length unit lattice constant bead diameter
Temperature kBT 1 1
Number density ρ 0.5/8 0.84
Linear box size L 256 ≤ 62
Number of monomers nmon 1048576 ≤ 196608
Largest chain length N 8192 1024
Mean bond length 〈|ln|〉 2.604 0.97
l = 〈l2n〉1/2 2.636 0.97
Effective bond length be 3.244 1.34
ρb3e 2.13 2.02
C∞ = (be/l)
2 1.52 1.91
lp = l(C∞ + 1)/2 3.32 1.41
24/π3/ρb3e 0.41 0.44
c1/ce 1.0 1.2
c2/ce 1.0 1.1
c3/ce 1.0 1.0
c4/ce 1.1 1.2
c5/ce 1.1 0.9
cP = c1(be/l)
2/8 0.078 0.124
Dimensionless compressibility g 0.245 0.08
Compression modulus v ≡ 1/gρ 66.7 14.9
vρ/b3eρ 0.96 1.8
TABLE II: Comparison of some static properties of dense BFM and BSM melts. The first six rows
indicate conventions and operational parameters. The effective bond length be and the swelling
coefficients cp (defined in Eq. (5)) are determined from the first five even moments of the segmental
size distribution. The dimensionless compressibility g = S(q → 0)/ρ has been obtained from the
total static structure factor S(q) = 1
∑nmon
k,l=1 〈exp(iq · (rk − rl))〉 in the zero wave vector limit
as shown at the end of Ref. [14]. The values indicated correspond to the asymptotic long chain
behavior. Properties of very small chains deviate slightly.
I ~ −3 I ~ −9
I ~ 45
I ~ −9−
r n m
FIG. 1: (Color online) Sketch of a polymer chain of length N in a dense melt in d = 3 dimensions.
As notations we use ri for the position vector of a monomer i, li = ri+1 − ri for its bond vector,
r = rm−rn for the end-to-end vector of the chain segment between the monomers n and m = n+s
and r = ||r|| for its length. Segment properties, such as the 2p-th moments
, are averaged over
all possible pairs of monomers (n,m) of a chain and over all chains. The second moment (p = 1)
is denoted Re(s) =
, the total chain end-to-end distance is Re(s = N − 1). The dashed
lines show the relevant graphs of the analytical perturbation calculation outlined in Sec. II B.
The numerical factors indicate for infinite chains (without chain end effects) the relative weights
contributing to the 1/
s-swelling of Re(s) indicated in Eq. (10).
a bdensity ρ
c(r,s)
correlation hole
Segmental
= const
R(s) Repulsion
FIG. 2: (Color online) Role of incompressibility and chain connectivity in dense polymer solutions
and melts. (a) Sketch of the segmental correlation hole of a marked chain segment of curvilin-
ear length s. Density fluctuations of chain segments must be correlated, since the total density
fluctuations (dashed line) are small. Consequently, a second chain segment feels an entropic re-
pulsion when both correlation holes start to overlap. (b) Self-similar pattern of nested segmental
correlation holes of decreasing strength u(s) ≈ s/ρR(s)3 ≈ ce/
s aligned along the backbone of
a reference chain. The large dashed circle represents the classical correlation hole of the total
chain (s ≈ N) [5]. This is the input of recent approaches to model polymer chains as soft spheres
[50, 52]. We argue that incompressibility on all scales and chain connectivity leads to a short
distance repulsion of the segmental correlation holes, which increases with decreasing s.
L06 (conserved topology)
L26,SS
L26,SS,DB
FIG. 3: (Color online) Diffusion time τe over the (root-mean-squared) chain end-to-end distance
Re(N − 1) as a function of chain length N for different versions of the Bond Fluctuation Model
(BFM). All data indicated are for the high number density (ρ = 0.5/8) corresponding to a polymer
melt with half the lattice sites being occupied. We have obtained τe = R
e(N−1)/6Ds from the self-
diffusion coefficient Ds measured from the free diffusion limit of the mean-squared displacement
of all monomers
δr(t)2
= 6Dst. Data from the classical BFM with topology conserving local
Monte Carlo (MC) moves in 6 spatial directions (L06) [26] are represented as stars. All other data
sets use topology violating local MC moves in 26 lattice directions (L26). If only local moves are
used, L26-dynamics is even at relatively short times perfectly Rouse like which allows the accurate
determination of Ds although the monomers possibly have not yet moved over Re(N − 1) for the
largest chain lengths considered. Additional slithering snake (SS) moves increase the efficiency of
the algorithm by approximately an order of magnitude (squares,bold line). The power law exponent
is changed from 2 to an empirical 1.62 (dashed line) if in addition we perform double-bridging (DB)
moves.
N=128
N=256
N=512
N=1024
N=2048
N=4096
N=8192
=3.244
Eq.(5)
FIG. 4: (Color online) Mean-squared segment size Re(s)
2/s vs. curvilinear distance s. We present
BFM data for different chain length N at number density ρ = 0.5/8. The averages are taken over
all possible monomer pairs (n,m = n+s). The statistics deteriorates, hence, for large s. Log-linear
coordinates are used to emphasize the power law swelling over several orders of magnitude of s.
The data approach the asymptotic limit (horizontal line) from below, i.e. the chains are swollen.
This behavior is well fitted by Eq. (5) for 1 ≪ s ≪ N (bold line). Non-monotonic behavior is found
for s → N , especially for small N . The dashed line indicates the measured total chain end-to-end
distances, be(N)
2 ≡ Re(N − 1)2/(N − 1) from Tab. I, showing even more pronounced deviations
from the asymptotic limit. The dash-dotted line compares this data with Eq. (19).
=3.235
=3.240
=3.244
=3.250
=3.255
BFM ρ=0.5/8, N=2048:
=3.244, c
=0.41=c
FIG. 5: (Color online) Replot of the mean-squared segment size as y = K1(s) = 1 − Re(s)2/b2es
vs. x = c1/
s, as suggested by Eq. (5), for different trial effective bond lengths be as indicated.
Only BFM chains of length N = 2048 are considered for clarity. This procedure is very sensitive
to the value chosen and allows for a precise determination. It assumes, however, that higher order
terms in the expansion of K1(s) may be neglected. The value be is confirmed from a similar test
for higher moments (Fig. 6).
x ~1/s
BFM b
=3.244
BSM b
=1.34
too small s !
FIG. 6: (Color online) Critial test of Eq. (5) where the rescaled moments y = Kp(s) of the segment
size distribution (defined in Eq. (1)) are plotted vs. x =
3(2pp!p)2
2(2p+1)!
. We consider the first five even
moments (p = 1, . . . , 5) for the BFM with N = 2048 and the BSM with N = 1024. Also indicated
is the rescaled radius of gyration, y = 5/8 (1 − 6R2g(s)/b2e(s + 1)), as a function of x = c1/
(filled circles). The BSM data has been shifted upwards for clarity. Without this shift a perfect
data collapse is found for both models and all moments. Keeping the same effective bond length
be for all moments of each model we fit for the swelling coefficients cp by rescaling the horizontal
axis. We find be ≈ 3.244 for the BFM and 1.34 for the BSM. If be is chosen correctly, all data
sets extrapolate linearly to zero for large s (x → 0). The swelling coefficients found are close the
theoretical prediction ce, as indicated in Tab. II. The plot demonstrates that the non-Gaussian
deviations scale as the segmental correlation hole, u(s) ∼ ce/
s and this for all moments as long
as x ≪ 1. The saturation at large x is due to the finite extensibility of short chain segments. Since
this effect becomes more marked for larger moments, the fit of be is best performed for p = 1.
u(s) c
Kλ(s)
λ=2: N=1024
λ=2: N=2048
λ=2: N=4096
λ=4: N=2048
λ=8: N=2048
λ=16: N=2048
Insufficient statistics & Finite chain length effects
⇒ ω = 3/2
FIG. 7: (Color online) Plot of Kλ(s) as a function of u(s)c1/ce ∼ 1/
s using the measured
u(s) ≡
24/π3s/ρRe(s)
3. For λ = 2 (corresponding to two segments being connected) BFM and
BSM data are compared. Several λ values are given for N = 2048 BFM chains. For chain segments
with 1 ≪ s ≪ N all data sets collapse on the bisection line confirming the so-called “recursion
relation” Kλ ≈ u proposed by Semenov and Johner [12]. The statistics becomes insufficient for
large s (left bottom corner). Systematic deviations arise for s → N due to additional finite-N
effects.
N=128
N=512
N=2048
N=4096
N=8192
Slope ω=3/2
Chain end effects
FIG. 8: (Color online) The bond-bond correlation function P (s)/cP as a function of the curvilinear
distance s. Various chain lengths are given for BFM. Provided that 1 ≪ s ≪ N , all data sets
collapse on the power law slope with exponent ω = 3/2 (bold line) as predicted by Eq. (23). The
dash-dotted curve P (s) ≈ exp(−s/1.5) shows that exponential behavior is only compatible with
very small chain lengths. The dashed lines correspond to the theoretical prediction, Eq. (24), for
short chains with N = 16, 32, 64 and 128 (from left to right).
BFM N=1024
BFM N=2048
BFM N=4096
BFM N=8192
BSM N=1024
y = 6x/5
y = 111x/35
y = 605x/105
small s !
Noise !
FIG. 9: (Color online) Non-Gaussian parameter αp(s) computed for the end-to-end distance of
chain segments as a function of ce/
s. Perfect data collapse for all chain lengths and both sim-
ulation models is obtained for each p. A linear relationship over nearly two orders of magnitude
is found as theoretically expected. Data for three moments (p = 2, 3, 4) are indicated showing
a systematic increase of non-Gaussianity with p. The data curvature for small s becomes more
pronounced for larger p.
N=256
N=512
N=1024
N=2048
N=4096
N=8192
m=n+s
FIG. 10: (Color online) Plot of Kxy(s) = 1−
averaged over all pairs of monomers
(n,m = n+s) and three different direction pairs as a function of ce/
s. As indicated by the sketch
at the bottom of the figure, Kxy(s) measures the correlation of the components of the segment
vector r. All data points collapse and show again a linear relationship Kxy ≈ u(s). Different
directions are therefore coupled! No curvature is observed over two orders of magnitude confirming
that higher order perturbation corrections are negligible. Noise cannot be neglected for large
s > 100 and finite segment-size effects are visible for s ≈ 1.
0.0 0.5 1.0 1.5 2.0
n=r/b
BFM s=32
BFM s=64
BFM s=128
BSM s=32
BSM s=64
BSM s=128
GaussEnhancem
entDepletion
FIG. 11: (Color online) Segment size distribution y = G(r, s)(bes
1/2)3 vs. n = r/bes
1/2 for several
s as indicated in the figure. Only data for BFM with N = 2048 and BSM with N = 1024 are
presented. (A similar plot can be achieved by renormalizing the axes using Re(s) instead of bes
1/2).
The bold line denotes the Gaussian behaviour y = (3/2π)3/2 exp(−3n2/2). One sees that compared
to this reference the measured distributions are depleted for small n ≪ 1 (where the data does not
scale) and enhanced for n ≈ 1.
0.0 0.5 1.0 1.5 2.0 2.5
n=r/b
BFM s=31
BFM s=63
BFM s=127
BSM s=32
BSM s=64
BSM s=128
f(n) from Eq. (3)
Enhancement
epletion
FIG. 12: (Color online) Deviation δG(r, s) = G(r, s) − G0(r, s) of the measured segmental
size distribution from the Gaussian behavior G0(r, s) expected from Flory’s hypothesis for sev-
eral s and both models as indicated in the figure. As suggested by Eq. (3), we have plotted
y = (δG(r, s)/G0(r, s))/(ce/
s) as a function of n = r/be
s. The Gaussian reference distribution
has been computed according to Eq. (2) for the measured effective bond length be. A close to
perfect data collapse is found for both models. This shows that the deviation scales linearly with
u(s) ≈ ce/
s, as expected. The bold line indicates the universal function of f(n) predicted by
Eq. (3).
0.0 0.5 1.0 1.5 2.0 2.5
m=r/R
BFM s=31
BFM s=63
BFM s=127
BSM s=32
BSM s=64
BSM s=128
f(m) from Eq. (29)
Enhancement
letio
FIG. 13: (Color online) Replot of the relative deviation of the measured segment size distribution,
y = (δG(r, s)/G0(r, s)))/(ce/
s), as a function of m = r/Re(s). The figure highlights that the
measured segment size is the only length scale relevant for describing the deviation from Flory’s
hypothesis. The same data sets and symbols are used as in the previous Fig. 12.
v(k) v(k) v(k)~ ~ ~
q q q q q
q−k q−k
FIG. 14: Interaction diagrams used in reciprocal space for the calculation of δG(q, t) in the scale
free limit. There exist three nonzero contributions to first-order perturbation, the first involving
two points inside the segment (first two lines of Eq. (A4)), the second one point inside and one
outside the segment (third line of Eq. (A4)) and the third one point on either side of the segment
(last line of Eq. (A4)). Momentum q flows from one correlated point to the other. Integrals are
performed over the momentum k. Dotted lines denote the effective interactions ṽ(k) given by
Eq. (9), bold lines the propagators which carry each a momentum q or q − k as indicated.
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[61] It is interesting to compare the numerical value I(∞) ≈ 1.59 obtained for the r.h.s of Eq. (16)
with the coefficients one would obtain by computing Eq. (15) either with the effective potential
ṽ(q) for infinite chains given by Eq. (9) or with the Padé approximation, Eq. (17). Within
these approximations of the full linear response formula, Eq. (14), the coefficients can be
obtained directly without numerical integration yielding overall similar values. In the first
case we obtain 15/8 ≈ 1.87 and in the second 11/8 ≈ 1.37. While the first value is clearly
not compatible with the measured end-to-end distances, the second yields a reasonable fit,
especially for small N < 1000, when the data is plotted as in Fig. 5. Ultimately, for very long
chains the correct coefficient should be 1.59 as is indicated by the dash-dotted line in Fig. 4.
[62] These topology non-conserving moves yield configurations which are not accessible with the
classical scheme with jumps in 6 directions only. Concerning the static properties we are
interested in this paper both system classes are practically equivalent. This has been confirmed
by comparing various static properties and by counting the number of monomers which become
“blocked” (in absolute space or with respect to an initial group of neighbor monomers) once
one returns to the original local scheme. Typically we find about 10 blocked monomers for a
system of 220 monomers. The relative difference of microstates is therefore tiny and irrelevant
for static properties. Care is needed, however, if the equilibrated configurations are used to
investigate the dynamics of the topology conserving BFM version. The same caveats arise for
the slithering snake and double-bridging moves.
[63] We have verified that the alternative definition 〈em=n+s · en〉 with en being the normalized
bond vector yields very similar results. This is due to the weak bond length fluctuations,
specifically at high densities, in both coarse-grained models under consideration. It is possi-
ble that other models show slightly different power law amplitudes cP depending on which
definition is taken.
Flory's ideality hypothesis revisited
Physical idea and sketch of the perturbation calculation
Scaling arguments
Perturbation calculation
Computational models and technical details
Bond fluctuation model
Bead spring model
Numerical results
The swelling of chain segments
Chain connectivity and recursion relation
Intrachain bond-bond correlations
Higher moments and associated coefficients
The segmental size distribution
Conclusion
Acknowledgments
Moments of the segmental size distribution and their generating function
References
|
0704.1621 | Stability Properties of Strongly Magnetized Spine Sheath Relativistic
Jets | to appear with shortened appendicies in The Astrophysical Journal
Stability Properties of Strongly Magnetized Spine Sheath Relativistic Jets
Philip E. Hardee
Department of Physics & Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA
[email protected]
ABSTRACT
The linearized relativistic magnetohydrodynamic (RMHD) equations describing a
uniform axially magnetized cylindrical relativistic jet spine embedded in a uniform axi-
ally magnetized relativistically moving sheath are derived. The displacement current is
retained in the equations so that effects associated with Alfvén wave propagation near
light speed can be studied. A dispersion relation for the normal modes is obtained. An-
alytical solutions for the normal modes in the low and high frequency limits are found
and a general stability condition is determined. A trans-Alfvénic and even a super-
Alfénic relativistic jet spine can be stable to velocity shear driven Kelvin-Helmholtz
modes. The resonance condition for maximum growth of the normal modes is obtained
in the kinetically and magnetically dominated regimes. Numerical solution of the dis-
persion relation verifies the analytical solutions and is used to study the regime of high
sound and Alfvén speeds.
Subject headings: galaxies: jets — gamma rays: bursts — ISM: jets and outflows —
methods: analytical — MHD — relativity — instabilities
1. Introduction
Relativistic jets are associated with active galactic nuclei and quasars (AGN), with black
hole binary star systems (microquasars), and are thought responsible for the gamma-ray bursts
(GRBs). In microquasar and AGN jets proper motions of intensity enhancements show mildly
superluminal for the microquasar jets ∼ 1.2 c (Mirabel & Rodriquez 1999), range from subluminal
(≪ c) to superluminal (. 6 c) along the M87 jet (Biretta et al. 1995, 1999), are up to ∼ 25 c along
the 3C 345 jet (Zensus et al. 1995; Steffen et al. 1995), and have inferred Lorentz factors γ > 100
in the GRBs (e.g., Piran 2005). The observed proper motions along microquasar and AGN jets
imply speeds from ∼ 0.9 c up to ∼ 0.999 c, and the speeds inferred for the GRBs are ∼ 0.99999 c.
Jets at the larger scales may be kinetically dominated and contain relatively weak magnetic
fields, e.g., equipartition between magnetic and gas pressure or less, but the possibility of much
stronger magnetic fields exists close to the acceleration and collimation region. Here general rel-
ativistic magnetohydrodynamic (GRMHD) simulations of jet formation (e.g., Koide et al. 2000;
Nishikawa et al. 2005; De Villiers, Hawley & Krolik 2003; De Villiers et al. 2005; Hawley & Krolik
2006; McKinney 2006; Mizuno et al. 2006) and earlier theoretical work (e.g., Lovelace 1976; Bland-
ford 1976; Blandford & Znajek 1977; Blandford & Payne 1982) invoke strong magnetic fields. In
addition to strong magnetic fields, GRMHD simulation studies of jet formation indicate that highly
collimated high speed jets driven by the magnetic fields threading the ergosphere may themselves
reside within a broader wind or sheath outflow driven by the magnetic fields anchored in the ac-
cretion disk (e.g., McKinney 2006; Hawley & Krolik 2006; Mizuno et al. 2006). This configuration
might additionally be surrounded by a less collimated accretion disk wind from the hot corona
(e.g., Nishikawa et al. 2005).
http://arxiv.org/abs/0704.1621v1
– 2 –
That relativistic jets may have jet-wind structure is indicated by recent observations of high
speed winds in several QSO’s with speeds, ∼ 0.1 − 0.4c, (Chartas, Brandt & Gallagher 2002,
Chartas et al. 2003; Pounds et al. 2003a; Pounds et al. 2003b; Reeves, O’Brien &Ward 2003). Other
observational evidence such as limb brightening has been interpreted as evidence for a slower external
sheath flow surrounding a faster jet spine, e.g., Mkn 501 (Giroletti et al. 2004), M 87 (Perlman et al.
2001), and a few other radio galaxy jets (e.g., Swain, Bridle & Baum 1998; Giovannini et al. 2001).
Additional circumstantial evidence such as the requirement for large Lorentz factors suggested by
the TeV BL Lacs when contrasted with much slower observed motions suggests the presence of
a spine-sheath morphology (Ghisellini, Tavecchio & Chiaberge 2005). At hundreds of kiloparsec
scales Siemignowska et al. (2007) have proposed a two component (spine-sheath) model to explain
the broad-band emission from the PKS 1127-145 jet. A spine-sheath jet structure has been proposed
based on theoretical arguments (e.g., Sol et al. 1989; Henri & Pelletier 1991; Laing 1996; Meier
2003). Similar type structure has been investigated in the context of GRB jets (e.g., Rossi, Lazzati
& Rees 2002; Lazzatti & Begelman 2005; Zhang, Wooseley & MacFadyen 2003; Zhang, Woosley &
Heger 2004; Morsony, Lazzati & Begelman 2006).
In order to study the effect of strong magnetic fields and the effect of a moving wind or
sheath around a jet or jet spine, I begin by adopting a simple system with no radial dependence of
quantities inside the jet spine and no radial dependence of quantities outside the jet in the sheath.
This “top hat” configuration with magnetic fields parallel to the flow can be described exactly
by the linearized relativistic magnetohydrodynamic (RMHD) equations. This system with no
magnetic and flow helicity is stable to current driven (CD) modes of instability (Istomin & Pariev
1994, 1996; Lyubarskii 1999). However, this system can be unstable to Kelvin-Helmholtz (KH)
modes of instability (Hardee 2004). This approach allows us to look at the potential KH modes
without complications arising from coexisting CD modes (see Baty, Keppens & Compte 2004) and
predictions can be verified by numerical simulations (Mizuno, Hardee & Nishikawa 2006).
This paper is organized as follows. In §2, I present the dispersion relation arising from a
normal mode analysis of the linearized RMHD equations. Analytical approximate solutions to
the dispersion relation for various limiting cases are given in §3. I verify the analytical solution
through numerical solution of the dispersion relation in §4. I summarize the stability results in
§5 and discuss the applicability of the present results in §6. Derivation of the linearized RMHD
equations is shown in Appendix A, derivation of the normal mode dispersion relation is presented
in Appendix B, and derivation of the analytical solutions is shown in Appendix C.
2. The RMHD Normal Mode Dispersion Relation
Let us analyze the stability of a spine-sheath system by modeling the jet spine as a cylinder
of radius R, having a uniform proper density, ρj, a uniform axial magnetic field, Bj = Bj,z, and a
uniform velocity, uj = uj,z. The external sheath is assumed to have a uniform proper density, ρe, a
uniform axial magnetic field, Be = Be,z, a uniform velocity ue = ue,z, and extends to infinity. The
sheath velocity corresponds to an outflow around the central spine if ue,z > 0 or represents backflow
when ue,z < 0. The jet spine is established in static total pressure balance with the external sheath
where the total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π, and the initial
equilibrium satisfies the zeroth order equations. Formally, the assumption of an infinite sheath
means that a dispersion relation could be derived in the reference frame of the sheath with results
transformed to the source/observer reference frame. However, it is not much more difficult to derive
a dispersion relation in the source/observer frame in which analytical solutions to the dispersion
relation take on simple revealing forms. Additionally, this approach lends itself to modeling the
propagation and appearance of jet structures viewed in the source/observer frame, e.g., helical
structures in the 3C 120 jet (Hardee, Walker & Gómez 2005).
The general approach to analyzing the time dependent properties of this system is to linearize
– 3 –
the ideal RMHD and Maxwell equations, where the density, velocity, pressure and magnetic field
are written as ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1, and
B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript
0. Additionally, the Lorentz factor γ2 = (γ0 + γ1)
2 ≃ γ20 + 2γ40u · v1/c2 where γ1 = γ30u · v1/c2.
The linearization is shown in Appendix A. In cylindrical geometry a random perturbation ρ1, v1
B1 and P1 can be considered to consist of Fourier components of the form
f1(r, φ, z, t) = f1(r) exp[i(kz ± nφ− ωt)] (1)
where flow is along the z-axis, and r is in the radial direction with the flow bounded by r = R. In
cylindrical geometry n is an integer azimuthal wavenumber, for n > 0 waves propagate at an angle
to the flow direction, and +n and −n give wave propagation in the clockwise and counter-clockwise
sense, respectively, when viewed in the flow direction. In equation (1) n = 0, 1, 2, 3, 4, etc.
correspond to pinching, helical, elliptical, triangular, rectangular, etc. normal mode distortions
of the jet, respectively. Propagation and growth or damping of the Fourier components can be
described by a dispersion relation of the form
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
. (2)
Derivation of this dispersion relation is given in Appendix B. In the dispersion relation Jn and H
are Bessel and Hankel functions, the primes denote derivatives of the Bessel and Hankel functions
with respect to their arguments. In equation (2)
χj ≡ γ2j γ2AjWj
̟2j − κ2jv2Aj
, (3a)
χe ≡ γ2eγ2AeWe
̟2e − κ2ev2Ae
, (3b)
β2j ≡
̟2j − κ2ja2j
̟2j − κ2jv2Aj
v2msj̟
j − κ2jv2Aja2j
, (4a)
β2e ≡
̟2ex − κ2ea2e
̟2e − κ2ev2Ae
v2mse̟
e − κ2ev2Aea2e
. (4b)
In equations (3a & 3b) and equations (4a & 4b) ̟2j,e ≡ (ω − kuj,e)
and κ2j,e ≡
k − ωuj,e/c2
γj,e ≡ (1 − u2j,e/c2)−1/2 is the flow Lorentz factor, γAj,e ≡ (1 − v2Aj,e/c2)−1/2 is the Alfvén Lorentz
factor, W ≡ ρ+[Γ/ (Γ− 1)]P/c2 is the enthalpy, a is the sound speed, vA is the Alfvén wave speed,
and vms is a magnetosonic speed. The sound speed is defined by
where 4/3 ≤ Γ ≤ 5/3 is the adiabatic index. The Alfvén wave speed is defined by
1 + V 2A/c
where V 2A ≡ B20/(4πW0). A magnetosonic speed corresponding to the fast magnetosonic speed for
propagation perpendicular to the magnetic field (e.g., Vlahakis & Königl 2003) is defined by
vms ≡
a2 + v2A − a2v2A/c2
a2/γ2A + v
– 4 –
3. Analytical Solutions to the Dispersion Relation
In this section analytical solutions to the dispersion relation in the low frequency limit, in the
fluid and magnetic limits at resonance (maximum growth), and in the high frequency limit are
summarized. The analytical solutions are derived in Appendix C.
3.1. Low Frequency Limit
Analytically each normal mode n contains a single fundamental/surface wave (ω −→ 0, k −→ 0,
ω/k > 0) solution and multiple body wave (ω −→ 0, k > 0, ω/k −→ 0) solutions that satisfy the
dispersion relation. In the low frequency limit the fundamental pinch mode (n = 0) solution is
given by
uj ± vw
1± vwuj/c2
where the pinch fundamental mode wave speed
v2w ≈ a2j
v2msj
v2msj
, (6)
δ ≡ −
̟2e − κ2ev2Ae
v2msj
with |δ| ∝
∣k2R2
∣ << 1. In this limit δ is complex and this mode consists of a growing and damped
wave pair. The imaginary part of the solution is vanishingly small in the low frequency limit. The
above form indicates that growth, which arises from the complex value of δ, will be reduced as
(v2Aj/v
msj) −→ 1. The unstable growing solution is associated with the backwards moving (in the
jet fluid reference frame) wave.
In the low frequency limit the surface helical, elliptical, and higher order normal modes (n > 0)
have a solution given by
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
where η ≡ γ2jWj
γ2eWe and a “surface” Alfvén speed is defined by
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
. (9)
In equation (9) note that the Alfvén Lorentz factor γ2Aj,e = 1 + V
Aj,e/c
2. Thus, the jet is stable to
n > 0 surface wave mode perturbations when
γ2j γ
e (uj − ue)
< γ2Ajγ
Ae +We/γ
) B2j +B
4πWjWe
. (10)
For example, with uj ≈ c >> ue, γ2e ≈ 1, γ2Aj >> γ2Ae ≈ 1, B2j >> B2e , and using γ2Aj =
1 +B2j /4πWjc
2 the jet is stable when
γ2j <
4πWec2
γ2Aj (11a)
– 5 –
or with Be = Bj , We = Wj , so that vA,j = vA,e, and with γA ≡ γA,e = γA,j the jet is stable when
γ2j γ
e (uj − ue)2 < 4γ2A(γ2A − 1)c2 . (11b)
Thus, the jet can remain stable to the surface wave modes even when the jet Lorentz factor exceeds
the Alfvén Lorentz factor.
In the low frequency limit the real part of the body wave solutions is given by
kR ≈ kminnmR ≡
v2msju
j − v2Aja2j
γ2j (u
j − a2j)(u2j − v2Aj)
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (12)
where n specifies the normal mode, m = 1, 2, 3, ... specify the first, second, third, etc. body wave
solutions, and
n(βjR)
n (βeR)
n (βeR)
In the absence of a significant external magnetic field and a significant external flow ǫn = 0 as
χe = γ
u2e − v2Ae
k2 = 0. In this low frequency limit the body wave solutions are either
purely real or damped, exist only when kminnmR has a positive real part, and with |ǫn| << 1 require
v2msju
j − v2Aja2j
γ2j (u
j − a2j )(u2j − v2Aj)
> 0 . (13)
Thus, the body modes can exist when the jet is supersonic and super-Alfvénic, i.e., u2j − a2j > 0
and u2j − v2Aj > 0, or in a limited velocity range given approximately by a2j > u2j > [γ2sj/(1+ γ2sj)]a2j
when v2Aj ≈ a2j , where γsj ≡ (1− a2j/c2)−1/2 is a sonic Lorentz factor.
3.2. Resonance
With the exception of the pinch fundamental mode which can have a relatively broad plateau
in the growth rate, all body modes, and all surface modes can have a distinct maximum in the
growth rate at some resonant frequency.
The resonance condition can be evaluated analytically in either the fluid limit where a >> VA
or in the magnetic limit where VA >> a. Note that in the magnetic limit, magnetic pressure
balance implies that Bj = Be. In these cases a necesary condition for resonance is that
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
, (14)
where vwj ≡ (aj , vAj) and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively. When this
condition is satisfied it can be shown that the wave speed at resonance is
vw ≈ v∗w ≡
γj(γwevwe)uj + γe(γwjvwj)ue
γj(γwevwe) + γe(γwjvwj)
where γw ≡ (1 − v2w/c2)−1/2 is the sonic or Alfvénic Lorentz factor accompanying vwj ≡ (aj , vAj)
and vwe ≡ (ae, vAe) in the fluid or magnetic limits, respectively.
– 6 –
The resonant wave speed and maximum growth rate occur at a frequency given by
ωR/vwe ≈ ω∗nmR/vwe ≡
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (vwe/v∗w − uevwe/c2)
. (16)
In equation (16) n specifies the normal mode, m = 0 specifies the surface wave, and m ≥ 1 specifies
the body waves. In the limit of insignificant sheath flow, ue = 0, and using eq. (15) for v
w in eq.
(16) allows the resonant frequency to be written as
ω∗nmRj/vwe =
(2n + 1)π/4 +mπ
v2we/u
j + 2
vwevwj/γju
v2wj/γ
)]1/2
and this predicts a resonant frequency that is primarily a function of the sound and Alfvén wave
speeds in the sheath. The effect of sheath flow is best illustrated by assuming comparable conditions
in the spine and sheath, γwjvwj ∼ γwevwe, and assuming that γjuj >> γeue in which case
ω∗nmRj/vwe ∼
(2n + 1)π/4 +mπ
1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e) /u2j
The term ue/uj in the denominator indicates that the resonant frequency increases as the shear
speed, uj − ue, declines. In the limit
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
the resonant frequency ω∗nmR/vwe −→ ∞.
The resonant wavelength is given by λ ≈ λ∗nm ≡ 2πv∗w/ω∗nm and can be calculated from
λ∗nm ≡
(2n+ 1)π/4 +mπ
(v∗w − ue)
vwe − (vweue/c2)v∗w
R . (17)
Equations (15 - 17) provide the proper functional dependence of the resonant wave speed, frequency
and wavelength provided (ue/uj)
2 << 1 and (vwe/uj)
2 << 1.
With the exception of the n = 0, m = 0, fundamental pinch mode, a maximum spatial growth
rate, kmaxI , is approximated by
kmaxI R ≈ k∗IR ≡ −
ln |R| , (18)
where
|R| ≈
4 (ω∗nmR/vwe)
(1− 2ue/uj) + (ln |R| /2)2
(ln |R| /2)2
. (19)
Equations (18) and (19) show that the maximum growth rate is primarily a function of the jet
sound, Alfvén and flow speed through vwj/γjuj, and secondarily a function of the sheath sound,
Alfvén and flow speed through (ω∗nmR/vwe)
(1− 2ue/uj).
– 7 –
I can illustrate the dependencies of the maximum growth rate on sound, Alfvén and flow speeds
by using
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− v2we/c2)− (v2we − u2e)/u2j
] × [(2n + 1)π/4 +mπ]2
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− v2we/u2j
× [(2n + 1)π/4 +mπ] . (20)
Thus, |R| increases as ω∗nm increases for higher order modes with larger n and larger m and this
result indicates an increase in the growth rate for larger n and larger m. When the sound or Alfvén
wave speed, vwe, increases |R| increases. This result indicates an increase in the growth rate at
the higher resonant frequency accompanying an increase in the sound or Alfvén wave speed in the
sheath.
The behavior of the maximum growth rate as the shear speed, uj−ue, declines is best illustrated
by considering the effect of an increasing wind speed where (v2we − u2e)/u2j << 1 is ignored. In this
|R|2 − 1
ln |R| ≈ 4 [(2n + 1)π/4 +mπ] (21)
and |R| will remain relatively independent of ω∗nm even as ω∗nm −→ ∞ as the shear speed decreases.
This result indicates a relatively constant resonant growth rate as the shear speed decreases.
In the fluid limit decline in the shear speed ultimately results in a decrease in the growth rate
and increase in the spatial growth length. This decline in the growth rate is also indicated by
equation (8) which, in the fluid limit, becomes
ηuj + ue
1 + η
± i η
1 + η
(uj − ue) . (22)
Equation (22) applies to frequencies below the resonant frequency ω∗nm and directly reveals the
decline in growth rates as uj − ue −→ 0.
In the magnetic limit the resonant frequency ω∗nmR/vAe −→ ∞ as
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
. (23)
Here equation (8) indicates that the jet is stable when
γ2j γ
e (uj − ue)
< V 2As ,
and the jet will be stable as ω∗nm −→ ∞ when
γ2j γ
1− ujue/c2
< 2γ2Ajγ
v2Ae + v
(vAj + vAe)
1 + vAjvAe/c
, (24)
where I have used an equality in equation (23) in equation (8) to obtain equation (24). Equation
(24) indicates that a high jet speed relative to the Alfvén wave speed is necessary for instability.
For example, if vA ≡ vAj = vAe and ue = 0, the jet is stable at high frequencies provided
γ2j <
1 + v2A/c
γ4A . (25a)
– 8 –
This high frequency condition is slightly different from the low frequency stabilization condition
found when vA ≡ vAj = vAe and ue = 0 from equation (11b)
γ2j (uj/c)
2 < 4γ2A(γ
A − 1) . (25b)
Note that eqs. (25a & 25b) are identical in the large Lorentz factor limit. Equations (25) predict
that stabilization at high frequencies occurs at somewhat higher jet speeds than stabilization at
lower frequencies. Determination of stabilization at intermediate frequencies requires numerical
solution of the dispersion relation. A non-negligable postive external flow requires even higher jet
speeds for the jet to be unstable. Thus, a strongly magnetized relativistic trans-Alfvénic jet is
predicted to be KH stable and a super-Alfvénic jet can be KH stable.
3.3. High Frequency Limit
Provided the condition, eq. (14), for resonance is met, the real part of the solutions to the
dispersion relation in the high frequency limit for fundamental, surface, and body modes is given
uj ± vwj
1± vwjuj/c2
. (26)
and describes sound waves vwj = aj or Alfvén waves vwj = vAj propagating with and against the
jet flow inside the jet. Unstable growing solutions are associated with the backwards moving (in
the jet fluid reference frame) wave but the growth rate is vanishingly small in this limit.
4. Numerical Solution of the Dispersion Relation
The detailed behavior of solutions within an order of magnitude of the resonant frequency
and for comparable sound and Alfvén wave speeds must be investigated by numerical solution of
the dispersion relation. Analytical solutions found in the previous section can be used for initial
estimates and to provide the functional behavior of solutions. Numerical solution of the dispersion
relation also allows a determination of the accuracy and applicability of the analytical expressions
in §3.
In this section pinch fundamental, helical surface and elliptical surface, and the associated
first body modes are investigated in the fluid, magnetic and magnetosonic regimes. These modes
are chosen as they have been identified with structure seen in relativistic hydrodynamic (RHD)
numerical simulations or tentatively identified with structures in resolved AGN jets. For example,
trailing shocks in a numerical simulation (Agudo et al. 2002) and in the 3C 120 jet (Gómez et al.
2001) have been identified with the first pinch body mode. The development of large scale helical
twisting of jets has been attributed to or may be associated with growth of the helical surface mode,
e.g., 3C 449 (Hardee 1981) and Cygnus A (Hardee 1996) Additionally, the development of twisted
filamentary structures has been attributed to helical and elliptical surface and first body modes,
e.g., 3C 273 (Lobanov & Zensus 2001), M87 (Lobanov, Hardee & Eilek 2003), 3C 120 (Hardee,
Walker & Gómez 2005), and have been studied in RHD numerical simulations, e.g., Hardee &
Hughes (2003); Perucho et al. (2006).
4.1. Fluid Limit
In this section the basic behavior of the pinch (F) fundamental, helical (S) surface and elliptical
(S) surface modes is investigated: (1) as a function of varying sound speed in the external sheath
or jet spine for a fixed sound speed in the jet spine or external sheath and no sheath flow, (2) as a
function of equal sound speeds in the jet spine and external sheath for no sheath flow, and (3) as
– 9 –
a function of sheath flow for a relatively high sound speed equal in jet spine and external sheath.
In general only growing solutions are shown and complexities associated with multiple crossing
solutions are not shown. For all solutions shown the jet spine Lorentz factor and speed are set
to γ = 2.5 and uj = 0.9165 c. Sound speeds are input directly with the only constant being the
sheath number density. Total pressure and spine density are quantities computed for the specified
sound speeds. The adiabatic index is chosen to be Γ = 13/9 when 0.1 ≤ aj,e/c ≤ 0.5 consistent
with relativistically hot electrons and cold protons (Synge 1957). For sound speeds aj,e ∼ c/
3 the
adiabatic index is set to Γ = 4/3. Solutions shown assume zero magnetic field. Test calculations
with magnetic fields giving magnetic pressures a few percent of the gas pressure and Alfvén wave
speeds an order of magnitude less than the sound speeds give almost identical results.
In Figure 1 solutions in the left column are for a fixed jet spine sound speed aj = 0.3 c and in
the right column are for a fixed external sheath sound speed ae = 0.3 c. The solutions shown in
Figure 1 confirm the accuracy of the low frequency solutions to the pinch fundamental mode, eqs.
(5 & 6), and the helical and elliptical surface modes, eq. (8). Note that fast or slow wave speeds
are possible at low frequencies depending on whether η ≃ (γjae/γeaj)2 in eq. (8) is much greater
or much less than one, respectively. The numerical solutions to the dispersion relation show that
the maximum growth rate is primarily a function of the jet spine sound speed and only secondarily
a function of the external sheath sound speed as indicated by eqs. (18 - 20). Where a distinct
supersonic resonance exists, the resonant frequency is primarily a function of the external sheath
sound speed as predicted from eq. (16). The analytical expression for the resonant frequency for the
helical and elliptical surface modes provides the correct functional variation to within a constant
multiplier provided ae ≤ c/
3 and aj < c/3. A dramatic increase in the resonant frequency and
modest increase in the growth rate for larger jet spine sound speeds indicates the transition to
transonic behavior. Equation (15) for the resonant wave speed and equation (17) for the resonant
wavelength also provide a reasonable approximation to the functional variations provided ae ≤ c/
and aj < c/3. These results confirm the resonant solutions found in §3.2. At frequencies more than
an order of magnitude above resonance the growth rate is greatly reduced and solutions approach
the high frequency limiting form given by eq. (26). Note that eq. (26) allows only relatively high
wave speeds at high frequencies because aj ≤ c/
In Figure 2 the behavior of solutions to the fundamental/surface (left column) and associated
first body mode (right column) shows how solutions change as the sound speed increases in both the
jet spine and external sheath. Here I illustrate the transition from supersonic to transonic behavior
for no flow in the sheath. At low frequencies the modes behave as predicted by the analytic
solutions given in §3.1. The solutions show the expected shift to a higher resonant frequency that
is primarily a function of the increased external sheath sound speed and an accompanying increase
in the resonant growth rate that is primarily a function of the increased jet spine sound speed. The
resonance disappears as sound speeds approach c/
3 as the jet becomes transonic as predicted
by the resonance condition in §3.2. In the transonic regime high frequency fundamental/surface
mode growth rates and wave speeds are identical with wave speeds given by eq. (26). Provided the
jet is sufficiently supersonic, i.e., aj,e < 0.5 c, the maximum growth rate of the first body mode
is greater than that of the pinch fundamental mode, is comparable to that of the helical surface
mode, and is less than that of the elliptical surface mode. A narrow damping peak shown for the
helical first body (B1) solution when aj,e = 0.4 c is indicative of complexities in the body mode
solution structure. In the transonic regime growth of the first body mode is less than that of the
pinch fundamental, helical surface and elliptical surface modes.
Figure 3 illustrates the behavior of fundamental/surface and first body modes as a function
of the sheath speed for equal sound speeds in spine and sheath of aj,e = 0.4 c. For this value
of the sound speeds a sheath speed ue = 0 provides a supersonic solution structure baseline. At
low frequencies the surface modes behave as predicted by eq. (8), and the wave speed rises as ue
– 10 –
Fig. 1.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes
for different sound speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow.
The real part of the wavenumber, krRj , is shown by the dashed lines and the imaginary part , kiRj , is shown by the
dash-dot lines as a function of the dimensionless angular frequency, ωRj/uj . For the pinch mode the vertical lines
indicate the maximum growth rate range. Otherwise, the vertical lines indicate the location of maximum growth.
Immediately under the dispersion relation solution panel is a panel that shows the relativistic wave speed, γwvw/c.
Line colors indicate the sound speed in units of c: (black) 0.10, (blue) 0.20, (cyan) 0.30, (green) 0.40, & (red) 0.577.
– 11 –
Fig. 2.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in spine and sheath and no sheath flow.
Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations
of the maximum growth rate are indicated by the vertical solid lines. A vertical arrow (helical B1) indicates a narrow
damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sound
speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.577.
– 12 –
Fig. 3.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and first body (right column) modes as a function of the sheath speed for equal sound speeds in spine and sheath.
Real and imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 1. Locations
of the maximum growth rate are indicated by the vertical solid lines. Vertical arrows (helical B1) indicate damping
features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath speed in
units of c: (black) 0.0, (blue) 0.20, (cyan) 0.35, (green) 0.40, & (red) 0.60.
– 13 –
increases. As ue increases the resonant frequency increases in accordance with eq. (16). On the
other hand, the growth rate at resonance does not vary significantly in accordance with eqs. (18 &
19). When the sheath speed exceeds the sound speed, solutions make a transition from supersonic
to transonic structure. Note that the transition point between supersonic and transonic behavior
is similar but not identical for the helical and elliptical surface modes, i.e., ocurs at a slightly
lower sheath speed for the elliptical mode. The first body modes also show an increase in resonant
frequency with little change in the maximum growth rate provided the sheath speed remains below
the sound speed. A significant damping feature in the helical first body (B1) panel, is found. While
a similar damping feature was not found for the pinch and elliptical first body mode, this does not
indicate a significant difference as the root finding technique does not find all structure associated
with the body modes. The body mode solution structure is complex with multiple solutions not
shown here and modest damping or growth can occur where solutions cross, e.g., Mizuno, Hardee
& Nishikawa (2006). When the sheath speed exceeds the sound speed the maximum body mode
growth rate delines significantly. This result is quite different from the transonic solution behavior
illustrated in Figure 2 when aj,e = 0.577 c for no sheath flow. Thus, sheath flow effects stability
of the relativistic jet beyond that accompanying an increase to the maximum sound speed in the
absence of sheath flow. The reduction in growth of the body modes in the presence of sheath flow
provides the relativistic jet equivalent of non-relativistic transonic/subsonic jet solution behavior.
At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26). Note that
the high frequency wave speeds are nearly independent of ue.
4.2. Magnetic Limit
In this subsection the basic behavior of pinch, helical and elliptical modes is investigated: (1)
as a function of varying Alfvén speed in the external sheath or jet spine for a fixed Alfvén speed
in the jet spine or external sheath and no sheath flow, (2) as a function of equal Alfvén speeds
in the jet spine and external sheath for no sheath flow, and (3) as a function of sheath speed for
a relatively high Alfvén speed equal in jet spine and external sheath. In general only growing
solutions are shown and complexities associated with multiple crossing solutions are not shown.
For all solutions shown the jet spine Lorentz factor and speed are set to γ = 2.5 and uj = 0.9165 c.
Alfvén speeds are on the order of two magnitudes larger than the sound speed and are determined
by varying the sound speeds but with a gas pressure fraction on the order of 0.01% of the total
pressure. Only the sheath number density is held constant. The adiabatic index is set to Γ = 5/3
when aj,e/c << 0.1 consistent with low gas pressures and temperatures.
The solutions shown in Figure 4 confirm the theoretical predictions in the magnetic limit with
behavior depending on the Alfvén speed like the behavior found for the sound speed (see Figure
1). The pinch fundamental mode (not shown) has a growth rate almost entirely dependent on
sound speeds and is negligable in the magnetic limit as predicted by eq. (6). In Figure 4 solutions
in the left column are for a fixed jet spine Alfvén speed vAj = 0.3 c and in the right column are
for a fixed external sheath Alfvén speed vAe = 0.3 c. The solutions shown confirm the accuracy
of the low frequency solutions for helical and elliptical surface modes given by eq. (8). Note that
low frequency wave speeds can be high or low depending on the values of η = γ2jWj/γ
eWe, VAe/γe
and VAj/γj . The numerical solutions to the dispersion relation show that the maximum growth
rate is primarily a function of the jet spine Alfvén speed and only secondarily a function of the
external sheath Alfvén speed as predicted by eqs. (18 - 20). The resonant frequency is primarily a
function of the external sheath Alfvén speed as predicted by eq. (16). The analytical expression for
the resonant frequency of the helical and elliptical surface modes provides the correct functional
variation to within a constant multiplier provided vAj,e < 0.5 c. Decrease in the growth rate
for jet sheath Alfvén speeds vAe > 0.5 c indicates the transition towards trans-Alfvénic behavior.
Equation (15) for the resonant wave speed and equation (17) for the resonant wavelength also
– 14 –
Fig. 4.— Solutions to the dispersion relation for pinch fundamental, helical surface and elliptical surface modes
for different Alfvén speeds in the sheath (left column) and in the spine (right column) are shown for no sheath flow.
Sound speeds are aj,e ∼ 0.01vAj,e. As in Figures 1 - 3, the real part of the wavenumber, krRj , is shown by the
dashed lines and the imaginary part , kiRj , is shown by the dash-dot lines as a function of the dimensionless angular
frequency, ωRj/uj . The vertical lines indicate the location of maximum growth. The underlying panel shows the
relativistic wave speed, γwvw/c. Line colors indicate the Alfvén speed in units of c: (black) 0.10, (cyan) 0.30, (green)
0.50, & (red) 0.80.
provide a reasonable approximation to the functional variations for vAj,e < 0.5 c. At frequencies
more than an order of magnitude above resonance the growth rate is greatly reduced and solutions
approach the high frequency limiting form given by eq. (26). The surface modes have relatively
slow wave speeds, γwvw/c < 1 at high frequencies when the Alfvén wave speed vAj > 0.5 c. Unlike
the fluid case, the helical and elliptical surface modes are stabilized for Alfvén speeds somewhat in
excess of vAj,e ∼ 0.8 c in accordance with eqs. (8 & 24).
In Figure 5 the behavior of solutions to the pinch fundamental mode is shown in addition
to the helical and elliptical surface (left column) and associated first body modes (right column)
and the figure shows how solutions change as the Alfvén speed increases in both the jet spine and
external sheath. The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to
– 15 –
Fig. 5.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in jet and sheath and no sheath flow.
Pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e. Real and
imaginary parts of the wavenumber as a function of angular frequency are shown as in Figure 4. Locations of the
maximum growth rate are indicated by the vertical solid lines. A vertical arrow (elliptical B1) indicates a narrow
damping feature. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the Alfvén
speed in units of c: (black) 0.10, (blue) 0.20, (green) 0.40, & (red) 0.60.
– 16 –
Fig. 6.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column)
and the first body (right column) modes are shown for equal sound speeds in jet and sheath for different sheath
flow speeds. The pinch fundamental mode sound speed is aj,e = 0.2 c. Sound speeds for all other cases are
aj,e ∼ 0.01vAj,e = 0.005 c. Real and imaginary parts of the wavenumber as a function of angular frequency are
shown as in Figure 4. Locations of the maximum growth rate are indicated by the vertical solid lines. A vertical
arrow indicates low frequency damping of the pinch B1 solutions. The underlying panel shows the relativistic wave
speed, γwvw/c. Line colors indicate the sheath speed in units of c: (black) 0.0, (blue) 0.20, (cyan) 0.30, (green) 0.40,
& (red) 0.60.
– 17 –
illustrate the mode behavior with increasing Alfvén speed. Sound speeds for all body modes and
for helical and elliptical surface modes are aj,e ∼ 0.01vAj,e. Here the transition from super-Alfvénic
towards trans-Alfvénic behavior for no flow in the sheath is illustrated. At low frequencies the
modes behave as predicted by the analytic solutions given in §3.1. The growth rate of the pinch
fundamental mode is reduced as the Alfvén speed increases as predicted by eq. (6). The surface
and body mode solutions show the expected shift to a higher resonant frequency that is primarily
a function of the increased sheath Alfvén speed and an accompanying increase in the resonant
growth rate that is primarily a function of the increased spine Alfvén speed. The resonance moves
to higher frequency but the maximum growth rate is reduced for Alfvén speeds vAj,e > 0.60 c
and all modes become stable at higher Alfvén speeds in accordance with eqs. (8 & 24). At high
frequencies wave speeds are given by eq. (26). Provided the jet is sufficiently super-Alfvénic, i.e.,
vAj,e < 0.6 c, the maximum growth rate of the first body mode is much greater than that of the
pinch fundamental mode, is comparable to that of the helical surface mode, and is less than that
of the elliptical surface mode. A narrow damping peak shown for the elliptical body mode (B1)
solution when vAj,e = 0.6 c indicated by the arrow is indicative of complexities in the body mode
solution structure.
Figure 6 illustrates the behavior of fundamental/surface and first body modes as a function
of the sheath speed for an equal Alfvén speed in spine and sheath of vAj,e = 0.5 c. For this value
of the Alfvén speeds a sheath speed ue = 0 provides a super-Alfvénic solution structure baseline.
The sound speed is aj,e = 0.2 c for the pinch fundamental mode panel in order to illustrate the
mode behavior with increasing sheath speed. Sound speeds for all other cases are aj,e ∼ 0.01vAj,e.
Solutions in the first pinch body mode panel show the damping solution as opposed to the purely
real solution at the lower frequencies (indicated by the arrow). At higher frequencies the body mode
is growing. At low frequencies the surface modes behave as predicted by the analytic solutions given
in §3.1 and the growth rate of the surface modes decreases as ue increases. Additionally, the growth
rate at resonance decreases as expected for this relatively high Alfvén speed as the sheath speed
increases. At the higher frequencies wave speeds are identical, with wave speeds given by eq. (26).
Note that the high frequency wave speeds are relatively independent of ue. When the velocity shear
speed drops to less than the “surface” Alfvén speed, see eq. (11b), the helical and elliptical surface
modes and the first body modes are stabilized. This surface and body mode mode stabilization
occurs when sheath speeds exceed ue ∼ 0.5 c. However, note that the maximum pinch fundmental
mode growth rate is insensitive to the sheath speed and remains unstable at ue = 0.6 c even when
all other modes are stabilized.
4.3. A High Sound and Alfvén Speed Magnetosonic Case
In this subsection the basic behavior of the pinch fundamental, helical surface, elliptical surface
and associated first body modes is illustrated for different sheath speeds. The sheath speeds span a
solution structure from supersonic to transonic but still super-Alfvénic flow. Here the sound speed
in jet spine and external sheath are set equal with aj,e = 0.577 c and Alfvén speeds are set equal
with vAj,e = 0.5 c. The solutions for this case are shown in Figure 7. With no sheath flow the
fundamental/surface and first body modes show a typical supersonic and super-Alfvénic structure
albeit the pinch fundamental mode now has a maximum growth rate comparable to the helical
and elliptical surface modes as a consequence of the high sound speed. The associated first body
modes also have maximum growth rates comparable to the fundamental/surface modes. Increase
in the sheath speed results in a decrease in the growth rate of the helical and elliptical surface
modes at low frequencies as predicted by eq. (8). The low frequency growth rate of the pinch
fundamental also declines with increasing sheath speed. The resonant frequency increases with
increasing sheath speed as expected from the analytical and numerical studies performed in the
fluid and magnetic limits and the fundamental/surface modes take on a transonic structure for
– 18 –
Fig. 7.— Solutions to the dispersion relation for pinch fundamental, helical surface, elliptical surface (left column),
and the associated first body (right column) modes are shown for a maximal spine and sheath sound speed, aj,e =
0.577 c, and a slightly smaller spine and sheath Alfvén speed, vAj,e = 0.5 c, for different sheath flow speeds. As in
previous figures the real part of the wavenumber, krRj , is shown by the dashed lines, the imaginary part , kiRj ,
is shown by the dash-dot lines, and the vertical lines indicate the location of maximum growth. Arrows indicate
damping features. The underlying panel shows the relativistic wave speed, γwvw/c. Line colors indicate the sheath
speed in units of c: (black) 0.0, (blue) 0.20, (green) 0.40, & (red) 0.50. Fast and slow refer to the faster and slower
moving solutions and the yellow extension indicates a damped solution.
– 19 –
sheath speeds 0.4 c ≤ ue ≤ 0.1 c.
At high frequencies the fundamental/surface modes exhibit very high growth rates provided
sheath flow remains below the Alfvén speed. On the other hand, the maximum growth rate of the
first body modes declines as the sheath speed increases and is reduced severely when ue > 0.1 c.
This behavior is similar to what is found for non-relativistic jets as flow enters the transonic and
super-Alfvénic regime (Hardee & Rosen 1999). Additional increase in the sheath flow speed to
ue > 0.4 c results in a decrease in the growth rate of the fundamental/surface modes. Solutions
for the helical and elliptical surface modes shown in Figure 7 for a sheath speed ue = 0.5 c equal
to the Alfvén speed illustrate some of the complexity associated with barely super-Alfvénic flow.
Here limited growth is associated with both the slow and fast helical and elliptical surface solution
pair. At slower sheath speeds in the super-Alfvénic regime growth is associated with the slow
surface solution, i.e., backwards moving in the jet fluid reference frame. The yellow dash-dot
line extension at higher frequencies in the helical and elliptical surface panels indicates a damped
solution. Solutions were very difficult to follow in this parameter regime and it is possible that
some solutions were not found. When the sheath speed ue > 0.5 c all modes are stabilized.
A choice of Alfvén speeds greater than sound speeds results in a more magnetic like solution
structure like that shown in §4.2. A choice of Alfvén speeds more than a factor of two less than sound
speeds produces a more fluid like solution structure like that shown in §4.1. The more complicated
solution structure illustrated in Figure 7 only occurs for a relatively narrow range of high sound
speeds with similar or slightly lesser Alfvén speeds. In general, the detailed solution structure for
situations in which sound and Alfvén speeds are comparable must by examined individually, e.g.,
Mizuno, Hardee & Nishikawa (2006), and further investigation of these cases is beyond the scope
of the present paper.
5. Summary
The analytical and numerical work performed here provides for the first time a detailed analysis
of the KH stability properities of a RMHD jet spine-sheath configuration that allows for relativistic
motions of the sheath, sound speeds up to c/
3, and, by keeping the displacement current in the
analysis, Alfvén wave speeds approaching lightspeed and large Alfvén Lorentz factors. In the fluid
limit, the present results confirm an earlier more restricted low frequency analytical and numerical
simulation study performed by Hardee & Hughes (2003). Provided the jet spine is super-sonic and
super-Alfvénic internally and also relative to the sheath, the helical, elliptical and higher order
surface modes and the pinch, helical, elliptical and higher order first body modes have a maximum
growth rate at a resonant frequency. The pinch fundamental growth rate is significant only when
the sound speeds, aj,e ∼ c/
3. In general, the first body mode maximum growth rate is: greater
than the pinch fundamental mode, slightly greater than the helical surface mode, slightly less than
the elliptical surface mode, and occurs at a higher frequency than the maximum growth rate for
the fundamental/surface mode.
The basic KH stability behavior as a function of spine-sheath parameters is indicated by the
analytic low frequency surface mode solution and by the behavior of the resonant frequency. The
analytic surface mode solution valid at frequencies below resonance is given by
± iωi
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
where
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
, (28)
– 20 –
and η ≡ γ2jWj
γ2eWe , V
A ≡ B2/4πW , W ≡ ρ + [Γ/ (Γ− 1)]P/c2 and γA ≡ (1 − v2A/c2)−1/2.
Equation (27) provides a temporal growth rate, ωi(k), and a wave speed, vw = ωr/k. The reciprocal
provides a spatial growth rate ki(ω), and growth length ℓ = k
i . Increase or decrease of the
growth rate, dependence on physical parameters and stabilization at frequencies/wavenumbers
below resonance is directly revealed by ωi in eq. (27). Note that higher jet Lorentz factors reduce
ωi through the dependence on η.
The resonant frequency is
(1− ue/v∗w)
2 − (vwe/v∗w − uevwe/c2)
, (29)
where v∗w is the wave speed at resonance, eq. (15). The resonant frequency increases as the sheath
sound or Alfvén wave speed, vwe ≡ (ae, vAe) increases and ω∗ −→ ∞ when the denominator
decreases to zero as
uj − ue
1− ujue/c2
vwj + vwe
1 + vwjvwe/c2
where vwj,e ≡ (aj,e, vAj,e) in the fluid and magnetic limits, respectively. Since eq. (27) applies below
resonance the overall behavior of the growth rate is indicated by ωi. Thus, growth rates decline to
zero as (uj − ue)2 − V 2As/γ2j γ2e −→ 0. The numerical analysis of the dispersion relation shows that
the pinch fundamental and all first body modes are comparably or more readily stabilized and thus
the jet is KH stable when
(uj − ue)2 − V 2As/γ2j γ2e < 0 . (30)
This stability condition takes on a particularly simple form when conditions in spine and sheath
are equal, i.e., Be = Bj, We = Wj, so that vA,j = vA,e, and with γA ≡ γA,e = γA,j
γ2j γ
e (uj − ue)
< 4γ2A
γ2A − 1
c2 (31)
indicates stability. This result implies that a trans-Alfvénic relativistic jet with γjuj & γAvA will
be KH stable, and that even a super-Alfvénic jet with γj >> γA can be KH stable.
6. Discussion
Formally, the present results and expressions apply only to magnetic fields parallel to an axial
spine-sheath flow in which conditions within the spine and within the sheath are independent of
radius and the sheath extends to infinity. A rapid decline in perturbation amplitudes in the sheath
as a function of radius, governed by the Hankel function in the dispersion relation, suggests that
the present results will apply to sheaths more than about three times the spine radius in thickness.
The relativistic jet is transonic in the absence of sheath flow only for spine and sheath sound
speeds ∼ c/
3. Only in this regime does the pinch fundamental have a significant growth rate and,
in general, we do not expect the pinch fundamental to grow significantly on relativistic jets. On
the other hand, the pinch first body mode can have a significant maximum growth rate and would
dominate any axisymmetric structure. The elliptical and higher order surface modes have increas-
ingly larger maximum growth rates at resonant frequencies higher than the helical surface mode,
and the maximum first body mode growth rates for helical and elliptical modes are comparable to
that of the surface modes. Nevertheless, we expect the helical surface mode to achieve the largest
amplitudes in the non-linear limit as a result of the reduced saturation amplitudes that accompany
the higher resonant frequency and shorter resonant wavelengths associated with the higher order
surface modes and all body modes.
– 21 –
In astrophysical jets we expect a toroidal magnetic field component, and possibly an ordered
helical structure and accompanying flow helicity. Jet rotation (e.g., Bodo et al. 1996), or a radial
velocity profile (e.g., Birkinshaw 1991) will modify the present results but will not stabilize the
helical mode. Two dimensional non-relativistic slab jet theoretical results, indicate that KH sta-
bilization occurs when the velocity shear projected on the wavevector is less than the projected
Alfvén speed (Hardee et al. 1992). In the work presented here magnetic and flow field are parallel
and project equally on the wavevector which for the helical (n = 1) and elliptical (n = 2) mode
lies at an angle θ = tan−1(n/kR) relative to the jet axis. Provided magnetic and flow helicity and
radial gradients in jet spine/sheath properties are not too large we expect the present results to
remain valid where uj,e and Bj,e refer to the poloidal velocity and field components.
KH driven normal mode structures move at less than the jet speed. The fundamental pinch
mode moves backwards in the jet frame at about the sound speed nearly independent of the sheath
properites and thus moves at nearly the jet speed in the source/observer frame. Low frequency and
long wavelength helical and higer order surface modes are advected with wave speed indicated by eq.
(27) and move slowly in the source/observer frame for light, i.e. η ≡ γ2jWj
γ2eWe < 1, and/or for
magnetically dominated flows. Higher frequency (above resonance) and shorter wavelength normal
mode structures move backwards in the jet frame at the sound/Alfvén wave speed, have a wave
speed nearly independent of the sheath properties, and can move slowly in the source/observer
frame only for magnetically dominated flows.
Where flow and magentic fields are parallel, current driven (CD) modes are stable (Isotomin
& Pariev 1994, 1996). Where magnetic and flow fields are helical CD modes can be unstable
(Lyubarskii 1999) in addition to the KH modes. CD and KH instability are expected to produce
helically twisted structure. However, the conditions for instability, the radial structure, the growth
rate and the pattern motions are different. For example, KH modes grow more rapidly when the
magnetic field is force-free (e.g., Appl 1996), and non-relativistic simulation work (e.g., Lery et
al. 2000; Baty & Keppens 2003; Nakamura & Meier 2004) indicates that CD driven structure is
internal to any spine-sheath interface and moves at the jet speed.
The differences between KH and CD instability can serve to identify the source of helical
structure on relativistic jets and allow determination of jet properties near to the central engine.
Perhaps the observation of relatively low proper motions in the TeV BL Lacs when intensity mod-
eling requires high flow Lorentz factors (Ghisellini et al. 2005) is an indication of a magnetically
dominated KH unstable spine-sheath configuration.
The author acknowledges partial support through National Space Science and Technology
Center (NSSTC/NASA) cooperative agreement NCC8-256 and by National Science Foundation
(NSF) award AST-0506666 to the University of Alabama.
A. Linearization of the RMHD Equations
In vector notation the relativistic MHD continuity equation, energy equation, and momentum
equation can be written as:
[γρ] +∇ · [γρv] = 0 , (A1)
γ2W − P
)− (v/c ·B)
γ2Wv +
v−(v ·B) B
= 0 , (A2)
v + v · ∇v
= −∇P − v
P + ρqE+
[j×B]
. (A3)
– 22 –
These equations along with Maxwell’s equations
∇ ·B = 0 ∇ ·E = 4πρq
∇×B = 1
E+ 4π
j ∇×E = −1
and assuming ideal MHD with comoving electric field equal to zero
E = −v ×B
provide the complete set of ideal RMHD equations. In the above W is the enthalpy, the Lorentz
factor γ = (1 − v · v/c2)−1/2, and ρ is the proper density. In what follows I will assume that the
effects of radiation can be ignored, the enthalpy is given by
W = ρ+
and the condition for isentropic flow is given by
+ v · ∇
= 0 .
The general approach to analyzing the time dependent properties of this system is to linearize
the ideal RMHD equations, where the density, velocity, pressure and magnetic field are written as
ρ = ρ0 + ρ1, v = u + v1 (we use v0 ≡ u for notational reasons), P = P0 + P1 E = E0 + E1, and
B = B0+B1, where subscript 1 refers to a perturbation to the equilibrium quantity with subscript
0. Additionally, W = W0 + W1, γ
2 = (γ0 + γ1)
2 ≃ γ20 + 2γ40u · v1/c2 and γ1 ≃ γ30u · v1/c2. It
is assumed that the initial equilibrium system satisfies the zero order equations. The linearized
continuity, energy and momentum equation become
[γ0ρ1 + γ1ρ0] +∇ · [γ0ρ1u+ γ0ρ0v1 + γ1ρ0u] = 0 , (A4)
γ20W1 − P1/c2 + 2γ40
u · v1/c2
γ20W1u+ 2γ
u · v1/c2
W0u+ γ
0W0v1
u · v1/c2
+ (1 + u2/c2)B0·B1 − (u ·B1/c+ v1·B0/c)u ·B0/c
2(B0·B1)u+B20v1 − (u ·B0)B1 − (u ·B1)B0 − (v1·B0)B0
= 0 ,
γ20W0
+ u · ∇v1
= −∇P1 −
(j0×B1) + (j1×B0)
. (A6)
The linearized Maxwell equations become:
∇ ·B1 = 0 ∇ ·E1 = 4πρq1
∇×B1 = 1c
j1 ∇×E1 = −1c
– 23 –
where I keep the displacement current in order to allow for strong magnetic fields and Alfvén wave
speeds comparable to lightspeed. Under the assumption of ideal MHD, the comoving electric field
is zero, the equilibrium charge density ρq,0 = 0, and the electric field
E1= −
u×B1 + v1×B0
is first order, the charge density ρq1 = (▽ ·E1) /4π is also first order, and the electrostatic force
term, ρq1E1, is second order and dropped from the linearized momentum equation. The condition
for isentropic perturbations becomes
P1 = ã
2ρ1 =
This basic set of linearized RMHD equations is similar to those found in Begelman (1998) but
allows a relativistic zeroth order velocity, i.e., v = u + v1 and u . c whereas Begelman allowed
only for relativistic first order motions, v1.
In what follows let us model a jet as a cylinder of radius R, having a uniform proper density,
ρj , a uniform axial magnetic field, Bj = Bz,j, and a uniform velocity, uj = uz,j. The external
medium is assumed to have a uniform proper density, ρe, a uniform axial magnetic field, Be = Bz,e,
and a uniform velocity, ue = uz,e. An external velocity could be the result of a wind or sheath
outflow around a central jet, ue > 0, or could represent backflow, ue < 0, in a cocoon surrounding
the jet. The jet is established in static total pressure balance with the external medium where the
total static uniform pressure is P ∗e ≡ Pe + B2e/8π = P ∗j ≡ Pj + B2j /8π. Under these assumptions
the linearized continuity equation becomes
[γ0ρ1 + γ1ρ0] + u
[γ0ρ1 + γ1ρ0] + γ0ρ0∇ · v1 = 0 . (A7)
The linearized energy equation becomes
γ20W1 −
+ 2γ40
γ20W1 + 2γ
+ γ20W0∇ · v1 = 0 . (A8)
This result for the linearized energy equation is found by noting that the magnetic terms in the
energy equation linearize to
Bz1 + u
− (uB0)∇ ·B1 +B20∇ · v1 −B20 ∂∂zvz1 =
∇ · v1 − ∂∂zvz1
∇ · v1 − ∂∂zvz1
where I have used
Bz1 + u
Bz1 = −
(rvr1) +
= −B0
∇ · v1 −
from ∂B1/∂t = ∇× (u×B1) +∇× (v1×B0). The linearized momentum equation becomes
γ20W0
v1 + u · ∇v1 − 14πc2γ2
v1×B0
−▽P1 − uc2
[(▽×B0)×B1 + (▽×B1)×B0] + 14πc2
– 24 –
where I have used
j0×B1
(∇×B0)×B1
j1×B0
(∇×B1)×B0
E1×B0 =
(∇×B1)×B0
v1×B0
×B0 ,
which includes the displacement current. The components of the linearized momentum equation
can be written as
γ20W0
vr1 + u
Br1 +
Br1 −
, (A10a)
γ20W0
vφ1 + u
Bφ1 +
Bφ1 −
(A10b)
γ20W0
vz1 + u
= − ∂
P1 (A10c)
where V 2A ≡ B20/(4πW0).
B. Normal Mode Dispersion Relation
In cylindrical geometry perturbations ρ1, v1, P1, andB1 can be considered to consist of Fourier
components of the form
f1(r, φ, z, t) = f1(r)e
i(kz±nφ−ωt)
where the flow is in the z direction, and r is in the radial direction with the jet bounded by r = R.
In cylindrical geometry n, an integer, is the azimuthal wavenumber, for n > 0 waves propagate at
an angle to the flow direction, where +n and −n refer to wave propagation in the clockwise and
counterclockwise sense, respectively, when viewed outwards along the flow direction. In general the
goal is to write a differential equation for the radial dependence of the total pressure perturbation
P ∗1 ≡ P1 + (B1 · B0)/4π = P ∗1 (r) exp[i(kz ± nφ − ωt)]. The differential equation can be obtained
from the energy equation by using the momentum equation and writing the velocity components
vr1, vφ1, vz1 in terms of P
1 , u,B0. The components of the linearized momentum equation (eqs.
A10a, b, & c) written in the form
γ20W0
vr1 + u
= − ∂
P ∗1 +
Br1 +
, (B1a)
γ20W0
vφ1 + u
P ∗1 +
Bφ1 +
, (B1b)
γ20W0
vz1 + u
P ∗1 +
Bz1 +
(B1c)
along with
B1 = −c(∇×E1) = ∇×(u×B1) +∇×(v1×B0)
– 25 –
are used to provide relations between Br1 and vr1, and Bφ1 and vφ1
Br1 + u
Br1 = B0
vr1 , (B2a)
Bφ1 + u
Bφ1 = B0
vφ1 , (B2b)
and to provide a relation between Bz1, vz1, and P
Bz1 + u
Bz1 =
vz1 + 2B0γ
vz1 + u
P ∗1 + u
P ∗1 .
(B2c)
To obtain equation (B2c) I have used
Bz1 + u
Bz1 = −
(rvr1) +
= −B0
▽ · v1 −
where
−γ20W0▽ · v1 = γ20
P1 + u
P1 + 2γ
vz1 + u
from the energy equation (eq. A8), and
W1 = ρ1 +
Using equations (B1a, b, & c) combined with
f1(r, φ, z, t) = −iωf1(r)ei(kz±nφ−ωt)
f1(r, φ, z, t) =
f1(r)e
i(kz±nφ−ωt)
f1(r, φ, z, t) = ±inf1(r)ei(kz±nφ−ωt)
f1(r, φ, z, t) = +ikf1(r)e
i(kz±nφ−ωt)
allows the velocity components to be written as
iγ20W0
ku− ω
vr1 = −
P ∗1 + i
k − ω u
Br1 , (B3a)
iγ20W0
ku− ω
vφ1 = −
P ∗1 + i
k − ω
Bφ1 , (B3b)
γ20W0 [ku− ω] vz1 = −
k − ω
P ∗1 +
k − ω
Bz1 . (B3c)
– 26 –
The perturbed magnetic field components from equations (B2a, b,& c) become
Br1 =
ku− ω
B0 , (B4a)
Bφ1 =
ku− ω
B0 , (B4b)
Bz1 =
(ku−ω)
k − ωu/c2
P ∗1 + γ
k − ωu/c2
+ (ku− ω) u
(ku− ω) + V 2A
(ku−ω)
− (k − ωu/c2) u
] B0 (B4c)
where I have used
k + 2γ20 (ku− ω)u/c2 = γ20
k − ωu/c2
+ (ku− ω)u/c2
γ20 (ku− ω)
ã−2 + Γ (Γ− 1)−1 c−2
+ ω/c2 = γ20
(ku− ω) /a2 −
k − ωu/c2
to obtain the expression for Bz1. Using equations (B4a, b, & c) for the perturbed magnetic field
components, I obtain the following relations between the perturbed velocity components v1 and
the total pressure perturbation P ∗1 :
vr1 ≡ Cr
P ∗1 = i
P ∗1 = i
(ku− ω)
γ20W0γ
(ku− ω)2 − (k − ωu/c2)2 v2A
P ∗1 , (B5a)
vφ1 ≡ CφP ∗1 = ∓
P ∗1 = ∓
(ku− ω)
γ20W0γ
(ku− ω)2 − (k − ωu/c2)2 v2A
]P ∗1 , (B5b)
vz1 ≡ CzP ∗1 = −
(ku− ω)
k − ωu/c2
γ20W0
(ku− ω)2 + γ2Av2A
(ku−ω)2
− (k − ωu/c2)2
]}P ∗1 . (B5c)
To obtain the above relationships I have used
(ku− ω)−
k − ωu/c2
ku− ω
= γ2A
(ku− ω)−
k − ωu/c2
(ku− ω)
in addition to
γ20 (ku− ω)
ã−2 + Γ (Γ− 1)−1 c−2
+ ω/c2 = γ20
(ku− ω) /a2 −
k − ωu/c2
where
v2A ≡
1 + V 2A/c
– 27 –
is the Alfvén wave speed and γ2A =
1− v2A/c2
is an Alfvénic Lorentz factor. Note that V 2A =
A and γ
A = 1 + V
2. Thus we have that
∇ · v1 = Cr ∂
P ∗1 +
P ∗1 +
P ∗1 + Cz
P ∗1 +
P ∗1 − n
P ∗1 + ikCzP
Using the energy equation (eq. A8) written in the form
−γ20W0▽ · v1 = γ20
P ∗1 + u
Bz1 + u
Bz1 + 2γ
vz1 + u
inserting
Bz1 + u
Bz1 = −B0
▽ · v1 −
and using vz1 = CzP
1 gives
∇ · v1 = −i Yγ2
(ku−ω)
(ku−ω)
]−1 [
2γ20 (ku− ω) uc2 −
(ku−ω)
where
Y = γ20
(ku− ω) /a2 −
k − ωu/c2
. (B8)
Setting equations (B6) and (B7) equal gives us a differential equation for P ∗1 in the form of Bessel’s
equation
P ∗1 + r
P ∗1 +
β2r2 − n2
P ∗1 = 0 (B9)
where
β2 = Y X
(ku−ω)
+kXCz +
(ku−ω)
]−1 [
2γ20 (ku− ω) uc2 −
(ku−ω)
XCz .
(B10)
I can simplify the expression for β2 by writing
(ku− ω) + V
(ku− ω)
γ20W0
(ku− ω) +
kCz +
2γ20 (ku− ω)
kY Cz
from which it follows that
(ku− ω) + V
(ku− ω)
γ20W0
Y + γ20 (ku− ω)
k − ωu/c2
+ (ku− ω) u/c2
– 28 –
where I have used 2γ20 (ku− ω)u/c2 = γ20
k − ωu/c2
+ (ku− ω)u/c2
− k. Substituting the ex-
pressions for X and Cz from equations (B5a, b, & c), and Y from equation (B8) and modest
algebraic manipulation yields
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2
(ku− ω)2 + (ku− ω)
k − ωu/c2
a2u/c2 −
(ku−ω)2
(k−ωu/c2)
a2+(ku−ω)(k−ωu/c2)ua2/c2
(a2+γ2Av
A)(ku−ω)
a2(k−ωu/c2)
(B11)
Additional regrouping provides the following form
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2
[(a2+γ2Av
A)(ku−ω)
a2(ku−ω)(k−ωu/c2)u/c2]
(ku−ω)2−(k−ωu/c2)
(a2+γ2Av
A)(ku−ω)
a2(k−ωu/c2)
} (B12)
from which I find that β2 can be written in the compact form:
̟2 − κ2a2
̟2 − κ2v2A
v2ms̟
2 − κ2v2Aa2
. (B13)
where ̟2 ≡ (ω − ku)2, κ2 ≡
k − ωu/c2
, and where the fast magnetosonic speed perpendicular
to the magentic field is given by (e.g., Vlahakis & Königl 2003)
vms ≡
a2 + v2A − a2v2A/c2
a2/γ2A + v
It is easily seen that this expression for β2 reduces to the relativistic pure fluid form
β2 −→
̟2 − κ2a2
= γ20
(ku− ω)2
k − ωu/c2
given in Hardee (2000) and that this expression for β2 reduces to the non-relativistic MHD form
β2 −→
̟2 − κ2a2
̟2 − κ2v2A
a2 + v2A
̟2 − κ2v2Aa2
(ku− ω)4
a2 + V 2A
(ku− ω)2 − k2V 2Aa2
where κ2 −→ k2 and v2A −→ V 2A given in Hardee, Clarke & Rosen (1997).
The solutions that are well behaved at jet center and at infinity are P ∗j1(r ≤ R) = CjJ±n(βjr),
and P ∗e1(r ≥ R) = CeH
±n(βer), respectively, where J±n and H
±n are the Bessel and Hankel
functions with arguments defined as
β2j ≡
̟2j − κ2ja2j
̟2j − κ2jv2Aj
v2msj̟
j − κ2jv2Aja2j
, (B14a)
– 29 –
β2e ≡
̟2e − κ2ea2e
̟2e − κ2ev2Ae
v2mse̟
e − κ2ev2Aea2e
, (B14b)
where̟2j,e ≡ (ω − kuj,e)
, κ2j,e ≡
k − ωuj,e/c2
, γ2j,e ≡
1− u2j,e/c2
and γ2Aj,e ≡
1− v2Aj,e/c2
The jet flow speed and external flow speed are positive if flow is in the +z direction.
The condition that the total pressure be continuous across the jet boundary requires that
CjJ±n(βjR) = CeH
±n(βeR) . (B15)
The first derivative of the total pressure is given by
P ∗1 = −iXvr1 .
and with
vr1 ≡
+ u · ∇
ξr = −i (ω − ku) ξr
where ξr is the fluid displacement in the radial direction it follows that
∂P ∗1
= − (ω − ku)Xξr . (B16)
The radial displacement of the jet and external medium must be equal at the jet boundary, i.e.,
r(R) = ξ
r(R), from which it follows that
− (ω − kuj)Xj
∂Jn(βjr)
∂ (βjr)
− (ω − kue)Xe
n (βer)
∂ (βer)
. (B17)
Inserting Cj and Ce in terms of the Bessel and Hankel functions leads to a dispersion relation
describing the propagation of Fourier components which can be written in the following form:
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
. (B18)
where the primes denote derivatives of the Bessel and Hankel functions with respect to their argu-
ments. The expressions
χj ≡ γ2j γ2AjWj
̟2j − κ2jv2Aj
(B19a)
χe ≡ γ2eγ2AeWe
̟2e − κ2ev2Ae
(B19b)
readily reduce to the non-relativistic form χ = ρ0[(ω − ku)2 − k2V 2A] where W0 −→ ρ0 given in
Hardee, Clarke & Rosen (1997). This dispersion relation describes the normal modes of a cylindrical
jet where n = 0, 1, 2, 3, 4, etc. involve pinching, helical, elliptical, triangular, rectangular, etc.
normal mode distortions of the jet, respectively.
– 30 –
C. Analytic Solutions and Approximations
Each normal mode n contains a fundamental/surface wave and multiple body wave solutions
to the dispersion relation. The low-frequency limiting form for the fundamental/surface modes are
obtained in the limit where ω −→ 0 and k −→ 0 but with ω/k 6= 0. In this limit the dispersion
relation for the fundamental (n = 0) and surface (n > 0) modes is given by
χj ≈ −12χe (βjR)
) + π
ǫ− iπ
n = 0 (C1)
χj ≈ −χe n > 0 (C2)
where in this limit βeR −→ 0 and βjR −→ 0, and I have used the small argument forms for the
Bessel and Hankel functions to write
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
(βeR) (βjR)
) + π
ǫ− iπ
n = 0
−βe/βj n > 0
where ǫ is Euler’s constant.
C.1. Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit
In the low frequency limit, dispersion relation solutions for the fundamental axisymmetric
pinch mode are obtained from equation (C1)
γ2j γ
̟2j − κ2jv2Aj
̟2e − κ2ev2Ae
) + π
ǫ− iπ
Here we have the trivial solution ̟2j − κ2jv2Aj = 0 with v2w = v2Aj and the more interesting zeroth
order solution
̟2j ≈ κ2j
v2Aja
v2msj
with wave speed in the proper frame given by
v2w = ̟
v2Aja
v2msj
. (C5)
To first order this magnetosonic wave solution (eq. C3) can be written as
̟2j [1− δ] ≃ κ2ja2j
v2msj
, (C6)
– 31 –
where
δ ≡ −
̟2e − κ2ev2Ae
v2msj
and δ is complex. Thus, in the low frequency limit this fundamental pinch mode (n = 0) solution
consists of a growing and damped wave pair with wave speed in the observer frame
uj ± vw
1± vwuj/c2
where
v2w ≃ a2j
v2msj
v2msj
. (C9)
Previous work has shown the unstable growing solution associated with the backwards moving (in
the jet fluid reference frame) wave.
C.2. Surface Modes (n > 0 ; m = 0) in the low frequency limit
In the low frequency limit the fundamental dispersion relation solution for all higher order
modes (n > 0) is most easily obtained from equation (C2) written in the form
γ2jWj
(ω − kuj)2 −
V 2Aj
k2 − ω2/c2
= −γ2eWe
(ω − kue)2 −
V 2Ae
k2 − ω2/c2
(C10)
where I have used χ ≡ γ20γ2AW0
̟2 − κ2v2A
= γ20W0
(ω − ku)2 −
k2 − ω2/c2
V 2A/γ
. The solu-
tion can be put in the form
[ηuj + ue]± iη1/2
(uj − ue)2 − V 2As/γ2j γ2e
(1 + V 2Ae/γ
2) + η(1 + V 2Aj/γ
(C11)
where
γ2jWj
γ2eWe
and a “surface” Alfvén speed is defined by
V 2As ≡
γ2AjWj + γ
) B2j +B
4πWjWe
The jet is stable to higher order n > 0 fundamental mode perturbations when
γ2j γ
e (uj − ue)
< γ2Ajγ
Ae +We/γ
) B2j +B
4πWjWe
. (C12)
Equation (C11) reduces to the relativistic fluid expression
ηuj + ue
1 + η
1 + η
(uj − ue) (C13a)
– 32 –
given in Hardee & Hughes (2003) equation (6a) where for pressure balance and equal adiabatic
index in jet and external medium η −→ γ2j ae/γ2eaj . Similarly equation (C11) reduces to the
non-relativistic MHD form
ηuj + ux
1 + η
1 + η
(uj − ue)2 − V 2As
(C13b)
given by Hardee & Rosen (2002) eq. (4) where V 2As −→ (ρj + ρe)
B2j +B
/ (4πρjρe) and η −→
ρj/ρe.
C.3. Body Modes (n ≥ 0 ; m ≥ 1) in the low frequency limit
In the low frequency limit the real part of the body wave solutions can be obtained in the limit
ω = 0, k 6= 0 where the dispersion relation can be written in the form
cos [βjR− (2n + 1)π/4] ≈ ǫn ≡
n(βjR)
n (βeR)
n (βeR)
. (C14)
Here I have assumed that the large argument form Jn(βjR) ≈ (2/πβjR)1/2 cos [βjR− (2n + 1)π/4]
applies. In the absence of a magnetic field and a flow surrounding the jet, χe = 0, ǫn = 0, and
solutions are found from βjR − (2n + 1)π/4 = ±mπ ± π/2, where m is an integer. Provided
ǫn << π/2 and θ ≈ cos−1 ǫn ≈ ± (π/2− ǫn), solutions can be found from βjR − (2n + 1)π/4 =
± [mπ + (π/2 ± ǫn)], where for ±ǫn the plus or minus sign is for m odd or even, respectively. In
the limit ω = 0
βjR ≈
γ2j (u
j − a2j)(u2j − v2Aj)
v2msju
j − v2Aja2j
kR , (C15)
and the solutions are given by
kR ≈ kminnmR ≡
v2msju
j − v2Aja2j
γ2j (u
j − a2j )(u2j − v2Aj)
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C16)
where I have set m −→ m+1 to be consistent with previous notation so m = 1 corresponds to the
first body mode.
In the limit a2j >> v
Aj equation (C16) reduces to the relativistic purely fluid form found in
Hardee & Hughes (2003)
kR ≈ kminnmR ≡
[(n+ 2m− 1/2)π/2 + (−1)mǫn]
M2j − 1
(C17a)
where M2j = u
j . In the limit v
Aj >> a
j equation (C16) becomes
kR ≈ kminnmR ≡
[(n+ 2m− 1/2)π/2 + (−1)mǫn]
M2Aj − 1
(C17b)
where M2Aj = u
– 33 –
Equation (C16) reduces to the non-relativistic MHD form found in Hardee & Rosen (2002)
kR ≈ kminnmR ≡
1−M2ms/M2AjM2j
× [(n+ 2m− 1/2)π/2 + (−1)mǫn] (C17c)
where M2ms = u
j + v
Aj) and I have used
γ2j (u
j − a2j)(u2j − v2Aj)
v2msju
j − v2Aja2j
= γ2j
a2+v2
1−M2ms/M2AjM2j
a2+v2
We note here that there is an error in equations (5) in previous articles in the treatment of the sign
on ǫn for even values of m.
C.4. The Resonance Condition
The resonance conditions are found by evaluating the transmittance, T , and reflectance, R, of
waves at the jet boundary where T = 1 +R. With the dispersion relation written as
n(βjR)
Jn(βjR)
n (βeR)
n (βeR)
(C18)
where Z = χ/β with
Z = γ2γ2AW
̟2 − κ2v2A
a2 + γ2Av
̟2 − γ2Aκ2v2Aa2
γ2γ2A (̟
2 − κ2a2)
̟2 − κ2v2A
(C19)
the reflectance
R = (Ze − Zj)/(Ze + Zj) . (C20)
For a fluid containing no magnetic field Z is a quantity related to the acoustic normal impedance
(Gill 1965). When Ze + Zj ≈ 0, R and T are large, and the reflected and transmitted waves have
an amplitude much larger than the incident wave.
C.4.1. The Fluid Limit (Alfvén speed ≪ sound speed)
For the case of a pure fluid
ζ2e + γ
, (C21a)
ζ2j + γ
, (C21b)
where χ/k2 = W
ζ2 + γ2κ2/k2
a2 and ζ ≡ β/k. For non-relativistic flows where (u2/c2)(ω/ku) <<
1, γ ≈ 1, and with adiabatic indicies Γj = Γe the reflectance
(ζe − ζj)(ζeζj − 1)
(ζe + ζj)(ζeζj + 1)
(C22)
– 34 –
and a supersonic resonance (Miles 1957) occurs when βe + βj = k(ζe + ζj) = 0. This supersonic
resonance corresponds to the maximum growth rate of the normal mode solutions to the dispersion
relation.
I now generalize the results in Hardee (2000) to include flow in the external medium relative
to the source/observer frame. Here Ze + Zj = 0 becomes
Γeζjχe + Γjζeχj = Γeζj
+ Γjζe
= 0 . (C23)
A necessary condition for resonance is ζj < 0 and ζe > 0, and on the real axis
uj − aj
1− ujaj/c2
ue + ae
1 + ueae/c2
. (C24)
It follows that the resonance only exists when
uj − aj
1− ujaj/c2
ue + ae
1 + ueae/c2
(C25a)
or equivalently
uj − ue
1− ujue/c2
aj + ae
1 + ajae/c2
. (C25b)
To find the resonant solution for the real part of the phase velocity I solve ζ2j = ε
2ζ2e where
here I set ε ≡ (Γjγ2j̟2j/k2a2j )/
= 1 so that
ζ2j = γ
(ω/k − uj)2
= ζ2e = γ
(ω/k − ue)2
. (C26)
The resulting quadratic equation can be written in the form
a2e/(γ
sj)− a2j/(γ2j γ2se)
)2 − 2
γ2j a
euj/γ
sj − γ2ea2jue/γ2se
γ2j a
u2j − a2j
− γ2ea2j
u2e − a2e
= 0 ,
(C27)
where I have used
γ2j a
− γ2ea2j
a2e/(γ
sj)− a2j/(γ2j γ2se)
and γ2s ≡
1− a2/c2
. The solutions to equation (C27) are given by
γ2j a
euj/γ
sj − γ2ea2jue/γ2se
a2e/(γ
sj)− a2j/(γ2j γ2se)
γseγsj
u2j − 2ujue + u2e
a2e/(γ
sj)− a2j/(γ2j γ2se)
] (C28)
with the resonant solution given by
v∗w =
(γseae)γjuj + (γsjaj)γeue
γj(γseae) + γe(γsjaj)
. (C29)
– 35 –
Inserting the resonant solution (eq. C29) into the expression for ε gives
0.695 ≤ ε2 =
≤ 1.44
where 2.78 ≤ Γ2γ4s ≤ 4. When aj = ae and Γj = Γe, ε2 = 1, and the resonant solution is exact. The
small range on ε (0.83 ≤ ε ≤ 1.2) suggests that this solution remains relatively robust for unequal
values of the sound speed and adiabatic index in the jet and external medium. In the absence of
an external flow the resonant solution
v∗w =
(γseae)γjuj
γj(γseae) + (γsjaj)
M2j − β2
M2j − β2
+ (M2e − β2)
is equivalent to the form given in Hardee (2000).
The resonant frequencies can be estimated using the large argument forms for the Bessel and
Hankel functions. In this limit the dispersion relation becomes
n(βjR)H
n (βeR)
Jn(βjR)H
n (βeR)
≈ i tan(βjR−
2n+ 1
. (C30)
From the dispersion relation with Ze + Zj ≈ 0, and (χj/βj)(βe/χe) = Zj/Ze ≈ βe/βj ≈ −1,
tan[βjR− (2n + 1)π/4]Re ≈ 0 on the real axis. It follows that
|βjR| ≈ |βeR| ≈ (2n+ 1)π/4 +mπ
can be used to obtain an estimate for the resonant frequencies from |βeR| ≈ (2n + 1)π/4 + mπ,
with result that the resonant frequencies are given by
ω∗nmR
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (ae/v∗w − ueae/c2)
. (C31)
In the absence of external flow, ue = 0, and for uj >> ae and 1 >> (kae/ω)
2 this expression
reduces to the form given in Hardee (2000).
When γj(γseae) >> γe(γsjaj), the resonant wave speed becomes v
w ≈ uj, ue/v∗w ≈ ue/uj and
provided ue << uj and ae << uj , the resonant frequency increases with increasing ue/uj and ae/uj
ω∗nmR
(2n + 1)π/4 +mπ
1− 2ue/uj(1− a2e/c2)− (a2e − u2e)/u2j
. (C32)
In general, the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w)
w − ueae/c2
)2 −→ 0. An
equivalent condition for (1− ue/v∗w)
w − ueae/c2
= 0 is
uj − ue
1− ujue/c2
aj + ae
1 + ajae/c2
, (C33)
and the resonance moves to higher frequencies with ω∗nm −→ ∞ when the“shear” speed drops below
a “surface” sound speed.
– 36 –
The behavior of the growth rate at resonance also can be found using the large argument forms
for the Bessel and Hankel functions. In this limit the reflectance can be written as
(Ze − Zj)
(Ze + Zj)
Jn(βjR)H
n (βeR)− J
n(βjR)H
n (βeR)
Jn(βjR)H
n (βeR) + J
n(βjR)H
n (βeR)
≈ exp[−2i(βjR−
2n+ 1
π)] , (C34)
2n+ 1
π ≈ i
ln |R| − φ
(C35)
where R ≡ |R| eiφ. It follows that
(βjR)I ≈
ln |R| (C36)
and since typically at resonance, |ω − kRuj | /aj >
∣kR − ωuj/c2
∣ I can approximate βj by
βj ≡ βRj + iβIj ≈ γj
(ω − kRuj)
− ikI
. (C37)
It follows that
(βjR)I ≈ −γj
kIR , (C38)
kIR ≈ −
ln |R| . (C39)
At resonance
(Ze − Zj)
(Ze + Zj)
βj − βe(χj/χe)
βj + βe(χj/χe)
βj − βe
βj + βe
≈ −2βe
βj + βe
(C40)
|R| ≈
βj − βe
βj + βe
−2βRe
βIj − βIe
βIj + β
(C41)
where I have used
βRj − βRe
≈ −2βRe from the resonance condition on the real axis. It follows
|R| ≈
(ω−kRue)
+ k2I
− γe ueae
− γeae
ω−kRue
ω−kRue
(C42)
where I have used
βe ≡ βRe + iβIe ≈ γe
(ω − kRue)
− ikI
ω − kRue
If I assume that γj(γseae) >> γe(γsjaj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1
|R| ≈
ω∗nmR
(1− 2ue/uj) + k2IR2
(1 + ue/uj)
(1 + ue/uj)
, (C43)
– 37 –
and since kIR ≈ − (aj/2γjuj) ln |R|
|R| ≈
ω∗nmR
(1− 2ue/uj) + [ln |R| /2]2
[ln |R| /2]2
. (C44)
From equation (C32)
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− a2e/c2)− a2e/u2j
] [(2n+ 1)π/4 +mπ]
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− a2e/u2j
[(2n + 1)π/4 +mπ] . (C45)
Formally |R| −→ ∞ as ω∗nm −→ ∞ when the jet speed drops below the “surface” sound speed
given by equation (C33). This result applies only to the surface modes and not to the body modes
as, in the fluid limit, the body modes do not exist when the jet speed drops below the jet sound
speed, see equation (C17). On the other hand, if say, a2e/u
j << 1, then
|R|2 − 1
ln |R| ≈ 4 [(2n + 1)π/4 +mπ] . (C46)
Formally |R| ≈ constant as ω∗nm −→ ∞ when the wind speed becomes comparable to the jet speed,
ue . uj , as must be the case for the velocity shear driven Kelvin-Helmholtz instability.
C.4.2. The Magnetic Limit (Alfvén speed ≫ sound speed)
For the magnetic limit in which magnetic pressure dominates gas pressure
Ze = γeγ
AeWevAe
̟2e − κ2ev2Ae
, (C47a)
Zj = γjγ
AjWjvAj
̟2j − κ2jv2Aj
, (C47b)
A necessary condition for resonance is
̟2e − κ2ev2Ae
> 0 and
̟2j − κ2jv2Aj
< 0 on the real axis
with result that Ze + Zj = 0 when
uj − vAj
1− ujvAj/c2
ue + vAe
1 + uevAe/c2
. (C48)
It follows that the resonance only exists when
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
. (C49)
This result is identical in form to the sonic case with sound speeds replaced by Alfvén wave speeds.
– 38 –
The resonant solution for the real part of the phase velocity is obtained from
Z2j = γ
̟2j −
k2 − ω2/c2
V 2Aj/γ
= Z2e = γ
̟2e −
k2 − ω2/c2
V 2Ae/γ
(C50)
where I have used γ2γ2A
̟2 − κ2v2A
k2 − ω2/c2
V 2A/γ
, and recall that v2A = V
The resulting quadratic equation can be written in the form
Aj − γ2eW 2e V 2Ae
)2 − 2
Ajuj − γ2eW 2e V 2Aeue
j − γ2eW 2e V 2Aeu2e
(C51)
where I have used
k2 − ω2/c2
V 2Aj/γ
j = γ
k2 − ω2/c2
V 2Ae/γ
because pressure balance in the magnetically dominated case requires WjV
Aj = WeV
Ae. The
solutions are given by
Ajuj − γ2eW 2e V 2Aeue ± γjγeWjWeVAjVAe (uj − ue)
Aj − γ2eW 2e V 2Ae
, (C52)
and the resonant solution becomes
v∗w =
γjWjVAjuj + γeWeVAeue
γjWjVAj + γeWeVAe
(γAevAe) γjuj + (γAjvAj) γeue
γj (γAevAe) + γe (γAjvAj)
(C53)
where I have used WVA = WV
A/ (γAvA), and WjV
Aj = WeV
Ae. This resonant solution has the
same form as the sonic case with sound speeds and sonic Lorentz factors replaced by Alfvén wave
speeds and Alfvénic Lorentz factors.
As in the sonic case the resonant frequencies are found from
|βeR| ≈ (2n + 1)π/4 +mπ
with result that the resonant frequencies are given by
ω∗nmR
(2n + 1)π/4 +mπ
(1− ue/v∗w)
2 − (vAe/v∗w − uevAe/c2)
. (C54)
When γj (γAevAe) >> γe (γAjvAj) the resonant wave speed becomes v
w ≈ uj and ue/v∗w ≈ ue/uj ,
and provided ue << uj and vAe << uj the resonant frequency increases with increasing ue/uj and
vAe/uj as
ω∗nmR
≈ (2n + 1)π/4 +mπ
1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j
. (C55)
Here the resonant frequency ω∗nm −→ ∞ as (1− ue/v∗w)
vAe/v
w − uevAe/c2
)2 −→ 0. An
equivalent condition for (1− ue/v∗w)
vAe/v
w − uevAe/c2
= 0 is
uj − ue
1− ujue/c2
vAj + vAe
1 + vAjvAe/c2
, (C56)
– 39 –
and the resonance moves to higher frequencies as the “shear” speed becomes trans-Alfvénic.
The behavior of the growth rate at resonance proceeds in the same manner as for the fluid
limit but with sound speeds replaced by Alfvén wave speeds. The resonant growth rate is now
given by
kIR ≈ −
ln |R| . (C57)
If I assume that γj(γAevAe) >> γe(γAjvAj), with resonant wave speed ω/kR ≈ uj and ue/uj << 1
|R| ≈
ω∗nmR
(1− 2ue/uj) + k2IR2
− vAe
(1 + ue/uj)
+ vAe
(1 + ue/uj)
, (C58)
From equation (C54)
ω∗nmR
(1− 2ue/uj) ≈
(1− 2ue/uj)
1− 2 (ue/uj) (1− v2Ae/c2)− (v2Ae − u2e)/u2j
] [(2n+ 1)π/4 +mπ]
and if say ue = 0, then
|R|2 − 1
ln |R| ≈ 4
1− v2Ae/u2j
[(2n + 1)π/4 +mπ] (C59)
and |R| increases as ω∗nm increases when the jet speed decreases. However, when the shear speed
drops below the “surface” Alfvén speed, see equations (C11 & C12) the jet is stable. This result
is quite different from the fluid limit where the jet remains unstable when the shear speed drops
below the “surface” sound speed. If I insert
uj − ue =
1− ujue/c2
1 + vAjvAe/c2
(vAj + vAe) .
from equation (C56) into equation (C12), it follows that the jet will be unstable when resonance
disappears only when
γ2j γ
1− ujue/c2
> 2γ2Ajγ
v2Ae + v
(vAj + vAe)
1 + vAjvAe/c
. (C60)
where I have used
v2Ae + v
Ae +We/γ
B2j +B
/ (4πWjWe)
as Be = Bj from
magnetic pressure balance. Formally |R| −→ ∞ as ω∗nm −→ ∞ only for jet Lorentz factors greatly
in excess of the Alfvénic Lorentz factor.
C.5. Wave modes at high frequency
To obtain the behavior of wave modes at high frequency I begin with the dispersion relation
written in the form
Jn(βjR)
n(βjR)
n (βeR)
n (βeR)
Jn(βjR)
∓Jn±1(βjR)± nβjRJn(βjR)
n−1(βeR)−
n (βeR)
n (βeR)
(C61)
– 40 –
and assume a large argument in the Hankel function with H
n (βeR) ≈ exp i [βeR− (2n+ 1) π/4]
and a small argument βjR << 1 in the Bessel function to write
J0(βjR)
−J1(βjR)
e−iπ/2 n = 0
e−iπ/2 n > 0
The small arguement form for the Bessel function gives J0(βjR)/J1(βjR) ≈ 2/βjR with result that
the dispersion relation becomes
βjR ≈
e−iπ/2
n = 0
e−iπ/2 n > 0
. (C62)
At high frequency and large wavenumber χj and χe, are proportional to k
2, βj and βe are pro-
portional to k, and βjR = ζjkR ∝ (kR)1/2 for n = 0. Thus, the internal solutions in the high
frequency and large wavenumber limit are given by βjR ≃ 0 and are found from
(kuj − ω)2 −
k − ωuj/c2
(kuj − ω)2 −
k − ωuj/c2
≈ 0 (C63)
and it follows that
uj ± aj
1± ujaj/c2
, (C64a)
uj ± vAj
1± ujvAj/c2
. (C64b)
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http://arxiv.org/abs/astro-ph/0611406
Introduction
The RMHD Normal Mode Dispersion Relation
Analytical Solutions to the Dispersion Relation
Low Frequency Limit
Resonance
High Frequency Limit
Numerical Solution of the Dispersion Relation
Fluid Limit
Magnetic Limit
A High Sound and Alfvén Speed Magnetosonic Case
Summary
Discussion
Linearization of the RMHD Equations
Normal Mode Dispersion Relation
Analytic Solutions and Approximations
Fundamental Pinch Mode (n = 0 ; m = 0) in the low frequency limit
Surface Modes (n > 0 ; m = 0) in the low frequency limit
Body Modes (n 0 ; m 1) in the low frequency limit
The Resonance Condition
The Fluid Limit (Alfvén speed sound speed)
The Magnetic Limit (Alfvén speed sound speed)
Wave modes at high frequency
|
0704.1622 | MATLAB codes for teaching quantum physics: Part 1 | MATLAB codes for teaching quantum physics: Part 1
R. Garcia,∗ A. Zozulya, and J. Stickney
Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609
(Dated: February 1, 2008)
Among the ideas to be conveyed to students in an introductory quantum course, we have the
pivotal idea championed by Dirac that functions correspond to column vectors (kets) and that
differential operators correspond to matrices (ket-bras) acting on those vectors. The MATLAB
(matrix-laboratory) programming environment is especially useful in conveying these concepts to
students because it is geared towards the type of matrix manipulations useful in solving introductory
quantum physics problems. In this article, we share MATLAB codes which have been developed at
WPI, focusing on 1D problems, to be used in conjunction with Griffiths’ introductory text.
Two key concepts underpinning quantum physics are
the Schrodinger equation and the Born probability equa-
tion. In 1930 Dirac introduced bra-ket notation for state
vectors and operators.1 This notation emphasized and
clarified the role of inner products and linear function
spaces in these two equations and is fundamental to
our modern understanding of quantum mechanics. The
Schrodinger equation tells us how the state Ψ of a particle
evolves in time. In bra-ket notation, it reads
|Ψ〉 = H |Ψ〉 (1)
where H is the Hamiltonian operator and |Ψ〉 is a ket or
column vector representing the quantum state of the par-
ticle. When a measurement of a physical quantity A is
made on a particle initially in the state Ψ, the Born equa-
tion provides a way to calculate the probability P (Ao)
that a particular result Ao is obtained from the measure-
ment. In bra-ket notation, it reads2
P (Ao) ∼ |〈Ao|Ψ〉|2 (2)
where if |Ao〉 is the state vector corresponding to the
particular result Ao having been measured, 〈Ao| = |Ao〉†
is the corresponding bra or row vector and 〈Ao|Ψ〉 is thus
the inner product between |Ao〉 and |Ψ〉. In the Dirac
formalism, the correspondence between the wavefunction
Ψ(~x) and the ket |Ψ〉 is set by the relation Ψ(~x) = 〈~x|Ψ〉,
where |~x〉 is the state vector corresponding to the particle
being located at ~x. Thus we regard Ψ(~x) as a component
of a state vector |Ψ〉, just as we usually3 regard ai = ı̂·~a
as a component of ~a along the direction ı̂. Similarly, we
think of the Hamiltonian operator as a matrix
d3~x |~x〉
Ψ(~x, t) + U(~x)
〈~x| (3)
acting on the space of kets.
While an expert will necessarily regard Eqs.(1-3) as a
great simplification when thinking of the content of quan-
tum physics, the novice often understandably reels under
the weight of the immense abstraction. We learn much
about student thinking from from the answers given by
our best students. For example, we find a common error
when studying 1D quantum mechanics is a student treat-
ing Ψ(x) and |Ψ〉 interchangeably, ignoring the fact that
the first is a scalar but the ket corresponds to a column
vector. For example, they may write incorrectly
〈p||Ψ〉|x〉 = |Ψ〉〈p|x〉 (incorrect!) (4)
or some similar abberation. To avoid these types of mis-
conceptions, a number of educators and textbook authors
have stressed incorporating a numerical calculation as-
pect to quantum courses.4,5,6,7,8,9 The motive is simple.
Anyone who has done numerical calculations can’t help
but regard a ket |Ψ〉 as a column vector, a bra 〈Ψ| as a
row vector and an operatorH as a matrix because that is
how they concretely represented in the computer. Intro-
ducing a computational aspect to the course provides one
further benefit: it gives the beginning quantum student
the sense that he or she is being empowered to solve real
problems that may not have simple, analytic solutions.
With these motivations in mind, we have developed
MATLAB codes10 for solving typical 1 D problems found
in the first part of a junior level quantum course based
on Griffith’s book.11 We chose MATLAB for our pro-
gramming environment because the MATLAB syntax is
especially simple for the typical matrix operations used
in 1D quantum mechanics problems and because of the
ease of plotting functions. While some MATLAB numeri-
cal recipes have previously been published by others,12,13
the exercises we share here are special because they em-
phasize simplicity and quantum pedagogy, not numerical
efficiency. Our point has been to provide exercises which
show students how to numerically solve 1 D problems in
such a way that emphasizes the column vector aspect of
kets, the row vector aspect of bras and the matrix aspect
of operators. Exercises using more efficient MATLAB
ODE solvers or finite-element techniques are omitted be-
cause they do not serve this immediate purpose. In part
II of this article, we hope to share MATLAB codes which
can be used in conjunction with teaching topics pertain-
ing to angular momentum and non-commuting observ-
ables.
http://arXiv.org/abs/0704.1622v1
I. FUNCTIONS AS VECTORS
To start students thinking of functions as column
vector-like objects, it is very useful to introduce them
to plotting and integrating functions in the MATLAB
environment. Interestingly enough, the plot command in
MATLAB takes vectors as its basic input element. As
shown in Program 1 below, to plot a function f(x) in
MATLAB, we first generate two vectors: a vector of x
values and a vector of y values where y = f(x). The com-
mand plot(x, y,′ r′) then generates a plot window con-
taining the points (xi, yi) displayed as red points (
′r′).
Having specified both x and y, to evaluate the definite
integral
ydx, we need only sum all the y values and
multiply by dx.
%****************************************************************
% Program 1: Numerical Integration and Plotting using MATLAB
%****************************************************************
N=1000000; % No. of points
L=500; % Range of x: from -L to L
x=linspace(-L,L,N)’; % Generate column vector with N
% x values ranging from -L to L
dx=x(2)-x(1); % Distance between adjacent points
% Alternative Trial functions:
% To select one, take out the comment command % at the beginning.
%y=exp(-x.^2/16); % Gaussian centered at x=0
%y=((2/pi)^0.5)*exp(-2*(x-1).^2); % Normed Gaussian at x=1
%y=(1-x.^2).^-1; % Symmetric fcn which blows up at x=1
%y=(cos(pi*x)).^2; % Cosine fcn
%y=exp(i*pi*x); % Complex exponential
%y=sin(pi*x/L).*cos(pi*x/L);% Product of sinx times cosx
%y=sin(pi*x/L).*sin(pi*x/L);% Product of sin(nx) times sin(mx)
%A=100; y=sin(x*A)./(pi*x); % Rep. of delta fcn
A=20; y=(sin(x*A).^2)./(pi*(A*x.^2));% Another rep. of delta fcn
% Observe: numerically a function y(x) is represented by a vector!
% Plot a vector/function
plot(x,y); % Plots vector y vs. x
%plot(x,real(y),’r’, x, imag(y), ’b’); % Plots real&imag y vs. x
axis([-2 2 0 7]); % Optimized axis parameters for sinx^2/pix^2
%axis([-2 2 -8 40]); % Optimized axis parameters for sinx/pix
% Numerical Integration
sum(y)*dx % Simple numerical integral of y
trapz(y)*dx % Integration using trapezoidal rule
II. DIFFERENTIAL OPERATORS AS
MATRICES
Just as f(x) is represented by a column vector |f〉 in
the computer, for numerical purposes a differential opera-
tor D̂ acting on f(x) is reresented by a matrixD that acts
on |f〉. As illustrated in Program 2, MATLAB provides
many useful, intuitive, well-documented commands for
generating matrices D that correspond to a given D̂.10
Two examples are the commands ones and diag. The
command ones(a, b) generates an a × b matrix of ones.
The command diag(A, n) generates a matrix with the el-
ements of the vector A placed along the nth diagonal and
zeros everywhere else. The central diagonal corresponds
to n = 0, the diagonal above the center one corresponds
to n = 1, etc...).
An exercise we suggest is for students to verify that
the derivative matrix is not Hermitian while the deriva-
tive matrix times the imaginary number i is. This can
be very valuable for promoting student understanding if
done in conjunction with the proof usually given for the
differential operator.
%****************************************************************
% Program 2: Calculate first and second derivative numerically
% showing how to write differential operator as a matrix
%****************************************************************
% Parameters for solving problem in the interval 0 < x < L
L = 2*pi; % Interval Length
N = 100; % No. of coordinate points
x = linspace(0,L,N)’; % Coordinate vector
dx = x(2) - x(1); % Coordinate step
% Two-point finite-difference representation of Derivative
D=(diag(ones((N-1),1),1)-diag(ones((N-1),1),-1))/(2*dx);
% Next modify D so that it is consistent with f(0) = f(L) = 0
D(1,1) = 0; D(1,2) = 0; D(2,1) = 0; % So that f(0) = 0
D(N,N-1) = 0; D(N-1,N) = 0; D(N,N) = 0; % So that f(L) = 0
% Three-point finite-difference representation of Laplacian
Lap = (-2*diag(ones(N,1),0) + diag(ones((N-1),1),1) ...
+ diag(ones((N-1),1),-1))/(dx^2);
% Next modify Lap so that it is consistent with f(0) = f(L) = 0
Lap(1,1) = 0; Lap(1,2) = 0; Lap(2,1) = 0; % So that f(0) = 0
Lap(N,N-1) = 0; Lap(N-1,N) = 0; Lap(N,N) = 0;% So that f(L) = 0
% To verify that D*f corresponds to taking the derivative of f
% and Lap*f corresponds to taking a second derviative of f,
% define f = sin(x) or choose your own f
f = sin(x);
% And try the following:
Df = D*f; Lapf = Lap*f;
plot(x,f,’b’,x,Df,’r’, x,Lapf,’g’);
axis([0 5 -1.1 1.1]); % Optimized axis parameters
% To display the matrix D on screen, simply type D and return ...
D % Displays the matrix D in the workspace
Lap % Displays the matrix Lap
III. INFINITE SQUARE WELL
When solving Eq. (1), the method of separation of vari-
ables entails that as an intermediate step we look for the
separable solutions
|ΨE(t)〉 = |ΨE(0)〉exp(−iEt/~) (5)
where |ΨE(0)〉 satisfies the time-independent Schrodinger
equation
H |ΨE(0)〉 = E |ΨE(0)〉. (6)
In solving Eq. (6) we are solving for the eigenvalues E
and eigenvectors |ΨE(0)〉 of H . In MATLAB, the com-
mand [V,E] = eig(H) does precisely this: it generates
two matrices. The first matrix V has as its columns the
eigenvectors |ΨE(0)〉. The second matrix E is a diago-
nal matrix with the eigenvalues Ei corresponding to the
eigenvectors |ΨEi(0)〉 placed along the central diagonal.
We can use the command E = diag(E) to convert this
matrix into a column vector. In Program 3, we solve for
the eigenfunctions and eigenvalues for the infinite square
well Hamiltonian. For brevity, we omit the commands
setting the parameters L,N, x, and dx.
%****************************************************************
% Program 3: Matrix representation of differential operators,
% Solving for Eigenvectors & Eigenvalues of Infinite Square Well
%****************************************************************
% For brevity we omit the commands setting the parameters L, N,
% x and dx; We also omit the commands defining the matrix Lap.
% These would be the same as in Program 2 above.
% Total Hamiltonian where hbar=1 and m=1
hbar = 1; m = 1; H = -(1/2)*(hbar^2/m)*Lap;
% Solve for eigenvector matrix V and eigenvalue matrix E of H
[V,E] = eig(H);
% Plot lowest 3 eigenfunctions
plot(x,V(:,3),’r’,x,V(:,4),’b’,x,V(:,5),’k’); shg;
E % display eigenvalue matrix
diag(E) % display a vector containing the eigenvalues
Note that in the MATLAB syntax the object V (:, 3)
specifies the column vector consisting of all the elements
in column 3 of matrix V . Similarly V (2, :) is the row
vector consisting of all elements in row 2 of V ; V (3, 1) is
the element at row 3, column 1 of V ; and V (2, 1 : 3) is
a row vector consisting of elements V (2, 1), V (2, 2) and
V (2, 3).
IV. ARBITRARY POTENTIALS
Numerical solution of Eq. (1) is not limited to any par-
ticular potential. Program 4 gives example MATLAB
codes solving the time independent Schrodinger equa-
tion for finite square well potentials, the harmonic os-
cillator potential and even for potentials that can only
solved numerically such as the quartic potential U = x4.
In order to minimize the amount of RAM required, the
codes shown make use of sparse matrices, where only
the non-zero elements of the matrices are stored. The
commands for sparse matrices are very similar to those
for non-sparse matrices. For example, the command
[V,E] = eigs(H,nmodes..) provides the nmodes lowest
energy eigenvectors V of of the sparse matrix H .
Fig. 1 shows the plot obtained from Program 4 for the
potential U = 1
·100 ·x2. Note that the 3 lowest energies
displayed in the figure are just as expected due to the
analytic formula
E = ~ω
with n = integer and ω =
= 10 rad/s.
V. A NOTE ON UNITS IN OUR PROGRAMS
When doing numerical calculations, it is important to
minimize the effect of rounding errors by choosing units
such that the variables used in the simulation are of the
%****************************************************************
% Program 4: Find several lowest eigenmodes V(x) and
% eigenenergies E of 1D Schrodinger equation
% -1/2*hbar^2/m(d2/dx2)V(x) + U(x)V(x) = EV(x)
% for arbitrary potentials U(x)
%****************************************************************
% Parameters for solving problem in the interval -L < x < L
% PARAMETERS:
L = 5; % Interval Length
N = 1000; % No of points
x = linspace(-L,L,N)’; % Coordinate vector
dx = x(2) - x(1); % Coordinate step
% POTENTIAL, choose one or make your own
U = 1/2*100*x.^(2); % quadratic harmonic oscillator potential
%U = 1/2*x.^(4); % quartic potential
% Finite square well of width 2w and depth given
%w = L/50;
%U = -500*(heaviside(x+w)-heaviside(x-w));
% Two finite square wells of width 2w and distance 2a apart
%w = L/50; a=3*w;
%U = -200*(heaviside(x+w-a) - heaviside(x-w-a) ...
% + heaviside(x+w+a) - heaviside(x-w+a));
% Three-point finite-difference representation of Laplacian
% using sparse matrices, where you save memory by only
% storing non-zero matrix elements
e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2;
% Total Hamiltonian
hbar = 1; m = 1; % constants for Hamiltonian
H = -1/2*(hbar^2/m)*Lap + spdiags(U,0,N,N);
% Find lowest nmodes eigenvectors and eigenvalues of sparse matrix
nmodes = 3; options.disp = 0;
[V,E] = eigs(H,nmodes,’sa’,options); % find eigs
[E,ind] = sort(diag(E));% convert E to vector and sort low to high
V = V(:,ind); % rearrange corresponding eigenvectors
% Generate plot of lowest energy eigenvectors V(x) and U(x)
Usc = U*max(abs(V(:)))/max(abs(U)); % rescale U for plotting
plot(x,V,x,Usc,’--k’); % plot V(x) and rescaled U(x)
% Add legend showing Energy of plotted V(x)
lgnd_str = [repmat(’E = ’,nmodes,1),num2str(E)];
legend(lgnd_str) % place lengend string on plot
order of unity. In the programs presented here, our fo-
cus being undergraduate physics students, we wanted to
avoid unnecessarily complicating matters. To make the
equations more familiar to the students, we explicitly left
constants such as ~ in the formulas and chose units such
that ~ = 1 and m = 1. We recognize that others may
have other opinions on how to address this issue. An al-
ternative approach used in research is to recast the equa-
tions in terms of dimensionless variables, for example by
rescaling the energy to make it dimensionless by express-
ing it in terms of some characteristic energy in the prob-
lem. In a more advanced course for graduate students or
in a course in numerical methods, such is an approach
which would be preferable.
VI. TIME DEPENDENT PHENOMENA
The separable solutions |ΨE(t)〉 are only a subset of
all possible solutions of Eq. (1). Fortunately, they are
complete set so that we can construct the general solution
−1.5 −1 −0.5 0 0.5 1 1.5
−0.05
x [m]
E = 4.99969
E = 14.9984
E = 24.9959
FIG. 1: Output of Program 4, which plots the energy eigen-
functions V (x) and a scaled version of the potential U(x) =
1/2 · 100 · x2. The corresponding energies displayed within
the figure legend, 4.99969, 14.9984 and 24.9959, are, within
rounding error, precisely those expected from Eq. (7) for the
three lowest-energy modes.
via the linear superposition
|Ψ(t)〉 =
aE |ΨE(0)〉exp(−iEt/~) (8)
where aE are constants and the sum is over all possible
values of E. The important difference between the sep-
arable solutions (5) and the general solution (8) is that
the probability densities derived from the general solu-
tions are time-dependent whereas those derived from the
separable solutions are not. A very apt demonstration
of this is provided in the Program 5 which calculates the
time-dependent probability density ρ(x, t) for a particle
trapped in a a pair of finite-square wells whose initial
state |Ψ(0)〉 is set equal to the the equally-weighted su-
perposition
|Ψ(0)〉 = 1√
(|ΨE0〉 + |ΨE1〉) (9)
of the ground state |ΨE0〉 and first excited state |ΨE1〉
of the double well system. As snapshots of the program
output show in Fig. 2, the particle is initially completely
localized in the rightmost well. However, due to E0 6= E1,
the probability density
ρ(x, t) =
[ |ψE0(x)|2 + |ψE1(x)|2 +
2|ψE0(x)||ψE1 (x)|cos2 ((E1 − E2)t/~) ] (10)
is time-dependent, oscillating between the ρ(x) that cor-
responds to the particle being entirely in the right well
ρ(x) = | |ψE0(x)| + |ψE1(x)| |
and ρ(x) for the particle being entirely in the left well
ρ(x) = | |ψE0(x)| − |ψE1(x)| |
. (12)
By observing the period with which ρ(x, t) oscillates
in the simulation output shown in Fig. 2 students can
verify that it is the same as the period of oscillation
2π~/(E1 − E2) expected from Eq. (10).
%****************************************************************
% Program 5: Calculate Probability Density as a function of time
% for a particle trapped in a double-well potential
%****************************************************************
% Potential due to two square wells of width 2w
% and a distance 2a apart
w = L/50; a = 3*w;
U = -100*( heaviside(x+w-a) - heaviside(x-w-a) ...
+ heaviside(x+w+a) - heaviside(x-w+a));
% Finite-difference representation of Laplacian and Hamiltonian,
% where hbar = m = 1.
e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2;
H = -(1/2)*Lap + spdiags(U,0,N,N);
% Find and sort lowest nmodes eigenvectors and eigenvalues of H
nmodes = 2; options.disp = 0; [V,E] = eigs(H,nmodes,’sa’,options);
[E,ind] = sort(diag(E));% convert E to vector and sort low to high
V = V(:,ind); % rearrange coresponding eigenvectors
% Rescale eigenvectors so that they are always
% positive at the center of the right well
for c = 1:nmodes
V(:,c) = V(:,c)/sign(V((3*N/4),c));
%****************************************************************
% Compute and display normalized prob. density rho(x,T)
%****************************************************************
% Parameters for solving the problem in the interval 0 < T < TF
TF = 10; % Length of time interval
NT = 100; % No. of time points
T = linspace(0,TF,NT); % Time vector
% Compute probability normalisation constant (at T=0)
psi_o = 0.5*V(:,1)+0.5*V(:,2); % wavefunction at T=0
sq_norm = psi_o’*psi_o*dx; % square norm = |<ff|ff>|^2
Usc = U*max(abs(V(:)))/max(abs(U)); % rescale U for plotting
% Compute and display rho(x,T) for each time T
for t=1:NT; % time index parameter for stepping through loop
% Compute wavefunction psi(x,T) and rho(x,T) at T=T(t)
psi = 0.5*V(:,1)*exp(-i*E(1)*T(t)) ...
+ 0.5*V(:,2)*exp(-i*E(2)*T(t));
rho = conj(psi).*psi/sq_norm; % normalized probability density
% Plot rho(x,T) and rescaled potential energy Usc
plot(x,rho,’o-k’, x, Usc,’.-b’); axis([-L/8 L/8 -1 6]);
lgnd_str = [repmat(’T = ’,1,1),num2str(T)];
text(-0.12,5.5,lgnd_str, ’FontSize’, 18);
xlabel(’x [m]’, ’FontSize’, 24);
ylabel(’probability density [1/m]’,’FontSize’, 24);
pause(0.05); % wait 0.05 seconds
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
T = 0s
x [m]
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
T = 2.8189s
x [m]
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
T = 5.6378s
x [m]
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
T = 8.4566s
x [m]
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
T = 11.2755s
x [m]
FIG. 2: The probability density ρ(x, t) output from Program
5 for a particle trapped in a pair of finite-height square well
potentials that are closely adjacent. The initial state of the
particle is chosen to be |Ψ(0)〉 ∼ |E0〉+ |E1〉. Shown is ρ(x, t)
plotted for times T = {0, 0.25, 0.5, 0.75, 1.0}×2π~/(E1 − E2).
VII. WAVEPACKETS AND STEP POTENTIALS
Wavepackets are another time-dependent phenomenon
encountered in undergraduate quantum mechanics for
which numerical solution techniques have been typically
advocated in the hopes of promoting intuitive acceptance
and understanding of approximations necessarily invoked
in more formal, analytic treatments. Program 6 calcu-
lates and displays the time evolution of a wavepacket for
one of two possible potentials, either U = 0 or a step
potential U = UoΘ(x − L). The initial wavepacket is
generated as the Fast Fourier Transform of a Gaussian
momentum distribution centered on a particular value
of the wavevector ko. Because the wavepacket is com-
posed of a distribution of different ks, different parts of
the wavepacket move with different speeds, which leads
to the wave packet spreading out in space as it moves.
While there is a distribution of velocities within the
wavepacket, two velocities in particular are useful in char-
acterizing it. The phase velocity vw = ω/k = E/p =
~ko/2m is the velocity of the plane wave component
which has wavevector ko. The group velocity vg = ~ko/m
is the velocity with which the expectation value < x >
moves and is the same as the classical particle velocity as-
sociated with the momentum p = ~k. Choosing U = 0,
students can modify this program to plot < x > vs t.
They can extract the group velocity from their numerical
simulation and observe that indeed vg = 2vw for a typ-
ical wave packet. Students can also observe that, while
vg matches the particle speed from classical mechanics,
the wavepacket spreads out as time elapses.
In Program 6, we propagate the wave function forward
via the formal solution
|Ψ(t)〉 = exp(−iHt/~)|Ψ(0)〉, (13)
where the Hamiltonian matrix H is in the exponential.
This solution is equivalent to Eq. (6), as as can be shown
by simple substitution. Moreover, MATLAB has no trou-
ble exponentiating the matrix that numerically repre-
senting the Hamiltonian operator as long as the matrix
is small enough to fit in the available computer memory.
Even more interestingly, students can use this method
to investigate scattering of wavepackets from various po-
tentials, including the step potential U = UoΘ(x−L/2).
In Fig. 3, we show the results of what happens as the
wavepacket impinges on the potential barrier. The pa-
rameters characterizing the initial wavepacket have been
deliberately chosen so that the wings do not fall outside
the simulation area and initially also do not overlap the
barrier on the right. If 〈E〉 ≪ Uo, the wavepacket is
completely reflected from the barrier. If 〈E〉 ≈ Uo, a
portion of the wave is is reflected and a portion is trans-
mitted through. If 〈E〉 ≫ Uo, almost all of the wave is
transmitted.
In Fig. 4 we compare the reflection probability R cal-
culated numerically using Program 6 with R calculated
by averaging the single-mode11 expression
R(E) =
E − Uo√
E − Uo
over the distribution of energies in the initial wavepacket.
While the numerically and analytically estimated R are
found to agree for large and small 〈E〉/Uo, there is a
noticeable discrepancy due to the shortcomings of the
%****************************************************************
% Program 6: Wavepacket propagation using exponential of H
%****************************************************************
% Parameters for solving the problem in the interval 0 < x < L
L = 100; % Interval Length
N = 400; % No of points
x = linspace(0,L,N)’; % Coordinate vector
dx = x(2) - x(1); % Coordinate step
% Parameters for making intial momentum space wavefunction phi(k)
ko = 2; % Peak momentum
a = 20; % Momentum width parameter
dk = 2*pi/L; % Momentum step
km=N*dk; % Momentum limit
k=linspace(0,+km,N)’; % Momentum vector
% Make psi(x,0) from Gaussian kspace wavefunction phi(k) using
% fast fourier transform :
phi = exp(-a*(k-ko).^2).*exp(-i*6*k.^2); % unnormalized phi(k)
psi = ifft(phi); % multiplies phi by expikx and integrates vs. x
psi = psi/sqrt(psi’*psi*dx); % normalize the psi(x,0)
% Expectation value of energy; e.g. for the parameters
% chosen above <E> = 2.062.
avgE = phi’*0.5*diag(k.^2,0)*phi*dk/(phi’*phi*dk);
% CHOOSE POTENTIAL U(X): Either U = 0 OR
% U = step potential of height avgE that is located at x=L/2
%U = 0*heaviside(x-(L/2)); % free particle wave packet evolution
U = avgE*heaviside(x-(L/2)); % scattering off step potential
% Finite-difference representation of Laplacian and Hamiltonian
e = ones(N,1); Lap = spdiags([e -2*e e],[-1 0 1],N,N)/dx^2;
H = -(1/2)*Lap + spdiags(U,0,N,N);
% Parameters for computing psi(x,T) at different times 0 < T < TF
NT = 200; % No. of time steps
TF = 29; T = linspace(0,TF,NT); % Time vector
dT = T(2)-T(1); % Time step
hbar = 1;
% Time displacement operator E=exp(-iHdT/hbar)
E = expm(-i*full(H)*dT/hbar); % time desplacement operator
%***************************************************************
% Simulate rho(x,T) and plot for each T
%***************************************************************
for t = 1:NT; % time index for loop
% calculate probability density rho(x,T)
psi = E*psi; % calculate new psi from old psi
rho = conj(psi).*psi; % rho(x,T)
plot(x,rho,’k’); % plot rho(x,T) vs. x
axis([0 L 0 0.15]); % set x,y axis parameters for plotting
xlabel(’x [m]’, ’FontSize’, 24);
ylabel(’probability density [1/m]’,’FontSize’, 24);
pause(0.05); % pause between each frame displayed
% Calculate Reflection probability
for a=1:N/2;
R=R+rho(a);
R=R*dx
numerical simulation for 〈E〉/Uo ≈ 1. This discrepancy
can be reduced significantly by increasing the number of
points in the simulation to 1600 but only at the cost of
significantly slowing down the speed of the computation.
For our purposes, the importance comparing the analyt-
ical and numerical calculations is that it gives student
a baseline from which to form an opinion or intuition
regarding the accuracy of Eq. (14).
0 10 20 30 40 50 60 70 80 90 100
x [m]
T = 0s
0 10 20 30 40 50 60 70 80 90 100
x [m]
T = 14.4271s
0 10 20 30 40 50 60 70 80 90 100
x [m]
T = 28.8543s
FIG. 3: Output of Program 6 showing a wavepacket encoun-
tering step potential of height ∼ 〈E〉 located at x/L = 0.5 at
different times.
VIII. CONCLUSIONS
One benefit of incorporating numerical simulation into
the teaching of quantum mechanics, as we have men-
tioned, is the development of student intuition. Another
is showing students that non-ideal, real-world problems
can be solved using the concepts they learn in the class-
room. However, our experimentation incorporating these
simulations in quantum physics at WPI during the past
year has shown us that the most important benefit is a
type of side-effect to doing numerical simulation: the ac-
ceptance on an intuitive level by the student that func-
tions are like vectors and differential operators are like
matrices. While in the present paper, we have only had
sufficient space to present the most illustrative MATLAB
��� ��� ��� ��� ��� ���
��
����
��
���
���
���
������������
�������������
�!��"�#
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FIG. 4: The reflection probability R vs. 〈E〉/Uo. The dashed
line is simply Eq. (14) where we subsitute E = 〈E〉, the solid
line is Eq. (14) averaged over the energy distribution in the
incident wavepacket, and the points are numerical results ob-
tained using Program 6, where the horizontal distance be-
tween points is σE/Uo where σE is the standard deviation of
the energy distribution in the wavepacket.
codes, our goal is to eventualy make available a more
complete set of polished codes is available for download-
ing either from the authors or directly from the file ex-
change at MATLAB Central.14
∗ Electronic address: [email protected]
1 P. A. M. Dirac, The Principles of Quantum Mechanics, 1st
ed., (Oxford University Press, 1930).
2 Born’s law is stated as a proportionality because an addi-
tional factor is necessary depending on the units of |Ψ〉
3 C. C. Silva and R. de Andrade Martins, “Polar and ax-
ial vectors versus quaternions,” Am. J. Phys. 70, 958-963
(2002).
4 R. W. Robinett, Quantum Mechanics: Classical Results,
Modern Systems, and Visualized Examples, (Oxford Uni-
versity Press, 1997).
5 H. Gould, “Computational physics and the undergraduate
curriculum,” Comput. Phys. Commun. 127, 610 (2000);
J. Tobochnik and H. Gould, “Teaching computational
physics to undergraduates,” in Ann. Rev. Compu. Phys.
IX, edited by D. Stauffer (World Scientific, Singapore,
2001), p. 275; H. Gould, J. Tobochnik, W. Christian, An
Introduction to Computer Simulation Methods: Applica-
tions to Physical Systems, (Benjamin Cummings, Upper
Saddle River, NJ, 2006) 3rd ed..
6 R. Spenser, “Teaching computational physics as a labora-
tory sequence,” Am. J. Phys. 73, 151-153 (2005).
7 D. Styer, “Common misconceptions regarding quantum
mechanics,” Am. J. Phys. 64, 31-34 (1996).
8 A. Goldberg, H. M. Schey, J. L. Schwartz, “Computer-
Generated Motion Pictures of One-Dimensional Quantum-
Mechanical Transmission and Reflection Phenomena,”
Am. J. Phys. 35, 177-186 (1967).
9 C. Singh, M. Belloni, and W. Christian, “Improving stu-
dents’ understanding of quantum mechanics,” Physics To-
day, August 2006, p. 43.
10 These MATLAB commands are explained in an exten-
sive on-line, tutorial within MATLAB and which is also
independently available on the MATHWORKS website,
http://www.mathworks.com/.
11 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd
Ed., Prentice Hall 2003.
12 A. Garcia, Numerical Methods for Physics, 2nd Ed., (Pren-
tice Hall, 1994).
13 G. Lindblad, “Quantum Mechanics
with MATLAB,” available on internet,
http://mathphys.physics.kth.se/schrodinger.html .
14 See the user file exchange at
http://www.mathworks.com/matlabcentral/.
mailto:[email protected]
http://www.mathworks.com/
http://mathphys.physics.kth.se/schrodinger.html
http://www.mathworks.com/matlabcentral/
|
0704.1623 | Nanodevices and Maxwell's Demon | 07041623rev_mod
arXiv:cond-mat/0704.1623
Lecture Notes in Nanoscale Science and Technology, Vol. 2,
Nanoscale Phenomena: Basic Science to Device Applications,
Eds. Z.K. Tang and P.Sheng, Springer (2008)
Nanodevices and Maxwell’s Demon
Supriyo Datta
School of Electrical and Computer Engineering,
NSF Network for Computational Nanotechnology,
Purdue Universtiy, West Lafayette, IN-47906, USA.
(Date: April 12, 2007)
In the last twenty years there has been significant progress in our understanding of quantum transport far
from equilibrium and a conceptual framework has emerged through a combination of the Landauer
approach with the non-equilibrium Green function (NEGF) method, which is now being widely used in the
analysis and design of nanoscale devices. It provides a unified description for all kinds of devices from
molecular conductors to carbon nanotubes to silicon transistors covering different transport regimes from
the ballistic to the diffusive limit. In this talk I use a simple version of this model to analyze a specially
designed device that could be called an electronic Maxwell’s demon, one that lets electrons go
preferentially in one direction over another. My objective is to illustrate the fundamental role of “contacts”
and “demons” in transport and energy conversion. The discussion is kept at an academic level steering clear
of real world details, but the illustrative devices we use are very much within the capabilities of present-day
technology. For example, recent experiments on thermoelectric effects in molecular conductors agree well
with the predictions from our model. The Maxwell’s demon device itself is very similar to the pentalayer
spin-torque device which has been studied by a number of groups though we are not aware of any
discussion of the possibility of using the device as a nanoscale heat engine or as a refrigerator as proposed
here. However, my objective is not to evaluate possible practical applications. Rather it is to introduce a
simple transparent model showing how out-of-equibrium demons suitably incorporated into nanodevices
can achieve energy conversion.
1. INTRODUCTION
Maxwell invented his famous demon to illustrate the
subtleties of the second law of thermodynamics and his
conjecture has inspired much discussion ever since [see for
example, Leff and Rex,1990,2003, Nikulov and Sheehan,
2004]. When the subject of thermodynamics was relatively
new, it was not clear that heat was a form of energy since
heat could never be converted entirely into useful work.
Indeed the second law asserts that none of it can be
converted to work if it is all available at a single
temperature. Heat engines can only function by operating
between two reservoirs at two different temperatures.
Maxwell’s demon is supposed to get around this
fundamental principle by creating a temperature
differential between two sides of a reservoir that is initially
at a uniform temperature. This is achieved by opening and
closing a little door separating the two sides at just the right
times to allow fast molecules (white) to cross to the left but
not the slow molecules (black, Fig.1a). As a result, faster
molecules crowd onto the left side making its temperature
higher than that of the right side.
Technology has now reached a point where one can think
of building an electronic Maxwell’s demon that can be
interposed between the two contacts (labeled source and
drain) of a nanoscale conductor (Fig.1b) to allow electrons
to flow preferentially in one direction so that a current will
flow in the external circuit even without any external
source of power. Such a device can be built (indeed one
could argue has already been built) though not surprisingly
it is expected to operate in conformity with the second law.
Here I would like to use this device simply to illustrate the
fundamental role of “contacts” and “demons” in transport
and energy conversion. I will try to keep the discussion at
an academic level steering clear of real world details. But it
should be noted that the illustrative devices we will discuss
are very much within the capabilities of present-day
technology. For example, recent pioneering experiments on
thermoelectric effects in molecular conductors [Reddy
et.al.2007] seem to agree well with the predictions from
our model [Paulsson and Datta,2003]. The device
discussed in Section 3 (Figs.5,6) incorporating an elastic
Maxwell’s demon (Datta 2005b,c) is being investigated
experimentally by introducing manganese impurities into a
GaAs spin-valve device with MnAs contacts [Saha
et.al.2006]. The device incorporating an inelastic demon
described in Section 4 (Figs.9,10) is very similar to the
pentalayer spin-torque device which has been studied by a
number of groups both experimentally [Fuchs et.al.2006,
Nanodevices and Maxwell’s demon
Supriyo Datta
(a) Maxwell’s demon (b) Electronic demon
Fig.1. (a) Maxwell’s demon opens and closes a trapdoor
to separate fast (white) molecules from slow (black)
molecules making the left warmer than the right, thus
creating a temperature differential without expending
energy . (b) Electronic Maxwell’s demon discussed in this
talk lets electrons go preferentially from right to left thus
creating a current in the external circuit without any
external source of energy.
Fig.2. Schematic representing the general approach used
to model nanoscale devices: (a) Simple version with
numbers γ , D used in this talk and (b) Complete version
with matrices Σ ,H (Adapted from Datta2005a).
Channel
Source Drain
<---- L ---->
Σ1 Σ2
µ1 µ2
µ1 µ2
1γ 2γ
D(E)
3 arXiv:cond-mat/0704.1623
[email protected]
Huai et.al.2004] and theoretically [Salahuddin and
Datta,2006a,b] though we are not aware of any discussion
of the possibility of using the device as a nanoscale heat
engine or as a refrigerator as proposed here. We leave it to
future work to assess whether these possibilities are of any
practical importance. Here my objective is to lay out a
simple transparent model showing how out-of-equilibrium
demons suitably incorporated into nanodevices can achieve
energy conversion.
In the last twenty years there has been significant progress
in our ability to tackle the problem of quantum transport far
from equilibrium and a conceptual framework has emerged
through a combination of the Landauer approach with the
non-equilibrium Green function (NEGF) method
[Datta,1989,1990, Meir and Wingreen,1992],which is
being widely used in the analysis and design of nanoscale
devices [see Datta,2005a and references therein]. It
provides a unified description for all kinds of devices from
molecular conductors to carbon nanotubes to silicon
transistors in terms of the Hamiltonian [H] describing the
channel, the self-energies [ Σ1,2] describing the connection
to the contacts and the [ Σ s] describing interactions inside
the channel (Fig.2b). In each case the details are very
different, but once these matrices (whose size depends on
the number of basis functions needed to describe the
channel) have been written down, the procedure for
obtaining quantities of interest such as current flow and
energy dissipation are the same regardless of the specifics
of the problem at hand. The model covers different
transport regimes from the ballistic to the diffusive limit
depending on the relative magnitudes of Σ1,2 and Σ s . In
this paper I will use a particularly simple version of this
approach (Fig.2a) where matrices like Σ1,2 and Σs are
replaced with numbers like γ1,2 and γ s having simple
physical interpretations:
γ1,2 /ℏ represents the rate of
transfer of channel electrons in and out of the contacts
while
γ s /ℏ represents the rate at which they interact with
any “demons” that inhabit the channel.
In the past it was common to have γ1,2 << γ s so that
transport was dominated by the interactions within the
channel, with contacts playing a minor enough role that
theorists seldom drew them prior to 1990! By contrast,
today’s nanodevices have reached a point where γ1,2 >>
γ s , placing them in what we could call the ballistic or
“Landauer limit”. An appealing feature of this limit is that
it physically separates the dynamics from the dissipation:
reversible dynamics dominates the channel while
dissipation dominates the contact. Usually these two
aspects of transport are conceptually so intertwined that it
is difficult to see how irreversibility is introduced into a
problem described by reversible dynamic equations
(Newton or Schrodinger) and this issue continues to spark
debate and discussion ever since the path-breaking work of
Boltzmann many years ago [see for example,
McQuarrie,1976].
Let me start with a brief summary of the basic framework
shown in Fig.2a that we call the bottom-up viewpoint
(Section 2). We will then use this approach to discuss a
specially designed device which is in the Landauer limit
except for a particularly simple version of Maxwell’s
demon that interacts with the channel electrons but does
not exchange any energy with them (Section 3).We then
consider a more sophisticated demon that exchanges
energy as well and show how it can be used to build
nanoscale heat engines and refrigerators (Section 4). We
conclude with a few words about entangled demons and
related conceptual issues that I believe need to be clarified
in order to take transport physics to its next level (Section
5). Readers may find the related video lectures posted on
the nanoHUB useful [Datta,2006] and I will be happy to
share the MATLAB codes used to generate the figures in
this paper.
2. BOTTOM-UP VIEWPOINT
Consider the device shown in Fig.1b without the “demon”
but with a voltage V applied across two contacts (labeled
“source” and “drain”) made to a conductor (“channel”).
How do we calculate the current I, as the length of the
channel L is made shorter and shorter, down to a few
atoms? This is not just an academic question since
experimentalists are actually making current measurements
through “channels” that are only a few atoms long. Indeed,
this is also a question of great interest from an applied
point of view, since every laptop computer contains about
one billion nanotransistors, each of which is basically a
conductor like the one in Figure 1b with L ~ 50 nm, but
with the demon replaced by a third terminal that can be
used to control the resistance of the channel.
As the channel length L is reduced from macroscopic
dimensions (~ millimeters) to atomic dimensions (~
nanometers), the nature of electron transport that is, current
flow, changes significantly (Fig.3). At one end, it is
described by a diffusion equation in which electrons are
viewed as particles that are repeatedly scattered by various
obstacles causing them to perform a “random walk” from
the source to the drain. The resistance obeys Ohm’s law: a
sample twice as long has twice the resistance. At the other
end, there is the regime of ballistic transport where the
resistance of a sample can be independent of length. Indeed
due to wavelike interference effects it is even possible for a
longer sample to have a lower resistance!
The subject of current flow is commonly approached using
a “top-down” viewpoint. Students start in high school from
the macroscopic limit (large L) and seldom reach the
atomic limit, except late in graduate school if at all. I
believe that this is primarily for historical reasons. After
all, twenty years ago, no one knew what the resistance was
for an atomic scale conductor, or if it even made sense to
Nanodevices and Maxwell’s demon
Supriyo Datta
Fig.3. Evolution of devices from the regime of diffusive
transport to ballistic transport as the channel length L is
scaled down from millimeters to nanometers.
Fig.4. Plot of energy current in a 1-D ballistic conductor
with G = e2 / h ≈ 40µS and an applied voltage of V = 0.05
volts. Energy dissipated is given by the drop in the energy
current, showing that the Joule heating V 2G = 0.1 µW is
divided equally between the two contact-channel
interfaces.
Diffusive
Transport
1 mm 100 µm 10 µm 1 µm 100 nm 10 nm 1 nm 1
Ballistic
Transport
Macroscopic
dimensions
Atomic
dimensions
<----- L ----->
0 0.2 0.4 0.6 0.8 1
Energy
Current
Normalized Distance Along Device
Source Channel Drain
Dissipationless channel
Newton’s law/
Schrodinger equation
Dissipation DissipationContacts assumed
to remain in equilibrium
5 arXiv:cond-mat/0704.1623
[email protected]
ask about its resistance But now that the bottom-line is
known, I believe that a “bottom-up” approach is needed if
only because nanoscale devices like the ones I want to talk
about look too complicated from the “top-down”
viewpoint.
In the top-down view we start by learning that the
conductance G=I/V is related to a material property called
conductivity σ through a relation that depends on the
sample geometry and for a rectangular conductor of cross-
section A and length L is given by G = σA /L . We then
learn that the conductivity is given by
σ = e2nτ /m
where e is the electronic charge, n is the electron density,
τ is the mean free time and m is the electron mass.
Unfortunately from this point of view it is very difficult to
understand the ballistic limit. Since electrons get from one
contact to the other without scattering it is not clear what
the mean free time τ is. Neither is it clear what one should
use for ‘n’ since it stands for the density of free electrons
and with molecular scale conductors it may not be clear
which electrons are free. Even the mass is not very clear
since the effective mass is deduced from the bandstructure
of an infinite periodic solid and cannot be defined for really
small conductors.
2.1. Conductance formula: the “bottom-up” version
A more transparent approach at least for small conductors
is provided by the bottom-up viewpoint [Datta, 2005a,
Chapter 1] which leads to an expression for conductance in
terms of two basic factors, namely the density of states D
around the equilibrium electrochemical potential and the
effective escape rate γ /ℏ from the channel into the
contacts ( ℏ = h / 2π , h being Planck’s constant):
(1a)
The escape rate appearing above is the series combination
of the escape rates associated with each contact:
(1b)
This is an expression that we can apply to the smallest of
conductors, even a hydrogen molecule. Although it looks
very different from the expression for conductivity
mentioned earlier, it is closer in spirit to another well-
known expression for the conductivity
(2)
in terms of the density of states per unit volume ˜ N and the
diffusion coefficient ˜ D . Indeed we could obtain Eq.(1a)
from Eq.(2) if we make the replacements
(3)
which look reasonable since the density of states for a large
conductor is expected to be proportional to the volume AL
and the time taken to escape from a diffusive channel is ~
˜ D /L2.
2.2 Current-voltage relation: without demons
The result cited above (Eq.(1a) is a linear response version
of a more general set of equations that can be used to
calculate the full current-voltage characteristics, which in
turn follow from the NEGF-Landauer formulation (Fig.2b).
For our purpose in this talk the simpler version (Fig.2a)
will be adequate and in this version the basic equations are
fairly intuitive:
(4)
These equations relate the currents per unit energy at
contacts 1 and 2 to the density of states D(E) , the Fermi
functions at the two contacts related to their
electrochemical potentials
(5)
and the distribution function f(E) inside the channel.
The total currents at the source and drain contacts are
obtained by integrating the corresponding energy resolved
currents
(6)
If the electrons in the channel do not interact with any
demons, we can simply set i1(E) = i2 (E) , calculate f(E)
and substitute back into Eq.(4) to obtain
(7)
The conductance expression stated earlier (Eq.(1a)) follows
from Eqs.(6) and (7) by noting that
(8)
G = (e2 / h) 2πD γ
γ = γ
D → ˜ N . AL and γ /ℏ → ˜ D / L2
i1(E) = (e /ℏ) γ1 D(E) [ f1 (E) − f (E)]
i2 (E) = (e /ℏ) γ 2 D(E) [ f (E) − f2 (E)]
f1,2 (E) =
1+ exp((E − µ1,2) / kBT1,2)
i1(E) = i2 (E)
= (e /ℏ)
γ1γ 2
γ1 + γ 2
D(E) [ f1(E) − f2 (E)]
dE [ f1(E) − f2 (E)] = µ1 − µ2∫
I1 = dE∫ i1(E) , I2 = dE∫ i2 (E)
σ = e2 ˜ N ˜ D
Nanodevices and Maxwell’s demon
Supriyo Datta
and assuming the density of states D(E) to be nearly
constant over the energy range of transport where
f1(E) − f2 (E) is significantly different from zero.
One-level conductor: Eq.(7) can be used more generally
even when the density of states has sharp structures in
energy. For example, for a very small conductor with just
one energy level in the energy range of interest, the density
of states is a “delta function” that is infinitely large at a
particular energy. Eqs.(6,7) then yield a current-voltage
characteristic as sketched below.
The maximum current is equal to
eγ1 / 2ℏ , assuming
γ 2= γ1. It might appear that the maximum conductance can
be infinitely large, since the voltage scale over which the
current rises is ~ kBT , so that dI/dV can increase without
limit as the temperature tends to zero. However, the
uncertainty principle requires that the escape rate of γ /ℏ
into the contacts from an energy level also broadens the
level by γ as shown below. This means that the voltage
scale over which the current rises is at least ~ (γ1 +γ2) /e
= 2γ1 /e , even at zero temperature.
This means that a small device has a maximum
conductance of
This rough estimate is not too far from the correct result
(9)
which is one of the seminal results of mesoscopic physics
that was not known before 1988. One could view this as a
consequence of the energy broadening required by the
uncertainty principle which comes out automatically in the
full Landauer-NEGF approach (Fig.2b), but has to be
inserted by hand into the simpler version we are using
where the density of states D(E) is an input parameter
(Fig.2a).
For our examples in this paper we will use a density of
states that is constant in the energy range of interest, for
which the current-voltage characteristic is basically linear.
2.3 Current-voltage relation: with demons
Defining is (E) as the “scattering current” induced by
interaction of the electrons with the “demon”, we can
write
(10)
This current can be modeled in general as a difference
between two processes one involving a loss of energy ε
from the demon and the other involving a gain of energy
by the demon.
(11))
The basic principle of equilibrium statistical mechanics
requires that if the demon is in equilibrium at some
temperature TD then the strength F (ε) of the energy loss
processes is related to the strength F (−ε) of energy gain
processes by the ratio:
(12)
With a little algebra one can show that this relation ensures
that if the electron distribution f(E) is given by an
equilibrium Fermi function with the same temperature T
is (E) = (e /ℏ) γ s D(E) dε∫ D(E +ε)
[F (ε) f (E) (1− f (E +ε)
− F (−ε) f (E +ε) (1− f (E)]
i1(E) = i2 (E) + is (E)
F (ε) = F (−ε) exp (−ε / kBTD )
-1 -0.5 0 0.5 1
Normalized voltage
Normalized
current
-1 -0.5 0 0.5 1
Normalized voltage
Normalized
current
eγ1 /2ℏ
Gmax = e
2 /h ≈ 25.8 KΩ
eγ1 / 2ℏ
eγ1 / 2ℏ µ1
Normalized voltage
Normalized
current
D(E)
-1 -0.5 0 0.5 1
7 arXiv:cond-mat/0704.1623
[email protected]
the two terms in Eq.(11) will cancel out. This result is
independent of the detailed shape of the function F (ε)
describing the spectrum of the demon, as long as Eq.(12) is
true. This means that if the demon is in equilibrium with
the electrons with the same temperature, there can be no
net flow of energy either to or from the demon. Indeed
one could view this as the basic principle of equilibrium
statistical mechanics and work backwards to obtain Eq.(12)
as the condition needed to ensure compliance with this
principle.
To summarize, if the electrons in the channel do not
interact with any “demons”, the current voltage
characteristics are obtained from Eq.(7) using the Fermi
functions from Eq.(5). For the more interesting case with
interactions, we solve for the distribution f(E) inside the
channel from Eqs.(4),(10) and (11) and then calculate the
currents. Usually the current flow is driven by an external
voltage that separates the electrochemical potentials µ1
and µ2 in Eq.(7). But thermoelectric currents driven by a
difference in temperatures T1 and T2 can also be
calculated from this model [Paulsson and Datta,2003] as
mentioned in the introduction. Our focus here is on a
different possibility for energy conversion, namely through
out-of-equilibrium demons.
2.4. Where is the heat?
Before we move on, let me say a few words about an
important conceptual issue that caused much argument in
the early days: Where is the heat dissipated in a ballistic
conductor? After all, if there is no demon to take up the
energy, there cannot be any dissipation inside the channel.
The answer is that the transiting electron appears as a hot
electron in the drain (right) contact and leaves behind a hot
hole in the source (left) contact (see Fig.4). The contacts
are immediately restored to their equilibrium states by
unspecified dissipative processes operative within each
contact. These processes can be quite complicated but are
usually incorporated surreptitiously into the model through
what appears to be an innocent boundary condition, namely
that the electrons in the contacts are always maintained in
thermal equilibrium described by Fermi distributions
(Eq.(5)) with electrochemical potentials µ1 and µ2 and
temperatures T1 and T2 .
To understand the spatial distribution of the dissipated
energy it is useful to look at the energy current which is
obtained by replacing the charge ‘e’ with the energy E of
the electron:
(13)
The energy currents at the source and drain contacts are
written simply as
(14)
assuming that the entire current flows around the
corresponding electrochemical potentials,
Fig.4 shows a spatial plot of the energy current from the
source end to the drain end for a uniform 1-D ballistic
conductor with a voltage of 50 mV applied across it. For a
conductor with no demon for electrons to exchange energy
with, i1(E) = i2 (E) making the energy current uniform
across the entire channel implying that no energy is
dissipated inside the channel. But the energy current
entering the source contact is larger than this value while
that leaving the drain contact is lower. Wherever the
energy current drops, it means that the rate at which energy
flows in is greater than the rate at which it flows out,
indicating a net energy dissipation. Clearly in this example,
0.05 µW is dissipated in each of the two contacts thus
accounting for the expected Joule heating given by V 2G =
0.1 µW.
Real conductors have distributed demons throughout the
channel so that dissipation occurs not just in the contacts
but in the channel as well. Indeed we commonly assume
the Joule heating to occur uniformly across a conductor.
But there are now experimental examples of nanoscale
conductors that would have been destroyed if all the heat
were dissipated internally and it is believed that the
conductors survive only because most of the heat is
dissipated in the contacts which are large enough to get rid
of it efficiently. The idealized model depicted in Fig.4 thus
may not be too far from real nanodevices of today.
As I mentioned in the introduction, what distinguishes the
Landauer model (Fig.4) is the physical separation of
dynamics and dissipation clearly showing that what makes
transport an irreversible process is the continual restoration
of the contacts back to equilibrium. Without this repeated
restoration, all flow would cease once a sufficient number
of electrons have transferred from the source to the drain.
Over a hundred years ago Boltzmann showed how pure
Newtonian dynamics could be supplemented to describe
transport processes, and his celebrated equation stands
today as the centerpiece for the flow of all kinds of
particles. Boltzmann’s approach too involved “repeated
restoration” through an assumption referred to as the
“Stohsslansatz” [see for example, McQuarrie 1976]
Today’s devices often involve Schrodinger dynamics in
place of Newtonian dynamics and the non-equilibrium
Green function (NEGF) method that we use (Fig.2b)
supplements the Schrodinger equation with similar
assumptions about the repeated restoration of the
I E s (E) = (µ1 /e) I1 ,
I E d (E) = (µ 2 /e) I2
IE1 = (1 /e) dE∫ E i1(E) ,
IE 2 = (1 /e) dE∫ E i2 (E)
Nanodevices and Maxwell’s demon
Supriyo Datta
surroundings that enter the evaluation of the various self-
energy functions Σ or the corresponding quantities γ in
the simpler model that we are using.
“Contacts” and “demons” are an integral part of all
devices, the most common demon being the phonon bath
for which the relation in Eq.(12) is satisfied by requiring
that energy loss ~ N and energy gain ~ (N+1), N being the
number of phonons given by the Bose Einstein factor if the
bath is maintained in equlibrium. Typically such demons
add channels for dissipation, but our purpose here is to
show how suitably engineered out-of-equilibrium demons
can act as sources of energy.
For this purpose, it is convenient to study a device
specially designed to accentuate the impact of the demon.
Usually the interactions with the demon do not have any
clear distinctive effect on the terminal current that can be
easily detected. But in this special device, ideally no
current flows unless the channel electrons interact with
the demon. Let me now describe how such a device can be
engineered.
3. ELASTIC DEMON
Let us start with a simple 1-D ballistic device but having
two rather special kinds of contacts. The source contact
allows one type of spin, say “black” (upspins, drawn
pointing to the right), to go in and out of the channel much
more easily than the other type, say “white” (downspins,
drawn pointing to the left, Fig.5a). Devices like this are
called spin-valves and are widely used to detect magnetic
fields in magnetic memory devices [see for example the
articles in Heinreich and Bland, 2004].
Although today’s spin-valves operate with contacts that are
far from perfect, since we are only trying to make a
conceptual point, let us simplify things by assuming
contacts that are perfect in their discrimination between the
two spins. One only lets black spins while the other only
lets white spins to go in and out of the channel. We then
have the situation shown in Fig.5b and no current can flow
since neither black nor white spins communicate with both
contacts. But if we introduce impurities into the channel
(the demon) with which electrons can interact and flip their
spin, then current flow should be possible as indicated:
black spins come in from the left, interact with impurities
to flip into white spins and go out the right contact
(Fig.5c).
Consider now what happens if the impurities are say all
white (Fig.6a). Electrons can now flow only when the bias
is such that the source injects and drain collects (positive
drain voltage), but not if the drain injects and the source
collects (negative drain voltage). This is because the source
injects black spins which interact with the white impurities,
flip into whites and exit through the drain. But the drain
only injects white spins which cannot interact with the
white impurities and cannot cross over into the source.
Similarly if the impurities are all black, current flows only
for negative drain voltage (Fig.6b). At non-zero
temperatures the cusps in the current-voltage
characteristics get smoothed out and we get the smoother
curves shown in Figs.6a,b.
Note the surprising feature of the plots at T=300K: there is
a non-zero current even at zero voltage! This I believe is
correct. Devices like those shown in Figs.6a,b with
polarized impurities could indeed be used to generate
power and one could view the system of impurities as a
Maxwell’s demon that lets electrons go preferentially from
source to drain or from drain to source. The second law is
in no danger, since the energy is extracted at the expense of
the entropy of the system of impurities that collectively
constitute the demon. Assuming the spins are all non-
interacting and it takes no energy to flip one, the polarized
state of the demon represents an unnatural low entropy
state. Every time an electron goes through, a spin gets
flipped raising the entropy and the flow will eventually
stop when demon has been restored to its natural
unpolarized state with the highest entropy of N I kB ln 2 ,
where N I is the number of impurities. To perpetuate the
flow an external source will have to spend the energy
needed to maintain the demon in its low entropy state.
3.1. Current versus voltage: Model summary
Let me now summarize the equations that I am using to
analyze structures like the one in Fig.6 quantitatively.
Basically it is an extension of the equations described
earlier (see Eq.(4), Section 2) to include the two spin
channels denoted by the subscripts ‘u’ (up or black) and ‘d’
(down or white):
(15)
(16)
For the scattering current caused by the demon we write
(see Eq.(11))
(17)
i1,u (E) = (e /ℏ) γ1,u Du (E) [ f1(E) − fu (E)]
i2,u (E) = (e /ℏ) γ 2,u Du (E) [ fu (E) − f2 (E)]
i1,d (E) = (e /ℏ) γ1,d Dd (E) [ f1(E) − fd (E)]
i2,d (E) = (e /ℏ) γ 2,d Dd (E) [ fd (E) − f2 (E)]
is,u (E) = − is,d (E)
= (e /ℏ) (2π J 2N I ) Du (E) Dd (E)
[Fd fu (E) (1− fd (E) − Fu fd (E) (1− fu (E)]
9 arXiv:cond-mat/0704.1623
[email protected]
Fig.5. Anti-parallel (AP) Spin-valve: (a) Physical
structure, (b) no current flows if the contacts can
discriminate between the two spins perfectly, (c) current
flow is possible if impurities are present to induce spin-
flip processes, Adapted from Datta, 2005b,c.
Source
Drain
Spin-flip
impurities
Source
Drain
(b (c)
Current
Voltage
(b) w/o spin-flip
(c) with spin-flip
DrainSource Channel
Nanodevices and Maxwell’s demon
Supriyo Datta
Fig.6. Perfect AP Spin-valve with (a) white (down spin,
drawn as pointing to the left) impurities and (b) black
(upspin, drawn as pointing to the right) impurities. Note
the non-zero current at zero voltage for non-zero
temperatures. Adapted from Datta, 2005b,c.
-0.1 -0.05 0 0.05 0.1
5x 10
Voltage
-0.1 -0.05 0 0.05 0.1
5x 10
Voltage
Current (A)
300K
300K
0 K 0 K
Source
Drain
(a) “White” impurities
Source
Drain
(b) “Black” impurities
11 arXiv:cond-mat/0704.1623
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Fig.7. An anti-parallel (AP) spin-valve with spin-flip
impurities and a simple equivalent circuit to help
visualize the equations we use to describe it, namely
Eqs.(15) through (17), Adapted from Datta, 2005b,c.
DrainSource
Magnetic
Impurities
down spin
up spin
g2,dg1,d
⇑ idem
Nanodevices and Maxwell’s demon
Supriyo Datta
noting that we are considering an elastic demon that can
neither absorb nor give up energy (ε =0). The first term
within parenthesis in Eq.(17) represents an up electron
flipping down by interacting with a down-impurity while
the second term represents a down electron flipping up by
interacting with an up-impurity. The strengths of the two
processes are proportional to the fractions Fd and Fu of
down and up-spin impurities.The overall strength of the
interaction is governed by the number of impurities N I and
the square of the matrix element J governing the electron-
impurity interaction.
Eqs.(15),(16) can be visualized in terms of an equivalent
circuit (Fig.7) if we think of the various f’s as “voltages”
since the currents are proportional to differences in ‘f’ just
as we expect for conductances
etc.
The scattering current (Eq.(17)) too could be represented
with a conductance gs if we set Fd = Fu = 0.5
(18a)
However, this is true only if the impurities are in
equilibrium, while the interesting current-voltage
characteristics shown in Fig.6 require an out-of-
equilibrium demon with Fd ≠ Fu . So we write the total
scattering current from Eq.(17) as a sum of two
components, one given by Eq.(18a) and another
proportional to ( Fd - Fu ) which we denote with a
subscript ‘dem’:
(18b)
Eqs.(16) and (17) can be solved to obtain the distribution
functions fu (E) and fd (E) by imposing the requirement
of overall current conservation (cf. Eq.(11)):
(19)
The currents are then calculated and integrated over energy
to obtain the terminal currents shown in Fig.6. We can also
find the energy currents using equations like Eqs.(14), (15)
and the results are shown in Fig.8 for Fd - Fu = -1 and for
Fd - Fu = 0 each with a voltage difference of 2 kBT =50
mV between the two terminals. With Fd - Fu = 0 the
direction of current is in keeping with an external battery
driving the device. But with Fd - Fu = -1, the external
current flows against the terminal voltage indicating that
the device is acting as a source of energy driving a load as
shown in the inset. This is also borne out by the energy
current flow which shows a step up at each interface
indicating that energy is being absorbed from the contacts
(~ 10 nW from each) and delivered to the external load.
But isn’t this exactly what the second law forbids? After all
if we could just absorb energy from our surroundings (the
contacts) and do useful things, there would be no energy
problem. The reason this device is able to perform this
impossible feat is that the impurities are assumed to be held
in a non-equilibrium state with very low entropy. A
collection of N I impurities can be unpolarized in 2
different ways having an entropy of S = N I kB ln 2 . But it
can be completely polarized ( Fd - Fu = ± 1) in only one
way with an entropy of S = 0. What this device does is to
exchange entropy for energy. Many believe that the
universe as a whole is evolving the same way, with
constant total energy, from a particularly low entropy state
continually towards a higher entropy one. But that is a
different matter.
To have our device continue delivering energy indefinitely
we will need an external source to maintain the impurities
in their low entropy state which will cost energy. The
details will depend on the actual mechanism used for the
purpose but we will not go into this. Note that if we do not
have such a mechanism, the current will die out as the
spins get unpolarized. This depolarization process can be
described by an equation of the form:
(20)
where ˜ τ 0 and si are related to the scattering current as
defined in Eqs.(18), while the additional time constant
represents processes unrelated to the channel electrons by
which impurities can relax their spins.
4. INELASTIC DEMON
We have argued above that although one can extract
energy from polarized impurities, energy is needed to keep
them in that state since their natural high entropy state is
the unpolarized one. It would seem that one way to keep
the spins naturally in a polarized state is to use a
nanomagnet, a collection of spins driven by a
ferromagnetic interaction that keeps them all pointed in the
is = (e /ℏ) (πJ
2NI ) Du (E)Dd (E) ( fd − fu )
idem = (Fd − Fu ) eNI / ˜ τ 0(E)
˜ τ 0(E )
= (1/ℏ) (πJ2) Du (E)Dd (E)
[ fd (1− fu ) + fu (1− fd )]
i1,u (E) = i2,u (E) + is,u (E)
i1,d (E) = i2,d (E) + is,d (E)
g1,u = (e
2 /ℏ) γ1,u Du
dP ∫=
13 arXiv:cond-mat/0704.1623
[email protected]
same direction. Could such a magnet remain polarized
naturally and enable us to extract energy from the contacts
Fig.8. Energy currents with Fd - Fu = -1 and with Fd -
Fu = 0 each with a voltage difference of 2 kBT =50 mV
between the two terminals. With Fd - Fu = 0 the direction
of current is in keeping with an external battery driving
the device. But with Fd - Fu = -1, the external current
flows against the terminal voltage indicating that the
device is acting as a source of energy driving a load as
shown in the inset. This is also borne out by the energy
current flow which shows a step up at each interface
indicating that energy is being absorbed from the contacts
(~ 10 nW from each) and delivered to the external load.
0 0.2 0.4 0.6 0.8 1
4x 10
Energy
Current
Normalized Distance Along Device
Source
Channel
Drain
0.05 V
Fd - Fu = 0
++++ 0.05 V ----
Drain Source
Fd - Fu = -1
Load
Resistor
Nanodevices and Maxwell’s demon
Supriyo Datta
forever? The answer can be “yes” if the magnet is at a
different temperature from the electrons. What we then
have is a heat engine working between two reservoirs (the
electrons and the magnet) at different temperatures and we
will show that its operation is in keeping with Carnot’s
principle as required by the second Law.
To model the interaction of the electrons with the
nanomagnet we need to modify the expression for the
scattering current (Eq.(17)) for it now takes energy to flip a
spin. We can write
(21)
where F (ε) denotes the magnon spectrum obeying the
general law stated earlier (see Eq.(12)) if the magnet has a
temperature TD . Eq.(21) can be solved along with
Eqs.(15,(16),(19) and (21) as before to obtain currents,
energy currents etc. But let us first try to get some insight
using simple approximations.
If we assume that the electron distribution functions fu (E)
and fd (E) are described by Fermi functions with
electrochemical potentials µu and µd respectively and
temperature T, and make use of Eq.(12) we can rewrite the
scattering current from Eq.(21) in the form
(22)
If we further assume the exponent to be small, so that 1 –
exp(-x) ≈ x , we can write
(23)
The first term represents a dissipative
current proportional to the potential
difference and can be represented by a
conductance like the gs in Fig.7,
while the second term is the demon
induced source term that can be
harnessed to do external work. It
vanishes when the demon temperature
TD equals the electron temperature T.
To be specific, let us assume that the demon is cooler than
the rest of the device (TD < T) so that
giving rise to a flow of electrons out of the upspin node
and back into the downspin node. If we use this to drive an
external load then µu − µd > 0, which will tend to reduce
the net current given by Eq.(23) and the maximum output
voltage one can get corresponds to “open circuit”
conditions with zero current:
(24)
This expression has a simple interpretation in terms of the
Carnot principle. Every time an electron flows around the
circuit giving up energy ε to the demon, it delivers energy
µu − µd to the load. All this energy µu − µd + ε is
absorbed from the contacts and the Carnot principle
requires that
Energy from contacts / T ≤ Energy to demon / TD
that is,
which is just a restatement of Eq.(24). Note that usual
derivations of the Carnot principle do not put the system
simultaneously in contact with two reservoirs at different
temperatures as we have done. Our treatment is closer in
spirit to the classic discussion of the ratchet and pawl in the
Feynman Lectures [Feynman et.al. 1963].
Fig.9 shows numerical results obtained by solving
Eqs.(21),(19),(15) and (16). The extracted power is a
maximum (Fig.9d) when the output voltage is somewhere
halfway between zero and the maximum output voltage of
80 mV (a few kBT ).Fig.9b shows the energy current
profile at an output voltage of 50 mV: energy is absorbed
from the source and drain contacts and given up partly to
is,u (E) = − is,d (E)
= (e /ℏ) (2π J 2N I ) dε∫ Du (E) Dd (E +ε)
[F (ε) fu (E) (1− fd (E +ε))
− F (−ε) fd (E +ε) (1− fu (E)]
is,u (E) = − is,d (E)
= (e /ℏ) (2π J 2N I ) dε∫ Du (E) Dd (E +ε)
F (ε) fu (E) (1− fd (E +ε)
F (−ε)
F (+ε)
fd (E +ε)
1− fd (E +ε)
1− fu (E)
fu (E)
= (e /ℏ) (2π J 2N I )
dε∫ Du (E) Dd (E +ε)F (ε) fu (E) (1− fd (E +ε)
1 − exp
µd −ε − µu
is,u = − is,d
µu − µd
is,u
< 0
is,u
µu − µd +ε
µu − µd
15 arXiv:cond-mat/0704.1623
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Fig.9. Heat engine: (a) AP Spin-valve with a cooled
magnet as Maxwell’s demon controlling the flow of
electrons. (b) Output current and output power versus
output voltage as the load is varied from short circuit
(V=0) to open circuit conditions (I=0). The spectrum of
the magnet is assumed to consist of a single energy ε =
2kBT = 50 meV. (c) Energy current profile assuming a
load such that the output voltage is 50 mV.
Output Voltage (V) Output Voltage (V)
Current (A) Power (W)
0 0.02 0.04 0.06 0.08
14x 10
0 0.02 0.04 0.06 0.08
2x 10
Normalized Distance Along Device
0 0.2 0.4 0.6 0.8 1
2x 10
Source Drain
“Demon”
Energy
Current
Drain
at 600 K
Source
at 600 K
“Magnet”
at 300 K
Nanodevices and Maxwell’s demon
Supriyo Datta
Fig.10. (a) Heat engine from Fig.9 operated as a
refrigerator by applying an external battery to inject
downspin (white) electrons from the drain that flip down
the thermally created upspins in the magnet, thus cooling
it. (b) Energy current profile showing that energy is
absorbed from the external battery and the demon and
dissipated in the source and drain contacts.
Fig.11. We have described spin-flip processes in terms of
a direct conversion from state A to state B. Quantum
mechanics, however, requires an intermediate state
consisting of a superposition of A and B before
wavefunction collapse reduces it to a B. This state may
have a significant role in devices with weak contacts and
strong interactions in the channel, requiring a model that
can deal with entangled states.
0 0.2 0.4 0.6 0.8 1
1.5 x 10
Normalized Distance Along Device
Energy
Current
Source Drain
“Demon”
Source
Drain
Source
Drain
Drain
at 300 K Source
at 300 K
“Magnet”
at 250 K
17 arXiv:cond-mat/0704.1623
[email protected]
the demon and the rest is delivered to the external load.
The efficiency (energy given to load / energy absorbed
from contact) is a maximum close to open circuit
conditions, but the energy delivered is very small at that
point as evident from Fig.9d.
As one might expect, one can also operate the same device
as a refrigerator by using an external source that seeks to
inject downspins (white) from the drain contact that flip
back thermally created upspins in the magnet thus cooling
it. It is evident from the energy current profile shown in
Fig.10 that in this case, energy is absorbed from the demon
and from the battery and given up to the source and drain
contacts.
5. ENTANGLED DEMON
In this talk I have tried to introduce a simple transparent
model showing how out-of-equibrium demons suitably
incorporated into nanodevices can achieve energy
conversion. At the same time this model illustrates the
fundamental role played by “contacts” and “demons” in
these processes. I would like to end by pointing out another
aspect of contacts that I believe is important in taking us to
our next level of understanding. The basic point can be
appreciated by considering a simple version of the spin
capacitor we started with (see Fig.5) but having just one
impurity (Fig.11). No current can flow in this structure
without spin-flip processes since the source injects black
(up) electrons while the drain only collects white (down)
electrons. But if the black electron interacts with the white
impurity(A) to produce a black impurity then the white
electron can be collected resulting in a flow of current.
The process of conversion
From A: Black electron ⊗ White impurity
To B: White electron ⊗ Black impurity
is incorporated into our model through the scattering
current (see Eq.(17)). A more complete quantum transport
model involving matrices (Fig.3a) rather than numbers
(Fig.3b) could be used to describe this effect, but the
essential underlying assumption in either case is that the
state of the electron-impurity system changes from A to B.
Quantum mechanics, however, paints a different picture of
the process involving an intermediate entangled state. It
says that the system goes from A into a state consisting of a
superposition of A and B and it is only when the electron is
collected by the drain that the wavefunction collapses to a
B.If the collection rates
γ1,2 /ℏ are much larger than the
interaction rate per impurity
γ s /ℏN I , we expect the
entangled state to play a minor role. But this may not be
true of devices with weak contacts and strong interactions
in the channel, requiring a model that can deal with
entangled states.
Entangled states are difficult to describe within the
conceptual framework we have been using where both the
electrons and the impurity are assumed to exist in
independent states. It is hard to describe a “conditional
state” where the electron is black if the impurity is white or
vice versa, let alone a superposition of the two. To account
for this entangled state we need to treat the electron and
impurity as one big system and write rate equations for it,
in the spirit of the many-electron rate equations widely
used to treat Coulomb blockade but the standard approach
[Beenakker 1991, Likharev 1999] needs to be extended to
include coherences and broadening. This is an area of
active research [see for example Braun et.al. 2004, Braig
and Brouwer 2005] where an adequate general approach
does not yet exist.
Actually correlated states (classical version of
entanglement) were an issue even before the advent of
quantum mechanics. Boltzmann ignored them through his
assumption of “molecular chaos” or “Stohsslansatz”,and it
is believed that it is precisely this assumption that leads to
irreversibility [see for example, McQuarrie 1976]. An
intervening entangled or correlated state is characteristic of
all “channel”-“contact" interactions, classical or quantum,
and the increase in entropy characteristic of irreversible
processes can be associated with the destruction or neglect
of the correlation and/or entanglement generated by the
interaction (see for example, Zhang 2007, Datta 2007).
This aspect is largely ignored in today’s transport theory,
just as even the presence of a contact was barely
acknowledged before the advent of mesoscopic physics.
But new experiments showing the effect of entanglements
on current flow are on the horizon and will hopefully lead
us to the next level of understanding.
This work was supported by the Office of Naval Research
under Grant No. N00014-06-1-0025 and the Network for
Computational Nanotechnology.
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Nanodevices and Maxwell’s demon
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0704.1624 | Complete Segal spaces arising from simplicial categories | COMPLETE SEGAL SPACES ARISING FROM SIMPLICIAL
CATEGORIES
JULIA E. BERGNER
Abstract. In this paper, we compare several functors which take simplicial
categories or model categories to complete Segal spaces, which are particularly
nice simplicial spaces which, like simplicial categories, can be considered to be
models for homotopy theories. We then give a characterization, up to weak
equivalence, of complete Segal spaces arising from these functors.
1. Introduction and Overview
The idea that simplicial categories, or categories enriched over simplicial sets,
model homotopy theories goes back to a series of several papers by Dwyer and
Kan [8], [10], [11], [12]. Taking the viewpoint that a model category, or more
generally a category with weak equivalences, can be considered to be a model for
a homotopy theory, they develop two methods to obtain from a model category a
simplicial category. This “simplicial localization” encodes higher-order structure
which is lost when we pass to the homotopy category associated to the model
category. Furthermore, they prove that, up to a natural notion of weak equivalence
of simplicial categories (see Definition 3.1), every simplicial category arises as the
simplicial localization of some category with weak equivalences [10, 2.1]. Thus, the
category of all (small) simplicial categories with these weak equivalences can be
regarded as the “homotopy theory of homotopy theories.”
This notion was mentioned briefly at the end of Dwyer and Spalinski’s introduc-
tion to model categories [13, 11.6], and was made precise by the author in [2, 1.1],
in which the category SC of small simplicial categories with these weak equivalences
is shown to have the structure of a model category.
However, this category is not practically useful for many purposes. Simplicial
categories are not particularly easy objects to work with, and the weak equivalences,
while natural generalizations of equivalences of categories, are difficult to identify.
Motivated by this problem, Rezk defines a model structure, which we denote CSS,
on the category of simplicial spaces, in which the fibrant-cofibrant objects are called
complete Segal spaces (see Definition 4.3). This model structure is especially nice,
in that the objects are just diagrams of simplicial sets and the weak equivalences,
at least between complete Segal spaces, are just levelwise weak equivalences of
simplicial sets. Furthermore, this model structure has the additional structures of
a simplicial model category and a monoidal model category. Because it is given by a
Date: October 25, 2018.
2000 Mathematics Subject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30,
18D20.
Key words and phrases. simplicial categories, model categories, complete Segal spaces, homo-
topy theories.
http://arxiv.org/abs/0704.1624v2
2 J.E. BERGNER
localization of a model structure on the category of simplicial spaces with levelwise
weak equivalences, it is an example of a presentation for a homotopy theory as
described by Dugger [7].
In his paper, Rezk defines two different functors, one from the category of sim-
plicial categories, and one from the category of model categories, to the category
of complete Segal spaces. Thus, he describes a relationship between simplicial cat-
egories and complete Segal spaces, but he does not give an inverse construction.
However, the author was able to show in [5] that the model categories SC and
CSS are Quillen equivalent to one another, and therefore are models for the same
homotopy theory. Thus, the hope is that we can answer questions about simplicial
categories by working with complete Segal spaces. This paper is the beginning of
that project.
However, the functor that we use to show that the two model categories are
Quillen equivalent is not the same as Rezk’s functor. Furthermore, the question
arises whether Rezk’s functor on model categories agrees with the composite of the
simplicial localization functor with his functor on simplicial categories. Rezk gives
the beginning of a proof that the two functors agree when the model category in
question has the additional structure of a simplicial model category. Our goal in
this paper is prove that, up to weak equivalence in CSS, the two functors from SC
to CSS are the same, that Rezk’s result holds for model categories which are not
necessarily simplicial, and that this result does imply that the functor on model
categories agrees with the one on their corresponding simplicial categories up to
weak equivalence.
We then go on to characterize, up to weak equivalence, the complete Segal space
arising from any simplicial category. It can be described at each level as the nerve
of the monoid of self weak equivalences of representing objects of the category.
We should mention that the model categories SC and CSS are only two of several
known models for the homotopy theory of homotopy theories. Our proof that the
two are Quillen equivalent actually uses two intermediate model structures SeCatc
and SeCatf on the category of Segal precategories, or simplicial spaces with a
discrete Space at level zero [5]. The fibrant-cofibrant objects in these structures are
known as Segal categories. Furthermore, Joyal and Tierney have shown that there is
a Quillen equivalence between each of these model structures and a model structure
QCat on the category of simplicial sets [17], [19]. The fibrant-cofibrant objects of
this model structure are known as quasi-categories and are generalizations of Kan
complexes [18]. One can compare our composite functor SC → CSS to one they
define which factors through QCat rather than through SeCatc and SeCatf . It is
a consequence of Joyal and Tierney’s work that these two different functors give
rise to weakly equivalent complete Segal spaces, as we will describe further in the
section on complete Segal spaces. An introduction to each of these model structures
and the Quillen equivalences can be found in the survey paper [4].
In [1], we use the results of this paper to consider the complete Segal spaces
arising from the homotopy fiber product construction for model categories, as de-
scribed by Toën in his work on derived Hall algebras [24]. It seems that results
relating diagrams of model categories to diagrams of complete Segal spaces will
prove to be useful.
COMPLETE SEGAL SPACES 3
Acknowledgments. I would like to thank Bill Dwyer, Charles Rezk, André Joyal,
and Myles Tierney for conversations about the work in this paper, as well as the
referee for helpful comments on the exposition.
2. Background on Model Categories and Simplicial Objects
In this section, we summarize some facts about model categories, simplicial sets,
and other simplicial objects that we need in the course of this paper.
A model category M is a category with three distinguished classes of mor-
phisms, fibrations, cofibrations, and weak equivalences. A morphism which is both
a (co)fibration and a weak equivalence is called an acyclic (co)fibration. The cate-
goryM with these choices of classes is required to satisfy five axioms [13, 3.3].
Axiom MC1 guarantees that M has small limits and colimits, so in particular
M has an initial object and a terminal object. An object X in a model category
M is fibrant if the unique map X → ∗ to the terminal object is a fibration. Dually,
X is cofibrant if the unique map from the initial object φ→ X is a cofibration.
The factorization axiom (MC5) can be applied in such a way that, given any
object X ofM, we can factor the map X → ∗ as a composite
∼ //X ′ //∗
of an acyclic cofibration followed by a fibration. In this case, X ′ is called a fibrant
replacement of X , since it is weakly equivalent to X and fibrant. This replacement
is not necessarily unique, but in all the model categories we consider here it can be
assumed to be functorial [16, 1.1.1]. Cofibrant replacements can be defined dually.
The structure of a model category enables us to invert the weak equivalences
formally in such a way that we still have a set, rather than a proper class, of mor-
phisms between any two objects. If we were merely to take the localizationW−1M,
there would be no guarantee that we would not have a proper class. However, with
the structure of a model category, we can define the homotopy category Ho(M) to
have the same objects asM, and as morphisms
HomHo(M)(X,Y ) = [X
cf , Y cf ]M,
where the right-hand side denotes homotopy classes of maps between fibrant-cofibrant
replacements of X and Y , respectively.
The standard notion of equivalence of model categories is given by the following
definitions. First, recall that an adjoint pair of functors F : C ⇆ D : G satisfies the
property that, for any objects X of C and Y of D, there is a natural isomorphism
ϕ : HomD(FX, Y )→ HomC(X,GY ).
The functor F is called the left adjoint and G the right adjoint [20, IV.1].
Definition 2.1. [16, 1.3.1] An adjoint pair of functors F : M ⇆ N : G between
model categories is a Quillen pair if F preserves cofibrations and G preserves fibra-
tions.
Definition 2.2. [16, 1.3.12] A Quillen pair of model categories is a Quillen equiv-
alence if if for all cofibrant X in M and fibrant Y in N , a map f : FX → Y is a
weak equivalence in D if and only if the map ϕf : X → GY is a weak equivalence
4 J.E. BERGNER
An important example of a model category is that of the standard model struc-
ture on the category of simplicial sets SSets. Recall that a simplicial set is a functor
X : ∆op → Sets, where ∆op is the opposite of the category ∆ of finite ordered sets
[n] = {0→ 1→ · · · → n} and order-preserving maps between them. We denote the
set X([n]) by Xn. In particular in X we have face maps di : Xn → Xn−1 and de-
generacy maps si : Xn → Xn+1 for each 0 ≤ i ≤ n, satisfying several compatibility
conditions. Three particularly useful examples of simplicial sets are the n-simplex
∆[n] and its boundary ∆̇[n] for each n ≥ 0, and the boundary with the kth face
removed, V [n, k], for each n ≥ 1 and 0 ≤ k ≤ n. Given a simplicial set X , we can
take its geometric realization |X |, which is a topological space [14, I.1].
In the standard model category structure on SSets, the weak equivalences are
the maps f : X → Y for which the geometric realization |f | : |X | → |Y | is a weak
homotopy equivalence of topological spaces [14, I.11.3]. In fact, this model structure
on simplicial sets is Quillen equivalent to the standard model structure on the
category of topological spaces [16, 3.6.7].
More generally, a simplicial object in a category C is a functor ∆op → C. The
two main examples which we consider in this paper are those of simplicial spaces
(also called bisimplicial sets), or functors ∆op → SSets, and simplicial groups. We
denote the category of simplicial spaces by SSets∆
A simplicial set X can be regarded as a simplicial space in two ways. It can be
considered a constant simplicial space with the simplicial set X at each level, and in
this case we will denote the constant simplicial set by cX or just X if no confusion
will arise. Alternatively, we can take the simplicial space, which we denote Xt, for
which (Xt)n is the discrete simplicial set Xn. The superscript t is meant to suggest
that this simplicial space is the “transpose” of the constant simplicial space.
A natural choice for the weak equivalences in the category SSets∆
is the class
of levelwise weak equivalences of simplicial sets. If we define the cofibrations to be
levelwise also, we obtain a model structure which is usually referred to as the Reedy
model structure on SSets∆
[22].
The Reedy model structure has the additional structure of a simplicial model
category. In particular, given any two objects X and Y of SSets∆
, there is a
mapping space, or simplicial set Map(X,Y ) satisfying compatibility conditions [15,
9.1.6]. In the case where X is cofibrant (as is true of all objects in the Reedy model
structure) and Y is fibrant, this choice of mapping space is homotopy invariant.
One way to obtain other model structures on the category of simplicial spaces is
to localize the Reedy structure with respect to a set of maps. While this process
works for much more general model categories [15, 3.3.1], we will focus here on this
particular case. Let S = {f : A → B} be a set of maps of simplicial spaces. A
Reedy fibrant simplicial space W is S-local if for each map f ∈ S, the induced map
Map(f,W ) : Map(B,W )→ Map(A,W )
is a weak equivalence of simplicial sets. A map g : X → Y is then an S-local
equivalence if for any S-local object W , the induced map
Map(g,W ) : Map(Y,W )→ Map(X,W )
is a weak equivalence of simplicial sets.
Theorem 2.3. [15, 4.1.1] There is a model structure LSSSets
on the category
of simplicial spaces in which
COMPLETE SEGAL SPACES 5
• the weak equivalences are the S-local equivalences,
• the cofibrations are levelwise cofibrations of simplicial sets, and
• the fibrant objects are the S-local objects.
Furthermore, this model category has the additional structure of a simplicial model
category.
We now turn to a few facts about simplicial groups, or functors from ∆op to the
category of groups. Given a simplicial group G, we can take its nerve, a simplicial
space with
nerve(G)n,m = Hom([m], Gn).
Taking the diagonal of this simplicial space, we obtain a simplicial set, also often
called the nerve of G.
From another perspective, for G a simplicial group (or, more generally, a sim-
plicial monoid), we can find a classifying complex of G, a simplicial set whose
geometric realization is the classifying space BG. A precise construction can be
made for this classifying space by the WG construction [14, V.4.4], [21]. However,
we are not so concerned here with the precise construction as with the fact that
such a classifying space exists, so for simplicity we will simply write BG for the
classifying complex of G.
3. Simplicial Categories and Simplicial Localizations
In this section, we consider simplicial categories and show how they arise from
Dwyer and Kan’s simplicial localization techniques. We then discuss model category
structures, first on the category of simplicial categories with a fixed object set, and
then on the category of all small simplicial categories.
First of all, we clarify some terminology. In this paper, by (small) “simplicial
category” we will mean a category with a set of objects and a simplicial set of
morphisms Map(x, y) between any two objects x and y, also known as a category
enriched over simplicial sets. This notion does not coincide with the more general
one of a simplicial object in the category of small categories, in which we would also
have a simplicial set of objects. Using this more general definition, if we impose
the additional condition that all face and degeneracy maps are the identity on the
objects, then we get our more restricted notion.
A simplicial category can be seen as a generalization of a category, since any
ordinary category can be regarded as a simplicial category with a discrete mapping
space. Given any simplicial category C, we can consider its category of components
π0C, which has the same objects as C and whose morphisms are given by
Homπ0C(x, y) = π0MapC(x, y).
The following definition of weak equivalence of simplicial categories is a natural
generalization of the notion of equivalence of categories.
Definition 3.1. A simplicial functor f : C → D is a Dwyer-Kan equivalence or
DK-equivalence if the following two conditions hold:
(1) For any objects x and y of C, the induced map Map(x, y) → Map(fx, fy)
is a weak equivalence of simplicial sets.
(2) The induced map on the categories of components π0f : π0C → π0D is an
equivalence of categories.
6 J.E. BERGNER
The idea of obtaining a simplicial category from a model categoryM goes back
to several papers of Dwyer and Kan [8], [11], [12]. In fact, they define two different
methods of doing so, the simplicial localization LM [12] and the hammock local-
ization LHM [11]. The first has the advantage of being easier to describe, while
the second is more convenient for making calculations.
It should be noted that these constructions can be made for more general cate-
gories with weak equivalences, and do not depend on the model structure if we are
willing to ignore the potential set-theoretic difficulties. However, as with the homo-
topy category construction, the hammock localization in particular can be defined
much more nicely when we have the additional structure of a model category.
We begin with the construction of the simplicial localization LM. Recall that,
given a category M with some choice of weak equivalences W , we denote the
localization ofM with respect toW byW−1M. This localization is obtained from
M by formally inverting the maps of W . Further, recall that, given a categoryM,
we denote by FM the free category on M, or category with the same objects as
M and morphisms freely generated by the non-identity morphisms of M. Note
in particular that there are natural functors FM → M and FM → F 2M [12,
2.4]. These functors can be used to define a simplicial resolution F∗M, which is a
simplicial category with the category F k+1M at level k [12, 2.5].
We can apply this same construction to the subcategoryW to obtain a simplicial
resolution F∗W . Using these two resolutions, we have the following definition.
Definition 3.2. [12, 4.1] The simplicial localization ofM with respect toW is the
localization (F∗W)
−1(F∗M). This simplicial localization is denoted L(M,W) or
simply LM.
The following result gives interesting information about the mapping spaces in
LM in the case where W is all ofM.
Proposition 3.3. [12, 5.5] Suppose that W =M and nerve(M) is connected.
(i) The simplicial localization LM is a simplicial groupoid, so for all objects x
and y, the simplicial sets MapLM(x, y) are all isomorphic. In particular, the
simplicial sets MapLM(x, x) are all isomorphic simplicial groups.
(ii) The classifying complex BMapLM(x, x) has the homotopy type of nerve(M),
and thus each simplicial set MapLM(x, y) has the homotopy type of the loop
space Ω(nerve(M)).
We now turn to the other construction, that of the hammock localization. Again,
letM be a category with a specified subcategory W of weak equivalences.
Definition 3.4. [11, 3.1] The hammock localization of M with respect to W ,
denoted LH(M,W), or simply LHM, is the simplicial category defined as follows:
(1) The simplicial category LHM has the same objects asM.
(2) Given objects X and Y of M, the simplicial set MapLHM(X,Y ) has as
k-simplices the reduced hammocks of width k and any length between X
COMPLETE SEGAL SPACES 7
and Y , or commutative diagrams of the form
· · · C0,n−1
· · · C1,n−1
~~~~~~~~~
CCCCCCCCC
Ck,1 Ck,2 · · · Ck,n−1
in which
(i) the length of the hammock is any integer n ≥ 0,
(ii) the vertical maps are all in W ,
(iii) in each column all the horizontal maps go the same direction, and if
they go to the left, then they are in W ,
(iv) the maps in adjacent columns go in opposite directions, and
(v) no column contains only identity maps.
Proposition 3.5. [8, 2.2] For a given model categoryM, the simplicial categories
LM and LHM are DK-equivalent.
We should add that the description of the hammock localization can be greatly
simplified if we make use of the model category structure on M. In this case,
Dwyer and Kan prove that it suffices to consider hammocks of length 3 such as the
following [8, §8]:
X C0,1
≃oo //C0,2 Y
≃oo .
Restricting to the category of simplicial categories with a fixed set O of ob-
jects, Dwyer and Kan prove the existence of a model structure on this category,
which we denote SCO [12, 7.2]. In this situation, the weak equivalences are the
DK-equivalences, but with the objects fixed the second condition follows imme-
diately from the first. The fibrations in this model structure are given by the
functors f : C → D inducing, for any objects x and y, fibrations of simplicial sets
MapC(x, y)→ MapD(x, y).
The cofibrations are then defined to be the maps with the left lifting property
with respect to the acyclic fibrations. However, they can be more precisely charac-
terized. To do so, we recall the definition of a free map of simplicial categories.
Definition 3.6. [12, 7.4] A map f : C → D in SCO is free if
(1) f is a monomorphism,
(2) if ∗ denotes the free product, then in each simplicial dimension k, the
category Dk admits a unique free factorization Dk = f(Ck) ∗ Fk, where Fk
is a free category, and
(3) for each k ≥ 0, all degeneracies of generators of Fk are generators of Fk+1.
8 J.E. BERGNER
Definition 3.7. [12, 7.5] A map f : C → D of simplicial categories is a strong
retract of a map f ′ : C → D′ if there exists a commutative diagram
>>}}}}}}}
id // D
Using these definitions, Dwyer and Kan prove the following result.
Proposition 3.8. [12, 7.6] The cofibrations of SCO are precisely the strong retracts
of free maps. In particular, a cofibrant simplicial category is a retract of a free
category.
This result can then be generalized to the category of all simplicial categories, in
which the DK-equivalences are the weak equivalences. If C is a simplicial category, a
morphism e ∈ HomC(a, b)0 is a homotopy equivalence if it becomes an isomorphism
in π0C.
Theorem 3.9. [2, 1.1] There is a model category structure SC on the category of
small simplicial categories in which
• the weak equivalences are the Dwyer-Kan equivalences, and
• the fibrations are the maps f : C → D satisfying the following two condi-
tions:
(i) For any objects a1 and a2 in C, the map
HomC(a1, a2)→ HomD(fa1, fa2)
is a fibration of simplicial sets.
(ii) For any object a1 in C, b in D, and homotopy equivalence e : fa1 → b
in D, there is an object a2 in C and homotopy equivalence d : a1 → a2
in C such that fd = e.
4. Complete Segal Spaces
Here we define complete Segal spaces and describe Rezk’s model structure on the
category of simplicial spaces, in which the complete Segal spaces are the fibrant-
cofibrant objects.
Recall that by a simplicial space we mean a simplicial object in the category
of simplicial sets, or functor ∆op → SSets. In section 2, we described the Reedy
model category structure on this category, in which both the weak equivalences
and cofibrations are defined levelwise. The model structure CSS is given by a
localization of this structure with respect to a set of maps.
We begin with the definition of a Segal space. In [23, 4.1], Rezk defines for each
0 ≤ i ≤ n − 1 a map αi : [1] → [n] in ∆ such that 0 7→ i and 1 7→ i + 1. There is
a corresponding map αi : ∆[1] → ∆[n]. Then for each n he defines the simplicial
space
G(n)t =
αi∆[1]
t ⊂ ∆[n]t.
COMPLETE SEGAL SPACES 9
Let X be a Reedy fibrant simplicial space. There is a weak equivalence of
simplicial sets
MapSSets∆opc (G(n)
t, X)→ X1 ×X0 · · · ×X0 X1︸ ︷︷ ︸
where the right hand side is the limit of the diagram
d0 // X0 X1
d1oo d0 // . . .
d0 // X0 X1
with n copies of X1.
Now, given any n, define the map ϕn : G(n)t → ∆[n]t to be the inclusion map.
Then for any Reedy fibrant simplicial space W there is a map
ϕn = MapSSets∆opc (ϕ
n,W ) : MapSSets∆opc (∆[n]
t,W )→ MapSSets∆opc (G(n)
t,W ).
More simply written, this map is
ϕn : Wn → W1 ×W0 · · · ×W0 W1︸ ︷︷ ︸
and is often called a Segal map. The Segal map is actually defined for any simplicial
space W , but here we assume Reedy fibrancy so that the mapping spaces involved
are homotopy invariant.
Definition 4.1. [23, 4.1] A Reedy fibrant simplicial space W is a Segal space if for
each n ≥ 2 the Segal map
ϕn : Wn →W1 ×W0 · · · ×W0 W1
is a weak equivalence of simplicial sets.
In fact, there is a model category structure SeSp on the category of simplicial
spaces in which the fibrant objects are precisely the Segal spaces [23, 7.1]. This
model structure is obtained from the Reedy structure via localization.
The idea is that in a Segal space there is a notion of “composition,” at least up
to homotopy. In fact, given a Segal space, we can sensibly use many categorical
notions. We summarize some of these ideas here; a detailed description is given by
Rezk [23]. The objects of a Segal space W are given by the set W0,0. Given the
(d1, d0) : W1 →W0 ×W0,
themapping space mapW (x, y) is given by the fiber of this map over (x, y). (The fact
that W is Reedy fibrant guarantees that this mapping space is homotopy invariant.)
Two maps f, g ∈ mapW (x, y)0 are homotopic if they lie in the same component of
the simplicial set mapW (x, y). Thus, we define the space of homotopy equivalences
Whoequiv ⊆W1 to consist of all the components containing homotopy equivalences.
Given any (x0, . . . , xn) ∈ W
0,0 , let mapW (x0, . . . , xn) denote the fiber of the
(α0, . . . , αn) : Wn →W
over (x0, . . . , xn). Consider the commutative diagram
Wn = Map(∆[n]
t,W )
ϕk //
Map(G(n)t,W )
wwppp
Wn+10
10 J.E. BERGNER
and notice that, since W is a Segal space, the horizontal arrow is a weak equivalence
and a fibration. In particular, this map induces an acyclic fibration on the fibers of
the two vertical arrows,
mapW (x0, . . . , xn)→ mapW (xn−1, xn)× · · · ×mapW (x0, x1).
Given f ∈ map(x, y)0 and g ∈ map(y, z)0, their composite is a lift of (g, f) ∈
map(y, z) × map(x, y) along ϕ2 to some k ∈ map(x, y, z)0. The result of this
composition is defined to be d1(k) ∈ map(x, z)0. It can be shown that any two
results are homotopic, so we can use g ◦ f unambiguously.
Then, the homotopy category of W , denoted Ho(W ), has as objects the elements
of the set W0,0, and
HomHo(W )(x, y) = π0mapW (x, y).
A homotopy equivalence in W is a 0-simplex of W1 whose image in Ho(W ) is an
isomorphism.
Definition 4.2. A map f : W → Z of Segal spaces is a Dwyer-Kan equivalence if
(1) for any objects x and y ofW , the induced map mapW (x, y)→ mapZ(fx, fy)
is a weak equivalence of simplicial sets, and
(2) the induced map Ho(W )→ Ho(Z) is an equivalence of categories.
Notice that the definition of these maps bears a striking resemblance to that
of the Dwyer-Kan equivalences between simplicial categories, hence the use of the
same name.
For a Segal space W , note that the degeneracy map s0 : W0 → W1 factors
through the space of homotopy equivalences Whoequiv, since the image of s0 consists
of “identity maps.” Given this fact, we are now able to give a definition of complete
Segal space.
Definition 4.3. [23, §6] A Segal space W is a complete Segal space if the map
W0 →Whoequiv given above is a weak equivalence of simplicial sets.
The idea behind this notion is that, although W0 is not required to be discrete,
as the objects are for a simplicial category, it is not heuristically too different from a
simplicial space with discrete 0-space. (This viewpoint is further confirmed by the
comparison of complete Segal spaces with Segal categories, which are essentially
the analogues of Segal spaces with discrete 0-space [5, 6.3].)
Now, we give a description of the model structure CSS. We do not give all the
details here, such as a description of an arbitrary weak equivalence, but refer the
interested reader to Rezk’s paper [23, §7].
Theorem 4.4. [23, 7.2] There is a model structure CSS on the category of simpli-
cial spaces, obtained as localization of the Reedy model structure, such that
(1) the fibrant objects are precisely the complete Segal spaces,
(2) the cofibrations are the monomorphisms; in particular, every object is cofi-
brant,
(3) the weak equivalences between Segal spaces are Dwyer-Kan equivalences,
(4) the weak equivalences between complete Segal spaces are levelwise weak
equivalences of simplicial sets.
COMPLETE SEGAL SPACES 11
Furthermore, CSS has the additional structure of a simplicial model category and
is cartesian closed.
The fact that CSS is cartesian closed allows us to consider, for any complete Segal
spaceW and simplicial space X , the complete Segal space WX . In particular, using
the simplicial structure, the simplicial set at level n is given by
(WX)n = Map(X ×∆[n]
t,W ).
If W is a (not necessarily complete) Segal space, then WX is again a Segal space;
in other words, the model category SeSp is also cartesian closed.
We denote the functorial fibrant replacement functor in CSS by LCSS . Thus,
given any simplicial space X , there is a weakly equivalent complete Segal space
LCSSX .
This model structure is connected to the model structure SC by a chain of Quillen
equivalences as follows. Each of these model categories is Quillen equivalent to a
model structure SeCatf on the category of Segal precategories. A Segal precategory
is a simplicial space X with X0 a discrete space. A Segal category is then a Segal
precategory with the Segal maps weak equivalences. In the model structure SeCatf ,
the fibrant objects are Segal categories, and so it is considered a Segal category
model structure on the category of Segal precategories. We have the following
chain of Quillen equivalences, with the left adjoint functors topmost:
SC ⇆ SeCatf ⇄ CSS.
The right adjoint SC → SeCatf is given by the nerve functor, and the left adjoint
SeCatf → CSS is given by the inclusion functor.
There is actually another chain of Quillen equivalences connecting the two model
structures; in this case, both SC and CSS are Quillen equivalent to Joyal’s model
structure QCat on the category of simplicial sets [18]. The fibrant objects in QCat
are quasi-categories, or simplicial sets K such that a dotted arrow lift exists making
the diagram
V [m, k] //
commute for any 0 < k < m. The chain of Quillen equivalences in this case is given
SC ⇄ QCat ⇆ CSS.
The right adjoint SC → QCat is given by Cordier and Porter’s coherent nerve
functor [6], [17, 2.10], and the right adjoint CSS → QCat is given given by sending
a simplicial space W to the simplicial set W∗,0 [19, 4.11]. It is a consequence of
work of Joyal [17, §1-2] and of Joyal and Tierney [19, §4-5] that the simplicial space
obtained from a simplicial category via these functors is weakly equivalent to the
one obtained from the composite functor described in the previous paragraph.
5. Obtaining Complete Segal Spaces from Simplicial Categories and
Model Categories
In this section, we describe several different ways of obtaining a complete Segal
space. First, we look at a particularly nice functor which Rezk uses to modify
12 J.E. BERGNER
the notion of a nerve of a category. Then we look at how this functor can be
generalized to one on any simplicial category, and how a similar idea can be used to
get a complete Segal space from any model category. We then consider the functors
used in the Quillen equivalences connecting SC and CSS.
Let us begin with Rezk’s classifying diagram construction, which associates to
any small category C a complete Segal space NC. First, we denote by nerve(C)
the ordinary nerve, which is the simplicial set given by (nerve(C))n = Hom([n], C).
Further, we denote by iso(C) the maximal subgroupoid of C, or subcategory of C
with all objects of C and whose only morphisms are the isomorphisms of C. By
C[n] we denote the category of functors [n]→ C, or the category whose objects are
n-chains of composable morphisms in C.
Definition 5.1. [23, 3.5] The classifying diagram NC is the simplicial space given
by (NC)n = nerve(iso(C
[n])).
Thus, (NC)0 is simply the nerve of iso(C), and (NC)1 is the nerve of the maxi-
mal subgroupoid of the morphism category of C. In particular, information about
invertible morphisms of C is encoded at level 0, while information about the other
morphisms of C does not appear until level 1.
Thus, the classifying diagram of a category can be regarded as a more refined
version of the nerve, since, unlike the ordinary nerve construction, it enables one
to recover information about whether morphisms are invertible or not. This con-
struction is also particularly useful for our purposes due to the following result.
Proposition 5.2. [23, 6.1] If C is a small category, then its classifying diagram
NC is a complete Segal space.
However, this construction, as defined above, cannot be used to assign a complete
Segal space to any simplicial category, since, beginning with level 1, we would have
homotopy invariance problems with a simplicial set of objects in C[1]. Rezk defines
an analogous functor, though, from the category of small simplicial categories which
is similar in spirit to the classifying diagram but avoids these difficulties.
Let I[m] denote the category with m + 1 objects and a single isomorphism
between any two objects, and let E(m) = nerve(I[m])t. If W is a Segal space and
X is any simplicial space, recall that WX denotes the internal hom object, which
is a Segal space. With these notations in place, we can give the definition of Rezk’s
completion functor. Let W be a Segal space. Then its completion Ŵ is defined as
a fibrant replacement in CSS of the simplicial space W̃ defined by
W̃n = diag([m] 7→ MapSSets∆op (E(m),W
∆[n]t)) = diag([m] 7→ (WE(m))n).
From a simplicial category C, then, we can takes its nerve to obtain a simplicial
space, followed by a fibrant replacement functor in the Segal space model structure,
to obtain a Segal space W . From W we can then pass to a complete Segal space
via this completion functor. We will denote this complete Segal space LC(W ), or
LC(C) where W comes from a simplicial category as just described.
The first important fact about this completion functor is that the completion
map iW : W → Ŵ = LC(W ) is not only a weak equivalence in the model category
CSS, but is also a Dwyer-Kan equivalence of Segal spaces [23, §14]. Furthermore,
this completion functor restricts nicely to the classifying diagram in the case where
C is a discrete category.
COMPLETE SEGAL SPACES 13
Proposition 5.3. [23, 14.2] If C is a discrete category, then LC(C) is isomorphic
to NC.
If we begin with a model category M with subcategory of weak equivalences
W , a functor analogous to the classifying diagram functor can be used to obtain a
complete Segal space. In this case, rather than taking the subcategory iso(M) of
isomorphisms ofM, we take the subcategory of weak equivalences, denoted we(M).
Thus, Rezk defines the classification diagram of (M,W), denoted N(M,W), by
N(M,W)n = nerve(we(M
[n])).
Unlike the classifying diagram, the classification diagram of a model category is not
necessarily a complete Segal space as stated, but taking a Reedy fibrant replacement
of it results in a complete Segal space, as we show in the next section.
Lastly, we have the two functors given by the two different chains of Quillen
equivalences between the model categories SC and CSS. As mentioned in the
previous section, these two functors are equivalent. In each case, the resulting
simplicial space is not Reedy fibrant in general, and so not a complete Segal space,
but applying the fibrant replacement functor LCSS results in a complete Segal space.
The first of these composite functors, in particular, is simple to describe ab-
stractly, as in the previous section, but it has a disadvantage over Rezk’s functor in
that it gives very little insight into what the resulting complete Segal space looks
like. In the next section, we prove that the two functors from SC to CSS result
in weakly equivalent complete Segal spaces, and that if we use Rezk’s classification
diagram construction we get a weakly equivalent complete Segal space to the one
we would obtain by taking the simplicial localization followed by his completion
functor. We then use Rezk’s functor to describe what the complete Segal space
corresponding to a simplicial category looks like.
6. Comparison of Functors from SC to CSS
Here we prove that each of the functors we have described all give rise to com-
plete Segal spaces weakly equivalent to those given in the previous section. We
begin by stating the result that establishes the equivalence between Rezk’s com-
pletion functor LC : SC → CSS and the functor arising from the chain of Quillen
equivalences factoring through SeCatf . Let LCSS denote the functorial fibrant re-
placement functor in CSS.
Theorem 6.1. If C is a simplicial category, then the complete Segal spaces LC(C)
and LCSS(nerve(C)) are weakly equivalent in CSS.
Proof. Let LS denote a fibrant replacement functor in the Segal space model struc-
ture SeSp on the category of simplicial spaces. The fact that the two functors
in question result in weakly equivalent complete Segal spaces can be shown by
considering the following chain of weak equivalences:
LCSS(nerve(C))← nerve(C)→ LSnerve(C)→ LC(C).
The map on the left is the localization functor in CSS and so is a weak equivalence
in CSS. The middle map is a weak equivalence in SeSp and therefore also a weak
equivalence in CSS, since the latter model category is a localization of the former.
The map on the right is Rezk’s completion, and it is a weak equivalence in CSS, as
given in the previous section. Therefore, the objects at the far left and right of this
14 J.E. BERGNER
zigzag, both of which are complete Segal spaces, are weakly equivalent as objects
of CSS. �
Now, we would like to compare either of these functors to the classifying diagram
construction for a model category M. In other words, we want to show that
N(M,W) is equivalent to LC(L
HM), where we first take the hammock localization
ofM to obtain a simplicial category, and then apply Rezk’s functor LC .
An initial problem here is that N(M,W) is not necessarily Reedy fibrant, and
so it is not necessarily a complete Segal space. We prove that a Reedy fibrant
replacement of it, denoted N(M,W)f , is in fact a complete Segal space in the
process of comparing the “mapping spaces” in this Reedy fibrant replacement to
the mapping spaces of the hammock localization LHM.
Theorem 6.2. Let M be a model category, and let W denote its subcategory of
weak equivalences. Then N(M,W)f , is a complete Segal space. Furthermore, for
any objects x, y of M, there is a weak equivalence of spaces mapN(M,W)f (x, y) ≃
MapLHM(x, y), and there is an equivalence of categories Ho(N(M,W)
f ) ≈ Ho(M).
This result was proved by Rezk in the case where M is a simplicial model
category [23, 8.3], namely, in the case where we do not need to pass to the simplicial
localization ofM to consider its function complexes. However, here we prove that,
as he conjectured [23, 8.4], the result continues to hold in this more general case.
We prove this theorem very similarly to the way Rezk proves it in the more
restricted case, using a proposition of Dwyer and Kan. To begin, we introduce
some terminology. LetM be a model category. A classification complex ofM, as
defined in [9, 1.2], is the nerve of any subcategory C ofM such that
(1) every map in C is a weak equivalence,
(2) if f : X → Y inM is a weak equivalence and either X or Y is in C, then f
is in C, and
(3) nerve(C) is homotopically small; i.e., each homotopy group of |nerve(C)| is
small [11, 2.2].
The special classification complex sc(X) of an object X in M is a connected
classification complex containing X .
LetM be a model category and X a fibrant-cofibrant object ofM. Denote by
Auth(X) the simplicial monoid of weak equivalences given by Auth
(X) in the
hammock localization LHM, and by BAuth(X) its classifying complex.
The following proposition was proved by Dwyer and Kan in [9, 2.3] in the case
thatM is a simplicial model category. However, the proof does not actually require
the simplicial structure; in fact, their proof is essentially the one given below, with
the extra step showing that the mapping spaces in the hammock localization are
equivalent to those given by the simplicial structure ofM [11, 4.8].
Proposition 6.3. Let X be an object of a model category M. The classifying
complex BAuth(X) is weakly equivalent to the special classification complex of X,
sc(X), and the two can be connected by a finite zig-zag of weak equivalences.
Proof. Let W be the subcategory of weak equivalences of M. Consider the con-
nected component of nerve(W) containing X . For the rest of this proof, we assume
that W is such that its nerve is connected. We further assume that nerve(W) is
homotopically small, taking an appropriate subcategory, as described in [11, 2.3],
if necessary.
COMPLETE SEGAL SPACES 15
In this case, by Proposition 3.3, the function complexes MapLW(X,X) are all iso-
morphic. Furthermore, by the same result, the classifying complex BMapLW(X,X)
has the homotopy type of nerve(W). Thus, we can take nerve(W) as sc(X).
Now, as in the statement of the proposition, we take Auth(X) to consist of the
components of MapLHM(X,X) which are invertible in π0mapL
HM(X,X). But,
by [11, 4.6(ii)], the map BMapLHW(X,X) → BAut
h(X) is a weak equivalence of
simplicial sets. Since LHW can be connected to LW by a finite string of weak
equivalences, it follows that so can MapLHW(X,X) and MapLW(X,X). Thus,
BMapLW(X,X) and BAut
h(X) can also be connected by such a string. It follows
that sc(X) has the same homotopy type as BAuth(X). �
Proof of Theorem 6.2. Consider the category M[n] of functors [n] → M. If M
is a model category, then M[n] can be given the structure of a model category
with the weak equivalences and fibrations given by levelwise weak equivalences and
fibrations inM. Given any map [m]→ [n], we obtain a functorM[n] →M[m].
Let Y = (y0 → y1 · · · → yn) be a fibrant-cofibrant object ofM
[n]. It restricts to
an object Y ′ = (y0 → y1 · · · → yn−1) inM
[n−1]. From this map, we obtain a map
of simplicial sets
BAuth
LHM[n](Y )→ BAut
LHM(yn)×BAut
LHM[n−1](Y
The homotopy fiber of this map is weakly equivalent to the union of those compo-
nents of MapLHM(yn−1, yn) containing the conjugates of the map fn−1 : yn−1 → yn,
or maps j ◦ fn−1 ◦ i, where i and j are self-homotopy equivalences.
Iterating this process, we can take the homotopy fiber of the map
BAuth
LHM[n](Y )→ BAut
LHM(yn)× · · · ×BAut
LHM(y0),
which is weakly equivalent to the union of the components of
MapLHM(yn−1, yn)× · · · ×MapLHM(y0, y1)
containing conjugates of the sequence of maps fi : yi → yi+1, 0 ≤ i ≤ n − 1.
However, applying Proposition 6.3 to the map in question shows that this simplicial
set is also the homotopy fiber of the map
sc(Y )→ sc(yn)× · · · × sc(y0).
Let U denote the simplicial space N(M,W) so that Un = nerve(we(M
[n])).
Then, let V be a Reedy fibrant replacement of U , from which we get weak equiva-
lences Un → Vn for all n ≥ 0.
For each n ≥ 0, there exists a map pn : Un → U
0 given by iterated face
maps to the “objects.” Then, for every (n+1)-tuple of objects (x0, x1, . . . , xn), the
homotopy fiber of pn over (x0, . . . , xn), given by
mapV (xn−1, xn)× · · · ×mapV (x0, x1),
is weakly equivalent to
MapLHM(x
n−1, x
n )× · · · ×MapLHM(x
0 , x
where xcf denotes a fibrant-cofibrant replacement of X inM. It follows that once
we take the Reedy fibrant replacement V of U , it is a Segal space.
Now, consider the set π0U0, which consists of the weak equivalence classes of
objects inM; it follows that π0V0 is an isomorphic set. Further, note that
HomHo(M)(x, y) = π0MapLHM(x
cf , ycf).
16 J.E. BERGNER
Thus, we have shown that Ho(M) is equivalent to Ho(V ).
It remains to show that V is a complete Segal space. Consider the space
Vhoequiv ⊆ V1, and define Uhoequiv to be the preimage of Vhoequiv under the natural
map U → V . Since V is a Reedy fibrant replacement for U , it suffices to show
that the complete Segal space condition holds, i.e., that U0 → Uhoequiv is a weak
equivalence of simplicial sets. Notice that Uhoequiv must consist precisely of the
components of U1 whose 0-simplices come from weak equivalences inM. In other
words, Uhoequiv = nerve(we(we(M))
[1]).
There is an adjoint pair of functors
F :M[1] ⇄M : G
for which F (x→ y) = x and G(x) = idx. This adjoint pair can be restricted to an
adjoint pair
F : nerve(we(we(M))[1]) ⇄ we(M) : G
which in turn induces a weak equivalence of simplicial sets on the nerves, Uhoequiv ≃
U0, which completes the proof. �
Now that we have proved that the mapping spaces and homotopy categories agree
for V and for LHM, it remains to show that they agree for LHM and LC(L
Theorem 6.4. Let M be a model category. For any x and y objects of LHM,
there is a weak equivalence of simplicial sets
MapLHM(x, y) ≃ mapLC(LHM)(x, y),
and there is an equivalence of categories π0L
HM≈ Ho(LC(L
HM)).
Note in particular that x and y are just objects of M, and that π0L
HM is
equivalent to the homotopy category Ho(M).
Proof. Given the hammock localization LHM of the model categoryM, we have
the following composite map of simplicial spaces:
X = nerve(LHM)→ Xf → LC(L
Here, Xf denotes a Reedy fibrant replacement of X . This composite is just Rezk’s
method for assigning the complete Segal space LC(L
HM) to the simplicial category
On the left-hand side, the mapping spaces of X = nerve(LHM) are precisely
those of LHM, by the definition of the nerve functor. In the nerve, one of these
mapping spaces, say mapX(x, y) for some objects x and y ofM, is given by the fiber
over (x, y) of the map (d1, d0) : X1 → X0×X0. Although these mapping spaces can
be defined for X , there is no reason that they are homotopy invariant. When we
take a Reedy fibrant replacement Xf of X , however, this map becomes a fibration,
and hence this fiber is actually a homotopy fiber and so homotopy invariant. For a
general simplicial space, we cannot assume that the mapping spaces of the Reedy
fibrant replacement are equivalent to the original ones. However, if the 0-space of
the simplicial space in question is discrete in degree zero, then the map above is
a fibration. Using an argument similar to the one in [5, §5], we can find a Reedy
fibrant replacement functor which leaves the 0-space discrete. While the space in
degree one might be changed in this process of passing to X
1 , it will still be weakly
equivalent X1. In particular, the mapping spaces in X
f will be weakly equivalent
to those in X .
COMPLETE SEGAL SPACES 17
Since the objects of Xf are just the objects of LHM, or the objects ofM, this
equivalence of mapping spaces gives us also an equivalence of homotopy categories.
The right-most map is the one defined by Rezk, iXf : X
f → X̂f , which takes a
Segal space to a complete Segal space. But, he defines this map in such a way that it
is in fact a Dwyer-Kan equivalence. In other words, it induces weak equivalences on
mapping spaces and an equivalence of homotopy categories. Thus, the composite
map induces equivalences on mapping spaces and an equivalence on homotopy
categories. �
7. A Characterization of Complete Segal Spaces Arising from
Simplicial Categories
In this section, we give a thorough description of the weak equivalence type of
complete Segal spaces which occur as images of Rezk’s functor from the category
of simplicial categories. We consider several different cases, beginning with ones for
which we can use the classifying diagram construction, i.e., discrete categories, and
then proceed to more general simplicial categories.
It should be noted that we are characterizing these complete Segal spaces up to
weak equivalence, and so the resulting descriptions are of the homotopy type of the
spaces in each simplicial degree. For example, in the case of a discrete category,
we describe the corresponding complete Segal space in terms of the isomorphism
classes of objects, rather than in terms of individual objects, in order to simplify
the description. One could just take all objects, and generally get much larger
spaces, if the more precise description were needed for the complete Segal space
corresponding to a given category.
Furthermore, notice that determining the homotopy type of the spaces in degrees
zero and one are sufficient to determine the homotopy type of all the spaces, since we
are considering Segal spaces. Thus, we focus our attention on these spaces, adding
in a few comments about how to continue the process with the higher-degree spaces.
7.1. Case 1: C is a discrete groupoid. If C = G is a group, then applying
Rezk’s classifying diagram construction results in a complete Segal space equivalent
to BG, i.e., the constant simplicial space which is the simplicial set BG at each
level. In particular, since all morphisms are invertible, we obtain essentially no new
information at level 1 that we didn’t have already at level 0.
Example 7.1. Let G = Z/2. Then (NG)0 is just the nerve, or BZ/2. Then (NG)1
has two 0-simplices, given by the two morphisms (elements) of G. However, these
two objects of G[1] are isomorphic, and the automorphism group of either one of
them is Z/2. Thus, (NG)1 is also equivalent to BZ/2.
If C has more than one object but only one isomorphism class of objects, we get
instead a simplicial space weakly equivalent to the constant simplicial space which
is BAut(x) at each level, for a representative object x. If C has more than one
isomorphism class 〈x〉, then the result will be weakly equivalent to the constant
simplicial space
〈x〉 BAut(x).
7.2. Case 2: C is a discrete category. Since in the classifying diagram NC,
(NC)0 picks out the isomorphisms of C only, we still essentially have
〈x〉 BAut(x)
18 J.E. BERGNER
at level 0. However, if C is not a groupoid, then there is new information at level
1. It instead looks like ∐
〈x〉,〈y〉
BAut(
Hom(x, y)α)
where the α index the isomorphism classes of elements of Hom(x, y). The subspace
of (NC)1 corresponding to 〈x〉, 〈y〉, denoted (NC)1(x, y), fits into a fibration
Hom(x, y)→ (NC)1(x, y)→ BAut(x)×BAut(y).
The space in dimension 2 is determined, then, by the spaces at levels 0 and 1.
The subspace corresponding to isomorphism classes of objects 〈x〉, 〈y〉, 〈z〉, denoted
(NC)2(x, y, z), fits into a fibration
Hom(x, y)×Hom(y, z)→ (NC)2(x, y, z)→ BAut(x) ×BAut(y)×BAut(z).
The whole space (NC)2, up to homotopy, looks like
〈x〉,〈y〉,〈z〉
〈α〉,〈β〉
Hom(x, y)α ×Hom(y, z)β
We could describe each (NC)n analogously.
Example 7.2. Let C denote the category with two objects and one nontrivial
morphism between them (· → ·). If {e} denotes the trivial group, then (NC)0 ≃
B{e} ∐ B{e} and (NC)1 ≃ B{e} ∐ B{e} ∐ B{e}. In particular, NC is not equiv-
alent to the classifying diagram of the trivial category with one object and one
morphism, which would be the constant simplicial space B{e}. However, note that
the nerves of these two categories are homotopy equivalent. Thus, we can see that
the classifying diagram is more refined than the nerve in distinguishing between
these two categories.
7.3. Case 3: C is a simplicial groupoid. First, consider the case where we have
a simplicial group G. Let Gn denote the group of n-simplices of G. Then
hocolim∆op(nerve(Gn)
t) = nerve(G).
Let LC denote Rezk’s completion functor which makes the nerve into a complete
Segal space. We claim that
LC(hocolim∆op(nerve(Gn)
t)) ≃ LC(hocolim∆opLC(nerve(Gn)
We actually prove the more general statement that, for any X = hocolim∆opXn,
LC(hocolim∆opXn) ≃ LC(hocolim∆opLCXn).
To prove this claim, first note that we have Rezk’s completion map
i : hocolim∆opXn → LC(hocolim∆opXn)
which is a weak equivalence. Furthermore, since in CSS any complete Segal space
Y is a local object and every object is cofibrant, we have a weak equivalence of
spaces
Map(LC(hocolim∆opXn), Y ) ≃Map(hocolim∆opXn, Y ).
COMPLETE SEGAL SPACES 19
So, for any complete Segal space Y , we have that
Map(LChocolim∆op(LCXn), Y ) ≃Map(hocolim∆op(LCXn), Y )
≃ holim∆Map(LCXn, Y )
≃ holim∆Map(Xn, Y )
≃Map(hocolim∆opXn, Y )
≃Map(LChocolim∆opXn, Y ).
Note that the above calculation depends on the fact that,
Map(hocolim∆opXn, Y ) ≃ holim∆Map(Xn, Y ),
which follows from working levelwise on simplicial sets.
Then, since Gn is a discrete group, completing its nerve is the same as taking
the classifying diagram NGn which, by case 1, is weakly equivalent to the constant
simplicial space BGn, denoted here cBGn. Thus we have:
LC(nerve(Gn)) ≃ LC [hocolim∆op(nerve(Gn))]
≃ LC [hocolim∆op(LC(nerve(Gn)))]
≃ LC [hocolim∆op(cBGn)]
≃ LC(BG)
≃ BG.
So, we obtain a simplicial space weakly equivalent to the constant simplicial
space with BG at each level. (Recall, however, that BG here is obtained by taking
the diagonal of the simplicial nerve, so it is not quite the identical case.) If we
have a simplicial groupoid, rather than a simplicial group, we obtain the analogous
result, replacing BG with ∐
BAut(x).
7.4. Case 4: C is a simplicial category with every morphism invertible up
to homotopy. Alternatively stated, this case covers the situation in which π0(C)
is a groupoid.
Recall that we have a model structure SCO on the category of categories with a
fixed object set O, in which the cofibrant objects are retracts of free objects. So,
taking a cofibrant replacement of C in this model category structure SCO essen-
tially gives a free replacement of C, denoted F (C), which is weakly equivalent to C.
(This cofibrant category can be obtained by taking a simplicial resolution F∗C and
then taking a diagonal [12, 6.1].) Now, taking the localization with respect to all
morphisms results in a simplicial groupoid. So, we have Dwyer-Kan equivalences
F (C)−1F (C) F (C)
≃oo ≃ //C
But, now F (C)−1F (C) is a simplicial groupoid weakly equivalent to C, so we have
now reduced this situation to case 3.
Note that, to write down a description of this complete Segal space in terms of the
original category C, we need to take isomorphism classes of objects in π0(C), or weak
equivalence classes, as well as self-maps which are invertible up to homotopy rather
than strict automorphisms. While we will still use 〈x〉 to denote the equivalence
class of a given object, we will use Auth(x) to signify homotopy automorphisms of
20 J.E. BERGNER
x. Thus, the complete Segal space corresponding to C in this case essentially looks
like ∐
BAuth(x)
at each level.
7.5. Case 5: C is any simplicial category. First consider the subcategory of C
containing all the objects of C and only the morphisms of C which are invertible up
to homotopy. Apply case 5 to get a complete Segal space, but take only the 0-space
of it, to be the 0-space of the desired complete Segal space.
To find the 1-space, first recall the definition of the completion functor as applied
to a Segal space W :
LC(W ) = LCSS(diag([m] 7→ (W
E(m))n)).
Recall further that (WE(m))n = Map(E(m)×∆[n]
t,W ). Thus, the Segal space we
obtain (before applying the functor LCSS) looks like
Map(E(0)×∆[0]t,W )⇐ Map(E(1)×∆[1]t,W ) ⇚ Map(E(2)×∆[2]t,W ) · · · .
If the Segal space W is a fibrant replacement of nerve(C), then the space at level 1
consists of diagrams
x′ // y
with the maps in the appropriate simplicial level.
For simplicity, we restrict to a given pair of objects x and y, representing given
equivalence classes. Consider the homotopy automorphisms of x and y. If they are
not all invertible, we take a cofibrant replacement and group completion as in case
4. So, without loss of generality, assume that Aut(x) and Aut(y) are simplicial
groups. Note that we have
Aut(x) = hocolim∆opAut(x)n
Aut(y) = hocolim∆opAut(y)n.
Now look at
Map(x, y) = hocolim∆opMap(x, y)n.
Consider for each n ≥ 0 the discrete category C(x, y)n which has as objects Map(x, y)n
and as morphisms pairs (α, β) of automorphisms in Aut(x)n × Aut(y)n making a
commutative square
with f, f ′ ∈Map(x, y)n.
Thus, the 1-space that we are interested in is also the 1-space of the complete
Segal space given by
LCSS(hocolim∆op(nerve(C(x, y)n))).
COMPLETE SEGAL SPACES 21
Using a straightforward argument about localization functors similar to the one in
case 3 (which can be found in [3, 4.1]), we can also apply the functor LCSS on the
inside to get an equivalent simplicial space
LCSS(hocolim∆op(LCSSnerve(C(x, y)n))).
But, since C(x, y)n is a discrete category, this space is just
LCSShocolim∆op(NC(x, y)n) ≃ hocolim∆op(NC(x, y)n).
Now, we restrict to the 1-space here, which is
hocolim∆op
〈x〉,〈y〉
Mapn(x, y)α
〈x〉,〈y〉
Map(x, y)α
As with the previous case, we can then go back and weaken to homotopy auto-
morphisms and equivalence classes of objects to consider categories before taking
a group completion, so our space looks like
〈x〉,〈y〉
BAuth
Map(x, y)α
We could then obtain the 2-space of our complete Segal space by considering
categories C(x, y, z)n defined similarly, and the description of the 2-space of the
classifying diagram of a discrete category as given in case 2.
We can summarize these results in the following theorem. For an object x of
a simplicial category C, let 〈x〉 denote the weak equivalence class of x in C, and
for a morphism α : x → y, let 〈α〉 denote the weak equivalence class of α in the
morphism category C[1]. Let Auth(x) denote the space of self-maps of x which are
invertible in π0C.
Theorem 7.3. Let C be a simplicial category. The complete Segal space corre-
sponding to C has the form
BAuth(x)⇐
〈x〉,〈y〉
BAuth
Map(x, y)α
⇚ · · · .
References
[1] J.E. Bergner, Homotopy fiber products of homotopy theories, in preparation.
[2] J.E. Bergner, A model category structure on the category of simplicial categories, Trans.
Amer. Math. Soc. 359 (2007), 2043-2058.
[3] J.E. Bergner, Simplicial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83.
[4] J.E. Bergner, A survey of (∞, 1)-categories, preprint available at math.AT/0610239.
[5] J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46
(2007), 397-436.
[6] J.M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams,
Math. Proc. Camb. Phil. Soc. (1986), 100, 65-90.
[7] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176.
[8] W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18
(1980), 17-35.
[9] W.G. Dwyer and D.M. Kan, A classification theorem for diagrams of simplicial sets, Topology
23(1984), 139-155.
[10] W.G. Dwyer and D.M. Kan, Equivalences between homotopy theories of diagrams, Algebraic
topology and algebraic K-theory (Princeton, N.J., 1983), 180–205, Ann. of Math. Stud., 113,
Princeton Univ. Press, Princeton, NJ, 1987.
http://arxiv.org/abs/math/0610239
22 J.E. BERGNER
[11] W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology 19 (1980),
427-440.
[12] W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17
(1980), no. 3, 267–284.
[13] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of
Algebraic Topology, Elsevier, 1995.
[14] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, vol. 174,
Birkhauser, 1999.
[15] Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and
Monographs 99, AMS, 2003.
[16] Mark Hovey, Model Categories, Mathematical Surveys and Monographs, 63. American Math-
ematical Society 1999.
[17] A. Joyal, Simplicial categories vs quasi-categories, in preparation.
[18] A. Joyal, The theory of quasi-categories I, in preparation.
[19] André Joyal and Myles Tierney, Quasi-categories vs Segal spaces, Contemp. Math. 431 (2007)
277-326.
[20] Saunders Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate
Texts in Mathematics 5, Springer-Verlag, 1997.
[21] J.P. May, Simplicial Objects in Algebraic Topology, University of Chicago Press, 1967.
[22] C.L. Reedy, Homotopy theory of model categories, unpublished manuscript, available at
http://www-math.mit.edu/∼psh.
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Soc. 353(3) (2001), 973-1007.
[24] Bertrand Toën, Derived Hall algebras, Duke Math. J. 135, no. 3 (2006), 587-615.
Kansas State University, 138 Cardwell Hall Manhattan, KS 66506
E-mail address: [email protected]
http://www-math.mit.edu/~psh
1. Introduction and Overview
2. Background on Model Categories and Simplicial Objects
3. Simplicial Categories and Simplicial Localizations
4. Complete Segal Spaces
5. Obtaining Complete Segal Spaces from Simplicial Categories and Model Categories
6. Comparison of Functors from SC to CSS
7. A Characterization of Complete Segal Spaces Arising from Simplicial Categories
7.1. Case 1: C is a discrete groupoid
7.2. Case 2: C is a discrete category
7.3. Case 3: C is a simplicial groupoid
7.4. Case 4: C is a simplicial category with every morphism invertible up to homotopy
7.5. Case 5: C is any simplicial category
References
|
0704.1625 | A Systematic Scan for 7-colourings of the Grid | A Systematic Scan for 7-colourings of the Grid
Markus Jalsenius
Department of Computer Science, University of Liverpool
Ashton Street, Liverpool, L69 3BX, United Kingdom
Kasper Pedersen
Department of Computer Science, University of Liverpool
Ashton Street, Liverpool, L69 3BX, United Kingdom
Abstract
We study the mixing time of a systematic scan Markov chain for sampling from
the uniform distribution on proper 7-colourings of a finite rectangular sub-grid
of the infinite square lattice, the grid. A systematic scan Markov chain cycles
through finite-size subsets of vertices in a deterministic order and updates the
colours assigned to the vertices of each subset. The systematic scan Markov chain
that we present cycles through subsets consisting of 2×2 sub-grids and updates the
colours assigned to the vertices using a procedure known as heat-bath. We give a
computer-assisted proof that this systematic scan Markov chain mixes in O(log n)
scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to
compute required couplings of colourings of 2×2 sub-grids. This is the first time the
mixing time of a systematic scan Markov chain on the grid has been shown to mix
for less than 8 colours. We also give partial results that underline the challenges of
proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings
of the grid by considering 2×3 and 3×3 sub-grids.
1 Introduction
This paper is concerned with sampling from the uniform distribution, π, on the set
of proper q-colourings of a finite-size rectangular grid. A q-colouring of a graph is an
assignment of a colour from a finite set of q distinct colours to each vertex and we say that
a colouring is a proper colouring if no two adjacent vertices are assigned the same colour.
Proper q-colourings of the grid correspond to the zero-temperature anti-ferromagnetic
q-state Potts model on the square lattice, a model of significant importance in statistical
physics (see for example Salas and Sokal [14]).
Sampling from π is computationally challenging, however it remains an important task
and it is frequently carried out in experimental work by physicists by simulating some
suitable random dynamics that converges to π. Ensuring that a dynamics converges to
π is generally straight forward, but obtaining good upper bounds on the number of steps
http://arxiv.org/abs/0704.1625v3
required for the dynamics to become sufficiently close to π is a much more difficult prob-
lem. Physicists are at times forced to “guess” (using some heuristic methods) the number
of steps required for their dynamics to be sufficiently close to the uniform distribution in
order to carry out their experiments. By establishing rigorous bounds on the convergence
rates (mixing time) of these dynamics computer scientists can provide underpinnings for
this type of experimental work and also allow a more structured approach to be taken.
Providing bounds on the mixing time of Markov chains is a well-studied problem in
theoretical computer science. However, the types of Markov chains frequently considered
by computer scientists do not always correspond to the dynamics usually used in the
experimental work by physicists. In computer science, the mixing time of various types
of random update Markov chains have been frequently analysed; notably on the grid by
Achlioptas, Molloy, Moore and van Bussel [1] and Goldberg, Martin and Paterson [9].
We say that a Markov chain on the set of colourings is a random update Markov chain
when one step of the the process consists of randomly selecting a set of vertices (often
a single vertex) and updating the colours assigned to those vertices according to some
well-defined distribution induced by π. Experimental work is, however, often carried out
by cycling through and updating the vertices (or subsets of vertices) in a deterministic
order. This type of dynamics has recently been studied by computer scientists in the
form of systematic scan Markov chains (systematic scan for short). For results regarding
systematic scan see for instance Dyer, Goldberg and Jerrum [5, 4] and Pedersen [12]
although these papers are not considering the grid specifically. It is important to note
that systematic scan remains a random process since the method used to update the
colour assigned to the selected set of vertices is a randomised procedure drawing from
some well-defined distribution induced by π.
In Section 3 we present a computer assisted proof that systematic scan mixes rapidly
when considering 7-colourings of the grid. Previously eight was the least number of
colours for which systematic scan on the grid was known to be rapidly mixing, due to
Pedersen [12], a result which we hence improve on in this paper. We will make use of
a recent result by Pedersen [12] to prove rapid mixing of systematic scan by bounding
the influence on a vertex (note that the literature traditionally talks about sites rather
than vertices). We will provide bounds on this influence parameter by using a heuristic
to mechanically construct sufficiently good couplings of proper colourings of a 2×2 sub-
grid. We will hence use a heuristic based computation in order to establish a rigorous
result about the mixing time of a systematic scan Markov chain. Finally, in Section 4,
we consider the possibility of proving rapid mixing of systematic scan for 6-colourings of
the grid by increasing the size of the sub-grids. We give lower bounds on the appropriate
influence parameter that imply that the proof technique we employ does not imply rapid
mixing of systematic scan for 6-colourings of the grid when using 2×2, 2×3 and 3×3
sub-grids.
1.1 Preliminaries and statement of results
Let Q = {1, . . . , 7} be the set of colours and V = {1, . . . , n} the set of vertices of a finite
rectangular grid G with toroidal boundary conditions. Working on the torus is common
practice as it avoids treating several technicalities regarding the vertices on the boundary
of a finite grid as special cases and hence lets us present the proof in a more “clean” way.
We point out however that these technicalities are straightforward to deal with (more
on this in Section 2). We formally say that a colouring σ of G is a function from V to
Q. Let Ω+ be the set of all colourings of G and Ω be the set of all proper q-colourings.
Then the distribution π, described earlier, is the uniform distribution on Ω. If σ ∈ Ω+
is a colouring and j ∈ V is a vertex then σj denotes the colour assigned to vertex j in
colouring σ. Furthermore, for a subset of vertices Λ ⊆ V and a colouring σ ∈ Ω+ we let
σΛ denote the colouring of the vertices in Λ under σ. For each vertex j ∈ V , let Sj denote
the set of pairs (σ, τ) ∈ Ω+ × Ω+ of colourings that only differ on the colour assigned to
vertex j, that is σi = τi for all i 6= j.
Let M be a Markov chain with state space Ω+ and stationary distribution π. Suppose
that the transition matrix of M is P . Then the mixing time from an initial colouring
σ ∈ Ω+ is the number of steps, that is applications of P , required for M to become
sufficiently close to π. Formally the mixing time of M from an initial colouring σ ∈ Ω+
is defined, as a function of the deviation ε from stationarity, by
Mixσ(M, ε) = min{t > 0 : dTV(P
t(σ, ·), π) ≤ ε}, (1)
where
dTV(θ1, θ2) =
|θ1(i)− θ2(i)| = max
|θ1(A)− θ2(A)| (2)
is the total variation distance between two distributions θ1 and θ2 on Ω
+. The mixing
time Mix(M, ε) of M is then obtained my maximising over all possible initial colourings
Mix(M, ε) = max
Mixσ(M, ε). (3)
We say that M is rapidly mixing if the mixing time of M is polynomial in n and log(ε−1).
We will make use of a recent result by Pedersen [12] to study the mixing time of a
systematic scan Markov chain for 7-colourings of the grid using block updates. We need
the following notation in order to define our systematic scan Markov chain. Define the
following set Θ = {Θ1, . . . ,Θm} of m blocks. Each block Θk ⊆ V is a 2×2 sub-grid and m
is the smallest integer such that
k=1Θk = V . For any block Θk and a pair of colourings
σ, τ ∈ Ω+ we write “σ = τ on Θk” if σi = τi for each i ∈ Θk and similarly “σ = τ off
Θk” if σi = τi for each i ∈ V \ Θk. We also let ∂Θk denote the set of vertices in V \ Θk
that are adjacent to some vertex in Θk, and we will refer to ∂Θk as the boundary of Θk.
Note from our previous definitions that σ∂Θk denotes the colouring of the boundary of
Θk under a colouring σ ∈ Ω
+. We will refer to σ∂Θk as a boundary colouring. Finally we
say that a 7-colouring of the 2×2 sub-grid Θk agrees with a boundary colouring σ∂Θk if
(1) no adjacent sites in Θk are assigned the same colour and (2) each vertex j ∈ Θk is
assigned a colour that is different to the colours of all boundary vertices adjacent to j.
For each block Θk and colouring σ ∈ Ω
+ let Ωk(σ) be the subset of Ω
+ such that
if σ′ ∈ Ωk(σ) then σ
′ = σ off Θk and σ
agrees with σ∂Θk . Let πk(σ) be the uniform
distribution on Ωk(σ). We then define P
[k] to be the transition matrix on the state space
Ω+ for performing a so-called heat-bath move on Θk. A heat-bath move on a block Θk,
given a colouring σ ∈ Ω+, is performed by drawing a new colouring from the distribution
πk(σ). Note in particular that applying P
[k] to a colouring σ ∈ Ω+ results in a colouring
σ′ ∈ Ω+ such that σ′ = σ off Θk and the colouring σ
of Θk is proper and agrees with
the colouring σ′∂Θk of the boundary of Θk (which is identical to σ∂Θk). We formally define
the following systematic scan Markov chain for 7-colourings of G, which systematically
performs heat-bath moves on 2×2 sub-grids, as follows. It is worth pointing out that this
holds for any ordering of the set of blocks.
Definition 1. The systematic scan dynamics for 7-colourings of G is a Markov chain
Mgrid with state space Ω
+ and transition matrix Pgrid = Π
It can be shown that the stationary distribution of Mgrid is π by considering the
construction of Pgrid. It is customary to refer to one application of Pgrid (that is updating
each block once) as one scan. One scan takes
|Θk| vertex updates and by construction
of Θ this sum is clearly of order O(n).
We will prove the following theorem and point out that this is the first proof of rapid
mixing of systematic scan for 7-colourings on the grid.
Theorem 2. Let Mgrid be the Markov chain from Definition 1 on 7-colourings of G.
Then the mixing time of Mgrid is
Mix(Mgrid, ε) ≤ 63 log(nε
−1). (4)
1.2 Context and related work
We now provide an overview of previous achievements for colourings of the grid. Previ-
ously it was known that systematic scan for q-colourings on general graphs with maximum
vertex degree ∆ mixes in O(logn) scans when q ≥ 2∆ due to Pedersen [12]. That result
is a hand-proof and uses block updates that updates the colour at each endpoint of an
edge during each step. Earlier Dyer et al. [4] had shown that a single-site systematic
scan Markov chain (where one vertex is updated at a time) mixes in O(logn) scans when
q > 2∆ and in O(n2 logn) scans when q = 2∆. It is hence well-established that system-
atic scan is rapidly mixing for q-colourings of the grid when q ≥ 8 but nothing has been
known about the mixing time for smaller q. The results of both Pedersen [12] and Dyer
et al. [4] bound the mixing time by studying the influence on a vertex. We will use that
technique in this paper as well, however we will construct the required couplings using a
heuristic. We defer the required definitions to Section 2 which also contains the proof of
Theorem 2.
Recent results have revealed that, in a single-site setting, one is not restricted use
the total influence on a vertex when analysing the mixing time of systematic scan by
bounding influence parameters. In a single-site setting one can define an n×n-matrix
whose entries are the influences that all vertices have on each other. Hayes [10] has
shown that providing a sufficiently small upper bound on the spectral gap of this matrix
implies rapid mixing of both systematic scan and random update. Dyer, Goldberg and
Jerrum [6] furthermore showed that an upper bound on any matrix norm also implies
rapid mixing of both types of Markov chains. These techniques are however not known
to apply to Markov chains using block moves. See the PhD thesis by Pedersen [13] for
more comprehensive review of the above results and for the difficulties in extending them
to cover block dynamics.
As random update Markov chains have received more attention than systematic scan
we also summarise some mixing results of interest regarding q-colourings of the grid (recall
that a random update Markov chain selects randomly a subset of sites to be updated at
each step). Achlioptas et al. [1] give a computer-assisted proof of mixing in O(n logn)
updates when q = 6 by considering blocks consisting of 2×3 sub-grids. Our computations
are similar in nature to the ones of Achlioptas et al. however their computations are not
sufficient to imply mixing of systematic scan as we will discuss in due course. More
recently Goldberg, Martin and Paterson [9] gave a hand-proof of mixing in O(n logn)
updates when q ≥ 7 using the technique of strong spatial mixing. Previously Salas and
Sokal [14] gave a computer-assisted proof of the q = 7 case, a result which was also
implied by another computer-assisted result due to Bubley, Dyer and Greenhill [3] that
applies to 4-regular triangle-free graphs. Finally it is worth pointing out that, in the
special case when q = 3, two complementary results of Luby, Randall and Sinclair [11]
and Goldberg, Martin and Paterson [8] give rapid mixing of random update.
2 Bounding the mixing time of systematic scan
This section will contain a proof of Theorem 2 although the proof of a crucial lemma,
which requires computer-assistance, is deferred to Section 3. We will bound the mixing
time of Mgrid by bounding the influence on a vertex, a parameter which we denote by
α and will define formally in due course. If α is sufficiently small then Theorem 2 from
Pedersen [12] implies that any systematic scan Markov chain, whose transition matrices
for updating each block satisfy two simple properties, mixes in O(logn) scans. For
completeness we restate this theorem (Theorem 3 below) and in the statement we let
M→ denote a systematic scan Markov chain whose transition matrices for each block
update satisfy the required properties.
Theorem 3. If α < 1 then the mixing time of M→ is
Mix(M→, ε) ≤
log(nε−1)
. (5)
For each block Θk the transition matrix P
[k] needs to satisfy the following two prop-
erties in order for Theorem 3 to apply.
1. If P [k](σ, τ) > 0 then σ = τ off Θk, and
2. π is invariant with respect to P [k].
It is pointed out in Pedersen [12] that if P [k] is a transition matrix performing a heat-bath
move then both of these properties are easily satisfied. Furthermore, it is pointed out
that when Ω is the set of proper colourings of a graph, then π is the uniform distribution
on Ω as we require. Since the transition matrices P [k] used in the definition of Mgrid
perform heat-bath updates we are hence able to use Theorem 3 to bound the mixing
time of Mgrid.
We are now ready to formally define the parameter α denoting the influence on a
vertex. For any pair of colourings (σ, τ) ∈ Si let Ψk(σ, τ) be a coupling of the distributions
induced by P [k](σ, ·) and P [k](τ, ·), namely πk(σ) and πk(τ) respectively. We remind
the reader that a coupling of two distributions π1 and π2 on state space Ω
+ is a joint
distribution Ω+ × Ω+ such that the marginal distributions are π1 and π2. For ease of
reference we also let pj(Ψk(σ, τ)) denote the probability that a vertex j ∈ Θk is assigned
a different colour in a pair of colourings drawn from some coupling Ψk(σ, τ). We then let
ρki,j = max
(σ,τ)∈Si
pj(Ψk(σ, τ)) (6)
be the influence of i on j under Θk. Finally the parameter α denoting the influence on
any vertex is defined as
α = max
ρki,j. (7)
Pedersen [12] actually defines α with a weight associated with each vertex, however as
we will not use weights in our proof we have omitted them from the above account. So, in
order to upper bound α we are required to upper bound the probability of a discrepancy
at each vertex j ∈ Θk under a coupling Ψk(σ, τ) of the distributions πk(σ) and πk(τ) for
any pair of colourings (σ, τ) ∈ Si that only differ at the colour of vertex i. Our main
task is hence to specify a coupling Ψk(σ, τ) of πk(σ) and πk(τ) for each pair of colourings
(σ, τ) ∈ Si and upper bound the probability of assigning a different colour to each vertex
in a pair of colourings drawn from that coupling.
Consider any block Θk and any pair of colourings (σ, τ) ∈ Si that differ only on the
colour assigned to some vertex i. Clearly the distribution on colourings of Θk, induced
by πk(σ) only depends on the boundary colouring σ∂Θk . Similarly, the distribution on
colourings of Θk, induced by πk(τ) depends only on τ∂Θk . If i 6∈ ∂Θk then the distributions
on the colourings of Θk, induced by πk(σ) and πk(τ), respectively, are the same and we
let Ψk(σ, τ) be the coupling in which any pair of colourings drawn from Ψk(σ, τ) agree
on Θk. That is, if the pair (σ
′, τ ′) of colourings are drawn from Ψk(σ, τ) then σ
′ = σ off
Θk, τ
′ = τ off Θk and σ
′ = τ ′ on Θk. This gives ρ
i,j = 0 for any i 6∈ ∂Θk and j ∈ Θk.
We now need to construct Ψk(σ, τ) for the case when i ∈ ∂Θk. For each j ∈ Θk
we need pj(Ψk(σ, τ)) to be sufficiently small in order to avoid ρ
i,j being too big. If the
ρki,j-values are too big the parameter α will be too big (that is greater than one) and we
cannot make use of Theorem 3 to show rapid mixing. Constructing Ψk(σ, τ) by hand such
that pj(Ψk(σ, τ)) is sufficiently small is a difficult task. It is, however, straight forward to
mechanically determine which colourings have positive measure in the distributions πk(σ)
and πk(τ) for a given pair of boundary colourings σ∂Θk and τ∂Θk . From these distributions
we can then use some suitable heuristic to construct a coupling that is good enough for
our purposes. We hence need to construct a specific coupling for each individual pair of
colourings differing only at a single vertex. In order to do this we will make use of the
following lemma, which is proved in Section 3.
Lemma 4. Let v1, . . . , v4 be the four vertices in a 2×2-block and z1, . . . , z8 be the boundary
vertices of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 7-
colourings of the boundary vertices such that Z and Z ′ agree on each vertex except on z1.
Let πZ and πZ′ be the uniform distributions on proper 7-colourings of the block that agree
with Z and Z ′, respectively. For i = 1, . . . , 4 let pvi(Ψ) denote the probability that the
colour of vertex vi differ in a pair of colourings drawn from a coupling Ψ of πZ and πZ′.
Then there exists a coupling Ψ such that pv1(Ψ) < 0.283, pv2(Ψ) < 0.079, pv3(Ψ) < 0.051
and pv4(Ψ) < 0.079.
z4 z5
Figure 1: General labeling of the vertices in a 2×2-block Θk and the vertices ∂Θk on the
boundary of the block.
i (b)
Figure 2: A 2×2-block Θk showing all eight positions of a vertex i ∈ ∂Θk on the boundary
of the block in relation to a vertex j ∈ Θk in the block.
Thus if i ∈ ∂Θk we let Ψk(σ, τ) be the coupling of πk(σ) and πk(τ) that draws the
colouring of Θk from the coupling Ψ in Lemma 4, where Z is the boundary colouring
obtained from σ∂Θk and Z
′ is obtained from τ∂Θk , and leaves the colour of the remaining
vertices, V \Θk, unchanged. That is, if the pair (σ
′, τ ′) of colourings are drawn from
Ψk(σ, τ) then σ
′ = σ off Θk, τ
′ = τ off Θk and the colourings of Θk in σ
′ and τ ′ are
drawn from the coupling Ψ in Lemma 4 (see the proof for details on how to construct
Ψ). It is straightforward to verify that this is indeed a coupling of πk(σ) and πk(τ). Note
that due to the symmetry of the 2×2-block, with respect to rotation and mirroring, we
can always label the vertices of Θk and ∂Θk such that label z1 in Figure 1 represents the
discrepancy vertex i on the boundary. Hence we can make use of Lemma 4 to compute
upper bounds on the parameters ρki,j. We summarise the ρ
i,j-values in the following
Corollary of Lemma 4. Note that due to the symmetry of the block we can assume that
vertex j ∈ Θk in the corollary is located in the bottom left corner, as Figure 2 shows.
Corollary 5. Let Θk be any 2×2-block, let j ∈ Θk be any vertex in the block and let
i ∈ ∂Θk be a vertex on the boundary of the block. Then
ρki,j = max
(σ,τ)∈Si
pj(Ψk(σ, τ)) <
0.283, if i and j as in Figure 2(a) or (b),
0.079, if i and j as in Figure 2(c) or (h),
0.051, if i and j as in Figure 2(e) or (f),
0.079, if i and j as in Figure 2(d) or (g).
If i /∈ ∂Θk is not on the boundary of the block then ρ
i,j = 0.
We can then use Corollary 5 to prove Theorem 2. The proof of Theorem 2 is given
here:
Proof of Theorem 2. Let αk,j =
ρki,j be the influence on j under Θk. We need αk,j to
be upper bounded by one for each block Θk and vertex j ∈ Θk in order to ensure that
α = maxk maxj∈Θk αk,j is less than one. Fix any block Θk and any vertex j ∈ Θk. A
vertex i ∈ ∂Θk on the boundary of the block can occupy eight different positions on the
boundary in relation to j as showed in Figure 2(a)–(h). Recall that we are working on
the torus, and hence every vertex on the boundary of the block will belong to G. Thus,
using the bounds from Corollary 5 we have
αk,j =
ρki,j < 2(0.283 + 0.079 + 0.051 + 0.079) = 0.984. (9)
Then α = maxk maxj∈Θk αk,j < maxk 0.984 = 0.984 < 1 and we obtain the stated bound
on the mixing time of Mgrid by Theorem 3.
We make the following remark. In the proof of Theorem 2 above, we assume that G
is a finite rectangular grid with toroidal boundary conditions. Hence, every block is a
2×2-sub-grid and each vertex on the block boundary belongs to V . We note that if G
is a finite rectangular grid without toroidal boundary conditions then some vertices on
the boundary ∂Θk of a block Θk might fall outside G. The sum in Equation (9) is over
boundary vertices i that do belong to V , and hence the number of terms in this sum is
reduced if some boundary vertices do not belong to V , making α smaller. Furthermore,
if G is a non-rectangular region of the grid then a block next to the boundary might be
smaller than 2×2 vertices. Suppose Θk is a block that is smaller than 2×2 vertices. Then
the vertices that are missing in order to make Θk a full 2×2-block are boundary vertices.
Suppose i ∈ ∂Θk belongs to V and i
′ ∈ ∂Θk does not belong to V . When constructing
couplings Ψk(σ, τ), where (σ, τ) ∈ Si, we must consider the vertex i
′ as “colourless”,
which would decrease the value of pki,j . A more rigorous analysis yields that our mixing
result with seven colours and 2×2-blocks holds for arbitrary finite regions G of the grid.
Of course we have yet to establish a proof of Lemma 4, and the rest of this paper will be
concerned with this. Our method of proof uses some ideas of Goldberg, Jalsenius, Martin
and Paterson [7] in so far as it is computer assisted and we will be focusing on minimising
the probability of assigning different colours to vertex v1 in the constructed couplings.
We will however be required to construct a coupling on the 2×2 sub-grid, rather than
establishing bounds on the disagreement probability of a vertex adjacent to the initial
discrepancy and then extending this to a coupling on the whole block recursively. Our
approach is similar to the one Achlioptas et al. [1] take, however we do not have the
option of constructing an “optimal” coupling using a suitable linear program (even when
feasible) since our probabilities will be maximised over all boundary colourings. The
crucial difference between the approaches is that Achlioptas et al. [1] are using path
coupling (see Bubley and Dyer [2]) as a proof technique which requires them to bound
the expected Hamming distance between a pair of colourings drawn from a coupling. This
in turn enables them to, for a given boundary colouring, specify an “optimal” coupling
which minimises Hamming distance. We are, however, required to bound the influence of
i on j for each boundary colouring and sum over the maximum of these influences. The
reason for this is the inherit maximisation over boundary colourings in the definition of
ρki,j as described above.
Finally it is worth mentioning that providing bounds on the expected Hamming dis-
tance is similar to showing that the influence of a vertex is small and it is known that
this condition implies rapid mixing of a random update Markov chain, see for example
Weitz [15]. In a single-site setting the condition “the influence of a vertex is small” also
implies rapid mixing of systematic scan (Dyer et al. [4]), however, in a block setting this
condition is not sufficient to give rapid mixing of systematic scan (Pedersen [13]), which
is why we need to bound the influence on a vertex.
3 Constructing the coupling by machine
In order to prove Lemma 4 we will construct a coupling Ψ of πZ and πZ′ for all pairs of
boundary colourings Z and Z ′ that are identical on all boundary vertices but vertex z1,
on which Z and Z ′ differ. For each coupling constructed we verify that the probabilities
pvi(Ψ), i = 1, . . . , 4, are within the bounds of the lemma. The method is well suited to be
carried out with the help of a computer and we have implemented a program in C to do
so. Before stating the proof of Lemma 4 we will discuss how a coupling can be represented
by an edge-weighted complete bipartite graph. We make use of this representation of Ψ
in the proof of the lemma.
3.1 Representing a coupling as a bipartite graph
Let S be a set of objects and let W be a set of |S| pairs (s, ws) such that s ∈ S and ws ≥ 0
is a non-negative value representing the weight of s. Each element s ∈ S is contained
in exactly one of the pairs in W . If the value ws is an integer (which it is in our case)
it can be regarded as the multiplicity of s in a multiset. The set W is referred to as a
weighted set of S. Let πS,W be the distribution on S such that the probability of s is
proportional to ws, where (s, ws) is a pair in W . More precisely, the probability of s in
πS,W is PrπS,W (s) = ws/
(t,wt)∈W
wt. For example, let W be a weighted set of S and let
S ′ ⊆ S be a subset of S. Assume the weight ws = 0 if s ∈ S\S
′ and ws = k if s ∈ S
where k > 0 is a positive constant. Then πS,W is the uniform distribution on S
The reason for introducing the notion of a weighted set is that it can be used when
specifying a coupling of two distributions. Let S be a set and let W and W ′ be two
weighted sets of S such that the sum of the weights in W equals the sum of the weights
in W ′. Let wtot denote this sum. That is, wtot =
(s,ws)∈W
(s′,w′
)∈W ′ w
s′. The
two weighted sets W and W ′ define two distributions πS,W and πS,W ′ on S. We want
to specify a coupling Ψ of πS,W and πS,W ′. Let K|S|,|S| be an edge-weighted complete
bipartite graph with vertex sets W and W ′. That is, for each pair (s, ws) ∈ W there
is an edge to every pair in W ′. Every edge e of K|S|,|S| has a weight we ≥ 0 such that
the following condition holds. Let (s, ws) be any pair in W ∪ W
′ and let E be the set
of all |S| edges incident to (s, ws). Then
we = ws. It follows that the sum of the
edge weights of all |S|2 edges in K|S|,|S| equals wtot, the sum of the weights in W (and
W ′). The idea is that K|S|,|S| represents a coupling Ψ of πS,W and πS,W ′. In order to
draw a pair of elements from Ψ we randomly select an edge e in K|S|,|S| proportional to
its weight. The endpoints of e represent the elements in S drawn from πS,W and πS,W ′.
More precisely, the probability of choosing edge e in K|S|,|S| with weight we is we/wtot.
If edge e = ((s, ws), (s
′, w′s′)) is chosen it means that we have drawn s from πS,W and s
from πS,W ′, the marginal distributions of Ψ.
The bipartite graph representation of a coupling will be used when we construct
couplings of colourings of 2×2-blocks in the proof of Lemma 4.
3.2 The proof of Lemma 4
Here is the proof of Lemma 4:
Proof of Lemma 4. Fix two colourings Z and Z ′ of the boundary that differ on vertex
z1. Let c be the colour of vertex z1 in Z and let c
′ 6= c be the colour of z1 in Z
′. Let CZ
and CZ′ be the two sets of proper 7-colourings of the block that agree with Z and Z
respectively. Let C+ be the set of all 7-colourings of the block. Let WZ and WZ′ be two
weighted sets of C+. The weights are assigned as follows.
• For the pair (σ, wσ) ∈ WZ let the weight wσ = |CZ′| if σ ∈ CZ , otherwise let wσ = 0.
• For the pair (σ, wσ) ∈ WZ′ let the weight wσ = |CZ | if σ ∈ CZ′, otherwise let
wσ = 0.
It follows from the assignment of the weights that the distribution πC+,WZ is the uniform
distribution on CZ . That is, πC+,WZ = πZ . Similarly, πC+,WZ′ is the uniform distribution
πZ′ on CZ′. Note that the sum of the weights is |CZ||CZ′| in both WZ and WZ′. Then
a coupling Ψ of πC+,WZ and πC+,WZ′ can be specified with an edge-weighted complete
bipartite graph K = K|C+|,|C+|. For a given valid assignment of the weights of the edges
of K, making K represent a coupling Ψ, we can compute the probabilities of having a
mismatch on a vertex vi of the block when two colourings are drawn from Ψ. Let E be
the set of all edges e = ((σ, wσ), (σ
′, w′σ′)) in K such that σ and σ
′ differ on vertex vi.
Then pvi(Ψ) =
e∈E we/|CZ||CZ′|.
In order to obtain sufficiently small upper bounds on pvi(Ψ) for the four vertices
v1, . . . , v4 in the block we would like to assign weights to the edges of K such that much
weight is assigned to edges between colourings that agree on many vertices in the block.
In general it is not clear exactly how to assign weights to the edges. For instance, if
we assign too much weight to edges between colourings that are identical on vertex v2
we might not be able to assign as much weight as we would like to on edges between
colourings that are identical on vertex v4. Thus, the probability of having a mismatch
on v4 would increase. Intuitively a good strategy would be to assign as much weight as
possible to edges between colourings that are identical on the whole block. This implies
that we try to assign as much weight as possible to edges between colourings that are
identical on vertex v1, the vertex adjacent to the discrepancy vertex z1 on the boundary.
If there is a mismatch on vertex v1 it should be a good idea to assign as much weight
as possible to edges between colourings that are identical on the whole block apart from
vertex v1. This idea leads to a heuristic in which the assignment of the edge weights is
divided into three phases. The exact procedure is described as follows.
In phase one we match identical colourings. For all colourings σ ∈ C+ of the block
the edge e = ((σ, wσ), (σ, w
σ)) in K will be given weight we = min(wσ, w
σ). That is, we
maximise the probability of drawing the same colouring σ from both πC+,WZ and πC+,WZ′ .
For the following two phases we define an ordering of the colourings in C+. We order
the colourings lexicographically with respect to the vertex order v3, v2, v4, v1. That is,
if the seven colours are 1, . . . , 7 the colouring of v3, v2, v4, v1 will start with 1, 1, 1, 1,
respectively. The next colouring will be 1, 1, 1, 2, and so on. This ordering of colourings
in C+ carries over to an ordering of the pairs in WZ and WZ′. That is, we order the pairs
(σ, wσ) in WZ with respect to the lexicographical ordering of σ. Similarly we order the
pairs in WZ′ . This ordering of the pairs will be important in the next two phases. It
provides some control of how colourings are being paired up in terms of the assignment
of the weights on edges between pairs. Edges will be considered with respect to this
ordering because choosing an arbitrary ordering of the edges would not necessarily result
in probabilities pvi(Ψ) that would be within the bounds of the lemma.
In the second phase we ignore the colour of vertex v1 and match colourings that are
identical on all of the remaining three vertices v2, v3 and v4. More precisely, for each
pair (σ, wσ) ∈ WZ , considered in the ordering explained above, we consider the edges
e = ((σ, wσ), (σ
′, w′σ′)) where σ and σ
′ are identical on all vertices but v1. The edges are
considered in the ordering of the second component (σ′, w′σ′) ∈ WZ′. We assign as much
weight as possible to e such that the total weight on edges incident to (σ, wσ) ∈ WZ does
not exceed wσ and such that the total weight on edges incident to (σ
′, w′σ′) ∈ WZ′ does
not exceed w′σ′ . Note that in the lexicographical ordering of the colourings, vertex v1 is
the least significant vertex and therefore the ordering provides some level of control of
pairing up colourings that are similar on the remaining three vertices. It turns out that
the resulting coupling is sufficiently good for proving the lemma.
In the third and last phase we assign the remaining weights on the edges. As in phase
two, for each pair (σ, wσ) ∈ WZ we consider the edges e = ((σ, wσ), (σ
′, w′σ′)). The pairs
and edges are considered in accordance with the ordering explained above. The difference
between the second and third phase is that now we do not have any restrictions on the
colourings σ and σ′. We assign as much weight as possible to e such that the total weight
on edges incident to (σ, wσ) ∈ WZ does not exceed wσ and such that the total weight on
edges incident to (σ′, w′σ′) ∈ WZ′ does not exceed w
σ′. After phase three we have assigned
all weights to the edges of K and hence K represents a coupling Ψ of πZ and πZ′.
From K we compute the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) as described
above. We have written a C-program which loops through all colourings Z and Z ′ of
the boundary of the block and constructs the bipartite graph K as described above.
For each boundary the probabilities pv1(Ψ), pv2(Ψ), pv3(Ψ) and pv4(Ψ) are successfully
verified to be within the bounds of the lemma. For details on the C-program, see
http://www.csc.liv.ac.uk/∼markus/systematicscan/.
4 Partial results for 6-colourings of the grid
In previous sections we have seen that systematic scan on the grid using 2×2-blocks and
seven colours mixes rapidly. An immediate question is whether we can do better and
show rapid mixing with six colours. This matter will be discussed in this section and we
will show that, even with bigger block sizes (up to 3×3), it is not possible to show rapid
mixing using the technique of this paper. More precisely, we will establish lower bounds
on the parameter α for 2×2-blocks, 2×3-blocks and 3×3-blocks. All three lower bounds
are greater than one and hence we cannot make use of Theorem 3 to show rapid mixing.
4.1 Establishing lower bounds for 2×2 blocks
We start by examining the 2×2-block again but this time with six colours. Lemma 4
provides upper bounds (under any colourings of the boundary) on the probabilities of
having discrepancies at each of the four vertices of the block when two 7-colourings are
drawn from the specified coupling. For six colours we will show lower bounds on these
probabilities under any coupling and a specified pair of boundary colourings. Once again,
let v1, . . . , v4 be the four vertices in a 2×2-block and let z1, . . . , z8 be the boundary vertices
of the block and let the labeling be as in Figure 1. Let Z and Z ′ be any two 6-colourings
of the boundary vertices that assign the same colour to each vertex except for z1. Let
πZ and πZ′ be the uniform distributions on the sets of proper 6-colourings of the block
that agree with Z and Z ′, respectively. Let Ψminvk (Z,Z
′) be a coupling of πZ and πZ′
that minimises pvk(Ψ). That is, pvk(Ψ) ≥ pvk(Ψ
(Z,Z ′)) for all couplings Ψ of πZ and
πZ′. Also let p
= maxZ,Z′ pvk(Ψ
(Z,Z ′)). We can hence say that there exist two
6-colourings Z and Z ′ of the boundary of a 2×2 block, that assign the same colour to
each vertex except for z1, such that pvk(Ψ) ≥ p
for any coupling Ψ of πZ and πZ′ . We
have the following lemma, which is proved by computation.
Lemma 6. Consider 6-colourings of the 2×2-block in Figure 1. Then plowv1 ≥ 0.379,
plowv2 ≥ 0.107, p
≥ 0.050 and plowv4 ≥ 0.107.
Proof. Fix one vertex vk in the block and fix two colourings Z and Z
′ of the boundary of
the block that differ only on the colour of vertex z1. Let CZ and CZ′ be the two sets of
proper 6-colourings of the block that agree with Z and Z ′, respectively. For c = 1, . . . , 6
let nc be the number of colourings in CZ in which vertex vk is assigned colour c. Similarly
let n′c be the number of colourings in CZ′ in which vertex vk is assigned colour c. It is
clear that the probability that vk is assigned colour c in a colouring σ
′ drawn from πZ
is PrπZ(σ
= c) = nc/|CZ|. For c = 1, . . . , 6 define mc = nc|CZ′|, m
c = n
c|CZ | and
M = |CZ||CZ′|. It follows that PrπZ(σ
= c) = mc/M and PrπZ′ (τ
= c) = m′c/M ,
where σ′ and τ ′ are colourings drawn from πZ and πZ′, respectively. Observe that the
quantities mc, m
c and M can be easily computed for a given pair of boundary colourings.
Now let Ψ be any coupling of πZ and πZ′ . It is easy to see that the probability that
vertex vk is coloured c in both colourings drawn from Ψ can be at most min(mc, m
c)/M .
Therefore, the probability of drawing two colourings from Ψ such that the colour of vertex
vk is the same in both colourings is at most
c=1,...,6
min(mc, m
c)/M , and the probability
of assigning different colours to vertex vk is at least pvk(Ψ) ≥ 1−
c=1,...,6min(mc, m
c)/M .
We have successfully verified the bounds in the statement of the lemma by maximising
the lower bound on pvk(Ψ) over all boundary colourings Z and Z
′ for each vertex vk
in the block. The computations are carried out with the help of a computer program
written in C. For details on the program, see http://www.csc.liv.ac.uk/∼markus/
systematicscan/.
For seven colours, Corollary 5 makes use of Lemma 4 to establish upper bounds on
the influence parameters ρki,j . These parameters are used in the proof of Theorem 2 to
obtain an upper bound on the parameter α. The upper bound on α is shown to be less
than one which implies rapid mixing for seven colours when applying Theorem 3. We can
use Lemma 6 to obtain lower bounds on the influence parameters ρki,j by completing the
coupling in a way analogous to the coupling in Corollary 5. This in turn will result in a
v1 v3v2z2
z4 z5 z6
z1 z9z10
v4 v5 v6
Figure 3: (a) General labeling of the vertices in a 2×3-block Θk and the vertices ∂Θk on
the boundary of the block. (b)–(c) All ten positions of a vertex i ∈ ∂Θk on the boundary
of the block in relation to a vertex j ∈ Θk in the corner of the block.
lower bound on the parameter α that is greater than one. That is, following the proof of
Theorem 2 and making use of Lemma 6, a lower bound on α will be
α ≥ 2(0.379 + 0.107 + 0.050 + 0.107) = 1.286 > 1. (10)
Hence we fail to show rapid mixing of systematic scan with six colours using 2×2-blocks.
4.2 Bigger blocks
We failed to show rapid mixing of systematic scan with six colours and 2×2-blocks and
we will now show that increasing the block size to both 2×3 and 3×3 will not be suf-
ficient either. Lemma 7 below considers 2×3-blocks and is analogous to Lemma 6. We
make use of the same notation as for Lemma 6, only the block is bigger and the label-
ing of the vertices is different (see Figure 3(a)). Lemma 7 is proved by computation
in the same way as Lemma 6. For details on the C-program used in the proof, see
http://www.csc.liv.ac.uk/∼markus/systematicscan/.
Lemma 7. Consider 6-colourings of the 2×3-block in Figure 3(a). Then plowv1 ≥ 0.3671,
plowv3 ≥ 0.0298, p
≥ 0.0997 and plowv6 ≥ 0.0174.
We will now use Lemma 7 to show that α > 1 for 2×3 blocks. Let Θk be any 2×3-
block and let j ∈ Θk be a vertex in a corner of the block. A vertex i ∈ ∂Θk on the
boundary of the block can occupy ten different positions on the boundary in relation to
j. See Figure 3(b) and (c). We can again determine lower bounds on the influences ρki,j
of i on j under Θk from Lemma 7. However, Lemma 7 provides lower bounds on ρ
i,j only
when i ∈ ∂Θk is adjacent to a corner vertex of the block, as in Figure 3(b). If i is located
as in Figure 3(c) we do not know more than that ρki,j is bounded from below by zero.
Nevertheless, the lower bound on α exceeds one. Let αk,j =
ρki,j be the influence on j
under Θk. Following the proof of Theorem 2 and using the lower bounds in Lemma 7 we
αk,j =
i in Fig. 3(b)
ρki,j +
i in Fig. 3(c)
ρki,j
≥ 2(0.3671 + 0.0298 + 0.0997 + 0.0174) = 1.028, (11)
where we set the lower bound on the second sum to zero. Now,
α = max
αk,j ≥ 1.028 > 1. (12)
v1 v3v2z2
z5 z6 z7
v4 v5 v6
z12 z11
v7 v8 v9
v1 v3v2
z6 z7 z8
z10v4 v5 v6
v7 v8 v9
z1 (c)
Figure 4: (a)–(b) General labeling of the vertices in a 3×3-block Θk and two different
labellings of the vertices ∂Θk on the boundary of the block. The discrepancy vertex
on the boundary has label z1. (b)–(c) All twelve positions of a vertex i ∈ ∂Θk on the
boundary of the block in relation to a vertex j ∈ Θk in the corner of the block.
Hence we cannot use Theorem 3 to show rapid mixing of systematic scan with six colours
and 2×3-blocks. It is interesting to note that considering 2×3-blocks was sufficient for
Achlioptas et al. [1] to prove mixing of a random update Markov chain for sampling
6-colourings of the grid.
Lastly, we increase the block size to 3×3 and show that a lower bound on α is still
greater than one. We have the following lemma which is proved by computation in the
same way as Lemmas 6 and 7. For details on the C-program used in the proof see
http://www.csc.liv.ac.uk/∼markus/systematicscan/.
Lemma 8. For 6-colourings of the 3×3-block with vertices labeled as in Figure 4(a) we
have plowv1 ≥ 0.3537, p
≥ 0.0245, plowv7 ≥ 0.0245 and p
≥ 0.0071. Furthermore,
for 6-colourings of the 3×3-block in Figure 4(b) we have plowv1 ≥ 0.0838, p
≥ 0.0838,
plowv7 ≥ 0.0138 and p
≥ 0.0138.
Note that Lemma 8 provides lower bounds on the probabilities of having a mismatch
on a corner vertex of the block when the discrepancy vertex on the boundary (labeled z1)
is adjacent to a corner vertex (Figure 4(a)) and adjacent to a middle vertex (Figure 4(b)).
Let Θk be any 3×3-block and let j ∈ Θk be a vertex in a corner of the block. A vertex
i ∈ ∂Θk on the boundary of the block can occupy twelve different positions on the
boundary in relation to j. See Figure 4(c) and (d). Analogous to Corollary 5 lower
bounds on the influences ρki,j of i on j under Θk can be determined from Lemma 8. Let
αk,j =
ρki,j be the influence on j under Θk. Following the proof of Theorem 2 and
using the lower bounds in Lemma 8 we have
αk,j =
i in Fig. 4(c)
ρki,j +
i in Fig. 4(d)
ρki,j
≥ 2(0.3537 + 0.0245 + 0.0245 + 0.0071) +
(0.0838 + 0.0838 + 0.0138 + 0.0138) = 1.0148. (13)
Thus, α = maxk maxj∈Θk αk,j ≥ 1.0148 > 1. Hence, we cannot use Theorem 3 to show
rapid mixing of systematic scan with six colours and 3×3-blocks.
A natural question is whether we can show rapid mixing using even bigger blocks. It
seems possible to do this although the computations rapidly become intractable as the
block size increases. Already with a 3×3-block the number of boundary colourings we
need to consider (after removing isomorphisms) is in excess of 106 and for each boundary
colouring there are more than 107 colourings of the block to consider. In addition to
simply generating the distributions on colourings of the block, the time it would take
to actually construct the required couplings, as we did in the proof of Lemma 4, would
also increase. Finally when using a larger block size, different positions of vertex j in
the block need to be considered whereas we could make use of to the symmetry of the
2×2-block to only consider one position of vertex j in the block. If different positions of
j have to be considered this has to be captured in the construction of the coupling and
would likely require more computations. The conclusion is that in order to show rapid
mixing for six colours of systematic scan on the grid we would most likely have to rely
on a different approach than the one presented in this paper.
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Introduction
Preliminaries and statement of results
Context and related work
Bounding the mixing time of systematic scan
Constructing the coupling by machine
Representing a coupling as a bipartite graph
The proof of Lemma ??
Partial results for 6-colourings of the grid
Establishing lower bounds for 22 blocks
Bigger blocks
|
0704.1626 | Magnetic exponents of two-dimensional Ising spin glasses | Magnetic exponents of two-dimensional Ising spin glasses
F. Liers1 and O. C. Martin2
Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969 Köln,Germany.
Univ Paris-Sud, UMR8626, LPTMS, Orsay, F-91405; CNRS, Orsay, F-91405, France.
(Dated: October 30, 2018)
The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using
exact ground state determination, we extract the properties of clusters flipped when increasing con-
tinuously a uniform field. We show that these clusters have many holes but otherwise have statistical
properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization ex-
ponent δ ≈ 1.30 ± 0.02 using lattice sizes up to 80 × 80; this is compatible with the droplet model
prediction δ = 1.282. The reason for previous disagreements stems from the need to analyze both
singular and analytic contributions in the low-field regime.
PACS numbers: 75.10.Nr, 75.40.-s, 75.40.Mg
Spin glasses [1, 2] have been the focus of much interest
because of their many remarkable features: they undergo
a subtle freezing transition as temperature is lowered,
their relaxational dynamics is slow (non-Arrhenius), they
exhibit ageing, memory effects, etc. Although there are
still some heated disputes concerning three-dimensional
spin glasses, the case of two dimensions is relatively
consensual, at least in the absence of a magnetic field.
Indeed, two recent studies [3, 4] found that the ther-
mal properties of two-dimensional Ising spin glasses with
Gaussian couplings agreed very well with the predictions
of the scaling/droplet pictures [5, 6]. Interestingly, the
situation in the presence of a magnetic field remains un-
clear; in particular, some Monte Carlo simulations [7] and
basically all ground state studies [8, 9, 10] seem to go
against the scaling/droplet pictures. Nevertheless, since
spin glasses often have large corrections to scaling, the
apparent disagreement with the droplet picture resulting
from these studies may be misleading and tests in one
dimension give credence to this claim [11].
In this study we use state of the art algorithms for de-
termining exact ground states in the presence of a mag-
netic field and treat significantly larger lattice sizes than
in previous work. By finding the precise points where
the ground states change as a function of the field, we
extract the excitations relevant in the presence of a field
which can then be compared to the zero-field droplets.
Although for small size lattices we agree with previous
studies, at our larger ones a careful analysis, taking into
account both the analytic and the singular terms, gives
excellent agreement with the droplet picture.
The model and its properties — We work on an L×L
square lattice having Ising spins on its sites and couplings
Jij on its bonds. The Hamiltonian is
H({σi}) ≡ −
Jijσiσj −B
σi (1)
The first sum runs over all nearest neighbor sites using
periodic boundary conditions to minimize finite size ef-
fects. The Jij are independent random variables of either
Gaussian or exponential distribution.
It is generally agreed that two-dimensional spin glasses
have a unique critical point at T = B = 0. There, the
free energy is non-analytic and in fact, standard argu-
ments [12] suggest that as T → 0 and B → 0 the free
energy goes as βF (L, β) ∼ βE0+Gs(TLyT , BLyB ) where
E0 is the ground-state energy, β the inverse temperature,
while yT and yB are the thermal and magnetic exponents.
Previous work when B = 0 is compatible with this form
and in fact also agrees with the scaling/droplet picture
of Ising spin glasses in which one has yT = −θ ≈ 0.282.
The stumbling block concerns the behavior when B 6= 0;
there, the droplet prediction in general dimension d is
yB = yT + d/2 (2)
but the numerical evidence for this is muddled at best. It
is thus worth reviewing the hypotheses assumed by the
droplet model so that they can be tested directly.
We begin with the fact that in any dimension d, a mag-
netic field destabilizes the ground state beyond a charac-
teristic length scale ξB. To see this, consider an infinites-
imal field and zero-field droplets of scale ℓ. These are
expected to be compact. The interfacial energy of such
droplets is O(ℓθ) while their total magnetization goes as
ℓd/2. The magnetic and interfacial energy are then bal-
anced when B reaches a value O(1/ℓd/2−θ): at that value
of the field, some of the droplets will flip and the ground
state will be destabilized. We then see that for each field
strength there is an associated magnetic length scale ξB
ξB ≈ B
−θ (3)
This leads to the identification yB = d/2−θ in agreement
with Eq. 2, giving yb ≈ 1.282 at d = 2.
The droplet model also predicts the scaling of the mag-
netization in the B → 0 limit via the exponent δ:
m(B) ∼ B1/δ (4)
If this form also holds for infinitesimal fields at finite
L, we can consider the field B∗ for which system-size
http://arxiv.org/abs/0704.1626v2
droplets flip; this happens when B = O(1/LyB) and then
the magnetization is O(L−d/2), the droplets having ran-
dom magnetizations. This leads to m(B∗) ∼ L−d/2 and
m(B∗) ∼ [1/LyB ]1/δ so that
dδ = 2yB (5)
Although the droplet model arguments are not proofs,
they seem quite convincing. Nevertheless, the numerical
studies measuring δ do not give good agreement with the
prediction δ = 1.282. For instance, using Monte Carlo at
“low enough” temperatures, Kinzel and Binder [7] find
δ ≈ 1.39. Since thermalization is difficult at low tem-
peratures, it is preferable to work directly with ground
states, at least when that is possible. This was done
by three independent groups [8, 9, 10] with increasing
power, leading to δ ≈ 1.48, δ ≈ 1.54 and δ ≈ 1.48. Taken
together, these studies show a real discrepancy with the
droplet prediction. To save the droplet model from this
thorny situation, one can appeal to large corrections to
scaling. Such potential effects have been considered [11]
in dimension one where it was shown that ξB was poorly
fitted by a pure power law unless the fields were very
small. Here we revisit the two-dimensional case to reveal
either the size of the corrections to scaling or a cause for
the break down of the droplet reasoning.
Computation of ground states — We determine the
exact ground state of the Hamiltonian (1) by computing
a maximum cut in the graph of interactions [13], a promi-
nent problem in combinatorial optimization. Whereas it
is polynomially solvable on two-dimensional grids with-
out a field and couplings bounded by a polynomial in the
size of the input, it is NP-hard with an external field. In
practice, we rely on a branch-and-cut algorithm [14, 15].
Let the ground state at a field B be denoted as
{σ(G)(B)}. To study the magnetization, we computed
the ground states at increasing values of B, in steps of
size 0.02. When focusing instead on the flipping clusters,
we had to determine the intervals in which the ground
state was constant and in what manner it changed when
going from one interval to the next. In Fig. 1 we show
the associated piecewise constant magnetization curves
for three samples of the disorder variables Jij at L = 10.
To get the sequence of intervals or break points associ-
ated with such a function exactly, we start by computing
the ground state in zero field. By applying postoptimal-
ity analysis from linear programming theory, we deter-
mine [10, 15] a range ∆B such that the ground state at a
field B remains the optimum in the interval [B,B+∆B].
We reoptimize at B+∆B+ ǫ, with ǫ being a sufficiently
small number. By repeatedly applying this procedure,
we get a new ground state configuration, and increase B
until all spins are aligned with the field. This procedure
works for system sizes in which the branch-and-cut pro-
gram can prove optimality without branching, i.e., with-
out dividing the problem into smaller sub-problems. If
the algorithm branches (this occurs only for the largest
0 0.5 1 1.5 2
sample 1
sample 2
sample 3
FIG. 1: Magnetization as a function of B for three typical
L = 10 samples.
system sizes studied here), we apply a divide-and-conquer
strategy for determining {σ(G)(B)} in an interval, say
[B1, B2]. For a fixed configuration the Hamiltonian (1) is
linear in the field, the slope being the system’s magneti-
zation. Let f1, f2 be the two linear functions associated
with {σ(G)(B1)} and {σ(G)(B2)}. If f1 and f2 are equal,
we are done. Otherwise, we determine the field B3 at
which the functions intersect and recursively solve the
problem in the intervals [B1, B3] and [B3, B2].
A typical sample at L = 80 requires about 2 hours of
cpu on a work station for determining the ground states
when B goes through the multiples of 0.02. The more
time consuming computation of the exact break points
takes about 4 hours on typical samples with L = 60,
but less than a minute if L ≤ 30 because the ground-
state determinations are fast and branching almost never
arises. For our work, we considered mainly the case of
Gaussian Jij , analyzing 2500 samples at L = 80, 5000
at L = 70, and from 2000 to 11000 instances for sizes
L = 60, 50, 40, 30, 24, 20, 14. We also analyzed a smaller
number of samples for Jij taken from an exponential
distribution; exponents showed no significant differences
when comparing to the Gaussian case.
The exponent δ — Given the Hamiltonian, it is easy
to see that for each sample the magnetization (density)
mJ(B) =
must be an increasing function of B. (The index J on the
magnetization is to recall that it depends on the disorder
realisation, but in the large L limit mJ is self averag-
ing; also, without loss of generality, we shall work with
B > 0.) At large fields mJ saturates to 1, while at low
fields, its growth law must be above a linear function of
B. Indeed, for continuous Jij , the distribution of local
fields has a finite density at zero and so small clusters
of spins will flip and will lead to a linear contribution to
the magnetization. A more singular behavior is in fact
predicted by the droplet model since δ > 1, indicating
that the system is anomalously sensitive to the magnetic
field perturbation.
If B is not too small, the convergence to the thermo-
dynamic limit (L → ∞) is rapid, and in fact one expects
exponential convergence in L/ξB. We should thus see an
envelope curve m(B) appear as L increases; to make a
power dependence on B manifest, we show in Fig. 2 a
log-log plot of the ratio m(B)/B1/δ where δ is set to its
droplet scaling value of 1.282. For that value of δ there
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
0.01 0.1 1
FIG. 2: Magnetization divided by B1/δ as a function of B; the
B1/1.45 line is to guide the eye. From top to bottom, L = 14,
20, 24, 30, 40, 50, 60, 70, and 80. Inset: m − χ1B divided
by B1/δ as a function of B. (Same L and symbols as in main
part of the figure.) In both cases, δ is set to its droplet model
value, δ = 1.282.
is not much indication that a flat region is developping
when L increases, while at L = 50 a direct fit to a power
gives δ = 1.45 (cf. line displayed in the figure to guide
the eye), as found in previous work [8, 9, 10]. The prob-
lem with this simple analysis is that m has both analytic
and non-analytic contributions; to lowest order we have
m = χ1B + χSB
1/δ + . . . (7)
Although χ1B is sub-dominant, it is far from negligible in
practice; for instance for it to contribute to less than 10%
of m, one would need B < (0.1χS/χ1)
1/0.282. This could
easily mean B < 10−3 for which there would be huge
finite size effects since L would then be much smaller than
the magnetic length ξB . We thus must take into account
the term χ1B; we have done this, adjusting χ1 so that
(m−χ1B)/B1/δ has an envelope as flat as possible. The
result is displayed in the inset of Fig. 2, showing that the
droplet scaling fits very well the data as long as the χ1B
term is included. In fact, direct fits to the form of Eq. 7
give δ in the range 1.28 to 1.32 depending on the sets of
L’s included in the fits.
The clusters that flip are like zero-field droplets —
The fundamental hypothesis in the droplet argument re-
lating δ or yB to θ is the fact that in an infinitesimal field
one flips droplets defined in zero field, droplets which are
compact and have random (except for the sign) magneti-
zations. We therefore now focus on the properties of the
actual clusters that are flipped at low fields.
At zero field, the droplet of lowest energy almost al-
ways is a single spin (this follows from the large number
of such droplets, in spite of their typically higher energy).
Thus as the field is turned on, the ground state changes
first mainly via single spin flips, and when large clusters
do flip (they finally do so but at larger fields), they nec-
essarily have many “holes” and thus do not correspond
exactly to zero-field droplets. This is not a problem for
the droplet argument as long as these clusters are com-
pact and have random magnetizations. To test this, we
1.14
1.16
1.18
1.22
1.24
10 20 30 40 50 60
FIG. 3: The cluster magnetization divided by the square root
of cluster volume — for the largest cluster flipped in each
sample — is insensitive to L. Inset: The clusters’s mean
surface scales as LdS with dS ≈ 1.32.
consider for each realization of the Jij disorder the largest
cluster that flips during the whole passage from B = 0
to B = ∞. According to the droplet picture, this clus-
ter should contain a number of spins V that scales as L2
(compactness) and have a total magnetization M that
scales as
V (randomness). This is confirmed by our
data where we find M/V ∼ 2/L; in Fig. 3 we plot the
disorder mean ofM/
V for increasing L; manifestly, this
mean is remarkably insensitive to L. Similar conclusions
apply to V/L2. For completeness, we show in the inset of
the figure that the surface of these clusters, defined as the
number of lattice bonds connecting them to their com-
plement, grows as LdS with dS ≈ 1.32; this is to be com-
pared to the value dS = 1.27 for zero-field droplets [16],
in spite of the fact that our clusters have holes. All in
all, we find that the clusters considered have statistical
properties that are completely compatible with those as-
sumed in the droplet scaling argument, thereby directly
validating the associated hypotheses.
The magnetic exponent yB and finite size scaling of the
magnetization — One can also measure the exponent
yB directly via the magnetic length which scales as ξB ∼
B−1/yB . For each sample, define B∗J as that field where
the ground state changes by the largest cluster of spins as
described in the previous paragraph. Since these clusters
involve a number of spins growing as L2, we can identify
J ) with L. Let B
∗ be the disorder average of B∗J ;
then B∗ ∼ L−yB from which we can estimate yB. We
find that a pure power with yB set to its value in the
droplet picture describes the data quite well; in the inset
of Fig. 4 we display the product L1.282B∗ as a function of
0.01 0.04 0.07
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
0.1 1 10
FIG. 4: Inset: Field B∗ times L1/δ as a function of 1/L shows
a limit at large L as expected in the droplet model (δ = 1.282).
Main figure: Data collapse plot exhibiting finite size scaling
of the singular part of the magnetization (L = 10, 14, 20, 24,
30, 40, 50, and 60).
1/L and see that the behavior is compatible with a large
L limit with O(1/L) finite size effects. Direct fits to the
form B∗(L) = uL−yB(1+v/L) give yBs in the range 1.28
to 1.30 depending on the points included in the fit.
Given the magnetic length, one can perform finite size
scaling (FSS) on the magnetization data m(B,L). Since
FSS applies to the singular part of an observable, we
should have a data collapse according to
m(B,L)− χ1B
m(B∗, L)− χ1B∗
= W (B/B∗) (8)
W being a universal function, W (0) = O(1) and W (x) ∼
x1/δ at large x. Using the value of χ1 previously deter-
mined, we display in Fig. 4 the associated data. The
collapse is excellent and we have checked that this also
holds when the Jij are drawn from an exponential distri-
bution. Added to the figure is the function x1/δ to guide
the eye (δ = 1.282 as predicted by the droplet model).
Conclusions — We have investigated the 2d Ising
spin glass with Gaussian and exponential couplings at
zero temperature as a function of the magnetic field.
The magnetization exponent δ can be measured; previ-
ous studies did not find good agreement with the droplet
model prediction δ = 1.282 because the analytic con-
tributions to the magnetization curve were mishandled,
while in this work we found instead 1.28 ≤ δ ≤ 1.32.
We also performed a direct measurement of the magnetic
length, obtaining for the associated exponent 1.28 ≤
yB ≤ 1.30, again in excellent agreement with the droplet
prediction. With this length we showed that finite size
scaling is realized without going to infinitesimal fields or
huge lattices. Finally, we validated the hypotheses un-
derlying the arguments of the droplet model inherent to
the in-field case; we find in particular that in the low-field
limit the spin clusters that are relevant are compact and
have random magnetizations. In summary, by combin-
ing improved computational techniques and greater care
in the analysis, we have lifted the discrepancy on the
magnetic exponents that has existed for over a decade
between numerics and droplet scaling.
We thank T. Jorg for helpful comments. The computa-
tions were performed on the cliot cluster of the Regional
Computing Center and on the scale cluster of E. Speck-
enmeyer’s group, both in Cologne. FL has been sup-
ported by the German Science Foundation in the projects
Ju 204/9 and Li 1675/1 and by the Marie Curie RTN
ADONET 504438 funded by the EU. This work was sup-
ported also by the EEC’s HPP under contract HPRN-
CT-2002-00307 (DYGLAGEMEM).
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|
0704.1627 | Thin elastic shells with variable thickness for lithospheric flexure of
one-plate planets | Thin elastic shells with variable thickness
for lithospheric flexure of one-plate planets
Mikael Beuthe
Royal Observatory of Belgium, Brussels, Belgium. E-mail: [email protected]
Abstract
Planetary topography can either be modeled as a load supported by the lithosphere, or as a
dynamical effect due to lithospheric flexure caused by mantle convection. In both cases the response
of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells.
On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the
flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell
have only been derived under the assumption of a shell of constant thickness. However local studies
of gravity and topography data suggest large variations in the thickness of the lithosphere. In this
article we obtain the scalar flexure equations governing the deformations of a thin spherical shell with
variable thickness or variable Young’s modulus. The resulting equations can be solved in succession,
except for a system of two simultaneous equations, the solutions of which are the transverse deflection
and an associated stress function. In order to include bottom loading generated by mantle convection,
we extend the method of stress functions to include loads with a toroidal tangential component. We
further show that toroidal tangential displacement always occurs if the shell thickness varies, even
in the absence of toroidal loads. We finally prove that the degree-one harmonic components of
the transverse deflection and of the toroidal tangential displacement are independent of the elastic
properties of the shell and are associated with translational and rotational freedom. While being
constrained by the static assumption, degree-one loads can deform the shell and generate stresses.
The flexure equations for a shell of variable thickness are useful not only for the prediction of the
gravity signal in local admittance studies, but also for the construction of stress maps in tectonic
analysis.
1 Introduction
Terrestrial planets are flattened spheres only at first sight: close-up views reveal a rich topography with
unique characteristics for each planet. Unity in diversity is found by studying the support mechanism
for topographic deviations from the hydrostatic planetary shape. A simple mechanism, called isostasy,
postulates mountains floating with iceberg-like roots in the higher density mantle. Another simple mech-
anism assumes that mountains stand on a rigid membrane called the mechanical lithosphere. The two
simple models predict very different gravity signals, from very weak in the first to very strong in the
second. The truth lies in-between: an encompassing model views topographic structures as loads on an
elastic shell with finite rigidity, called the elastic lithosphere (the elastic lithosphere is a subset of the
mechanical lithosphere). The rigidity depends on the elastic properties of the rocks and on the apparent
elastic thickness of the lithosphere. The latter parameter is the main objective of many studies, since its
value can be related to the lithospheric composition and temperature. Another important application of
the model of lithospheric flexure is the determination of stress maps, which can then be compared with
the observed distribution of tectonic features.
http://arxiv.org/abs/0704.1627v2
Besides the important assumption of elasticity, the model of lithospheric flexure is often simplified
by two approximations. The first one states that the area to be analyzed is sufficiently small so that the
curved lithospheric shell can be modeled as a flat plate. The second approximation states that the shell
(or plate) is thin, which means that deformations are small and that the elastic thickness is small with
respect to the wavelength of the load.
On Earth, the model of lithospheric flexure has been very successful for the understanding of the
topography of the oceanic plates, whereas the analysis of continental plates is fraught with difficulties
due to the very old and complex structure of the continents. As far as we know, plate tectonics do not
occur at the present time on other terrestrial planets, which are deemed one-plate planets [Solomon ,
1978], although the term single-shell would fit better because of the curvature. A single shell can support
loads of much larger extent and greater weight, the best example of which is the huge Tharsis volcanic
formation covering a large portion of Mars. Indeed such a load cannot be supported by the bending
moments present in a thin flat plate, whereas it can be supported by stresses tangent to the shell: the
shell acts as a membrane.
The first application of thin shell theory to a planet was done by Brotchie and Silvester [1969] for
the lithosphere of the Earth (see also Brotchie [1971]). However the approximation of flat plate theory
was seen to be sufficient when it was understood that the lithosphere of the Earth is broken into several
plates [Tanimoto, 1998]. Brotchie and Silvester [1969] do not consider tangential loads and their flexure
equation only includes dominant terms in derivatives; their equation is thus a special case of the equations
of Kraus [1967] and Vlasov [1964] discussed below. The articles reviewed hereafter use the thin shell
theory of Kraus or Vlasov unless mentioned otherwise.
On the Moon, Solomon and Head [1979] used Brotchie’s equation to study displacement and stress
in mare basins. Turcotte et al. [1981] estimated gravity-topography ratios for the mascons and discussed
the type of stress supporting topographic loads. Arkani-Hamed [1998] modeled the support of mascons
with Brotchie’s equation. Sugano and Heki [2004] and Crosby and McKenzie [2005] estimated the elastic
thickness of the lithosphere from Lunar Prospector data.
On Mars, the dominance of the Tharsis rise in the topography led to numerous applications of
the theory of thin elastic shells to lithospheric flexure. Thurber and Toksöz [1978], Comer et al. [1985],
Hall et al. [1986] and Janle and Jannsen [1986] used Brotchie’s equation to estimate the lithospheric
thickness under Martian volcanoes. Turcotte et al. [1981] studied the transition between bending and
membrane regimes. Willemann and Turcotte [1982] analyzed the lithospheric support of Tharsis. Sleep and Phillips
[1985] analyzed the membrane stress distribution on the whole surface. Banerdt et al. [1992], Banerdt and Golombek
[2000] and Phillips et al. [2001] used a model of lithospheric flexure including membrane support, bending
stresses and tangent loads [Banerdt , 1986] in order to study the global stress distribution. Arkani-Hamed
[2000] determined the elastic thickness beneath large volcanoes with Brotchie’s equation whereas Johnson et al.
[2000] estimated the elastic thickness beneath the North Polar Cap. Using local admittance analysis with
spatiospectral methods, McGovern et al. [2002] determined the elastic thickness at various locations [see
also McGovern et al., 2004]. McKenzie et al. [2002] made local admittance analyses of line-of-sight grav-
ity data both with flat plates and with spherical shell models. Turcotte et al. [2002] used the spherical
shell formula for a one-dimensional wavelet analysis of the admittance in order to determine the average
elastic thickness of the lithosphere. Zhong and Roberts [2003] and Lowry and Zhong [2003] studied the
support of the Tharsis rise with an hybrid model including the flexure of a thin elastic shell as well as the
internal loading of a thermal plume in the mantle. Belleguic et al. [2005] determined the elastic thickness
and the density beneath large volcanoes. Searls et al. [2006] investigated the elastic thickness and the
density beneath the Utopia and Hellas basins.
Venus is considered as a one-plate planet but does not have giant volcanic or tectonic structures
comparable to Tharsis. The spherical shell model has thus not been used as often for Venus as for Mars.
Banerdt [1986] studied the global stress distribution. Janle and Jannsen [1988] and Johnson and Sandwell
[1994] used Brotchie’s equation to estimate the lithospheric thickness in various locations. Sandwell et al.
[1997] computed global strain trajectories for comparison with observed tectonics. Lawrence and Phillips
[2003] inverted the admittance in order to estimate the elastic thickness and mantle density anomalies
over two lowland regions and one volcanic rise. Anderson and Smrekar [2006] established a global map
of the elastic thickness based on local admittance analysis. Mercury’s topography and gravity fields are
not yet known well enough to warrant the application of a thin elastic shell model. We refer to Wieczorek
[2007] for a review of recent results regarding the lithosphere of terrestrial planets.
The elastic thickness of the lithosphere is not at all homogeneous over the surface of a planet. For
example, McGovern et al. [2004] (for Mars) and Anderson and Smrekar [2006] (for Venus) find litho-
spheric thickness variations of more than 100 km. The former study explains the variation in lithospheric
thickness in terms of different epochs of loading, as the lithosphere is thickening with time. Other studies
however inferred that spatial variations in lithospheric thickness on Mars are as important as temporal
variations [Solomon and Head , 1982, 1990; Comer et al., 1985]. Thin spherical shell models have always
been applied with a constant elastic thickness for the whole lithosphere. Local studies are done by win-
dowing both the data (gravity and topography) and the model predictions for gravity [Simons et al.,
1997; Wieczorek and Simons , 2005]. The assumptions behind these methods are that the elastic thick-
ness is constant within the window and that the area outside the window can be neglected. The first
assumption is of course true for a small enough window, but the size of the window is limited from below
by the resolution of the data [Wieczorek and Simons , 2005]. Even if the first assumption were true, the
second assumption is violated in two ways (unless the elastic thickness is spatially constant). First, the
deformation of the shell within the window as well as the associated stress field are both modified if the
elastic thickness is changed in the area outside the window. Second, the value of the predicted gravity
field within the window depends on the shell deflection outside the window.
These reasons make it interesting to develop a model of the lithospheric flexure for a spherical shell
of variable thickness. Although a full inversion of the gravity and topography data is impractical with
such a model because of the huge size of the parameter space, other applications are of high interest. For
example a two-stage inversion can be considered: in the first stage a constant elastic thickness is assumed,
and the resulting values are used in the second stage as a starting point for an inversion with variable
elastic thickness (the parameter space can also be constrained with an a priori). Moreover this model
can be used to produce synthetic data and thus allows us to check the validity of inversions assuming a
constant elastic thickness. Finally, stress and strain fields can be computed for given variations of the
elastic thickness, with the aim of comparing stress and strain maps with tectonic features.
General equations governing the deformations of a thin elastic shell have been given by various
authors [e.g. Love, 1944; Vlasov , 1964; Kraus, 1967]. However the possibility of variable shell thickness
is only considered at the early stage where the strain-displacement relationships, Hooke’s law and the
equilibrium equations are separately derived for the thin shell. The combination of these three sets of
equations into a unique equation for the transverse deflection is made under the restriction of constant
elastic thickness, ‘owing to the analytical complications which would otherwise arise’ [see Kraus, 1967,
p. 199].
This article is dedicated to the derivation of the minimum set of equations governing the deformations
of a thin elastic spherical shell with variable thickness. Using Kraus’ method of stress functions, we find
that the transverse deflection is the solution of a simultaneous system of two differential equations of the
fourth order. Contrary to the case of constant thickness, these equations cannot be combined due to
the presence of products of derivatives of the thickness and derivatives of the deflection or of the stress
function. In order to include bottom loading generated by mantle convection, we extend the method of
stress functions to include toroidal tangential loads, which were not considered by Kraus [1967]. Non-
toroidal loading is for example generated by tangential lithostatic forces whereas toroidal loading could
be due to mantle flow generating drag at the base of the lithosphere (mantle flow also produces non-
toroidal loading). With applications to tectonics in mind, we derive the equations relating the tangential
displacements and the stresses to the transverse deflection and the stress functions. We further show
that toroidal tangential displacement occurs even if there is no toroidal loading (unless the shell thickness
is constant). Finally we prove three properties specific to the degree-one harmonic components: (1)
the degree-one transverse deflection and the degree-one toroidal tangential displacement drop from the
elasticity equations because they represent rigid displacements of the whole shell, (2) the transverse and
tangential components of degree-one loads are related so that the shell does not accelerate, (3) degree-
one loads can deform the shell and generate stresses. Though our aim is to introduce a variable shell
thickness, the final equations are also valid for a variable Young’s modulus (Poisson’s ratio must be kept
constant).
Another way to take into account variations of the lithospheric thickness consists in treating the
lithosphere as a three-dimensional spherical solid that is elastic [Métivier et al., 2006] or viscoelastic [e.g.
Zhong et al., 2003; Latychev et al., 2005; Wang and Wu, 2006]. The resulting equations are exact (no
thin shell approximation) and can be solved with finite element methods. The thin shell assumption
is probably satisfied for known planetary lithospheres; in case of doubt, it is advisable to compare the
results of thin shell theory with three-dimensional models assuming constant elastic thickness: static
thick shell models [e.g. Banerdt et al., 1982; Janes and Melosh , 1990; Arkani-Hamed and Riendler , 2002]
or time-dependent viscoelastic models [e.g. Zhong and Zuber , 2000; Zhong , 2002]. The advantage of thin
shell equations is their two-dimensional character, making them much easier to program and quicker to
solve on a computer. Solving faster either gives access to finer two-dimensional grids or allows to examine
a larger parameter space.
We choose to work within the formalism of differential calculus on curved surfaces, without which
the final equations would be cumbersome. Actually the only tool used in this formalism is the covariant
derivative, which can be seen by geophysicists as just a way of combining several terms into one ‘deriva-
tive’. All necessary formulas are given in the Appendix. The possibility of writing a differential equation
in terms of covariant derivatives (in a tensorial form) also provides a consistency check. The presence of
derivatives that cannot be included into covariant derivatives is simply forbidden. This is not without
meaning for differential equations of the fourth order including products of derivatives.
In section 2, we show how to obtain the strain-displacement relationships, Hooke’s law and the
equilibrium equations for a thin spherical shell. These equations are available in the literature for the
general case of a thin shell [e.g. Kraus, 1967], but we derive them for the spherical case in a simpler way,
starting directly with the metric for the spherical shell. We examine in detail the various approximations
made to obtain the thin shell theory of flexure, refraining until the end from taking the ‘thin shell’
quantitative limit in order to ascertain its influence on the final equations. In section 3, we use the
method of stress functions to obtain the flexure equations governing the displacements and the stresses.
In section 4, we give the final form of the flexure equations in the thin shell approximation. We also study
the covariance and the degree-one projection of the flexure equations. In section 5, we examine various
limit cases in which the flexure equations take a simpler form: the membrane limit, the Euclidean limit
and the limit of constant thickness.
2 Fundamental equations of elasticity
2.1 Three-dimensional elasticity theory
Linear elasticity theory is based on three sets of equations. We directly state them in tensorial form for
an isotropic material, since they are derived in Cartesian coordinates in many books [e.g. Ranalli , 1987;
Synge and Schild , 1978]. Recall that, in tensorial notation, there is an implicit summation on indices
that are repeated on the same side of an equation. The first set of equations includes strain-displacement
relationships:
ǫij =
(ui,j + uj,i) , (1)
where ǫij is the infinitesimal strain tensor and ui are the finite displacements. The ‘comma’ notation
denotes the spatial derivative (see Appendix 7.2).
The second set includes the constitutive equations of elasticity, or Hooke’s law, relating the strain
tensor and the stress tensor σij :
σij = λ ǫ δij + 2Gǫij , (2)
where ǫ = ǫ11 + ǫ22 + ǫ33 and δij = 1 if i= j, otherwise it equals zero. The parameter λ is known as the
first Lamé constant. The parameter G is known as the second Lamé constant, or the shear modulus, or
the modulus of rigidity. Boundary conditions are given by
σij nj = Ti , (3)
with nj being the normal unit vector of the surface element and Ti being the surface force per unit area.
The third set includes equations of motion which reduce to equilibrium equations for stresses if the
problem is static:
σij,j = 0 . (4)
Body forces, such as gravity, are assumed to be absent. Both strain and stress tensors are symmetric:
ǫij = ǫji and σij = σji.
These three-dimensional equations do not yet have the right form for the description of the defor-
mations of a two-dimensional spherical shell. Various methods have been used to generate appropriate
equations. Love [1944] and Timoshenko and Woinowsky-Krieger [1964] derive strain-displacements and
equilibrium equations directly on the surface of the sphere (the latter only for the special cases of no
bending or axisymmetrical loading). Sokolnikoff [1956] derives strain-displacement equations in three-
dimensional curvilinear coordinates using an arbitrary diagonal metric but states without proof the equi-
librium equations in curvilinear coordinates. Kraus [1967] uses Sokolnikoff’s form of strain-displacement
equations and derives equilibrium equations for an arbitrary two-dimensional surface using Hamilton’s
principle (i.e. virtual displacements).
Instead of directly deriving equations on the two-dimensional surface of the sphere, we will first
obtain their form in three-dimensional curvilinear coordinates and then restrict them to the surface of
the sphere. The first step can elegantly be done through the use of tensors [Synge and Schild , 1978].
Equations (1)-(4) are tensorial with respect to orthogonal transformations, but not with respect to other
coordinate transforms (one reason being the presence of usual derivatives). In other words, they are
only valid in Cartesian coordinates. In a three-dimensional Euclidean space, tensorial equations have
a simplified form in Cartesian coordinates because supplementary terms that make them tensorial with
respect to arbitrary coordinate transformations are zero. The missing terms can be reconstructed by using
a set of rules, such as the replacement of usual derivatives by covariant derivatives and the substitution
of tensorial contraction to sum on components. Correspondence rules lead to the following three sets of
equations:
ǫij =
ui|j + uj|i
, (5)
σij = λ ǫ gij + 2Gǫij , (6)
gjk σij|k = 0 , (7)
where ǫ = gkl ǫkl. The notation ui|j denotes the covariant derivative of ui (see Appendix 7.2). The
metric and its inverse are noted gij and g
ij , respectively. Tensorial components cannot be expressed in
a normalized basis (except for Cartesian coordinates) which is more common for physical interpretation
(see Appendix 7.1). Covariant components in equations (5)-(7) are related to components defined in a
normalized basis (written with a hat) by:
gii ûi ,
ǫij =
giigjj ǫ̂ij ,
σij =
giigjj σ̂ij ,
where there is no implicit summation on repeated indices.
In the next section we will introduce additional assumptions in order to restrict the equations to the
two-dimensional surface of a spherical shell.
2.2 Spherical shell
2.2.1 Assumptions of the thin shell theory
Suppose that the two first coordinates are the colatitude θ and longitude ϕ on the surface of the sphere,
whereas the third coordinate ζ is radial. R is the shell radius.
Assumptions of the thin shell theory are [see Kraus, 1967, chap. 2.2]:
1. The shell is thin (say less than one tenth of the radius of the sphere).
2. The deflections of the shell are small.
3. The transverse normal stress is negligible: σζζ = 0.
4. Normals to the reference surface of the shell remain normal to it and undergo no change of length
during deformation: ǫθζ = ǫϕζ = ǫζζ = 0.
The second assumption allows us to use linear equations to describe the deflections. The third and fourth
assumptions are not fully consistent: we refer to Kraus [1967] for more details. We will relax them in
the derivation of the equations for the deflection of a spherical shell. The crucial assumption is σζζ = 0
which is essential for the restriction of Hooke’s law to the two-dimensional shell. As we will see later, σζζ
cannot be zero since it is related to the non-zero transverse load (besides the fact that it is incompatible
with a vanishing transverse strain). What is absolutely necessary is that σζζ ≪ σii for i = (θ, ϕ). In
section 5.3.3, we will show that this condition is satisfied if the wavelength of the load is much larger
than the thickness of the shell.
The reference surface is the middle surface of the shell. With the aim of integrating out the third
coordinate, a coordinate system is chosen so that the radial coordinate ζ is zero on the reference surface.
The metric is given by
ds2 = (R+ ζ)
dθ2 + sin2 θ dϕ2
+ dζ2 . (8)
Christoffel symbols necessary for the computation of the covariant derivatives are given in Appendix 7.3.
2.2.2 Strain-displacement equations
With the metric (8), the strain-displacement equations (5) become
ǫ̂θθ =
(ûθ,θ + ûζ) ,
ǫ̂ϕϕ =
(csc θ ûϕ,ϕ + cot θ ûθ + ûζ) ,
ǫ̂θϕ =
(csc θ ûθ,ϕ − cot θ ûϕ + ûϕ,θ) , (9)
ǫ̂ζζ = ûζ,ζ ,
ǫ̂θζ =
((R + ζ) ûθ,ζ − ûθ + ûζ,θ) ,
ǫ̂ϕζ =
((R + ζ) ûϕ,ζ − ûϕ + csc θ ûζ,ϕ) ,
where all quantities are given in a normalized basis. The fourth assumption of the thin shell theory
implies that the displacements are linearly distributed across the thickness of the shell, with the transverse
displacement being constant. Displacements can thus be expanded to first order in ζ:
(ûθ, ûϕ, ûζ) = (vθ + ζβθ, vϕ + ζβϕ, w) . (10)
The coefficients (vθ , vϕ, w, βθ, βϕ) are independent of ζ: (vθ, vϕ, w) represent the components of the
displacement vector of a point on the reference surface, whereas (βθ, βϕ) represent the rotations of tangents
to the reference surface oriented along the tangent axes. We determine βθ and βϕ by applying the fourth
assumption of the thin shell theory and the expansion (10) to the last two strain-displacement equations
(vθ − w,θ) ,
(vϕ − csc θ w,ϕ) . (11)
The substitution of the expansion (10) into the fourth strain-displacement equation (ǫ̂ζζ = ûζ,ζ) leads to
the condition ǫ̂ζζ = 0, as postulated by the thin shell theory.
After the insertion of the expansion formulas (10) and (11), the first three strain-displacement equa-
tions (9) become
ǫ̂θθ = ǫ
1 + ζ/R
κ0θ ,
ǫ̂ϕϕ = ǫ
1 + ζ/R
κ0ϕ , (12)
2 ǫ̂θϕ = γ
1 + ζ/R
The extensional strains ǫ0θ, ǫ
ϕ and γ
θϕ are defined by
ǫ0θ =
(vθ,θ + w) ,
ǫ0ϕ =
(csc θ vϕ,ϕ + cot θ vθ + w) , (13)
γ0θϕ =
(vϕ,θ − cot θ vϕ + csc θ vθ,ϕ) .
They represent the normal and shearing strains of the reference surface. The flexural strains κ0θ, κ
ϕ and
τ0 are given by
κ0θ = −
O1 w ,
κ0ϕ = −
O2 w , (14)
τ0 = −
O3 w .
They represent the changes in curvature and the torsion of the reference surface during deformation
[Kraus, 1967]. The differential operators O1,2,3 are defined by
+ 1 ,
O2 = csc2 θ
+ cot θ
+ 1 , (15)
O3 = csc θ
− cot θ
The zero upper index in (ǫ0θ, ǫ
θϕ, κ
0) refers to the reference surface and appears in order to
follow Kraus’ notation.
In Love’s theory, one makes the approximation 1
∼= 1R in equations (12). However this does not
simplify the calculations when the shell is a sphere because it has only one radius of curvature (for a
shell with two radii of curvature, this approximation is a great simplification). Our choice to keep the
factor 1
leads to the same results as the theory of Flügge-Lur’e-Byrne, explained in Kraus [1967,
chap. 3.3a] or Novozhilov [1964, p. 53], in which this factor is expanded up to second order. In any
case the choice between the approximation or the expansion of this factor does not affect the equations
(derived in the next section) relating the stress and moment resultants to the strains: they are the same
is approximated to zeroth order, expanded to second order or fully kept.
2.2.3 Hooke’s law
When the metric is diagonal and the basis is normalized, Hooke’s law (6) becomes
σ̂ii = λ
ǫ̂kk + 2G ǫ̂ii (i = j) ,
σ̂ij = 2G ǫ̂ij (i 6= j) .
There is no implicit summation on repeated indices.
The third assumption of the thin shell theory, σ̂ζζ = 0, can be used to eliminate ǫ̂ζζ from Hooke’s
σ̂θθ =
1− ν2
(ǫ̂θθ + ν ǫ̂ϕϕ) ,
σ̂ϕϕ =
1− ν2
(ǫ̂ϕϕ + ν ǫ̂θθ) .
Young’s modulus E and Poisson’s ratio ν are related to Lamé parameters by
(1 + ν)(1 − 2ν)
2(1 + ν)
In principle, the fourth assumption of the thin shell theory, ǫ̂θζ = ǫ̂ϕζ = 0, leads to σ̂θζ = σ̂ϕζ = 0 but
non-vanishing values must be retained for purposes of equilibrium.
Figure 1: Stress resultants and stress couples acting on a small element of the shell. The directions of the
stress resultants (simple arrows) and the rotation sense of the stress couples (double arrows) correspond
to positive components (tensile stress is positive). Loads (qθ, qϕ, q) act on the reference surface.
The substitution of the expansion (12) into the thin shell approximation of Hooke’s law gives
σ̂θθ =
1− ν2
ǫ0θ + νǫ
1 + ζ/R
κ0θ + ν κ
σ̂ϕϕ =
1− ν2
ǫ0ϕ + νǫ
1 + ζ/R
κ0ϕ + ν κ
, (16)
σ̂θϕ =
2(1 + ν)
γ0θϕ +
1 + ζ/R
with ǫ0θ, ǫ
θϕ, κ
ϕ and τ
0 defined by equations (13) and (14). The expressions for σ̂θζ and σ̂ϕζ will
not be needed.
We now integrate the stress distributions across the thickness h of the shell (see Figure 1). The stress
resultants and couples obtained in this way are defined per unit of arc length on the reference surface:
∫ h/2
σ̂ii (1 + ζ/R) dζ (i = θ, ϕ) ,
Nθϕ = Nϕθ =
∫ h/2
σ̂θϕ (1 + ζ/R) dζ ,
∫ h/2
σ̂iζ (1 + ζ/R) dζ (i = θ, ϕ) , (17)
∫ h/2
σ̂ii (1 + ζ/R) ζ dζ (i = θ, ϕ) ,
Mθϕ = Mϕθ =
∫ h/2
σ̂θϕ (1 + ζ/R) ζ dζ .
We evaluate these integrals with the expansion (16). The tangential stress resultants are
Nθ = K
ǫ0θ + ν ǫ
Nϕ = K
ǫ0ϕ + ν ǫ
, (18)
Nθϕ = K
γ0θϕ .
Explicit expressions for the transverse shearing stress resultants Qi are not needed since these quantities
will be determined from the equilibrium equations. The moment resultants are
Mθ = D
κ0θ + ν κ
ǫ0θ + ν ǫ
Mϕ = D
κ0ϕ + ν κ
ǫ0ϕ + ν ǫ
, (19)
Mθϕ = D
The extensional rigidity K and the bending rigidity D are defined by
1− ν2
, (20)
12(1− ν2)
. (21)
Their dimensionless ratio ξ is a large number,
ξ = R2
, (22)
the inverse of which will serve as an expansion parameter for thin shell theory.
2.2.4 Equilibrium equations
With the metric (8), the components θ, ϕ and ζ of the equilibrium equations (7) respectively become
(R+ ζ)
(sin θ σ̂θθ),θ + σ̂θϕ,ϕ − cos θ σ̂ϕϕ + sin θ σ̂ζθ
+ sin θ
(R+ ζ)
= 0 ,
(R + ζ)
(sin θ σ̂θϕ),θ + σ̂ϕϕ,ϕ + cos θ σ̂θϕ + sin θ σ̂ζϕ
+ sin θ
(R+ ζ)
= 0 ,
(R + ζ)
(sin θ σ̂θζ),θ + σ̂ϕζ,ϕ − sin θ (σ̂θθ + σ̂ϕϕ)
+ sin θ
(R+ ζ)
= 0 ,
where the equations have been multiplied by sin θ (R + ζ), (R + ζ) and sin θ (R + ζ)2, respectively. The
stress components are given in a normalized basis.
The integration on ζ of these three equations in the range [−h/2, h/2] yields the equilibrium equations
for the forces:
(sin θ Nθ),θ +Nθϕ,ϕ − cos θ Nϕ + sin θ Qθ +R sin θ qθ = 0 , (23)
(sin θNθϕ),θ +Nϕ,ϕ + cos θ Nθϕ + sin θ Qϕ +R sin θ qϕ = 0 , (24)
(sin θ Qθ),θ +Qϕ,ϕ − sin θ (Nθ +Nϕ)−R sin θ q = 0 , (25)
where qθ and qϕ are the components of the tangential load vector per unit area of the reference surface:
(R+ ζ)
= R2 qi (i = θ, ϕ) .
We choose the convention that tensile stresses are positive (see Figure 1). The transverse load per unit
area of the reference surface is noted q and is taken to be positive toward the center of the sphere:
(R+ ζ)
= −R2 q . (26)
The first two equilibrium equations for the stresses can also be multiplied by ζ before the integration
to yield the equilibrium equations for the moments:
(sin θMθ),θ +Mθϕ,ϕ − cos θMϕ −R sin θ Qθ = 0 , (27)
(sin θMθϕ),θ +Mϕ,ϕ + cos θMθϕ −R sin θ Qϕ = 0 . (28)
We have neglected small terms in
ζ (R+ ζ)
where i = (θ, ϕ). A third equilibrium equation for
the moments exists but has the form of an identity: Mθϕ =Mϕθ.
3 Resolution
3.1 Available methods
At this stage the elastic theory for a thin spherical shell involves 17 equations: six strain-displacement
relationships (13)-14), six stress-strain relations (18)-(19) making Hooke’s law, and five equilibrium equa-
tions (23)-(23) and (27)-(28). There are 17 unknowns: six strain components (ǫ0θ, ǫ
θϕ, κ
three displacements (w, vθ , vϕ), three tangential stress resultants (Nθ, Nϕ, Nθϕ), two transverse shearing
stress resultants (Qθ, Qϕ), and three moment resultants (Mθ,Mϕ,Mθϕ). The three equations (16) are
also needed if the tangential stresses (σ̂θθ, σ̂θϕ, σ̂ϕϕ) are required. The quantities of primary interest to
us are the transverse deflection and the tangential stresses (sometimes tangential strain is preferred, as
in Sandwell et al. [1997] or Banerdt and Golombek [2000]). We thus want to find the minimum set of
equations that must be solved to determine these quantities.
We are aware of two methods of resolution [Novozhilov , 1964, p. 66]. In the first one, we insert the
strain-displacement relationships into Hooke’s law, and substitute in turn Hooke’s law into the equilibrium
equations. This method yields three simultaneous differential equations for the displacements. Once the
displacements are known, it is possible to compute the strains and the stresses. The second method
supplements the equilibrium equations with the equations of compatibility [Novozhilov , 1964, p. 27] that
relate the partial derivatives of the strain components. It is then convenient to introduce the so-called
stress functions [Kraus , 1967, p. 243], without direct physical interpretation, which serve to define the
stress resultants without introducing the tangential displacements. Equations relating the transverse
displacement and the stress functions are then found by applying the third equation of equilibrium and
the third equation of compatibility (it is also possible to use all three equations of compatibility in order
to directly solve for the stress and moment resultants). Once the transverse displacement and the stress
functions are known, stresses can be computed.
If the shell thickness is constant, the deformations of a thin spherical shell can be completely calcu-
lated with both methods. If the shell thickness is variable, the three equations governing displacements,
obtained with the first method, cannot be decoupled and are not easy to solve. Kraus’ method with
stress functions leads to a system of three equations (relating the transverse displacement and the two
stress functions), in which the first equation is decoupled and solved before the other two. This method
thus provides a system of equations much easier to solve and will be chosen in this article.
When solving the equations, one usually assumes from the beginning the large ξ limit, i.e. 1+ ξ ∼= ξ
where ξ is defined by equation (22). We will only take this limit at the end of the resolution. This
procedure will not complicate the computations, since we have to compute anyway many new terms
because of the variable shell thickness.
3.2 Differential operators
We will repeatedly encounter the operators Oi which intervene in the expressions (14) for the flexural
strains (κ0θ, κ
0). Since we are looking for scalar equations, we need to find out how the operators Oi
can be combined in order to yield scalar expressions, i.e. expressions that are invariant with respect to
changes of coordinates on the sphere.
The first thing is to relate the Oi to tensorial operators. Starting from the covariant derivatives on the
sphere ∇i, we construct the following tensorial differential operators of the second degree in derivatives:
Dij = ∇i∇j + gij , (29)
where ∇i denotes the covariant derivative (see Appendix 7.2). These operators give zero when applied
on spherical harmonics of degree one (considered as scalars):
Dij Y1m = 0 (m = −1, 0, 1) . (30)
This property can be explicitly checked on the spherical harmonics (109) with the metric and the formulas
for the double covariant derivatives given in Appendix 7.4.
In two-dimensional spherical coordinates (θ, ϕ), the three operators Oi defined by equations (15) are
related to the operators Dij acting on a scalar function f through
O1 f = Dθθ f ,
O2 f = csc2 θDϕϕ f , (31)
O3 f = csc θDθϕ f .
The operators O1,2,3f actually correspond to normalized Dijf , i.e. Dijf/(
giigjj).
The usual derivatives of the operators Oi satisfy the useful identities (112)-(113) which are proven
in Appendix 7.7. These identities are the consequence of the path dependence of the parallel transport
of vectors on the curved surface of the sphere.
Invariant expressions are built by contracting all indices of the differential operators in their tensorial
form. The indices can be contracted with the inverse metric gij or with the antisymmetric tensor εij (see
Appendix 7.4), which should not be confused with the strain tensor ǫij . In the following, a and b are
scalar functions on the sphere.
At degree 2, the only non-zero contraction of the Dij is related to the Laplacian (104):
∆′a ≡ gij Dij a
= (∆+ 2) a (32)
= (O1 +O2) a
= a,θ,θ + cot θ a,θ + csc
2 θ a,ϕ,ϕ + 2 a .
At degree 4, a scalar expression symmetric in (a, b) is given by
A(a ; b) ≡ [∆′a][∆′b]− [Dij a][Dij b] ,
= [∆ a][∆ b]− [∇i∇j a][∇i∇j b] + [∆ a] b+ a [∆ b] + 2 a b (33)
= [O1 a][O2 b] + [O2 a][O1 b]− 2 [O3 a][O3 b]
= (a,θ,θ + a)
csc2 θ b,ϕ,ϕ + cot θ b,θ + b
csc2 θ a,ϕ,ϕ + cot θ a,θ + a
(b,θ,θ + b)
−2 csc2 θ (a,θ,ϕ − cot θ a,ϕ) (b,θ,ϕ − cot θ b,ϕ) .
where upper indices are raised with the inverse metric: Dij = gikgjlDkl. The action of an operator does
not extend beyond the brackets enclosing it.
If a is constant, A(a ; b) = a∆′b. It is useful to define an associated operator A0 that gives zero if
its first argument is constant:
A0(a ; b) = A(a ; b)− a [∆′b] . (34)
A scalar expression of degree 4 antisymmetric in (a, b) is given by
B1(a ; b) ≡ gij εkl [Dik a] [Djl b]
= gij εkl [∇i∇k a] [∇j∇l b] (35)
= [(O1 −O2) a] [O3b]− [O3a] [(O1 −O2) b]
= csc θ
a,θ,θ − csc2 θ a,ϕ,ϕ − cot θ a,θ
(b,θ,ϕ − cot θ b,ϕ)
− csc θ (a,θ,ϕ − cot θ a,ϕ)
b,θ,θ − csc2 θ b,ϕ,ϕ − cot θ b,θ
We will also need another operator of degree 4:
B2(a ; b) ≡ εij [∇ia ] [∇j ∆′b]
= csc θ
a,θ [∆
′b],ϕ − a,ϕ [∆
′b],θ
. (36)
The sum of the operators B1 and B2 is noted B:
B(a ; b) = B1(a ; b) + B2(a ; b) . (37)
If either a or b is constant, B(a ; b) = 0.
The operators A and B have an interesting property proven in Appendix 7.8: for arbitrary scalar
functions a and b, A(a ; b) and B(a ; b) do not have a degree-one term in their spherical harmonic expansion.
This is not true of A0, B1 and B2.
3.3 Transverse displacement
3.3.1 Resolution of the equations of equilibrium
The first step consists in finding expressions for the moment resultants (Mθ,Mϕ,Mθϕ) in terms of the
transverse displacement and the stress resultants. The extensional strains (ǫ0θ, ǫ
θϕ) can be eliminated
from the equations for stress and moment resultants (18)-(19). The flexural strains (κ0θ, κ
0) depend
on the transverse displacement w through equations (14). We thus obtain
Mθ = −
(O1 + νO2)w +
Mϕ = −
(O2 + νO1)w +
Nϕ , (38)
Mθϕ = −
(1− ν)O3 w +
Nθϕ ,
where the parameter ξ is defined by equation (22).
The second step consists in solving the equilibrium equations for moments in order to find the
transverse shearing stress resultants (Qθ, Qϕ). We substitute expressions (38) into equations (27)-(28).
Knowing that the stress resultants satisfy the equilibrium equations (23)-(24), we obtain new expressions
for Qθ and Qϕ (identities (112)-(113) are helpful):
Qθ = −
(D∆′w),θ − (1− η)Rqθ
(1− ν) (D,θ O2 − csc θD,ϕO3) w −
(η,θNθ + csc θ η,ϕNθϕ) , (39)
Qϕ = −
csc θ (D∆′w),ϕ − (1− η)Rqϕ
(1− ν) (−D,θ O3 + csc θD,ϕ O1) w −
(η,θ Nθϕ + csc θ η,ϕNϕ) . (40)
The operator ∆′ is defined by equation (32) and η is a parameter close to 1:
1 + ξ
. (41)
The third step consists in finding expressions for the stress resultants (Nθ, Nϕ, Nθϕ) in terms of
stress functions by solving the first two equilibrium equations (23)-(24). Let us define the following linear
combinations of the stress and moment resultants:
(Pθ, Pϕ, Pθϕ) =
Mθ, Nϕ +
Mϕ, Nθϕ +
. (42)
We observe that these linear combinations satisfy simplified equations of equilibrium:
(sin θ Pθ),θ + Pθϕ,ϕ − cos θ Pϕ +R sin θ qθ = 0 ,
(sin θ Pθϕ),θ + Pϕ,ϕ + cos θ Pθϕ +R sin θ qϕ = 0 . (43)
Comparing these equilibrium equations with the identities (112)-(113), we see that the homogeneous
equations (i.e. equations (43) with a zero tangential load qθ = qϕ = 0) are always satisfied if
(Pθ, Pϕ, Pθϕ) = (O2,O1,−O3)F , (44)
where F is an auxiliary function called stress function. For the moment, this function is completely
arbitrary apart from being scalar and differentiable.
Particular solutions of the full equations (43) can be found if we express the tangential load qT =
qθθ̂ + qϕϕ̂ in terms of the surface gradient of a scalar potential Ω (consoidal or poloidal component) and
the surface curl of a vector potential V r̂ (toroidal component):
qT = −
∇̄Ω + 1
∇̄ × (V r̂) . (45)
Surface operators are defined in Appendix 7.5, where the terms consoidal/poloidal are also discussed.
The covariant components of qT are (qθ, sin θqϕ) and can be expressed as − 1RΩ,i +
gjkεikV,j , which
gives
qθ = −
Ω,θ +
R sin θ
V,ϕ ,
sin θ qϕ = −
Ω,ϕ −
sin θ
V,θ . (46)
If the tangential load is consoidal (V = 0), a particular solution of equations (43) is given by
(Pθ, Pϕ, Pθϕ) = (1, 1, 0)Ω . (47)
If the tangential load is toroidal (Ω = 0), a particular solution of equations (43) is given by
(Pθ, Pϕ, Pθϕ) = (2O3,−2O3,O2 −O1)H , (48)
where we have introduced a second stress function H which satisfies the constraint
∆′H = −V + V0 , (49)
where V0 is a constant (identities (112)-(113) are useful). This equation allows us to determine the stress
function H if the toroidal source V is known.
The general solution of the equations (43) is given by the sum of the general solution (44) of the
homogeneous equations and the two particular solutions (47)-(48) of the full equations:
(Pθ, Pϕ, Pθϕ) = (O2F +Ω + 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) . (50)
The stress resultants (Nθ, Nϕ, Nθϕ) can now be obtained from (Pθ, Pϕ, Pθϕ) by using equations (42) and
(38):
(Nθ, Nϕ, Nθϕ) = η (Pθ, Pϕ, Pθϕ) + η
(O1 + νO2,O2 + νO1, (1− ν)O3)w , (51)
which finally give
Nθ = η
O2 F +
(∆′ − (1− ν)O2)w +Ω+ 2O3H
Nϕ = η
O1 F +
(∆′ − (1− ν)O1)w +Ω− 2O3H
, (52)
Nθϕ = η
−O3 F + (1 − ν)
O3 w + (O2 −O1)H
The fourth step consists in expressing the third equation of equilibrium (25) in terms of the transverse
displacement w and the stress functions F and H . For this purpose, it is handy to express the transverse
shearing stress resultants (Qθ, Qϕ) in terms of (w,F,H). We thus substitute Nϕ, Nθ, Nθϕ, given by
equations (52), into the expressions for Qθ and Qϕ, given by equations (39)-(40):
Qθ = −
(ηD∆′w),θ +
(ηD),θ O2 − csc θ (ηD),ϕ O3
− (η,θ O2 − csc θ η,ϕO3)F + Ω,θ − (ηΩ),θ − csc θ V,ϕ
− 2 (η,θ O3 + csc θ η,ϕ O2)H + csc θ (η,ϕ ∆′H − η(∆′H),ϕ) , (53)
Qϕ = −
csc θ (ηD∆′w),ϕ +
− (ηD),θ O3 + csc θ (ηD),ϕ O1
− (−η,θ O3 + csc θ η,ϕ O1)F + csc θ (Ω,ϕ − (ηΩ),ϕ) + V,θ
+2 (η,θ O1 + csc θ η,ϕ O3)H − (η,θ ∆′H − η(∆′H),θ) . (54)
The various terms present in Qθ and Qϕ can be classified into generic types according to their
differential structure. Each generic type will contribute in a characteristic way to the third equation
of equilibrium. It is worthwhile to compute the generic contribution of each type before using the full
expressions of the stress resultants with their multiple terms. Identities (112)-(113) are helpful in this
calculation. We thus write the third equation of equilibrium (25) as follows:
I (Qθ ;Qϕ ; 0)− (Nθ +Nϕ)−Rq = 0 , (55)
where the differential operator I is defined for arbitrary expressions (X,Y, Z) by
I(X ;Y ;Z) = csc θ (sin θX),θ + csc θ Y,ϕ − cot θ Z . (56)
This operator must be evaluated for the following generic types present in (Qθ, Qϕ):
I (a,θ ; csc θ a,ϕ ; 0) = ∆a ,
I (− csc θ a,ϕ ; a,θ ; 0) = 0 ,
I (a,θ O2b− csc θ a,ϕ O3b ;−a,θ O3b+ csc θ a,ϕO1b ; 0) = A0(a ; b) , (57)
I (a,θ O3b+ csc θ a,ϕ O2b ;−a,θ O1b− csc θ a,ϕO3b ; 0) = B1(a ; b) ,
I (− csc θ (a,ϕ∆′b − a[∆′b],ϕ) ; a,θ ∆′b− a[∆′b],θ ; 0) = 2B2(a ; b) ,
where a and b are scalar functions. The operators A0, B1 and B2 are defined in section 3.2.
We now substitute (Qθ, Qϕ) and (Nϕ, Nθ) into the third equation of equilibrium (55) and use formulas
(57). We thus obtain the first of the differential equations that relate w and the stress functions F and
∆′ (ηD∆′w)− (1 − ν)A(ηD ;w) +R3 A(η ;F ) + 2R3 B(η ;H) = −R4 q +R3 (∆Ω−∆′(ηΩ)) . (58)
The operators ∆′, A and B are defined in section 3.2.
3.3.2 Compatibility relation
A second equation relating the transverse displacement and the stress functions comes from the compat-
ibility relation which is derived by eliminating (vθ, vϕ) from the strain-displacement equations (13):
sin θ γ0θϕ,ϕ
sin2 θ ǫ0ϕ,θ
+ ǫ0θ,ϕ,ϕ − sin θ cos θ ǫ0θ,θ + 2 sin2 θ ǫ0θ −
sin2 θ
∆′w . (59)
The strain components are related to the stress resultants through Hooke’s law (18). The substitution of
equations (18) into the compatibility equation (59) gives
∆′ (α (Nθ +Nϕ))−
∆′w − (1 + ν)J (αNθ ;αNϕ ;αNθϕ) = 0 . (60)
where α is the reciprocal of the extensional rigidity:
α ≡ 1
K(1− ν2)
. (61)
For arbitrary expressions (X,Y, Z), J (X ;Y ;Z) is defined by
J (X ;Y ;Z) = csc2 θ
sin2 θX,θ
+ Y,ϕ,ϕ + 2 (sin θ Z,ϕ),θ
− cot θ Y,θ + 2 Y . (62)
As in the case of the third equation of equilibrium, it is practical to classify the terms present in
(Nθ, Nϕ, Nθϕ) into generic types and evaluate separately their contribution to the equation of compati-
bility. There are three types of terms for which we must evaluate the operator J (identities (112)-(115)
are helpful in this calculation):
J (a ; a ; 0) = ∆′a ,
J (aO2b ; aO1b ;−aO3b) = A(a ; b) , (63)
J (2aO3b ;−2aO3b ; a (O2 −O1)b) = 2B(a ; b) ,
where a and b are scalar functions. The operators ∆′, A and B are defined in section 3.2.
We now evaluate J in equation (60) with expressions (52) and formulas (63):
J (αNθ ;αNϕ ;αNθϕ) = A(ηα ;F ) +
∆′ (ηαD∆′w)− 1
(1− ν)A(ηαD ;w)
+ ∆′ (ηαΩ) + 2B(ηα ;H) . (64)
The term A(ηαD ;w) can be rewritten with the help of the following equality:
1− ν2
A(ηαD ;w) = ∆′w −A(η ;w) , (65)
since (1 − ν2)ηαD/R2 = η/ξ and (η/ξ),i = −η,i.
We finally substitute Nθ + Nϕ, given by equations (52), into the compatibility equation (60) and
use expressions (64)-(65). We thus obtain the second of the differential equations that relate w and the
stress functions F and H :
∆′ (ηα∆′F )− (1 + ν)A(ηα ;F )−
A(η ;w) − 2(1 + ν)B(ηα ;H) = −(1− ν)∆′(ηαΩ) . (66)
3.4 Tangential displacements
Assuming that the flexure equations (58) and (66) for the transverse displacement w and the stress
functions (F,H) have been solved, we now show how to calculate the tangential displacements.
In analogy with the decomposition of the tangential load in equations (45)-(46), the tangential
displacement can be separated into consoidal and toroidal components:
v = ∇̄S + ∇̄ × (T r̂) , (67)
where S and T are the consoidal and toroidal scalars, respectively. The covariant components of v are
(vθ, sin θ vϕ) and can be expressed as S,i + g
jkεikT,j (see Appendix 7.5), which gives
vθ = S,θ + csc θ T,ϕ ,
sin θ vϕ = S,ϕ − sin θ T,θ . (68)
The strain-displacement equations (13) become
ǫ0θ =
((O1 − 1)S +O3 T + w) ,
ǫ0ϕ =
((O2 − 1)S −O3 T + w) , (69)
γ0θϕ =
(2O3S + (O2 −O1)T ) .
The stress resultants (Nθ, Nϕ, Nθϕ) given by equations (18) become
(∆S + (1 + ν)w − (1− ν) ((O2 − 1)S −O3 T )) ,
(∆S + (1 + ν)w − (1− ν) ((O1 − 1)S +O3 T )) , (70)
Nθϕ =
(2O3 S + (O2 −O1)T ) .
The toroidal potential T cancels in the sum Nθ +Nϕ:
Nθ +Nϕ =
(1 + ν) (∆S + 2w) .
The consoidal displacement potential S can thus be related to (w,F,Ω) by using expressions (52) for the
stress resultants:
∆S = Rηα (1− ν) (∆′F + 2Ω) + η
∆′w − 2w , (71)
where α is defined by equation (61).
It is more difficult to extract the toroidal displacement potential T . When the shell thickness is
constant, decoupled equations for the displacements can be found by suitable differentiation and combi-
nation of the three equilibrium equations (23)-(25) for the stress resultants. This method does not work
if the shell thickness is variable because the resulting equations are coupled. The trick consists in relating
the tangential displacements to (w,F,H) by the way of equations similar to the homogeneous part of the
first two equilibrium equations, but with (Nθ, Nϕ, Nθϕ) replaced by
(Nθ, Nϕ, Nθϕ), so that derivatives
do not mix with derivatives of T . We will thus calculate the following expression in two different
ways (from equations (52) and (70)):
= csc2 θ (−X,ϕ + sin θ Y,θ) ,
where X and Y are defined by
X = sin θ I
Nθϕ ;
Y = sin θ I
Nθϕ ;
Nϕ ;−
The operator I is defined by equation (56).
As before, it is easier to begin with the evaluation of the operator Z for generic contributions:
Z(a ; a ; 0) = 0 ,
Z(aO2b ; aO1b ;−aO3b) = −B(a ; b) ,
Z(2aO3b ;−2aO3b ; a(O2 −O1)b) = 2A(a ; b)−∆′ (a∆′b) ,
On the one hand, the evaluation Z for (Nθ, Nϕ, Nθϕ) given by equations (70) gives
−1− ν
∆∆′ T .
On the other hand, the evaluation of Z for (Nθ, Nϕ, Nθϕ) given by equations (52) gives
+ (1− ν)B
+ 2RA
The equality of the two previous formulas yields the sought equation for T :
∆∆′ T = 2R (1 + ν) (B (ηα ;F )− 2A (ηα ;H) + ∆′ (ηα∆′H)) + 2B (η ;w) . (72)
We have used the relation (η/ξ),i = −η,i.
The equation for T shows that toroidal displacement always occurs when the shell thickness varies.
The right-hand side of equation (72) only vanishes when two conditions are met: (1) there is no toroidal
source (so that the stress function H vanishes) and (2) the shell thickness is constant (so that the terms
in B vanish).
3.5 Stresses
Stresses can be computed from (w,F,H) and Ω by substituting equations (14), (18) and (52) into equa-
tions (16):
σ̂θθ =
(O2 F +Ω+ 2O3H) +
R(1− ν2)
(∆′ − (1− ν)O2)w ,
σ̂ϕϕ =
(O1 F +Ω− 2O3H) +
R(1− ν2)
(∆′ − (1− ν)O1)w , (73)
σ̂θϕ =
(−O3 F + (O2 −O1)H) +
R(1 + ν)
O3w .
Stresses at the surface are obtained by setting ζ = h/2.
4 Flexure equations and their properties
4.1 Thin shell approximation
The flexure equations derived in section 3 already include several assumptions of the thin shell theory,
but not yet the first one stating that the shell is thin. Of course, the three other assumptions can be
seen to be consequences of the first one [see Kraus, 1967, chap. 2.2], but we have not imposed in a
quantitative way the thinness condition on the equations. We thus impose the limit of small h/R or,
equivalently, the limit of large ξ (defined by equation (22)) on the flexure equations for (H,F,w, S, T ).
This procedure amounts to expand η ≈ 1 − 1/ξ (neglecting terms in 1/ξ wherever appropriate) and to
neglect the derivatives of η in equations (49), (58), (66), (71) and (72). We thus obtain the final flexure
equations for the displacements (w, S, T ) and for the stress functions (F,H):
∆′H = −V + V0 , (74)
∆′ (D∆′w)− (1 − ν)A(D ;w) +R3∆′F = −R4 q − 2R3Ω+ R
∆Ω , (75)
∆′ (α∆′F )− (1 + ν)A(α ;F )− 1
∆′w = −(1− ν)∆′(αΩ) + 2(1 + ν)B(α ;H) , (76)
∆S = Rα (1 − ν) (∆′F + 2Ω) + 1
∆′w − 2w , (77)
∆∆′ T = 2R (1 + ν) (B (α ;F )− 2A (α ;H) + ∆′ (α∆′H)) . (78)
Recall that the differential operators ∆′, A and B are defined by equations (32), (33) and (37). The
potential pairs (Ω, V ) and (S, T ) are related to the tangential load and displacement by equations (45)
and (67), respectively. In the second equation, the term R3∆Ω/ξ has been kept since it could be large if
Ω has a short wavelength. For the same reason, the term ∆′w/ξ has been kept in the fourth equation.
In the third and fifth equations, the terms depending on H belong to the right-hand sides since they
can be considered as a source once H has been calculated from the first equation. The same can be said
of the terms depending on w and F in the fourth and fifth equations: they are supposed to be known from
the simultaneous resolution of the second and third equations. The difficulty in solving the equations
thus lies with the two flexure equations for w and F which are linear with non-constant coefficients (all
other equations are linear - in their unknowns - with constant coefficients). Once these two core equations
have been solved, all other quantities are easily derived from them.
The bending rigidity D, defined by equation (21), characterizes the bending regime: the shell locally
bends in a similar way as a flat plate undergoing small deflections with negligible stretching. The pa-
rameter α, defined by equation (61), is the reciprocal of the extensional rigidity K and characterizes the
membrane regime of the shell in which bending moments are negligible and the load is mainly supported
by internal stresses tangent to the shell. Since D and K are respectively proportional to the third and
first power of the shell thickness, the membrane regime (in which the D-depending terms are neglected)
is obtained in the limit of an extremely thin shell. This observation and the fact that such a shell, lacking
rigidity, cannot support bending moments justify the use of the term ‘membrane’.
The weight of the various terms in the equations depends on two competing factors: the magnitude
of the coefficient multiplying the derivative and the number of derivatives. On the one hand, a coefficient
containing D will be smaller than a coefficient containing α−1 since αD/R2 ∼ ξ−1 is a small number
(see equation (22)). On the other hand, a large number of derivatives will increase the weight of the term
if the derived function has a small wavelength. The transition between the membrane and the bending
regimes thus depends on the wavelength of the load: if the load has a large wavelength (with respect to
the shell radius), the flexure of the shell will be well described by equations without the terms depending
on D (see section 5.1), whereas the flexure under loads of small wavelength is well described by equations
keeping only the D-depending terms with the largest number of derivatives (see section 5.2).
Stress functions are associated with membrane stretching and give negligible contributions in the
bending regime. Formulas (52) show that F (respectively H) plays the role of potential for the stress
resultants in the membrane regime when the load is transversal (respectively tangential toroidal). There
is no stress function associated with the tangential consoidal component of the load because it is identical
to Ω, the consoidal potential of the load.
The variation of the shell thickness has two effects. First, it couples the spherical harmonic modes
that are solutions to the equations for a shell of constant thickness. Second, the toroidal part remains
intertwined with the transversal and consoidal parts, whereas it decouples if the thickness is constant.
For example, the toroidal load is a source for the transverse deflection through the term B in the third
flexure equation. Furthermore, the stress function F is a source for the toroidal displacement in the fifth
flexure equation.
For numerical computations, the flexure equations (74)-(78) obtained in the thin shell approximation
(to which we can add the equations (73) for the stresses) are adequate. For this purpose, it is not useful
to keep small terms in 1/ξ since the theory rests on assumptions only true for a thin shell. In the rest of
the article, we will continue to work with the equations (49), (58), (66), (71) and (72) for more generality.
4.2 Covariance
Because of their tensorial form, the flexure equations (74)-(78) are covariant; this is also true of the more
general equations (49), (58), (66), (71) and (72). This property means that their form is valid in all
systems of coordinates on the sphere, though the tensor components (the covariant derivatives of scalar
functions, the metric and the antisymmetric tensor) will have a different expression in each system. The
scalar functions will also have a different dependence on the coordinates in each system. The covariance
of the final equations was expected. We indeed started with tensorial equations in section 2.1; their
restriction to the sphere in principle respected the tensoriality with respect to changes of coordinates
on the sphere. However the covariance of the two-dimensional theory was not made explicit until we
obtained the final equations. This property is thus a strong constraint on the form of the solution and
a check of its validity, though only necessary and not sufficient (other covariant terms may have been
ignored).
Another advantage of the covariant form is the facility to express the final equations in different
systems of coordinates (even non-orthogonal ones) with the aim of solving them. For example, the fi-
nite difference method in spherical coordinates (θ, ϕ) suffers from a very irregular grid and from pole
singularities. These problems can be avoided with the ‘cubed sphere’ coordinate system [Ronchi et al.,
1996]. Operators including covariant derivatives (here the Laplacian and the operators A and B) can ex-
pressed in any system of coordinates whose metric is known. Christoffel symbols and tensorial differential
operators can be computed with symbolic mathematical software.
4.3 Degree one
Displacements of degree one require special consideration. All differential operators acting on (w,F,H)
(that is ∆′, A and B) in the flexure equations (58) and (66) can be expressed in terms of the operators
Dij = ∇i∇j+gij (see section 3.2). The Dij have the interesting property that they give zero when acting
on spherical harmonics of degree one (see equation (30)). Therefore the degree-one terms in the spherical
harmonic expansion of w vanish from the flexure equations. The magnitude of the transverse deflection
of degree one neither depends on the load nor on the elastic properties of the spherical shell.
More generally, the homogeneous (q = Ω = V = 0) flexure equations (49), (58) and (66) are satisfied
by w and F being both of degree one. According to equations (71)-(72), the corresponding tangential
displacement is constrained by ∆S = −2w and ∆∆′T = 0, so that S and T are also of degree one,
with S = w. These conditions lead to vanishing strains (see equations (69)) which indicate a rigid
displacement. The total displacement is then given by
u = w r̂+ ∇̄w + ∇̄ × (T r̂) ,
where w and T are of degree one. In Appendix 7.6, we show that the first two terms represent a
rigid translation whereas the last term represents a rigid rotation. As expected, stresses vanish for
such displacements (see equation (73)). This freedom in translating or rotating the solution reflects the
freedom in the choice of the reference frame (in practice the reference frame is centered at the center of
the undeformed shell). The same freedom of translation is also found in the theory of deformations of a
spherical, radially stratified, gravitating solid [e.g. Farrell , 1972; Greff-Lefftz and Legros , 1997; Blewitt ,
2003].
What can we say about degree-one loading? Let us first examine what happens when the flexure
equations are projected on the spherical harmonics of degree one. Since the operator ∆′ annihilates the
degree one in any spherical harmonic expansion, terms of the form ∆′f vanish when they are projected
on the spherical harmonics of degree one. Moreover, the operators A and B also vanish in this projection
since they do not contain a degree-one term (see equations (116)-(117) of Appendix 7.8). Therefore the
degree-one component of the flexure equation (66) is identically zero, whereas the degree-one component
of the flexure equation (58) is
q + ∇̄ · qT
Yi = 0 (i = x, y, z) , (79)
where dω = sin θ dθ dϕ and Yi are the real spherical harmonics of degree one. The integral is taken over
the whole spherical surface. We have used the relation ∆Ω = −R ∇̄ ·qT derived from equations (45) and
(106).
The first term in the integrand of equation (79) is the projection on the Cartesian axes (x̂, ŷ, ẑ) of
the vector field q r̂:
(q Yx, q Yy, q Yz) = (q r̂ · x̂, q r̂ · ŷ, q r̂ · ẑ) ,
where we have used formulas (109) for the spherical harmonics.
The second term in equation (79) can be rewritten with the identity (102) and Gauss’ theorem (107):
∇̄ · qT
Yi = −
dω qT · ∇̄Yi (i = x, y, z) ,
where the Yi are considered as scalars. Since qT is orthogonal to r̂, the integrand is the projection of qT
on the Cartesian axes (x̂, ŷ, ẑ):
qT · ∇̄Yx,qT · ∇̄Yy,qT · ∇̄Yz
= (qT · x̂,qT · ŷ,qT · ẑ) ,
where we have used formulas (110) for the gradients of the spherical harmonics.
Recalling that the transverse load q was defined positive towards the center of the sphere (see equation
(26)), we define a total load vector q = −q r̂+qT . With the above results, we can rewrite the degree-one
projection (79) as
dω (q · x̂ ,q · ŷ ,q · ẑ) = (0, 0, 0) , (80)
which means that the integral (over the whole spherical surface) of the projection on the coordinate axes
of the total load vector vanishes. This result is the consequence of the static assumption in the equations
of motion (7), since a non-zero sum of the external forces would accelerate the sphere.
In practice, degree-one loads on planetary surfaces are essentially due to mass redistribution [Greff-Lefftz and Legros ,
1997] and have a tangential consoidal component (for example the gravitational force is not directed to-
ward the center of figure of the shell). If the shell thickness is variable, a non-zero Ω of degree one will
induce degrees higher than one in w and in S. If the shell thickness is constant, the degree-one load
drops from the flexure equations (58)-(66) and w is not affected. However, the degree-one Ω generates
(assuming a constant shell thickness) a degree-one tangential displacement through equation (71), so that
S 6= w. Whether the shell thickness is variable or not, a degree-one Ω thus generates a total displacement
which is not only a translation but also a tangential deformation [Blewitt , 2003], in which case stresses
do not vanish as shown by equations (73).
5 Limit cases
5.1 Membrane limit
A shell is in a membrane state of stress if bending moments (Mθ,Mϕ,Mθϕ) can be neglected, in analogy
with a membrane which cannot support bending moments. Equations (19) show that this is true if the
bending rigidity vanishes: D = 0. Consistency with equation (22) imposes the limit of infinite ξ (or
η = 1). With these approximations, the first flexure equation (58) for (w,F,H) becomes
∆′F = −Rq − 2Ω . (81)
The second flexure equation (66) for (w,F,H) becomes
∆′w = ∆′ (α∆′F )− (1 + ν)A(α ;F )− 2(1 + ν)B(α ;H) + (1− ν)∆′(αΩ) . (82)
If q is independent of w, F and w can be successively determined with spherical harmonic transforms
from equations (81) and (82) (though the right-hand side of the latter equation must be computed with
another method). If q has a linear dependence in w (such as when the sphere is filled with a fluid), w
can be eliminated between equations (81) and (82), so that F and w can also be computed in succession
(however the equation for F cannot be solved by a spherical harmonic transform). H is supposed to be
known since equation (49) is not modified and can be solved with spherical harmonics.
The stresses are obtained from equations (73) with the additional approximation of neglecting the
term in ζ:
(σ̂θθ, σ̂ϕϕ, σ̂θϕ) =
(O2F +Ω+ 2O3H,O1F +Ω− 2O3H,−O3F + (O2 −O1)H) .
Bending moments play a small role if the load has a large wavelength. In practice, the threshold at
which bending moments become significant can be evaluated from the constant thickness equation for w
(see section 5.3.1). One must compare the magnitudes of the terms in D and 1/α in the left-hand side
of equation (88). Bending moments are negligible (i.e. the term in D) if the spherical harmonic degree ℓ
of the transverse displacement w is such that
ℓ . k
, (83)
where k = (12(1− ν2))1/4 ≈ 1.8 if we take ν = 1/4. This threshold is about 10 for a planet with a radius
of 3400 km and a lithospheric thickness equal to 100 km. Flexure equations in the membrane limit of a
shell with constant thickness have been used by Sleep and Phillips [1985] to study the lithospheric stress
in the Tharsis region of the planet Mars.
5.2 Euclidean limit
The equation for the deflection of a rectangular plate with variable thickness has been derived by
Timoshenko and Woinowsky-Krieger [1964, p. 173]. We will check here that the Euclidean limit of
our equations gives the same answer.
Let us define the coordinates (x, y) = (Rϕ,R θ′), where θ′ = θ/2 − θ is the latitude. We work in a
small latitude band around the equator (so that θ′ ≪ 1) and in the limit of large spherical radius R and
large ξ (η = 1). Under this change of coordinate, each derivative introduces a factor R so that terms with
the largest number of derivatives dominate. In particular, covariant derivatives can be approximated by
usual derivatives. The surface Laplacian (104) can be approximated as follows:
∆ ≈ R2
≡ R2∆e .
We assume that there are no tangential loads (Ω = V = H = 0). The flexure equations (75)-(76) for the
transverse displacement become:
R4∆e (D∆ew)− (1 − ν)R4 Ae(D ;w) +R5∆eF = −R4 q , (84)
R4 ∆e (α∆eF )− (1 + ν)R4 Ae(α ;F )−R ∆ew = 0 , (85)
where the operator Ae is defined by
Ae(a ; b) = (∆e a) (∆e b)− a,i,j b,i,j
Equation (85) gives a relation between the magnitudes of F and w:
O(R3 ∆eF ) ∼ O(w/α) .
In the large R limit, equation (84) thus becomes
∆e (D∆ew) − (1− ν)Ae(D ;w) = −q ,
which is the equation derived by Timoshenko and Woinowsky-Krieger [1964]. This equation has been
used by Stark et al. [2003], Kirby and Swain [2004] and Pérez-Gussinyé et al. [2004] for the local analysis
of the lithosphere of the Earth. Its one-dimensional version, in which Ae vanishes, has been used by
Sandwell [1984] to describe the flexure of the oceanic lithosphere on Earth and by Stewart and Watts
[1997] to model the flexure at mountain ranges.
5.3 Shell with constant thickness
5.3.1 Displacements
If the thickness of the shell is constant, the toroidal part of the tangential displacement decouples. The
terms in B indeed drop from the flexure equations (58) and (66) so that the equations for (w,F ) depend
only on (q,Ω):
ηD∆′∆′w − (1− ν) ηD∆′w + ηR3 ∆′F = −R4 q +R3 ((1 − η)∆Ω− 2ηΩ) , (86)
∆′∆′F − (1 + ν)∆′F − 1
∆′w = −(1− ν)∆′Ω , (87)
where we used the property A(a ; b) = a∆′b valid for constant a. We eliminate F from these equations
and obtain a sixth order equation relating w to (q,Ω):
ηD∆∆′∆′w +
∆′w = −R4 (∆′ − 1− ν) q +R3
1 + ξ
∆′ − 1− ν
∆Ω . (88)
The elimination of ∆′F between equations (77) and (86) gives an equation relating the consoidal
displacement potential S to (w, q,Ω):
∆S = − 1
1 + ν
1 + ξ
∆∆′w − 2w − (1 − ν)R2α q + 1− ν
1 + ξ
Rα∆Ω .
Terms not including a Laplacian can be eliminated with equation (88), so that we obtain an explicit
solution for S in terms of (w, q,Ω):
1 + ξ
1− ν2
(∆ + 1 + ν)∆′w + w +R2α q − Rα
1 + ξ
(∆− ξ(1 + ν))Ω , (89)
where the integration constant has been set to zero.
Equations (49) and (72) give an equation for the toroidal displacement potential:
∆′ T = −2ηRα (1 + ν)V , (90)
where the integration constant has been set to zero. We have assumed that ∆V 6= 0, otherwise we get
∆′T = 0.
The differential equations given in this section can be solved with spherical harmonics so that the co-
efficients of the spherical harmonic expansions of (w, S, T ) can be expressed in terms of the corresponding
coefficients of the loads (q,Ω, V ) (see Kraus [1967], Turcotte et al. [1981], Banerdt [1986]).
5.3.2 Comparison with the literature
We now compare our equations for a shell of constant thickness with those found in the literature. The
formulas of Banerdt [1986] (taken from the work of Vlasov [1964]) are the most general:
∆3 + 4∆2
(∆ + 2)w = −R4 (∆ + 1− ν) q +R3
∆− 1− ν
∆Ω , (91)
(∆ + 2)χ =
Dξ(1− ν)
∆V . (92)
Banerdt’s notation is slightly different: his formulas are obtained with the substitutions ξ → ψ, Ω → RΩ
and V → RV . The normal rotation χ is proportional to the radial component (in a normalized basis) of
the curl of the tangential displacement:
(∇× v)r̂ .
The curl ∇× v is related to our surface curl (103) by
∇× v = ∇̄ × v + csc θ
(sin θv̂ϕ),θ − v̂θ,ϕ
With the formulas (68) and (104), we get ∇× v = −∆T r̂ so that equation (92) becomes
∆′ T = −2Rα(1 + ν)V . (93)
We see that Banerdt’s equations (91) and (93) coincide with our equations (88) and (90) in the limit
of large ξ (η = 1), with one exception: the bending term for w is written D(∆3 + 4∆2)w instead of
D∆∆′∆′w = (∆3 + 4∆2 + 4∆)w. This error has propagated in many articles and is of consequence for
the degree-one harmonic component, since it violates the static assumption and spoils the translation
invariance discussed in section 4.3. The impact on higher degrees is negligible. Because of this mistake,
many authors give a separate treatment to the first harmonic degree. Banerdt also gives formulas for the
tangential displacements in terms of consoidal and toroidal scalars (A,B) corresponding to our scalars
(S, T ): his formula (A10) is equivalent to our equation (89) in the limit of large ξ.
If we ignore temperature effects, Kraus’ first equation for (w,F ) is equivalent to our equation
(86) in the limit of large ξ, whereas his second equation for (w,F ) is equivalent to the combination
eq.(87)+ 1+ν
eq.(86) in the limit of large ξ [see Kraus , 1967, eq. 6.54h and 6.55d]. Note that the
definition of Kraus’ stress function F [Kraus, 1967, p. 243] differs from ours:
FKraus = F − k(1 − ν)
with k = 1. This freedom of redefining F for arbitrary k remains as long as D is constant. The flexure
equation for w, equation (88), is unaffected so that the solution for w is unchanged. In the final step,
Kraus makes a mistake when combining the two equations for (w,F ) and thus obtains a flexure equation
for w with the same error as in equation (91). Kraus does not include toroidal loading. The flexure
equation of Turcotte et al. [1981] is taken from Kraus [1967] without the tangential loading and is the
same as equation (91) with Ω = 0.
The flexure equation of Brotchie and Silvester [1969] is given in our notation by
D∆2w +
w = −R4 q , (94)
where q includes their term γw describing the response of the enclosed liquid. This equation can be
obtained from our equation (88) as follows: keep only the derivatives of the highest order in each term,
set Ω = 0, take the limit of large ξ (η = 1) and integrate. Brotchie and Silvester choose to work in
the approximation of a shallow shell and with axisymmetrical loading, solving their equation in polar
coordinates with Bessel-Kelvin functions. The reduction to fourth order in equation (94), the shallow
shell approximation and the axisymmetrical assumption are not justified nowadays since the full equation
(88) can be quickly solved with computer-generated spherical harmonics.
The contraction due to a transverse load of degree 0, w = −R2α(1 − ν)q/2, is equivalent to the
radial displacement computed by Love in the limit of a thin shell [Love, 1944, p.142]. However ad-
ditional assumptions about the initial state of stress and the internal density changes are necessary
[Willemann and Turcotte, 1982] so that the degree 0 is usually excluded from the analysis.
5.3.3 Breakdown of the third assumption of thin shell theory
With the spherical harmonic solutions of the equations for a shell of constant thickness, it is possible to
check the thin shell assumption stating that the transverse normal stress is negligible with respect to the
tangential normal stress. The magnitude of the former can be estimated by the load q (see definition
(26)) whereas the magnitude of the latter can be approximated with formulas (73) evaluated on the outer
surface:
(σ̂θθ + σ̂ϕϕ) |h
∆′F − Eh
4R2(1− ν)
∆′w .
where we have assumed the absence of tangential loads (Ω = 0) and the limit of large ξ. We can relate
σT to q by using the solution in spherical harmonics of equations (87) and (88). Since the thin shell
assumption is expected to fail for a load of sufficiently small wavelength, we assume that the spherical
harmonic degree ℓ is large. Assuming ℓ≫ 1, we obtain
(∆′F )ℓm ≈
wℓm ,
wℓm ≈ −
1 + ℓ
ξ(1−ν2)
qℓm ,
where the spherical harmonic coefficients are indexed by their degree ℓ and their order m. If the shell is
not in a membrane state of stress (see equation (83)), ℓ2 > 2R/h so that σT can be approximated by
(σT )lm ≈
ξ(1 + ν)
qlm .
The thin shell assumption holds if q < σT , that is if
3(1 + ν)
or λ >
3(1 + ν)
h , (95)
where λ is the load wavelength (λ ≈ 2πR/ℓ). We have
3(1 + ν) ≈ 1.9 and 2π/
3(1 + ν) ≈ 3.2 if we
take ν = 1/4. This condition on λ is consistent with the transition zone between the thin and thick shell
responses analyzed in Janes and Melosh [1990] and Zhong and Zuber [2000], but does not coincide with
the constraint given in Willemann and Turcotte [1982], which is ℓ < 2π
R/h (this last condition looks
more like the threshold (83) for the membrane regime).
Though the stress distribution is affected, the limit (95) on the degree ℓ is not important for the
displacements, since they tend to zero at small wavelengths. Therefore the theory does not break down
at short wavelength if one is interested in the computation of the gravity field associated to the transverse
deflection of the lithosphere.
6 Conclusion
The principal results of this article are the five flexure equations (74)-(78) governing the three displace-
ments of the thin spherical shell and the two auxiliary stress functions. Stresses are derived quantities
which can be obtained from equations (73). The shell thickness and Young’s modulus can vary, but
Poisson’s ratio must be constant. The loads acting on the shell can be of any type since we extend the
method of stress functions to include not only transverse and consoidal tangential loads, but also toroidal
tangential loads. The flexure equations can be solved one after the other, except the two equations
(75)-(76) for the transverse deflection w and the stress function F , which must be simultaneously solved.
Tangential loading is usually neglected when solving for the deflection because of its small effect. In that
case, it is sufficient to solve the two equations (75)-(76) with Ω = H = 0:
∆′ (D∆′w)− (1− ν)A(D ;w) +R3∆′F = −R4 q ,
∆′ (α∆′F )− (1 + ν)A(α ;F )− 1
∆′w = 0 .
However tangential loading must be taken into account when computing stress fields [Banerdt , 1986].
In the long-wavelength limit (i.e. membrane regime), all equations can be solved one after the other
because it is possible to solve for F before solving for the transverse deflection. If a small part of the shell
is considered, the flexure equations reduce to the equations governing the deflection of a flat plate with
variable thickness. If the shell thickness is constant, the flexure equations reduce to equations available
in the literature which can be completely solved with spherical harmonics. Our rigorous treatment
of the thin shell approximation has clarified the effect of the shell thickness on the flexure equations.
We emphasize the need to use the correct form for the equations (without the common mistake in the
differential operator acting on w) in order to have the correct properties for the degree-one deflection and
degree-one load.
We have also obtained two general properties of the flexure equations. First we have shown that
there is always a toroidal component in the tangential displacement if the shell thickness is variable.
Second we have proven that the degree-one harmonic components of the transverse deflection and of the
toroidal component of the tangential displacement do not depend on the elastic properties of the shell.
This property reflects the freedom under translations and rotations of the reference frame. Besides we
have shown that degree-one loads are constrained by the static assumption but can deform the shell and
generate stresses.
This article was dedicated to the theoretical treatment of the flexure of a thin elastic shell with
variable thickness. While the special case of constant thickness admits an analytical solution in terms of
spherical harmonics, the general flexure equations must be solved with numerical methods such as finite
differences, finite elements or pseudospectral methods. In a forthcoming paper, we will give a practical
method of solution and discuss applications to real cases.
Acknowledgments
M. Beuthe is supported by a PRODEX grant of the Belgian Science Federal Policy. The author thanks
Tim Van Hoolst for his help and Jeanne De Jaegher for useful comments. Special thanks are due to
Patrick Wu for his constructive criticisms which helped to improve the manuscript.
7 Appendix
7.1 Covariant, contravariant and normalized components
Tensors can be defined by their transformation law under changes of coordinates. The two types of
tensor components, namely covariant and contravariant components, transform in a reciprocal way under
changes of coordinates. Tensor components cannot be expressed in a normalized basis: the space must
have a coordinate vector basis (for contravariant components) and a dual basis (for covariant components)
which are not normalized.
The only exception is a flat space with Cartesian coordinates, where covariant, contravariant and
normalized components are identical. Since the metric is the scalar product of the elements of the
coordinate vector basis, the covariant components are related to components defined in a normalized
basis (written with a hat) by
gii ûi ,
whereas the relation for contravariant components is
ûi .
The normalized Cartesian basis (x̂, ŷ, ẑ) is related to the normalized basis for spherical coordinates
(r̂, θ̂, ϕ̂) by
x̂ = cos θ cosϕ θ̂ − sinϕ ϕ̂+ sin θ cosϕ r̂ ,
ŷ = cos θ sinϕ θ̂ + cosϕ ϕ̂+ sin θ sinϕ r̂ , (96)
ẑ = − sin θ θ̂ + cos θ r̂ .
7.2 Covariant derivatives
Usual derivatives are indicated by a ‘comma’:
vi,j =
Covariant derivatives (defined below) are indicated by a ‘bar’ or by the operator ∇i:
vi|j = ∇j vi .
The former notation emphasizes the tensorial character of the covariant derivative since the covariant
derivative adds a covariant index to the vector. The latter notation is more adapted when we are interested
by the properties of the operator.
The covariant derivative of a scalar function f is equal to the usual derivative, f|i = f,i, and is itself
a covariant vector: f|i = vi. Covariant derivatives on covariant and contravariant vector components are
defined by
vi|j = vi,j − Γkij vk , (97)
vi |j = v
,j + Γ
k , (98)
where the summation on repeated indices is implicit. The symbols Γkij are the Christoffel symbols of the
second kind [Synge and Schild , 1978]. Their expressions for the metrics used in this article are given in
sections 7.3 and 7.4.
Covariant differentiation of higher order tensors is explained in Synge and Schild [1978] but we only
need the rule for a covariant tensor of second order:
σij|k = σij,k − Γlik σlj − Γljk σil .
If some of the indices of the tensor are contravariant, the rule is changed according to equation (98). The
covariant derivatives of the metric and of the inverse metric are zero: gij|k = 0 and g
7.3 Three-dimensional spherical geometry
The geometry of a thin spherical shell of average radius R can be described with coordinates θ, ϕ and
ζ, respectively representing the colatitude, longitude and radial coordinates. The radial coordinate ζ is
zero on the reference surface (i.e. the sphere of radius R) of the shell. The non-zero components of the
metric are given by
gθθ = (R + ζ)
gϕϕ = (R + ζ)
2 sin2 θ ,
gζζ = 1 .
The non-zero Christoffel symbols are given by
θθ = −(R+ ζ) ,
Γζϕϕ = −(R+ ζ) sin2 θ ,
Γθζθ = Γ
θζ = Γ
ζϕ = Γ
Γθϕϕ = − sin θ cos θ ,
ϕθ = Γ
θϕ = cot θ .
7.4 Two-dimensional spherical geometry
If θ and ϕ respectively represent the colatitude and longitude coordinates, the non-zero components of
the metric on the surface of the sphere are given by
gθθ = 1 ,
gϕϕ = sin
2 θ . (99)
The non-zero Christoffel symbols are given by
Γθϕϕ = − sin θ cos θ ,
ϕθ = Γ
θϕ = cot θ .
The double covariant derivatives of a scalar function f are thus given by
f|θ|θ = f,θ,θ ,
f|θ|ϕ = f|ϕ|θ = f,θ,ϕ − cot θ f,ϕ ,
f|ϕ|ϕ = f,ϕ,ϕ + sin θ cos θ f,θ .
An antisymmetric tensor εij is defined by
εij ≡
det gij ε̄ij ,
where ε̄ij is the antisymmetric symbol invariant under coordinate transformations: ε̄θϕ = −ε̄ϕθ = 1,
ε̄θθ = ε̄ϕϕ = 0 (ε̄ij is usually called a tensor density; Synge and Schild [1978] call it a relative tensor of
weight -1). The non-zero covariant components of εij are given for the metric of the spherical surface by
εθϕ = −εϕθ = sin θ .
The non-zero contravariant components, εij = gikgjlεkl, are given by
εθϕ = −εϕθ = csc θ .
The covariant derivative of the tensor εij is zero: εij|k = 0.
7.5 Gradient, divergence, curl and Laplacian
Various differential operators on the surface of the sphere can be constructed with covariant derivatives.
In this section, f and t are scalar functions defined on the sphere and v is a vector tangent to the sphere.
Backus [1986] gives more details on surface operators and on Helmholtz’s theorem.
As mentioned in Appendix 7.2, the covariant derivative of a scalar function f defined on the sphere
is a covariant vector tangent to the sphere whose components are f,θ and f,ϕ. The surface gradient of f
is the same vector with its components expressed in the normalized basis (θ̂, ϕ̂):
∇̄f = f,θ θ̂ + csc θ f,ϕ ϕ̂ . (100)
The contraction of the covariant derivative with the components of a vector v yields a scalar:
vi |i = v
,θ + cot θ v
θ + vϕ,ϕ .
The surface divergence is the corresponding operation on the vector with its components expressed in the
normalized basis (θ̂, ϕ̂):
∇̄ · v = csc θ
(sin θ v̂θ),θ + v̂ϕ,ϕ
. (101)
Since the result is a scalar, vi
= ∇̄ · v. A useful identity is
∇̄ · (f v) = ∇̄f · v + f ∇̄ · v . (102)
The contraction of the antisymmetric tensor εij with the covariant derivative of a scalar t yields the
covariant components of a vector v:
vi = g
jk εik t,j .
The components are given for the metric (99) by vθ = csc θ t,ϕ and vϕ = − sin θ t,θ. If t is considered
as the radial component of the radial vector t = t r̂ (the covariant radial component is equal to the
normalized one), vi are the non-zero covariant components of the three-dimensional curl of t, which is
tangent to the sphere. This fact justifies the definition of the surface curl of t, which is equal to the
vector v but with components given in the normalized basis (θ̂, ϕ̂):
∇̄ × t = csc θ t,ϕ θ̂ − t,θ ϕ̂ . (103)
The contraction of the double covariant derivative acting on a scalar f defines the surface Laplacian:
∆f = gij f|i|j
= f,θ,θ + cot θ f,θ + csc
2 θ f,ϕ,ϕ . (104)
The surface Laplacian can also be seen as the composition of the surface divergence with the surface
gradient: ∆f = ∇̄ · ∇̄f .
According to Helmholtz’s theorem, a vector tangent to the sphere can be written as the sum of the
surface gradient of a scalar f and the surface curl of a radial vector t r̂:
v = ∇̄f + ∇̄ × (t r̂) . (105)
While t is always called the toroidal scalar (or potential) for v, there is no standard terminology for f .
Backus [1986] calls f the consoidal scalar for v. Some authors [e.g. Banerdt , 1986] call f the poloidal
potential for v. The origin of this use lies in the theory of mantle convection, in which plate tectonics are
assumed to be driven by mantle flow. Under the assumption of an incompressible mantle fluid, the velocity
field of the fluid is solenoidal, i.e. its 3-dimensional divergence vanishes. In such a case, the velocity field
can be decomposed into a poloidal part (∇×∇×(P r̂)) and a toroidal part (∇×(Q r̂)), where differential
operators are 3-dimensional [Backus , 1986]. If the velocity field is tangent to the spherical surface, the
poloidal component at the surface is also the consoidal component [Forte and Peltier , 1987]. However the
fields for which we use Helmholtz’s theorem, i.e. the tangential surface load and the tangential surface
displacement, do not belong to 3-dimensional solenoidal vector fields. We thus prefer to use the term
‘consoidal’.
The surface divergence of v depends only on the consoidal scalar f :
∇̄ · v = ∆f . (106)
The two-dimensional version of Gauss theorem is
dω ∇̄ · v = 0 . (107)
where dω = sin θ dθ dϕ and the integral is taken over the whole spherical surface. It can be proven with
formula (101).
7.6 Rigid displacements
At the surface of a sphere subjected to deformation, the displacement u of a point can be expressed with
the help of Helmholtz’s theorem (105) in terms of three scalar functions (w, S, T ) depending on θ and ϕ:
u = w r̂+ ∇̄S + ∇̄ × (T r̂) . (108)
Strains (and stresses) vanish for rigid displacements. Equations (69) show that strains vanish when
(w, S, T ) are of degree one, with S = w (recall that the operatorsOi annihilate the degree one). Assuming
these conditions, we now show that utransl = w r̂ + ∇̄w represents a rigid translation whereas urot =
∇̄ × (t r̂) represents a rigid rotation of the sphere. We choose as basis the real spherical harmonics of
degree one which form the components of the radial unit vector in Cartesian coordinates:
(Yx, Yy, Yz) = (sin θ cosϕ, sin θ sinϕ, cos θ)
= (x̂, ŷ, ẑ) · r̂ . (109)
We need the surface gradient of the real spherical harmonics which can be computed with formulas (96)
and (100):
∇̄Yx, ∇̄Yy, ∇̄Yz
= (x̂− sin θ cosϕ r̂, ŷ − sin θ sinϕ r̂, ẑ− cos θ r̂) . (110)
If the expansion of w in the degree-one basis is w = a Yx + b Yy + c Yz, then
w r̂+ ∇̄w = a x̂+ b ŷ + c ẑ ,
so that utransl is indeed a rigid translation of the sphere.
If the expansion of T in the degree-one basis is T = a′Yx + b
′Yy + c
′Yz, then
∇̄ × (T r̂) = a′
− sinϕ θ̂ − cos θ cosϕ ϕ̂
cosϕ θ̂ − cos θ sinϕ ϕ̂
+ c′ sin θ ϕ̂ ,
so that urot includes a rigid rotation of the sphere, with (a
′, b′, c′) being the angles of rotation around
the axes (x̂, ŷ, ẑ), respectively. Though urot seems to include a uniform radial expansion, one should
recall that linearized strain-displacement equations are not valid for large displacements. Since strains
vanish, the radial expansion is not physical and urot represents a pure rotation. Finite deformations are
for example discussed in Love [1944][pp. 66-73] and Sokolnikoff [1956][pp. 29-33].
7.7 Differential identities for the operators Oi
The differential operators Oi defined by equations (15) satisfy differential identities useful when obtaining
the flexure equations. They are special cases of differential identities valid in curved spaces. The presence
of curvature makes the parallel transport of vectors path-dependent; this property quantifies the curvature
of space and can be expressed as the lack of commutativity of the covariant derivatives of a vector v:
vi|j|k − vi|k|j = Riljk vl , (111)
where Riljk are the covariant components of the Riemann tensor. On the sphere, the Riemann tensor
has only one independent component that is non-zero, Rθϕθϕ = − sin2 θ. Other components are related
by the symmetries Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ.
The substitution of f,i to vi in the commutation relation (111) provides two differential identities
satisfied by double covariant derivatives acting on scalar functions:
csc θ f|ϕ|ϕ
− csc θ f|ϕ|θ,ϕ − cos θ f|θ|θ + sin θ f,θ = 0 ,
f|θ|θ,ϕ − f|ϕ|θ,θ − cot θ f|ϕ|θ + f,ϕ = 0 .
The replacement in the above equations of the double covariant derivatives by the normalized dif-
ferential operators (31) yields the following identities:
(sin θO2f),θ − (O3f),ϕ − cos θO1f = 0 (I1) , (112)
(sin θO3f),θ − (O1f),ϕ + cos θO3f = 0 (I2) . (113)
These identities can also be directly checked with the definitions (15) of the operators Oi.
The identities (I1)-(I2) can be differentiated to generate identities of higher order. A first useful
identity is obtained from sin θ(I1),θ − (I2),ϕ = 0:
csc2 θ
sin2 θ (O2f),θ
+ (O1f),ϕ,ϕ − 2
sin θ (O3f),ϕ
− cot θ (O1f),θ + 2O1f = ∆
′f . (114)
A second useful identity is obtained from (I1),ϕ + sin θ(I2),θ = 0:
csc2 θ
sin2 θ (O3f),θ
− (O3f),ϕ,ϕ +
sin θ ((O2 −O1) f),ϕ
+ cot θ (O3f),θ − 2O3f = 0 . (115)
7.8 No degree one in operators A and B
We want to prove that the operators A and B defined by equations (33) and (37) do not have any
degree-one term in their spherical harmonic expansion:
dω A(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (116)
dω B(a ; b)Y ∗1p = 0 (p = −1, 0, 1) , (117)
where (a, b) are arbitrary scalar functions on the sphere, dω = sin θ dθ dϕ and the integral is taken over
the whole spherical surface.
This property is not a straightforward consequence of constructing A and B with Dij as a building
block. Although A and B can be factored into terms without degree one (such as Dija or ∆′a), the
product of the factors may contain degree-one terms in its spherical harmonic expansion.
Without loss of generality, we can prove the above identities with the arguments (a, b) being spherical
harmonics of given degree and order. The general result is then obtained by superposition. Let a and b
be spherical harmonics of order m and n: a ∼ eimϕ and b ∼ einϕ (we will not use their harmonic degree
in the proof). All derivatives with respect to ϕ in the operators A and B can then be replaced with the
rules a,ϕ → ima and b,ϕ → inb. The integral over ϕ in equations (116)-(117) gives
dϕ ei(m+n−p)ϕ = 2π δm+n−p,0 ,
so that the integral is zero unless n = p−m.
First consider the case p = 0 (n = −m), that is the projection on the zonal spherical harmonic of
degree one. We thus have to calculate
dθ A0 and
dθ B0 with
A0 ≡ sin θ cos θA(a ; b) ,
B0 ≡ sin θ cos θ B(a ; b) .
The trick consists in rewriting the integrands as total derivatives:
cos2 θ a,θ b,θ + cot θ
sin2 θ −m2
(ab),θ +
sin2 θ +m2 csc2 θ cos 2θ
B0 = −im
cos θ a,θ b,θ + sin θ a b,θ − csc θ cos2 θ a,θ b+ cos θ a b,θ,θ
The sought integrals are thus given by
dθ A0 =
a,θ b,θ −m2 cot θ (ab),θ +m2 csc2 θ ab
dθ B0 = −im
cos θ a,θ b,θ − csc θ a,θ b+ cos θ a b,θ,θ
where we have dropped the terms containing at least one power of sin θ which vanish at the limits; we
have also replaced cos2 θ and cos 2θ by their value at the limits. The remaining terms can be evaluated
by recalling the dependence in sin θ of the spherical harmonics: a = (sin θ)|m| a0 and b = (sin θ)
|m| b0,
where a0 and b0 are polynomials in cos θ. The only non-zero terms at the limits of the integrals are those
for |m| = 1, in which case we have at the limits: a,θb,θ = a0b0, cot θ (ab),θ = 2a0b0, csc2 θ ab = a0b0,
csc θ a,θb = cos θ a0b0, ab,θ,θ = 0. However these terms cancel in the sums so that the integrals vanish for
all m. This completes the proof for the case p = 0.
Now consider the case p = ±1 (n = −m ± 1), that is the projections on the sectoral spherical
harmonics of degree one. We thus have to calculate
dθ A±1 and
dθ B±1 with
A±1 ≡ sin2 θA(a ; b) ,
B±1 ≡ sin2 θB(a ; b) .
We again write the integrands as total derivatives:
A±1 =
sin θ cos θ a,θ b,θ −
sin θ cos θ +m2
(ab),θ −
cos2 θ ∓ 2m
a,θ b
+ sin2 θ a b,θ + 2m(m∓ 1) cot θ ab
B±1 = −i
(m∓ 1) sin θ a,θ b,θ +m sin θ a b,θ,θ − (m∓ 1) cos θ a,θ b
−m cos θ a b,θ −m(1∓m) csc θ ab
The sought integrals are thus given by
dθ A±1 =
−m2 (ab),θ − (1∓ 2m) a,θ b+ 2m(m∓ 1) cot θ ab
dθ B±1 = i
(m∓ 1) cos θ a,θ b+m cos θ a b,θ +m(1∓m) csc θ ab
where we have dropped the terms containing at least one power of sin θ and replaced cos2 θ by its value
at the limits. The remaining terms can be evaluated as in the case p = 0, but with a = (sin θ)|m| a0 and
b = (sin θ)|m∓1| b0. All terms give zero at the limits of the integrals for all values of m. This completes
the proof for the case p = ±1. We have thus proven the identities (116)-(117).
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Introduction
Fundamental equations of elasticity
Three-dimensional elasticity theory
Spherical shell
Assumptions of the thin shell theory
Strain-displacement equations
Hooke's law
Equilibrium equations
Resolution
Available methods
Differential operators
Transverse displacement
Resolution of the equations of equilibrium
Compatibility relation
Tangential displacements
Stresses
Flexure equations and their properties
Thin shell approximation
Covariance
Degree one
Limit cases
Membrane limit
Euclidean limit
Shell with constant thickness
Displacements
Comparison with the literature
Breakdown of the third assumption of thin shell theory
Conclusion
Appendix
Covariant, contravariant and normalized components
Covariant derivatives
Three-dimensional spherical geometry
Two-dimensional spherical geometry
Gradient, divergence, curl and Laplacian
Rigid displacements
Differential identities for the operators Oi
No degree one in operators A and B
|
0704.1628 | Detection of single electron spin resonance in a double quantum dot | arXiv:0704.1628v1 [cond-mat.mes-hall] 12 Apr 2007
Detection of single electron spin resonance in a double quantum dot
F. H. L. Koppens,∗ C. Buizert, I. T. Vink, K.C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen
Spin-dependent transport measurements through a double quantum dot are a valuable tool for
detecting both the coherent evolution of the spin state of a single electron as well as the hybridization
of two-electron spin states. In this paper, we discuss a model that describes the transport cycle in
this regime, including the effects of an oscillating magnetic field (causing electron spin resonance)
and the effective nuclear fields on the spin states in the two dots. We numerically calculate the
current flow due to the induced spin flips via electron spin resonance and we study the detector
efficiency for a range of parameters. The experimental data are compared with the model and we
find a reasonable agreement.
A. Introduction
Recently, coherent spin rotations of a single electron
were demonstrated in a double quantum dot device [1].
In this system, spin-flips of an electron in the dot were in-
duced via an oscillating magnetic field (electron spin res-
onance or ESR) and detected through a spin-dependent
transition of the electron to another dot, which already
contained one additional electron. This detection scheme
is an extension of the proposal for ESR detection in a sin-
gle quantum dot by Engel and Loss [2]. Briefly, the device
can be operated (in a spin blockade regime [3]) such that
the electron in the left dot can only move to the right dot
if a spin flip in one of the two dots is induced via ESR.
From the right dot, the electron exits to the right reser-
voir and another electron enters the left dot from the left
reservoir. A continuous repetition of this transition will
result in a net current flow.
Compared to the single dot detection scheme [2], us-
ing the double-dot as the detector has two major advan-
tages. First, the experiment can be performed at a lower
static magnetic field and consequently with lower, tech-
nically less demanding, excitation frequencies. Second,
the spin detection is rather insensitive to unwanted oscil-
lating electric fields, because the relevant dot levels can
be positioned far from the Fermi energies of the leads.
These electric fields are unavoidably generated together
with the oscillating magnetic field as well.
The drawback of the double-dot detector is that spin
detection is based on the projection in the two-electron
singlet-triplet basis, while the aim is to detect single spin
rotations. However, this detection is still possible be-
cause the electrons in the two dots experience different
effective nuclear fields. This is due to the hyperfine inter-
action of the electron spins with the (roughly 106) nuclear
spins in the host semiconductor material of each quan-
tum dot [4–11]. In order to provide more insight in this
double-dot ESR detection scheme for single spin rota-
tions, it is necessary to analyze the coherent evolution of
the two-electron spin states together with the transitions
∗Electronic address: [email protected]; Kavli Institute of
NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands
in the transport cycle.
In this paper, we discuss a model that describes the
transport cycle in the spin blockade regime while includ-
ing the coherent coupling between the two dots, and the
influence of the static and oscillating magnetic field to-
gether with the effective nuclear fields on the electron
spin states. The aim is to understand how effectively
single spin resonance will affect the measured quantity
in the experiment, namely the current flow in the spin
blockade regime. The organization of this paper is as
follows. First, we will explain the transport cycle and
the mechanism that causes spin blockade. Next, we will
briefly discuss the static system Hamiltonian and the
mixing of the two-electron spin states by the effective
nuclear field. Then we add an oscillating magnetic field
to this Hamiltonian, that forms -together with the double
dot tunnelling processes- the basis of the rate equations
that describe how the density matrix of the two-electron
spin states evolves in time. The current flow, calculated
from the steady state solution of the density operator,
is then analyzed for different coherent coupling values,
magnitudes of the oscillating magnetic field, in combina-
tion with different effective nuclear fields in the two dots.
This provides further insight in the optimal conditions
for spin-flip detection with a double quantum dot.
B. Spin blockade
In the spin-blockade regime, the double-dot is tuned
such that one electron always resides in the right dot,
and a second electron can tunnel from the left reservoir
through the left and right dots, to the right reservoir [3].
This current-carrying cycle can be described with the
occupations (m, n) of the left and right dots: (1, 1) →
(0, 2) → (0, 1) → (1, 1). When an electron enters the
left dot and forms a double-dot singlet state S11 with the
electron in the right dot (S = |↑↓〉 − |↓↑〉, normalization
omitted for brevity), it is possible for the left electron
to move to the right dot, because the right dot singlet
state S02 is energetically accessible. Next, one electron
tunnels from the right dot to the right lead and another
electron can again tunnel into the left dot. If, however,
the two electrons form a double-dot triplet state T11, the
left electron cannot move to the right dot, as the right
http://arxiv.org/abs/0704.1628v1
dot triplet state T02 is much higher in energy (due to
the relatively large exchange splitting in a single dot).
The electron can also not move back to the lead and
therefore further current flow is blocked as soon as any
of the (double-dot) triplet states is formed (Fig. 1a,b).
Spin blockade only occurs if at least one of the eigen-
states of the system Hamiltonian is a pure triplet state.
If processes are present that induce transitions from all
the three (1,1) triplet states to the (1,1) singlet state,
spin blockade is lifted and a current will flow. As we will
see below, the presence of the nuclear spins in the host
semiconductor can give rise to these kind of transitions.
This can be seen most easily by adding the effect of the
hyperfine interaction to the system Hamiltonian.
C. System Hamiltonian
The system Hamiltonian is most conveniently writ-
ten in the two-electron singlet-triplet basis with the
quantization-axis in the z-direction. The basis states are
S11, T
11, T
11, T
11 and S02. The subscript m,n denotes
the dot occupancy. We exclude the T02 state from the
model, because this state is energetically inaccessible and
therefore does not play an important role in the trans-
DLR>>t DLR=0
S - S11 02
S - S11 02
-10 -5 0 5 10
DLR/t
b) a)
FIG. 1: a) A schematic of the double dot and the electro-
chemical potentials (energy relative to the (0,1) state) of the
relevant two-electron spin states. For ∆LR > t, transitions
from the S11 state to the S02 state are possible via inelastic
relaxation with rate Γin. Spin blockade occurs when one of
the T i11 states is occupied. b) Similar schematic for ∆LR = 0,
where the singlet states are hybridized. Also in this case,
spin blockade occurs when one of T i11 states is occupied. c)
Energy levels as a function of detuning. At ∆LR = 0, the
singlet states hybridize into bonding and anti-bonding states.
The splitting between the triplets states corresponds to the
Zeeman energy gµBBext.
port cycle. Furthermore, we neglect the thermal energy
kT in the description, which is justified when the bias
over the two dots is much larger than kT . The system
Hamiltonian is given by
H0 = − ∆LR|S02〉〈S02|+ t
|S11〉〈S02|+ |S02〉〈S11|
− gµBBext
|T+11〉〈T
11| − |T
11〉〈T
, (1)
where ∆LR is the energy difference between the |S11〉
and |S02〉 state (level detuning, see Fig.1a), t is the tun-
nel coupling between the |S11〉 and |S02〉 states, Bext the
external magnetic field in the z-direction and Sz
L(R) the
spin operator along z for the left (right) electron. The
eigenstates of the Hamiltonian (1) for finite external field
are shown in figure 1c. For ∆LR < t, the tunnel coupling
t causes an anti-crossing of the |S11〉 and |S02〉 states. For
∆LR < 0, transport is blocked by Coulomb blockade (i.e.
the final state |S02〉 is at a higher energy than the initial
state S11). For ∆LR ≥ 0, transport will be blocked when
one of the three triplet states becomes occupied (spin
blockade). In Fig.1a and b, we distinguish two regimes:
∆LR > t where the (exchange) energy splitting between
T 011 and S11 is negligibly small and transitions from S11
to S02 occur via inelastic relaxation with rate Γin and the
energy. A different regime holds for ∆LR < t, where S11
is coherently coupled with S02 giving rise to a finite (ex-
change) splitting between T 011 and the hybridized singlet
states. We will return to this distinction in the discussion
below.
D. Singlet-triplet mixing by the nuclear spins
The effect of the hyperfine interaction with the nuclear
spins can be studied [12] by adding a static (frozen) ef-
fective nuclear field BLN (B
N ) at the left (right) dot to
the system Hamiltonian:
Hnucl = −gµB
N · SL +BRN · SR
= −gµB(BLN −BRN ) · (SL − SR)/2
−gµB(BLN +BRN ) · (SL + SR)/2. (2)
For the sake of convenience, we separate the inhomoge-
neous and homogeneous contribution, for reasons which
we will discuss later. Considering the nuclear field as
static is justified since the tunnel rates and electron spin
dynamics are expected to be much faster than the dy-
namics of the nuclear system [10, 13, 14]. Therefore,
we will treat the Hamiltonian as time-independent. The
effect of nuclear reorientation will be included later by
ensemble averaging.
We will show now that triplet states mix with the S11
state if the nuclear field is different in the two dots (in
all three directions). This mixing will lift spin blockade,
detectable as a finite current running through the dots for
∆LR ≥ 0. The effective nuclear field can be decomposed
-30 -20 -10 0 10 20 30
Magnetic field (mT)
Bext=0 B » Bext ND
-10 -5 0 5 10
10 S02
a +b +g +kS T T T11 11 11 11
-10 -5 0 5 10
a +kS T11 11
DLR/t DLR/t
FIG. 2: a) Observed current flow in the inelastic transport
regime (gµB∆LR ≫ t) due to singlet-triplet mixing by the
nuclei. b) Electrochemical potentials in the presence of Hnucl
(t ∼ ∆BN ). Singlet and triplet eigenstates are denoted by
red and blue lines respectively. Hybridized states (of sin-
glet and triplet) are denoted by dotted purple lines. For
gµBBext ≫ t, gµB∆BN , the split-off triplets (T
and T−
) are
hardly perturbed and current flow is blocked when they be-
come occupied. Parameters: t = 0.2µeV, gµBBN,L=(0.1,0,-
0.1)µeV, gµBBN,R=(-0.1,-0.2,-0.2)µeV and gµBBext=2µeV.
in a homogeneous and an inhomogeneous part (see right-
hand side of (2)). The homogeneous part simply adds
vectorially to the external field Bext, changing slightly
the Zeeman splitting and preferred spin orientation of the
triplet states. The inhomogeneous part ∆BN ≡ BLN −
N on the other hand couples the triplet states to the
singlet state, as can be seen readily by combining the
spin operators in the following way
SxL − SxR =
|S11〉〈T−11| − |S11〉〈T
11| + h.c.
i|S11〉〈T−11| − i|S11〉〈T
11| + h.c.
SzL − SzR =
|S11〉〈T 011|+ |T 011〉〈S11|
. (3)
The first two expressions reveal that the inhomogeneous
field in the transverse plane ∆BxN , ∆B
mixes the |T+11〉
and |T−11〉 states with the |S11〉. The longitudinal com-
ponent ∆BzN mixes |T 011〉 with |S11〉 (third expression).
The degree of mixing between two states will depend
strongly on the energy difference between them [5]. In
the case of gµBBext, t < gµB
〉, the three triplet
states are close in energy to the |S11〉 state. Their
intermixing will be strong, lifting spin blockade. For
gµBBext ≫ t, gµB
〉 the |T+11〉 and |T
11〉 states
are split off in energy by an amount of gµBBext. Con-
sequently the perturbation of these states caused by the
nuclei will be small. Although the |T 011〉 remains mixed
with the |S11〉 state, the occupation of one of the two
split-off triplet states can block the flow through the sys-
The effect of nuclear mixing is shown in Fig. 2 [5].
The observed current flow through the system is typi-
cally in the order of a few hundreds of fA (Fig. 2a).
At zero field, where the mixing is strongest, the current
flow is largest. Increasing the field gradually restores
spin blockade. Fig. 2b shows the energy levels for zero
and finite external field. The theoretical calculations of
the nuclear-spin mediated current flow (obtained from
a master equation approach) are discussed in references
[12, 15].
E. Oscillating magnetic field and rate equations
So far, we have seen that the occurrence of transitions
between singlet and triplet spin states are detectable as
a small current in the spin blockade regime. We will
now discuss how this lifting of spin blockade can also be
used to detect single spin rotations, induced via electron
spin resonance. The basic idea is the following. The
basic idea is the following. If the system is blocked in
e.g. | ↑〉| ↑〉, and the driving field rotates e.g. the left
spin, then transitions are induced to the state | ↓〉| ↑〉.
This state contains a singlet component and therefore a
probability for the electron to move to the right dot and
right lead. Inducing single spin rotations can therefore
lift spin blockade.
However, together with the driving field, the spin tran-
sitions are much more complicated due to the interplay of
different processes: spin resonance of the two spins, inter-
action with the nuclear fields, spin state hybridization by
coherent dot coupling and inelastic transitions from the
S(1,1) state to the S(0,2) state. In order to understand
the interplay of these processes, we will first model the
system with a time-dependent Hamiltonian and a den-
sity matrix approach. Next, we will discuss the physical
interpretation of the simulation results.
The Hamiltonian now also contains a term with an os-
cillating magnetic field in the x-direction with amplitude
Hac(t) = gµBBac sin(ωτ) · (SxL + SxR). (4)
We assume that Bac is equal in both dots, which is a
reasonable approximation in the experiment (from simu-
lations we find that the difference of Bac is 20% at most
[1]). We assume Bext ≫ BN , Bac, which allows applica-
tion of the rotating wave approximation [16]. Therefore,
we will define B1 ≡ 12Bac, which is in the rotating frame
the relevant driving field for the ESR process.
In order to study the effect of ESR and the nuclear
fields that are involved in the transport cycle, we will
construct rate equations that include the unitary evo-
lution of the spins in the dots governed by the time-
dependent Hamiltonian. This approach is based on the
model of reference [12], where the Hamiltonian contained
only time-independent terms. Seven states are involved
in the transport cycle, namely the three (1,1) triplets
|T i11〉, the double and single dot singlet states |S11〉 and
|S02〉 and the two (0,1) states | ↑01〉 and | ↓01〉, making
the density operator a 7× 7 matrix. The rate equations
based on the time-independent Hamiltonian are given in
[12]. These are constructed from the term that gives the
unitary evolution of the system governed by the Hamilto-
nian (H = H0 +Hac) dρ̂k/dτ = − i~ 〈k|[H, ρ̂]|k〉, together
with terms that account for incoherent tunnelling pro-
cesses between the states. The rate equations for the
diagonal elements are given by
= − i
〈T+11|[H, ρ̂]|T
ρ̂↑01
= − i
〈T−11|[H, ρ̂]|T
ρ̂↓01
dρ̂T 0
〈T 011|[H, ρ̂]|T 011〉+
ρ̂↑01 + ρ̂↓01
dρ̂S11
〈S11|[H, ρ̂]|S11〉+
ρ̂↑01 + ρ̂↓01
− Γinρ̂S11
dρ̂S02
〈S02|[H, ρ̂]|S02〉+ Γinρ̂S11 − ΓRρ̂S02
dρ̂↑01
ρ̂S02 − ΓLρ̂↑01
dρ̂↓01
ρ̂S02 − ΓLρ̂↓01 (5)
The rate equations for the off-diagonal elements are
0 2 4 6 8 10
Time ( s)m
transport via S02
RF on
Spin blockade
(0, / )
FIG. 3: Time evolution of the diagonal elements of the density
matrix for one particular nuclear configuration. Parameters:
~ω = gµB100mT, Bext =100 mT, B
N,x,y,z =(0,0,2.2) mT,
BRN,x,y,z =(0,0,0), B1 = 1.3 mT, ΓL = 73 MHz, ΓR = 73
MHz, ~Γin = gµBB
N,z and ∆LR=200µeV, t=0.3 µeV.
given by
dρ̂jk
= − i
〈j|[H, ρ̂]|k〉 − 1
Γj + Γk
ρ̂jk (6)
where the indices j, k ∈
T i11, S11, S02, ↑01, ↓01
label the
states available to the system. The tunneling/projection
rates Γj equal Γin and ΓR for the |S11〉 and |S02〉 states
respectively, and equal zero for the other 5 states. The
first term on the right-hand side describes the unitary
evolution of the system, while the second term describes
a loss of coherence due to the finite lifetime of the sin-
glet states. This is the first source of decoherence in our
model. The second one is the inhomogeneous broaden-
ing due to the interaction with the nuclear system. We
do not consider other sources of decoherence, as they are
expected to occur on much larger timescales.
Because we added a time-dependent term to the
Hamiltonian (the oscillating field), we numerically calcu-
late the time evolution of ρ̂(t), treating the Hamiltonian
as stationary on the timescale ∆τ ≪ 2π/ω. To reduce the
simulation time, we use the steady state solution ρ̂τ→∞
in the absence of the oscillating magnetic field as the ini-
tial state ρ̂(τ = 0) for the time evolution. At τ = 0 the
oscillating field is turned on and the system evolves to-
wards a dynamic equilibrium on a timescale set by the
inverse of the slowest tunnelling rate Γ. This new equi-
librium distribution of populations is used to calculate
the current flow, which is proportional to the occupation
of the |S02〉 state (I = eΓRρ̂S02). An example of the time
evolution of the density matrix elements is shown in Fig.
3. The figure clearly reveals that the blockade is lifted
when the oscillating field is applied. This is visible as an
increase of the occupation of the |S02〉 state.
In order to simulate the measured current flow we have
to consider the fact that the measurements are taken
with a sampling rate of 1 Hz. As the timescale of the
nuclear dynamics is believed to be much faster than 1
Hz [10, 13, 14], we expect each datapoint to be an in-
tegration of the response over many configurations of
the nuclei. The effect of the evolving nuclear system is
included in the calculations by averaging the different
values of the (calculated) current flow obtained for each
frozen configuration. These configurations are randomly
sampled from a gaussian distribution of nuclear fields in
the left and right dot (similar as in [12]). Because the
electron in the two dots interact with different nuclear
spins, the isotropic gaussian distributions in the two dots
are uncorrelated, such that
〉 and
〈B2N,x〉 = 〈B2N,y〉 = 〈B2N,z〉. For the sake of convenience
we define
〉 and σN,z =
-100 -50 0 50 100
B (mT)ext
-g B =hm wB ext g B =hm wB ext
Simulation: Inelastic transport ( )DLR>>t
h )G m sin B N/(g
B (mT)ext
-g B =hm wB ext g B =hm wB ext
-100 -50 0 50 100
Simulation: Resonant transport
t/(gm sB N)
( )DLR=0
0 50-50 100-100
RF at 460 MHz
P~-16dBm
RF off
-g B =hm wB ext g B =hm wB ext
B (mT)ext
Experimental data
FIG. 4: a). Calculated average current flow in the inelastic
transport regime. Parameters: ~ω = gµB100mT, Bext =100
mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, t=0.3 µeV
and ∆LR = 200 µeV. Results are similar for any value for t,
provided that ∆LR ≫ t. b) Calculated average current flow
in the resonant transport regime at zero detuning for differ-
ent values of t. Parameters: ~ω = gµB100mT, σN =2.2mT,
B1 = 1.3mT, ΓL,R = 73 MHz, Γin = 0 and ∆LR = 0. Av-
eraged over 400 nuclear configurations for t/(gµBσN ) > 0.5
and 60 configurations for t/(gµBσN) = 0.5. Simulation car-
ried out for positive magnetic fields only; values shown for
negative fields are equal to results obtained for positive field.
c) Experimental data from Ref [1] with (curve offset by 100
fA for clarity) and without oscillating magnetic field. The fre-
quency of the oscillating magnetic field is 460 MHz and the
applied power is -16dBm. Simulation carried out for posi-
tive magnetic fields only; values shown for negative fields are
equal to results obtained for positive field.
F. Simulation results and physical picture
An example of the calculated (average) current flow
as a function of Bext (Fig. 4a,b) shows a (split) peak
around zero magnetic field and two satellite peaks for
Bext = ±~ω/(gµB), where the spin resonance condition
is satisfied. This (split) peak atBext = 0 is due to singlet-
triplet mixing by the inhomogeneous nuclear field, and
the splitting depends on the tunnel coupling, similar as
the observations in [5]. The response from the induced
spin flips via the driving field is visible for the both in-
elastic and resonant transport regime, and the current
flow has comparable magnitude to the peak at Bext = 0.
The satellite peaks are also visible in the experimental
data from [1] (also shown here in Fig. 4), although the
shape and width of the satellite peaks are different, as
we will discuss later.
We want to stress that the ESR satellite peaks only
appear when an inhomogeneous nuclear field is present
in the simulations. In other words, for ∆BN = 0 and
B1 equal in both dots, spin rotations are induced in both
dots at the same time and at the same rate. Starting,
for example, from the state |T+11〉 = | ↑↑〉 transitions are
induced to the state | ↓↓〉 via the intermediate state | ↑
+ ↓〉|↑ + ↓〉/
2 = (|T+11〉+|T
11〉+2|T 011〉)/
2. No mixing
with the singlet state takes place (the evolution is in the
triplet-subspace) and no current will therefore flow.
The ESR sattelite peaks are visible for both resonant
and inelastic transport regime (Figs. 4a,b). For the res-
onant transport regime, we see that for t/σN < 5 the
sattelite peak increases in height when increasing t, sim-
ply because the coupling between the two singlet states
increases. However, further increasing t reduces the sig-
nal, and this is because the exchange splitting then plays
a more important role. Namely, increasing the exchange
splitting reduces the mixing between the T 011 state with
the hybridized singlet state by the nuclear field gradi-
ent. This mixing is a crucial element for detecting the
induced rotations of one of the two electron spins. In
the inelastic transport regime, this exchange splitting is
negligibly small and therefore the height of the sattelite
peak depends only on Γin and the driving field B1.
A study of the height of the satellite peak as a function
of B1 reveals a non-monotonous behaviour, which can
be seen in Fig. 5a. The physical picture behind this
behavior is most easily sketched by distinguishing three
regimes:
1. For B1 < σN,z, for most of the nuclear configu-
rations the spin in at most one of the two dots
is on resonance, so spins are flipped in either the
left or right dot. In that case transitions are in-
duced from e.g. | ↑↑〉 to | ↑↓〉 = |S11〉 + |T 011〉 or
|↓↑〉 = |S11〉 − |T 011〉. The resulting current flow ini-
tially increases quadratically with B1, as one would
normally expect (Fig. 5a).
2. For B1 ≫ σN,z, for most of the nuclear configu-
rations two spins are rotated simultaneously due
B1 (mT)
0 0.2 0.4 0.6 0.8 2 3 4 5 6 7 8
Experimental data
Detuning ~360 eVm
Detuning ~320 eVm
Model
sN,z=1.3mT
0.10.01 1 10
B1 N,z/ 2s
FIG. 5: Height and width of the ESR satellite peak. a) Cir-
cles: calculated ESR peak height as a function of driving am-
plitude B1. Parameters: ~ω = gµB100mT, Bext =100 mT,
σN =2.2mT, ΓL,R = 73 MHz, t=0.3 µeV, ~Γin = gµBσN
and ∆LR = 200 µeV. Lines are the current measurements
for 2 different values of ∆LR. The measurements show time-
dependent (telegraph type) behavior. Therefore, the curves
are obtained by repeating sweeps of B1 and then selecting
the largest current value for each value of B1. b) Calculated
width of the ESR satellite peaks as a function of B1. For
small ESR power the peak is broadened by the random nu-
clear fluctuations, at high powers it is broadened by B1.
to power broadening of the Rabi resonance. The
stronger B1, the more the transitions occur only in
the triplet subspace (the driving field B1 that ro-
tates two spins dominates the S−T0 mixing by the
nuclear spins). As a result, the current decreases
for increasing B1.
3. If B1 ∼ σN,z the situation is more complex because
both processes (rotation of 2 spins simultaneously
and transitions from T 011 to S11) are effective. We
find that if both processes occur with comparable
rates, the overall transition rate to the singlet state
is highest. This is the reason why the current has
a maximum at B1 ≈ σN,z (Fig. 5a).
The experimental data of the ESR satellite peak height
(normalized by the zero-field current flow) for two dif-
ferent values of ∆LR are shown in Fig. 5a. In order to
compare the experimental results with the model we have
estimated the rate Γin from the measured current flow at
Bext = 0 (we found similar values for both curves). The
agreement of the experimental data with the model is
reasonable, as it shows the expected quadratic increase
with B1, as well as a comparable peak height. However,
we see that variations of the level detuning ∆LR can re-
sult in considerable differences of the measured ESR peak
height. We have two possible explanations for the devi-
ations of the experimental data with the model. First,
we have found experimental signatures of dynamic nu-
clear polarization when the ESR resonance condition was
fulfilled. We expect that this is due to feedback of the
electron transport on the nuclear spins (similar to that
discussed in [11, 15, 17]), although the exact processes
are not (yet) fully understood. Second, unwanted electric
fields affect the electron tunnelling processes, but are not
taken into account in the model. We expect that these
electric fields will not change the location and width of
the ESR sattelite peaks because this field does not couple
the spin states. It is however possible that the height of
the satellite peak is altered by the electric field because if
can affect the coupling between the S(0,2) with the S(1,1)
state.
Finally, we discuss the width of the ESR satellite peak
(Fig. 5b). If the inelastic tunnelling process between
the dots (with rate Γin) and B1 are both smaller than
σN,z, the ESR peak (obtained from simulations) is broad-
ened by the statistical fluctuations of the effective nuclear
field. For high B1, the width approaches asymptotically
the line with slope 1 (see Fig. 5b). In this regime, the
peak is broadened by the RF amplitude B1. In the ex-
periment [1], the shape of the satellite peak was different
(flat on top with sharp edges) than expected from the
model. Furthermore, the FWHM was larger than ex-
pected from just σN,z. We attribute this to feedback of
the ESR-induced current flow on the nuclear spin bath.
As a result, a clear FWHM increase with B1 could not
be observed.
It should be noted that in the simulation the central
peak is broader than the satellite peaks. From study-
ing the influence of various parameters in the model, we
conclude that the greater width of the central peak is
caused by the tranverse nuclear field fluctuations (BN,x
and BN,y), which broaden the central peak but not the
ESR satellite peaks.
We conclude that the model discussed here qualita-
tively agrees with the main features that were observed
in the double dot transport measurements that aims at
detecting (continuous wave) ESR of a single electron spin.
The details of the ESR satellite peak height and width
do not agree quantitatively with the model. We believe
these deviations can be attributed to unwanted electric
fields and feedback of the electron transport on the nu-
clear spin polarization. Improving the understanding of
these feedback mechanisms remains interesting for future
investigation as it might point towards a direction to mit-
igate the decoherence of the electron spin [12, 18].
Acknowledgments
This study was supported by the Dutch Organization
for Fundamental Research on Matter (FOM), the Nether-
lands Organization for Scientific Research (NWO) and
the Defense Advanced Research Projects Agency Quan-
tum Information Science and Technology program.
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|
0704.1629 | Donor type semiconductor at low temperature as maser active medium | Microsoft Word - Maser
Donor type semiconductor at low temperature
as maser active medium
Yuri Kornyushin
Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne, CH-3975
In some semiconductors donor impurity atoms can attract additional
electrons, forming negative donor impurity ions. Thus we have 3 energy
levels for electrons: zero energy levels at the bottom of the conductivity
band, negative energy levels of the bounded electrons of the negative
donor impurity ions, and deeper negative energy levels of the outer
electrons of the neutral donor impurity atoms. So the donor impurity
atoms could serve as active centres for a maser. The maximum achievable
relative population is 0.5. Typical wavelength of the generated oscillation
is 0.14 mm; three level scheme could be realized at rather low
temperatures, considerably lower than 6 K.
Let us consider some donor impurity atom in a donor-type semiconductor having
one more outer electron than the host atom of a semiconductor (e.g., the host silicon
atom). In the simplest case of a non-degenerate standard conductivity band the
equation of a motion of a superfluous electron is the same as that for the electron in a
hydrogen atom [1]. The bond energy at that is as follows [1]:
Ebd = (m0e
)(m/m0
), (1)
where m0 is the free electron mass, m is the effective electron mass in semiconductor
conductivity band, e is the electron charge, and is the Planck constant divided by 2.
Comparative to the bond energy in a hydrogen atom (see, e.g., [2]) the right-hand
part of Eq. (1) contains additional factor (m/m0
). At m = 0.1m0 and = 12 [1] this
factor, (m/m0
) = 6.94410
Hydrogen atom can attract an additional electron, forming negative hydrogen ion
[3]. The electron affinity of a free electron to a hydrogen atom is Eah = 0.754 eV [3].
The bond energy in a hydrogen atom, Ebh = (m0e
) = 13.598 eV [3]. Taking into
account that in a donor semiconductor we have an additional factor (m/m0
), we have
the affinity of a conductivity band electron to a donor impurity atom Ead =
0.754(m/m0
) eV and Ebd = 13.598(m/m0
) eV. At m = 0.1m0 and = 12 [1] we have
Ead = 5.23610
eV = 6.076 K and Ebd = 9.44310
So at temperature considerably lower than Ead (about 6 K here) we have donor
atoms acting as active maser centres. We have three energy levels of electrons: zero
energy levels at the bottom of the conductivity band, negative energy levels of
electron, forming negatively charged donor ions, and deeper negative energy levels of
the outer electron of the neutral donor impurity atoms. So we can pump some outer
electrons of some neutral donor impurity atoms to the conductivity band. At low
enough temperature these electrons will form negative impurity ions with some other
neutral donor impurity atoms, thus forming highly populated levels above the ground
state level, Ebd. When high population of the upper levels is achieved, the frequency,
Ebd Ead = 8.9210
eV = 2.1410
(1/s) = 0.14 mm, could be generated or
amplified. It is rather obvious that the maximum concentration of the negative donor
impurity ions, which could be achieved, is 0.5nd (nd is the number of the donor
impurity atoms per unit volume of a semiconductor).
References
1. A. L. Efros. Semiconductors, in: Encyclopaedic Dictionary Solid State Physics, V.
2 (Kiev: Naukova Dumka, 1998), p. 91 (in Russian).
2. L. D. Landau and E. M. Lifshits, Quantum Mechanics (Oxford: Pergamon, 1986).
3. 1988 CRC Handbook of Chemistry and Physics, ed. R. C. Weast (Boca Raton, FL:
CRC).
|
0704.1630 | Exciting the Magnetosphere of the Magnetar CXOU J164710.2-455216 in
Westerlund 1 | Mon. Not. R. Astron. Soc. 000, 1–6 (2007) Printed 29 October 2018 (MN LATEX style file v2.2)
Exciting the Magnetosphere of the Magnetar
CXOU J164710.2-455216 in Westerlund 1
M. P. Muno,1 B. M. Gaensler,2,3 J. S. Clark,3,4 R. de Grijs,5 D. Pooley,6,7
I. R. Stevens,8 & S. F. Portegies Zwart9,10
1Space Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125; [email protected]
2School of Physics A29, The University of Sydney, NSW 2006, Australia
3Harvard-Smithsonian Center for Astrophysics, 60 Garden St. Cambridge, MA 02138
4Department of Physics & Astronomy, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
5Department of Physics & Astronomy, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, U.K.
6Chandra Fellow
7Astronomy Department, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA
8School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
9Astronomical Institute ’Anton Pannekoek’ Kruislaan 403, 1098SJ Amsterdam, the Netherlands
10Section Computational Science Kruislaan 403, 1098SJ Amsterdam, the Netherlands
Accepted 2007 March 15. Received 2007 February 13; in original form 2006 November 25
ABSTRACT
We describe XMM-Newton observations taken 4.3 days prior to and 1.5 days subse-
quent to two remarkable events that were detected with Swift on 2006 September 21
from the candidate magnetar CXOU J164710.2-455216: (1) a 20 ms burst with an en-
ergy of 1037 erg (15–150 keV), and (2) a rapid spin-down (glitch) with ∆P/P ∼ −10−4.
We find that the luminosity of the pulsar increased by a factor of 100 in the interval
between observations, from 1×1033 to 1×1035 erg s−1 (0.5–8.0 keV), and that its spec-
trum hardened. The pulsed count rate increased by a factor of 10 (0.5–8.0 keV), but
the fractional rms amplitude of the pulses decreased from 65 to 11 per cent, and their
profile changed from being single-peaked to exhibiting three peaks. Similar changes
have been observed from other magnetars in response to outbursts, such as that of 1E
2259+586 in 2002 June. We suggest that a plastic deformation of the neutron star’s
crust induced a very slight twist in the external magnetic field, which in turn generated
currents in the magnetosphere that were the direct cause of the X-ray outburst.
Key words: stars: neutron — pulsar: individual (CXOU J164710.2-455216) — X-
rays: bursts — stars: magnetic fields
1 INTRODUCTION
Young, isolated neutron stars come in a variety of mani-
festations, including ordinary radio pulsars, compact cen-
tral objects in supernova remnants, soft gamma repeaters
(SGRs), and anomalous X-ray pulsars (AXPs). The latter
two classes of source share long rotational periods (P=5–10
s), rapid spin-down rates (Ṗ&10−12 s s−1), X-ray luminosi-
ties (LX&10
33 erg s−1) that exceed their spin-down power,
and the frequent production of second-long soft gamma-ray
bursts (Woods & Thompson 2006). These properties sug-
gest that they are magnetars, neutron stars powered by the
unwinding of extremely strong (B&1015 G) internal mag-
netic fields (Thompson & Duncan 1995, 1996).
The phenomenology associated with magnetars is
thought to be driven by how the unwinding internal fields
interact with the crusts of the neutron stars, which in turn
determines the geometries of the external magnetic fields
(Thompson & Duncan 1995, 1996; Thompson, Lyutikov,
& Kulkarni 2002). In some cases, the crusts respond to
the unwinding fields plastically, and the energy is gradu-
ally deposited into the magnetospheres. This causes tran-
sient ‘active periods,’ in which the persistent fluxes in-
crease on timescales of weeks to years (Woods et al. 2004;
Gotthelf et al. 2004). Fractures may also occur in the crust,
which generate waves in the external fields, and in turn pro-
duce sudden soft gamma-ray ‘bursts’ with energies up to
1041 erg (Göğüş et al. 2001; Gavriil, Kaspi, & Woods 2002).
In the most extreme cases, instabilities can rearrange the
entire external magnetic field, producing ‘giant flares’ with
energies of 1044 −1046 erg (Hurley et al. 1999; Palmer et al.
2005; Hurley et al. 2005). Finally, changes in the coupling
between the bulk of the crust and a superfluid component
appear to change the crust’s angular momentum, as is sug-
gested by both secular variations in the spin down rates
c© 2007 RAS
http://arxiv.org/abs/0704.1630v1
2 M. P. Muno et al.
Figure 1. Images of the counts received by the EPIC-pn in the
0.5–10 keV band on 2006 September 16 (left) and 22 (right).
The images are centred on the core of the star cluster Wester-
lund 1 (α, δ = 251.h76792 –45.◦84972 [J2000]). In addition to
the AXP CXOU J164710.2-455216, also visible in the images are
three bright OB/WR stars. The blank strips in the image are
gaps between the chips in the detector array.
on time-scales of weeks (Gavriil & Kaspi 2004; Woods et al.
2006) or sudden, day-long episodes of spin-up (‘glitches’)
or spin-down (Woods et al. 1999; Gavriil & Kaspi 2003;
Dall’Osso et al. 2003; Kaspi et al. 2003; Woods et al. 2004).
Unfortunately, the frequent, sensitive monitoring observa-
tions that are required to identify transient active periods,
to detect bursts, and to track the rotation of these pulsars
have not always been available. Therefore, in many cases
the causal connections between these phenomena have been
unclear (e.g., Gavriil & Kaspi 2003; Woods et al. 2005).
Here we report XMM-Newton observations of the
10.6 s X-ray pulsar, CXOU J164710.2-455216 (Muno et al.
2006), that bracketed a series of events that occurred near
2006 September 21. Near this time, Swift detected a soft
gamma-ray burst (Krimm et al. 2006) and a glitch with
∆P/P ∼ −10−4 (Israel et al. 2007). These events con-
firm our original hypothesis that this source is a magnetar
(Muno et al. 2006). We find that during the interval between
our two XMM-Newton observations, there were also dra-
matic changes in the luminosity, spectrum, and pulse profile
of CXOU J164710.2-455216. We compare these to changes
observed during active periods from other magnetars, and
discuss the implications for the interaction between the mag-
netic fields and crusts of the these neutron stars.
2 OBSERVATIONS
As part of the guest observer programme, XMM-Newton ob-
served CXOU J164710.2-455216 for 46 ks starting on 2005
September 16 at 18:59:38 (UTC). Fortuitously, 4.3 days
later, on 2006 September 21 at 01:34:53 (UTC), the Swift
Burst Alert Telescope (BAT) detected a 20 ms burst from
the direction of Westerlund 1 (Krimm et al. 2006), with an
energy of 3×1037 erg (15–150 keV; for a distance D=5 kpc;
Clark et al. 2005). In response, the director of XMM-Newton
carried out an observation lasting 30 ks beginning 1.5 days
later on 2006 September 22 at 12:40:27 (UTC). We analysed
the XMM-Newton observations in order to study changes in
the X-ray flux, spectrum, and pulse profile.
We analysed the data taken with the European Photon
Figure 2. Phase-averaged spectra of CXOU J164710.2-455216
taken on 2006 September 16 and 22 (top panels), in units of de-
tector counts. Models for the spectra are shown with a solid line:
a single absorbed blackbody on September 16, and two absorbed
blackbodies on September 22. For the latter spectrum, the cool
and hot blackbodies are indicated with the dotted and dashed
lines, respectively. The bottom panels display the difference be-
tween the data and the models, in units of the 1σ uncertainty on
the data. There are systematic residuals at low energies in the
September 22 spectrum, but these are not significant enough to
affect the overall model.
Imaging Camera (EPIC). For most of the timing and spec-
tral analysis, we used data taken with 73.4 ms time resolu-
tion using the pn array. The data from the MOS arrays were
taken with 2.4 s time resolution, which was inadequate for
studying the profile of this 10.6 s pulsar. Moreover, the data
suffered from pile-up during the second observation, when
the source was bright (see below). Therefore, we only used
the MOS data to generate spectra for the first observation.
We processed the observation data files using the stan-
dard tools (epchain and emchain) from the Science Anal-
ysis Software version 7.0. The events were filtered in the
standard manner, and we adjusted the arrival times of the
events to the Solar System barycentre. Images from the
EPIC-pn data are displayed in Figure 1. Comparing the
data from before and after the Swift burst, we find that
CXOU J164710.2-455216 increased in count rate by a factor
of 80 (0.5–8.0 keV).
Next, we extracted pulse-phase-averaged spectra from
within 15′′ of the location of CXOU J164710.2-455216 (α,
δ = 251.h79250, –45.◦87136 [J2000]). Estimates of the back-
ground were extracted from a 30′′ circular region that was
located 1.′5 west of the source region. We obtained the detec-
tor response and effective area using standard tools (rmfgen
and arfgen). The EPIC-pn spectra are displayed in Figure 2.
We modeled these spectra using XSPEC version 12.2.1.
We first assumed that the spectra could be described as
blackbody emission absorbed by interstellar gas and scat-
tered by dust. This model was acceptable for the observa-
tions before the burst on September 16 (χ2/ν = 59.4/67),
but was inconsistent with the data from September 22 (χ2/ν
= 2255/1136). For the later observation, we could model the
spectrum with two continuum components, either the sum
of two blackbodies, or a blackbody plus power law contin-
uum. We assumed that the interstellar absorption column
toward the source did not change between observations. The
spectral parameters, fluxes, and luminosities for the above
models are listed in Table 1. For completeness, we also list
c© 2007 RAS, MNRAS 000, 1–6
Exciting a Magnetar’s Fields 3
Table 1. Spectral Models for CXOU J164710.2-455216
2005 2006
May–Jun Sep 16 Sep 22
Two Blackbodies
NH (10
22 cm−2) 1.28 1.28 1.28(2)
kT1 (keV) 0.60(1) 0.54(1) 0.67(1)
Abb,1 (km
2) 0.09(1) 0.08(1) 3.62(2)
kT2 (keV) . . . . . . 1.7(1)
Abb,2 (km
2) . . . . . . 0.021(6)
FX (10
−13 erg cm−2 s−1) 2.3 1.5 215.7
LX (10
33 erg s−1) 1.4 1.0 109.7
Blackbody Plus Power Law
NH (10
22 cm−2) 1.44 1.44 1.44(1)
kT1 (keV) 0.58(2) 0.52(1) 0.68(1)
Abb,1 (km
2) 0.11(1) 0.11(1) 2.87(3)
Γ . . . . . . 2.07(4)
NΓ (10
−3 cm−2 s−1 keV−1) . . . . . . 3.7(9)
FX (10
−13 erg cm−2 s−1) 2.3 1.5 214.1
LX (10
33 erg s−1) 1.5 1.1 130.3
The reduced chi-squared for both joint fits were 1423/1298. The
interstellar absorption was assumed not to have changed over
the course of these observations. To compute the area of the
blackbody emission, we assumed D=5 kpc. NΓ is the photon
flux density of the power law at 1 keV. Uncertainties are 1σ, for
one degree of freedom. Fluxes are in the 0.5–8.0 keV band.
parameters from models of the spectra taken with Chandra
during 2005 May and June (Muno et al. 2006).
For both models, we found that the luminosity was a
factor of 100 higher (0.5–8.0 keV) 1.5 days after the burst
than it was 4.3 days before the burst. The increase in flux
was largely because the area of the ≈0.5 keV blackbody
increased from 0.1 km2 before the burst to ≈3 km2 after
the burst. It also resulted from the prominence of the hard
component after the burst. Modeled as a kT=1.7 keV black-
body, it produced 26 per cent of the observed flux on 2006
September 22 (18 per cent of the absorption-correction flux).
Modeled as a Γ=2.07 power law, it produced 50 per cent of
the observed flux (70 per cent of the intrinsic flux; 0.5-8.0
keV). If we add these components to our models for the
spectra taken on 2006 September 16, we find that their frac-
tional contribution to the observed flux was lower: <15 per
cent for the blackbody, and <35 per cent for the power law.
To identify pulsations from CXOU J164710.2-455216,
we computed Fourier periodograms using the Rayleigh
statistic. (A search for pulsations from other point sources
in the field revealed no other pulsars.) This provided an
initial estimate of the pulse period, which we then refined
by computing pulse profiles from non-overlapping 5000 s
intervals during each observation, measuring their phases
by cross-correlating them with the average pulse profile
from each observation, and modeling the differences be-
tween the assumed and measured phases using first-order
polynomials. The best-fitting periods were 10.61065(7) s
and 10.61064(8) s for 2006 September 16 and 22, respec-
tively. These values are within 1.5σ of the periods measured
in 2005 May and June, 10.6112(4) s and 10.6107(1) s, re-
spectively (Muno et al. 2006). The reference epochs of the
pulse maxima for the two observations in 2006 September
were 53994.786313(2) and 54000.526588(1) (MJD, Barycen-
tre Dynamical Time). Monitoring observations taken with
Figure 3. Pulse profiles of CXOU J164710.2-455216 taken on
2006 September 16 (top panels) and 2006 September 22 (bottom
panels), and in three energy bands: 0.5–2.0 keV (left panels), 2.0–
3.5 keV (middle panels), and 3.5–7.0 keV (right panels). Two
identical cycles are repeated in each panel. The dashed line in the
top panel represents the background count rate.
Swift reveal that a glitch with a fractional period change of
∆P/P ∼ −10−4 occurred between these two observations; a
discussion of this result is presented in (Israel et al. 2007).
We used these ephemerides to compute the pulse pro-
files in the full band of 0.5–8.0 keV, and three sub-bands:
0.5–2.0 keV, 2.0–3.5 keV, and 3.5–7.0 keV. The root-mean-
squared (rms) amplitudes of the pulsations in the full band
(0.5–8.0 keV) increased from 0.02 count s−1 before the burst,
to 0.29 count s−1 after the burst. At the same time, the frac-
tional rms amplitudes declined from 64 per cent before the
burst to 11 per cent after the burst. Moreover, the pulse pro-
file changed dramatically after the burst, as can be seen in
the profiles from the sub-bands displayed in Figure 3. Before
the burst, the pulse at all energies was single peaked, and
the differences in the pulse profile as a function of energy
are not very pronounced. After the burst, the pulse in the
full band displayed three distinct peaks, and a dependence
on energy developed. Specifically, in the 3.5–7.0 keV band,
the third peak was absent and the flux between the first two
peaks (phases 0.1–0.3) was larger, so that the overall profile
was more sinusoidal at high energies than at low.
We examined whether phase-resolved spectroscopy
could provide any insight into the origin of the pulses. Un-
fortunately, CXOU J164710.2-455216 was too faint on 2006
September 16 to generate spectra for all but the peak of
the pulse. We did examine phase-resolved spectra for 2006
September 22, but found no systematic trend relating the
spectral parameters with the intensity as a function of phase.
Finally, we searched for bursts by examining the time
series of events recorded by the EPIC-pn. We found no ev-
idence for bursts producing more than 4 counts within the
73.4 ms frame time, which placed an upper limit to their ob-
served fluence of 3×10−11 erg cm−2 (for a Γ=1.8 power law;
Krimm et al. 2006), or an energy of <2× 1035 erg (0.5–8.0
keV; D=5 kpc).
c© 2007 RAS, MNRAS 000, 1–6
4 M. P. Muno et al.
3 DISCUSSION
In the 5.8 days between our two XMM-Newton obser-
vations of CXOU J164710.2-455216, a remarkable set of
events occurred. First, the phase-averaged luminosity of
CXOU J164710.2-455216 increased by a factor of ∼100,
from LX = 1 × 10
33 to LX = 1 × 10
35 erg s−1 (0.5–
8.0 keV; Fig. 1; Campana & Israel 2006), and the spec-
trum hardened (Table 1). Energetically, this is the most
important feature of this active period. In the 1.5 days af-
ter the burst, if we conservatively assume the persistent
flux from CXOU J164710.2-455216 was constant, the to-
tal energy released was ∼1040 erg (0.5–8.0 keV). Second,
a 20 ms long burst with an energy of 3 × 1037 erg (15–
150 keV) was detected from this source with the BAT on
board Swift (Krimm et al. 2006). Third, a glitch was ob-
served in the spin period of the pulsar, with ∆P/P ∼ −10−4
(Israel et al. 2007). Fourth, the pulse profile changed from
having a simple, single-peaked structure, to exhibiting three
distinct peaks with pronounced energy dependence (Fig. 3).
Similar changes in the fluxes, spectra, and timing properties
of magnetars have been observed before, but the combina-
tion observed from CXOU J164710.2-455216 is unique.
It is common for the persistent luminosities of mag-
netars to vary on time scales of weeks to years. The per-
sistent luminosities from the SGRs 1900+14 (Woods et al.
2001) and 1806–20 (Woods et al. 2006) and the bright
AXPs 1E 1048.1–5937 (Gavriil & Kaspi 2004; Tiengo et al.
2005) and 1E 2259+586 (Woods et al. 2004) have been
observed to vary by factors of 2–3 around ∼1034 − 1035
erg s−1 (0.5–10 keV). The luminosities of SGR 1627–
41 (Kouveliotou et al. 2003) and the transient AXP XTE
J1810–597 (Ibrahim et al. 2004; Gotthelf et al. 2004) have
been observed to increase by factors of 100, from ∼1033
to ∼1035 erg s−1 (0.5–10 keV). The larger luminosities,
∼1035 erg s−1, appear to be a rough upper envelope for
the persistent 0.5–8.0 keV fluxes of magnetars (not count-
ing bursts and giant flares). Indeed, the active period from
CXOU J164710.2-455216 also had LX ≈ 10
35 erg s−1 (0.5–
8.0 keV). This persistent flux is generally assumed to be
produced because the unwinding internal fields induce grad-
ual, plastic deformations in the crust and external magnetic
fields, which in turn heats the surface or magnetosphere
(Thompson & Duncan 1995, 1996). Therefore, the increase
in the flux from CXOU J164710.2-455216 demonstrates that
either the unwinding of the internal fields, or the response of
the crust to that unwinding, is intermittent and can activate
in .5 days.
The active periods from magnetars are often accompa-
nied by second-long bursts. These bursts are the hallmarks
of SGRs, and during their active periods hundreds will oc-
cur over the course of a year with energies of up to 1041
erg (2–60 keV; Göğüş et al. 2001).The bursts detected from
AXPs have all been weaker, with peak energies of .1038
erg (2–60 keV). In the AXPs XTE J1810–597 (Woods et al.
2005) and 1E 1048.1–5937 (Gavriil, Kaspi, & Woods 2006),
the bursts that have been detected are infrequent and rela-
tively isolated. In 1E 2259+586 (Woods et al. 2004), a series
of bursts were detected during an 11 ks observation that oc-
curred within 7 days of the start of an active period in 2002
June. The burst detected from CXOU J164710.2-455216 re-
sembles those from 1E 2259+586, in that it occurred very
near the start of an active period. The energy of the burst
(3×1037 erg; 15–150 keV) is trivial compared to that released
as persistent flux (&1040 erg; 0.5–8.0 keV), so it is probably
not a trigger, but a symptom of the active period. Under the
magnetar model, the bursts that accompany the active pe-
riods are caused by fractures that occur in the crust. These
fractures inject into the magnetosphere currents that are
unstable to to wave motion, which quickly generates hot, X-
ray emitting plasma (Thompson & Duncan 1995, 1996). It
is reasonable to expect that such fractures would be stronger
and occur more frequently when the persistent flux is higher,
because the crust is already under stress.
Variations in the spin-down rates have been observed
from several luminous (LX & 10
34 erg s−1; 0.5–8.0 keV)
magnetars. Torque variations have been detected from
1E 1048.1–5937 (Gavriil & Kaspi 2004) and SGR 1806–20
(Woods et al. 2006), in association with their active periods.
Sudden period changes have been seen in three cases. Two
glitches have been detected from 1RXS J170849–400910
with ∆P/P ∼ −1 × 10−6 and −6 × 10−6 (Gavriil & Kaspi
2003; Dall’Osso et al. 2003). Neither were associated with
active periods, but the monitoring observations were sparse,
so one could have been missed (Dall’Osso et al. 2003). One
glitch accompanied the 2002 June active period of 1E
2259+586 in which the spin period decreased by ∆P/P ∼
−10−6 (Kaspi et al. 2003; Woods et al. 2004). Finally, a dra-
matic episode of spin-down occurred near the time of a
1044 erg (3–100 keV) giant flare from SGR 1900+14, with
∆P/P ∼ 10−4 (Woods et al. 1999). This is of comparable
magnitude to the glitch from CXOU J164710.2-455216, al-
beit of the opposite sign (Israel et al. 2007).
The glitch appears to have been a major energetic
component of the outburst from CXOU J164710.2-455216.
Glitches are ascribed to sudden changes in the moments of
inertia of the neutron stars that occur when crustal move-
ments change how superfluid in the interior is coupled to
the bulk of the crust (e.g., Dall’Osso et al. 2003; Kaspi et al.
2003). The change in rotational energy during the glitch, as-
suming most of the star rotates as a solid body, is on order
∆Erot ∼ IΩ∆Ω, where I∼10
45 g cm2 is the moment of in-
ertia of a neutron star with mass M=1.4 M⊙ and radius
R=1 km. For CXOU J164710.2-455216 Ω=0.6 rad s−1 and
∆Ω=6×10−5 rad s−1, so ∆Erot∼10
40 erg. However, a larger
input of energy into the stellar interior may be required to
unpin the superfluid vortices and initiate the glitch, ∼1042
erg (e.g., Link & Epstein 1996; Thompson et al. 2000). In
contrast, the radiative output of CXOU J164710.2-455216 in
the first week of this active period was only ∼1040 erg (0.5–
8.0 keV). Whereas for the giant flare from SGR 1900+14 and
the 2002 June active period from 1E 2259+586 it appeared
that most of the energy was radiated away from the mag-
netosphere (Thompson et al. 2000; Woods et al. 2004), for
CXOU J164710.2-455216 most of the energy was probably
input into the interior of the neutron star.
The change in the pulse profile of
CXOU J164710.2-455216 is also difficult to understand
from an energetic standpoint. Changes in the qualitative
shape of the pulse profiles (as opposed to changes in the
pulsed fraction) have only been seen previously from three
sources. For 1E 2259+586, the profile before the 2002 June
burst exhibited two distinct peaks, whereas after the burst
the phases between the peaks contained more flux, so that
c© 2007 RAS, MNRAS 000, 1–6
Exciting a Magnetar’s Fields 5
part of the profile resembled a single plateau of emission
(Woods et al. 2004). This change is minor compared to that
from CXOU J164710.2-455216 in Figure 3. Large changes in
the harmonic structure of the pulse profile have only been
observed in response to the giant flares from SGRs. For
SGR 1900+14 the profile had three peaks before the flare
in 1998, and a single peak during and after (Woods et al.
2001). For SGR 1806–20, the opposite change occurred in
2004: it shifted from having a simple, single-pulsed profile
to having multiple peaks (Woods et al. 2006).
For the SGRs, the changes in the pulse profiles are
thought to occur because the multipole structure of the ex-
ternal magnetic fields are rearranged. This is reasonable,
because the giant flares release a significant fraction of the
energy in the external fields. For a dipole, this would be
B2extR
∼ 1045 G, where we take Bext∼10
14 G, and
R∼10 km (Woods et al. 1999; Hurley et al. 2005). However,
for CXOU J164710.2-455216, and to a lesser degree for 1E
2259+586, it is unreasonable to suggest that active periods
releasing only ∼1040 erg of X-rays resulted from a significant
rearrangement of the exterior magnetic fields.
Instead, we suggest that a change occurred in the dis-
tribution of currents in the magnetosphere. We hypothesize
that the emission in quiescence is thermal emission from
the cooling neutron star, which emerges through a hot spot
where the opacity of the highly-magnetized atmosphere is
lowest (Heyl & Hernquist 1998). A single hot spot on the
surface could explain the single-peaked, fully modulated
(≈70 per cent rms) pulse in quiescence (Özel, Psaltiz, &
Kaspi 2001). We suggest that the active period was initi-
ated when a very small twist was imparted to the magnetic
field by plastic motions of the crust. Currents formed to
compensate for this twist, which heated the surface of the
star and resonantly scattered the emission from its surface
(Table 1). Both of these would contribute to creating the
complex pulse profile (Thompson et al. 2002). If our sce-
nario is correct, when this source returns to quiescence, the
pulse should regain its single-peaked profile.
4 CONCLUSIONS
We have examined the X-ray luminosity, spectrum, and
pulse profile of CXOU J164710.2-455216 before and after
an interval during which Swift detected a soft gamma-ray
burst and a timing glitch from the source. The energy radi-
ated from the exterior was too small to have resulted from
a significant rearrangement of the external magnetic fields
of CXOU J164710.2-455216. Instead, the dramatic change
in the pulse profile indicates that the underlying emission
mechanism changed. Before the burst, the X-ray emission
was probably powered by the thermal energy of the star,
whereas afterwards it was powered by currents in the mag-
netosphere. Moreover, the glitch required an energy at least
as large as the energy released as X-rays, &1040 erg, which
suggests that much of the energy of this event was input
into the interior of the neutron star. Future X-ray observa-
tions of this source will reveal the duration and duty cycle
of this active period, which would constrain the amount of
energy input into the interior. This could help answer why
the emission, which is thought to be produced as the inter-
nal fields of magnetars unwind, can remain inactive for years
and then suddenly turn on in a few days.
ACKNOWLEDGMENTS
We thank N. Schartel for providing the discretionary obser-
vation, G. Israel for sharing the results of the Swift observa-
tions, and the referee for helpful comments. MPM was sup-
ported by the NASA XMM Guest Observer Facility; BMG
by a Federation Fellowship from the Australian Research
Council and an Alfred P. Sloan Research Fellowship; and
SFPZ by the Royal Dutch Academy of Arts and Sciences.
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c© 2007 RAS, MNRAS 000, 1–6
http://arxiv.org/abs/astro-ph/0703684
6 M. P. Muno et al.
Woods P. M., Kouveliotou C., Finger M. H., Göğüş E.,
Wilson C. A., Patel S. K., Hurley K., Swank J. H. 2006,
ApJ, 654, 470
Woods P. M., Thompson C. 2006, in Compact Stellar X-
ray Sources, eds. W. Lewin, M. van der Klis, Cambridge
University Press, 547
c© 2007 RAS, MNRAS 000, 1–6
Introduction
Observations
Discussion
Conclusions
|
0704.1631 | Further Evidence that the Redshifts of AGN Galaxies May Contain
Intrinsic Components | Further Evidence that the Redshifts of AGN Galaxies May
Contain Intrinsic Components
M.B. Bell1
ABSTRACT
In the decreasing intrinsic redshift (DIR) model galaxies are assumed to be
born as compact objects that have been ejected with large intrinsic redshift com-
ponents, zi, out of the nuclei of mature AGN galaxies. As young AGN galaxies
(quasars) they are initially several magnitudes sub-luminous to mature galaxies
but their luminosity gradually increases over 108 yrs, as zi decreases and they
evolve into mature AGN galaxies (BLLacs, Seyferts and radio galaxies). Evi-
dence presented here that on a logz-mv plot the bright edge of the AGN galaxy
distribution at z = 0.1 is unquestionably several magnitudes sub-luminous to the
brightest radio galaxies is then strong support for this model and makes it likely
that the high-redshift AGN galaxies (quasars) are also sub-luminous, having sim-
ply been pushed above the radio galaxies on a logz-mv plot by the presence of a
large intrinsic component in their redshifts. An increase in luminosity below z =
0.06 is also seen. It is associated in the DIR model with an increase in luminosity
as the sources mature but, if real, is difficult to interpret in the cosmological
redshift (CR) model since at this low redshift it is unlikely to be associated with
a higher star formation rate or an increase in the material used to build galax-
ies. Whether it might be possible in the CR model to explain these results by
selection effects is also examined.
Subject headings: galaxies: active - galaxies: distances and redshifts - galaxies:
quasars: general
1. Introduction
Because the belief that the redshift of quasars is cosmological has become so entrenched,
and the consequences now of it being wrong are so enormous, astronomers are very reluctant
1Herzberg Institute of Astrophysics, National Research Council of Canada, 100 Sussex Drive, Ottawa,
ON, Canada K1A 0R6; [email protected]
http://arxiv.org/abs/0704.1631v2
– 2 –
to consider other possibilities. However, there is increasing evidence that some galaxies may
form around compact, seed objects ejected with a large intrinsic redshift component from
the nuclei of mature active galaxies. In this model, as the intrinsic component decreases
the compact objects evolve into mature active galaxies in a time frame of a few times 108
yrs (Arp 1997, 1998, 1999; Bell 2002a,b,c,d, 2004, 2006; Bell and McDiarmid 2006, 2007;
Burbidge 1999; Galianni et al. 2005; Lopéz-Corredoira and Gutiérrez 2006). In the DIR
model radio galaxies represent the end of the AGN galaxy evolutionary sequence, where
most of the intrinsic redshift component has disappeared and their luminosity has peaked.
Only then can these objects be detected to large cosmological distances and can it be seen
that they are good standard candles. There is every reason to assume that at each stage
of their evolution (at each zi value) they will also be good standard candles. In this paper
AGN refers to the active nucleus and AGN galaxy refers to the nucleus plus host galaxy.
It was recently demonstrated that the high redshift AGN galaxies detected to date
appear to have a mean distance near 300 Mpc (Bell 2004), and therefore few beyond ∼ 500
Mpc will have been detected. However, in the DIR model it is assumed that this birthing
process through compact object ejection has taken place at all cosmological epochs and that
those galaxies that were born in the early universe still survive today, even though they
will have almost certainly evolved beyond the mature AGN galaxy (radio galaxy) stage.
Although they may no longer contain active nuclei, by this point in their evolution their
redshifts will contain only a very small intrinsic redshift component. This remnant intrinsic
redshift is observed to-day in common spiral galaxies (Tifft 1996, 1997; Bell and Comeau
2003; Bell, Comeau and Russell 2004), and the local Hubble constant is found to be Ho
= 58 km s−1 Mpc−1 when the intrinsic components are removed (Bell and Comeau 2003;
Bell, Comeau and Russell 2004). This value is smaller than the value (Ho = 72) obtained
by Freedman et al. (2001) before removal of the intrinsic components. In most respects the
DIR model is perfectly compatible with the standard Big Bang model of the Universe. It
differs mainly in the way galaxies are born and the claim that in this model at least the radio
galaxies pass through an initial short-lived AGN period (108 yrs) in which their redshifts
contain an intrinsic component that quickly disappears. After that, as they evolve through
the next 1010 years they can be used as they are today, to study cosmology. Although there
is now a considerable amount of evidence supporting the DIR model, there are also some
well-known arguments against this model that have been raised by those who support the
CR model (e.g. the Lyman forest, lensing by intervening galaxies, etc.). An explanation
of these arguments in the DIR model can be found in the Discussion section of a previous
paper (Bell 2004).
In the CR model the location of high-redshift AGN galaxies (quasars) on a logz-mv plot
can be explained by the presence of a non-thermal component superimposed on their optical
– 3 –
luminosity. In the DIR model their location on this plot is explained by the presence of a
non-cosmological redshift component superimposed on their redshift. This paper uses an
updated logz-mv plot containing over 100,000 AGN galaxies to compare the most luminous
radio galaxies and first-ranked cluster galaxies at each redshift to the high luminosity edge
of the AGN galaxy distribution in an attempt to see which model (CR or DIR model) can
best explain the data. In this paper the standard candle (constant luminosity) slope is used
as a reference to make luminosity comparisons at a given redshift. This is shown as a dashed
line in Fig 1 and a solid line in Fig 2. Luminosity increases to the left.
2. The Data
A logz-mv plot for those radio sources with measured redshifts that were detected in
the 1 Jy radio survey (Stickel et al. 1994) is presented in Fig 1. The quasars are plotted as
filled circles and the radio galaxies as open squares. As discussed above, in the DIR model
the radio galaxies are the objects that high-redshift quasars and other AGN galaxies evolve
into when their intrinsic redshift component has largely disappeared. In Fig 1, first-ranked
cluster galaxies (Sandage 1972a; Kristian et al. 1978) are indicated by the dashed line. The
most luminous radio galaxies, like first-ranked cluster galaxies, are clearly good standard
candles to large cosmological distances, and their redshifts must then be cosmological, as
expected in both the CR and DIR models since any intrinsic redshift component will have
almost completely disappeared.
All the sources listed as quasars and active galaxies in the updated Véron-Cetty/Véron
catalogue (Véron-Cetty and Véron 2006) (hereafter VCVcat) are plotted in Fig 2. Since the
VCVcat is made up of AGN galaxies from many different surveys, there will undoubtedly
be differences in the selection criteria involved. However, since AGN galaxies are easily
distinguishable from other types of galaxies, the normally strict selection criteria are not
required in this case to obtain a source sample that is made up almost entirely of AGN
galaxies. In that sense the VCVcat is probably the most complete sample of AGN galaxies
available to-day. Because the source distribution in the plot in Fig 2 is continuous, the
sources listed as quasars and AGN are clearly the same, and there is therefore no reason
to separate them into two different categories as was done arbitrarily in the VCVcat. This
should not be too surprising since they have long been lumped together in unification models
(Antonucci 1993). In Fig 2 the abrupt decrease in the number of sources for 0.5 < z < 3
and mv > 21 is explained by a faint magnitude cut-off near mv = 21m. It cannot affect the
conclusions drawn here because at each redshift we are only comparing the bright, or high
luminosity, edge of the source distribution (where the source density increases sharply when
– 4 –
moving from bright to faint). For example, in Fig 2, at z = 0.03, 0.06, 0.15 and 1, the high
luminosity edge of the AGN galaxy distribution is at mv = 14, 15, 18, 17, respectively.
However, some surveys have had other observer, or program-imposed limits applied
that can also affect the bright edge of the source distribution and this is discussed in more
detail in Section 3.1. The slope change in the high-redshift tail (z > 3) may be due to
uncertainties in converting to visual magnitudes and/or to large k-dimming effects that have
been unaccounted for. Whatever the cause, it will also not affect the arguments presented
here that only apply to sources at lower redshifts.
3. Discussion
In Fig 1, the large triangle shows where the quasars would be located in the DIR
model if the intrinsic component in their redshifts could be removed. All must lie below
the radio galaxies. In this plot there are no AGN galaxies below the radio galaxies, and
it is therefore easy to conclude that quasars are at the distance implied by their redshifts
and are therefore super-luminous to first ranked cluster galaxies at all epochs. This was the
conclusion drawn by Sandage (1972b, see his Fig 4) from a plot similar to Fig 1. Sandage
argued that since no quasars lie to the right (fainter) of the radio galaxy distribution, this
can be understood if a quasar consists of a normal, strong radio galaxy with a non-thermal
component superimposed on its optical luminosity. He concluded from this evidence that
quasars redshifts are cosmological.
In Fig 2 many of the high redshift quasars are also located above the radio galaxies,
however, here most of the low- and intermediate-redshift AGN galaxies fall below the radio
galaxy line. This is what is expected in the DIR model where AGN galaxies are born sub-
luminous and reach their most luminous point when the intrinsic redshift component has
disappeared. They must therefore all fall below the mature galaxy line. If those detected to
date are all nearer than ∼ 500 Mpc (Bell 2004) most will also be located below the dashed
line at z = 0.1 in Fig 2. This is what is seen in Fig 2 when the intrinsic component is
small. The fact that low-redshift AGN galaxies are located below this line when the intrinsic
component is too small to push them above it, suggests strongly that it is only the intrinsic
component present in the high redshift sources that has pushed these sources above the radio
galaxies. This argument is also supported by the shape of the plot in Fig 2, which starts out
flat near z = 0.06, steepening gradually to z = 0.2 and then more rapidly to high redshifts.
This conclusion is further supported by the fact that the zi ∼ 0 AGN galaxies (radio galaxies)
are good standard candles, and there is therefore no reason to think that the other AGN
galaxies will not be, for a given intrinsic redshift value.
– 5 –
Because almost all of the AGN galaxies are less luminous than the highest luminosity
radio galaxies and first-ranked cluster galaxies at redshifts below z ∼ 0.3, the explanation
proposed by Sandage (1972b) can no longer be valid. Quasars cannot be normal radio
galaxies, or even Seyferts, with a non-thermal optical component superimposed. In fact,
since the high luminosity edge of the AGN galaxy distribution in Fig 2 is ∼ 3 mag fainter
than the high luminosity edge of the radio galaxies at z = 0.1, if quasars are sub-luminous
galaxies brightened by a superimposed non-thermal optical component, at z = 2 this su-
perimposed component would have to increase the optical luminosity of the source by up
to ∼ 9 magnitudes. This could even get worse at higher redshifts when k-dimming effects
are included, which would make the standard model involving a superimposed non-thermal
nuclear component increasingly difficult to believe.
In the CR model the peak in quasar activity (luminosity and number) near z = 2 is
assumed to be associated with a period when the star formation rate was higher than at
present, and because there was more raw material around to make galaxies. In Fig 2, not only
does the high luminosity edge of the AGN galaxies get intrinsically much fainter towards low
redshifts (moving further to the right relative to the standard candle slope), below z ∼ 0.3
this decrease in luminosity begins to slow down. Below z = 0.1 their luminosity begins to
increase again, eventually approaching that of the brightest radio galaxies. How is this to
be explained in the CR model when we can no longer use the argument that there is more
raw material around? This is one of the questions that will need to be addressed if the CR
model is to continue to be favored, since this increase is exactly what is predicted in the DIR
model as the AGN galaxies mature into radio galaxies. One possible explanation in the CR
model is discussed in the following section.
3.1. Selection Effects in the Data
Although in a sample like VCVcat it is difficult to take into account all of the selection
effects that might be active, since the Sloan Digital Sky Survey (SDSS) sources are likely to
make up the largest single portion of the sample the target selection process in that survey
is worth examining. First, the survey is sensitive to all redshifts lower than z = 5.8, and the
overall completeness is expected to be over 90% (Richards et al. 2002). Extended sources
were also targeted as low-redshift quasar candidates in order to investigate the evolution
of AGN at the faint end of the luminosity function. During the color selection process no
distinction was made between quasars and the less luminous Seyfert nuclei. Objects that had
the colors of low-redshift AGN galaxies were targeted even if they were resolved. This policy
was in contrast to some other quasar surveys that reject extended objects, thereby imposing
– 6 –
a lower limit to the redshift distribution of the survey (Richards et al. 2002). In addition
to selecting normal quasars, the selection algorithm also makes it sensitive to atypical AGN
such as broad absorption line quasars and heavily reddened quasars (Richards et al. 2002).
In addition to the detection limit set by the sensitivity of the observing system the
SDSS also contains two additional observer, or program-imposed, limits. One of these was
a faint-edge limit at i∗ = 19.1m, and the other was a bright-edge cut-off at i∗ = 15m. The
reasons why these limits were imposed can be found in Richards et al. (2002). Although
color-selected quasar candidates below z = 3 were only targeted to a Galactic extinction-
corrected i∗ magnitude of 19.1, as noted above, since we are only examining the bright edge
of the logz-mv plot, this faint edge limit is not expected to have affected the results. However,
the bright edge cut-off at i∗ = 15m could have affected the shape of the bright edge of the
logz-mv plot and this needs to be examined more closely.
In Fig 2, for 0.7 < z < 3 it is possible that the bright edge cut-off could have prevented
the detection of some of the brighter sources, although if many were missed we might expect
to see some evidence of a sharp cut-off along the bright edge similar to that seen at mv ∼ 21m.
None is seen. Furthermore, since the bright edge of the distribution between z = 0.1 and z
= 0.5 is at least 1 magnitude fainter than many sources detected at the higher redshifts it
seems unlikely that the i∗ = 15 limit could have significantly affected the bright edge of the
distribution in this redshift range. In fact, it is apparent from Fig 2 of Schneider et al. (2007)
(which is a plot of the i magnitude of the 77,429 objects in the SDSS Fifth Data Release
quasar catalogue versus redshift) that in the SDSS catalogue it is unlikely that many sources
were missed at any redshift because of the cut-off at i = 15.
It is also worth noting that the sources that lie outside the limits imposed in the
SDSS have not been discarded. SDSS photometry for those objects brighter than i∗ =
15 is sufficiently accurate that they can be used in follow-up studies should the need arise.
(Richards et al. 2002).
In Fig 2 there is also an increase seen in the number of AGN galaxies as z increases.
Such an increase is expected in the CR model where the redshift is distance related and
where it would be due to the increasing volume of space sampled as z increases. This would
then support the CR model. However, it needs to be kept in mind also that if a bright
edge cut-off is affecting the shape of the bright edge of the source distribution, it would
presumably also have created this increase in source number with redshift by preventing the
detection of many more of the bright sources at low redshifts. In the DIR model, where
the redshift of AGN galaxies is age related, the number density of sources as a function of
cosmological redshift can only be determined after the intrinsic component is removed.
– 7 –
This paper examines the AGN galaxies listed in the VCVcat and draws conclusions
based on that sample. It contains the quasars found in the SDSS that were available at the
time the catalog was prepared, and approximately 11,000 Seyferts and BLLacs, but whether
the current VCVcat contains many AGN galaxies found in the SDSS galaxy survey is unclear.
Hao et al. (2005a,b) have pointed out that although the color selection technique used in
the SDSS is very efficient, selecting AGN galaxies is a complex process and requires that the
optical luminosity of the active nucleus be at least comparable to the luminosity of the host
galaxy for the color to be distinctive. Thus the color selection systematically misses AGN
galaxies with less luminous nuclei at low redshift. If mainly faint sources at redshifts below
z = 0.08 were missed, it is conceivable that the bright edge currently visible near mv ∼ 14
might have been created by the selection process. In this case there might be no luminosity
increase below z = 0.1, which would be more easily explained in the CR model. However,
if the VCVcat does not contain many AGN galaxies found in the SDSS galaxy survey this
would not be a problem here. Furthermore, in the DIR model, where the luminosity of the
host galaxy is predicted to increase as it matures, presumably bright AGN galaxies as well
as faint ones could be missed if the host galaxy has brightened significantly so as to swamp
the nucleus. Also, in Fig 2, the bend in the distribution towards higher luminosities near z =
0.07 and mv = 18 does seem to point to a real increase in the luminosity at lower redshifts.
However, if this sample is incomplete at low redshifts for a particular magnitude range, the
conclusions drawn here may change when a more complete sample becomes available.
Also, if AGN galaxies at vastly different redshifts are to be compared, as here, it is
important that the optical magnitude of the entire galaxy be used and not simply that of
the nucleus. It is the total magnitude that has invariably been used for high redshift quasars
because of the difficulty of separating the nuclear and host contributions. Hao et al. (2005b)
point out that, in attempting to obtain the luminosity function of the active nucleus, it is
important that it not be contaminated by the host galaxy. Since the brightening predicted
in the DIR model below z = 0.1 is due to the host galaxy maturing and increasing in
luminosity, the contribution from the entire host galaxy must be included in the magnitudes
used in the logz-mv plot if the brightening is to be detected. Although the luminosity of
the nucleus may be adequate in determining the luminosity function of the active nucleus in
the CR model, because of the complex process required to identify AGN galaxies (Hao et al.
2005b), obviously great care will be required in obtaining the magnitudes of low redshift
AGN galaxies if they are to be used in logz-mv plots.
– 8 –
4. Conclusion
The most luminous radio galaxies and first-ranked cluster galaxies have been compared
here to the high luminosity edge of the AGN galaxy distribution on a logz-mv plot. It is
found that while the radio galaxies and cluster galaxies are good standard candles at all
epochs, the luminosity of the AGN galaxies varies significantly from one epoch to another.
Compared to the comparison galaxies the AGN galaxies are found to be super-luminous at
high redshifts, but become sub-luminous as the redshift decreases. These new results show
that below z = 0.3 the rate of luminosity decrease begins to slow down and below z = 0.1
the luminosity begins to increase again. Although their apparent super-luminous nature at
high z can be explained by a higher star formation rate, and the fact that there might have
been more raw material around to make galaxies at that epoch, a luminosity increase below
z = 0.1 is more difficult to explain when these arguments are unlikely explanations. It is
therefore concluded here that the evidence favors the argument that the high redshift AGN
galaxies (quasars) that do lie above the mature galaxy line on a logz-mv plot have all been
pushed there because of a large intrinsic component in their redshifts and not because they
have a superimposed non-thermal component that is many magnitudes brighter than that
seen in radio galaxies. All AGN galaxies then will be sub-luminous to mature galaxies, as
predicted in the DIR model. For a given intrinsic redshift component, all are likely also
to be good standard candles. Finally, if it turns out that many faint AGN galaxies at low
redshifts have been missed in a particular magnitude range the conclusion that the bright
edge of the logz-mv plot increases again in this redshift range may need to be re-evaluated.
Such an effect might be introduced by the selection effect discussed by Hao et al. (2005a,b),
but only if the VCVcat contains many of the SDSS AGN galaxies, as explained in Sec 3.1.
I wish to thank two anonymous referees for suggestions on how this paper might be
improved. I also thank Dr. D. McDiarmid for helpful comments.
– 9 –
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This preprint was prepared with the AAS LATEX macros v5.2.
– 11 –
Fig. 1.— Plot of redshift versus optical magnitude for quasars (filled circles) and radio
galaxies (open squares) from (Stickel et al. 1994). The dashed line represents brightest clus-
ter galaxies from (Sandage 1972a; Kristian et al. 1978). See text for an explanation of the
triangle.
– 12 –
Fig. 2.— Logz-mv plot of all 106,958 sources listed as quasars and active galaxies in the
Véron-Cetty-Véron catalogue. The solid line indicates first-ranked clusters from Sandage
(1972a); Kristian et al. (1978). The dashed line indicates the maximum distance for high-
redshift AGN detected to date from (Bell 2004). In the DIR model any AGN that lie above
this line have been pushed there by the presence of an intrinsic redshift component.
Introduction
The Data
Discussion
Selection Effects in the Data
Conclusion
|
0704.1632 | Semiclassical scattering amplitude at the maximum point of the potential | SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM
POINT OF THE POTENTIAL
IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Abstract. We compute the scattering amplitude for Schrödinger operators at a critical
energy level, corresponding to the maximum point of the potential. We follow [30], using
Isozaki-Kitada’s representation formula for the scattering amplitude, together with results
from [5] in order to analyze the contribution of trapped trajectories.
Contents
1. Introduction 2
2. Assumptions and main results 4
3. Proof of the main resolvent estimate 10
4. Representation of the Scattering Amplitude 16
5. Computations before the critical point 19
5.1. Computation of u− in the incoming region 19
5.2. Computation of u− along γ
6. Computation of u− at the critical point 23
6.1. Study of the transport equations for the phases 25
6.2. Taylor expansions of ϕ+ and ϕ
6.3. Asymptotics near the critical point for the trajectories 34
6.4. Computation of the ϕkj ’s 36
7. Computations after the critical point 43
7.1. Stationary phase expansion in the outgoing region 43
7.2. Behaviour of the phase function Φ 47
Date: April 12, 2007.
2000 Mathematics Subject Classification. 81U20,35P25,35B38,35C20.
Key words and phrases. Scattering amplitude, critical energy, Schrödinger equation.
Acknowledgments: We would like to thank Johannes Sjöstrand for helpful discussions during the prepa-
ration of this paper. The first author also thanks Victor Ivrii for supporting visits to Université Paris Sud,
Orsay, and the Department of Mathematics at Orsay for the extended hospitality.
http://arxiv.org/abs/0704.1632v2
2 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
7.3. Integration with respect to time 48
Appendix A. Proof of Proposition 2.5 53
Appendix B. A lower bound for the resolvent 53
Appendix C. Lagrangian manifolds which are transverse to Λ± 55
Appendix D. Asymptotic behaviour of certain integrals 57
References 60
1. Introduction
We study the semiclassical behavior of scattering amplitude at energy E > 0 for Schrö-
dinger operators
(1.1) P (x, hD) = −h
∆ + V (x)
where V is a real valued C∞ function on Rn, which vanishes at infinity. We shall suppose
here that E is close to a critical energy level E0 for P , which corresponds to a non-degenerate
global maximum of the potential. Here, we address the case where this maximum is unique.
Let us recall that, if V (x) = O(〈x〉−ρ) for some ρ > (n + 1)/2, then for any ω 6= θ ∈ Sn−1
and E > 0, the problem
P (x, hD)u = Eu,
u(x, h) = ei
2Ex·ω/h +A(ω, θ,E, h)e
2E|x|/h
|x|(n−1)/2
+ o(|x|(1−n)/2) as x→ +∞, x|x| = θ,
has a unique solution. The scattering amplitude at energy E for the incoming direction ω
and the outgoing direction θ is the real number A(ω, θ,E, h).
For potentials that are not decaying that fast at infinity, it is not that easy to write down a
stationary formula for the scattering amplitude: If V (x) = O(〈x〉−ρ) for some ρ > 1, one can
define the scattering matrix at energy E using wave operators (see Section 4 below). Then,
writing
(1.2) S(E, h) = Id− 2iπT (E, h),
one can see that T (E, h) is a compact operator on L2(Sn−1), which kernel T (ω, θ,E, h) is
smooth out of the diagonal in Sn−1×Sn−1. Then, the scattering amplitude is given for θ 6= ω,
(1.3) A(ω, θ,E, h) = c(E))h(n−1)/2T (ω, θ,E, h),
where
(1.4) c(E) = −2π(2E)−
4 (2π)
2 e−i
(n−3)π
We proceed here as in [30], where D. Robert and H. Tamura have studied the semiclassical
behavior of the scattering amplitude for short range potentials at a non-trapping energy E .
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 3
An energy E is said to be non-trapping when K(E), the trapped set K(E) at energy E, is
empty. This trapped set is defined as
(1.5) K(E) =
(x, ξ) ∈ p−1(E), exp(tHp)(x, ξ) 6→ ∞ as t→ ±∞
where Hp is the Hamiltonian vector field associated to the principal symbol p(x, ξ) =
V (x) of the operator P . Notice that the scattering amplitude has been first studied, in the
semiclassical regime, by B. Vainberg [32] and Y. Protas [27] in the case of compactly supported
potential, and for non-trapping energies, where they obtained the same type of result.
Under the non-trapping assumption, and some other non-degeneracy condition (in fact our
assumption (A4) below), D. Robert and H. Tamura have shown that the scattering amplitude
has an asymptotic expansion with respect to h. The non-degeneracy assumption implies in
particular that there is a finite number N∞ of classical trajectories for the Hamiltonian p,
with asymptotic direction ω for t→ −∞ and asymptotic direction θ as t→ +∞. Robert and
Tamura’s result is the following asymptotic expansion for the scattering amplitude:
(1.6) A(ω, θ,E, h) =
iS∞j /h
aj,m(ω, θ,E)h
m +O(h∞), h→ 0,
where S∞j is the classical action along the corresponding trajectory. Also, they have computed
the first term in this expansion, showing that it can be given in terms of quantities attached
to the corresponding classical trajectory only.
There are also some few works concerning the scattering amplitude when the non-trapping
assumption is not fulfilled. In his paper [24], L. Michel has shown that, if there is no trapped
trajectory with incoming direction ω and outgoing direction θ (see the discussion after (2.6)
below), and if there is a complex neighborhood of E of size ∼ hN for some N ∈ N possibly
large, which is free of resonances, then A(ω, θ,E, h) is still given by Robert and Tamura’s
formula. The potential is also supposed to be analytic in a sector out of a compact set, and
the assumption on the existence of a resonance free domain around E amounts to an estimate
on boundary value of the meromorphic extension of the truncated resolvent of the for
(1.7) ‖χ(P − (E ± i0))−1χ‖ = O(h−N ), χ ∈ C∞0 (Rn).
Of course, these assumptions allow the existence of a non-empty trapped set.
In [2] and [3], the first author has shown that at non-trapping energies or in L. Michel’s
setting, the scattering amplitude is an h-Fourier Integral Operator associated to a natural
scattering relation. These results imply that the scattering amplitude admits an asymptotic
expansion even without the non-degeneracy assumption, and in the sense of oscillatory inte-
grals. In particular, the expansion (1.6) is recovered under the non-degeneracy assumption
and as an oscillatory integral.
In [21], A. Lahmar-Benbernou and A. Martinez have computed the scattering amplitude
at energy E ∼ E0, in the case where the trapped set K(E0) consists in one single point
corresponding to a local minimum of the potential (a well in the island situation). In that
case, the estimate (1.7) is not true, and their result is obtained through a construction of the
resonant states.
In the present work, we compute the scattering amplitude at energy E ∼ E0 in the case
where the trapped set K(E0) corresponds to the unique global maximum of the potential.
4 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
The one-dimensional case has been studied in [28, 14, 15], with specific techniques, and we
consider here the general n > 1 dimensional case.
Notice that J. Sjöstrand in [31], and P. Briet, J.-M. Combes and P. Duclos in [7, 8] have
described the resonances close to E0 in the case where V is analytic in a sector around R
From their result, it follows that Michel’s assumption on the existence of a not too small
resonance-free neighborhood of E0 is satisfied. However, we show below (see Proposition 2.5)
that for any ω ∈ Sn−1, there is at least one half-trapped trajectory with incoming direction
ω, so that L. Michel’s result never applies here.
Here, we do not assume analyticity for V . We compute the contributions to the scattering
amplitude arising from the classical trajectories reaching the unstable equilibrium point, which
corresponds to the top of the potential barrier. At the quantum level, tunnel effect occurs,
which permits the particle to pass through this point. Our computation here relies heavily
on [5], where a precise description of this phenomena has been obtained. In a forthcoming
paper, we shall show that in this case also, the scattering amplitude is an h-Fourier Integral
Operator.
This paper is organized in the following way. In Section 2, we describe our assumptions, and
state our main results: a resolvent estimate, and the asymptotic expansion of the scattering
amplitude in the semiclassical regime. Section 3 is devoted to the proof of the resolvent
estimate, from which we deduce in Section 4 estimates similar to those in [30]. In that
section, we also recall briefly the representation formula for the scattering amplitude proved
by Isozaki and Kitada, and introduce notations from [30]. The computation of the asymptotic
expansion of the scattering amplitude is conducted in sections 5, 6 and 7, following the classical
trajectories. Eventually, we have put in four appendices the proofs of some side results or
technicalities.
2. Assumptions and main results
We suppose that the potential V satisfies the following assumptions
(A1) V is a C∞ function on Rn, and, for some ρ > 1,
∂αV (x) = O(〈x〉−ρ−|α|).
(A2) V has a non-degenerate maximum point at x = 0, with E0 = V (0) > 0 and
∇2V (0) =
. . .
, 0 < λ1 ≤ λ2 ≤ . . . ≤ λn.
(A3) The trapped set at energy E0 is K(E0) = {(0, 0)}.
Notice that the assumptions (A1)–(A3) imply that V has an absolute global maximum
at x = 0. Indeed, if L = {x 6= 0; V (x) ≥ E0} was non empty, the geodesic, for the
Agmon distance (E0 − V (x))1/2+ dx, between 0 and L would be the projection of a trapped
bicharacteristic (see [1, Theorem 3.7.7]).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 5
As in D. Robert and H. Tamura in [30], one of the key ingredient for the study of the
scattering amplitude is a suitable estimate for the resolvent. Using the ideas in [5, Section 4],
we have obtained the following result, that we think to be of independent interest.
Theorem 2.1. Suppose assumptions (A1), (A2) and (A3) hold, and let α > 1
be a fixed
real number. We have
(2.1) ‖P − (E ± i0))−1‖α,−α . h−1| lnh|,
uniformly for |E − E0| ≤ δ, with δ > 0 small enough. Here ‖Q‖α,β denotes the norm of the
bounded operator Q from L2(〈x〉α dx) to L2(〈x〉β dx).
Moreover, we prove in the Appendix B that our estimate is not far from optimal. Indeed,
we have the
Proposition 2.2. Under the assumptions (A1) and (A2), we have
(2.2) ‖(P − E0 ± i0)−1‖α,−α & h−1
| lnh|.
We would like to mention that in the case of a closed hyperbolic orbit, the same upper
bound has been obtained by N. Burq [9] in the analytic category, and in a recent paper [11]
by H. Christianson in the C∞ setting.
As a matter of fact, in the present setting, S. Nakamura has proved in [26] an O(h−2)
bound for the resolvent. Nakamura’s estimate would be sufficient for our proof of Theorem
2.6, but it is not sharp enough for the computation of the total scattering cross section along
the lines of D. Robert and H. Tamura in [29]. In that paper, the proof relies on a bound
O(h−1) for the resolvent, but it is easy to see that an estimate like O(h−1−ε) for any small
enough ε > 0 is sufficient. If we denote
(2.3) σ(ω,E0, h) =
|A(ω, θ,E, h)|2dθ,
the total scattering cross-section, and following D. Robert and H. Tamura’s work, our resolvent
estimates gives the
Theorem 2.3. Suppose assumptions (A1), (A2) and (A3) hold, and that ρ > n+1
, n ≥ 2.
If |E − E0| < δ for some δ > 0 small enough, then
(2.4) σ(ω,E, h) = 4
2−1(2E)−1/2h−1
V (y + sω)ds
dy +O(h−(n−1)/(ρ−1)).
Now we state our assumptions concerning the classical trajectories associated with the
Hamiltonian p, that is curves t 7→ γ(t, x, ξ) = exp(tHp)(x, ξ) for some initial data (x, ξ) ∈
T ∗Rn. Let us recall that, thanks to the decay of V at infinity, for given α ∈ Sn−1 and
z ∈ α⊥ ∼ Rn−1 (the impact plane), there is a unique bicharacteristic curve
(2.5) γ±(t, z, α,E) = (x±(t, z, α,E), ξ±(t, z, α,E))
such that
(2.6)
|x±(t, z, α,E) −
2Eαt− z| = 0,
|ξ±(t, z, α,E) −
2Eα| = 0.
6 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We shall denote by Λ−ω the set of points in T
n lying on trajectories going to infinity with
direction ω as t → −∞, and Λ+θ the set of those which lie on trajectories going to infinity
with direction θ as t→ +∞:
(2.7)
Λ−ω =
γ−(t, z, ω,E) ∈ T ∗Rn, z ∈ ω⊥, t ∈ R
Λ+θ =
γ+(t, z, θ, E) ∈ T ∗Rn, z ∈ θ⊥, t ∈ R
We shall see that Λ−ω and Λ
θ are in fact Lagrangian submanifolds of T
Under the assumptions (A1), (A2) and (A3) there are only two possible behaviors for
x±(t, z, α,E) as t→ ∓∞: either it escapes to ∞, or it goes to 0.
First we state our assumptions for the first kind of trajectories. For these, we also have,
for some ξ∞(z, ω,E),
ξ−(t, z, ω) = ξ∞(z, ω,E),
and we shall say that the trajectory γ−(t, z, ω,E) has initial direction ω and final direction
θ = ξ∞(z, ω,E)/2
E. As in [30] we shall suppose that there is only a finite number of
trajectories with initial direction ω and final direction θ. This assumption can be given in
terms of the angular density
(2.8) σ̂(z) = |det(ξ∞(z, ω,E), ∂z1ξ∞(z, ω,E), . . . , ∂zn−1ξ∞(z, ω,E))|.
Definition 2.4. The outgoing direction θ ∈ Sn−1 is called regular for the incoming direction
ω ∈ Sn−1, or ω-regular, if θ 6= ω and, for all z′ ∈ ω⊥ with ξ∞(z′, ω,E) = 2
Eθ, the map
ω⊥ ∋ z 7→ ξ∞(z, ω,E) ∈ Sn−1 is non-degenerate at z′, i.e. σ̂(z′) 6= 0.
We fix the incoming direction ω ∈ Sn−1, and we assume that
(A4) the direction θ ∈ Sn−1 is ω-regular.
Then, one can show that Λ−ω ∩ Λ+θ is a finite set of Hamiltonian trajectories (γ∞j )1≤j≤N∞ ,
γ∞j (t) = γ
∞(t, z∞j ) = (x
j (t), ξ
j (t)), with transverse intersection along each of these curves.
We turn to trapped trajectories. Let us notice that the linearization Fp at (0, 0) of the
Hamilton vector field Hp has eigenvalues −λn, . . . ,−λ1, λ1, . . . , λn. Thus (0, 0) is a hyper-
bolic fixed point for Hp, and the stable/unstable manifold Theorem gives the existence of
a stable incoming Lagrangian manifold Λ− and a stable outgoing Lagrangian manifold Λ+
characterized by
(2.9) Λ± = {(x, ξ) ∈ T ∗Rn, exp(tHp)(x, ξ) → 0 as t→ ∓∞} .
In this paper, we shall describe the contribution to the scattering amplitude of the trapped
trajectories, that is those going from infinity to the fixed point (0, 0). We have proved in
Appendix A the following result, which shows that there are always such trajectories.
Proposition 2.5. For every ω, θ ∈ Sn−1, we have
(2.10) Λ−ω ∩ Λ− 6= ∅ and Λ+θ ∩ Λ+ 6= ∅.
We suppose that
(A5) Λ−ω and Λ− (resp. Λ
and Λ+) intersect transversally.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 7
Under this assumption, Λ−ω ∩ Λ− and Λ+θ ∩ Λ+ are finite sets of bicharacteristic curves. We
denote them, respectively,
(2.11) γ−k : t 7→ γ
−(t, z−k ) = (x
k (t), ξ
−(t)), 1 ≤ k ≤ N−,
(2.12) γ+ℓ : t 7→ γ
+(t, z+ℓ ) = (x
+(t), ξ+(t)), 1 ≤ ℓ ≤ N+.
Here, the z−
(resp. the z+
) belong to ω⊥ (resp. θ⊥) and determine the corresponding curve
by (2.6).
We recall from [18, Section 3] (see also [5, Section 5]), that each integral curve γ±(t) =
(x±(t), ξ±(t)) ∈ Λ± satisfies, in the sense of expandible functions (see Definition 6.1 below),
(2.13) γ±(t) ∼
γ±j (t)e
±µj t, as t→ ∓∞,
where µ1 = λ1 < µ2 < . . . is the strictly increasing sequence of linear combinations over N of
the λj’s. Here, the functions γ
j : R → R2n are polynomials, that we write
(2.14) γ±j (t) =
M ′j∑
γ±j,mt
Considering the base space projection of these trajectories, we denote
(2.15) x±(t) ∼
g±j (t)e
±µj t, as t→ ∓∞, g±j (t) =
M ′j∑
g±j,mt
Let us denote ̂ the (only) integer such that µb = 2λ1. We prove in Proposition 6.11 below that
if j < ̂, then M ′j = 0, or more precisely, that g
j (t) = g
j is a constant vector in Ker(Fp∓λj).
We also have M ′
≤ 1, and g−
can be computed in terms of g−1 .
In this paper, concerning the incoming trajectories, we shall assume that,
(A6) For each k ∈ {1, . . . , N−}, g−1 (z
) 6= 0.
Finally, we state our assumptions for the outgoing trajectories γ+ℓ ⊂ Λ+ ∩Λθ+. First of all,
it is easy to see, using Hartman’s linearization theorem, that there exists always a m ∈ N
such that g+m(z
ℓ ) 6= 0. We denote
(2.16) ℓℓℓ = ℓℓℓ(ℓ) = min{m, g+m(z+ℓ ) 6= 0}
the smallest of these m’s. We know that µℓℓℓ is one of the λj’s, and that M
ℓℓℓ = 0.
In [5], we have been able to describe the branching process between an incoming curve
γ− ⊂ Λ− and an outgoing curve γ+ ⊂ Λ+ provided 〈g−1 |g
1 〉 6= 0 (see the definition for
Λ̃+(ρ−) before [5, Theorem 2.6]). Here, for the computation of the scattering amplitude, we
can relax a lot this assumption, and analyze the branching in other cases that we describe
now. Let us denote, for a given pair of paths (γ−(z−k ), γ
+(z+ℓ )) in (Λ
ω ∩ Λ−)× (Λ+θ ∩ Λ+),
(2.17) M2(k, ℓ) = −
j∈I1(2λ1)
α,β∈I2(λ1)
βV (0)
(g−1 (z
αV (0)
(g+1 (z
8 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
M1(k, ℓ) =−
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))j + (g
(z−))j(g
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
Cα,β,(2.18)
where
Cα,β =− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j(4λ
1 − λ2j)
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0).(2.19)
Here, we have set I1 = {1, . . . , n}, 1j = (δij)i=1,...,n ∈ Nn and
(2.20) Im(µ) = {β ∈ Nn, β = 1k1 + · · ·+ 1km with λk1 = · · · = λkm = µ},
the set of multi-indices β of length |β| = m with each index of its non-vanishing components
in the set {j ∈ N, λj = µ}. We also denote Im ⊂ Nn the set of multi-indices of length m.
We will suppose that
(A7) For each pair of paths (γ−(z−k ), γ
+(z+ℓ )), k ∈ {1, . . . , N−}, ℓ ∈ {1, . . . , N+}, one of the
three following cases occurs:
(a) The set
m < ̂, 〈g−m(z−k )|g+m(z
ℓ )〉 6= 0
is not empty. Then we denote
k = min
m < ̂, 〈g−m(z−k )|g
)〉 6= 0
(b) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, and M2(k, ℓ) 6= 0.
(c) For all m < ̂, we have 〈g−m(z−k )|g+m(z
ℓ )〉 = 0, M2(k, ℓ) = 0 and M1(k, ℓ) 6= 0.
As one could expect (see [30], [28] or [15]), action integrals appear in our formula for the
scattering amplitude. We shall denote
S∞j =
(|ξ∞j (t)|2 − 2E0)dt, j ∈ {1, . . . , N∞},(2.21)
S−k =
|ξ−k (t)|
2 − 2E01t<0
dt, k ∈ {1, . . . , N−},(2.22)
(t)|2 − 2E01t>0
dt, ℓ ∈ {1, . . . , N+},(2.23)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 9
and ν∞j , ν
ℓ , ν
k the Maslov indexes of the curves γ
j , γ
ℓ , γ
k respectively. Let also
D−k = limt→+∞
∣∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
∣∣∣ e−(Σλj−2λ1)t,(2.24)
D+ℓ = limt→−∞
∣∣∣ det
∂x+(t, z, ω,E0)
∂(t, z)
∣∣∣ e(Σλj−2λℓℓℓ)t,(2.25)
be the Maslov determinants for γ−
, and γ+
respectively. We show below that 0 < D−
+∞. Eventually we set
(2.26) Σ(E, h) =
− iE − E0
Then, the main result of this paper is the
Theorem 2.6. Suppose assumptions (A1) to (A7) hold, and that E ∈ R is such that
E −E0 = O(h). Then
A(ω, θ,E, h) =
Aregj (ω, θ,E, h) +
Asingk,ℓ (ω, θ,E, h) +O(h
∞),(2.27)
where
(2.28) Aregj (ω, θ,E, h) = e
iS∞j /h
j,m(ω, θ,E)h
j,0 (ω, θ,E) =
−iν∞j π/2
σ̂(zj)1/2
Moreover we have
• In case (a)
Asingk,ℓ (ω, θ,E, h) = e
k,ℓ,m(ω, θ,E, lnh)h
(Σ(E)+bµm)/µk−1/2,(2.29)
where the a
k,ℓ,m
(ω, θ,E, ln h) are polynomials with respect to lnh, and
k,ℓ,0(ω, θ,E, ln h) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
× e−iν
π/2e−iν
π/2(D−
)−1/2
× |g−1 (z
k )| |g
ℓℓℓ (z
(z−k )
(z+ℓ )
〉)−Σ(E)/µk .(2.30)
• In case (b)
(2.31) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| lnh|Σ(E)/λ1
(1 + o(1)),
10 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
)| |g+
− iM2(k, ℓ)
)−Σ(E)/2λ1
.(2.32)
• In case (c)
(2.33) Asingk,ℓ (ω, θ,E, h) = e
k,ℓ (ω, θ,E)
hΣ(E)/2λ1−1/2
| ln h|Σ(E)/2λ1
(1 + o(1)),
where
k,ℓ (ω, θ,E) =
π1−n/2
ei(nπ/4−π/2)
)−1/2
(Σ(E)
(2λ1λℓℓℓ)
3/2(2λ1)
Σ(E)/2λ1−1
× e−iν
π/2e−iν
π/2(D−k D
× |g−1 (z
k )| |g
ℓℓℓ (z
− iM1(k, ℓ)
)−Σ(E)/2λ1
.(2.34)
Here, the µ̂j are the linear combinations over N of the λk’s and λk − λ1’s, and the function
z 7→ z−Σ(E)/µk is defined on C\]−∞, 0] and real positive on ]0,+∞[.
Of course the assumption that 〈g−1 |g
1 〉 6= 0 (a subcase of (a)) is generic. Without the
assumption (A4), the regular part Areg of the scattering amplitude has an integral rep-
resentation as in [3]. When the assumption (A7) is not fulfilled, that is when the terms
corresponding to the µj with j ≤ ̂ do not contribute, we don’t know if the scattering
amplitude can be given only in terms of the g±1 ’s and of the derivatives of the potential.
3. Proof of the main resolvent estimate
Here we prove Theorem 2.1 using Mourre’s Theory. We start with the construction of an
escape function close to the stationary point (0, 0) in the spirit of [10] and [5]. Since Λ+ and
Λ− are Lagrangian manifolds, one can choose local symplectic coordinates (y, η) such that
(3.1) p(x, ξ) = B(y, η)y · η,
where (y, η) 7→ B(y, η) is a C∞ mapping from a neighborhood of (0,0) in T ∗Rn to the space
Mn(R) of n× n matrices with real entries, such that,
(3.2) B(0, 0) =
. . .
We denote U a unitary Fourier Integral Operator (FIO) microlocally defined in a neighborhood
of (0, 0), which canonical transformation is the map (x, ξ) 7→ (y, η), and we set
(3.3) P̂ = UPU∗.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 11
Here the FIO U∗ is the adjoint of U , and we have UU∗ = Id+O(h∞) and U∗U = Id+O(h∞)
microlocally near (0, 0). Then P̂ is a pseudodifferential operator, with a real (modulo O(h∞))
symbol p̂(y, η) =
j p̂j(y, η)h
j , such that
(3.4) p̂0 = B(y, η)y · η.
We set B1 = Oph(b1),
(3.5) b1(y, η) =
χ̃2(y, η),
where M > 1 will be fixed later and χ̃1 ≺ χ̃2 ∈ C∞0 (T ∗Rn) with χ̃1 = 1 near (0, 0). In what
follows, we will assume that hM < 1. In particular, b1 ∈ S1/2(| ln h|). Here and in what
follows, we use the usual notation for classes of symbols. For m an order function, a function
a(x, ξ, h) ∈ C∞(T ∗Rn) belongs to Sδh(m) when
(3.6) ∀α ∈ N2n, ∃Cα > 0, ∀h ∈]0, 1], |∂αx,ξa(x, ξ, h)| ≤ Cαh−δ|α|m(x, ξ).
Let us also recall that, if a ∈ Sα(1) and b ∈ Sβ(1), with α, β < 1/2, we have
(3.7)
Oph(a),Oph(b)
= Oph
ih{b, a}
+ h3(1−α−β) Oph(r),
with r ∈ Smin(α,β)(1): In particular the term of order 2 vanishes.
Hence, we have here
(3.8) [B1, P̂ ] = Oph
ih{p̂0, b1}
+ | lnh|h3/2 Oph(rM ),
with rM ∈ S1/2(1). The semi-norms of rM may depend on M . We have
(3.9) {p̂0, b1} = c1 + c2,
{p̂0, χ̃2}(3.10)
p̂0, ln
By + (∂ηB)y · η
hM + y2
Bη + (∂yB)y · η
hM + η2
χ̃2.(3.11)
The symbols c1 ∈ S1/2(| lnh|), c2 ∈ S1/2(1) satisfy supp(c1) ⊂ supp(∇χ̃2). Let ϕ̃ ∈
C∞0 (T
n) be a function such that ϕ̃ = 0 near (0, 0) and ϕ̃ = 1 near the support of ∇χ̃2. We
Oph(c1) =Oph(ϕ̃)Oph(c1)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃)Oph(ϕ̃) +O(h∞)
≥− C1h| ln h|Oph(ϕ̃2) +O(h2| ln h|),(3.12)
for some C1 > 0. On the other hand, using [5, (4.96)–(4.97)], we get
(3.13) Oph(c2) ≥ εM−1 Oph(χ̃1) +O(M−2),
12 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
for some ε > 0. With the notation A1 = U
∗B1U , the formulas (3.8), (3.9), (3.12) and (3.13)
imply
−i[A1, P ] =− iU∗[B1, P ]U
≥εhM−1U∗Oph(χ̃1)U − C1h| lnh|U∗ Oph(ϕ̃2)U
+O(hM−2) +OM (h3/2| lnh|).(3.14)
If κ is the canonical transformation associated to U , then χj = χ̃j ◦ κ, j = 1, 2 and ϕ = ϕ̃ ◦ κ
are C∞0 (T
∗(Rn), [0, 1]) functions which satisfy χ1 = 1 near (0, 0) and ϕ = 0 near (0, 0). Using
Egorov’s Theorem, (3.14) becomes
(3.15) − i[A1, P ] ≥ εhM−1 Oph(χ1)− C1h| lnh|Oph(ϕ) +O(hM−2) +OM (h3/2| lnh|).
Now, we build an escape function outside of supp(χ1) as in [22]. Let 1(0,0) ≺ χ0 ≺
χ1 ≺ χ2 ≺ χ3 ≺ χ4 ≺ χ5 be C∞0 (T ∗(Rn), [0, 1]) functions with ϕ ≺ χ4. We define a3 =
g(ξ)(1−χ3(x, ξ))x ·ξ where g ∈ C∞0 (Rn) satisfies 1p−1([E0−δ,E0+δ]) ≺ g. Using [6, Lemma 3.1],
we can find a bounded, C∞ function a2(x, ξ) such that
(3.16) Hpa2 ≥
0 for all (x, ξ) ∈ p−1([E0 − δ,E0 + δ]),
1 for all (x, ξ) ∈ supp(χ4 − χ0) ∩ p−1([E0 − δ,E0 + δ]),
and we set A2 = Oph(a2χ5). We denote
(3.17) A = A1 + C2| lnh|A2 + | lnh|A3,
where C2 > 1 will be fixed later. Now let ψ̃ ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]) with ψ̃ = 1 near
E0. We recall that ψ̃(P ) is a classical pseudodifferential operator of class Ψ
0(〈ξ〉−∞) with
principal symbol ψ̃(p). Then, from (3.15), we obtain
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1ψ̃(P )Oph(χ1)ψ̃(P )− C1h| ln h|ψ̃(P )Oph(ϕ)ψ̃(P )
+ C2h| lnh|Oph
ψ̃2(p)(χ4 − χ0)
+ C2h| ln h|Oph
ψ̃2(p)a2Hpχ5
+ h| ln h|Oph
ψ̃2(p)(ξ2 − x · ∇V )(1 − χ3)
+ h| ln h|Oph
ψ̃2(p)x · ξHp(gχ3)
+O(hM−2) +OM (h3/2| lnh|).(3.18)
From (A1), we have x·∇V (x) → 0 as x→ ∞. In particular, if χ3 is equal to 1 in a sufficiently
large zone, we have
(3.19) ψ̃2(p)(ξ2 − x · ∇V )(1− χ3) ≥ E0ψ̃2(p)(1− χ3).
If C2 > 0 is large enough, the G̊arding inequality implies
(3.20)
C2 Oph
ψ̃2(p)(χ4 − χ0)
−C1 Oph
ψ̃2(p)ϕ
ψ̃2(p)x · ξHp(gχ3)
≥ Oph
ψ̃2(p)(χ4 − χ0)
+O(h).
As in [22], we take χ5(x) = χ̃5(µx) with µ small and χ̃5 ∈ C∞0 ([E0 − δ,E0 + δ], [0, 1]). Since
a2 is bounded, we get
(3.21)
∣∣C2ψ̃2(p)a2Hpχ5
∣∣ ≤ µC2‖a2‖L∞‖Hpχ̃5‖L∞ . µ.
Therefore, if µ is small enough, (3.19) implies
(3.22) Oph
ψ̃2(p)(ξ2−x ·∇V )(1−χ3)
+C2 Oph
ψ̃2(p)a2Hpχ5
ψ̃2(p)(1−χ3)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 13
Then (3.18), (3.20), (3.22) and the G̊arding inequality give
−iψ̃(P )[A,P ]ψ̃(P ) ≥εhM−1 Oph
ψ̃2(p)χ1
+ h| ln h|Oph
ψ̃2(p)(χ4 − χ0)
h| ln h|Oph
ψ̃2(p)(1 − χ3)
+O(hM−2) +OM (h3/2| lnh|)
≥εhM−1 Oph
ψ̃2(p)
+O(hM−2) +OM (h3/2| ln h|).(3.23)
Choosing M large enough and 1E0 ≺ ψ ≺ ψ̃, we have proved the
Lemma 3.1. Let M be large enough and ψ ∈ C∞0 ([E0− δ,E0+ δ]), δ > 0 small enough, with
ψ = 1 near E0. Then, we have
(3.24) − iψ(P )[A,P ]ψ(P ) ≥ εh−1ψ2(P ).
Moreover
(3.25) [A,P ] = O(h| ln h|).
From the properties of the support of the χj, we have
[[P,A], A] =[[P,A1], A1] + C2| lnh|[[P,A1], A2]
+ C2| ln h|[[P,A2], A1] + C22 | lnh|2[[P,A2], A2] + C2| lnh|2[[P,A2], A3]
+ C2| ln h|2[[P,A3], A2] + | lnh|2[[P,A3], A3] +O(h∞).(3.26)
We also know that P ∈ Ψ0(〈ξ〉2), A2 ∈ Ψ0(〈ξ〉−∞) and A3 ∈ Ψ0(〈x〉〈ξ〉−∞). Then, we can
show that all the terms in (3.26) with j, k = 2, 3 satisfy
(3.27) [[P,Aj ], Ak] ∈ Ψ0(h2).
On the other hand,
(3.28) [[P,A1], A2] = U
∗[[P̂ , B1], UA2U
∗]U +O(h∞),
with UA2U
∗ ∈ Ψ0(1). From (3.8) – (3.11), we have [P̂ , B1] ∈ Ψ1/2(h| ln h|) and then
(3.29) [[P,A1], A2] = O(h3/2| lnh|).
The term [[P,A2], A1] gives the same type of contribution. It remains to study
(3.30) [[P,A1], A1] = U
∗[[P̂ , B1], B1]U +O(h∞).
Let χ̃3 ∈ C∞0 (T ∗Rn), [0, 1]) with χ̃2 ≺ χ̃3 and
(3.31) f =
χ̃3(y, η) ∈ S1/2(| ln h|).
Then, with a remainder rM ∈ S1/2(1) which differs from line to line,
i[P̂ , B1] =hOph
f{χ̃2, p̂0}+ c2
− h3/2| ln h|Oph(rM )
=hOph(f)Oph({χ̃2, p̂0}) + hOph(c2) + h3/2| lnh|Oph(rM ).(3.32)
14 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In particular, since [P̂ , B1] ∈ Ψ1/2(h| ln h|), c2 ∈ S1/2(1) and f ∈ S1/2(| lnh|),
[[P̂ , B1], B1] =[[P̂ , B1],Oph(fχ̃2)]
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(fχ̃2)]− ih[Oph(c2),Oph(fχ̃2)]
+O(h3/2| ln h|2)
=− ih[Oph(f)Oph({χ̃2, p̂0}),Oph(f)Oph(χ̃2)] +O(h| ln h|)
=− ihOph(f)[Oph({χ̃2, p̂0}),Oph(f)]Oph(χ̃2)
− ih[Oph(f),Oph(f)]Oph({χ̃2, p̂0})Oph(χ̃2)
− ihOph(f)Oph(f)[Oph({χ̃2, p̂0}),Oph(χ̃2)]
− ihOph(f)[Oph(f),Oph(χ̃2)]Oph({χ̃2, p̂0}) +O(h| ln h|)
=O(h| ln h|).(3.33)
From (3.26), (3.27), (3.29) and (3.33), we get
(3.34) [[P,A], A] = O(h| lnh|).
As a matter of fact, using [5], one can show that [[P,A], A] = O(h). Now we can use the
following proposition which is an adaptation of the limiting absorption principle of Mourre
[25] (see also [12, Theorem 4.9], [19, Proposition 2.1] and [4, Theorem 7.4.1]).
Proposition 3.2. Let (P,D(P )) and (A,D(A)) be self-adjoint operators on a separable
Hilbert space H. Assume the following assumptions:
i) P is of class C2(A). Recall that P is of class Cr(A) if there exists z ∈ C \ σ(P ) such
(3.35) R ∋ t→ eitA(P − z)−1e−itA,
is Cr for the strong topology of L(H).
ii) The form [P,A] defined on D(A) ∩D(P ) extends to a bounded operator on H and
(3.36) ‖[P,A]‖ . β.
iii) The form [[P,A],A] defined on D(A) extends to a bounded operator on H and
(3.37) ‖[[P,A],A]‖ . γ.
iv) There exist a compact interval I ⊂ R and g ∈ C∞0 (R) with 1I ≺ g such that
(3.38) ig(P )[P,A]g(P ) & γg2(P ).
v) β2 . γ . 1.
Then, for all α > 1/2, limε→0〈A〉−α(P − E ± iε)−1〈A〉−α exists and
(3.39)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . γ−1,
uniformly for E ∈ I.
Remark 3.3. From Theorem 6.2.10 of [4], we have the following useful characterization of
the regularity C2(A). Assume that (ii) and (iv) hold. Then, P is of class C2(A) if and only
if, for some z ∈ C \ σ(P ), the set {u ∈ D(A); (P − z)−1u ∈ D(A) and (P − z)−1u ∈ D(A)}
is a core for A.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 15
Proof. The proof follows the work of Hislop and Nakamura [19]. For ε > 0, we define M2 =
ig(P )[P,A]g(P ) and Gε(z) = (P − iεM2 − z)−1 which is analytic for Re z ∈ I and Im z > 0.
Following [12, Lemma 4.14] with (3.35)), we get
(3.40) ‖g(P )Gε(z)ϕ‖ . (εγ)−1/2|(ϕ,Gε(z)ϕ)|1/2,
(3.41) ‖(1− g(P ))Gε(z)‖ . 1 + εβ‖Gε(z)‖,
and then
(3.42) ‖Gε(z)‖ . (εγ)−1,
for ε < ε0 with ε0 small enough, but independent on β, γ.
As in [19], let Dε = (1 + |A|)−α(1 + ε|A|)α−1 for α ∈]1/2, 1] and Fε(z) = DεGε(z)Dε. Of
course, from (3.42),
(3.43) ‖Fε(z)‖ . (εγ)−1,
and (3.40) and (3.41) with ϕ = Dεψ give
(3.44) ‖Gε(z)Dε‖ . 1 + (εγ)−1/2‖Fε‖1/2.
The derivative of Fε(z) is given by (see [12, Lemma 4.15])
(3.45) ∂εFε(z) = iDεGεM
2GεDε = Q0 +Q1 +Q2 +Q3,
Q0 =(α− 1)|A|(1 + |A|)−α(1 + ε|A|)α−2Gε(z)Dε
+ (α− 1)DεGε(z)|A|(1 + |A|)−α(1 + ε|A|)α−2(3.46)
Q1 =DεGε(1− g(P ))[P,A](1 − g(P ))GεDε(3.47)
Q2 =DεGε(1− g(P ))[P,A]g(P )GεDε +DεGεg(P )[P,A](1 − g(P ))GεDε(3.48)
Q3 =−DεGε[P,A]GεDε.(3.49)
From (3.44), we obtain
(3.50) ‖Q0‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
and from (3.36), v) of Proposition 3.2, (3.41) and (3.42), we get
(3.51) ‖Q1‖ . γ−1.
Using in addition (3.44), we obtain
(3.52) ‖Q2‖ . 1 + (εγ)−1/2‖Fε‖1/2.
Now we write Q3 = Q4 +Q5 with
Q4 = −DεGε[P − iεM2 − z,A]GεDε(3.53)
Q5 = −iεDεGε[M2,A]GεDε.(3.54)
For Q4, we have the estimate
(3.55) ‖Q4‖ . εα−1
1 + (εγ)−1/2‖Fε‖1/2
On the other hand, (3.36), (3.37) and v) imply
(3.56) ‖[M2,A]‖ . γ.
16 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then (3.44) gives
(3.57) ‖Q5‖ . 1 + ‖Fε‖.
Using the estimates on the Qj, we get
(3.58) ‖∂εFε‖ . εα−1
γ−1 + (εγ)−1/2‖Fε‖1/2 + ‖Fε‖
Using (3.43) and integrating (3.37) N times with respect to ε, we get
(3.59) ‖Fε‖ . γ−1
1 + ε2α(1−2
−N )−1),
so that, for N large enough,
(3.60) lim sup
‖〈A〉−α(P − E ± iδ)−1〈A〉−α‖ . γ−1.
Using, as in [19], that z 7→ F0(z) is Hölder continuous, we prove the existence of the limit
limIm z→0 F0(z) for Re z ∈ I and the proposition follows from (3.60). �
From Lemma 3.1 and (3.34), we can apply Proposition 3.2 with A = A/| ln h|, β = h and
γ = h/| ln h|. Therefore we have the estimate
(3.61)
∥∥〈A〉−α(P − E ± i0)−1〈A〉−α
∥∥ . h−1| ln h|,
for E ∈ [E0 − δ,E0 + δ]. As usual, we have
(3.62) ‖〈x〉−α〈A〉α‖ = O(1),
for α ≥ 0. Indeed, (3.62) is clear for α ∈ 2N, and the general case follows by complex
interpolation. Then, (3.61) and (3.26) imply Theorem 2.1.
4. Representation of the Scattering Amplitude
As in [30], our starting point for the computation of the scattering amplitude is the rep-
resentation given by Isozaki and Kitada in [20]. We recall briefly their formula, that they
obtained writing parametrices for the wave operators W± as Fourier Integral Operators, tak-
ing advantage of the well-known intertwining property W±P = P0W±, P = P0 + V . The
wave operators are defined by
(4.1) W± = s− lim
eitP/he−itP0/h,
where the limit exist thanks to the short-range assumption (A1). The scattering operator
is by definition S = (W+)∗W−, and the scattering matrix S(E, h) is then given by the
decompostion of S with respect to the spectral measure of P0 = −h2∆. Now we recall briefly
the discussion in [30, Section 1,2] (see also [3]), and we start with some notations.
If Ω is an open subset of T ∗Rn , we denote by Am(Ω) the class of symbols a such that
(x, ξ) 7→ a(x, ξ, h) belongs to C∞(Ω) and
(4.2)
∣∣∣∂αx ∂
ξ a(x, ξ)
∣∣∣ ≤ Cαβ〈x〉m−|α|〈ξ〉−L, for all L > 0, (x, ξ) ∈ Ω, (α, β) ∈ Nd × Nd.
We also denote by
(4.3) Γ±(R, d, σ) =
(x, ξ) ∈ Rn × Rn : |x| > R, 1
< |ξ| < d,± cos(x, ξ) > ±σ
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 17
with R > 1, d > 1, σ ∈ (−1, 1), and cos(x, ξ) = 〈x,ξ〉|x| |ξ| , the outgoing and incoming subsets
of T ∗Rn, respectively. Eventually, for α > 1
, we denote the bounded operator F0(E, h) :
L2α(R
n) → L2(Sn−1) given by
(4.4) (F0(E, h)f) (ω) = (2πh)−
2 (2E)
2E〈ω,x〉f(x)dx,E > 0.
Isozaki and Kitada have constructed phase functions Φ± and symbols a± and b± such that,
for some R0 >> 0, 1 < d4 < d3 < d2 < d1 < d0, and 0 < σ4 < σ3 < σ2 < σ1 < σ0 < 1:
i) Φ± ∈ C∞(T ∗Rn) solve the eikonal equation
(4.5)
|∇xΦ±(x, ξ)|2 + V (x) =
in (x, ξ) ∈ Γ±(R0, d0,±σ0), respectively.
ii) (x, ξ) 7→ Φ±(x, ξ) − x · ξ ∈ A0 (Γ±(R0, d0,±σ0)) .
iii) For all (x, ξ) ∈ T ∗Rn
(4.6)
∂xj∂ξk
(x, ξ) − δjk
∣∣∣∣ < ε(R0),
where δjk is the Kronecker delta and ε(R0) → 0 as R0 → +∞.
iv) a± ∼
j=0 h
ja±j , where a±j ∈ A−j(Γ±(3R0, d1,∓σ1)), supp a±j ⊂ Γ±(3R0, d1,∓σ1),
a±j solve
(4.7) 〈∇xΦ±|∇xa±0〉+
(∆xΦ±) a±0 = 0
(4.8) 〈∇xΦ±|∇xa±j〉+
(∆xΦ±) a±j =
∆xa±j−1, j ≥ 1,
with the conditions at infinity
(4.9) a±0 → 1, a±j → 0, j ≥ 1, as |x| → ∞.
in Γ±(2R0, d2,∓σ2), and solve (4.7) and (4.8) in Γ±(4R0, d1,∓σ2).
v) b± ∼
j=0 h
jb±j, where b±j ∈ A−j(Γ±(5R0, d3,±σ4), supp b±j ⊂ Γ±(5R0, d3,±σ4),
b±j solve (4.7) and (4.8) with the conditions at infinity (4.9) in Γ±(6R0, d4,±σ3), and
solve (4.7) and (4.8) in Γ±(6R0, d3,±σ3).
For a symbol c and a phase function ϕ, we denote by Ih(c, ϕ) the oscillatory integral
(4.10) Ih(c, ϕ) =
(2πh)n
(ϕ(x,ξ)−〈y,ξ〉)c(x, ξ)dξ
and we set
(4.11)
K±a(h) = P (h)Ih(a±,Φ±)− Ih(a±,Φ±)P0(h),
K±b(h) = P (h)Ih(b±,Φ±)− Ih(b±,Φ±)P0(h).
The operator T (E, h) for E ∈] 1
[ is then given by (see [20, Theorem 3.3])
(4.12) T (E, h) = T+1(E, h) + T−1(E, h) − T2(E, h),
18 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where
(4.13) T±1(E, h) = F0(E, h)Ih(a±,Φ±)∗K±b(h)F∗0 (E, h)
(4.14) T2(E, h) = F0(E, h)K∗+a(h)R(E + i0, h) (K+b(h) +K−b(h))F∗0 (E, h),
where we denote from now on R(E ± i0, h) = (P − (E ± i0))−1.
Writing explicitly their kernel, it is easy to see, by a non-stationary phase argument, that
the operators T±1 are O(h∞) when θ 6= ω. Therefore we have
(4.15) A(ω, θ,E, h) = −c(E)h(n−1)/2T2(ω, θ,E, h) +O(h∞),
where c(E) is given in (1.4).
As in [30], we shall use our resolvent estimate (Theorem 2.1) in a particular form. It was
noticed by L. Michel in [24, Proposition 3.1] that, in the present trapping case, the following
proposition follows easily from the corresponding one in the non-trapping setting. Indeed, if
ϕ is a compactly supported smooth function, it is clear that P̃ = −h2∆+ (1− ϕ(x/R))V (x)
satisfies the non-trapping assumption for R large enough, thanks to the decay of V at ∞.
Writing [30, Lemma 2.3] for P̃ , one gets the
Proposition 4.1. Let ω± ∈ A0 has support in Γ±(R, d, σ±) for R > R0. For E ∈ [E0 −
δ,E0 + δ], we have
(i) For any α > 1/2 and M > 1, then, for any ε > 0,
(4.16) ‖R(E ± i0, h)ω±(x, hDx)‖−α+M,−α = O(h−3−ε).
(ii) If σ+ > σ−, then for any α≫ 1,
(4.17) ‖ω∓(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
(iii) If ω(x, ξ) ∈ A0 has support in |x| < (9/10)R, then for any α≫ 1
(4.18) ‖ω(x, hDx)R(E ± i0, h)ω±(x, hDx)‖−α+δ,−α = O(h∞).
Then we can follow line by line the discussion after Lemma 2.1 of D. Robert and H. Tamura,
and we obtain (see Equations 2.2-2.4 there):
(4.19) A(ω, θ,E, h) = c(E)h−(n+1)/2〈R(E + i0, h)g−eiψ−/h, g+eiψ+/h〉+O(h∞),
where
(4.20) g± = e
−iψ±/h[χ±, P ]a±(x, h)e
iψ±/h,
(4.21) ψ+(x) = Φ+(x,
2Eθ), ψ−(x) = Φ−(x,
2Eω).
Moreover the functions χ± are C
n) functions such that χ± = 1 on some ball B(0, R±),
with support in B(0, R± + 1).
Eventually, we shall need the following version of Egorov’s Theorem, which is also used in
Robert and Tamura’s paper.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 19
Proposition 4.2 ([30, Proposition 3.1]). Let ω(x, ξ) ∈ A0 be of compact support. Assume
that, for some fixed t ∈ R, ωt is a function in A0 which vanishes in a small neighborhood of
{(x, ξ); (x, ξ) = exp(tHp)(y, η), (y, η) ∈ suppω}.
‖Oph(ωt)e−itP/hOph(ω)‖−α,α = O(h∞),
for any α ≫ 1. Moreover, the order relation is uniform in t when t ranges over a compact
interval of R.
In the three next sections, we prove Theorem 2.6 using (4.19). We set
(4.22) u− = u
− = R(E + i0, h)g−eiψ−/h,
and our proof consists in the computation of u− in different region of the phase space, following
the classical trajectories γ∞j , or γ
k and γ
ℓ . It is important to notice that we have (P−E)u− =
0 out of the support of g−.
5. Computations before the critical point
5.1. Computation of u− in the incoming region.
We start with the computation of u− in an incoming region which contains the micro-
support of g−. Notice that, thanks to Theorem 2.1, 〈x〉−αu−(x) is a semiclassical family of
distributions for α > 1/2.
Lemma 5.1. Let P be a Schrödinger operator as in (1.1) satisfying only (A1). Suppose that
I is a compact interval of ]0,+∞[, and d > 0 is such that I ⊂] 1
[. Suppose also that
0 < σ+ < 1, R is large enough and K ⊂ T ∗Rn is a compact subset of {|x| > R} ∩ p−1(I).
Then there exists T0 > 0 such that, if ρ ∈ K and t > T0,
(5.1) exp(tHp)(ρ) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
Proof. Let δ > 0. From the construction of C. Gérard and J. Sjöstrand [17], there exists a
function G(x, ξ) ∈ C∞(R2n) such that,
(HpG)(x, ξ) ≥ 0 for all (x, ξ) ∈ p−1(]
[),(5.2)
(HpG)(x, ξ) > 2E(1− δ) for |x| > R0 and p(x, ξ) = E ∈]
[,(5.3)
G(x, ξ) = x · ξ for |x| > R0.(5.4)
Let ρ ∈ K, and γ(t) = (x(t), ξ(t)) = exp(tHp)(ρ) be the corresponding Hamiltonian curve.
We distinguish between 2 cases:
1) For all t > 0, we have |x(t)| > R0.
Then G(γ(t)) > 2E(1 − δ)t+G(ρ) and, for t > T1 with T1 large enough,
(5.5) G(γ(t)) > 2 sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
20 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
By continuity, there exists a neighborhood U of γ such that, for all γ̃ ∈ U , we have
(5.6) G(γ̃(T1)) > sup
x∈B(0,R0)
p(x,ξ)∈I
G(x, ξ).
Since G is non-decreasing on γ̃(t), we have |x̃(t)| > R0 for all t > T1, and then
(5.7) G(γ̃(t)) > 2E(1 − δ)(t − T1) +G(γ̃(T1)) > 2E(1 − δ)t − C.
On the other hand, by uniformly finite propagation, we have |x̃(t)| <
2E(1+ δ)t+C. From
(5.7), we get |x̃(t)| > 1
t− C for all γ̃ ∈ U , and then |ξ̃(t)| =
2E + ot→∞(1). In particular,
the previous estimates gives
(5.8) |x(t)| > R/2,
(5.9) cos
x̃, ξ̃
(t) >
2E(1− δ)t− C
2E(1 + δ)t+ C)(
2E + ot→∞(1))
1 + δ
+ ot→∞(1) > 1− 3δ,
for t > T0 with T0 large enough but independent on γ̃ ∈ U . Thus, for t > T0 and γ̃ ∈ U , we
(5.10) γ̃(t) ∈ Γ+(R/2, d, σ+),
with σ+ = 1− 3δ.
2) There exist T2 > 0 such that |x(T2)| = R0.
Then there exists V a neighborhood of γ such that, for all γ̃ ∈ V, we have |x̃(T2)| < 2R0. Let
t > T2.
a) If |x̃(t)| ≤ R/2, then γ̃(t) ∈ B(0, R/2) × Rn.
b) Assume now |x̃(t)| > R/2. Denote by T3 (> T2) the last time (before t) such that
|x̃(T3)| = 2R0. Then
G(γ̃(t)) >2E(1 − δ)(t− T3) +G(γ̃(T3))(5.11)
>2E(1 − δ)(t− T3)− C,(5.12)
where C depend only on R0. On the other hand, the have |x̃(t)| <
2E(1 + δ)(t − T3) + C
(where the constant C depend only on R0). Then,
(5.13) t− T3 >
|x̃(t)|√
2E(1 + δ)
2E(1 + δ)
(5.14) |ξ̃(t)| =
2E + oR→∞(1),
x̃, ξ̃
(t) >
2E(1− δ)|x̃(t)|
|x̃(t)|(
2E(1 + δ))(
2E + oR→∞(1))
+O(R−1)
1 + δ
+ oR→∞(1) > 1− 2δ + oR→∞(1).(5.15)
So, if R is large enough, γ̃(t) ∈ Γ+(R/2, d, σ+), σ+ = 1− 3δ.
Then a) and b) imply that, for all γ̃ ∈ V and t > T0 := T2, we have
(5.16) γ̃(t) ∈ Γ+(R/2, d, σ+) ∪ (B(0, R/2) × Rn).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 21
The lemma follows from (5.10), (5.16) and a compactness argument. �
Recall that the microsupport of g−(x)e
iψ−(x)/h ∈ C∞0 (Rn) is contained in Γ−(R−, d1, σ1).
Let ω−(x, ξ) ∈ A0 with ω− = 1 near Γ−(R−/2, d1, σ1) and supp(ω−) ⊂ Γ−(R−/3, d0, σ0).
Using the identity
(5.17) u− =
e−it(P−E)/h(g−e
iψ−/h)dt+R(E + i0, h)e−iT (P−E)/h(g−eiψ−/h),
and Proposition 4.1, Proposition 4.2 and Lemma 5.1, we get
(5.18) Oph(ω−)u− = Oph(ω−)
e−it(P−E)/h(g−e
iψ−/h)dt+O(h∞),
for some T > 0 large enough. In particular,
(5.19) MS(Oph(ω−)u−) ⊂ Λ−ω ∩ (B(0, R− + 1)× Rn).
5.2. Computation of u− along γ
Now we want to compute u− microlocally along a trajectory γ
k . We recall that γ
k is a
bicharacteristic curve (x−k (t), ξ
k (t)) such that (x
k (t), ξ
k (t)) → (0, 0) as t → +∞, and such
that, as t→ −∞,
(5.20)
|x−k (t)−
2E0ωt− z−k | → 0,
2E0ω| → 0.
If R− is large enough, a− solves (4.7) and (4.8) microlocally near γ
k ∩ MS(g−eiψ−/h). In
particular, microlocally near γ−k ∩ Γ−(R−/2, d1, σ1) ∩ (B(0, R−)× Rn), u− is given by (5.18)
e−it(P−E)/h([χ−, P ]a−e
iψ−/h)dt+O(h∞)
e−it(P−E)/h(χ−(P − E)a−eiψ−/h)dt
e−it(P−E)/h((P − E)χ−a−eiψ−/h)dt+O(h∞)
(P − E)e−it(P−E)/h(χ−a−eiψ−/h)dt+O(h∞)
=(P − E)R(E + i0, h)a−eiψ−/h +O(h∞)
iψ−/h +O(h∞).(5.21)
Now, using (5.21), and the fact that u− is a semiclassical distribution satisfying
(5.22) (P − E)u− = 0,
we can compute u− microlocally near γ
∩ B(0, R−) using Maslov’s theory (see [23] for
more details). Moreover, it is proved in Proposition C.1 (see also [5, Lemma 5.8]) that the
Lagrangian manifold Λ−ω has a nice projection with respect to x in a neighborhood of γ
close
to (0, 0). Then, in such a neighborhood, u− is given by
(5.23) u−(x) = a−(x, h)e
π/2eiψ−(x)/h,
22 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where ν−k denotes the Maslov index of γ
k , and ψ− satisfies the usual eikonal equation
(5.24) p(x,∇ψ−) = E0.
Here, to the contrary of (4.21), we have written E = E0 + zh with z = O(1), and we choose
to work with z in the amplitudes instead of the phases. As usual, we have
(5.25) ∂t(ψ−(x
(t))) = ∇ψ−(x−k (t)) · ∂tx
(t) = ∇ψ−(x−k (t)) · ξ
(t) = |ξ−
(t)|2,
so that
(5.26) ψ−(x
(t)) = ψ−(x
(s)) +
(u)|2du
We also have ψ−(x
k (s)) = (
2E0ωs+ z
k ) ·
2E0ω + o(1) as s→ −∞, and then
(5.27) ψ−(x
k (t)) = 2E0s+
|ξ−k (u)|
2du+ o(1), s→ −∞.
We have obtained in particular that
(5.28) ψ−(x
k (t)) =
|ξ−k (u)|
2−2E01u<0 du =
|ξ−k (u)|
2−V (x−k (u))+E0 sgn(u) du.
We turn to the computation of the symbol. The function a−(x, h) ∼
k=0 a−,k(x)h
satisfies the usual transport equations:
(5.29)
∇ψ− · ∇a−,0 +
(∆ψ− − 2iz)a−,0 = 0,
∇ψ− · ∇a−,k +
(∆ψ− − 2iz)a−,k = i
∆a−,k−1, k ≥ 1,
In particular, we get for the principal symbol
(5.30) ∂t(a−,0(x
(t))) = ∇a−,0(x−k (t)) · ξ
(t) = ∇a−,0(x−k (t)) · ∇ψ−(x
(t)),
so that,
(5.31) ∂t(a−,0(x
k (t))) = −
∆ψ−(x
k (t))− 2iz
a−,0(x
k (t))
and then
(5.32) a−,0(x
k (t)) = a−,0(x
k (s)) exp
∆ψ−(x−(u)) du + i(t− s)z
On the other hand, from [30, Lemma 4.3], based on Maslov theory, we have
(5.33) a−,0(x
k (t)) = (2E0)
1/4D−k (t)
−1/2eitz ,
where
(5.34) D−
(t) =
∣∣ det
∂x−(t, z, ω,E0)
∂(t, z)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 23
6. Computation of u− at the critical point
Now we use the results of [5] to get a representation of u− in a whole neighborhood of
the critical point. Indeed we saw already that (P − E)u− = 0 out of the support of g−, in
particular in a neighborhood of the critical point. First, we need to recall some terminology
of [18] and [5].
We recall from Section 2 that (µj)j≥0 is the strictly growing sequence of linear combinations
over N of the λj’s. Let u(t, x) be a function defined on [0,+∞[×U , U ⊂ Rm.
Definition 6.1. We say that u : [0,+∞[×U → R, a smooth function, is expandible, if, for
any N ∈ N, ε > 0, α, β ∈ N1+m,
(6.1) ∂αt ∂
u(t, x)−
uj(t, x)e
−µj t
e−(µN+1−ε)t
for a sequence of (uj)j smooth functions, which are polynomials in t. We shall write
u(t, x) ∼
uj(t, x)e
−µj t,
when (6.1) holds.
We say that f(t, x) = Õ(e−µt) if for all α, β ∈ N1+m and ε > 0 we have
(6.2) ∂αt ∂
xf(t, x) = O(e−(µ−ε)t).
Definition 6.2. We say that u(t, x, h), a smooth function, is of class SA,B if, for any ε > 0,
α, β ∈ N1+m,
(6.3) ∂αt ∂
xu(t, x, h) = O
hAe−(B−ε)t
Let S∞,B =
A SA,B. We say that u(t, x, h) is a classical expandible function of order (A,B),
if, for any K ∈ N,
(6.4) u(t, x, h) −
uk(t, x)h
k ∈ SK+1,B,
for a sequence of (uk)k expandible functions. We shall write
u(t, x, h) ∼
uk(t, x)h
in that case.
Since the intersection between Λ−ω and Λ− is transverse along the trajectories γ
k ), and
since g−1 (z
k ) 6= 0, Theorem 2.1 and Theorem 5.4 of [5] implies that one can write, microlocally
near (0, 0),
(6.5) u− =
∫ N−∑
αk(t, x, h)eiϕ
k(t,x)/hdt,
24 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where the αk(t, x, h)’s are classical expandible functions in S0,2ReΣ(E):
(6.6)
αk(t, x, h) ∼
αkm(t, x)h
αkm(t, x) ∼
αkm,j(t, x)e
−2(Σ(E)+µj )t,
and where the αkm,j(t, x)’s are polynomial with respect to t. We recall from (2.26) that, for
E = E0 + hz,
(6.7) Σ(E) =
− iz.
Following line by line Section 6 of [5], we obtain (see [5, (6.26)])
αk0,0(0) = e
iπ/4(2λ1)
3/2e−iν
π/2|g(γ−k )|(D
−1/2(2E0)
1/4.(6.8)
Notice that from (5.32) and Proposition C.1, we have 0 < D−
< +∞.
From [5, Section 5], we recall that the phases ϕk(t, x) satisfies the eikonal equation
(6.9) ∂tϕ
k + p(x,∇xϕk) = E0,
and that they have the asymptotic expansion
(6.10) ϕk(t, x) ∼
ϕkj,m(x)t
me−µjt,
with Mkj < +∞. In the following, we denote
(6.11) ϕkj (t, x) =
ϕkj,m(x)t
and the first ϕkj ’s are of the form
ϕk0(t, x) =ϕ+(x) + ck(6.12)
ϕk1(t, x) =− 2λ1g−(z−k ) · x+O(x
2),(6.13)
where ck ∈ R is the constant depending on k given by
(6.14) ck = “ψ−(0)” = lim
k (t)) = S
thanks to (5.28) (see also [5, Lemma 5.10]). Moreover ϕ+ is the generating function of the
outgoing stable Lagrangian manifold Λ+ with ϕ+(0) = 0. We have
(6.15) ϕ+(x) =
x2j +O(x3).
The fact that ϕk1(t, x) does not depend on t and the expression (6.13) follows also from
Corollary 6.6 and (6.109).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 25
6.1. Study of the transport equations for the phases.
Now, we examine the equations satisfied by the functions ϕkj (t, x), defined in (6.10), for
the integers j ≤ ̂ (recall that ̂ is defined by µb = 2λ1). For clearer notations, we omit the
superscript k until further notice.
Let us recall that the function ϕ(t, x) satisfies the eikonal equation (6.9), which implies
(see (6.10))
(6.16)
e−µjtϕj,m(x)(−µjtm+mtm−1)+
∇ϕj,m(x)tme−µj t
+V (x) ∼ E0,
and then
e−µjtϕj,m(x)(−µjtm +mtm−1) +
∇ϕj,m∇ϕe,em(x)e−(µj+µe)ttm+ em
+V (x) ∼ E0.(6.17)
When µj < 2λ1, the double product of the previous formula provides a term of the form e
if and only if µj = 0 or µe = 0. In particular, the term in e
−µjt in (6.17) gives
(6.18)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm = 0.
When µj = 2λ1, one gets also a term in e
−2λ1t for µj = µe = λ1 and then
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+(x) ·
∇ϕj,m(x)tm
tm+ em∇ϕ1,m(x)∇ϕ1, em(x) = 0.(6.19)
We denote
(6.20) L = ∇ϕ+(x) · ∇
the vector field that appears in (6.18) and (6.19). We set also L0 =
j λjxj∂j its linear part
at x = 0, and we begin with the study of the solution of
(6.21) (L− µ)f = g,
with µ ∈ R and f , g ∈ C∞(Rn). First of all, we show that it is sufficient to solve (6.21) for
formal series.
Proposition 6.3. Let g ∈ C∞(Rn) and g0 the the Taylor expansion of g at 0. For each
formal series f0 such that (L−µ)f0 = g0, there exists one and only one function f ∈ C∞(Rn)
defined near 0 such that f = f0 +O(x∞) and
(6.22) (L− µ)f = g,
near 0.
26 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proof. Let f̃0 be a C
∞ function having f0 has Taylor expansion at 0. With the notation
f = f̃0 + r, the problem (6.22) is equivalent to find r = O(x∞) with
(6.23) (L− µ)r = g − (L− µ)f̃0 = r̃,
where r̃ ∈ C∞ has g0 − (L− µ)f0 = 0 as Taylor expansion at 0. Let y(t, x) be the solution of
(6.24)
∂ty(t, x) = ∇ϕ+(y(t, x)),
y(0, x) = x.
Thus, (6.23) is equivalent to
(6.25) r(x) =
e−µsr̃(y(s, x))ds + e−µtr(y(t, x)).
Since r(x), r̃(x) = O(x∞) and y(s, x) = O(eλ1t|x|) for t < 0, the functions e−µtr(y(t, x)),
e−µtr̃(y(t, x)) are O(eNt) as t→ −∞ for all N > 0. Then
(6.26) r(x) =
e−µsr̃(y(s, x))ds,
and r(x) = O(x∞). The uniqueness follows and it is enough to prove that r given by (6.26)
is C∞. We have
(6.27) ∂t(∇xy) = (∇2xϕ+(y))(∇xy),
and since ∇2xϕ+ is bounded, there exists C > 0 such that
(6.28) |∇xy(t, x)| . e−Ct,
has t → −∞. Then, e−µs(∇r̃)(y(s, x))(∂jy(t, x)) = O(eNt) as t → −∞ for all N > 0 and
∂jr(x) =
−µs(∇r̃)(y(s, x))(∂jy(t, x))ds. The derivatives of order greater than 1 can be
treated the same way. �
We denote
(6.29) Lµ = L− µ : CJxK → CJxK,
where we use the standard notation CJxK for formal series, and CpJxK for formal series of
degree ≥ p. We notice that
(6.30) Lµx
α = (L0 − µ)xα + C|α|+1JxK = (λ · α− µ)xα +C|α|+1JxK.
Recall that Iℓ(µ) has been defined in (2.20). The number of elements in Iℓ(µ) will be denoted
(6.31) nℓ(µ) = #Iℓ(µ).
One has for example n2(µ) =
n1(µ)(n1(µ)+1)
Proposition 6.4. Suppose µ ∈]0, 2λ1[. With the above notations, one has KerLµ⊕ ImLµ =
CJxK. More precisely:
i) The kernel of Lµ has dimension n1(µ), and one can find a basis (Ej1 , . . . , Ejn1(µ)
KerLµ such that Ej(x) = xj + C2JxK, j ∈ I1(µ).
ii) A formal series F = F0 +
Fjxj + C2JxK belongs to ImLµ if and only if Fj = 0 for
all j ∈ I1(µ).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 27
Remark 6.5. Thanks to Propostion 6.3, the same result is true for germs of C∞ functions
at 0. Notice that when µ 6= µj for all j, Lµ is invertible.
Proof. For a given F =
α Fαx
α ∈ CJxK, we look for solutions E =
α ∈ CJxK to the
equation
(6.32) Lµ
The calculus of the term of order x0 in (6.32) leads to the equation
(6.33) E0 = −
With this value for E0, (6.32) becomes, using again (6.30),
(6.34)
|α|=1
(λ · α− µ)Eαxα =
|α|=1
α + C2JxK.
We have two cases:
If α /∈ I1(µ), one should have
(6.35) Eα =
λ · α− µ.
If α ∈ I1(µ), the formula (6.34) becomes Fα = 0. In that case, the corresponding Eα can
be chosen arbitrarily.
Now suppose that the Eα are fixed for any |α| ≤ n− 1 (with n ≥ 2), and such that
(6.36) Lµ
|α|≤n−1
α + CnJxK.
We can write (6.32) as
(6.37) Lµ
|α|=n
α − Lµ
|α|≤n−1
+ Cn+1JxK,
or, using again (6.30),
(6.38)
|α|=n
(λ · α− µ)Eαxα =
|α|≤n
α − Lµ
|α|≤n−1
+Cn+1JxK.
Since |α| ≥ 2, one has λ · α ≥ 2λ1 > µ, so that (6.38) determines by induction all the Eα’s
for |α| = n in a unique way. �
Corollary 6.6. If j < ̂, the function ϕj(t, x) does not depend on t, i.e. we have Mj = 0.
Proof. Suppose that Mj ≥ 1, then (6.18) gives the system
(6.39)
(L− µj)ϕj,Mj = 0,
(L− µj)ϕj,Mj−1 = −Mjϕj,Mj ,
with ϕj,Mj 6= 0. But this would imply that ϕj,Mj ∈ KerLµ ∩ ImLµ, a contradiction. �
28 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
As a consequence, for j < ̂, the equation (6.18) on ϕj reduces to
(6.40) (L− µj)ϕj,0 = 0,
and, from Proposition 6.4, we get that
(6.41) ϕj(t, x) = ϕj,0(x) =
k∈I1(µ)
dj,kxk +O(x2).
Now we pass to the case j = ̂, and we study (6.19). First of all, we have seen that ϕ1 does
not depend on t, so that this equation can be written
(6.42)
ϕj,m(x)(−µjtm +mtm−1) +∇ϕ+ ·
∇ϕj,m(x)tm +
∣∣∇ϕ1(x)
∣∣2 = 0.
As for the study of (6.18), we begin with that of (6.21), now in the case where µ = 2λ1.
We denote Ψ : Rn1(2λ1) −→ Rn2(λ1) the linear map given by
(6.43) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α(L− µ)xβ
α∈I2(λ1)
and we set
(6.44) n(Ψ) = dimKerΨ.
Recalling that L = ∇ϕ+(x) · ∇, we see that
(6.45) Ψ(Eβ1 , . . . , Eβn1(2λ1)
β∈I1(2λ1)
∂α∂βϕ+(0)
α∈I2(λ1)
More generally, for any |α| = 2, we denote
(6.46) Ψα((Eβ)β∈I1(2λ1)) =
β∈I1(2λ1)
∂α∂βϕ+(0)
Then, at the level of formal series, we have the
Proposition 6.7. Suppose µ = 2λ1. Then
i) KerLµ has dimension n2(λ1) + n(Ψ).
ii) A formal series F =
α Fαx
α belongs to ImLµ if and only if
∀α ∈ I1(2λ1), Fα = 0,(6.47)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ.(6.48)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 29
iii) If F ∈ ImLµ, any formal series E =
α with LµE = F satisfies
F0,(6.49)
λ · α− 2λ1
Fα, for α ∈ I1 \ I1(2λ1),(6.50)
β∈I1(2λ1)
|β|=1
β /∈I1(2λ1)
∂β∂αϕ+(0)
2λ1 − λ · β
α∈I2(λ1)
.(6.51)
Moreover for α ∈ I2 \ I2(λ1), one has
(6.52) Eα =
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
2λ1 − λ · β
∂α+βϕ+(0)
Last, E is completely determined by F and a choice of the Eα for |α| ≤ 2 such that
(6.49)– (6.52) are satisfied.
iv) KerLµ ∩ Im(Lµ)2 = {0}.
Proof. For a given F =
α Fαx
α we look for a E =
α such that L2λ1E = F . First of
all, we must have
(6.53) E0 = −
When this is true, we get
(6.54)
|α|=1
Eα(L0 − 2λ1)xα =
|α|=1
Fα(L− 2λ1)xα + C2JxK,
and we obtain as necessary condition that Fα = 0 for any α ∈ I1(2λ1). So far, the Eα for
α ∈ I1(2λ1) can be chosen arbitrarily, and we must have
(6.55) Eα =
λ · α− 2λ1
, α ∈ I2 \ I1(2λ1).
We suppose that (6.53) and (6.55) hold. Then we should have
(6.56)
|α|=2
Eα(L0−2λ1)xα =
|α|=2
|α|=1
α/∈I1(2λ1)
|α|=1
Eα(L−2λ1)xα
+C3JxK.
Notice that the second term in the R.H.S of (6.56) belongs to C2JxK thanks to (6.55). Again,
we have to cases:
• When α ∈ I2(λ1), the corresponding Eα can be chosen arbitrarily, but one must have
|β|=1
∂α(L− 2λ1)xβ
|x=0(6.57)
=Ψα((Eβ)β∈I1(2λ1)) +
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.58)
30 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
and this, with (6.55), gives (6.51).
• When |α| = 2, α /∈ I2(λ1), one obtains
λ · α− 2λ1
|β|=1
∂α(L− 2λ1)xβ
λ · α− 2λ1
Fα −Ψα((Eβ)β∈I1(2λ1))−
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
,(6.59)
and this, with (6.55), gives (6.52).
Now suppose that (6.53), (6.55), (6.57) and (6.59) hold, and that we have chosen a value
for the free variables Eα for α ∈ I1(2λ1)∪I2(λ1). Thanks to the fact that λ ·α 6= 2λ1 for any
α ∈ Nn with |α| = 3, we see as in the proof of Propostion 6.4, that the equation (6.54) has a
unique solution, and the points (i), (ii) and (iii) follows easily.
We prove the last point of the proposition, and we suppose that
(6.60) E =
α ∈ KerLµ ∩ Im(Lµ)2.
First, we have E ∈ KerLµ ∩ ImLµ. Thus, E0 = 0 by (6.49), Eα = 0 for α ∈ I1(2λ1) by
(6.47), and Eα = 0 for α ∈ I1 \ I1(2λ1) by (6.50). Last, since LµE = 0, we also have Eα = 0
for α ∈ I2 \ I2(λ1), and finally,
(6.61) E =
α∈I2(λ1)
α + C3JxK.
Moreover, one can write E = LµG for some G ∈ ImLµ. Since E0 = 0, we must have G0 = 0.
Since G ∈ ImLµ, by (6.47), we have Gα = 0 for α ∈ I1(2λ1). Finally, since Eα = 0 for
|α| = 1, α /∈ I1(2λ1), the same is true for the corresponding Gα, and
(6.62) G =
|α|≥2
Then, since Lµx
α = 0+C3[x] for α ∈ I2(λ1), we obtain Eα = 0 for α ∈ I2(λ1). As above, we
then get that, for |α| ≥ 3, Eα = 0, and this ends the proof. �
Corollary 6.8. We always have Mb ≤ 2. If, in addition, λk 6= 2λ1 for all k ∈ {1, . . . , n}, then
Mb ≤ 1.
Proof. Suppose that Mb ≥ 3. Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.63)
(L− µb)ϕb,Mb −1 = −Mbϕb,Mb(6.64)
(L− µb)ϕb,Mb −2 = −(Mb − 1)ϕb,Mb −1,(6.65)
with ϕb,Mb 6= 0. Notice that we have used the fact that Mb − 2 > 0 in (6.65). But this gives
ϕb,Mb ∈ Ker(L − µb) and (L − µb)2ϕb,Mb−2 = Mb(Mb − 1)ϕb,Mb , so that ϕb,Mb ∈ Im(L− µb)2.
This contradicts point (iv) of Proposition 6.7.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 31
Now we suppose that λk 6= 2λ1 for all k ∈ {1, . . . n}, that is I1(2λ1) = ∅, and that Mb = 2.
Then (6.42) gives
(L− µb)ϕb,Mb = 0(6.66)
(L− µb)ϕb,Mb −1 = −Mb ϕb,Mb(6.67)
with ϕb,Mb 6= 0. Therefore we have ϕb,Mb ∈ KerLµb ∩ ImLµb , and we get the same conclusion
as in (6.61): ϕb,Mb(x) = O(x2). Then, we write
(6.68) ϕb,Mb = (L− µb)g,
and we see, as in (6.62), that g = O(x2), here because I1(2λ1) = ∅. Finally, we conclude also
that ϕb,Mb = 0, a contradiction. �
6.2. Taylor expansions of ϕ+ and ϕ
Now we compute the Taylor expansions of the leading terms with respect to t, of the phase
functions ϕ(t, x) = ϕk(t, x).
Lemma 6.9. The smooth function ϕ+(x) =
x2j +O(x3) satisfies
(6.69) ∂αϕ+(0) = −
λ · α∂
αV (0),
for |α| = 3, and
(6.70) ∂αϕ+(0) = −
2(λ · α)
β,γ∈I2
α=β+γ
β! γ!
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
λ · α∂
αV (0),
for |α| = 4, where α, β, γ ∈ Nn are multi-indices.
Proof. The smooth function x 7→ ϕ+(x) is defined in a neighborhood of 0, and it is charac-
terized (up to a constant: we have chosen ϕ+(0) = 0) by
(6.71)
p(x,∇ϕ+(x)) =
|∇ϕ+(x)|2 + V (x) = 0
∇ϕ+(x) = (λjxj)j=1,...,n +O(x
The Taylor expansion of ϕ+ at x = 0 is
(6.72) ϕ+(x) =
x2j +
|α|=3,4
∂αϕ+(0)
xα +O(x5),
and we have
(6.73) ∂jϕ+(x) = λjxj +
|α|=3,4
∂αϕ+(0)
xα−1j +O(x4).
32 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Therefore
|∇ϕ+(x)|2 =
j + 2
|α|=3
)∂αϕ+(0)
xα + 2
|α|=4
)∂αϕ+(0)
|α|=3
∂αϕ+(0)
xα−1j
+O(x5).(6.74)
Let us compute further the last term in (6.74):
|α|=3
∂αϕ+(0)
xα−1j
|β|,|γ|=3
∂βϕ+(0)
∂γϕ+(0)
xβ+γ−21j
|α|=4
α=β+γ
|β|,|γ|=2
βϕ+(0)
γϕ+(0)
·(6.75)
Writing the Taylor expansion of V at x = 0 as
(6.76) V (x) =
x2j +
|α|=3,4
∂αV (0)
xα +O(x5),
and using the eikonal equation (6.71), we obtain first, for any α ∈ Nn with |α| = 3,
(6.77) ∂αϕ+(0) = −
λ · α∂
αV (0).
Then, (6.74) and (6.75) give
(6.78) ∂αϕ+(0) = −
λ · α∂
αV (0) − 1
2(λ · α)
β,γ∈I2
α=β+γ
βV (0)
λj + λ · β
γV (0)
λj + λ · γ
for |α| = 4. �
Now we pass to the function ϕ1. This function is a solution, in a neighborhood of x = 0,
of the transport equation
(6.79) Lϕ1(x) = λ1ϕ1(x),
where L is given in (6.20).
Lemma 6.10. The C∞ function ϕ1(x) = −2λ1g−1 (z
) · x+O(x2) satisfies
(6.80) ∂αϕ1(0) =
2λ1α!
(λ1 − λ · α)(λ1 + λ · α)
αV (0)
g−1 (z
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 33
for |α| = 2, and
∂αϕ1(0) =−
λ1 − λ · α
k∈I1(λ1),j∈I1
β,γ∈I2
α+1j=β+γ
βV (0)
λj + λ · β
γV (0)
(λ1 − λ · γ)(λ1 + λ · γ)
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
k∈I1,j∈I1(λ1)
β,γ∈I2
1j+α=β+γ
(α+ 1j)!
βV (0)
λk + λ · β
γV (0)
λk + λ · γ
g−1 (z
(λ1 − λ · α)(λ1 + λ · α)
j∈I1(λ1)
αV (0)
g−1 (z
.(6.81)
for |α| = 3.
Proof. We write
(6.82) ϕ1(x) =
ajxj +
|α|=2,3
α +O(x4),
and Lemma 6.9 together with (6.73) give all the coefficients in the expansion
(6.83) ∇ϕ+(x) =
λjxj +
|α|=2,3
Aj,αx
α +O(x4)
j=1,...,n
In fact, we have
(6.84) Aj,α =
∂α+1jϕ+(0)
and aα =
∂αϕ1(0)
We get
Lϕ1(x) =
∂jϕ+(x)∂jϕ1(x)
ajλjxj +
|α|=2
αjλjaα + ajAj,α
|α|=3
αjλjaαx
|β|=|γ|=2
Aj,βγjaγx
β+γ−1j +
|α|=3
ajAj,αx
+O(x4)
ajλjxj +
|α|=2
λ · α aα +
Aj,αaj
|α|=3
λ · α aα +
α=β+γ−1j
|β|,|γ|=2
Aj,βγjaγ + ajAj,α
xα +O(x4).(6.85)
34 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (6.79) gives, for all α ∈ Nn with |α| = 2,
(6.86) aα =
λ1 − λ · α
Aj,αaj ,
and, for all α ∈ Nn with |α| = 3,
(6.87) aα =
λ1 − λ · α
β,γ∈I2
α+1j=β+γ
γjAj,βaγ + ajAj,α
Then, the lemma follows from (6.84). �
6.3. Asymptotics near the critical point for the trajectories.
The informations obtained so far are not sufficient for the computation of the ϕj ’s. We
shall obtain here some more knowledge by studying the behaviour of the incoming trajectory
γ−(t) as t → +∞. We recall from [18, Section 3] (see also [5, Section 5]), that the curve
γ−(t) = (x−(t), ξ−(t)) ∈ Λ− ∩ Λ−ω satisfy, in the sense of expandible functions,
(6.88) γ−(t) =
M ′j∑
γ−j,mt
me−µjt,
Notice that we continue to omit the subscript k for γ−k = (x
k , ξ
k ), z
k , . . . Writing also
(6.89) x−(t) ∼
g−j,m(t, z−)e
−µj t, g−j (z
−, t) =
M ′j∑
g−j,m(z
−)tm,
for some integers M ′j, we know that g
−) = g−1,0(z
−) 6= 0. Since ξ−(t) = ∂tx−(t), we have
(6.90) ξ−(t) ∼
M ′j∑
g−j,m(z
−)(−µjtm +mtm−1)e−µjt.
Proposition 6.11. If j < ̂, then M ′j = 0. We also have M
≤ 1, and M ′
= 0 when
I1(2λ1) 6= ∅. Moreover
(6.91) (g−
|α|=2
∂α+βV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
and, for |β| = 1, β /∈ I1(2λ1),
(6.92) (g−
(2λ1 + λ · β)(2λ1 − λ · β)
|α|=2
∂α+βV (0)
(g−1 (z
−))α.
Proof. First of all, since ∂tγ
−(t) = Hp(γ
−(t)), we can write
(6.93) ∂tγ
−(t) = Fp(γ
−(t)) +O(t2M ′1e−2λ1t),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 35
where
(6.94) Fp = d(0,0)Hp =
, Λ2 = diag(λ21, . . . , λ
We obtain
(6.95)
1≤j<b
M ′j∑
(Fp + µj)γ
1≤j<b
M ′j∑
γ−j,mmt
m−1e−µj t.
Now suppose j < ̂ and M ′j ≥ 1. We get, for this j, for some γ
j,M ′j
6= 0,
(6.96)
(Fp + µj)γ
j,M ′j
(Fp + µj)γ
j,M ′j−1
=M ′jγ
j,M ′j
so that Ker(Fp + µj) ∩ Im(Fp + µj) 6= {0}. Since Fp is a diagonizable matrix, this can easily
be seen to be a contradiction.
Now we pass to the study of M ′
. So far we have obtained that
(6.97) γ−(t) =
1≤j<b
γ−j e
−µjt +
tme−2λ1t +O(tCe−µb+1t),
and we can write
(6.98) Hp(x, ξ) =
|α|=2
∂α∇V (0)
xα +O(x3)
.
Thus we have
(6.99) Hp(γ
−(t)) = Fp
γ−j e
−µjt +
tme−2λ1t
+ e−2λ1tA(γ−1 ) +O(e
−(2λ1+ε)t),
where, noticing that µj + µj′ = 2λ1 if and only if j = j
′ = 1,
(6.100) A(γ−1 ) =
|α|=2
∂α∇V (0)
(g−1 )
.
For the terms of order e−2λ1t, we have, since ∂tγ
−(t) = Hp(γ
−(t)),
(6.101) (Fp + 2λ1)
mtm−1 −A(γ−1 ).
Thus, if we suppose that M ′
≥ 2, we obtain
(6.102)
(Fp + 2λ1)γ
b,M ′
(Fp + 2λ1)γ
b,M ′
b,M ′
36 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Then again we have γ−
b,M ′
∈ Ker(Fp + 2λ1) ∩ Im(Fp + 2λ1), a contradiction.
Eventually, if λj 6= 2λ1 for all j, then Ker(Fp + 2λ1) = {0}. Therefore, if we suppose that
b = 1, we see that γb,1 6= 0 satisfies the first equation in (6.102) and we get a contradiction.
Now we compute γ−
(t) = γ−
t+ γ−
. We have
(6.103)
(Fp + 2λ1)γ
(Fp + 2λ1)γ
−A(γ−1 ),
and we see that γ−
= Πγ−
= ΠA(γ−1 ), where Π is the projection on the eigenspace of
Fp associated to −2λ1. We denote by ej = (δi,j ⊗ 0)i=1,...,n and εj = (0 ⊗ δi,j)i=1,...,n for
j = 1, . . . , n, so that (e1, . . . en, ε1, . . . , εn) is the canonical basis of R
2n = T(0,0)T
n. Then it
is easy to check that , for all j, v±j = ej±λj1εj is an eigenvector of Fp for the eigenvalue ±λj .
In the basis {e1, ε1, . . . , en, εn} the projector Π is block diagonal and, if Kj = Vect(ej , εj), we
(6.104) Π|Kj
1/2 −1/4λ1
−λ1 1/2
for j ∈ I1(2λ1),
0 for j /∈ I1(2λ1).
Therefore, we obtain
(6.105) (g−
|α|=2
∂β∂αV (0)
(g−1 (z
−))α for β ∈ I1(2λ1),
0 for β /∈ I1(2λ1).
Now suppose that k /∈ I1(2λ1). Then the second equality in (6.103) restricted to Kk gives
(6.106)
2λ1 1
λ2k 2λ1
Πkγb,0 = −ΠkA(γ−1 ),
where Πk denotes the projection onto Kk. Solving this system, one gets
(6.107) (g−
4λ21 − λ2k
ΠxΠkA(γ
and, together with (6.100), this ends the proof of Proposition 6.11. �
6.4. Computation of the ϕkj ’s.
Here we compute the ϕkj ’s for j ≤ ̂. We still omit the superscript k. From [5], we know
that ξ−(t) = ∇xϕ
t, x−(t)
, so that, using (6.41),
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
+∇ϕb,2(0)t2e−2λ1t +∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t).(6.108)
Since ϕ+ = −ϕ− and ξ− ∈ Λ−, we have ∇ϕ+(x−(t)) = −ξ−(t), and we obtain first, by (6.90),
(6.109) ∇ϕj(0) = −2µjg−j (z
for 1 ≤ j < ̂.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 37
Now we study ϕb(t, x) = ϕb,0(x) + tϕb,1(x) + t
2ϕb,2(x) when I1(2λ1) 6= ∅. It follows from
(6.108) that we have
(6.110)
− 4λ1g−b,1(z
−) = ∇ϕb,1(0),
− 4λ1g−b,0(z
−) + 2g−
(z−) = ∇ϕb,0(0) +∇2ϕ1(0)g−1 (z
On the other hand, we have seen that, by (6.19), the functions ϕb,2, ϕb,1 and ϕb,0 satisfy
(6.111)
(L− 2λ1)ϕb,2 = 0,
(L− 2λ1)ϕb,1 = −2ϕb,2,
(L− 2λ1)ϕb,2 = −ϕb,1 −
|∇ϕ1(0)|2.
In particular ϕb,2 ∈ Ker(L− 2λ1) ∩ Im(L− 2λ1) so that (see (6.61)),
(6.112) ϕb,2(x) =
α∈I2(λ1)
c2,αx
α +O(x3).
Going back to (6.108), we notice that we obtain now
ξ−(t) =∇ϕ+(x−(t)) +∇ϕ1(x−(t))e−λ1t +
2≤j<b
∇ϕj(0)e−µj t
∇ϕb,1(0)te−2λ1t +∇ϕb,0(0)e−2λ1t + Õ(e−µb+1t),(6.113)
and this equality is consistent with Proposition 6.11.
Then, (6.49) and (6.50) give
(6.114) ϕb,1(x) =
α∈I1(2λ1)
c1,αx
|α|=2
c1,αx
α +O(x3),
and, by (6.51), we have
(6.115) Ψ((c1,β)β∈I1(2λ1)) = (−2c2,α)α∈I2(λ1).
By (6.52), we also have for |α| = 2, α /∈ I2(λ1),
(6.116) c1,α =
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
c1,β .
The function ϕb,0(x) =
|α|≤2 c0,αx
α +O(x3) satisfies (see (6.42))
(6.117) (L− 2λ1)ϕb,0 = −ϕb,1 −
∣∣∇ϕ1(x)
First of all, the compatibility condition (6.47) gives
(6.118) ∀α ∈ I1(2λ1), c1,α = −∇ϕ1(0) · ∂α∇ϕ1(0),
so that in particular, by (6.115), the function ϕb,2 is known up to O(x3) terms:
(6.119) ∀α ∈ I2(λ1), c2,α =
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0),
38 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
(6.120) ∀α /∈ I2(λ1), |α| = 2, c1,α = −
2λ1 − λ · α
β∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0).
Now (6.49) and (6.50) give
(6.121) c0,0 = ϕb,0(0) =
|∇ϕ1(0)|2,
(6.122) ∀α /∈ I1(2λ1), |α| = 1, c0,α =
2λ1 − λ · α
∇ϕ1(0) · ∂α∇ϕ1(0).
From the other compatibility condition (6.48), we know that
c1,α +
∇ϕ1(0) · ∂α∇ϕ1(0) +
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
α∈I2(λ1)
∈ ImΨ,(6.123)
and, from (6.51), we obtain a relation between the (c0,β)β∈I1(2λ1) and the (c1,α)α∈I2(λ1),
namely
∀α ∈ I2(λ1), c1,α =−
∂α∇ϕ1(0) · ∇ϕ1(0)−
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β −
|β|=1
β/∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
·(6.124)
Using the second equation in (6.110), we obtain, for |β| = 1,
(6.125) c0,β = −4λ1(g−b,0(z
−))β + 2(g−
(z−))β − ∂β∇ϕ1(0) · g−1 (z
At this point, we have computed the functions ϕb,1(x) and ϕb,2(x) up to O(x3), in terms
of derivatives of ϕ+ and ϕ1, and of the g
(z−). We shall now use the expressions we have
obtained in Section 6.2 and in Section 6.3 to give these functions in terms of g−1 and of
derivatives of V only.
First of all, by (6.112), (6.119), Lemma (6.9) and Lemma (6.10), we obtain
ϕb,2(x) =−
γ∈I1(2λ1)
α,β∈I2(λ1)
∂β+γV (0)
(g−1 (z
∂α+γV (0)
+O(x3).(6.126)
Then we have
(6.127) ϕb,1(x) = −4λ1g−b,1(z
−) · x+
α∈I2(λ1)
c1,αx
|α|=2
α/∈I2(λ1)
c1,αx
α +O(x3),
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 39
where the c1,α are given by (6.124) and (6.125) for α ∈ I2(λ1), and by (6.120) for α /∈ I2(λ1).
• For |α| = 2, α /∈ I2(λ1), we obtain by (6.116), Lemma 6.9 and Lemma 6.10,
c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
β∈I1(2λ1)
∂α+βV (0)
(λ1 + λj)(3λ1 + λj)
β∇V (0) · g−1 (z−)(g
−))j .(6.128)
Since (g−1 (z
−))j = 0 but for j ∈ I1(λ1), we get, changing notations a bit,
(6.129) c1,α =
(2λ1 + λ · α)(2λ1 − λ · α)
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β.
• Now we compute c1,α for α ∈ I2(λ1).
For the last term in the R.H.S. of (6.124), we obtain
|β|=1
β /∈I1(2λ1)
∂α+βϕ+(0)
∇ϕ1(0) · ∂β∇ϕ1(0)
2λ1 − λ · β
γ∈I1\I1(2λ1)
β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.130)
Using (6.91) and (6.125), we have also
β∈I1(2λ1)
∂α+βϕ+(0)
c0,β =
γ∈I1(2λ1)
∂α+γV (0)
(z−))γ +
γ∈I1(2λ1)
β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))β .(6.131)
40 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We pass to the computation of − 1
∂α∇ϕ1(0) · ∇ϕ1(0) for α ∈ I2(λ1). We obtain
∂α∇ϕ1(0) · ∇ϕ1(0) = −
β∈I2(λ1)
∂α+βV (0)
(g−1 (z
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
((α+ 1p)j + 1)(αp + 1)
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
−))j(g
+ 2λ1
j,p,k=1
β,γ∈I2
β+γ=α+1p+1j
(αp + 1)γj
(λ1 − λ · γ)(λ1 + λ · γ)(λj + λ · β)
β+1jV (0)
∂γ+1kV (0)
(g−1 (z
−))k(g
= I + II + III.
(6.132)
Writing δ = 1j + 1p, we get
(6.133) II = −1
β,γ,δ∈I2
β+γ=α+δ
(α+ δ)!
(λk + λ · β)(λk + λ · γ)
∂β+1kV (0)
∂γ+1kV (0)
(g−1 (z
α! δ!
Since δ ∈ I2(λ1) (otherwise (g−1 (z−))δ = 0), we have β, γ ∈ I2(λ1) and, changing notations a
(6.134) II = −1
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
In the last term III, we can suppose that γ = 1j+1q for some q ∈ {1, . . . , n}. Then γj = γ!
and, writing β = 1a + 1b we have
III = λ1
j,k,p=1
(αp + 1)(g
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
(αp + 1)
(λ1 − λj − λq)(λ1 + λj + λq)(λj + λa + λb)
∂j,a,bV (0)∂j,q,kV (0).(6.135)
Since α ∈ I2(λ1) and 1p ∈ I1(λ1) (otherwise (g−1 (z−))p = 0), we have 1a, 1b, 1q ∈ I1(λ1) so
that we can write
(6.136)
III = −
j,k,p=1
(αp + 1)
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 41
Now it is easy to check, noticing that (α+ 1p)k ∈ {1, 2, 3, 4} and examining each case, that
(6.137)
a,b,q∈I1
1a+1b+1q=α+1p
∂j,a,bV (0)∂j,q,kV (0) =
(α+ 1p)k
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).
Therefore, we have
III = −1
j,k,p=1
(α + 1p + 1k)!
λj(2λ1 + λj)2
(g−1 (z
−))k(g
a,b,c,d∈I1
1a+1b+1c+1d=α+1p+1k
∂j,a,bV (0)∂j,c,dV (0).(6.138)
Eventually, setting β = 1p + 1k, γ = 1a + 1b and δ = 1c + 1d, we get
(6.139)
III = −
β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
We are left with the computation of
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0) = −
β,γ∈I1(λ1)
β+γ=α
βϕ1(0) · ∂j∂γϕ1(0)
λ2j(2λ1 + λj)
β,γ∈I1(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ.(6.140)
At this point, we notice that
α∈I2(λ1)
β,γ∈I1(λ1)
β+γ=α
∂β∇ϕ1(0) · ∂γ∇ϕ1(0)xα
λ2j (2λ1 + λj)
β,γ∈I1(λ1)
α∈I2(λ1)
β+γ=α
k,ℓ=1
∂j∂k∂
βV (0)(g−1 (z
−))k∂j∂ℓ∂
γV (0)(g−1 (z
−))ℓ x
λ2j (2λ1 + λj)
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
αV (0)
βV (0)
xα(g−1 (z
}(6.141)
42 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
From (6.124), (6.130), (6.131) (6.139), and (6.141), we finally obtain that
α∈I2(λ1)
c1,αx
γ∈I1\I1(2λ1)
α,β∈I2(λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
γ∈I1(2λ1)
α,β∈I2(λ1)
∂α+γV (0)
∂β+γV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
∂α+βV (0)
(g−1 (z
−))βxα
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2
γ+δ=α+β
(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α+ β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λj(2λ1 + λj)2
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
(α + β)!
γ,δ∈I2(λ1)
γ+δ=α+β
λ2j(2λ1 + λj)
γV (0)
δV (0)
(g−1 (z
α,β∈I2(λ1)
λ2j(2λ1 + λj)
αV (0)
βV (0)
xα(g−1 (z
−))β ,
(6.142)
or, more simply,
α∈I2(λ1)
c1,αx
α = −
γ∈I1(2λ1)
α∈I2(λ1)
∂α+γV (0)
(z−))γxα +
α,β∈I2(λ1)
(g−1 (z
γ∈I1\I1(2λ1)
(2λ1 − λ · γ)(λ · γ)(2λ1 + λ · γ)2
∂α+γV (0)∂β+γV (0)
γ∈I1(2λ1)
∂α+γV (0)∂β+γV (0)− ∂α+βV (0)
− (α+ β)!
γ,δ∈I2
γ+δ=α+β
γV (0)
δV (0)
λ2j (2λ1 + λj)
αV (0)∂j∂
βV (0)
.(6.143)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 43
7. Computations after the critical point
7.1. Stationary phase expansion in the outgoing region.
Now we compute the scattering amplitude starting from (4.19). First of all, we change the
cut-off function χ+ so that the support of the right hand side of the scalar product in (4.19)
is close to (0, 0).
ℓχ+ = 0
supp(∇χ+)
χ+ = 1
ℓeχ+ = 0
supp(∇eχ+)
eχ+ = 1
Figure 1. The support of χ+ and χ̃+ in T
Using Maslov’s theory, we construct a function v+ which coincides with a+(x, h)e
iψ+(x)/h
out of a small neighborhood of
∩ (B(0, R+ +1)×Rn) and such that v+ is a solution of
(P−E)v+ = 0 microlocally near
. Let χ̃+(x, ξ) ∈ C∞(T ∗Rn) such that χ̃+(x, ξ) = χ+(x)
out of a small enough neighborhood of
∩ (B(0, R++1)×Rn). In particular, (P −E)v+
is microlocally 0 near the support of χ+ − χ̃+. So, we have
〈u−, [χ+, P ]v+〉 =〈u−, [Op(χ̃+), P ]v+〉+ 〈u−, (χ+ −Op(χ̃+))(P − E)v+〉
− 〈(P − E)u−, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞)− 〈g−eiψ−/h, (χ+ −Op(χ̃+))v+〉
=〈u−, [Op(χ̃+), P ]v+〉+O(h∞),(7.1)
since the microsupport of g−e
iψ−/h and χ+− χ̃+ are disjoint. Thus, the scattering amplitude
is given by
(7.2) A(ω, θ,E, h) = c(E)h−(n+1)/2〈u−, [Op(χ̃+), P ]v+〉+O(h∞).
Now we will prove that, modulo O(h∞), the only contribution to the scattering amplitude
in (7.2) comes from the values of the functions u− and v+ microlocally on the trajectories γ
and γ∞j . From (5.18), the fact that u− = O(h−C) and (P −E)u− = 0 microlocally out of the
microsupport of g−e
−iψ−/h, and the usual propagation of singularities theorem, we get
(7.3) MS(u−) ⊂ Λ−ω ∪ Λ+.
Moreover, we have
(7.4) MS(v+) ⊂ Λ+θ .
44 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Now, let f∞j (resp. f
ℓ ) be C
n) functions with support in a small enough neighborhood
of γ∞j (resp. γ
ℓ ∩MS(v+)) such that f∞j = 1 (resp. f
k = 1) in a neighborhood of γ
j (resp.
γ+ℓ ∩MS(v+)). In particular, we assume that all these functions have disjoint support. Since
u− and v+ have disjoint microsupport out of the support of the f
j and the f
ℓ , we have
A(ω, θ,E, h) =c(E)h−(n+1)/2
〈Op(f∞j )u−,Op(f∞j )[Op(χ̃+), P ]v+〉
+ c(E)h−(n+1)/2
〈Op(f+ℓ )u−,Op(f
ℓ )[Op(χ̃+), P ]v+〉+O(h
=Areg +Asing.(7.5)
Concerning the terms which contain f∞j , Areg, we are exactly in the same setting as in [30,
Section 4]. The computation there gives
(7.6) Areg =
j,m(ω, θ,E)h
iS∞j /h +O(h∞).
Now we compute Asing. Proceeding as in Section 5.2 for u−, one can show that v+ can be
written as
(7.7) v+(x) = a+(x, h)e
π/2eiψ+(x)/h,
microlocally near any ρ ∈ γ+
close enough to (0, 0). Here ν+
is the Maslov index of γ+
. The
phase ψ+ and the classical symbol a+ satisfy the usual eikonal and transport equations. In
particular, as in (5.28) and (5.33), we have
(7.8)
ℓ (t)) = −
|ξ+ℓ (u)|
2 − 2E01u>0 du = −
|ξ+ℓ (u)|
2 − V (x+ℓ (u)) −E0 sgn(u) du,
and a+(x, h) ∼
m a+,m(x)h
m with
(7.9) a+,0(x
ℓ (t)) = (2E0)
1/4(D+ℓ (t))
−1/2eitz,
where
(7.10) D+ℓ (t) =
∣∣ det
∂x+(t, z, θ, E0)
∂(t, z)
We can chose χ̃+ so that the microsupport of the symbol of Op(f
ℓ )[Op(χ̃+), P ] is contained
in a vicinity of such a point ρ ∈ γ+ℓ (see Figure 1). Then, microlocally near ρ, we have
(7.11) Op(f+
)[Op(χ̃+), P ]v+ = ã+(x, h)e
π/2eiψ+(x)/h,
(7.12) ã+(x, h) =
ã+,m(x)h
(7.13) ã+,0(x) = −i{χ̃+, p}(x,∇ψ+(x))a+,0(x).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 45
From [5, Section 5], the Lagrangian manifold
{(x,∇xϕk(t, x)); ∂tϕk(t, x) = 0},
coincides with Λ−ω . In particular, since MS(v+) ⊂ Λ+θ and since there is no curve γ∞(z∞j )
sufficiently closed to the critical point, the finite times in (6.5) give a contribution O(h∞)
to the scattering amplitude (4.19). In view of the equations (6.5), (6.12) and (7.11), the
principal contribution of Asing will come from the intersection of the manifolds Λ+θ and Λ+.
Recall that, from (A5), the manifolds Λ+θ and Λ+ intersect transversally along γ
In particular, to compute Asing, we can apply the method of stationary phase in the
directions that are transverse to γ+
. For each ℓ, after a linear and orthonormal change of
variables, we can assume that g+
) is collinear to the xℓℓℓ–direction, and that V (x) satisfies
(A2). We denote Hℓxℓℓℓ = {y = (y1, . . . , yn) ∈ R
n; yℓℓℓ = xℓℓℓ} the hyperplane orthogonal to
(0, . . . , 0, xℓℓℓ, 0, . . . , 0).
We shall compute Asing in the case where there is only one incoming curve γ−
and one
outgoing curve γ+
. In the general case, Asing is simply given by the sum over k and ℓ of such
contributions. Using (4.19), (6.5) and (7.11), we can write
Asing =c(E)h
−(n+1)/2
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dx
c(E)h−(n+1)/2√
y∈Hxℓℓℓ
k(t,x)−ψ+(x))/hαk(t, x, h)ã+(x, h)e
π/2dt dy dxℓℓℓ.(7.14)
Let Φ(y) = ϕk(t, xℓℓℓ, y) − ψ+(xℓℓℓ, y) be the phase function in (7.14). From (6.10)–(6.13), we
can write
(7.15) Φ(y) = S−k + (ϕ+ − ψ+)(xℓℓℓ, y) + ψ̃(t, xℓℓℓ, y),
where ψ̃ = O(e−λ1t) is an expandible function. Since the manifolds Λ+
and Λ+ intersect
transversally along γ+
, the phase function y → (ϕ+−ψ+)(xℓℓℓ, y) has a non degenerate critical
point yℓ(xℓℓℓ) ∈ Hℓxℓℓℓ ∩ Πxγ
, and xℓℓℓ 7→ yℓ(xℓℓℓ) is C∞ for xℓℓℓ 6= 0. Then, from the implicit
function theorem, the function Φ has a unique critical point yℓ(t, xℓℓℓ) ∈ Hℓxℓℓℓ for t large enough
depending on xℓℓℓ. The function (t, xℓℓℓ) 7→ yℓ(t, xℓℓℓ) is expandible and we have
(7.16) yℓ(t, xℓℓℓ) = y
ℓ(xℓℓℓ)−Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
e−µ1t + Õ
e−µ2t
As a consequence, Φ
yℓ(t, xℓℓℓ)
is also expandible.
Since ϕ+ and ψ+ satisfy the same eikonal equation, we get (see (5.25))
(7.17) ∂t(ϕ+ − ψ+)(x+ℓ (t)) = |ξ
(t)|2 − |ξ+
(t)|2 = 0.
46 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Thus, (ϕ+ − ψ+)(yℓ(xℓℓℓ)) does not depend of xℓℓℓ and is equal to
(ϕ+ − ψ+)(yℓ(xℓℓℓ)) = lim
(ϕ+ − ψ+)(x+ℓ (t))
|ξ+ℓ (s)|
2 − 2E01s>0 ds
(s)|2 − V (x+
(s))− E0 sgn(s) ds
,(7.18)
where we have used (7.8). Therefore, the phase function Φ at the critical point yℓ(t, xℓℓℓ) is
equal to
yℓ(t, xℓℓℓ)
µm≤2λ1
t, yℓ(xℓℓℓ)
e−µmt
Hess(ϕ+ − ψ+)−1
yℓ(xℓℓℓ)
yℓ(xℓℓℓ)
· ∇ϕ1
yℓ(xℓℓℓ)
e−2µ1t + Õ(e−eµt),(7.19)
where µ̃ is the first of the µj’s such that µj > 2λ1.
Using the method of the stationary phase for the integration with respect to y ∈ Hℓxℓℓℓ in
(7.14), we get
(7.20) Asing = c(E)h
−(n+1)/2
(2πh)(n−1)/2
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ(t, xℓℓℓ, h) dt dxℓℓℓ +O(h∞).
TheO(h∞) term follows from the fact that the error term stemming from the stationary phase
method can be integrated with respect to time t, since αk ∈ S0,2ReΣ(E), with ReΣ(E) > 0
(see the beginning of Section 6). The symbol f ℓ(t, xℓℓℓ, h) is a classical expandible function of
order S1,2ReΣ(E) in the sense of Definition 6.2:
(7.21) f ℓ(t, xℓℓℓ, h) ∼
f ℓm(t, xℓℓℓ, lnh)h
where the f ℓm are polynomials with respect to lnh and
(7.22) f ℓ0(t, xℓℓℓ, lnh) = α
t, yℓ(t, xℓℓℓ)
ã+,0
yℓ(t, xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(t,xℓℓℓ))π/4
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2
Using Proposition C.1, we compute the Hessian of Φ, and we get
yℓ(xℓℓℓ)
=diag(−λ1, . . . ,−λℓℓℓ−1, λℓℓℓ,−λℓℓℓ+1, . . . ,−λn) + o(1),
yℓ(xℓℓℓ)
=diag(λ1, . . . , λn) + o(1).
Then, for xℓℓℓ small enough and t large enough depending on xℓℓℓ, we have
∣∣detΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
)∣∣1/2 =
j 6=ℓℓℓ
2λj + o(1),(7.23)
sgnΦ′′|Hℓxℓℓℓ
yℓ(t, xℓℓℓ)
= n− 1,(7.24)
as xℓℓℓ goes to 0.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 47
7.2. Behaviour of the phase function Φ.
Suppose that j ∈ N is such that j < ̂. From (6.40), we have
(7.25) ϕkj (x
(s0)) = e
−µj(s−s0)ϕkj (x
(s)).
Combining (6.41) with (6.109), we get
ϕkj (x
ℓ (s0)) =e
µjs0e−µjs
− 2µj〈g−j (z
k )|g
ℓ 〉)e
µjs +O(e2λ1s)
=− 2µj
g−j (z
∣∣g+j (z
eµjs0 .(7.26)
We suppose first that we are in the case (a) of assumption (A7). Then, (7.19) becomes
(7.27) Φ
yℓ(t, xℓℓℓ)
− 2µk
eµks(xℓℓℓ)e−µkt + Õ(e−µk+1t).
Here s(xℓℓℓ) is such that x
(s(xℓℓℓ)) = x
ℓ(xℓℓℓ) and the Õ(e−µk+1t) is in fact expandible, uniformly
with respect to xℓℓℓ when xℓℓℓ varies in a compact set avoiding 0.
Suppose now that we are in the case (b) of assumption (A7). Of course, from (7.26), we
have ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂. On the other hand, Corollary 6.8 and (6.111) imply
(7.28) ϕk
b,2(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,2(x
ℓ (s)).
Combining this with (6.126), we get
b,2(x
(s0)) =e
2λ1s0e−2λ1s
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s
+O(e3λ1s)
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s0 .(7.29)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k − S
j∈I1(2λ1)
α,β∈I2(λ1)
∂α+1jV (0)
∂β+1jV (0)
g−1 (z
g+1 (z
e2λ1s(xℓℓℓ)
× t2e−2λ1t +O(te−2λ1t)
=S−k + S
ℓ +M2(k, ℓ)t
2e−2λ1t +O(te−2λ1t).(7.30)
As in (7.27), the term O(te−2λ1t) is in fact expandible uniformly with respect to xℓℓℓ when xℓℓℓ
varies in a compact set avoiding 0.
Eventually, we suppose that we are in the case (c) of assumption (A7). Then we obtain
from (7.26) and (7.29) that ϕj
yℓ(xℓℓℓ)
= 0 for all j < ̂ and ϕb,2
yℓ(xℓℓℓ)
= 0. With the last
identity in mind, Equation (6.111) on ϕk
implies
(7.31) ϕk
b,1(x
ℓ (s0)) = e
−2λ1(s−s0)ϕk
b,1(x
ℓ (s)).
48 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
In order to compute ϕk
b,1(x
ℓ (s)), we put the expansion (2.15) for x
ℓ (s) (with Proposition 6.11
in mind) into (6.127). The third term in (6.127) will be, at least, O(e(µ2+µ1)s) = o(e2λ1s).
Thank to (6.91) and thanks to the fact that M2(k, ℓ) = 0, the first term in (6.127) will give
no contribution of order se2λ1s and will be of the form
(7.32) − 4λ1g−b,1(z
−) · x+ℓ (s) = −
α∈I2(λ1)
αV (0)
(g−1 (z
−))α(g+
(z+))je
2λ1s + Õ(eµb+1s)
It remains to study the contribution the second term in (6.127), as given in (6.143). As
previously, the first term of the third line in (6.143) will give a term of order o(e2λ1s). The
other terms will contribute to the order e2λ1s for
α∈I2(λ1)
αV (0)
(z−))j(g
+))α +
α,β∈I2(λ1)
(g−1 (z
(g+1 (z
− ∂α+βV (0) +
j∈I1\I1(2λ1)
λ2j (4λ
1 − λ2j )
∂α+γV (0)∂β+γV (0)
γ,δ∈I2(λ1)
γ+δ=α+β
(γ + δ)!
γ! δ!
γV (0)∂j∂
δV (0)
.(7.33)
Thus, combining (7.32) and (7.33), the discussion above leads to
b,1(x
ℓ (s0)) =e
2λ1s0e−2λ1s
M1(k, ℓ)e2λ1s + o(e2λ1s)
=M1(k, ℓ)e2λ1s0 .(7.34)
In particular, (7.19) becomes, in that case,
yℓ(t, xℓℓℓ)
=S−k + S
ℓ +M1(k, ℓ)e
2λ1s(xℓℓℓ)te−2λ1t +O(e−2λ1t).(7.35)
As above, the O(e−2λ1t) is expandible uniformly with respect to the variable xℓℓℓ when xℓℓℓ varies
in a compact set avoiding 0.
7.3. Integration with respect to time.
Now we perform the integration with respect to time t in (7.20). We follow the ideas of
[18, Section 5] and [5, Section 6]. Since yℓ(t, xℓℓℓ) is expandible (see (7.16)), and since Φ is C
outside of xℓℓℓ = 0, the symbol f
ℓ(t, xℓℓℓ, h) is expandible.
We compute only the contribution of the principal symbol (with respect to h) of f ℓ, since
the other terms can be treated the same way, and the remainder term will give a contribution
O(h∞) to the scattering amplitude. In other word, we compute
(7.36) Asing0 =
c(E)h−(n+1)/2√
(2πh)(n−1)/2h
eiΦ(y
ℓ(t,xℓℓℓ))/hf ℓ0(t, xℓℓℓ) dt dxℓℓℓ +O(h∞).
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 49
First, we assume that we are in the case (a) of the assumption (A7). In that case, Ψ is
given by (7.27). For xℓℓℓ fixed in a compact set outside from 0, we set
yℓ(t, xℓℓℓ)
− (S−k + S
=− 2µk
eµks(xℓℓℓ)e−µkt +R(t, xℓℓℓ),(7.37)
and we perform the change of variable t→ τ in (7.36), and we assume for a moment that
(7.38)
Here R(t, xℓℓℓ) = Õ(e−µk+1t) is expandible. As in [18, Section 5] and [5, Section 6], we get
e−t ∼
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−1/µkτ1/µk
τ bµj/µkbj(− ln τ, xℓℓℓ)
(7.39)
t ∼− 1
ln τ +
− 2µk
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
τ bµj/µkbj(− ln τ, xℓℓℓ)(7.40)
τ bµj/µkbj(− ln τ, xℓℓℓ),(7.41)
where the bj ’s change from line to line. These expansions are valid in the following sense:
Definition 7.1. Let f(τ, y) be defined on ]0, ε[×U where U ⊂ Rm. We say that f = Ô(g(τ))
(resp. f = ô(g(τ))), where g(τ) is a non-negative function defined in ]0, ε[ if and only if for
all α ∈ N and β ∈ Nm,
(7.42) (τ∂τ )
α∂βy f(τ, y) = O(g(τ)),
(resp. o(g(τ))) for all (τ, y) ∈]0, ε[×U .
Thus, an expression like f ∼
j=1 τ
bµj/µkfj(− ln τ, xℓℓℓ), where fj(− ln τ, xℓℓℓ) is a polynomial
with respect to ln τ , like in (7.39)–(7.41), means that, for all J ∈ N,
(7.43) f(τ, x)−
τ bµj/µkfj(− ln τ, xℓℓℓ) = Ô(τ bµJ/µk).
We shall say that such symbols f are called expandible near 0.
Since f ℓ0(t, xℓℓℓ, h) is expandible (see Definition 6.1) with respect to t, this symbol is also
expandible near 0 with respect to τ in the previous sense. In particular, we get
(7.44) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
τ (Σ(E)+cµj)/µk f̃ ℓ0,j(− ln τ, xℓℓℓ),
where the f̃ ℓ0,j’s are polynomials with respect to ln τ . The principal symbol f̃
0,0 is independent
on ln τ and we have
(7.45) f̃ ℓ0,0(xℓℓℓ) =
− 2µk
eµks(xℓℓℓ)
)−Σ(E)/µkf ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.46) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
50 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t of each term in the right hand side of (7.44),
modulo a term O(h∞) (see (D.3)–(D.4) in Lemma D.1). Then, we get
(7.47) Asing0 =
c(E)h−1/2
(2π)1−n/2
f̂j(ln h)h
(Σ(E)+bµj )/µk ,
where f̂j(lnh) is a polynomial in respect to lnh. f̂0 does not depend on h and we have
(7.48) f̂0 = Γ(Σ(E)/µk)(−i)−Σ(E)/µk
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ.
To finish the proof, it remains to perform the integration with respect to xℓℓℓ in (7.48). From
(7.22) and (7.45), it becomes
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
eµks(xℓℓℓ)
)−Σ(E)/µk
× α0,0(yℓ
xℓℓℓ)
ã+,0
yℓ(xℓℓℓ)
π/2 e
i sgnΦ′′
|Hℓxℓℓℓ
(yℓ(xℓℓℓ))π/4
∣∣ detΦ′′|Hℓxℓℓℓ
yℓ(xℓℓℓ)
)∣∣1/2
dxℓℓℓ.(7.49)
Now we make the change of variable xℓℓℓ 7→ s given by yℓ(xℓℓℓ) = x+ℓ (s) (then s(xℓℓℓ) = s). In
particular,
(7.50) dxℓℓℓ = ∂s(x
ℓ,ℓℓℓ(s))ds = λℓℓℓ|g
ℓℓℓ (z
ℓ )|e
λℓℓℓs(1 + o(1))ds,
as s→ −∞. In this setting, we get
(7.51) α0,0(x
(s)) = α0,0(0)(1 + o(1)),
as s→ −∞, where α0,0(0) is given in (6.8). We also have, from (7.9) and (7.13),
(7.52) ã+,0(x
ℓ (s)) = −i∂s
χ̃+(γ
ℓ (s))
(2E0)
1/4(D+ℓ )
−1/2eisz.
Then, putting (7.23), (7.24), (7.50), (7.51) and (7.52) in (7.49), we obtain
f̂0 =Γ(Σ(E)/µk)
(z−k )
(z+ℓ )
〉)−Σ(E)/µkα0,0(0)∂s
χ̃+(γ
ℓ (s))
i(n−1)π/4
j 6=ℓℓℓ
λℓℓℓ|g+ℓℓℓ (z
)|(2E0)1/4(D+ℓ )
−1/2eisze−Σ(E)seλℓℓℓs(1 + o(1)) ds
i(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓ|g+ℓℓℓ (z
)|Γ(Σ(E)/µk)
〉)−Σ(E)/µk
× e−iν
π/2α0,0(0)(2E0)
1/4(D+ℓ )
χ̃+(γ
ℓ (s))
(1 + o(1)) ds.(7.53)
Here the o(1) does not depend on χ̃+. Now, we choose a family of cut-off functions (χ̃
+)j∈N
such that the support of ∂t
goes to −∞ as j → +∞. We also assume that
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 51
ℓ (t))
is non negative (see Figure 1). Then
f̂0 =−
ei(n+1)π/4
j 6=ℓℓℓ
)−1/2
λℓℓℓΓ(Σ(E)/µk)e
π/2eiπ/4(2λ1)
3/2e−iν
× |g−1 (z
)| |g+
〉)−Σ(E)/µk(7.54)
(2E0)
1/2(D−
)−1/2 × (1 + o(1)).(7.55)
as j → +∞. Since f̂0 is also independent of χ̃+, we obtain Theorem 2.6 from (7.47) and
(7.48), in the case (a) and under the assumption (7.38). When 〈g−
(z−k )|g
(z+ℓ )〉 > 0, we set
τ as the opposite of the R.H.S. of (7.37), and we obtain the result along the same lines (see
Remark D.2).
Now we assume that we are in the case (b) of the assumption (A7). In that case, the
phase function Ψ is given by (7.30). For xℓℓℓ fixed in a compact set outside from 0, we set,
mimicking (7.37),
yℓ(t, xℓℓℓ)
− (S−k + S
=M2(k, ℓ)e2λ1s(xℓℓℓ)t2e−2λ1t +R(t, xℓℓℓ)(7.56)
where R(t, xℓℓℓ) = O(te−2λ1t) is expandible with respect to t. As above, we assume that
M2(k, ℓ) is positive (the other case can be studied the same way).
Following (7.39), we want to write s := e−t as a function of τ . Since t 7→ τ(t) is expandible
with respect to t, we have
(7.57) τ = M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ)),
where r(s, xℓℓℓ) = ô(1). In particular, ∂sτ > 0 for s positive small enough and then, for ε > 0
small enough, s 7→ τ(s) is invertible for 0 < s < ε. We denote s(τ) the inverse of this function.
We look for s(τ) of the form
(7.58) s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u(τ, xℓℓℓ)
(− ln τ)1/λ1
where u(τ, xℓℓℓ) has to be determined. Using (7.57), the equation on u is
τ =M2(k, ℓ)e2λ1s(xℓℓℓ)(ln s)2s2λ1(1 + r(s, xℓℓℓ))
=τu2λ1
(2λ1)
−2M2(k, ℓ)e2λ1s(xℓℓℓ)
+ 2λ1
− 2ln(− ln τ)
1 + r
(2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 u
(− ln τ)1/λ1
, xℓℓℓ
=τF (τ, u, xℓℓℓ),(7.59)
where F = u2λ1(1 + r̃(τ, u, xℓℓℓ)) and r̃ = ô(1) for u close to 1 (here (u, xℓℓℓ) are the variables y
in Definition 7.1). In other word, to find u, we have to solve F (t, u, xℓℓℓ) = 1.
First we remark that u 7→ F (τ, u, xℓℓℓ) is real-valued and continuous. Since, for δ > 0 and
τ small enough, F (τ, 1 − δ, xℓℓℓ) < 1 < (τ, 1 + δ, xℓℓℓ), there exists u ∈ [1 − δ, 1 + δ] such that
F (τ, 1 + δ, xℓℓℓ) = 1. Thank to the discussion before (7.58), the function s(τ) is of the form
(7.58) with u(τ, xℓℓℓ) ∈ [1− δ, 1 + δ], for τ small enough.
52 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
For τ > 0, the function F is C∞ and, since r̃ = ô(1), we have
(7.60) ∂u
F (τ, u, xℓℓℓ)− 1
(u(τ, xℓℓℓ)) = 2λ1u
2λ1−1(1 + oτ (1)) > λ1,
for τ small enough. The notation oτ (1) means a term which goes to 0 as τ goes to 0. Here
we have used the fact that u(τ, xℓℓℓ) is close to 1. In particular, the implicit function theorem
implies that u(τ, xℓℓℓ) is C
We write u = 1 + v(τ, xℓℓℓ) and we known that v ∈ C∞ and v = oτ (1). Differentiating the
equality
(7.61) 1 = F (τ, u(τ, xℓℓℓ), xℓℓℓ) =
u(τ, xℓℓℓ)
)2λ1(
1 + r̃(τ, u(τ, xℓℓℓ), xℓℓℓ)
one can show that v = ô(1). Thus we have
e−t =s(τ) = (2λ1)
M2(k, ℓ)e2λ1s(xℓℓℓ)
)1/2λ1 1 + r̂(τ, xℓℓℓ)
(− ln τ)1/λ1
,(7.62)
t =− ln τ
(1 + r̂(τ, xℓℓℓ)),(7.63)
+ r̂(τ, xℓℓℓ),(7.64)
where r̂(τ, xℓℓℓ) = ô(1) change from line to line.
Since f ℓ0(t, xℓℓℓ, h) is expandible with respect to t, we get, from (7.62)–(7.64),
(7.65) f̃ ℓ0(τ, xℓℓℓ) = −f ℓ0(t, xℓℓℓ)τ
= τΣ(E)/2λ1(− ln τ)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) + r̂(τ, xℓℓℓ)
where r̂ = ô(1) and
(7.66) f̃ ℓ0,0(xℓℓℓ) = (2λ1)
Σ(E)/λ1−1
M2(k, ℓ)e2λ1s(xℓℓℓ)
)−Σ(E)/2λ1
f ℓ0,0(xℓℓℓ).
In that case, (7.36) becomes
(7.67) Asing0 =
c(E)h−1/2
(2π)1−n/2
∫∫ +∞
eiτ/hf̃ ℓ0(τ, xℓℓℓ)
dxℓℓℓ +O(h∞).
Note that f̃ ℓ0(τ, xℓℓℓ) has in fact a compact support with respect to τ . Now, using Lemma D.1,
we can perform the integration with respect to t in (7.67), modulo an error term given by
(D.3)–(D.4) in Lemma D.1. Then, we get
Asing0 =
c(E)h−1/2
(2π)1−n/2
)/hΓ(Σ(E)/2λ1)(−i)−Σ(E)/2λ1
× hΣ(E)/2λ1(− lnh)−Σ(E)/λ1
f̃ ℓ0,0(xℓℓℓ) dxℓℓℓ + o(1)
,(7.68)
as h goes to 0. The rest of the proof follows that of (7.55).
At last, the proof of Theorem 2.6 in the case (c) can be obtained along the same lines, and
we omit it.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 53
Appendix A. Proof of Proposition 2.5
We prove that Λ+θ ∩Λ+ 6= ∅. From the assumption (A2), the Lagrangian manifold Λ+ can
be described, near (0, 0) ∈ T ∗(Rn), as
(A.1) Λ+ = {(x, ξ); x = ∇ϕ̃+(ξ)},
for |ξ| < 2ε, with ε > 0 small enough. For η ∈ Sn−1, let (x(t, η), ξ(t, η)) be the bicharacteristic
curve with initial condition (ϕ̃(εη), εη). We have
(A.2) Λ+ = {(x(t, η), ξ(t, η)); t ∈ R, η ∈ Sn−1} ∪ {(0, 0)}.
The function ξ(t, η) is continuous on R× Sn−1. From the classical scattering theory (see [13,
Section 1.3]), we know that this function ξ(t, η) converges uniformly to
(A.3) ξ(∞, η) := lim
ξ(t, η),
as t→ +∞ and ξ(∞, η) ∈
2E Sn−1.
Then, the function
(A.4) F (t, η) =
1−t , η)
|ξ( t
1−t , η)|
is well defined for 0 ≤ t ≤ 1 with the convention F (1, η) = ξ(∞, η)/
2E. Here we used that
|ξ(t, η)| 6= 0 for each t ∈ [0,+∞], η ∈ Sn−1. The previous properties of ξ(t, η) imply the
continuity of F (t, η) on [0, 1] × Sn−1.
From (A.2), to prove that Λ+θ ∩ Λ+ 6= ∅ for all θ ∈ Sn−1, it is enough (equivalent) to show
the surjectivity of η → F (1, η). But if η → F (1, η) is not onto, then ImF (1, ·) ⊂ Sn−1 \ {a
point}. And since Sn−1 \ {a point} is a contractible space, F (1, ·) is homotopic to a constant
(A.5) f : Sn−1 → Sn−1.
On the other hand, F : [0, 1]× Sn−1 −→ Sn−1 gives a homotopy between F (0, ·) = IdSn−1 and
F (1, ·). In particular, we have
(A.6) 1 = deg(F (0, ·)) = deg(F (1, ·)) = deg(f(·)) = 0,
which is impossible (see [16, Section 23] for more details).
Appendix B. A lower bound for the resolvent
Let χ ∈ C∞(]0,+∞[) be a non-decreasing function such that
(B.1) χ(x) =
x for 0 < x < 1
2 for 2 < x,
Let also ϕ ∈ C∞0 (R) an even function such that 0 ≤ ϕ ≤ 1, 1[−1,1] ≺ ϕ, and suppϕ ⊂ [−2, 2].
We set
(B.2) u(x) =
j/2hϕ
|xj |1/2
uj(x),
54 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
where 0 < α < 2β will be fixed later on. The uj ’s are of course C
∞ functions, and we have
(B.3) (P − E0)u = −
∆u(x)−
x2ju(x) +O(x3u(x)).
Lemma B.1. For any h small enough, we have
hβn| lnh|n/2 . ‖u‖L2(Rn) . hβn| lnh|n/2,(B.4)
∥∥|x|3u(x)
L2(Rn)
. h3αhβn| lnh|n/2.(B.5)
Proof. First of all, the second estimate follow easily from the first one: we have
∥∥|x|3u(x)
∥∥2 =
|x|6|u(x)|2dx . h6α‖u‖2,
since u vanishes if |x| > 2hα. Thanks to the fact that u is a product of n functions of one
variable, it is enough to estimate
dt = 2
∫ 2hα
We have
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
so that
dt ≤ I ≤ 2
∫ 2hα
dt+ 2
4 dt.
The first estimate follows from the fact that 2β − α > 0, once we have noticed that
∫ Ahα
dt = h2β
(2β − α)| ln h|+ α lnA
On the other hand, we have
∆u(x)−
x2ju(x) =
j 6=k
uj(xj)
u′′k(xk)−
x2kuk(xk)
From Lemma B.1, we get
∥∥(P − E0)u
∥∥ .hβ(n−1)| lnh|(n−1)/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3αhβn| lnh|n/2
h−β | lnh|−1/2 sup
1≤k≤n
∥∥h2u′′k(t) + λ
2uk(t)
∥∥+ h3α
‖u‖.(B.6)
We also have
(B.7) h2u′′k(t) + λkt
2uk(t) = e
h2v′′h(t) + ihλk(2t∂t + 1)vh(t)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 55
where we have set vh(t) = ϕ
|t|1/2
. Notice that the right hand side of (B.7) is an even
function, so that we only have to consider t > 0. The point here, is that we have, for t > 0,
(B.8) (2t∂t + 1)
= − h
2 if 0 < t <
O(1) if h
< t < h2β ,
0 if h2β < t.
Therefore, we obtain
∥∥(2t∂t + 1)vh
∥∥2 =2
∫ 2hα
(2t∂t + 1)
|t|1/2
∫ 2hα
|t|1/2
∫ h2β
∫ 2hα
|t|1/2
dt . h2β .(B.9)
On the other hand, an easy computation gives, still for t > 0,
v′′h(t) =h
−2αϕ′′
4t5/2
.(B.10)
Computing the L2–norm of each of these terms as in Lemma B.1 and (B.9), we obtain
(B.11) ‖h2v′′h‖ . h2+β−2α + h2+β−2α + h2−3β + h2−3β ,
and, eventually, from (B.6), (B.7), (B.9) and (B.11),
∥∥(P − E0)u
h−β | lnh|−1/2
h1+β + h2+β−2α + h2−3β
+ h3α
Therefore we obtain Proposition 2.2 if we can find α > 0 and β > 0 such that
2− 2α > 1, 2− 4β > 1, 3α > 1 and 2β > α,
and one can check that α = 5/12 and β = 11/48 satisfies these four inequalities.
Appendix C. Lagrangian manifolds which are transverse to Λ±
Let Λ ⊂ p−1(E0) be a Lagrangian manifold such that Λ ∩ Λ− is transverse along a Hamil-
tonian curve γ(t) = (x(t), ξ(t)). Then, where exists a 6= 0 and ν ∈ {1, . . . , n} such that
(C.1) γ(t) = (a+O(e−εt))e−λν t,
as t→ +∞. The vector a is an eigenvector of
(C.2)
V ′′(0) 0
for the eigenvalue λν . Thus, up to a linear change of variable in R
n, we can always assume
that Πxa is collinear to the xν–direction. The goal of this section is to prove the following
geometric result.
56 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Proposition C.1. For t large enough, Λ projects nicely on Rnx near γ(t). In particular,
there exists ψ ∈ C∞(Rn) defined near Πxγ, unique up to a constant, such that Λ = Λψ :=
{(x,∇ψ(x)); x ∈ Rn}. Moreover, we have
(C.3) ψ′′(x(t)) =
. . .
. . .
+O(e−εt),
as t→ +∞.
Remark C.2. The same result hold in the outgoing region: If γ = Λ ∩ Λ+ is transverse,
Λ projects nicely on Rnx near γ(t), t → −∞. Then Λ = Λψ for some function ψ satisfying
ψ′′(x(t)) = diag(−λ1, . . . ,−λν−1, λν ,−λν+1, . . . ,−λn) +O(eεt).
Proof. We follow the proof of [18, Lemma 2.1]. There exist symplectic local coordinates (y, η)
centered at (0, 0) such that Λ− (resp. Λ+) is given by y = 0 (resp. η = 0) and
(ξj + λjxj) +O((x, ξ)2),(C.4)
(ξj − λjxj) +O((x, ξ)2).(C.5)
Then, p(x, ξ) = A(y, η)y · η with A0 := A(0, 0) = diag(λ1, . . . , λn).
(C.6)
A0 +O(e−λ1t) 0
O(e−λ1t) A0 +O(e−λ1t)
We denote by U(t, s) the linear operator such that U(t, s)δ solves (C.6) with U(s, s) = Id.
Since Λ∩Λ− = γ is transverse, there exists En−1(t0) ⊂ Tγ(t0)Λ, a vector space of dimension
n − 1 disjoint from Tγ(t0)Λ−. For convenience, we set En(t0) = En−1(t0) ⊕ Rv for some
v /∈ Tγ(t0)Λ + Tγ(t0)Λ−. Let E•(t) = U(t, t0)E•(t0). From [18, Lemma 2.1], there exists Bt =
O(e−λ1t) such that En(t) is given by δη = Btδx. Now, if δ ∈ En−1(t), we have σ(Hp, δ) = 0
since En−1(t)⊕ RHp = Tγ(t)Λ and Λ is a Lagrangian manifold. From (C.1), we have
(C.7) Hp(γ(t)) = γ̇(t) = −λν(ãeην +O(e−εt))e−λν t,
where eην is the basis vector corresponding to ην and then
(C.8) 0 = σ(eλν tHp, δ) = λν ãδyν +O(e−εt)|δ|.
It follows that δ ∈ En−1(t) if and only if (δyν , δη) = B̃tδy′ with B̃t = O(e−εt). Using Tγ(t)Λ =
En−1(t)⊕ RHp, we obtain that Tγ(t)Λ has a basis formed of vector fj(t) such that
fj =eyj +O(e−εt) for j 6= ν(C.9)
fν =eην +O(e−εt).(C.10)
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 57
In the (x, ξ)-coordinates, Tγ(t)Λ has a basis formed of vector f̃j(t) of the form
f̃j =eξj + λjexj +O(e−εt) for j 6= ν(C.11)
f̃ν =eξν − λjexν +O(e−εt),(C.12)
and the lemma follows. �
Appendix D. Asymptotic behaviour of certain integrals
Lemma D.1. Let α ∈ C, Reα > 0, β ∈ R and χ ∈ C∞0 (]−∞, 1/2[) be such that χ = 1 near
0. As λ goes to +∞, we have
(D.1)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)).
Moreover, if β ∈ N, we get
(D.2)
eiλttα(− ln t)βχ(t) dt
= (−iλ)−α
(j)(α)(−1)j
ln(−iλ)
+O(λ−∞).
Finally, if s(t) ∈ C∞(]0,+∞[) satisfies
(D.3) |∂jt s(t)| = o
tα−j(− ln t)β
for all j ∈ N and t→ 0, then
(D.4)
eiλts(t)χ(t)
(ln λ)βλ−α
Here (−iλ)−α = eiαπ/2λ−α and ln(−iλ) = lnλ− iπ/2.
Remark D.2. Notice that one obtains the behaviour of these quantities as λ → −∞ by
taking the complex conjugate in these expressions.
Proof. We begin with (D.2) and assume first that β = 0. Then, we can write
eiλttαχ(t)
= lim
ei(λ+iε)ttαχ(t)
= lim
I1(α, ε) + I2(α, ε)
,(D.5)
where
I1(α, ε) =
e−(ε−iλ)ttα
,(D.6)
I2(α, ε) =
ei(λ+iε)ttα(1− χ(t)) dt
·(D.7)
It is clear that
(D.8) I1(α, ε) = (ε− iλ)−αΓ(α),
where z−α is well defined on C\]−∞, 0] and real positive on ]0,+∞[. In particular
(D.9) lim
I1(α, ε) = (−iλ)−αΓ(α).
58 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Concerning, I2(α, ε), we remark that r(t, α) = t
α−1(1− χ(t)) is a symbol which satisfies
(D.10) |∂jt ∂kαr(t, α)| . 〈t〉Reα−1−j〈ln t〉k,
for all j, k ∈ N uniformly for t ∈ [0,+∞[ and α in a compact set of {Re z > 0}. Then, making
integration by parts in (D.7), we obtain
(D.11) I2(α, ε) =
(ε− iλ)j
e(iλ−ε)t∂jt r(t, α) dt,
for all j ∈ N. Now, if j is large enough (j > Reα), ∂jt r(t, α) is integrable in time uniformly
with respect to ε. In particular, for such j,
(D.12) lim
I2(α, ε) = e
ijπ/2λ−j
eiλt∂
t r(t, α) dt,
and then (see (D.10) or Cauchy’s formula)
∂kα lim
I2(α, ε) =e
ijπ/2λ−j
eiλt∂
αr(t, α) dt
=O(λ−∞),(D.13)
for all k ∈ N. Then we obtain (D.2) for β = 0. To obtain the result for β ∈ N, it is enough
to see that
eiλttα(ln t)βχ(t)
eiλttαχ(t)
(−iλ)−αΓ(α)
+ ∂βα lim
I2(α, ε)
=(−iλ)−α
(j)(α)
− ln(−iλ)
+O(λ−∞),(D.14)
from (D.13). Thus, (D.2) is proved.
Let u be a function C∞(]0,+∞[) be such that
(D.15) |∂jt u(t)| . tReα−j(− ln t)β,
near 0. Let ϕ ∈ C∞(R) such that ϕ = 1 for t < 1 and ϕ = 0 for t > 2. For δ > 0, we have
(D.16)
eiλtu(t)χ(t)
1−ϕ(t/δ)
= (−iλ)−N
eiλt∂Nt
u(t)χ(t)
1−ϕ(t/δ)
for all N .
If one of the derivatives falls on 1−ϕ(t/δ), the support of this contribution is inside [δ, 2δ].
Therefore, the corresponding term will be bounded by δReα−N−1(ln δ)β and will contribute
like δReα−N (− ln δ)β to the integral.
If ones of the derivatives falls on χ(t), the support of the integrand will be a compact set
outside of 0 and then this function will be O(1). The contribution to the integral of such
term will be like 1.
SEMICLASSICAL SCATTERING AMPLITUDE AT THE MAXIMUM POINT OF THE POTENTIAL 59
If all the derivatives fall on u(t)t−1, we corresponding term will satisfies
eiλt∂Nt
u(t)t−1
1− ϕ(t/δ)
dt =O(1)
tReα−1−N (− ln t)β(1− χ(t))dt
.(− ln δ)βδReα−N ,(D.17)
for N large enough (N > Reα).
From this 3 cases, we deduce
(D.18)
eiλtu(t)χ(t)
1− ϕ(t/δ)
(− ln δ)βδα−Nλ−N
Taking δ = (ελ)−1, we get
(D.19)
eiλtu(t)χ(t)
1− ϕ(t/δ)
ε(lnλ)βλ−α
as λ→ +∞.
We now assume (D.3), and we want to prove (D.4). Since, for t small enough
(D.20) tReα−1(− ln t)β .
tReα(− ln t)β
we get
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ =oδ→0(1)
tReα−1(− ln t)βdt
=oδ→0(1)δ
Re α(− ln δ)β .(D.21)
Here oδ→0(1) stands for a term which goes to 0 as δ goes to 0. If δ = (ελ)
−1, we get
(D.22)
eiλts(t)χ(t)ϕ(t/δ)
∣∣∣ = oλ→+∞(1)λ−α(ln λ)β,
when λ → +∞ and ε fixed. Taking ε small enough in (D.19), and then λ large enough in
(D.22), we get (D.4).
We are left with (D.1). We need to compute
(D.23) I =
eiλttα(− ln t)βϕ(t/δ) dt
Performing the change of variable s = λt, we get
I =λ−α
∫ 2/ε
eissα(ln λ− ln s)βϕ(εs) ds
=(lnλ)βλ−α
∫ 2/ε
eissα(1− ln s/ lnλ)βϕ(εs) ds
.(D.24)
60 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
We remark that, in the previous equation, − ln s/ lnλ > − ln(2/ε)/ ln λ > −1/2 for λ large
enough. Using (1 + u)β = 1 +O(|u|+ |u|max(1,β)) for u > −1/2, we get
I =(lnλ)βλ−α
∫ 2/ε
eissαϕ(εs)
+ (lnλ)βλ−α
∫ 2/ε
sReαO
( | ln s|
( | ln s|
)max(1,β))
ϕ(εs)
=(lnλ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
.(D.25)
Note that the Oε in (D.25) depends on ε.
Then, using (D.19), (D.25) and (D.19) again, we get
eiλttα(− ln t)βχ(t) dt
=I +O
ε(ln λ)βλ−α
=(ln λ)β
eiλttα(− ln t)βϕ(t/δ) dt
(lnλ)β−1λ−α
ε(lnλ)βλ−α
=(ln λ)β
eiλttα(− ln t)β dt
ε(lnλ)βλ−α
(lnλ)β−1λ−α
ε(ln λ)βλ−α
.(D.26)
Choosing ε small enough, then λ large enough, and using (D.2) with β = 0 to compute the
first term, we obtain
(D.27)
eiλttα(− ln t)βχ(t) dt
= Γ(α)(ln λ)β(−iλ)−α(1 + o(1)),
and this finishes the proof for (D.1). �
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http://arxiv.org/abs/math/0602069
62 IVANA ALEXANDROVA, JEAN-FRANÇOIS BONY, AND THIERRY RAMOND
Ivana Alexandrova, Department of Mathematics, East Carolina University, Greenville, NC
27858, USA
E-mail address: [email protected]
Jean-François Bony, Institut de Mathématiques de Bordeaux, (UMR CNRS 5251), Université
de Bordeaux I, 33405 Talence, France
E-mail address: [email protected]
Thierry Ramond, Mathématiques, Université Paris Sud, (UMR CNRS 8628), 91405 Orsay,
France
E-mail address: [email protected]
1. Introduction
2. Assumptions and main results
3. Proof of the main resolvent estimate
4. Representation of the Scattering Amplitude
5. Computations before the critical point
5.1. Computation of u- in the incoming region
5.2. Computation of u- along -k
6. Computation of u- at the critical point
6.1. Study of the transport equations for the phases
6.2. Taylor expansions of + and k1
6.3. Asymptotics near the critical point for the trajectories
6.4. Computation of the jk's
7. Computations after the critical point
7.1. Stationary phase expansion in the outgoing region
7.2. Behaviour of the phase function
7.3. Integration with respect to time
Appendix A. Proof of Proposition 2.5
Appendix B. A lower bound for the resolvent
Appendix C. Lagrangian manifolds which are transverse to
Appendix D. Asymptotic behaviour of certain integrals
References
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