Symmetry Breaking Study with Deformed Ensembles
Symmetry Breaking Study with Deformed Ensembles ∗
J. X. de Carvalho1,2, M. S. Hussein†1,2, M. P. Pato2 and A. J. Sargeant2
1Max-Planck-Institut für Physik komplexer Systeme
Nöthnitzer Straβe 38, D-01187 Dresden, Germany
2Instituto de F́ısica, Universidade de São Paulo
C.P. 66318, 05315-970 São Paulo, S.P., Brazil
A random matrix model to describe the coupling of m-fold symmetry is con-
structed. The particular threefold case is used to analyze data on eigenfrequencies
of elastomechanical vibration of an anisotropic quartz block. It is suggested that
such experimental/theoretical study may supply powerful means to discern intrinsic
symmetries of physical systems.
The standard ensembles of Random Matrix Theory (RMT) [1] have had wide appli-
cation in the description of the statistical properties of eigenvalues and eigenfunctions of
complex many-body systems. Other ensembles have also been introduced [2], in order
to cover situations that depart from universality classes of RMT. One such class of en-
sembles is the so-called Deformed Gaussian Orthogonal Ensemble (DGOE) [3, 4, 5, 6]
that proved to be particularly useful when one wants to study the breaking of a discrete
symmetry in a many-body system such as the atomic nucleus.
In fact, the use of spectral statistics as a probe of symmetries in physical systems
has been a subject of intensive experimental and theoretical investigation following the
pioneering work of Bohigas, Giannoni and Schmit [7] which showed that the quantal
behaviour of classically chaotic systems exhibits the predictions supplied by the RMT.
Examples of symmetry breaking in physical systems that have been studied include nuclei
[8, 9], atoms [10, 11] and mesoscopic devices such as quantum dots [12].
In the case of nuclei, the Mitchell group at the Triangle Universities Nuclear Labo-
ratory [8, 9], studied the effect of isospin symmetry breaking, in odd-odd nuclei such as
26Al. They detected the breakdown of this important symmetry by the applications of
two statistics: the short-range, nearest neighbor level spacing distribution (NND) and
the long range Dyson’s ∆-statistics [8, 9]. These results were well described by a DGOE
in which a pair of diagonal blocks is coupled. The strength of the coupling needed to
account for the symmetry breaking can be traced to the average matrix element of the
Coulomb interaction responsible for this discrete symmetry breaking [4, 13]. The justi-
fication for the use of block matrices to describe the statistics of a superposition of R
spectra with different values of the conserved quatum number can be traced to Refs.
[1, 14]. In the case of non-interacting spectra, i.e. if the quantum number is exactly
conserved, the answer is a superposition of the R spectra. Since the level repulsion is
present in each one of the R spectra, their superposition does not show this feature.
Thus, we can say that for each spectra of states of a given value of the quantum number,
one attaches a random matrix (GOE). For R spectra each of which has a given value of
the conserved quantum number, one would have an R × R block diagonal matrix. Each
∗Supported in part by the CNPq and FAPESP (Brazil).
†Martin Gutzwiller Fellow, 2007/2008.
http://arxiv.org/abs/0704.1262v2
block matrix will have a dimension dictated by the number state of that spectra. If the
quantum number is not conserved then the R × R block matrix acquires non-diagonal
matrices that measure the degree of the breaking of the associated symmetry. This idea
was employed by Guhr and Weidenmüller [13] and Hussein and Pato [3] to discuss isopin
violation in the nucleus 26Al. In reference [3], the random block matrix model was called
the Deformed Gaussian Orthogonal Ensemble (DGOE).
In order to study transitions amongst universal classes of ensembles such as order-
chaos (Poisson→GOE), symmetry violation transitions (2GOE→1GOE), experiments
on physical systems are more complicated due to the difficulty of tuning the interaction
(except, e.g. in highly excited atoms where the application of a magnetic field allows
the study of GOE-GUE transitions). To simulate the microscopic physical systems, one
relies on analog computers such as microwave cavities, pioneered by A. Richter and
collaborators [15] and acoustic resonators of Ellegaard and collaborators[16, 17, 18]. It
is worth mentioning at this point that the first to draw attention to the applicability of
RMT to accoustic waves in physical system was Weaver [19].
In the experiment of Ellegaard et al. [17] what was measured were eigenfrequencies
of the elastomechanical vibrations of an anisotropic crystal block with a D3 point-group
symmetry. The rectangular crystal block employed by Ellegard was so prepared as to
have only a two-fold flip symmetry retained. Then, to all effects, the quartz specimen
resembles a system of two three-dimensional Sinai billiards. The statistical treatment
of the eigenfrequencies of such a block would follow that of the superposition of two
uncoupled GOE’s.
Then, by removing octants of progressively larger radius from a corner of the crystal
block this remnant two-fold symmetry was gradually broken. The spectral statistics show
a transition towards fully a chaotic system as the octant radius increases. What was then
seen was that the measured NND is compatible with a two block DGOE description but
the ∆-statistics was discrepant. This discrepancy was attributed to pseudo integrable
behavior and this explanation was later implemented with the result that the long-range
behavior was fitted at the cost, however, of loosing the previous agreement shown by the
NND[23].
Here we reanalyse this experiment following the simpler idea of extending the DGOE
matrix model [5] to consider the coupling of three instead of two GOE’s [6]. We show that
within this extension both, the short- and the long-range statistics, are reasonably fitted
suggesting that the assumption of the reduction of the complex symmetries of anisotropic
quartz block may not be correct. Our findings have the potential of supplying very precise
means of testing details of symmetry breaking in pysical systems.
To define the ensembles of random matrices we are going to work with, we recall
the construction based on the Maximum Entropy Principle [3], that leads to a random
Hamiltonian which can be cast into the form
H = H0 + λH1, (1)
where the block diagonal H0 is a matrix made of m uncoupled GOE blocks and λ (0 ≤
λ ≤ 1) is the parameter that controls the coupling among the blocks represented by the
H1 off-diagonal blocks. For λ = 1, the H1 part completes the H0 part and H = H
These two matrices H0 and H1 are better expressed introducing the following m pro-
jection operators
| j >< j |, (2)
where Ii defines the domain of variation of the row and column indexes associated with
ith diagonal block of size Mi. Since we are specifically interested in the transition from
a set of m uncoupled GOE’s to a single GOE, we use the above projectors to generalize
our previous model [3, 4] by writing
GOEPi (3)
GOEQi (4)
where Qi = 1− Pi. It is easily verified that H = HGOE for λ = 1.
The joint probability distribution of matrix elements can be put in the form [3, 20]
P (H,α, β) = Z−1N exp
−αtrH2 − βtrH2
with the parameter λ being given in terms of α and β by
λ = (1 +
)−1/2. (6)
Statistical measures of the completely uncoupled m blocks have been derived. They
show that level repulsion disappears which can be understood since eigenvalues from
different blocks behave independently. In fact, as m increases the Poisson statistics are
gradually approached. In the interpolating situation of partial coupling, some approxi-
mate analytical results have been derived. In Ref. [20], for instance, it has been found
that the density ρ(E) for arbitrary λ and m is given by
ρ(E) =
ρi(E) (7)
where
ρi(E) =
a2i − E2, | E |≤ a
0, | E |> a
is Wigner’s semi-circle law with a =
N/α and
a2i = a
1− Mi
. (9)
Eq. (5) can be used to calculate exactly analytically the NND for 2 × 2 and 3 × 3
matrices [6]. For the 2× 2 case the DGOE, Eq. (5), gives
P2×2(s, β) = αs exp
) exp
, (10)
where I0 is the modified Bessel function, whose asymptotic form is
I0(x) →
. (11)
Thus,there is no level repulsion for β → ∞, P2×2(s,∞) = 2πα exp
, which repre-
sents the 2x2 Poisson distribution where the usual exponential is replaced by a Gaussian.
The prefactor is just 1 if 2α is taken to be π. In the opposite limit, β → 0, I0(x) ≈ 1−x2/4
and one obtains the Wigner distribution,
P2×2(s, β → 0) ≈
s (12)
Note that the parameter λ of eq (6) is 0 if β is ∞ and 1 if β is 0.
For higher dimensions Eq. (5) can only be used for numerical simulations. This is what
we are now reporting, using 2 and 3 bolck matrices of sizes 105 x 105 and 70 x 70 each,
respectively. The size of the whole matrix is 210 x 210. Further, we take an ensemble of
1000 elements and fix α to be 1. We apply our model to analyse the eigenfrequency data
of the elastomechanical vibrations of an anisotropic quartz block used in [17].
In this reference in order to break the flip symmetry of the crystal block gradually
they removed an octant of a sphere of varying size at one of the corners. The rectangular
quartz block has the dimensions 14 × 25 × 40mm3. The radii of the spheres containing
the octants are r = 0.0, 0.5, 0.8, 1.1, 1.4 and 1.7mm representing figures (a)− (f). Figs.
1x and 2x of Ref. [17] correspond to an octant of a huge sphere of radius r = 10.0mm,
whose center is inside the crystal and close to one of the corners. They found 1424, 1414,
1424, 1414, 1424 and 1419 frequency eigenmodes in cases (a)− (f), respectively. These
eigenfrequecies were measured in the frequency range between 600 and 900 kHz. Thus
the average spacing between the modes is about 214Hz. The histograms and circles in
the two figures of Ref. [17] represent the short-range nearest-neighbor distributions P (s)
(Fig. 1) and the long range ∆3(L) statistics (Fig. 2). In our DGOE simulation the
unfolding of the calculated spectra is performed with the DGOE density given by Eq.
(7) above.
In figures 1 and 2, we show the results of our simlulations as compared to the data
of Ellegaard et al. [17] for the spacing distribution and in figures 3 and 4 the long
range correlation exemplified by the spectral rigidity ∆3(L). We simulate the gradual
breaking of the 2- or 3-fold symmetry by changing the value of the parameter λ above.
We see clearly that in so far as the ∆3(L) is concerned a 3-GOE description works much
better than a 2-GOE one. It is clear, however that both descriptions fall below the data,
specially at large L. We shall analyse this discrepancy in the following using the missing
level effect[21].
It is often the case that there are some missing levels in the statistical sample analysed.
Such a situation was addressed recently by Bohigas and Pato [21] who have shown that
if g fraction of the levels or eigenfrequencies is missing, the ∆3(L) becomes
(L) = g
+ (1− g)2∆3
. (13)
The presence of the linear term, even if small, could explain the large L behavior of the
measured ∆3(L). We call this effect the Missing Level (ML) effect. Another possible
deviation of ∆3 could arise from the presence of pseudo-integrable effect (PI) [22, 23].
This also modifies ∆3 by adding a Poisson term just like Eq. (13). In the following we
show that there is no need for the PI effects to explain the large-L data on the ∆3 if the
ML effect is taken into account.
We take a study case Figs. 3b and 4b which correspond to r = 0.5mm and where 1414
frequency eigenvalues were found. We consider this a potential ML case and take for ∆3,
the expression given in Eq. (13) and apply to our simulations. We find perfect fit to the
data, if g is taken to be 0.1, namely only 90% of the eigenfrequencies were in fact taken
into account in the statistical analysis. There is, threfore, room to account much better
for all cases (Fig. 2a, 2c, . . . ) by appropiately choosing the correponding value of g. We
have also verified that if a 2GOE description is used, namely, m = 2 , then an account of
the large-L behaviour of ∆3 can also be obtained if a much larger number of levels were
missing in the sample. In our particular case of Fig. 2b, we obtained g = 0.3. This is 3
times larger than the ML needed in the 3GOE description. We consider the large value
of g needed in the 2GOE description, much too large to conform to the reported data in
Ref [17]. Figure 5 summarizes our the above.
It is therefore clear that the 3GOE description of the spectral rigidity of the eifen-
frequency spectra of [17] for the crystal block does work very well if a small fraction of
the levels is taken to be missing, without resort to pseudointegrable trajectories or levels
that do not feel the symmetry breaking [23]. On the other hand, the 2GOE description,
which does as good as the 3GOE one in fitting the measured P (s), fails dramatically in
accounting for the spectral rigidity, even if as much as 30 per cent of the levels are taken
as missing.
In conclusion, a random matrix model to describe the coupling of m-fold symmetry is
constructed. The particular threefold case is used to analyse data on eigenfrequencies of
elastomechanical vibration of a anisotropic quartz block. By properly taking into account
the ML effect we have shown that the quartz block could very well be described by 3
uncoupled GOE’s , which are gradually coupled by the breaking of the three-fold sym-
metry (through the gradual removal of octants of increasing sizes), till a 1GOE situation
is attained. This, therefore, indicates that the unperturbed quartz block may possess
another symmetry, besides the flip one. A preliminary version of the formal aspect of
this work has previously appeared in [24].
[1] M.L. Mehta, Random Matrices 2nd Edition (Academic Press, Boston, 1991); T.A. Brody
et al., Rev. Mod. Phys. 53, 385 (1981); T. Guhr, A. Müller-Groeling and A. Weidenmüller,
Phys. Rep. 299, 189 (1998).
[2] F.J. Dyson, J. Math. Phys. 3, 1191 (1962).
[3] M. S. Hussein, and M. P. Pato, Phys. Rev. Lett. 70,
1089 (1993).
[4] M. S. Hussein, and M. P. Pato, Phys. Rev. C 47, 2401
(1993).
[5] M. S. Hussein, and M. P. Pato, Phys. Rev. Lett. 80, 1003 (1998).
[6] C. E. Carneiro, M. S. Hussein and M. P. Pato, in H. A. Cerdeira, R. Ramaswamy, M. C.
Gutzwiller and G. Casati (eds.) Quantum Chaos, p. 190 (World Scientific, Singapore)
(1991).
[7] O. Bohigas, M. J. Giannoni and C. Schmit, Phys. Rev. Lett. 52, 1 (1984). See also O.
Bohigas and M. J. Giannoni, in Mathematical and Computational Methods in Nuclear
Physics, edited by J. S. DeHesa, J. M. Gomez and A. Polls, Lecture Notes in Physics Vol.
209 (Springer-Verlag, New York, 1984).
[8] G. E. Mitchell, E. G. Bilpuch, P. M. Endt, and F. J. Shriner, Phys. Rev. Lett. 61, 1473
(1988).
[9] A.A. Adams, G.E. Mitchell, and J.F. Shriner, Jr., Phys. Lett. B 422, 13 (1998).
[10] B. D. Simons, A. Hashimoto, M. Courtney, D. Kleppner and B. L. Altshuler, Phys. Rev.
Lett. 71, 2899 (1993).
[11] G. R. Welch, M. M. Kash, C-h Iu, L. Hsu and D. Kleppner, Phys. Rev. Lett. 62, 893 (
1989).
[12] See, e.g. Y. Alhassid, Rev. Mod. Phys. 72, 895 (2000) and references therein.
[13] T. Guhr and, H.A. Weidenmüller. Ann. Phys. (NY), 199, 412 (1990).
[14] N. Resenzweig and C. E. Porter, Phys. Rev. 120, 1698 (1960).
[15] H.-D. Graf, H. L. Harney, H. Lengeler, C. H. Lewnkopf, C. Rangacharyulu, A. Richter, P.
Schardt and H. A. Weidenmuller, Phys. Rev. Lett. 69, 1296(1992); H. Alt, H. -D. Graf, H. L.
Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt and H. A. Weidenmuller, Phys.
Rev. Lett. 74, 62 (1995); H. Alt, C. Dembowski, H. -D. Graf, R. Hofferbert, H. Rehfeld,
A. Richter, R. Schuhmann and Weiland, Phys. Rev. Lett. 79, 1029 (1997); H. Alt, C. I.
Brabosa, H. -D. Graf, T. Guhr, H. L. Harney, R. Hofferbert, H. Rehfeld and A. Richter,
Phys. Rev. Lett. 81, 4847 (1998); C. Dembowski, H. -D. Graf, A. Heine, R. Hofferbert, H.
Rehfeld and A. Richter, Phys. Rev. Lett. 84, 867 (2000); C. Dembowski, H. -D. Graf. H.
L. Harney, A. Heine, W. D. Heiss, H. Rehfeld and A. Richter, Phys. Rev. Lett. 86, 787
(2001); C. Dembowski, H. -D. Graf, A. Heine, T. Hesse, H. Rehfeld and A. Richter, Phys.
Rev. Lett. 86, 3284 (2001); C. Dembowski, B. Dietz, H. -D. Graf, A. Heine, T. Papenbrock,
A. Richter and C. Richter, Phys. Rev. Lett. 89, 064101-1 (2002); C. Dembowski, B. Dietz,
A. Heine, F. Leyvraz, M. Miski-Oglu, A. Richter and T. H. Seligman, Phys. Rev. Lett.
90, 014102-1 (2003); C. Dembowski, B. Dietz, H. -D. Graf, H. L. Harney, A. Heine, W.
D. Heiss and A. Richter, Phys. Rev. Lett. 90, 034101-1 (2003); C. Dembowski, B. Dietz,
T. Friedrich, H. -D. Graf, A. Heine, C. Mejia-Monasterio, M. Miski-Oglu, A. Richter and
T. H. Seligman, Phys. Rev. Lett. 93, 134102-1 (2004); B. Dietz, T. Guhr, H. L. Harney
and A. Richter, Phys. Rev. Lett. 96, 254101 (2006); E. Bogomolny, B. Dietz, T. Friedrich,
M. Miski-Oglu, A. Richter, F. Schafer and C. Schmit, Phys. Rev. Lett. 97, 254102 (2006);
B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schafer and H. A.
Weidenmuller, Phys. Rev. Lett. 98, 074103 (2007).
[16] C. Ellegaard, T. Guhr, K. Lindemann, H. Q. Lorensen, J. Nygard and M. Oxborrow, Phys.
Rev. Lett. 75, 1546 (1995).
[17] C. Ellegaard, T. Guhr, K. Lindemann, J. Nygard and M. Oxborrow, Phys. Rev. Lett. 77,
4918 (1996).
[18] P. Bertelsen, C. Ellegaard, T. Guhr, M. Oxborrow and K. Schaadt, Phys. Rev.Lett. 83,
2171 (1999).
[19] R. L. Weaver, J. Acoustic. Soc. Am. 85, 1005 (1989).
[20] A. C. Bertuola, J. X. de Carvalho, M. S. Hussein, M. P. Pato, and A. J. Sargeant, Phys.
Rev. E 71, 036117 (2005).
[21] O. Bohigas and M. P. Pato, Phys. Lett. B, 595, 171 (2004).
[22] D. Biswas and S. R. Jain, Phys. Rev. A 42, 3170 (1990).
[23] A. Abd El-Hady, A. Y. Abul-Magd, and M. H. Simbel, J. Phys. A 35, 2361 (2002).
[24] M. S. Hussein, J. X. de Carvalho, M. P. Pato and A. J. Sargeant, Few-Body Systems, 38,
209 (2006).
P(s) d
0 1 2 3 4
FIG. 1: Nearest Neighbour Distributions. Histograms show data (a)-(x) from Ref. [17]. Thick
histograms show the three coupled GOE fits to the data carried out using the DGOE numerical
simulations using Eq. (5). Also shown as the full thin line the three uncoulped GOE P (s).
In graph (x) the dotted line is the Poisson distribution, the dashed line is the two uncoupled
GOE P (s). The very thin line is Wigner distribution which is hidden behind histograms. The
values of λ that adjust the data are 0.0032, 0.0071, 0.0158, 0.0250, 0.0333, 0.9950, 1.000 for cases
(a)-(x). See text for details.
0 1 2 3 4
FIG. 2: Nearest Neighbour Distributions. Histograms show data (a)-(x) from Ref. [17]. Thick
histograms show the two coupled GOE fits to the data carried out using the DGOE numerical
simulations using Eq. (5). Also shown as the full thin line the two uncoulped GOE P (s). In
graph (x) the dotted line is the Poisson distribution, the dashed line is three uncoupled GOE
P (s). The very thin line is Wigner distribution which is hidden behind histograms. The values
of λ that adjust the data are 0.000, 0.0258, 0.0200, 0.0400, 0.0705, 0.0600, 1.000 for cases (a)-(x).
See text for details.
∆3(L)
0 10 20 30 40 50
FIG. 3: Spectral Rigidities. The thick lines are the DGOE simulation for the three coupled
GOE’s.The same values of λ as in Fig. 1 were used. The thin lines correspond the three
uncoupled GOE’s case. The data points are from Ref. [17]. See text for details.
0.8 a
0 10 20 30 40 50
FIG. 4: Spectral Rigidities. The thick lines are the DGOE simulation for the two coupled
GOE’s. The same values of λ as in Fig. 2 were used. The thin lines correspond the two
uncoupled GOE’s case. The data points are from Ref. [17]. See text for details.
0 10 20 30 40 50
∆3(L)
2GOE, g=0.30
3GOE, g=0.10
FIG. 5: The ML effect. The data points correspond to case (b) of Ref. [17], r = 0.5mm. The
full line corresponds to our three coupled GOE’s fit with λ = 0.0071, figure 3b and g = 0.10.
The dashed line corresponds to our two coupled GOE’s fit with λ = 0.0258, figure 4b and
g = 0.30. See text for details.
	References
