Spin coupling in zigzag Wigner crystals
Spin coupling in zigzag Wigner crystals
A. D. Klironomos,1,2 J. S. Meyer,2 T. Hikihara,3 and K. A. Matveev1
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
(Dated: October 28, 2018)
We consider interacting electrons in a quantum wire in the case of a shallow confining potential and
low electron density. In a certain range of densities, the electrons form a two-row (zigzag) Wigner
crystal whose spin properties are determined by nearest and next-nearest neighbor exchange as well
as by three- and four-particle ring exchange processes. The phase diagram of the resulting zigzag
spin chain has regions of complete spin polarization and partial spin polarization in addition to
a number of unpolarized phases, including antiferromagnetism and dimer order as well as a novel
phase generated by the four-particle ring exchange.
PACS numbers: 73.21.Hb,71.10.Pm
I. INTRODUCTION
The deviations of the conductance from perfect quanti-
zation in integer multiples of G0 = 2e
2/h observed in bal-
listic quantum wires at low electron densities have gener-
ated great experimental and theoretical interest in recent
years.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27
These conductance anomalies manifest themselves as
quasi-plateaus in the conductance as a function of
gate voltage at about 0.5 to 0.7 of the conductance
quantum G0, depending on the device. Although most
experiments are performed with electrons in GaAs
wires,1,2,3,4,5,6,7,8,9,10,11 a similar “0.7 structure” was
recently observed in devices formed in two-dimensional
hole systems.12,13,14 It is widely accepted that the origin
of the quasi-plateau lies in correlation effects, but a
complete understanding of this phenomenon remains
elusive.
Although some alternative interpretations have been
proposed,11,26,27 most commonly the experimental find-
ings are attributed to non-trivial spin properties of quan-
tum wires.1,4,5,6,7,8,9,10,14,15,16,17,18,19,20,21,22,23,24,25 In a
truly one-dimensional geometry the spin coupling is rel-
atively simple: electron spins are coupled antiferromag-
netically, and the low energy properties of the system
are described by the Luttinger liquid theory. The pic-
ture may change dramatically when transverse displace-
ments of electrons are important and the system be-
comes quasi-one-dimensional. In particular, the spon-
taneous spin polarization of the ground state, which was
proposed1,6,9,10,14,15,16 as a possible origin of the conduc-
tance anomalies, is forbidden in one dimension,28 but
allowed in this case.
The electron system in a quantum wire undergoes
a transition from a one-dimensional to a quasi-one-
dimensional state when the energy of quantization in the
confining potential is no longer large compared to other
important energy scales. In this paper we consider the
spin properties of a quantum wire with shallow confin-
ing potential. In such a wire the electron system be-
comes quasi-one-dimensional while the electron density
is still very low, and thus the interactions between elec-
trons are effectively strong. At very low densities, elec-
trons in the wire form a one-dimensional structure with
short-range crystalline order—the so-called Wigner crys-
tal. As the density increases, strong Coulomb interac-
tions cause deviations from one-dimensionality creating
a quasi-one-dimensional zigzag crystal with dramatically
different spin properties. In particular, ring exchanges
will be shown to play an essential role.
We find several interesting spin structures in the
zigzag crystal. In a sufficiently shallow confining po-
tential, in a certain range of electron densities, the 3-
particle ring exchange dominates and leads to a fully
spin-polarized ground state. At higher electron densities,
and/or in a somewhat stronger confining potential, the
4-particle ring exchange becomes important. We study
the phase diagram of the corresponding spin chain us-
ing the method of exact diagonalization, and find that
the 4-particle ring exchange gives rise to novel phases,
including one of partial spin polarization.
The paper is organized as follows. The formation of a
Wigner crystal in a quantum wire and its evolution into
a zigzag chain as a function of electron density are dis-
cussed in Sec. II. Spin interactions in a zigzag Wigner
crystal which arise through 2-particle as well as ring ex-
changes are introduced in Sec. III. The numerical calcu-
lation of the relevant exchange constants is presented in
Sec. IV. The results of the numerical calculation estab-
lish the existence of a ferromagnetic phase at intermedi-
ate densities and the dominance of the 4-particle ring ex-
change at high densities. Subsequently, a detailed study
of the zigzag chain with 4-particle ring exchange is pre-
sented in Sec. V. An attempt to construct the phase dia-
gram for a realistic quantum wire as a function of electron
density and interaction strength is presented in Sec. VI.
The paper concludes with a discussion of the relation of
our work to recent experiments, given in Sec. VII. A
brief summary of some of our results has been reported
previously in Ref. 29.
http://arxiv.org/abs/0704.0776v1
0 0.05 0.1 0.15 0.2 0.25
(a) ν=0.70
(b) ν=0.90
(c) ν=1.75
FIG. 1: Wigner crystal of electrons in a quantum wire. The
structure as determined by the dimensionless distance be-
tween rows d/r0 depends on the parameter ν proportional
to electron density (see text). As density grows, the one-
dimensional crystal (a) gives way to a zigzag chain (b,c).
II. WIGNER CRYSTALS IN QUANTUM WIRES
We consider a long quantum wire in which the elec-
trons are confined by some smooth potential in the direc-
tion transverse to the wire axis. Assuming a quadratic
dispersion and zero temperature, the kinetic energy of
an electron is typically of the order of the Fermi en-
ergy EF = (π~n)
2/8m, whereas the Coulomb interaction
energy is of the order of e2n/ǫ. Here, n is the (one-
dimensional) density of electrons, ǫ is the dielectric con-
stant of the host material, and m is the effective electron
mass. As the density of electrons is lowered, Coulomb
interactions become increasingly more important, and at
n ≪ a−1B they dominate over the kinetic energy, where
the Bohr radius is given as aB = ~
2ǫ/me2. (In GaAs its
value is approximately aB ≈100Å.)
In this low-density limit, the electrons can be treated
as classical particles. They will minimize their mutual
Coulomb repulsion by occupying equidistant positions
along the wire, forming a structure with short-range crys-
talline order—the so-called Wigner crystal, Fig. 1(a).
Unlike in higher dimensions, the long-range order in a
one-dimensional Wigner crystal is smeared by quantum
fluctuations, and only weak density correlations remain
at large distances.30 However, as will be shown in the
following sections, the coupling of electron spins is con-
trolled by electron interactions at distances of order 1/n,
where the picture of a one-dimensional Wigner crystal is
applicable. Henceforth, we speak of a Wigner crystal in
a quantum wire with this important distinction in mind.
Upon increasing the density, the inter-electron distance
diminishes, and the resulting stronger electron repulsion
will eventually overcome the confining potential Vconf ,
transforming the classical one-dimensional Wigner crys-
tal into a staggered or zigzag chain31,32, as depicted in
Fig. 1(b,c). From the comparison of the Coulomb inter-
action energy Vint(r) = e
2/ǫr with the confining potential
an important characteristic length scale emerges. Indeed,
the transition from the one-dimensional Wigner crystal
to the zigzag chain is expected to take place when dis-
tances between electrons are of the order of the scale r0
such that Vconf(r0) = Vint(r0).
It is therefore necessary to identify the electron equi-
librium configuration as a function of density. In order
to proceed in a quantitative way we consider a specific
model, namely a quantum wire with a parabolic confining
potential Vconf(y) = mΩ
2y2/2, where Ω is the frequency
of harmonic oscillations in the potential Vconf(y). Within
that model the characteristic length scale r0 is given as
2e2/ǫmΩ2
. (1)
It is convenient for the following discussion to measure
lengths in units of r0. To that respect we introduce a
dimensionless density
ν = nr0. (2)
Then minimization of the energy with respect to the
electron configuration31,32 reveals that a one-dimensional
crystal is stable for densities ν < 0.78, whereas a zigzag
chain forms at intermediate densities 0.78 < ν < 1.75.
(If density is further increased, structures with larger
numbers of rows appear.31,32) The distance d between
rows grows with density as shown in Fig. 1. Note that at
ν ≈ 1.46 the equilateral configuration is achieved. There-
fore, at higher densities—and in a curious contradiction
in terms—the distance between next-nearest neighbors is
smaller than the distance between nearest neighbors (see
Fig. 1(c)).
III. SPIN EXCHANGE
In order to introduce spin interactions in the Wigner
crystal, it is necessary to go beyond the classical limit.
In quantum mechanics spin interactions arise due to ex-
change processes in which electrons switch positions by
tunneling through the potential barrier that separates
them. The tunneling barrier is created by the exchanging
particles as well as all other electrons in the wire. The re-
sulting exchange energy is exponentially small compared
to the Fermi energy EF . Furthermore, as a result of
the exponential decay of the tunneling amplitude with
distance, only nearest neighbor exchange is relevant in a
one-dimensional crystal. Thus, the one-dimensional crys-
tal is described by the Heisenberg Hamiltonian H1 =∑
j J1SjSj+1, where the exchange constant J1 is posi-
tive and has been studied in detail recently.24,33,34,35 The
exchange constant being positive leads to a spin-singlet
ground state with quasi-long-range antiferromagnetic or-
der, in accordance with the Lieb-Mattis theorem.28
The zigzag chain introduced in the previous section
displays much richer spin physics. As the distance be-
tween the two rows increases as a function of density, the
distance between next-nearest neighbors becomes com-
parable to and eventually even smaller than the distance
between nearest neighbors, as illustrated in Fig. 1(b,c).
Consequently, the next-nearest neighbor exchange con-
stant J2 may be comparable to or larger than the nearest
neighbor exchange constant J1. Drawing intuition from
studies of the two-dimensional Wigner crystal,36,37,38,39
one comes to a further important realization regarding
the physics of the zigzag chain: in addition to 2-particle
exchange processes, ring exchange processes, in which
three or more particles exchange positions in a cyclic
fashion, have to be considered in this geometry.
It has long been established that, due to symmetry
properties of the ground state wavefunctions, ring ex-
changes of an even number of fermions favor antiferro-
magnetism, while those of an odd number of fermions
favor ferromagnetism.40 In a zigzag chain, the Hamilto-
nian reads
J1Pj j+1 + J2Pj j+2 − J3(Pj j+1 j+2 + Pj+2 j+1 j)
+J4(Pj j+1 j+3 j+2 + Pj+2 j+3 j+1 j)− . . .
, (3)
where Pj1...jl denotes the cyclic permutation operator of l
spins. Here the exchange constants are defined such that
all Jl > 0. Furthermore, only the dominant l-particle ex-
changes are shown. A more familiar form of the Hamilto-
nian in terms of spin operators is obtained by noting that
Pij =
+ 2SiSj and Pj1...jl = Pj1j2Pj2j3 . . . Pjl−1jl .
Using spin operators and considering the two-spin ex-
changes one obtains the Hamiltonian
H12 =
(J1SjSj+1 + J2SjSj+2) . (4)
The competition between the nearest neighbor and next-
nearest neighbor exchanges becomes the source of frus-
tration of the antiferromagnetic spin order and eventu-
ally leads to a gapped dimerized ground state at J2 >
0.24J1.
41,42,43,44
The simplest ring exchange involves three particles and
is therefore ferromagnetic. Including the 3-particle ring
exchange J3 in addition to the 2-particle exchanges, the
Hamiltonian of the corresponding spin chain retains a
simple form. The 3-particle ring exchange does not in-
troduce a new type of coupling, but rather modifies the
2-particle exchange constants.40 For a zigzag crystal we
find the effective 2-particle exchange constants
J̃1 = J1 − 2J3, (5)
J̃2 = J2 − J3. (6)
Thus the total Hamiltonian has the form
H123 =
J̃1SjSj+1 + J̃2SjSj+2
, (7)
where J̃1 and J̃2 can have either sign.
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J 2~ 0
FM AF
Dimers
FIG. 2: The phase diagram including nearest neighbor, next-
nearest neighbor, and 3-particle ring exchanges. The effective
couplings eJ1 and eJ2 are defined in the text. The shaded region
between the dimer and ferromagnetic phases corresponds to
the exotic phase predicted in Ref. 48.
Consequently, regions of negative (i.e. ferromagnetic)
nearest and/or next-nearest neighbor coupling become
accessible. The phase diagram of the Heisenberg spin
chain (7) with both positive and negative couplings has
been studied extensively.41,42,43,44,45,46,47,48,49,50 In ad-
dition to the antiferromagnetic and dimer phases dis-
cussed earlier, a ferromagnetic phase exists for J̃1 <
min{0,−4J̃2}.46 An exotic phase called the chiral-biaxial-
nematic phase has been predicted48 to appear for J̃1 < 0
and −0.25 < J̃2/J̃1 < −0.38. However, the nature of the
system in this parameter region is still controversial. The
phase diagram is drawn in Fig. 2.
Thus, depending on the relative magnitudes of the var-
ious exchange constants, different phases are realized.
Extensive studies of the two-dimensional Wigner crys-
tal have shown that, at low densities (or strong interac-
tions), the 3-particle ring exchange dominates over the
2-particle exchanges. As a result, the two-dimensional
Wigner crystal becomes ferromagnetic at sufficiently
strong interactions.36,39 Given that the electrons in a
two-dimensional Wigner crystal form a triangular lat-
tice, by analogy, one should expect a similar effect in
the zigzag chain at densities where the electrons form ap-
proximately equilateral triangles. More specifically, upon
increasing the density and consequently the distance be-
tween rows, one would expect the system to undergo a
phase transition from an antiferromagnetic to a ferromag-
netic phase. To establish this scenario conclusively, the
various exchange energies in the zigzag crystal have to be
determined. The system differs from the two-dimensional
crystal in two important aspects. (i) The electrons are
subject to a confining potential as opposed to the flat
background in the two-dimensional case. Even more im-
portantly, (ii) the electron configuration depends on den-
sity, cf. Fig. 1, as opposed to the ideal triangular lattice
in two dimensions. In the following section, we proceed
with a numerical study of the exchange energies for the
specific configurations of the zigzag Wigner crystal in a
parabolic confining potential.
IV. SEMICLASSICAL EVALUATION OF THE
EXCHANGE CONSTANTS
The effective strength of interactions is usually de-
scribed by the interaction parameter rs which measures
the relative magnitude of the interaction energy and the
kinetic energy and is of order the distance between elec-
trons measured in units of the Bohr radius. For quan-
tum wires, it is more appropriate to use the parameter
rΩ = r0/aB, which takes into account the confining po-
tential. Within our model, the interaction parameter rΩ
rΩ = 2
2ǫ2~2
. (8)
For rΩ ≫ 1, strong interactions dominate the physics
of the system, and a semiclassical description is appli-
cable. In order to calculate the various exchange con-
stants, we use the standard instanton method, success-
fully employed in the study of the two-dimensional36,37,38
and one-dimensional34,35 Wigner crystal. Within this
approach, the exchange constants are given by Jl =
J∗l exp (−Sl/~). Here Sl is the value of the Euclidean
(imaginary time) action, evaluated along the classical ex-
change path. By measuring length and time in units of r0
and T =
2/Ω, respectively, the action S[{rj(τ)}] can be
rewritten in the form S = ~η
rΩ, where the functional
η[{rj(τ)}] =
+ y2j
|rj−ri|
 (9)
is dimensionless.
Thus, we find the exchange constants in the form
Jl = J
l exp (−ηl
rΩ), (10)
where the dimensionless coefficients ηl depend only on
the electron configuration (cf. Fig. 1) or, equivalently,
on the density ν. The exponents ηl are calculated nu-
merically for each type of exchange by minimizing the
action (9) with respect to the instanton trajectories of
the exchanging electrons. This procedure is mathemati-
cally equivalent to solving a set of coupled, second order
in the imaginary time τ , differential equations for the
trajectories rj(τ). The boundary conditions at τ = ±∞
are, respectively, the original equilibrium configuration
and the configuration where the electrons have exchanged
positions according to the exchange process considered.
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
FIG. 3: The exponents η1, η2, η3, and η4 as functions of the
dimensionless density ν.
ν η1 η2 η3 η4
1.0 1.050 2.427 1.254 1.712
1.1 1.161 2.169 1.261 1.605
1.2 1.255 1.952 1.275 1.532
1.3 1.337 1.754 1.287 1.469
1.4 1.406 1.566 1.293 1.398
1.5 1.456 1.376 1.278 1.299
1.6 1.471 1.169 1.215 1.135
1.7 1.391 0.901 1.022 0.784
TABLE I: The numerically calculated values of the density
dependent exponents ηl, see Eq. (10). The computation was
carried out including 12 moving spectator particles on either
side of the exchanging particles. Corrections to all ηl from
the remaining spectators do not exceed 0.1%.
In the simplest approximation only the exchanging
electrons are included in the calculation while all other
electrons, being frozen in place, create the background
potential. It turns out, however, that it is important
to take into account the motion of “spectators”—the
electrons in the crystal to the left and to the right of
the exchanging particles—during the exchange process.
The results presented here are obtained by successively
adding more spectators on both sides until the values ηl
converge. We find that including 12 moving spectators
on either side of the exchanging particles determines the
exponents to an accuracy better than 0.1%.
Figure 3 shows the calculated exponents for various ex-
changes as a function of dimensionless density ν and the
corresponding values are reported in Table I. At strong
interactions (rΩ ≫ 1), the exchange with the smallest
value of ηl is clearly dominant, and the prefactor J
l is of
secondary importance to our argument. At low densities,
when the zigzag chain is still close to one-dimensional, J1
(c) J3
(b) J(a) J 2
(d) J4
FIG. 4: The calculated particle trajectories for various ex-
changes at a representative density ν = 1.5. It is evident that
only a few near neighbors of the exchanging particles move
appreciably.
is the largest exchange constant, and the spin physics is
controlled by the nearest-neighbor exchange. In an inter-
mediate density regime, when the electron configuration
is close to equilateral triangles, the 3-particle ring ex-
change dominates. Thus, the numerical calculation con-
firms our original expectation, and a transition from an
antiferromagnetic to a ferromagnetic state takes place
upon increasing the density. Surprisingly, however, at
even higher densities the 4-particle ring exchange is the
dominant process. The role of the 4-particle ring ex-
change and the phase diagram of the associated zigzag
spin chain will be the subject of the following section.
More complicated exchanges have also been computed,
namely multi-particle (l ≥ 5) ring exchanges as well as
exchanges involving more distant neighbors. However,
the exchanges displayed in Fig. 3 were found to be the
dominant ones.29
It is important to note here that spectators contribute
to our results in an essential way. Allowing spectators to
move results not only in quantitative changes (namely a
reduction of the initially overestimated values ηl) but in
qualitative changes as well: at high densities, the dom-
inance of the 4-particle ring exchange J4 over the next-
nearest neighbor exchange J2 is obtained only if specta-
tors are taken into account. In particular, it is necessary
to include at least 6 moving spectators on each side of the
exchanging particles for J4 to take over at high densities.
The considerable effect that the spectators have on
the values of the exponents raises the question whether
a short-ranged interaction potential might cause further
quantitative or qualitative changes to the physical pic-
ture. In order to investigate that possibility we have
repeated the entire calculation for a modified Coulomb
interaction of the form
V (x) =
x2 + (2d)2
. (11)
This particular interaction accounts for the presence of a
metal gate, modeled by a conducting plane at a distance
d from the crystal. The gate screens the bare Coulomb
potential, modifying the electron-electron interaction at
long distances. Our calculation shows that this modifica-
tion affects the values of the exponents only weakly, even
when the gate is placed at a distance from the crystal
comparable to the inter-particle spacing. Qualitatively,
the physical picture remains the same, with the order of
dominance of the various exchanges unaffected through-
out the range of densities.
At the same time, it is particularly noteworthy that
(both for the screened and unscreened interaction) the
contribution of the spectator electrons saturates rapidly
as their number is increased. This is an indication that
the destruction of long-range order in the quasi-one-
dimensional Wigner crystal by quantum fluctuations will
not affect our conclusions. Figure 4 shows the particle
trajectories for the dominant exchanges at a represen-
tative density of ν = 1.5. The trajectories of both the
exchanging particles and a subset of the spectators are
shown, and their relative displacements can be readily
compared.
V. FOUR-PARTICLE RING EXCHANGE
We have shown in the preceding section that in a cer-
tain range of densities, the 4-particle ring exchange dom-
inates. Unlike the 3-particle exchange, the 4-particle
ring exchange not only modifies the nearest and next-
nearest neighbor exchange constants, but, in addition,
introduces more complicated spin interactions.40 For the
zigzag chain, we find
H4 = J4
SjSj+l + 2
(SjSj+1)(Sj+2Sj+3)
+(SjSj+2)(Sj+1Sj+3)− (SjSj+3)(Sj+1Sj+2)
. (12)
Not much is known about the physics of zigzag spin
chains with interactions of this type. We have stud-
ied this particular system described by the Hamiltonian
H = H123 + H4 using exact diagonalization, consider-
ing systems of N = 12, 16, 20, 24 sites. Periodic bound-
ary conditions have been imposed, and we have employed
the well-known Lanczos algorithm to calculate a few low-
energy eigenstates.
Figure 5 shows the total spin S of the ground state
as a function of the effective couplings J̃1/J4 and J̃2/J4
for the largest system considered, one with N = 24 sites.
The darkest region corresponding to maximal total spin is
the ferromagnetic phase, which occurs for large negative
couplings in direct analogy to the phase diagram for the
system without four-spin interactions (see Fig. 2). For all
system sizes that we have considered, the obtained phase
boundary is almost independent of the system size and
agrees very well with the conditions for ferromagnetism
J̃1 + 2J4 < 0, (13)
J̃1 + 4J̃2 + 10J4 < 0, (14)
FIG. 5: Total spin S of the ground state for a chain of N = 24
sites as a function of the effective couplings eJ1/J4 and eJ2/J4.
derived by treating the four-spin terms in the Hamilto-
nian (12) on a mean field level near the ferromagnetic
state.
A new phase of partial spin polarization appears adja-
cent to the ferromagnetic phase. The partially polarized
phase possesses a ground state total spin of S = 2 for
N = 12, S = 2 or 4 for N = 16, 20, and S = 4 for
N = 24; it appears that total spin of one third of the
saturated magnetization N/2 prevails throughout most
of that phase. The phase persists, to a significant extent,
in range and form as N increases. Therefore, we believe
it is not a finite size effect. We note here that it has been
shown rigorously that a model described by a Hamilto-
nian having a similar form to ours also exhibits a ground
state with partial spin polarization.51 On the other hand,
the scattered points corresponding to non-zero total spin
in the first quadrant (J̃1, J̃2 > 0) appear to shift posi-
tion as N increases and the size of the total spin remains
small, S ≤ 2, for all system sizes considered. We cannot
ascertain at this point whether they persist in a larger
system.
At large values of |J̃1|/J4 and |J̃2|/J4, one would ex-
pect to recover the phases present in the absence of J4.
Thus, the large white area in Fig. 5 corresponding to
total spin S = 0 should contain the antiferromagnetic
phase, analogues of the dimer phases observed in the
system without four-spin interactions, and possibly en-
tirely new phases as well. In order to distinguish between
these phases, we first calculate the overlap between the
ground state wavefunctions in our model and the ones
representing the dimer and antiferromagnetic phases in
the well-studied model with J4 = 0. The representative
ground state wavefunctions are obtained for the chain
with J4 = 0 and typical parameter sets of (J̃1, J̃2) cho-
sen deep in the dimer and antiferromagnetic phases of
the phase diagram shown in Fig. 2. The results for the
−6 −4 −2 0 2 4 6
−6 −4 −2 0 2 4 6
8 1.0
FIG. 6: Overlaps of the ground state wavefunctions in the
presence of the 4-particle ring exchange with the wavefunc-
tions representing (a) the dimer and (b) the antiferromag-
netic phase for J4 = 0. The representative ground states
(a) and (b) are obtained for ( eJ1, eJ2, J4) = (1, 10, 0) and
( eJ1, eJ2, J4) = (1,−10, 0), respectively.
chain with N = 24 sites are shown in Fig. 6. As can
be seen from the figure, the ground states for a broad
region of large positive J̃2/J4 have a significant overlap
with the representative ground state of the dimer phase
while the ground states for large positive J̃1/J4 and/or
negative J̃2/J4 resemble very much the one belonging to
the antiferromagnetic phase. This behavior indicates the
appearance of the expected dimer and antiferromagnetic
phases for large effective couplings |J̃1|/J4 and |J̃2|/J4.
We have confirmed the existence of these phases in the
corresponding parameter regimes by studying the associ-
ated structure factors.
In order to study and clarify the properties of the sys-
tem in more detail, we have calculated the excitation
energies
∆En(S,Q) = En(S,Q)− Egs, (15)
where En(S,Q) is the energy of n-th lowest level in the
subspace characterized by the total spin S and the mo-
−2 0 2 4 6
J2 / J4
J1 /J4 = 2, N = 24
: (0, 0)
: (0, π)
: (1, π)
FIG. 7: Excitation energies ∆En(S,Q) in the system of N =
24 sites for eJ1/J4 = 2 as functions of eJ2/J4. The two-lowest
levels are plotted for the subspaces of (S,Q) = (0, 0) and (0, π)
while only the lowest one is shown for all other subspaces.
The energies for (S,Q) = (0, 0), (0, π), and (1, π) are plotted
by thick solid, dotted, and dashed curves, respectively. The
energies of the levels belonging to other subspaces are shown
by thin gray curves.
mentum Q, and Egs is the ground state energy. Figure 7
shows the results for the system of size N = 24, obtained
along the vertical line in the phase diagram given by
J̃1/J4 = 2. At large positive J̃2/J4, the ground and first-
excited states belong to the subspace (S,Q) = (0, 0) and
(0, π), respectively.52 These states are expected to form
the ground state doublet of the dimer phase in the ther-
modynamic limit. For J̃2/J4 > (J̃2/J4)c,dim ∼ 3.5, one of
the dimer doublet states is the ground state and the sys-
tem is in the dimer phase. At smaller J̃2/J4, both states
of the dimer doublet shift upward and move away from
the low-energy regime, while other states decrease steeply
in energy and eventually become the ground state. We
therefore take the point (J̃2/J4)c,dim as the boundary of
the dimer phase. After the transition, the system enters
a region with exotic ground states and a large number of
low-lying excitations. We have numerically checked that
these exotic ground states have no or, at most, negligibly
small overlap with the ground state of either the dimer or
antiferromagnetic phases. When J̃2/J4 decreases further,
the exotic states leave the low-energy regime and the
system predictably enters the antiferromagnetic phase,
which occurs for J̃2/J4 < (J̃2/J4)c,AF ∼ 0.1.
Performing the same type of analysis for several pa-
rameter lines, we can estimate the phase boundaries
(J̃2/J4)c,dim and (J̃2/J4)c,AF as functions of J̃1/J4. In
the limit of large negative coupling J̃1/J4 → −∞, the
boundary of the dimer phase (J̃2/J4)c,dim approaches the
line J̃1 = −0.38J̃2, suggesting a smooth connection to
the behavior for J̃1 < 0 and J4 = 0 (cf. Ref. 48). In
−5 0 5
Dimers
FIG. 8: The phase diagram of the Heisenberg chain including
nearest neighbor, next-nearest neighbor, and 4-particle ring
exchanges. The expected phases consist of a ferromagnetic
and an antiferromagnetic phase as well as a dimer phase. In
addition, a novel region (4P ) dominated by the 4-particle ring
exchange appears. The latter includes a phase of partial spin
polarization (M). Triangles, squares and circles correspond
to the boundaries obtained for N = 16, 20, and 24 sites, re-
spectively. We note that although the phase of partial spin
polarization persists as the system size is increased, its bound-
ary with the 4P phase has a rather irregular size dependence
and is represented approximately in the figure.
a similar fashion, at large positive coupling J̃1/J4, we
find no indication for the appearance of exotic phases
after J̃1/J4 ≥ 6; the data of the energy spectrum and
the wavefunction overlaps show essentially the same be-
haviors as those at J̃1/J4 → ∞. We therefore conclude
that there occurs a direct transition between the dimer
and antiferromagnetic phases and estimate the transition
line using the method of level spectroscopy, according
to which the transition point is determined by the level
crossing between the first-excited states in the dimer and
antiferromagnetic phases.43
Combining all these phase boundaries and including
the boundaries of the ferromagnetic and partially spin
polarized phases which were obtained using the total spin
of the ground state as a criterion, we determine the phase
diagram in the J̃1/J4 versus J̃2/J4 plane. The result is
shown in Fig. 8. The phase diagram has similarities to
the one obtained without the four-spin interaction term,
see Fig. 2. In particular, the expected ferromagnetic, an-
tiferromagnetic, and dimer phases appear for large values
of the effective couplings, |J̃1|/J4 and |J̃2|/J4. But more
importantly, at not too large values of the effective cou-
plings, new phases appear as a direct result of the new
interaction term. We can identify a phase with partial
spin polarization and a region occupied by one or sev-
eral novel phases with total spin S = 0. In the region
where J4 dominates, the ground state has no similarity
at the level of wavefunctions with that of the conventional
phases. It is important to note that the region occupied
by the new phases becomes broader as the system size
N grows, indicating that it survives even in the thermo-
dynamic limit. From the analysis of the wavefunction
overlaps between the ground states, there are strong in-
dications that the novel unpolarized region might consist
of several different phases. Unfortunately, it has proven
difficult to clarify the nature of the new phases and, in
particular, discover the order parameters that character-
ize them based solely on the analysis of small systems.
Therefore, the issue is relegated to future studies. In
the absence of detailed understanding of its properties,
we collectively dub the region of the phase diagram the
“4P” phase.
VI. PHASE DIAGRAM FOR REALISTIC
QUANTUM WIRES
Having identified possible phases of the zigzag chain,
the most interesting question is which of the various
phases appearing in the phase diagram Fig. 8 are ac-
cessible in quantum wires. At finite rΩ, the calculations
of the exchange constants discussed in Sec. IV have to be
completed in an important way by computing the prefac-
tors Jl in Eq. (10). To that effect it is necessary to take
into account Gaussian fluctuations around the classical
exchange path. We employ the method introduced by
Voelker and Chakravarty38 which, for the sake of com-
pleteness, is outlined in the Appendix . The prefactors
have the form
J∗l =
AlFl rΩ
, (16)
where Fl is density dependent. The factor Al is used to
account for multiple classical trajectories corresponding
to the same exchange process (see Appendix). Table II
contains the values of Fl we calculated for the various
exchanges considered in this work. Note that, in order
to achieve a comparable level of convergence, a more ac-
curate determination of the instanton trajectories was
required for the calculation of the prefactors J∗l than for
the calculation of the exponents ηl. By including up to 28
moving spectators on either side of the exchanging par-
ticles, we have been able to achieve an accuracy better
than 2%.
We are now in a position to map out the areas of the
phase diagram of Fig. 8 that are encountered as one tra-
verses the density region of interest for a given rΩ. The
resulting phase diagram obtained with the calculated ex-
change energies is shown in Fig. 9. Since the semiclassical
approximation is applicable only at rΩ ≫ 1, we do not
extend the phase diagram to values of rΩ < 10. It turns
out that the spin polarized phases are only realized at
rΩ & 50. On the other hand, the novel “4P” phase is
ν F1 F2 F3 F4
1.0 1.12 ≃ 6 1.22 2.44
1.1 1.04 ≃ 4 1.03 1.73
1.2 1.05 2.38 0.97 1.28
1.3 1.08 1.86 0.97 1.15
1.4 1.19 1.71 1.02 1.13
1.5 1.40 1.63 1.14 1.18
1.6 1.80 1.51 1.26 1.19
1.7 2.07 1.07 0.81 0.50
TABLE II: The numerically calculated values of the density
dependent part Fl of the exchange energy prefactor J
, see
Eq. (16), calculated with mobile spectators. For all the num-
bers reported, the accuracy is better than 2%, except for F2 at
ν = 1.0, 1.1, for which extrapolated values with an estimated
error of ∼ 10% are shown.
1.1 1.2 1.3 1.4 1.5 1.6
AF 4P
FIG. 9: The phase diagram as a function of the dimension-
less density ν and interaction strength rΩ. The various phases
were obtained by first calculating the effective couplings eJ1/J4
and eJ2/J4 for a given point; subsequently, the correspond-
ing phase was determined utilizing the calculated boundaries
shown in Fig. 8 for a system of N = 24 sites.
expected to appear in a certain density range as long as
rΩ ≫ 1.
VII. DISCUSSION
In the preceding sections we have studied the coupling
of spins of electrons forming a zigzag Wigner crystal in a
parabolic confining potential. We have found that apart
from the 2-particle exchange couplings between the near-
est and next-nearest neighbor spins, the 3- and 4-particle
ring exchange processes have to be taken into account.
At relatively low electron densities, when the transverse
displacement of electrons is small compared to the dis-
tance between particles, Fig. 1(b), the nearest-neighbor
2-particle exchange dominates. In this regime the spins
form an antiferromagnetic ground state, with low-energy
excitations described by the Tomonaga-Luttinger theory.
At relatively high densities, when the transverse displace-
ments are large, Fig. 1(c), the 4-particle ring exchange
processes dominate. Since the ring exchange processes in-
volving even numbers of particles favor spin-unpolarized
states, the ground state of the system in this regime has
zero total spin. Finally, if the confining potential is suf-
ficiently shallow, so that the parameter rΩ & 50, there
is an intermediate density range in which the 3-particle
exchange processes are important, and the ground state
is spontaneously spin-polarized. These results are sum-
marized in Fig. 9.
We expect that the zigzag Wigner crystal state can be
realized in quantum wires. In order for the zigzag crys-
tal to form the confining potential of the wire should be
rather shallow, so that large values rΩ ≫ 1 of the pa-
rameter (8) could be achieved. The exact shape of the
confining potential in existing wires is not well known.
Using the quoted value of subband spacing ∼ 20 meV
we estimate that the parameter rΩ is of order unity in
cleaved-edge-overgrowth wires.53 The confining potential
in split-gate quantum wires tends to be more shallow.
For a typical value 1 meV of subband spacing we es-
timate rΩ ≈ 6. Finally, for p-type quantum wires13,54
with subband spacing ∼ 300 µeV we estimate rΩ ≈ 20.
These hole systems are the most promising devices for
observation of the zigzag Wigner crystal.
Given the relatively modest values of rΩ . 20 in the ex-
isting quantum wire structures, we do not expect that the
spontaneously spin-polarized ground state will be easily
observed in experiments. Instead, we expect that as the
density of charge carriers is increased, a transition from
antiferromagnetism to a state dominated by 4-particle
ring exchanges will occur. We have found that the ground
state in this phase has a complicated size dependence,
which makes it very difficult to identify its nature by
exact diagonalization of finite-size chains. To fully un-
derstand the spin properties in the high density regime,
further studies of zigzag ladders with ring exchange cou-
pling are needed.
Acknowledgments
We acknowledge helpful discussions with A. Läuchli
and T. Momoi. This work was supported by the U. S.
Department of Energy, Office of Science, under Contract
No. DE-AC02-06CH11357. T.H. was supported in part
by a Grant-in-Aid from the Ministry of Education, Cul-
ture, Sports, Science and Technology (MEXT) of Japan
(Grant Nos. 16740213 and 18043003). Part of the calcu-
lations were performed at the Ohio Supercomputer Cen-
ter thanks to a grant of computing time.
APPENDIX: CALCULATION OF THE
PREFACTORS
In order to find the prefactors J∗l in the expressions
for the exchange constants, fluctuations around the in-
stanton trajectory have to be taken into account. The
Euclidean (imaginary time) path integral for the propa-
gator G(R1,R2;T ) = 〈R1|e−TH |R2〉 can be written as
G(R1,R2;T ) =
∫ R(T )=R2
R(0)=R1
DR e−
S[R], (A.1)
where the Euclidean action is given by
S[R] =
+ V (R)
. (A.2)
Here R is a M -dimensional position vector, where M/2
is the total number of moving particles, including the
exchanging particles as well as the spectators. In the
semiclassical limit, the integral is dominated by the clas-
sical path Rcl(τ) that extremizes the action S for a
given exchange process. (The exponents η are given as
η = S[Rcl]/(~
rΩ).) The Gaussian quantum fluctua-
tions about the classical path can be taken into account
by defining fluctuation coordinates u(τ) ≡ R(τ)−Rcl(τ)
and subsequently expanding the action to second order.
We obtain for the propagator
G(R1,R2;T ) = F [Rcl]e
S[Rcl], (A.3)
F [Rcl] =
∫ u(T )=0
u(0)=0
Du(τ) e− 1~ δS[u(τ)], (A.4)
δS[u(τ)] =
2(τ) + uT (τ)H(τ)u(τ)
, (A.5)
Hkp(τ) =
∂2V (R)
∂Rk∂Rp
R=Rcl(τ)
. (A.6)
In the preceding formulas, R1 and R2 correspond to two
configurations of electrons that minimize the electrostatic
potential V (R) describing electron-electron interactions
as well as the confining potential. The exchange constant
is related to the ratio of the propagator for a particular
exchange process R1 → R2, divided by the propagator
for the trivial path Rcl(τ) = R1:
F [Rcl]
F [R1]
S[Rcl]. (A.7)
We start from the expression for the propagator in the
semiclassical limit and proceed by partitioning the time
interval [0, T ] into N subintervals (τ0, τ1), (τ1, τ2), . . . ,
(τN−1, τN ), with τ0 = 0, τN = T . The partition is cho-
sen sufficiently fine as to enable the approximation that
in each subinterval, the Hessian matrix H(τ) of the sec-
ond derivative of the potential can be considered time
independent, H(τ) ≃ H(τν) ≡ Hν . (In what follows, we
use the convention that for the fluctuation coordinates,
superscripts denote time subinterval, while subscripts de-
note spatial coordinate.) Subsequently the path integral
is calculated as a product of path integrals over the par-
titioned interval. Moreover, each individual path inte-
gral is that of a multidimensional harmonic oscillator,
for which analytic results exist. We then have
F [Rcl] =
du1 G1(u
1,u0; τ1 − τ0) . . .
duN−1 GN−1(u
N−1,uN−2; τN−1 − τN−2)GN (uN ,uN−1; τN − τN−1), (A.8)
and the propagator for each subinterval is
ν ,uν−1; τν − τν−1) =
∫ u(τν)=uν
u(τν−1)=uν−1
Du(τ) exp
2(τ) + uT (τ)Hνu(τ)
. (A.9)
Within each imaginary time subinterval, we define or-
thonormal eigenvectors qνµ =
k=1 U
k. The unitary
matrix Uν is such that Hν = UνΛν(Uν)T , with Λ a diag-
onal matrix of eigenvalues (λνµ)
2, µ = 1 . . .M , where M
is the number of spatial coordinates. Then one immedi-
ately obtains
ν ,qν−1; τν − τν−1) =
∫ q(τν)=qν
q(τν−1)=qν−1
Dq(τ) exp
2(τ) + qT (τ)Λνq(τ)
= F̄ [qcl]e
δS[qcl],
(A.10)
where qcl is the classical trajectory connecting q
ν−1 and qν . Considering the fluctuation part first, we obtain an
elementary path integral
F̄ [qcl] =
∫ q(τν)=0
q(τν−1)=0
Dq(τ) exp
dτ qT (τ)
, (A.11)
where
Bνµ =
~ sinh(λνµ∆τν)
, (A.12)
and ∆τν = τn − τn−1. The exponent δS[qcl] can now be
calculated explicitly
δS[qcl]
[(qνµ)
cl + (q
cl] cosh(λ
µ∆τν)
−2(qνµ)cl(qν−1µ )cl
. (A.13)
The subscript “cl” used for notational clarity will be sub-
sequently dropped from all expressions. With some addi-
tional algebra, the remaining integral is easily evaluated.
With the following definitions
Γνkp =
~ tanh(λνµ∆τν)
Uνpµ (A.14)
∆νkp =
~ sinh(λνµ∆τν)
Uνpµ, (A.15)
we find
F [Rcl] = (2π)
, (A.16)
where the M(N − 1) ×M(N − 1) matrix Ωνλkp has com-
ponents
Ωνλkp = (Γ
kp + Γ
kp )δ
ν,λ −∆νkpδν,(λ+1) −∆λkpδν,(λ−1).
(A.17)
The calculation of F [R1] is carried out in an identical
manner and the subscript “0” will be used to distinguish
the results pertaining to that calculation.
Finally, one has to account for the existence of an
eigenvalue of the matrix Ω which is identically zero in
the continuum limit and corresponds to the zero mode
associated with uniform translation of the instanton in
imaginary time. The procedure is standard55 and we
simply report the result for the prefactor here. One ob-
tains
G = T
Bνµ,0
detΩ0
det′ Ω
, (A.18)
where the primed determinant implies the exclusion of
the eigenvalue corresponding to the zero mode. Revert-
ing to the system of units used in this work, the prefactor
of the exchange energy is given by
J∗l =
Al rΩ
Bνµ,0
detΩ0
det′ Ω
(A.19)
The additional factor Al is used to account for multiple
classical trajectories corresponding to the same exchange
process, as happens for the case of nearest and next-
nearest neighbor exchanges (i.e., A1 = A2 = 2, whereas
Al = 1 for l ≥ 3).
The numerical implementation of the method outlined
above is straightforward. In particular, the quantity that
needs to be numerically calculated, once for each type of
exchange at all densities of interest, is
Bνµ,0
detΩ0
det′ Ω
. (A.20)
We note here that the eigenvalue corresponding to the
zero mode is easily calculated with the same procedure
used by Voelker and Chakravarty38. In the definition of
the prefactor, see Eqs. (A.4) and (A.5), one replacesH(τ)
with H(τ)− λ, with λ a free parameter. Subsequently, a
numerical search for the smallest eigenvalue that results
in 1/F (λ) = 0 is carried out. The smallest eigenvalue
corresponds to the zero mode, and for a finite partition
of the imaginary time interval it is a small but finite
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