arXiv:1203.1624v2 [cond-mat.quant-gas] 11 Jul 2012
Anomalous Behavior of Spin Systems with Dipolar Interactions
D. Peter1, S. M¨ uller2, S. Wessel3, and H. P. B¨ uchler1
1Institute for Theoretical Physics III, University of Stuttgart, Stuttgart, Germany
2Max Planck Institute for Physics, 80805 Munich, Germany and
3Institute for Theoretical Solid State Physics, JARA-FIT, and JARA-HPC,
RWTH Aachen University, Otto-Blumenthal Strasse 26, D-52056 Aachen, Germany
(Dated: July 12, 2012)
We study the properties of spin systems realized by cold polar molecules interacting via dipole-
dipole interactions in two dimensions. Using a spin wave theory, that allows for the full treatment
of the characteristic long-distance tail of the dipolar interaction, we find several anomalous features
in the ground state correlations and the spin wave excitation spectrum, which are absent in their
counterparts with short-range interaction. The most striking consequence is the existence of true
long-range order at finite temperature for a two dimensional phase with a broken U(1) symmetry.
PACS numbers: 67.85.-d, 75.10.Jm, 75.30.Ds, 05.30.Jp
The foundation for understanding the behavior and
properties of quantum matter is based on models with
short-range interactions. Experimental progress in cool-
ing polar molecules  and atomic gases with large mag-
netic dipole moments  has however increased the in-
terest in systems with strong dipole-dipole interactions.
While many properties of quantum systems with dipole-
dipole interactions derive from our understanding of sys-
tems with short-range interactions, the dipole-dipole in-
teraction can give rise to phenomena not present in their
short-range counterparts. Prominent examples are the
description of dipolar Bose-Einstein condensates, where
the contribution of the dipolar interaction cannot be in-
cluded in the s-wave scattering length , and the ab-
sence of a first order phase transition with a jump in
the density . In this letter, we demonstrate anomalous
behavior in two dimensional spin systems with dipolar in-
teractions realized by polar molecules in optical lattices.
A remarkable property of cold polar molecules con-
fined into two dimensions is the potential formation of
a crystalline phase for strong dipole-dipole interactions
[5, 6]. In contrast to a Wigner crystal with Coulomb
interactions , the crystalline phase exhibits the con-
ventional behavior expected for a crystal realized with
a short-range repulsion and the characteristic 1/r3be-
havior of the dipole interaction can be truncated at dis-
tances involving several interparticle separations. Several
strongly correlated phases have been predicted, which be-
have in analogy to systems with interactions extending
over a finite range, such as a Haldane phase , super-
solids [9, 10], pair supersolids in bilayer systems , va-
lence bond solids , as well as p-wave superfluidity ,
and self-assembled structures in multilayer setups .
On the other hand, it has recently been demonstrated
that polar molecules in optical lattices are also suitable
for emulating quantum phases of two dimensional spin
Here, we demonstrate that such spin models with
dipole-dipole interactions exhibit several anomalous fea-
tures, which are not present in their short-range coun-
terparts. The analysis is based on analytical spin wave
theory, which allows for the full treatment of the 1/r3
tail of the dipole-dipole interactions. We find that the
excitation spectrum exhibits anomalous behavior at low
momenta, which gives rise to unconventional dynamic
properties of the spin wave excitations.
we derive from this anomalous behavior the existence of
a long-range ordered ferromagnetic phase at finite tem-
peratures; this finding is consistent with the well-known
Mermin-Wagner theorem as the latter does not exclude
order for interactions with a 1/rαtail, where α ≤ 4 [18–
20]. Finally, we show that the dipole-dipole interaction
gives rise to algebraic correlations even in gapped ground
states, in agreement with recent predictions [21, 22].
FIG. 1. (a) Mean-field phase diagram for the XXZ model with
dipolar interactions, where tanθ is the ratio between the XY
and the Ising spin couplings. (b) Ground state energy per
particle: the dashed lines show the mean-field predictions,
while the solid lines include the contributions from the spin
waves.At the critical values θc and˜θc, the ground state
energy exhibits the jump ∆ec ≈ 0.14J and ∆˜ ec ≈ 0.06J,
indicating the potential formation of an intermediate phase.
We focus on a set up of polar molecules confined into
two dimensions in a square lattice, with each lattice site
filled by one polar molecule. A static electric field applied
along the z direction splits the rotation levels, and allows
us to define a spin 1/2 system by selecting two states in
the rotational manifold. Then, the Hamiltonian reduces
to a XXZ model with dipole-dipole interaction between
the spins 
Here, the first term accounts for the static dipole-dipole
interaction between the different rotational levels with
strength J cosθ , while the last term describes the virtual
exchange of a microwave photon between the two polar
molecules with strength J sinθ, and a denotes the lattice
spacing. The dependence of the couplings J and θ on
the microscopic parameters is discussed in Ref. [16, 17,
23] and the one-dimensional version of this model has
recently been studied in Ref. .
Before analyzing this spin model on the square lat-
tice, we present a summary of the phase diagram for
its counterpart with nearest neighbor interactions only.
Then, the phase diagram is highly symmetric and ex-
hibits four different phases: (i) an Ising antiferromag-
netic phase (I-AF) for −π/4 < θ < π/4 with an excita-
tion gap, (ii) an XY antiferromagnetic phase (XY-AF) for
π/4 < θ < 3π/4 with a linear excitation spectrum, (iii)
an Ising ferromagnetic phase (I-F) for 3π/4 < θ < 5π/4
with an excitation gap, and finally (iv) a XY ferromag-
netic phase (XY-F) for 5π/4 < θ < 7π/4 with a linear
Next, we analyze the modifications of the phase dia-
gram due to dipole-dipole interactions between the spins
within mean-field theory. The main influence is the re-
duction of the stability for the antiferromagnetic phases,
as the next nearest neighbor interaction introduces a
weak frustration to the system. The ground state en-
ergy per lattice site within mean-field reduces to eI-AF=
J cosθ ǫK/4 and eXY-AF= J sinθ ǫK/4 for the antiferro-
magnetic phases. The summation over the dipole interac-
tion reduces to a dimensionless parameter ǫK≈ −2.646,
which is related to the dipolar dispersion
at the corner of the Brillouin zone K = (π/a,π/a).
In turn, the ferromagnetic phases are enhanced with
a mean-field energy eI-F = J cosθ ǫ0/4 and eXY-F =
J sinθ ǫ0/4 with ǫ0 ≈ 9.033. The modifications to the
phase diagram are shown in Fig. 1: first, the Heisenberg
points at θ = π/4,5π/4 are protected by the SU(2) sym-
metry and still provide the transition between the Ising
and the XY phases. However, the transitions from the
ferromagnetic towards the antiferromagnetic phase are
shifted to the values θc = arctan(ǫK/ǫ0) ≈ −0.1π and
˜θc= π + arctan(ǫ0/ǫK) ≈ 0.6π.
The dipole dispersion ǫq in Eq. (2) converges very
slowly due to the characteristic power law decay of the
dipole-dipole interaction.It is this slow decay, which
will give rise to several peculiar properties of the system.
Therefore, we continue first with a detailed discussion of
this dipolar dispersion. The precise determination of ǫq
is most conveniently performed using an Ewald summa-
tion , which transforms the summation over the slowly
converging terms with algebraic decay into a summation
of exponential factors, i.e.,
ǫq= −2πa|q|erfc(a|q|/2√π) + 4π
with erfc(x) the complementary error function. The im-
portant feature of the dipole dispersion is captured by
the first term in Eq. (3), which gives rise to a linear and
nonanalytic behavior ǫq ∼ ǫ0− 2πa|q| for small values
q ≪ 1/a, while all remaining terms are analytic. It is
this linear part, which will give rise to several uncon-
ventional properties of spin systems in 2D with dipolar
interactions, and is a consequence of the slow decay of
the dipole-dipole interaction. The summation in the last
term converges very quickly and guarantees the periodic-
ity of the dipolar dispersion. The quantitative behavior
is shown in Fig. 2a, and the numerical efficient determi-
nation provides ǫ0 ≈ 9.033, and ǫK = (1/√2 − 1)ǫ0 ≈
Next, we analyze the excitation spectrum above the
mean-field ground states within a spin wave analysis.
The spin wave analysis is well established [25, 26], and its
application for a spin system with dipolar interaction is
straightforward. Details of the calculation are presented
in the supplementary material for the antiferromagnetic
XY phase. The results are summarized in Table I, and
shown in Fig. 2. In the following, we present a detailed
discussion for each of the four ordered phases.
FIG. 2. Spin wave excitations with Γ = (0,0), M = (0,π/2),
and K = (π/a,π/a) for different θ angles. (a) Spectrum of
the I-F phase which also shows the behavior of the dipolar
dispersion ǫq for θ = −3π/4, see red line. (b-d) Spectrum for
the XY-F, I-AF and XY-AF phases. Each red line is a critical
excitation spectrum indicating an instability.
Ising ferromagnetic phase: The ferromagnetic mean-
field ground state is twofold degenerate with all spins
Ground state αSpin wave excitation spectrum Eα
Ground state energy per spin eα
v0Eα(q) =J cosθǫ0
XY-FJ?sinθ(ǫq− ǫ0)(cosθǫq− sinθǫ0)
J?(sinθǫq+K− cosθǫK)(sinθǫq− cosθǫK)
J?sinθ(ǫq+K− ǫK)(cosθǫq− sinθǫK)
TABLE I. Spin wave excitation spectrum Eα
qand ground state energy eα.
either point up or down, and is the exact ground state
for θ = π, i.e., |G? =?
spectrum reduces to EI-F
q, see Table I. The spin waves
exhibit an excitation gap ∆: (i) approaching the Heisen-
berg point at θ = −3π/4, the excitation gap vanishes,
indicating the instability towards the XY ferromagnet,
(ii) in turn, for antiferromagnetic XY couplings, the gap
is minimal at K, vanishes at the mean-field transition
point˜θcand drives an instability towards the formation
of antiferromagnetic ordering.
In contrast to any short-range ferromagnetic spin
model, the dispersion relation EI-F
small momenta, but rather exhibits a linear behavior,
q ∼ EI-F
which is a consequence of the dipolar interaction in the
system. This anomalous behavior strongly influences the
dynamics of the spin waves. The dynamical behavior of
a single localized spin excitation is shown in Fig. 3a for a
Gaussian initial state. In order to probe the linear part in
the dispersion relation, the width σ of the localization is
much larger than the lattice spacing a, and therefore, the
dynamics is well described by a continuum description.
Instead of the conventional quantum mechanical spread-
ing, one finds a ballistic expansion of a cylindrical wave
packet with velocity c. In addition, the dipole-dipole
interaction also strongly influences the correlation func-
tion. Within conventional perturbation theory, we find
algebraic correlations ?Sx
decay of correlations even in gapped systems is a pecu-
liar property of spin models with long-range interactions
XY-ferromagnetic phase: Here, the spins are aligned in
the xy plane. Within the spin wave analysis, we obtain
the excitation spectrum EXY-F
state energy eXY-F. In the low momentum regime, the
dispersion relation behaves as EXY-F
to the well known linear Goldstone modes for the broken
U(1) symmetry. This anomalous behavior is a peculiar
property of the dipolar interaction, and the most cru-
cial consequence is the existence of long-range order for
the continuous broken symmetry at finite temperatures
even in two dimensions . This property follows im-
i|↓?i. Within the spin wave anal-
ysis, the ground state is not modified and the excitation
is not quadratic for
0 + ?c|q| with velocity c = −2πaJ sinθ/?,
j? ∼ 1/|r|3. This algebraic
and the modified ground
?|q|, in contrast
scribed by the Gaussian wave packet ψ0(r) = e−|r|2/2σ2/√πσ2
with σ ≫ a in the continuum description. (a) For a linear
dispersion c|q| in the I-F phase, the dynamics is described
by cylindrical symmetric wave packets (see inset) traveling
with velocity c, instead of the conventional quantum mechan-
ical spreading for massive systems.
dispersion with α?|q| in the XY-F phase, the behavior at
|ψ(x,τ)|2= ξ(x/τ − 1/2)/τ2(see inset) using rescaled time
τ = tα/√σ and space x = |r|/σ coordinates. It describes a
cylindrical symmetric wave front with velocity α√σ.
Time evolution for localized spin excitations de-
(b) For an anomalous
long times t ≫√σα reduces to a scaling function ξ(z) via
mediately from the above spin wave analysis: the order
parameter reduces to m ≡ ∆m − 1/2 = ?Sx
∆m accounts for the suppression of the order parameter
by quantum fluctuations. Within spin wave theory, it
reduces to (∆m = ?a†
This expression is finite and small: at T = 0, the inte-
grand behaves as ∼ 1/?|q| and we find a suppression of
corrections due to quantum fluctuations is a good justifi-
cation for the validity of the spin wave analysis. On the
other hand, even at finite temperatures, the low momen-
tum behavior of the integrand takes the form ∼ T/|q|,
and provides a finite contribution in contrast to a con-
ventional Goldstone mode, which provides a logarithmic
The appearance of a long-range order at a finite tem-
perature for a ground state with a broken U(1) sym-
metry is a peculiar feature of dipole-dipole interactions,
which renders the system more mean-field like. The sys-
?cosθǫq+ sinθ(ǫq− 2ǫ0)
the order ∆m ≈ 0.008 at θ = −π/2. The smallness of this
T = 0
0 < T < Tc
Tc < T
j? − m2∼ |r|−3/2
TABLE II. Correlation functions in the XY-F phase predicted
by the spin wave analysis and high temperature expansion.
tem therefore exhibits a finite temperature transition at
a critical temperature Tcinto a disordered phase; such a
behavior is consistent with the classical XY model with
dipolar interactions . The correlation functions deter-
mined within spin wave theory and a high temperature
expansion are summarized in Table II. Note, that the spin
wave analysis neglects the influence of vortices. This is
well justified here, as the dipolar interactions give rise
to a confining of vortices, i.e., the interaction potential
between a vortex–antivortex pair increases linearly with
the separation between the vortices.
The spin wave dynamics caused by the anomalous dis-
persion relation ∼
wave packet of width σ. Interestingly, the propagation
velocity of the wave packets is proportional to√σ and
thus faster for broad wave packets, in contrast to the
usual dispersion dynamics. This is a consequence of the
group velocity vq∼ 1/?|q| which is large for the small
Ising antiferromagnetic phase: Next, we focus on the
antiferromagnetic phases and start with the I-AF ground
state.Again, the ground state is twofold degenerate
on bipartite lattices. We choose the ground state with
spin up on sublattice A and spin down on sublattice B,
i.e., |G? =?
and we obtain the spin wave excitation spectrum EI-AF
and ground state energy eI-AF, see Table I. The sys-
tem exhibits an excitation gap as expected for a system
with a broken Z2 symmetry.
teractions give rise to an anomalous behavior at small
momenta similar to the ferromagnetic Ising phase with
spin waves at low momenta is analogous to the Ising fer-
romagnet, see Fig. 3. Within spin wave theory, we obtain
that the antiferromagnetic correlations ?(−1)i−jSβ
and the ferromagnetic correlations ?Sβ
the power law ∼ 1/|r|3with β = x,y,z determined
by the characteristic behavior of the dipole-dipole in-
teraction. The excitation gap vanishes approaching the
mean-field critical point θc towards the XY- ferromag-
netic phase, and also while approaching the Heisenberg
point at θ = π/4. For the latter, the qualitative behavior
of the excitation spectrum changes drastically within a
very narrow range of θ, see Fig. 2c.
XY antiferromagnetic phase: Finally, we analyze the
properties of the antiferromagnetic XY phase. In con-
?|q| is shown in Fig. 3b for a Gaussian
momentum components involved in the broad wave pack-
j∈B|↓?j. The spin wave analy-
sis is straightforward (see supplementary information),
However, the dipole in-
∼ −sinθ|q|. Consequently, the dynamics of
j? decay with
trast to the ferromagnetic XY phase, the excitation spec-
exhibits the conventional linear Goldstone
mode, see Fig. 2d. This can be understood, as the an-
tiferromagnetic ordering introduces a cancellation of the
dipolar interactions, and provides a behavior in analogy
to its short-range counter part: true long-range order
exists only at T = 0, while at finite temperature the
system exhibits quasi long-range order and eventually
undergoes a Kosterlitz-Thouless transition for increasing
temperature. Nevertheless, the dipole-dipole interactions
give rise to the characteristic algebraic correlations, e.g.,
verse spin correlation at zero temperature.
Finally, we comment on the transitions between the
different phases. The spin wave analysis predicts, that
the excitation spectrum for each phase becomes unsta-
ble at the mean-field critical points: For the Heisenberg
points at θ = π/4,5π/4, such a behavior is expected due
to the enhanced symmetry and one indeed finds, that at
the critical point, the excitation spectrum from the Ising
phase coincides with the spectrum from the XY ground
state. Consequently, the spin waves provide the same
contribution to the ground state energy, see Fig. 1b. In
turn, at the instability points θc and˜θc, the excitation
spectrum from the antiferromagnetic phase is different
from the spectrum for the ferromagnetic F phase. Con-
sequently, the ground state energy within the spin wave
analysis exhibits a jump, see Fig. 1a.
ior is an indication for the appearance of an intermediate
phase. However, this question cannot be conclusively an-
swered within the presented analysis due to the limited
validity of spin wave theory close to the transition points.
However, the appearance of a first order phase transition
can be excluded by arguments similar to Ref. .
We thank M. Hermele for helpful discussions. Support
by the Deutsche Forschungsgemeinschaft (DFG) within
SFB / TRR 21 and National Science Foundation under
Grant No. NSF PHY05-51164 is acknowledged.
j? ∼ 1/|r|3for the antiferromagnetic trans-
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We present the derivation of the spin wave excitation
spectrum within the spin wave analysis. The basic ap-
proach is to start with the ground states exhibiting per-
fect order, which are the correct ground states for the
classical model at the four points θ = 0,±π/2,π. Then,
we introduce bosonic creation and annihilation operators
creating a spin excitation above the ground state accord-
ing to the Holstein-Primakoff transformation. The spin
Hamiltonian then reduces to a Bose-Hubbard model. In
lowest order, we can ignore the interactions between the
bosonic particles, and obtain a quadratic Hamiltonian
in the bosonic operators, which is diagonalized using a
Bogoliubov-Valantin transformation. The latter trans-
formation deforms the ground state and introduces fluc-
tuations into the system.
In the following, we demonstrate the spin wave analy-
sis for the most revealing case: the antiferromagnetic XY
phase. The generalization to the other ground states is
straightforward. Without loss of generality, we choose
the antiferromagnetic order to point along the x di-
rection. The square lattice is bipartite, and we de-
note the two sublattices as A and B. Then, the anti-
ferromagnetic mean-field ground-state is given by |G? =
and spins on sublattice B pointing in the positive x di-
rection (Sx|→? = ?/2|→?). Excitations on sublattice A
are created by flipping a spin with the ladder operator
Sx+= Sz−iSy, while excitations on sublattice B are cre-
ated via Sx−= Sz+iSy. We apply a Holstein-Primakoff
transformation to bosonic operators
ing in the negative x direction, i.e., Sx|←? = −?/2|←?,
j∈B|→?jwith spins on sublattice A point-
where the phase eiKRi
sublattice-dependent sign with K = (π/a,π/a). The fac-
tor ϕ(ni) = 1 − ni is introduced to guarantee bosonic
commutation relations for the operators ai. Here, we are
interested in the leading order of the spin wave expansion,
and can therefore set ϕ(ni) ≈ 1. The bosonic operators
= e−iKRi accounts for the
ai= (Sz+ iSyeiKRi)/?,a†
i= (Sz− iSyeiKRi)/?,
and the number operator ni= a†
Expanding the spin Hamiltonian in terms of the
bosonic operators leads to a Bose-Hubbard Hamiltonian
for the spin wave excitations. In leading order, we can
neglect the interactions between the bosons and obtain