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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 [1] and atomic gases with large mag-

netic dipole moments [2] 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 [3], and the ab-

sence of a first order phase transition with a jump in

the density [4]. 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 [7], 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 [8], super-

solids [9, 10], pair supersolids in bilayer systems [11], va-

lence bond solids [12], as well as p-wave superfluidity [13],

and self-assembled structures in multilayer setups [14].

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

models [15–17].

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].

Remarkably,

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

Page 2

2

to a XXZ model with dipole-dipole interaction between

the spins [16]

H =Ja3

?2

?

i?=j

cosθ Sz

iSz

j+ sinθ?Sx

|Ri− Rj|3

iSx

j+ Sy

iSy

j

?

. (1)

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. [24].

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

excitation spectrum.

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

ǫq=

?

j?=0

eiRjq a3

|Rj|3

(2)

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 [7], 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

?∞

?

+ λ2e−πλ|Ri|2

e−a2|q|2

4π

−1

3

?

(3)

+2π

?

i?=0

1

dλ

λ3/2

?

e−πλ(

Ri

a+aq

2π)

2

a2

+iRiq

?

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 ≈

−2.646

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

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3

Ground state αSpin wave excitation spectrum Eα

q

Ground state energy per spin eα

I-FJ(sinθǫq− cosθǫ0)

3J cosθǫ0

4

+1

2

?

dq

v0Eα(q) =J cosθǫ0

+1

2

+1

2

+1

2

4

XY-FJ?sinθ(ǫq− ǫ0)(cosθǫq− sinθǫ0)

J?(sinθǫq+K− cosθǫK)(sinθǫq− cosθǫK)

J?sinθ(ǫq+K− ǫK)(cosθǫq− sinθǫK)

3J sinθǫ0

4

3J cosθǫK

4

3J sinθǫK

4

?

?

?

dq

v0Eα

dq

v0Eα

dq

v0Eα

q

I-AF

q

XY-AF

q

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,

i.e., EI-F

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

[21, 22].

XY-ferromagnetic phase: Here, the spins are aligned in

the xy plane. Within the spin wave analysis, we obtain

the excitation spectrum EXY-F

q

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 [19]. This property follows im-

i|↓?i. Within the spin wave anal-

ysis, the ground state is not modified and the excitation

q

is not quadratic for

0 + ?c|q| with velocity c = −2πaJ sinθ/?,

iSx

j? ∼ 1/|r|3. This algebraic

and the modified ground

q

∼

?|q|, in contrast

FIG. 3.

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

divergence.

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-

i?/?, where

iai?)

∆m=

dq

v0

?cosθǫq+ sinθ(ǫq− 2ǫ0)

4Eq

coth

?Eq

2T

?

−1

2

?

.

the order ∆m ≈ 0.008 at θ = −π/2. The smallness of this

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4

correlation function

?Sz

?Sy

T = 0

∼ |r|−5/2

0 < T < Tc

∼ |r|−3

∼ |r|−1

Tc < T

∼ |r|−3

∼ |r|−3

iSz

iSx

j?

iSy

j+ Sx

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 [19]. 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

ets.

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

EI-AF

q

−EI-AF

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-

i∈A|↑?i

?

j∈B|↓?j. The spin wave analy-

sis is straightforward (see supplementary information),

q

However, the dipole in-

0

∼ −sinθ|q|. Consequently, the dynamics of

iSβ

j?

iSβ

j? decay with

trast to the ferromagnetic XY phase, the excitation spec-

trum EXY-AF

q

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.,

?(−1)i−jSz

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. [4].

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.

iSz

j? ∼ 1/|r|3for the antiferromagnetic trans-

Such a behav-

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SUPPLEMENTARY INFORMATION

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

i∈A|←?i

ing in the negative x direction, i.e., Sx|←? = −?/2|←?,

?

j∈B|→?jwith spins on sublattice A point-

Sz

i=?

2(ai+ a†

i)ϕ(ni),Sy

i=?

2i(ai− a†

i)eiKRiϕ(ni),

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

reduce to

= e−iKRi accounts for the

ai= (Sz+ iSyeiKRi)/?,a†

i= (Sz− iSyeiKRi)/?,

iai=1

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

2+ SxeiKRi/?.

H

J

= sinθǫK

?

3N

4

−1

2

?

?

|Rij/a|3

i

?

a†

iai+ aia†

i

??

ia†

(4)

+1

4

?

i?=j

χij

?

a†

iaj+ aia†

j

+ ηij

?

aiaj+ a†

j

?

Page 6

6

with Rij = Ri− Rj, N the number of lattice sites,

and the coupling the terms χij = cosθ + sinθeiKRij

and ηij= cosθ −sinθeiKRijincluding the antiferromag-

netic ordering. Introducing the Fourier representation

ai=?

1

4

q

qaqe−iqRi/√N, the terms involving the bosonic

operators in Eq. (4) reduce to

?

?

(cosθ ǫq+ sinθǫq+K− 2sinθǫK)

?

a†

qaq+ aqa†

q

?

+(cosθǫq− sinθǫq+K)

?

aqa−q+ a†

qa†

−q

??

.

The diagonalization of this Hamiltonian is straightfor-

ward using a standard Bogoliubov transformation with

b†

q− vqa−q. Then, the Hamiltonian takes the

form

q= uqa†

H =3JN sinθǫK

4

+

?

q

EXY-AF

q

?

b†

qbq+1

2

?

(5)

with the spin-wave excitation spectrum EXY-AF

tion, the coefficients for the Bogoliubov transformation

are given by

q

. In addi-

uq,vq= ±

?

1

2

?cosθǫq+ sinθ (ǫq+K− 2ǫK)

2Eq

± 1

?

(6)

with Eq≡ EXY-AF

that the transformation is canonical. In addition, the

ground state obeys the condition bq|vac? = 0, and the

ground state energy per spin at zero temperature T = 0

reduces to eXY-AF, see Table I.

We are now able to check the validity of the spin

wave approach self-consistently: the deformation of the

ground state by the spin wave analysis provides a sup-

pression of the antiferromagnetic order m ≡ ∆m −1

?Sx

∆m =

v0

q

/J. The property u2

q− v2

q= 1 asserts

2=

ieiKRi?/?, and thus

?

dq

?a†

?

qaq

?=

?exp(EXY-AF

?

dq

v0

?

v2

q+ (2v2

q+ 1)fq

?

, (7)

where fq =

for the thermal occupation of the spin waves. At zero

?b†

qbq

=

q

/T) − 1?−1accounts

temperature T = 0, this expression converges and we

obtain ∆m ≈ 0.03 for θ =˜θcas well as ∆m = 0.39 for

θ ≈

momentum behavior of the integrand scales as |q|−2, and

therefore ∆m diverges logarithmically: the long range

order is destroyed by the thermal spin wave fluctuations,

and gives rise to the well-known quasi long-range order

in analogy to short-range XY models.

Finally, the spin wave analysis also allows us to ana-

lyze the correlation functions cαα(Rij) = ?Sα

Using the translational invariance of our system, the cor-

relation functions reduce to

π

4. In turn, at finite temperatures T > 0, the low

iSα

jeiKRij?.

cαα(r) =

?

−q?. Combining the Holstein Pri-

dq

v0

cαα(q + K)e−iqr

(8)

with cαα(q) = ?Sα

makoff transformation Eq. (4) and the Bogoliubov trans-

formation the correlation functions can be expanded in

terms of the coefficients uqand vq

qSα

czz(q + K) =1

4(uq+K+ vq+K)2coth

?JEq

?

2T

?

,

cyy(q + K) =1

4(uq− vq)2coth

?JEq

2T

. (9)

The long distance behavior |r| → ∞ of the correlation

function is determined by the low momentum behavior

of the above expressions

(uq+K+ vq+K)2∼ |q| + const.

(uq− vq)2∼

1

|q|

(10)

and describes the leading non-analytic part. The latter

can be replaced using the following relation, which de-

rives via an Ewald summation,

|q|γ∼

?

j?=0

eiqRj

|Rj|2+γ

(11)

for |q| → 0 and γ > −2; (for γ = 0,2,4,... the left side

is replaced by |q|γlog|q|). At zero temperature T = 0,

the integration in Eq. (8) is straightforward and provides

the scaling behavior czz(r) ∼ |r|−3and cyy(r) ∼ |r|−1.