A.J. Stoffel and M. Gulacsi- Finite Temperature Properties of a Supersolid: a RPA Approach

8/3/2019 A.J. Stoffel and M. Gulacsi- Finite Temperature Properties of a Supersolid: a RPA Approach

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arXiv:0911.169

8v2

[cond-mat.oth

er]29Nov2009

EPJ manuscript No.(will be inserted by the editor)

Finite Temperature Properties of a Supersolid: a RPA Approach

A.J. Stoffel1,2 and M. Gulacsi1,2

1 Max-Planck-Institute for the Physics of Complex Systems, D-01187 Dresden, Germany2 Nonlinear Physics Centre, Australian National University, Canberra, ACT 0200, Australia

Received: date / Revised version: date

Abstract. We study in random-phase approximation the newly discovered supersolid phase of 4He andpresent in detail its finite temperature properties. 4He is described within a hard-core quantum lattice gasmodel, with nearest and next-nearest neighbour interactions taken into account. We rigorously calculateall pair correlation functions in a cumulant decoupling scheme. Our results support the importance of thevacancies in the supersolid phase. We show that in a supersolid the net vacancy density remains constantas function of temperature, contrary to the thermal activation theory. We also analyzed in detail thethermodynamic properties of a supersolid, calculated the jump in the specific heat which compares well tothe recent experiments.

PACS. 05.30.Jp Boson systems 67.80.-s Quantum solids 67.80.bd Superfluidity in solid 4He, supersolid4He 75.10.Jm Quantized spin models

1 Introduction

One of the biggest accomplishments of theoretical con-densed matter physics is the ability to classify various

phases and phase transitions by their mathematical or-der. These mathematical orders are usually expressed byorder parameters reflecting certain limiting behaviours oftwo particle correlation functions. This concept makes itpossible to describe the properties of physically very dif-ferent systems in a common language and establish a linkbetween them.Without this concept, the counter-intuitive idea of a su-persolid, i.e. a solid that also exhibits a superflow, wouldnot have been conceivable. In this language supersolidity,firstly proposed by Andreev and Lifshitz[1] in 1969 andpicked up by Leggett and Chester in 1970[3, 2] is the clas-sification of systems that simultaneously exhibit diagonaland off-diagonal long range order. Yet the idea of a super-

solid was reluctantly received by the scientific community,because early experiments failed to detect any such effectin Helium-4. Apart from John Goodkind[4] who had pre-viously seen suspicious signals in the ultrasound signals ofsolid helium, physicists were surprised when in 2004 Kimand Chan [5,6] announced the discovery of supersolid he-lium. Although equipped with a head start of more than30 years the theoretical understanding of the supersolidstate lags vastly behind, not least because supersolidityas we observe it today is nothing like what the pioneersin early 1970s had anticipated.It is now evident that defectons and impurities play a cru-cial role in the formation of the supersolid state. However,the data and the results of various experiments draw a

rather complex picture of the supersolid phase. The nor-mal solid to supersolid transition is not of the usual first orsecond order type transition but bears remarkable resem-blance to the Kosterlitz Thouless transition well-known intwo dimensional systems. Furthermore annealing experi-ments show that 3He impurities play a significant role butthe data remain all but conclusive, as the measured su-perfluid density varies by order of magnitudes among thedifferent groups. Recently a change in the shear modulusof solid helium was found and the measured signal almostidentically mimics the superfluid density measured by Kimand Chan [5,6]. Some popular theories give plausible ex-planations of some aspects of the matter. That, such asthe vortex liquid theory suggested by Phil Anderson [9],is in good agreement with properties of the phase tran-sition; theories based on defection networks are capableof describing the change in the shear modulus. However,we feel that a satisfying and comprehensive theory is still

missing. Part of the problem stems from the complexity ofthe models. Many models are only at sufficient accuracysolvable with numerical Monte Carlo methods. The re-sults, doubtlessly useful, are seldom intuitive and the lackof analytical results do not meet our notion of the under-standing of a phenomenon. Other approaches on the otherhand are limited in their analytical significance and ratherrepresent a phenomenological ansatz.In this work we attempt to fill a part of this gap in the the-oretical understanding of the supersolid phase. We presenta theory of supersolidity in a Quantum-Lattice Gas model(QLG) beyond simple mean-field approaches. Followingthe approach of K.S. Liu and M.E. Fisher[12] we map theQLG model to the anisotropic Heisenberg model (aHM).
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2 A.J. Stoffel, M. Gulacsi: Finite Temperature Properties of a Supersolid: a RPA Approach

The method of Greens functions proves to be very suc-cessful in the description of ferromagnetic and antiferro-magnetic phases and we use this method to investigate thesupersolid phase which corresponds to a canted antiferro-magnetic phase. Applying the Random-phase Approxima-

tion (RPA), we broke down using cumulant decoupling,the higher order Greens functions to obtain a closed setof solvable equations. This method gives a fully quantummechanical and analytical solution of a supersolid phase.We will see that quantum fluctuations have a significanteffect on the net vacancy density of the supersolid and wewill also see that the superfluid state is unstable againstzero-energy quasi-particle excitations. We also derive im-portant thermodynamic properties of this model and de-rive formulas for interesting properties such as the powerlaw exponents and the jump of the specific heat across thenormal solid to supersolid line.The paper is organized as follows: In Section 2 we intro-

duce the generic Hamiltonian of a bosonic many body sys-tem and discretize it to a quantum lattice gas model. Thismodel is equivalent to the anisotropic Heisenberg modelin an external field and we will identify the correspondingphases in Section 3. In Section 4 we re-derive the classicalmean-field solution and discuss briefly their significancebefore we in Section 5 recapitulate the Greens functionsfor the anisotropic Heisenberg model in the random-phaseapproximation at zero temperature. The ground state ofthe system is obtained by solving the corresponding self-consistency equation. In the following two sections we de-rive basic thermodynamic properties as well as ordinarydifferential equations to calculate the first and second or-der phase transition lines. In Section 8 we analyse the

quasi-particle energy spectrum and in the last section wecalculate phase diagrams for various parameter sets andanalyse the properties of the supersolid phase and the cor-responding transitions.

2 Generic Hamiltonian and AnisotropicHeisenberg Model

If we neglect possible 3He impurities supersolid Helium-4is a purely bosonic system whose dynamics and structureare governed by a generic bosonic Hamiltonian:

H =

d3x(x)( 12m 2 + )(x)

+1

2

d3xd3x(x)(x)V(x x)(x)(x)

(1)

Here (x) are the particle creation operators and (x)destruction operators and obey the usual bosonic commu-tation relations. Hamiltonians such as in Eq. (1) are not,due to the vast size of the many body Hilbert-space, di-agonalisable even for simple potentials V(x). The accom-plishment of any successful theory is to find an approxi-mation that sustains the crucial mechanism while reduc-ing the mathematical complexity to a traceable level. Here

we follow the method of Matsubara and Matsuda [14] whosuccessfully introduced the Quantum Lattice Gas (QLG)model to describe superfluid Helium. We believe that thequantum lattice gas model is particularly useful for super-solids as the spatial discretization of this model serves as a

natural frame for the crystal lattice of (super)-solid heliumand a bipartite lattice elegantly simplifies the problem ofbreaking translational invariance symmetry for states thatexhibit diagonal long range order.According to Matsubara and Tsuneto [13] the generic Hamil-tonian Eq. (1) in the discrete lattice model reads:

H =

i

ni +

ij

uij(ai a

j )(ai aj) +

ij

Vij ninj

(2)

Here uij are solely finite for nearest neighbor and nextnearest neighbor hopping and otherwise zero. The val-

ues of unn and unnn are such that the kinetic energy isisotropic up to the 4th order. As atoms do not penetrateeach other there can exist only one atom at a time on a lat-tice site and consequently a and a are the creation andannihilation operators of a hard core boson commutingon different lattice sites, but obey the anti-commutatorrelations on identical sites. Equation (2) is the Hubbardmodel in 3 dimensions for hard core bosons. Neither fullybosonic nor fermionic, the lack of Wicks theorem inhibitsthe application of pertubative field theory though hardcore bosonic systems are algebraic identical to spin sys-tems. A simple transformation, where the bosonic opera-tors are substituted by spin-1/2 operators generating theSU(2) Lie algebra, maps the present QLG model onto the

anisotropic Heisenberg model:

H = hz

i

Szi +

ij

Jij Szi S

zj +

ij

Jij (Sxi S

xj + S

yi S

yj )(3)

Here the correspondence between the QLG parametersand the spin coupling constants is given by:

Jij = Vij

Jij = 2uij

hz = +

j

Jij

j

Jij (4)

3 Phases

The self-consistency equations in the random-phase ap-proximation as we will derive in a later chapter are verylengthy and therefore it is our primary goal to keep thepresent model as simple as possible while still being able todescribe the crucial physics. For this reason we will definethe anisotropic Heisenberg model on a bipartite BCC lat-tice which consists of two interpenetrating SC sub-latticesas Figure 1 shows. In this lattice geometry the perfectlysolid phase is composed of a fully occupied (on-site) sub-lattice A while sublattice B refers to the empty interstitialand is consequently vacant. As there is no spatial density


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A.J. Stoffel, M. Gulacsi: Finite Temperature Properties of a Supersolid: a RPA Approach 3

A

B

Fig. 1. The BCC lattice consists of two interpenetrating SCsub-lattices. In the perfect solid phase one sub-lattice (i.e. sub-lattice A) serves as on-site centers and is occupied while sub-lattice B represents the empty interstitial and is vacant (Forsimplicity, we only present the two dimensional case).

Table 1. Possible magnetic phases and the correspondingphases of the Hubbard model. All Phases are defined by theirlong range order. The columns, from left to right, are the spinconfigurations, magnetic phases, ODLRO, DLRO and corre-sponding 4He phases, respectively.

FE No No Normal Liquid

CFE Yes No Superfluid

CAF Yes Yes Sup ersolid

AF No Yes Normal Solid

variation in the liquid phases both sublattices are equallyoccupied, and the mean occupation number simply cor-responds to the particle density. We are aware that theabove choice of SC sublattices does not reflect the true 4Hecrystal structure which is hexagonal closed-packed (HCP).Nevertheless we believe that crucial physical propertiessuch as phase transition and critical constants do not de-pend on the specific geometry as long as no other effectssuch as frustration are evoked.Table 3 charts the various magnetic phases and identifies

the corresponding phases of the4

He system. Accordingto their spin configurations we identify the four magneticphases to be of ferromagnetic (FE), canted ferromagnetic(CFE), canted anti-ferromagnetic (CAF) and antiferro-magntic (AF) orders. The off-diagonal long range orderparameter m1 and the diagonal long range order m2 are:

m1 = SxA + S

xB

m2 = SzA S

zB (5)

which readily identify the corresponding phases of the He-lium system.

0.5 1 1.5 2T [K]

2

4

6

8

hz

CAF

CFE

FE

AF

Fig. 2. The phase diagram for J1 = 1.498, J2 = 0.562,

J1

= 3.899 and J2

= 1.782 as calculated by MF.

4 Mean-Field Solution

In our previous work [10] we have already re-derived theclassical mean-field solution of the anisotropic Heisenbergmodel at zero temperature. As this model provides an easyand intuitive access to the model we extend the approx-imation to finite temperatures as was done by K.S Liuand M.E. Fisher [12] and briefly discuss its solution andphase diagram. The mean-field Hamiltonian is obtained bysubstituting the spin- 1

2operators with their expectation

values:

HMF = hz(SzA + SzB)2J

1 S

zAS

zB J

2 (S

zAS

zA + S

zBS

zB)

2J1 SxAS

xB J

2 (S

xAS

xA + S

xBS

xB)

(6)

Here J1 = q1J

iA,jB, J

2 = q2J

iA,jA , J

1 =

q1JiA,jB and J2 = q1J

iA,jA where q1 = 6 and

q2 = 8 are the number of nearest and next nearest neigh-bors. The mean value of Sy drops out as the randomlybroken symmetry Sx Sy (ODLRO) allows for Sy = 0.The standard method of deriving the corresponding self-consistency equations is to minimize the Helmholtzs Free

energy F = H T S. The entropy S is given by the pseudospin entropy of the system:

S = 1

2[(

1

2+ SA)ln(

1

2+ SA) + (

1

2 SA)ln(

1

2 SA)

+(1

2+ SB)ln(

1

2+ SB) + (

1

2 SB)ln(

1

2 SB)](7)

where SA =

SzA2 + SxA

2 and SB =

SzB 2 + SxB

2.In the canted anti-ferromagnetic and the canted ferro-magnetic states there are 4 self-consistency equations inthe ferromagnetic and anti-ferromagnetic phases whereSx = Sy = 0 they are reduced by two. These equationsare readily obtained by differentiating the free energy with


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respect to Sz and Sx respectively. The resulting equa-tions can be rearranged to yield:

SzA

2 + SxA2 =

tanh(A)

2SzB

2 + SxB2 = tanh(

B)2

(8)

hz + 2SzA(J2 J

2 ) + 2SzBJ

1 = 2J

1

SxB

SxASzA

hz + 2SzB(J2 J

2 ) + 2SzAJ

1 = 2J

1

SxA

SxBSzB(9)

where

A = [(2J1 S

xB + 2J

2 S

xA)

2 +

(2J1 S

zB + 2J

2 S

zA + h

z)2]12

B = [(2J1 S

xA + 2J

2 S

xB)

2 +

(2JSzB + 2JSzA + h

z)2]12 (10)

The two equations (Eq. (9)) are dismissed in the ferromag-netic and the anti-ferromagnetic phases as the transversalfields SxA and S

xB become zero. At zero temperature

T = 0 the free energy and H coincide. This allows us todeduce the phases at absolute zero in the limiting cases(limits of hz) from Equation (6). In the limit of hz the Hamiltonian reduces to an effective one particle model:

H = hz(SzA + SzB) (11)

Consequently the system will be in the energetically favor-

able ferromagnetic phase. In the opposite limit, h

z

0and with antiferromagnetic coupling J1 < 0 the term of

nearest neighbor interaction

H = J1 SzASzB (12)

is the only term that significantly lowers the energy. There-fore the ground-state is the anti-ferromagnetic state. AtT=0 and a suitable choice of parameters the canted fer-romagnetic and the canted anti-ferromagnetic phases arerealized in-between these limits as seen in Figure 2. How-ever with increasing temperature the regions of cantedferromagnetic and canted anti-ferromagnetic phases de-plete and at sufficiently high temperatures only the anti-

ferromagnetic and ferromagnetic phases survive. As wecan see from the phase diagram in Figure 2 most tran-sitions are of second order. Only for parameter regimeswhere the CAF does not appear the resulting CFE-AFis of first order. The boundary lines are defined by ordi-nary differential equations and generally need to be calcu-lated numerically. Nonetheless there exist analytical ex-pressions for the locations of the transitions at absolutezero as derived by Matsuda and Tsuneto [13]. The ferro-magnetic to canted ferromagnetic transition point is de-termined by Equations (9) if we set SzA = SzB =

1

2

and SxA = SxB = 0:

hzF ECF E

= J1

+ J2

J

1

J

2

(13)

Equally the canted anti-ferromagnetic to anti-ferromagnetictransition is defined by SzA = SzB =

1

2and SxA =

SxB = 0:

h

z

CAFAF =

(J

1 + J

2 J

2 )2

(J

1 )2

(14)The canted ferromagnetic and the canted anti-ferromagneticphases coexist where the order parameter of the DLRO,m1 = SzA SzB approaches zero. We replace SzAand SzB in Equation (9) with m1 and m2 = SzA+SzBand retain only linear terms ofm1. Subtracting and addingboth equations respectively yields:

hz + 2m2(J2 J

2 + J

1 ) = 2J

1 m2

2m1(J2 J

2 J

1 ) = 2J

1 m1

4m22 + 1

4m22 1(15)

We used that SzA2 + SxA2 = 14 at T=0. The solu-tion of these two equations determine the correspondingtransition point which is given by:

hzCF ECAF =J1 + J

2 J

1 J

2

J1 + J

2 + J

1 J

2

(J

1 + J

2 J

2 )

2 (J1 )2 (16)

5 Greens Functions

Although the classical mean-field theory is quite insight-ful and gives an accurate description of the variously or-

dered phases its fails to take quantum fluctuations andquasi-particle excitations into account. Hence, in order toovercome these shortcomings and to derive a fully quan-tum mechanical approximation we solve the anisotropicHeisenberg model in the random-phase approximation whichis based on the Greens function technique. At finite tem-perature the retarded and advanced Tyablikov [15,16] com-mutator Greens function defined in real time are:

GijRet(t) = i(t)|[Si (t), S

j ]|

GijAdv (t) = i(t)|[Si (t), S

j ]| (17)

The average involves the usual quantum mechanical as

well as thermal averages. The most successful technique ofsolving many body Greens function involves the methodof equation of motion which is given by:

itGxyijRet

(t) = (t)[Sxi , Syj ] i(t)[[S

xi , H], S

yj ]

itGxyijAdv

(t) = (t)[Sxi , Syj ] + i(t)[[S

xi , H], S

yj ]

(18)

The commutator [Sxi , H] can be eliminated by using theHeisenberg equation of motion giving rise to higher, thirdorder Greens functions on the RHS. In order to obtain aclosed set of equations we apply the cumulant decouplingprocedure which splits up the third order differential equa-tion into products of single operator expectation values


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and two spin Greens functions. The cumulant decoupling[17] is based on the assumption that the last term of thefollowing equality is negligible:

ABC =

ABC + BAC

+CAB 2ABC

+(A A)(B B)(C C) (19)

The set of differential equations is not explicitly dependentof the temperature, i.e. temperature dependence solelycomes with thermal averaging of the single operator expec-tation values. Therefore the solution is formally identicalto the zero temperature solution and the detailed deriva-tion of the Greens function in their full form can be foundin Ref. [11].The averages of the spin operator, appearing in the cu-mulant decoupling scheme, have to be determined self-consistently. Two self-consistent equations can be derivedfrom correlation functions corresponding to the Greensfunctions. The self-consistency equations of the cantedferromagnetic (superfluid) and canted antiferromagnetic(supersolid) phases are structurally different from the fer-romagnetic (normal solid) and antiferromagnetic (normalfluid) phases, as the off-diagonal long range order increasesthe number of degrees of freedom by two and hence twoadditional conditions, resulting from the analytical prop-erties of the commutator Greens functions, apply. Thus,the self-consistency equations of the canted phases, canbe written as a function of the temperature, the externalmagnetic field and the spins in x-direction:

FcA(SxA, SxB, hz, T) = 0

FcB(SxA, S

xB, h

z, T) = 0 (20)

similar the self-consistency equations for the ferromag-netic and anti-ferromagnetic phases:

FncA (SzA, S

zB, h

z, T) = 0

FncB (SzA, S

zB, h

z, T) = 0 (21)

These equations involve a 3 dimensional integral over thefirst Brillouin zone. As explained in the work [10] on thezero temperature formalism this integral can be reducedto a two dimensional integral by introducing a general-ized density of state (DOS) and gives the model a widerapplicability.

6 Thermodynamic Properties

The relation between then QLG and the anisotropic Heisen-berg model is such that the chemical potential corre-sponds to the external field hz, i.e. the grand canonicalpartition function in the QLG corresponds to the canoni-cal partition function in the anisotropic Heisenberg modelwhere the number of spins is fixed. Consequently, thermo-dynamic potentials of the two models are related as:

QLG

N = Heisenberg

(22)

Here refers to an arbitrary thermodynamic potential.In the same way as the ground state minimizes the in-ternal energy U = H at absolute zero, the free energyF = U T S is minimized at finite temperatures. We wishto stress that the internal energy in the present approxi-

mation cannot be derived accurately from the expectationvalue of the Hamiltonian in the following way:

U = hz

i

Szi +

ij

Jij S

zi S

zj

+

ij

Jij (Sxi S

xj + S

yi S

yj ) (23)

The cumulant decoupling, though a good approximationto the anisotropic Heisenberg model, is also an exact solu-tion of an unknown effective Hamiltonian Heff. Thereforethermodynamically consistent results are only obtained ifthe effective Hamiltonian is substituted in the equation

above, i.e. U = Heff. Here, as we do not know the ex-plicit form of this effective model, we have to integrate thefree energy from thermodynamic relations:

dF =SzA + S

zB

2dhz + SdT (24)

This equation allows us to select the ground state in re-gions where two or more phases exist self-consistently ac-cording to the random-phase approximation. The entropyof the spin system is given by

S =

d3k

1

1 + exp((k))log(

1

1 + exp((k)))

+1

1 + exp((k)) log(1

1 + exp((k)) )

(25)

This formula is of purely combinatorial origin and reflectsthe fact that the hard-core boson system is equivalent to afermionic system given by the Jordan-Wigner transforma-tion. The (k) terms refer to the energies of the spin-waveexcitations. In the solid phases both branches have to betaken into account. The entropy of the anisotropic Heisen-berg model given for a fixed number of spins correspondsto the entropy of the QLG model at a constant volume.Therefore, in order to obtain the usual configurational en-tropy of the QLG, we have to divide by the number ofparticles per unit cell: (Sconf =

2SnA+nB

).

Here nA = 1/2 SzA and nB = 1/2 SzB are the particleoccupation numbers on lattice sites A and B. As the na-ture of the normal solid to supersolid phase transition isnot yet satisfactorily understood recent experiments [19]have focused on the behavior of the specific heat acrossthe transition line in the hope of shedding light on thematter. The specific heat at constant temperature andconstant pressure respectively are given by:

CV = T

Sconf

T

T,V,N

, Cp = T

Sconf

T

T,P,N

(26)Although the external magnetic field hz in the spin modelis an observable the corresponding quantity in the QLG


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model, namely the chemical potential is not. Therefore weare also interested in attaining a formula for the pressureassociated with a certain chemical potential. The relation-ship is most easily derived from the following Maxwell re-lation:

P

T,V

=

N

V

T,

=#lattice sites

V(1 ) (27)

where := SzA + SzB. Note that, using this equation in

order to obtain the pressure at any specific temperaturewe need a reference point, i.e. a chemical potential wherethe corresponding pressure is known. This point is givenby which corresponds to N 0 and, hence,P 0. Consequently, in order to obtain the pressure fora specific chemical potential we have to integrate overthe interval [, ].

7 Boundary Lines

The state of the system at any point in T and hz is givenby the self-consistency equations and the free energy ascan be derived from Eq. (24). Nevertheless the resultingcomputations come at high computational cost and there-fore it seems most feasible to derive ODEs which deter-mine the first and second order transition lines. First wewill derive the ordinary differential equations which definethe more abundant second order transitions. The normalfluid (FE) and the normal solid (CFE) phases are deter-mined by Equations (21) and the supersolid (CAF) and

superfluid (CFE) are defined by Equations (20) and con-dition Equation (9). Consequently on the SS-NS and SF-NF transition line, where both the normal (FE and CFE)and the super (CFE and CAF) phases coexists followingequations hold:

FNA (SzA, SzB , hz, T) = 0

FNB (SzA, SzB , hz, T) = 0

hz + 2SzA(J2 J

2 ) + 2SzBJ

1 = 2J

1

SxB

SxASzA

hz + 2SzB(J2 J

2 ) + 2SzAJ

1 = 2J

1

SxA

SxBSzB(28)

On the SS-NS boundary line the quotient SxA/SzA isnot known a priori and therefore we eliminate it in theequation above yielding:

FNA (SzA, SzB, hz, T) = 0

FNB (SzA, SzB, hz, T) = 0

f(SzA, SzB, hz)

:= (hz + 2SzA(J2 (0) J

2 ) + 2SzBJ

1 )

(hz + 2SzB(J2 J

2 ) + 2SzAJ

1 ) (2J

1 SzB)

2 = 0(29)

We introduce a variable s which parametrizes the bound-ary curve. If we for instance choose ds=dT we get the

following set of ordinary differential equations, definingthe NF-SF and the NS-SS transition lines:

FNA

SzA FN

A

SzB FN

A

hzFN

A

T

FNB

SzA

FNB

SzB

FNB

hz

FNB

TfSzA

fSzB

fhz 0

0 0 0 1

SzA

sSzB

shzs

Ts

=

00

01

(30)

Upon crossing the SF-SS transition line, coming from thesuperfluid phase the set of possible solutions branches offinto two phases, the supersolid and a non-physical (com-plex valued) superfluid phase. Therefore any matrix of or-dinary differential equations will render a singularity andconsequently we have to approach the transition line in alimiting process:

FSA (SxA, SxB , hz, T) = 0

FSB (SxA, SxB , hz, T) = 0

lim0(SxA Sxb ) = 0 (31)

The resulting ODE is

FSA

SxAFS

A

SxB FS

A

hzFS

A

T

FSB

SxAFS

B

SxB FS

B

hzFS

B

T

1 1 0 00 0 0 1

SxA

sSxB

shz

sTs

=

001

(32)

which defines the superfluid to supersolid transition. Asmentioned previously, there are certain parameter regimeswhere not all four possible phases are appearing and con-sequently a first order transition (mostly superfluid to su-persolid) will occur. In the previous chapter we have seen

that such a transition line is difficult to locate. However,a tricritical point frequently appears in the phase diagramand at this point the first order transition evolves into asecond order transition. This tricritical point can be takenas a initial value for a differential equation defining thecorresponding first order transition line.The relevant ODE may be derived from a Clausius Clapey-ron type equation. On the transition line both phases haveequal free energy. Hence:

S

(SzA + SzB)=

T

hz(33)

where S refers to the spin entropy as derived in the pre-vious section (Eq. (25)) and refers to the difference ofeither entropy or spin mean-field of the superfluid and thenormal solid phases. This equation together with ds = dTand the total derivative of the two self-consistency equa-tions for the normal solid and one equation for the super-fluid phase form a set of 5 ODEs determining Sx in thesuperfluid phase and SzA and SzB in the normal solidphase along the boundary line in the T hz plane:

FS

Sx0 0 F

S

hzFS

T

0FN

A

SzA FN

A

SzB FN

A

hzFN

A

T

0FN

B

SzA FN

B

SzB FN

B

hzFN

B

T

0 0 0 S SzA+B

0 0 0 0 1

(34)


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8 Excitation spectrum

The superfluid phase features, due to spontaneously bro-ken U(1) symmetry, the well know gapless Goldstone bosons,i.e. linear phonons. The supersolid phase additionally ex-

hibits a second, gapped branch which is due to the breakdown of discrete translational symmetry.Figure 4 reveals a zero frequency mode in the supersolidphase at [100] of the first Brillouin zone and consequentlythe superfluid to supersolid phase transition is character-ized by a collapsing roton minimum at [100]. The disper-sion relation in the superfluid (CFE) phase is given by:

(k) = 2{(J1 (1(k) 1) + J2 (2(k) 1))

Sz2(J1 (1(k) 1) + J

2 (2(k) 1))

Sx2(J1 + J

2 J

11(k) J

2 2(k))

}1/2 (35)

From this equation we can see that the energy possiblygoes to zero at [100] (corresponds to 1(k) = 1 and2(k) = 1) when following condition is fulfilled:

J1 + J2 + J

1 J

2 < 0 (36)

Hence we obtain a further condition (supplementary toEq. (16)) for the existence of the superfluid to supersolidtransition.Equation (35) allows for the existence of a second regionof the reciprocal lattice space where the dispersion rela-tion might go soft. For 1(k) = 0 and 2(k) = 1 whichcorresponds to [111] we obtain following condition:

2J2

J1 + J2 + J

2

> 0 (37)

It is also interesting to study the behavior of the excitationspectrum with increasing temperature. In a conventionalsuperfluid the long wave-length behavior is given by:

(k) =no(T)V(0)

mk (38)

Here V(0) is the interaction potential at zero momentumand m is the particles mass. The density of the conden-sate no(T) typically decreases with increasing temperatureand for that reason we expect lower energies with increas-

ing temperature. In Figure 3 we can see that the quasi-particle energies indeed decrease with increasing temper-ature. Apart from the region in the vicinity of [100] theenergies at higher temperatures lie significantly lower thanthose ones closer to absolute zero. This is important as itwill contribute to the variation of thermodynamic quan-tities such as the entropy or the specific heat. Figure 4depicts the variation of the excitation spectrum with in-creasing temperature in the supersolid phase. In this phasethe low lying phonon branch mostly depletes with increas-ing temperature, although there exist a region between[110] and [111] where the zero temperature spectrum issignificantly higher. Contrary to the phonon branch thegapped mode lifts the energy around the long wave length

k

[100] [110] [111][000] [000]

Fig. 3. Excitation spectrum in the superfluid phase for for

J1 = 1.498K, J

2 = 0.562K, J

1

= 3.899K, J2

= 1.782Kand hz = 3. Solid line refers to T = 0, dotted to T = 0.5,

dashed line to T = 1 and the long dashed line to T = 1.3.

k

[000] [100] [110] [111] [000]

Fig. 4. Excitation spectrum in the supersolid phase for for

J1 = 1.498K, J

2 = 0.562K, J

1

= 3.899K, J2

= 1.782Kand hz = 0.8. The Solid lines refer to T = 0 and the dashedlines to T = 1.

limit [000] and around [100]. In comparison with the super-fluid dispersion relation the excitation spectrum changesits form/shape rather than scaling down with increasingtemperature as in the superfluid phase. We observe that

the supersolid phase exhibits a more complex and diversestructure than the superfluid or normal solid phases alone.

9 Discussion

9.1 On Finite Temperature Properties

In this section we will discuss finite temperature propertiesof the anisotropic Heisenberg model and its solution inthe random-phase approximation. In order to be able tocompare the temperature dependence of the model in therandom-phase approximation with the classical mean-field


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0 0.5 1 1.5 2T [K]

0

2

4

6

8

hz

SF (CFE)

NF (FE)

SS (CAF)NS (AF)

Fig. 5. The phase diagram for J1 = 1.498K, J2 = 0.562K,

J1

= 3.899K and J2

= 1.782K in the RPA (solid lines)and in classical MF (dashed lines). MF overestimates the tem-

perature by about 30%.

approximation we chose the set of parameters that wasextensively scrutinized by Liu and Fisher [12]:

J1 = 1.498K

J2 = 0.562K

J1 = 3.899K

J2 = 1.782K (39)

As mentioned in the previous section, Liu and Fisher [12]

have chosen this set of parameters because it provides ar-guably the best fit to the phase diagram of Helium-4. Sincewe believe that the validity of the quantum lattice gasmodel is too limited to appropriately reproduce the be-havior of Helium-4 over the whole range of temperatureand pressure we do not discuss most properties in thepressure-temperature space but rather present the majorpart of the results in the more comprehensible magneticfield -temperature coordinates. Only where the theory canbe compared to relevant experimental data, such as theheat capacity at constant pressure we work in the corre-sponding representation.The phase diagram of the anisotropic Heisenberg model inT and hz coordinates is given in Figure 5. The dashed lines

correspond to the phase diagram of the mean-field ap-proximation. We see that the diagrams are quantitativelyquite similar but in the mean-field solution the tempera-ture is somewhat overestimated giving an approximately30% higher temperature for the tetra-critical point. Asmentioned before the critical magnetic fields hzc , due toquantum fluctuations, are lower in the random-phase ap-proximation. This effect is most distinctive on the super-solid to superfluid transition line as there the deletion ofthe spin magnitude is strongly pronounced.The net vacancy density , density of vacancies minus den-sity of interstitials sparked interest as the question arose[18] as to whether the number of vacancies follow predic-tions of thermal activation theory or are due to the effects

0 0.5 1 1.5 2 2.5 3T/Tc

0

0.02

0.04

0.06

0.08

netvacancydens

ity

Fig. 6. Net vacancy density for four different pressures. Thedashed line shows the curve expected if the normal solid wouldexist down to zero temperature

0.5 1 1.5h

z

0

0.05

0.1

0.15

0.2

=Sz A+Sz B

Fig. 7. Net vacancy density as a function of the external mag-netic field (equates to the chemical potential) for four differenttemperatures. The solid line refers to T=0K, the dashed lineto T=0.4K, the long dashed line to T=0.7K and the dottedline refers to T=1K.

of an incommensurate crystal. Figure 6 shows the net va-cancy density in the supersolid and the normal solid phaseat constant pressures. In isobar curves, curves of constantpressure, the magnetic field hz is controlled through Equa-

tion (27). We see that the net vacancy density is nearlyconstant in the supersolid phase. Only in the normal solidphase the net vacancy density increases exponentially withincreasing temperature in total agreement with classicalthermal activation theory. The almost temperature inde-pendent behavior of the net vacancy density in the su-persolid phase is an important finding of the quantumlattice gas model and should be observable in high reso-lution experiments if this effect is real. In Figure ( 7) weplotted the net vacancy density as a function of hz (orequivalently the chemical potential) across the normalsolid, the supersolid and the superfluid phases for varioustemperatures, namely T = 0K, T = 0.4K, T = 0.7Kand T = 1.0K. The term net vacancy density does not


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0 0.2 0.4 0.6 0.8T/Tc

0

0.01

0.02

0.03

0.04

-F/K

0.11e-07

1e-06

1e-05

0.0001

0.001

0.01

Fig. 8. Free energy in the supersolid phase. The leading con-tribution comes from a T4.2-term. The inset shows the freeenergy in double logarithmic scale. The dashed line is the fit

to the T4.2

-term. The leading correction comes from a T5

-term,indicated by the long dashed line.

Fig. 9. [Color online] 3D plot of the spin entropy. The fourphases are clearly distinguishable as the entropy is non-smoothacross the transition lines.

0.5 1 1.5 2T/Tc

0.2

0.4

0.6

0.8

1

1.2

1.4

CP

Fig. 10. Specific heat at constant pressure.

0 0.025 0.05 0.075 0.1 0.125

= Sz

A+S

z

B

0

0.5

1

1.5

T[K]

Fig. 11. Curves show the net vacancy density as a function oftemperature on the supersolid to normal solid transition lines.The solid line refers to set 1: J1 = 1.498K, J

2 = 0.562K,

J1

= 3.899K and J2

= 1.782K and the dashed line to set

2: J1 = 0.5K, J2 = 0.5K, J

1

= 2.0K and J2

= 0.5K.

have a physical meaning in the superfluid phase and herethe quantity rather refers to the particle density of thefluid. The particle density in the superfluid phase is lin-ear in hz and independent of the temperature T, whichfollows immediately from condition Eq. (9) as SxA/S

xB

is equal to one in the fluid phase. In the supersolid phasethe dependence of the net vacancy density on the chemi-cal potential is stronger than in the superfluid phase. Thechemical potential (magnetic field hz) is roughly inversely

proportional to the pressure. Meaning that the superfluidphase exhibits a higher compressibility than the supersolidphase, which is a quite remarkable result. Interestingly thenet vacancy density in the supersolid phase also increaseslinearly with the chemical potential. This is due to theratio of the superfluid order parameter SxA/S

xB varies

as the square root of the magnetic field in the vicinity ofthe transition:

SxA/SxB 1

hz hzCAFAF (40)

The exponent 12

is typical for mean-field type approxima-tions and appears close to all transition lines. Figure 11shows the net vacancy density of the model on the su-

persolid to normal solid transition line as a function oftemperature, and also reveals the mean-field type squareroot law dependence.At zero temperature in the solid phase the net vacancydensity is equal to zero, hence the crystal is incompress-ible. In real systems compressibility usually occurs as aresult of change in the lattice constant a, characterized bythe Grueneisen parameter. The quantum lattice gas modeldoes not take this effect into account since the lattice con-stant is treated as a constant. However measurements haveshown that the lattice constant in the (supersolid) heliumis almost a constant indicating that the net vacancy den-sity is the crucial parameter. Also of interest are the freeenergy and the entropy. Figure (8) shows the free energy of


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the anisotropic Heisenberg model in the supersolid (CAF)phase at constant pressure. At low T the free energy usu-ally follows a power law:

F T (41)

The coefficient is a universal property of the modelwhich is constant over the entire regime of a phase. Theexponent is most easily acquired in the log-log plot, shownin the inset of the plot. In this logarithmic scale is givenby the slope of the curve and the leading contribution isgiven by, approximately

= 4.2 (42)

in the supersolid region. This is close in value to the usualT4-term attributed to the linear phonon modes, and fol-lows from Equation (25) with (k) linear in k. Here theT4-term is solely due to the superfluid mode and in realsolids an additional T4-term contribution, accounting forthe lattice phonon-modes, will appear. The logarithmicplot also reveals the leading correction to the free energygiven by = 5.The non-configurational entropy of the system over the en-tire range of temperature and magnetic field hz is given inFigure 9. All four phases are visible and as expected froma thermodynamically equilibrated system the entropy ismonotonically increasing with respect to temperature.Figure 10 depicts the configurational specific heat at con-stant pressure in the supersolid and the normal solid phase.The jump in the specific heat at the critical temperatureTc, in agreement with the second order phase transition,appears to be smeared out due to numerical inaccuracies

as the specific heat is the second derivation of the freeenergy which had to be integrated of the interval [hz, ].The jump in the specific heat may also be calculated fromfollowing formula which is an analogy to the Clausius-Clapeyron equation:

2F

hz2

dhz2

dT2

T L

+ 2F

hzT

dhz

dT

T L

+ 2F

T2= 0

(43)

Ch = T 2F

T2

= T (SzA + SzB)

hzhz

T2

T L

+T

(SzA + SzB)

T

hzT

T L

(44)

As we have CP = Ch T2F

Thz( hzT )P, we have for the

specific heat at constant pressure:

CP = T

(SzA + SzB)

hz

hzT

2T L

+T

(SzA + SzB)

T

hzT

T L

hzT

P

(45)

0 0.1 0.2 0.3 0.4 0.5 0.6T [K]

0.8

1

1.2

1.4

1.6

1.8

2

hz

0 0.1 0.2 0.3

0.84

0.85

0.86

0.87

NF

SF

NS

SS


J1

= 1.0K and J2

= 0.5K. The supersolid phase vanishesbelow T=0.323K. The resulting SF-NS transition is first order.

For the values corresponding to Figure 10 we obtain anestimated jump of 0.02 which is in good agreement withthe curve.

9.2 First Order Boundary Lines

In parameter regimes where the supersolid phase does notappear in certain temperature regions there consequentlyappears a first order phase transition between the super-fluid and the normal solid phase. Liu and Fisher [12] com-pared the free energies of the competing phases to estab-

lish the transition line. This procedure is not applicablein the random-phase approximation, as was outlined inthe section on thermodynamic properties. Other than themean-field approximation where the Hamiltonian is givenby Equation (6) the effective Hamiltonian of the random-phase approximation is not known. Therefore we haveto integrate the first order transition line from a Clau-sius Clapeyron like equation as derived in the section onBoundary lines. A set of parameters which exhibits sucha first order transition at low temperatures is given by:

J1 = 0.5

J2 = 0.5

J1 = 1.0J2 = 0.5 (46)

The corresponding phase diagram is shown in Figure 12.According to mean-field Eq. (14) and Eq. (16) the sec-ond order superfluid to supersolid transition as well asthe supersolid to normal solid transition is at absolutezero at hz = 0.86603, implying that the supersolid phasedoes not exist at zero temperature. At higher tempera-tures (above T > 0.323K) the supersolid phase does exist.At T = 0.323K where the SF-SS and the SS-NS transi-tion lines intersect there occurs a tricritical point. On thistricritical point the superfluid and normal solid phases co-exist (as well as the supersolid phase) and the first order


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0 0.5 1 1.5 2T [K]

0.8

1

1.2

1.4

1.6

hz

NF

SF

SS

NStricritical points


J1

= 1.2K and J2

= 2.4K. The supersolid phase does notappear above T=1.785K. The two tricritical points are con-

nected by a first order SF-NS phase transition.

phase transition line can be calculated from the ordinarydifferential equation as has been derived in the section onboundary lines (Eq. (34)). The tricritical point is as suchthe starting point for the integration of the ODE.There also exist a regime of parameters where the super-solid phase is suppressed at higher temperatures as can beseen in Figure 13 which corresponds to:

J1 = 0.38

J2 = 2.5

J

1 = 1.2J2 = 2.4 (47)

The SF-SS and the SS-NS transition lines converge beforethe NF-SF line is reached, hence no tetracritical pointsuch as in Figure 5 is present. The two tricritical pointsare connected by a first order superfluid to normal solidtransition line.

9.3 Beyond the Model

In Section 8 we have seen that the supersolid phase ap-pears if and only if the roton dip at 1 = 1 and 2 = 1,

i.e. [001] of the first Brillouin zone collapses. Additionallywe have seen that the model also allows for a collapsingminimum at [111] if condition Eq. (37) is met. A set ofparameters that fulfills this condition is given by:

J1 = 0.5K

J2 = 0.5K

J1 = 2K

J2 = 1.5K (48)

Note that the nearest neighbor and the next nearest neigh-bor constants in this configuration J1 and J

2 , corre-

sponding to the kinetic energy, are relatively weak and

0 0.2 0.4 0.6 0.8T [K]

0

1

2

3

4

5

hz

NF

SF

unknown phase

Fig. 14. Phase diagram for J1 = 0.5K,J2 = 0.5K, J

1

=

2K and J2

= 1.5K. The superfluid phase becomes unstabledue to a imaginary quasi-particle spectrum (dashed line). RPA

does not yield any stable phase beneath that line.

are both of equal strength, leading to a highly anisotropickinetic energy. Figure 14 shows the corresponding phasediagram according to the random-phase approximation.The normal fluid to superfluid transition line starts at

hz = 4.5 (49)

at absolute zero and decreases with increasing tempera-ture. According to Eq. (14) and Eq. (16) the critical exter-nal fields hz defining the superfluid to supersolid and thesupersolid to normal solid transitions are, due to negativevalues under the square root, imaginary and hence phys-

ically not relevant. Consequently in the classical mean-field approximation the superfluid phase extends down tohz = 0 and a first order superfluid to normal solid tran-

sition does not occur as the relatively large negative J2

increases the free energy of a possible solid phase.The random-phase approximation however draws a slightlydifferent picture. Analogous to the classical mean-fieldsolution the random-phase approximation also yields aphase transition near hz = 4.5. But unlike the classicalmean-fields solution, the superfluid phase here does notsurvive all the way down to hz = 0. Due to the partic-ular choice of parameters the superfluid phase becomesunstable at around hz = 2; i.e the quasi-particle spectrum

turns imaginary at 1(K) = 0 and 2(k) = 1 ([111]). In-terestingly beyond this line no other stable phase exists inthe present approach; there is no set of spin fields SxA,SxB, SzA and SzB that solves the self-consistencyequations (20) or (21).To understand why the present approach breaks down andhow the phase below hz = 2 might look like we first inves-tigate the physical meaning of the collapsing roton mini-mum at [100] (equivalently [010] and [001]) leading to thesupersolid phase consisting of two SC sub-lattices as al-ready analyzed in the previous sections: The roton dip at[100] corresponds to a density wave given by:

cos(2x/a) + cos(2y/a) + cos(2z/a)

3(50)


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Fig. 15. Two dimensional projection of the lattice structureof a (super)-solid phase on three sub-lattices triggered by acollapse of the roton minimum at [111] of the first Brillouinzone.

This density wave takes on value one on sub-lattice A andminus one on sub-lattice B, hence it reproduces the peri-odicity of the supersolid crystal.In the same way a collapsing roton dip on the main diag-

onals [111] as given here refers to following density wave:

cos(x/a) cos(y/a)cos(z/a) (51)

This density wave yields zero on sub-lattice B and alter-natingly one and minus one on sub-lattice A (neighborshave opposite signs). Consequently this phase refers to a(super)-solid phase exhibiting three sub-lattices A, B andC where the mean-fields take on three different values (seeFig. 15). It would be quite interesting to study the pos-sibilities and properties of such a phase and we leave asfuture work the extension of the present approach to ac-count for a three sub-lattice phase and the investigationits properties.

10 Conclusion

In this paper we extended the mean-field theory of theQLG model by Liu and Fisher [12] at finite temperaturesand employed a random-phase approximation, where wederived Greens functions using the method of equation ofmotion. We applied the cumulant decoupling procedureto split up emergent third order Greens functions in theEoM. In comparison to the MF theory employed by Liuand Fisher [12] the RPA is a fully quantum mechanicallysolution of the QLG model and therefore takes quantumfluctuations as well as quasi-particle excitations into ac-

count. The computational results show that these quasi-particle excitations are capable to render the superfluidphase unstable and thus evoke a phase transition. Quan-tum fluctuations account for additional vacancies and in-terstitials even at zero temperature and most interestinglythe net vacancy density is altered in the supersolid phase.A potential shortcoming of the RPA is the lack of knowl-edge of the effective Hamiltonian and therefore the inter-nal and free energy. Usually the free energy is needed tocompute first order transition lines as the ground state isgiven by the lowest energy state. We bypassed this obsta-cle by deriving a Clausius-Clapeyron like equation whichdefines the first order superfluid to normal solid transi-tion line. The entropy which is an input parameter of

this equation is calculated from the spin wave excitationspectrum. The jump in the specific heat across the sec-ond order superfluid to supersolid transition line revealsinformation about the nature of the transition. Howeverthe specific heat is the second derivative of the free en-

ergy and thus the jump is smeared out by the numericalcalculations. Consequently we derived an equation whichgives an optional estimation of the jump and is in goodagreement with the numerical estimate. Most importantour theory predicts a net vacancy density which, in thesupersolid phase is significantly different from thermal ac-tivation theory. In the normal solid phase the net vacancydensity roughly follows the predictions of thermal acti-vation theory, although quantum mechanical effects givea measurable contribution. Across the phase transition,however, in the supersolid phase the net vacancy densitystays rather constant as T increases.

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A.J. Stoffel and M. Gulacsi- Finite Temperature Properties of a Supersolid: a RPA Approach

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Transcript of A.J. Stoffel and M. Gulacsi- Finite Temperature Properties of a Supersolid: a RPA Approach