PHYSICAL REVIEW A 89, 063614 (2014)

Spin-orbit-coupled Bose gases at finite temperatures

Renyuan Liao,1 Zhi-Gao Huang,1 Xiu-Min Lin,1 and Oleksandr Fialko2

1College of Physics and Energy, Fujian Normal University, Fuzhou 350108, China2Institute of Natural and Mathematical Sciences and Centre for Theoretical Chemistry and Physics,

Massey University, Auckland 0632, New Zealand(Received 30 December 2013; published 20 June 2014)

Spin-orbit coupling is predicted to have a dramatic effect on thermal properties of a two-component atomic Bosegas. We show that in three spatial dimensions it lowers the critical temperature of condensation and enhancesthermal depletion of the condensate fraction. In two dimensions we show that spin-orbit coupling destroyssuperfluidity at any finite temperature, modifying dramatically the cerebrated Berezinskii-Kosterlitz-Thoulessscenario. We explain this by the increase of the number of low-energy states induced by spin-orbit coupling,enhancing the role of quantum fluctuations.

DOI: 10.1103/PhysRevA.89.063614 PACS number(s): 67.85.Fg, 03.75.Mn, 05.30.Jp, 67.85.Jk

I. INTRODUCTION

There are numerous phenomena in a wide range of quantumsystems, ranging from condensed matter to atomic and nuclearphysics, where spin-orbit coupling (SOC) plays an importantrole. A recently discovered class of topological insulators,quantum spin Hall effect [1], and Majorana fermions [2]rely on SOC and are expected to retain their quantumproperties up to room temperature. However, the electronicsystems cannot be easily controlled and the details of SOCare usually not known. Therefore, it is a difficult task tomanipulate such systems. In contrast, ultracold atoms havebeen demonstrated to be a remarkable platform for emulationof various condensed-matter phenomena due to their ability tobe easily manipulated at will [3]. The pioneering experimentalrealization of synthetic gauge fields and SOC [4] is defininga new dimension in exploring quantum many-body systemswith ultracold atomic gases. The engineered SOC (with equalRashba and Dresselhaus strength) in a neutral atomic Bose-Einstein condensate was achieved by dressing two atomic spinstates with a pair of lasers. It allows one to study the richphysics of SOC effects in bosonic systems [5]. Recently, meth-ods to generate Rashba-type SOC have been suggested [6]. Itsrealization will make it possible to study rich ground-statephysics proposed in fermionic [7] and bosonic systems [8], ofwhich many properties have no condensed-matter analogs.

Spin-orbit coupling leads to a huge degeneracy of theground state of a single particle [9]. This enhances the role ofquantum fluctuations making condensation of noninteractingbosons not possible [9,10]. However, it has been shownthat interactions among atoms stabilize condensation [11].The role of quantum fluctuations is especially essential intwo dimensions, destroying condensation but not necessar-ily superfluidity. This yields, in particular, the celebratedBerezinskii-Kosterlitz-Thouless (BKT) phase transition in twodimensions with a critical temperature separating superfluidand normal phases. How do the quantum fluctuations in thepresence of SOC affect the BKT phenomenon and thermalproperties of a Bose condensate? These questions shall beaddressed in this paper. Previous theoretical studies have beenfocused mainly on the ground-state properties of interactingSOC quantum gases, leaving the experimentally relevantphysics at finite temperatures intact. In the light of the recent

advances of generating SOC in neutral gases and surginginterest in thermal Bose gases with a special emphasis onthe BKT physics [12], our present study is an interesting andurgent task.

According to the Mermin-Wagner theorem, long-rangeorder at finite temperature does not exist in one spatialdimension. In this paper we present a study of a SOCtwo-component atomic Bose gas at finite temperature in twoand three spatial dimensions. The interplay between quantumand thermal fluctuations in the presence of SOC is shown toyield dramatic modifications of the familiar physics. First, byresorting to the Popov approximation, we develop a formalismsuitable for treating the system at finite temperature in threedimensions. Within this formalism we find that the SOCgreatly suppresses the critical temperature of condensation andenhances thermal depletion of the condensate. We then derivean effective theory suitable to study the celebrated BKT phasetransition in two dimensions. The BKT transition temperature,in contrast to the previous case, is shown to drop to zero in thepresence of SOC.

II. THREE-DIMENSIONAL CASE

We consider a three-dimensional homogeneous two-component Bose gas with an isotropic in-plane (x-y plane)Rashba spin-orbit coupling, described by the following grandcanonical Hamiltonian in real space:

H =∫

dr

[∑σ

ψ†σ

(−�

2∇2

2m− μ

)ψσ + (ψ†

↑Rψ↓ + H.c.)

+∑

σ

gσσ

2(ψ†

σψσ )2 + g↑↓ψ†↑ψ↑ψ

†↓ψ↓

]. (1)

Here ψσ is a Bose field satisfying the usual commutationrelation [ψσ (r),ψ†

σ ′(r′)] = i�δσσ ′δ(r − r′) and the spin indexσ = ↑,↓ denotes two pseudospin states of the Bose gas withatomic mass m. The interparticle interaction gσσ ′ is related tothe two-body scattering length aσσ ′ as gσσ ′ = 4π�

2aσσ ′/m.For simplicity, we shall assume that the interactions betweenlike-spin particles are the same, g↑↑ = g↓↓ = g. The chemicalpotential μ is introduced to fix the total particle numberdensity. The Rashba spin-orbit coupling is described by the

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LIAO, HUANG, LIN, AND FIALKO PHYSICAL REVIEW A 89, 063614 (2014)

operator R = λ(px − ipy), with λ being the coupling strength.Throughout the rest of this paper, we set � = 2m = kB = 1 anddefine n1/3 as a momentum scale and n2/3 as an energy scale.For the system to be weakly interacting, we set g = 0.1n−1/3.

The Hamiltonian of a noninteracting system is diagonalizedin the helicity basis after the Fourier transform of the fieldsψσ (r) = 1/

√L3

∑q ψσ (q) exp(−iq · r). The gas is assumed

to be in a box with size L. This results in two branches ofspectrum E±

q = q2 − μ ± λq⊥, where q⊥ is the magnitude ofthe in-plane momentum. The lowest-energy state is thereforeinfinitely degenerate, sitting on the circle q⊥ = λ/2 in theplane qz = 0. This increases the low-energy density of stateswith dramatic implications on the thermal properties of theBose gas to be explored below. For an interacting system,an earlier mean-field study [13] found that there exist theplane-wave (PW) phase for g � g↑↓ and the striped phasefor g < g↑↓. The PW phase is the result of condensation ata single momentum state breaking explicitly the rotationalsymmetry, while the striped phase represents a superpositionof two condensates at two opposite momenta.

Within the framework of the functional field in-tegral, the partition function of the system is [14]Z = ∫

D[ψ∗,ψ] exp(−S[ψ∗,ψ]), with the action S =∫ β

0 dτ [∫

dr∑

σ ψ∗σ ∂τψσ + H (ψ∗,ψ)], where β = 1/T is the

inverse temperature. Here, for simplicity, we restrict ourselfto study the PW phase, as an analogous treatment of thestriped phase is more involved. Our choice of the PW phaseis also motivated by the fact that the striped phase has notyet been realized experimentally. We further assume that thecondensation occurs at momentum κ = (λ/2,0,0). Withoutloss of generality, the condensate wave function can be chosenas (φ0↑,φ0↓) = √

n0(1,−1)eiλx/2, with n0 being the condensatedensity for either species. We split the Bose field into themean-field part φ0σ and the fluctuating part φqσ as ψqσ =φ0σ δqκ + φqσ . After substitution, the action can be formallywritten as S = S0 + Sf , where S0 = βL3[−2(κ2 + μ)n0 +(g + g↑↓)n2

0] is the mean-field contribution and Sf denotes acontribution from fluctuating fields. At this point, the action is

exact. However, it contains terms of cubic and quartic ordersin fluctuating fields. To deal with such an action, one mustresort to some sort of approximation. The celebrated Bogoli-ubov approximation for Bose-Einstein condensation (BEC)is valid strictly at zero temperature. At finite temperatures,the self-consistent Hartree-Fock-Bogoliubov approximationgives a gapped spectrum [15], violating the Hugenholtz-Pinestheorem [16] and the Goldstone theorem [17], which resultsfrom the spontaneous symmetry breaking of U(1) gaugesymmetry. We choose the Popov theory [18], which yieldsa gapless spectrum and therefore is more suitable for treatingfinite-temperature Bose gases. The mean-field treatment is areliable theoretical scheme yielding good qualitative results inthe weakly interacting regime [19]. The detailed propertiesat the critical regime are likely to be nontrivial and needto be addressed by more sophisticated techniques such asrenormalization-group theory, which is beyond the purposeof this work.

Under the Popov approximation, which takes into ac-count interactions between excitations [20], the termswith three and four fluctuating fields in the actionare approximated as follows (neglecting anomalous aver-age): φ∗

σφσφσ ≈ 2〈φ∗σφσ 〉φσ , (φ∗

σφσ )2 ≈ 4〈φ∗σφσ 〉φ∗

σφσ , and|φ↑φ↓|2 ≈ 〈φ∗

↑φ↑〉φ∗↓φ↓ + 〈φ∗

↓φ↓〉φ∗↑φ↑. Within the Popov ap-

proximation we also require the first-order term to vanish,which fixes the chemical potential as μ = μp = −κ2 +(g + g↑↓)n0 + (2g + g↑↓)nf , where nf ≡ 〈φ∗

σφσ 〉 is the den-sity of particles of either component excited out of thecondensate. We define a four-dimensional column vec-tor �qn = (φκ+q,n↑,φκ+q,n↓,φ∗

κ−q,n↑,φ∗κ−q,n↓), whose compo-

nents are defined through the Fourier transform φσ (r,t) =1/

√L3β

∑q,n φq,nσ exp(−iq · r − iwnt), where wn = 2nπ/β

is the bosonic Matsubara frequencies. Retaining terms ofzeroth and quadratic orders in the fluctuating fields, we canthen bring the fluctuating part of the action into the compactform Sf ≈ ∑

q,n12�∗

qnG−1(q,iwn)�Tqn − β

∑q ε−q, where

εq = (q + κ)2 − μ + (2g + g↑↓)(n0 + nf ) and G−1(q,iwn) isthe inverse Green’s function defined as follows:

G−1(q,iwn) =

⎛⎜⎜⎝

−iwn + εq Rκ+q − g↑↓n0 gn0 −g↑↓n0

R∗κ+q − g↑↓n0 −iwn + εq −g↑↓n0 gn0

gn0 −g↑↓n0 iwn + ε−q R∗κ−q − g↑↓n0

−g↑↓n0 gn0 Rκ−q − g↑↓ iwn + ε−q

⎞⎟⎟⎠. (2)

The poles of the Green’s function give the spectrum of elementary excitations ωqs . It can be found by replacing iωn with ωqs andsolving the secular equation detG−1(q,ωqs) = 0. The index s denotes different solutions of the secular equation. We retain onlytwo positive solutions in the following. The spectrum is used to study thermodynamic properties of the system. The Gaussian actionpermits direct integration over the fluctuating fields to yield the grand potential − lnZ/β = S0/β + �f , where the second term

�f =∑q,s

(ln (1 − e−βωqs )

2β+ ωqs − εq

2+ (g2 + g2

↑↓)n20

2q2

)(3)

is the result of the Gaussian integration. The density ofparticles excited out of the condensates for either speciescan be easily evaluated as nf = 1/(2L3)(∂�f /∂μ)μ=μp

. Atthe critical temperature the condensed density n0 = 0 andnf is equal to the density of the total number of particlesn. In this special case, it is straightforward to calculate

analytically the poles of the Green’s function and obtainthe following secular equation determining the criticaltemperature:

n = 1

2L3

∑q,s=±

1

exp{[

q2z + (q⊥ + sλ/2)2

]/Tc

} − 1. (4)

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SPIN-ORBIT-COUPLED BOSE GASES AT FINITE . . . PHYSICAL REVIEW A 89, 063614 (2014)

FIG. 1. (Color online) Critical temperature of condensation asa function of the SOC strength. The SOC decreases the criticaltemperature due to an increase of the low-energy density of stateson the circle q⊥ = λ/2. Here T 0

c is the critical temperature for anoninteracting Bose system without SOC. We have set g↑↓ = g.

In the absence of SOC, the two terms are identical andwe get the usual number equation to determine the criticaltemperature of condensation. The biggest contribution tothe sum over the momentum is from the vicinity of thepoint |q| = 0. Spin-orbit coupling changes the situation.Now the biggest contribution to the sum comes from thecircle qz = 0,q⊥ = λ/2 with much higher weight than in theprevious case. The critical temperature Tc must be decreasedfor fixed density. More qualitatively, the critical temperaturein the thermodynamic limit is estimated as

Tc ≈ T 0c

g3/2(1)

g3/2(z). (5)

Here T 0c = 4π [n/g3/2(1)]2/3 is the critical temperature for a

noninteracting Bose gas without SOC, z = exp (−λ2/4Tc),and gp(z) = ∑∞

l=1 zl/ lp [21]. As shown in Fig. 1, the criticaltemperature Tc indeed decreases rapidly as the SOC strengthincreases.

The situation is more complicated for temperatures smallerthan the critical temperature. The secular equation shouldbe supplemented with the equation for the total densityn0 + nf = n. By solving the two equations we obtain thecondensate density n0 for a fixed total density of atoms n.Being an intrinsic property of a Bose-Einstein condensate,the condensate fraction n0/n provides key information aboutthe robustness of the superfluid state. It is shown in Fig. 2,where three typical SOC couplings (λ = 0, λ/n1/3 = 0.2, andλ/n1/3 = 0.4) are chosen for comparison. In the absence ofSOC, the condensate fraction decreases gradually to zero as thetemperature reaches T 0

c , since under the Popov approximation,the critical temperature of a weakly interacting Bose gas isidentical to that of a noninteracting one [22]. The effect ofSOC on the condensate fraction is more pronounced when thetemperature gets closer to the transition temperature Tc. We canexplain this by a similar argument leading to the reduction ofthe critical temperature: At a fixed temperature the thermalcomponent benefits mainly from the low-energy states onthe circle q⊥ = λ/2, while the condensed component benefits

FIG. 2. (Color online) Condensate fraction as a function of tem-perature for various spin-orbit-coupling strengths λ. The effectof SOC on the thermal depletion is more pronounced at highertemperatures since the thermal component benefits more from thelow-energy states than the condensed one. We have set g↑↓ = g.

mainly from a single momentum at qx = λ/2. Therefore, SOCeffects the thermal component more than the condensed one,leading to the more pronounced thermal depletion at highertemperatures.

III. TWO-DIMENSIONAL CASE

We turn now to study the effects of SOC on the systemin two spatial dimensions. In the absence of SOC there is aquasi-long-range order in two dimensions at sufficiently lowtemperatures with a correlation function decaying accordingto some power law. At higher temperatures the correlationfunction decays exponentially, where the algebraic order isdestroyed by proliferation of vortices. The BKT separatesthese two regimes [23]. Since there is no condensation now,the Popov approximation is not applicable. Here we derive alow-energy effective theory suitable to probe the correlationfunction. Anticipating that phase degrees of freedom play anessential role now, we adopt the density-phase representationof the field operators ψσ = ρ

1/2σ eiθσ . The Hamiltonian for

Rashba-type spin-orbit coupling HR is written as

HR = λ

∫dr

[(ρ↑ρ↓

)1/2

∂xρ↓ sin(θ↓ − θ↑)

+ 2(ρ↑ρ↓)1/2∂xθ↓ cos(θ↓ − θ↑)

]

+ λ

∫dr

[(ρ↓ρ↑

)1/2

∂yρ↑ cos(θ↓ − θ↑)

+ 2(ρ↑ρ↓)1/2∂yθ↑ sin(θ↓ − θ↑)

]. (6)

The fields are then separated again into a mean-field partand a fluctuating part as ρσ = ρ0 + ρσ and θσ = θ0σ + θσ .Here ρ0 is usually called a quasicondensate density and theexplicit breaking of the rotational symmetry by SOC resultsin θ0↑ = θ0↓ − π = λx/2. Retaining terms of zeroth andquadratic order in the fluctuating fields, the action is split into

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LIAO, HUANG, LIN, AND FIALKO PHYSICAL REVIEW A 89, 063614 (2014)

two parts S ≈ S0 + Sg , where similarly to the previous caseS0 = βL2[−2(λ2/4 + μ)ρ0 + (g + g↑↓)ρ2

0 ] is the mean-fieldcontribution. The Gaussian action Sg contains the fluctuatingfields up to second order:

Sg =∫

dτ

∫dr

[∑σ

(iρσ ∂τ θσ + (∇ρσ )2

4ρ0+ ρ0(∇ θσ )2

+ λρσ ∂x θσ + gρ2σ

)+ 2g↑↓ρ↑ρ↓

]+ SR,

SR = λ

∫dτ

∫dr

(−ρ↓∂xθ↑ − ρ↑∂xθ↓ − ρ↓

2ρ0∂yρ↑

− 2ρ0θ↓∂yθ↑ + ρ0λ

2(θ↓ − θ↑)2 + λ

8ρ0(ρ↑ − ρ↓)

). (7)

At this point, the Gaussian action is exact and it containsfluctuations of fast modes ρσ and fluctuations of slow modesθσ . To distill a low-energy effective theory, one neglects thespatial dependence of the density fluctuations. Integrating outsuch density fluctuations, one obtains an action solely in termsof phase degrees of freedom

Sg =∫

dτ

∫dr

(− (∂τ θ+)2

2(g + g↑↓)+ (∂τ θ− + i2λ∂xθ−)2

2(g − g↑↓) + λ2/(2ρ0)

+ ρ0

2[(∇θ+)2 + (∇θ−)2] + ρ0λθ−∂yθ+ + ρ0

2λ2θ2

−

),

(8)

where θ± = θ↑ ± θ↓. In the absence of SOC, the above actiondescribes two decoupled quantum XY models representingtwo branches of phase fluctuations with a linear spectrum inmomentum space yielding the quasi-long-range order. Thecorresponding correlation functions C±(r) = 〈e[θ±(r)−θ±(0)]〉 ∝r−T/ρ0(g±g↑↓) decay according to the power law rather thanexponentially [23]. In the presence of SOC, we notice thatthe field θ− describes a massive fluctuation as the action Sg

contains the term ρ0λ2θ2

−/2. Physically, θ− denotes the twistbetween the phase fields of the two pseudospin componentsand the coefficient ρ0λ

2/2 acts as the role of a mass. To studythe fields separately, we integrate out one of the fields. Thisincorporates their contribution on each others’ low-energydynamics. Integration over the field θ+ renormalizes thegradient terms leaving the mass term intact. The field θ−stays massive and its fluctuations have negligible effect onlow energies. They do not destroy the long-range order.We thus turn our attention to the field θ+. To capture theessential physics at low energy, we integrate out the massivefield θ− by assuming homogeneous temporal fluctuations,yielding the effective action for the θ+ sector (we defineg± = g ± g↑↓)

Sg[θ+] =∫

dτ

∫dr

[− (∂τ θ+)2

2g++ ρ0

2(∇θ+)2 − ρ0

2(∂yθ+)

×(

1 − 16∂2x

4ρ0g− + λ2− ∇2

λ2

)−1

(∂yθ+)

]. (9)

By employing the gradient expansion, namely,

1

1 − 16∂2x

4ρ0g−+λ2 − ∇2

λ2

≈ 1 + 16∂2x

4ρ0g− + λ2+ ∇2

λ2, (10)

we finally obtain the effective action for the θ+ sector as

Sg = 1

2

∫dτ

∫dr

(− (∂τ θ+)2

g++ ρ0(∂xθ+)2 + ρ0

λ2

(∂2y θ+

)2)

.

(11)

The first two terms in the action are the part of the familiarXY model. The third term is the manifestation of thebroken rotational symmetry and it adds a twist. To seethis we calculate the spectrum of elementary excitations.We first Fourier transform the action for it to becomeSg = 1/2

∑q,n(w2

n/g+ + ρ0q2x + ρ0q

4y/λ

2)|θ+(q,wn)|2. Thephase fluctuations in the Fourier space can be easily evaluated,〈|θ+(q,wn)|2〉= g+(w2

n + w2q)−1. The poles of this expression

give the low-energy spectrum of the symmetric phasewq =

√g+ρ0(q2

x + q4y/λ

2). It is clear that now the energycarried by elementary excitations of the system does notscale linearly as in the XY model case. As a result, lessenergy is required to excite low-energy modes as comparedto the XY model case, enhancing the role of quantumfluctuations. The Landau criterion for the superfluid criticalvelocity gives minq(wq/q) = 0, implying SOC destroyssuperfluidity in two dimensions. This is substantiated by theform of the correlation function, which reads C+(r) ≈ e−I

with I = ∑q,n〈|θ+(q,wn)|2〉(1 − eiq·r)/2β. The asymptotic

behavior of it for large separations along the x direction isI ≈ T

√λ|x|/2πg+ρ0 and along the y direction it is I ≈

T λ|y|/2g+ρ0. Hence, at any finite temperature, the correlationfunction decays exponentially along both x and y directions,in stark contrast to the power law exhibited by the conventionalXY model in the low-temperature phase. The superfluid orderin two dimensions is predicted to be destroyed by SOC.

In the conventional XY model the disappearance of thesuperfluid order is associated with the proliferation of vortices.To provide an intuitive understanding of the proliferationof vortices at any finite temperature in our system, wewrite the distortion field as u = �∇θ+. Any two-dimensionaldistortion can be written as u = �∇φ − �∇ × (zψ) with ψ(r) =∑

i ni ln(|r − ri |) [24]. Here �∇φ describes a potential flowand ψ behaves like the potential due to a set of charges 2πni

located at ri . One can easily verify that the topological chargecan be related to �∇ × u via∮

u · dl =∫

�∇ × u · zd2r = 2π∑

i

ni . (12)

We have ∂xθ+ = ∂xφ − ∂yψ and ∂yθ+ = ∂yφ + ∂xψ . Theenergy part of the low-energy action becomes

E = ρ0

2

∫d2r

((∂xθ )2 + 1

λ2

(∂2y θ

)2)

= ρ0

2

∫d2r

((∂xφ − ∂yψ)2 + 1

λ2

(∂2yφ + ∂xyψ

)2)

. (13)

This implies that the spin-wave part described by the field φ iscoupled to the vortex part described by the field ψ . As a result

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SPIN-ORBIT-COUPLED BOSE GASES AT FINITE . . . PHYSICAL REVIEW A 89, 063614 (2014)

of this coupling, the spin-density waves carry away energyfrom the vortex sector. This can be seen by integrating out thespin-wave field φ, which produces a second-order contributionin the vortex field ψ with negative sign. Minimization of thefree energy of a vortex F = E − T S implies a reduction ofthe critical temperature assuming that the entropy of puttinga vortex into the system does not change. It is interestingthat for a vortex-antivortex pair the contribution to the energyfrom the spin-wave sector translates into an interaction termwith positive sign, i.e., repulsion. The spin waves thereforescreen the Coulomb attraction between a vortex and antivortex.All this provides an intuitive understanding of the criticaltemperature reduction of vortex proliferation in accordancewith the more careful quantitative analysis presented earlier.

IV. SUMMARY

We predicted dramatic implications of SOC on the thermalproperties of atomic Bose gases. We showed how a simplechange in a single-particle spectrum induced by SOC maylead to complete destruction of the quasi-long-range order intwo dimensions modifying dramatically the celebrated BKTscenario while retaining true long-range order of the BEC inthree dimensions. This is explained by the role of quantumfluctuations amplified by SOC. In particular, we found thatSOC reduces the critical temperature of condensation and

enhances the thermal depletion of the condensate, while in twodimensions it destroys superfluidity at any finite temperature.Our results may shed light on the microscopic mechanismof the BKT transition. The BKT physics has been underactive study in various two-dimensional systems for morethan 40 years since its first discovery [23]. It is believed tobe associated with dissociation of bound vortex-antivortexpairs around the critical temperature. In contrast, our resultsseem to show that there are no bound vortex pairs in twodimensions at any finite temperature in the presence of SOC.This indicates that unexplored physics in two dimensionsmight take place. We thus hope that our work will contributeto a deeper understanding of thermal quantum fluids such asthe fundamental relationship between BEC and superfluidityand stimulate experiments to verify our predictions by meansof modern quantum technology [6,12].

ACKNOWLEDGMENTS

We would like to thank Joachim Brand for useful discus-sions. This work was supported by the NBPRC under GrantNo. 2011CBA00200, the NSFC under Grants No. 11274064and No. 61275215, and NCET-13-0734. O.F. was supportedby the Marsden Fund Council through government funding(Contract No. MAU1205), administrated by the Royal Societyof New Zealand.

[1] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959(2010); M. Z. Hasan and C. L. Kane, ibid. 82, 3045 (2010);X. L. Qi and S. C. Zhang, ibid. 83, 1057 (2011).

[2] V. Mourik, K. Zuo, S. M. Frlov, S. R. Plissard, E. P. A. M.Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012).

[3] D. Jaksch and P. Zoller, Ann. Phys. (NY) 315, 52 (2005).[4] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature

(London) 471, 83 (2011); M. Aidelsburger, M. Atala,S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev.Lett. 107, 255301 (2011); P. Wang, Z.-Q. Yu, Z. Fu, J. Miao,L. Huang, S. Chai, H. Zhai, and J. Zhang, ibid. 109, 095301(2012); L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,W. S. Bakr, and M. W. Zwierlein, ibid. 109, 095302 (2012).

[5] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A 78,023616 (2008); T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107,150403 (2011); Y. Li, L. P. Pitaevskii, and S. Stringari, ibid.108, 225301 (2012); Y. Li, G. I. Martone, L. P. Pitaevskii, andS. Stringari, ibid. 110, 235302 (2013).

[6] B. M. Anderson, G. Juzeliunas, V. M. Galitski, and I. B.Spielman, Phys. Rev. Lett. 108, 235301 (2012); B. M. Anderson,I. B. Spielman, and G. Juzeliunas, ibid. 111, 125301 (2013).

[7] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. Rev. B 84,014512 (2011); M. Iskin and A. L. Subasi, Phys. Rev. Lett. 107,050402 (2011); M. Gong, S. Tewari, and C. Zhang, ibid. 107,195303 (2011); H. Hu, L. Jiang, X.-J. Liu, and H. Pu, ibid. 107,195304 (2011); Z.-Q. Yu and H. Zhai, ibid. 107, 195305 (2011);R. Liao, Y. Yi-Xiang, and W. M. Liu, ibid. 108, 080406 (2012).

[8] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401(2011); H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, ibid.108, 010402 (2012); T. Kawakami, T. Mizushima, M. Nitta, and

K. Machida, ibid. 109, 015301 (2012); Z. F. Xu, Y. Kawaguchi,L. You, and M. Ueda, Phys. Rev. A 86, 033628 (2012); X. Zhou,Y. Li, Z. Cai, and C. Wu, J. Phys. B 46, 134001 (2013).

[9] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012).[10] Q. Zhou and X. Cui, Phys. Rev. Lett. 110, 140407 (2013).[11] R. Barnett, S. Powell, T. Grass, M. Lewenstein, and

S. Das Sarma, Phys. Rev. A 85, 023615 (2012); X. Cui andQ. Zhou, ibid. 87, 031604 (2013); T. Ozawa and G. Baym,Phys. Rev. Lett. 109, 025301 (2012); R. Liao, Z.-G. Huang,X.-M. Lin, and W.-M. Liu, Phys. Rev. A 87, 043605 (2013).

[12] T. Donner, S. Ritter, T. Bourdel, A. Ottl, M. Kohl, andT. Esslinger, Science 315, 1556 (2007); P. Clade, C. Ryu,A. Ramanathan, K. Helmerson, and W. D. Phillips, Phys. Rev.Lett. 102, 170401 (2009); C. Huang, X. Zhang, N. Gemelke,and C. Chin, Nature (London) 470, 236 (2011); R. Debuquois,L. Chomaz, T. Yefsah, J. Leonard, J. Beugno, C. Weitenberg,and J. Dalibard, Nat. Phys. 8, 645 (2012); J. Y. Choi, S. W. Seo,and Y. I. Shin, Phys. Rev. Lett. 110, 175302 (2013); L. C. Ha,C. L. Hung, X. Zhang, U. Eismann, S. K. Tung, and C. Chin,ibid. 110, 145302 (2013); J.-Y. Zhang, S.-C. Ji, L. Zhang, Z.-D.Du, W. Zheng, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan,Nat. Phys. 10, 314 (2014).

[13] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev. Lett. 105,160403 (2010).

[14] A. Altland and B. Simons, Condensed Matter Field Theory(Cambridge University Press, Cambridge, 2006).

[15] A. Griffin, Phys. Rev. B 53, 9341 (1996).[16] N. Hugenholtz and D. Pines, Phys. Rev. 116, 489

(1959).[17] J. Goldstone, Phys. Rev. 127, 965 (1962).

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LIAO, HUANG, LIN, AND FIALKO PHYSICAL REVIEW A 89, 063614 (2014)

[18] V. Popov, Functional Integrals in Quantum Field Theory andStatistical Physics (Reidel, Dordrecht, 2001).

[19] S. Sachdev, Quantum Phase Transitions (Cambridge UniversityPress, Cambridge, 2011).

[20] J. O. Anderson, Rev. Mod. Phys. 76, 599 (2004).[21] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation

(Oxford University Press, New York, 2003).

[22] C. Pethick and H. Smith, Bose-Einstein Condensationin Dilute Gases (Cambridge University Press, Cambridge,2008).

[23] J. V. Jose, 40 Years of Berezinskii-Kosterlitz-Thouless Theory(World Scientific, Singapore, 2013).

[24] M. Kardar, Stastical Physics of Fields (Cambridge UniversityPress, Cambridge, 2007).

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