PRA 85, 053611 (2012)

PHYSICAL REVIEW A 85, 053611 (2012)

Finite-temperature phase diagram of a spin-1 Bose gas

Yuki Kawaguchi,1 Nguyen Thanh Phuc,1 and P. Blair Blakie21Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

2Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin, New Zealand(Received 31 January 2012; published 10 May 2012)

We formulate a self-consistent Hartree-Fock theory for a spin-1 Bose gas at finite temperature and apply it tocharacterization of the phase diagram. We find that spin coherence between thermal atoms in different magneticsublevels develops via coherent collisions with the condensed atoms, and is a crucial factor in determining thephase diagram. We develop analytical expressions to characterize the interaction- and temperature-dependentshifts of the phase boundaries.

DOI: 10.1103/PhysRevA.85.053611 PACS number(s): 03.75.Mn, 05.30.Jp, 03.75.Hh

I. INTRODUCTION

A key feature of a system with spin internal degreesof freedom is that the atoms can condense into a rangeof phases, characterized by various spin order parameters,dependent upon the nature of the interactions and the externalmagnetic field (e.g., see Fig. 1). The seminal theory forthe spin-1 Bose gas was developed in 1998 [2,3] and soonafter realized in experiments [4,5]. Aspects of the equilibriumphase diagram were initially observed in Ref. [4], and morerecently experiments have used external fields to investigatethe dynamical properties of this system (e.g., see Refs. [69]),including quenches between phases [10,11].

Several theoretical treatments within mean-field approx-imations have considered the equilibrium properties of acondensed spin-1 Bose gas at finite temperature [1216].Natu and Mueller have predicted that, for sufficiently largespin-dependent interaction strength, pairing or spontaneousmagnetization will occur at slightly higher temperature thanthe condensation transition [17]. In the two-dimensional (2D)regime, where condensation is expected to be suppressed, thefinite-temperature phase diagram has recently been elucidated[18,19].

This paper investigates the finite-temperature phase di-agram of the spin-1 Bose gas, including both linear andquadratic Zeeman effects, which were not fully consideredin previous work [1214,16]. Figure 1 shows the mean-fieldphase diagram at T = 0 drawn in the parameter space ofthe linear (p) and quadratic (q) Zeeman energies [1,4]. Weinvestigate how the phase boundaries in Fig. 1 change as thetemperature increases, using a Hartree-Fock (HF) mean-fieldtheory. Although HF theory is the simplest many-body theory,it forms an important building block for more advancedmany-body theories, and for comparison to other types ofcalculations. A key feature of our theory is the inclusion of spincoherence between noncondensate (thermal atoms) in differentmagnetic sublevels. We find that when the condensate is in astate of spontaneously broken spin rotational symmetry (aboutthe direction of the applied field), i.e., in the antiferromagneticand broken-axisymmetry phases, the spin coherence betweennoncondensed atoms also develops via coherent collisionswith the condensed atoms. Moreover, the noncondensate spincoherence has a large effect on the phase boundaries inthe finite-temperature regime. We derive analytical relations

between the shifts in the phase boundaries and noncondensatespin density or spin coherence, which agree well withthe full numerical results. These analytical results furnishadditional insight into how the thermal fluctuations influencethe condensate order and directly show the importance of thenoncondensate spin coherence.

Finally, we note that HF calculations are generally expectedto provide a good qualitative description of the interactingsystem. Indeed, HF theory accurately describes a range ofthermodynamic measurements made on the scalar three-dimensional Bose gas (e.g., see [2022]). However, the spinorsituation is much less clear. Our recent work [16] suggeststhat the spinor gas, in the regime of current experiments with87Rb, is strongly interacting, in the sense that the correctionsto the Bogoliubov theory are nonperturbative. There alsoremain a number of open questions about the explanation ofcurrent experiments (e.g., see [8,11]) and what role thermalfluctuations, dipole-dipole interactions, or nonequilibriumeffects play. The work we present here provides an importantstep toward achieving a more complete understanding ofthermal effects in the spinor Bose gas.

II. BASIC FORMALISM

We consider a spin-1 Bose gas confined in an opticalpotential U (r) and subject to a uniform magnetic field alongz. The single-particle description of the atoms is provided bythe Hamiltonian

(h0)ij =[

h222M

+ U (r) pi + qi2]ij , (1)

where p and q are the coefficients of the linear and quadraticZeeman terms, respectively, the subscripts i,j = 1,0, + 1,refer to the magnetic sublevels of the atoms, and M is theatomic mass. The value of q is tunable independently of p,using an off-resonant microwave field [23].

Introducing spinor field operators i(r), thecold-atom Hamiltonian, including interactions, is given

053611-11050-2947/2012/85(5)/053611(11) 2012 American Physical Society

KAWAGUCHI, PHUC, AND BLAKIE PHYSICAL REVIEW A 85, 053611 (2012)

p 2 = 2 c1n q

p = q + 12c 1n

1/2

1

1 P

FM

FM

AFMq/c1n

p/c1n

( a ) c1 > 0 , T = 0 (b) c1 < 0 , T = 0p = q

p2=q 22|c1|n q

2

P

FM

FM

BAq/|c 1|n

p/|c1|n

FIG. 1. (Color online) The T = 0 phase diagram of a spin-1 Bosegas for cases where the spin-dependent interaction is (a) antiferro-magnetic (c1 > 0) and (b) ferromagnetic (c1 < 0). The vertical andhorizontal axes are the linear and quadratic Zeeman energies (seetext) in units of |c1|n, where n is the total number density (whichis identical to the condensate number density at T = 0). The phasesshown are (FM) ferromagnetic, (P) polar, (AFM) antiferromagnetic,and (BA) broken-axisymmetry phases (see Sec. IV A and Refs. [1,4]).The rotational symmetry about the direction of the applied field isspontaneously broken in the AFM and BA phases.

by [2,3]

H =

dr

{i,j

[i (r)(h0)ij j (r)

+ c02

i (r) j (r) j (r) i(r)

]

+ c12

,i,j,k,l

(f)ij (f)kl i (r) k (r) l(r) j (r)}, (2)

where = x, y, or z specifies the spin components, with fbeing the 3 3 spin-1 matrices. The parameters c0 and c1are referred to as the spin-independent and spin-dependentinteraction parameters, respectively, and are given by c0 =4h2(a0 + 2a2)/3M , c1 = 4h2(a2 a0)/3M , with aS (S =0,2) being the s-wave scattering length for the scatteringchannel of total spin S.

III. HARTREE-FOCK THEORY

A. General inhomogeneous theoryThe basic mean-field approach is to assume that when

there is a condensate in the system the field operator can bedecomposed as

i(r) = i(r) + i(r), (3)where i(r) is a classical field describing the condensate andthe fluctuation operator i(r) describes the noncondensatemodes. The HF equations can be derived by using a variationalapproach to minimize the free energy (e.g., see Appendix Aand Refs. [24,25]). Key to this approach is the factorization ofthe expectation value of the interaction terms into expressionsinvolving products of first-order correlation functions:

i (r) j (r) = ncij (r) + nncij (r), (4)

where we have introduced the notation ncij (r) i (r)j (r)and nncij (r) i (r) j (r) for the condensate and nonconden-sate one-body density matrices, respectively.1 We emphasizethat in the presence of a condensate, nncij (r) may havenonzero off-diagonal elements, that is, may exhibit partialphase coherence between thermal atoms in different magneticsublevels. Since the Hamiltonian (2) is invariant under a spinrotation about the z axis, nncij (r) should be diagonal in thenormal phase (without pairing or ferromagnetic order [17]) sothat the system is invariant under spin rotations. In a condensedphase, however, if the condensate spontaneously breaks therotational symmetry in spin space, the noncondensate is alsodistributed inhomogeneously in spin space due to coherentcollisions between condensed and noncondensed atoms. Non-condensate spin coherence was experimentally observed in atwo-component Bose gas [26].

The generalized Gross-Pitaevskii equation (GPE) for thecondensate is (see Appendix A)

i(r) =j

Lijj (r), (5)

where

Lij = (h0)ij + c0(nc + nnc)ij + c0nncji

+ c1

[(F c + F nc

)(f)ij +k,l

(f)ik(f)lj nnclk]

(6)is the Gross-Pitaevskii matrix operator, and

nc(r) =

i

ncii(r), (7)

F c (r) =i,j

(f)ij ncij (r), (8)

nnc(r) =

i

nncii (r), (9)

F nc (r) =i,j

(f)ij nncij (r) (10)

are the number and spin densities associated with the con-densed and noncondensed atoms.

The HF grand-canonical Hamiltonian for the nonconden-sate is given by (see Appendix A)

KHF =

dri,j

i Aij (r) j , (11)

where

Aij = Lij ij + c0ncji + c1k,l

(f)ik(f)lj nclk, (12)

i.e., differing from the condensate operator Lij by the inclusionof the exchange interactions with the condensate.

1The full one-body density matrix also retains off-diagonal positionarguments; however, these are not needed to formulate HF theory fora gas with contact interactions. In this work we will use the termoff-diagonal in reference to the spin indices.

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FINITE-TEMPERATURE PHASE DIAGRAM OF A SPIN-1 . . . PHYSICAL REVIEW A 85, 053611 (2012)

By finding the eigenvalues () and eigenvectors [u()j (r)]of Aij , i.e.,

u()i (r) =

j

Aij (r)u()j (r), (13)

normalized so thati

dr u

()i (r)u()i (r) = , (14)

the noncondensate density matrix is given by

nncij (r) =

u()i (r)u()j (r)n, (15)

where n = 1/[exp() 1] is the Bose-Einstein distributionfunction with = 1/(kBT ).

B. Specialization to the uniform systemFor the purpose of studying the finite-temperature phase

diagram we now discuss the specialization of the HF formalismto a uniform system. In this case U (r) 0 and the mean fields(ncij and nncij ) are spatially independent. The condensate occursin the zero-momentum spatial mode, and the generalized GPE(5) reduces to the nonlinear algebraic equation

i =j

Lij j , (16)

where

Lij = (pi + qi2)ij + c0[(nc + nnc)ij + nncji]

+ c1

[(F c + F nc

)(f)ij +k,l

(f)ik(f)lj nnclk].

(17)The excited modes have plane-wave spatial dependence:

u()j (r) = u()j eikr, (18)

where u()j is a constant spinor (and is independent of k) andwe have adopted the notation {,k}, with k a wave vectorand an index to distinguish between modes.

The HF Hamiltonian takes the form

Aij = h222M

+Aij , (19)

where

Aij = Lij ij + c0ncji + c1,k,l

(f)ik(f)lj nclk (20)

is a constant matrix. Notably the spatial and spin parts inEq. (19) are decoupled and can be treated separately [allowingus to use the excited mode of the form given in Eq. (18)].Diagonalizing Aij , we obtain the three eigenvectors u()j withrespective eigenvalues , and hence the excitation spectrumis given by

k = h2k2

2M+ . (21)

To evaluate the noncondensate one-body density matrix we set (2 )3

dk in Eq. (15) and obtain

nncij =3

=1u

()i u

()j

Li3/2(e )3dB

, (22)

where dB = h/

2MkBT is the thermal de Broglie wave-length and Li (z)

t=1 z

t/t is the polylogarithm. We notethat for the thermal cloud to saturate, and hence condensationto occur, at least one of the eigenvalues must approach zeroat the condensation temperature.

IV. RESULTS

The effect of the thermal cloud on the condensate is, in gen-eral, quite complicated and requires the full self-consistent cal-culation. We numerically solve the coupled Gross-Pitaevskiiand HF equations self-consistently in the temperature rangeof T = (00.5)T0, where T0 is the condensation temperatureof an ideal scalar gas with the same total number density.Because there are three internal states, the condensationtemperature of an ideal spin-1 gas at p = q = 0 is reducedto T spinorc = (1/3)2/3T0 0.48T0. For 87Rb and 23Na gases (inthe F = 1 hyperfine multiplet) the spin-dependent interactionis small relative to the spin-independent interaction (c0 102|c1|); however, for generality we explore larger valuesof up to c1/c0 = 0.5, which might be realizable with newspecies of atoms, or using magnetic or optical manipulation ofinteratomic interactions.

A. Identification of phasesFor the system we consider here of a spin-1 Bose gas subject

to a magnetic field, a variety of phases arise and are wellcharacterized for the T = 0 case (see Fig. 1). These phasesare identified according to the properties of the condensateorder parameter (1,0,1). Here, we briefly summarize thedefining characteristics of each phase and discuss how weidentify these phases in our HF calculations (for more detailson the definition and properties of these phases, see Ref. [1]).

Ferromagnetic (FM) phase. The condensate order param-eter is of the form (nc,0,0) for p > 0. In this phase thecondensate is fully magnetized along the direction of theapplied field, i.e.,

F c = 0 and F cz/nc = 1, (23)

where F c = [(F cx )2 + (F cy )2]1/2 is the transverse spin density.Antiferromagnetic (AFM) phase. The condensate order

parameter is of the form (nc1,1,0,nc1,1). In this phasethe condensate is partially magnetized along the direction ofthe applied field, i.e.,

F c = 0 and 0 < F cz/nc < 1. (24)

Polar (P) phase. The condensate order parameter is of theform (0,nc,0). In this phase the condensate is unmagnetized,i.e.,

F c = 0 and F cz/nc = 0. (25)

Broken-axisymmetry (BA) phase. The condensate orderparameter is of the form (nc1,1,nc0,0,nc1,1) (see Ap-pendix C 1 and Ref. [1] for more details). In this phase

053611-3


AFM

FM

FIG. 2. (Color) Results of the HF calculation for antiferromag-netic interactions with c1/c0 = 0.05. (a) Temperature dependence ofthe phase diagram in (q,p) space, where the FM-P and AFM-P phaseboundaries are independent of temperature. The region of the AFMphase shrinks as the temperature increases. The longitudinal magne-tization per atom of (b) the condensate and (c) the noncondensate atT/T0 = 0.1. The transverse magnetizations are always zero for bothcondensed and noncondensed atoms.

the condensate is partially magnetized but tilts against thedirection of the applied field, i.e.,

F c > 0. (26)We use the conditions (23)(26) to identify the phase of

any self-consistent solution we obtain to the HF equations.Obtaining precise equality is not possible in finite-precisionnumerical calculations, and in practice we identify each phasewhen the appropriate equality (or inequality) is satisfied toone part in 104 (e.g., we identify the ferromagnetic phase byrequiring F cz /nc 0.9999).

B. Antiferromagnetic interactions1. Numerical results

The results for c1/c0 = 0.05 are summarized in Fig. 2.Figure 2(a) shows the temperature dependence of the q-pphase diagram. The region of the P phase is unchanged,whereas the AFMFM phase boundary moves downward asthe temperature increases. Figures 2(b) and 2(c) are plots of thelongitudinal magnetizations of condensate and noncondensate,respectively, at T/T0 = 0.1. When the condensate is in theP phase, the noncondensate is magnetized in the z directiondue to the linear Zeeman effect. On the other hand, whenthe condensate is magnetized in the z direction (i.e., in theFM and AFM phases), the noncondensate is magnetizedantiparallel to the condensate. This is because the condensate

mainly occupies the lowest Zeeman sublevel (i = 1) in thesephases, and therefore the residual noncondensate atoms preferto populate the other spin states. This can be understood asfollows: The noncondensed atoms in spin states different fromthe condensate interact with the condensate only via the direct(Hartree) term; in contrast, it is of higher energetic cost fornoncondensate atoms to occupy the same spin state as thecondensate because both the direct (Hartree) and exchange(Fock) terms contribute.

2. AFM-FM phase boundaryHere, we focus on the temperature dependence of the linear

Zeeman energypb that specifies the AFM-FM phase boundary.The order parameter for the AFM phase is given by

1

0

1

=

nc1,10

nc1,1

, (27)

where we can choose 1 as positive real numbers withoutloss of generality, because the phases of 1 can be removedby a gauge transformation and a spin rotation about the zaxis. In other words, both the gauge transformation and spinrotation symmetries are spontaneously broken in the AFMphase. Sincencij has the off-diagonal elementsnc1,1 = nc1,1 =

nc1,1nc1,1, n

ncij , in general, has the off-diagonal components

nnc =

nnc1,1 0(nnc1,1

)0 nnc0,0 0nnc1,1 0 nnc1,1

. (28)

The generalized GPE (16) for the AFM phase reduces to(

p Cnnc1,1C(nnc1,1

)p

)(1

1

)= 0, (29)

where

= (q + c0n + c1nnc0,0 + C+

nnc1,1 + nnc1,12

), (30)

p = p c1F cz c0 + 3c1

2F ncz , (31)

C = c0 c1, (32)

with n = nc + nnc. At T = 0, Eq. (29) has an AFM solution(1 = 0) when p = 0, that is, p = c1F cz . From the fact thatF cz = nc at the AFM-FM phase boundary, pb at T = 0 is givenby

pb

c1n= 1. (33)

At T = 0, the condition that Eq. (29) has a nontrivialsolution determines . Substituting and the solution of(1,1) into the HF equations and solving self-consistently,we obtain the following relation among p, F cz , and F ncz in the

053611-4


AFM phase:

p = 3C+ 2C4

F cz +4C+ 3C

4F ncz

12

(C+F cz CF ncz

2

)2+ C2F cz F ncz . (34)

The detailed derivation of Eq. (34) is given in Appendix B.Using the fact that F cz = nc at the AFM-FM phase boundary,and expanding Eq. (34) in terms of F ncz /nc, the phase boundaryis approximated as

pb

c1n= n

c

n+ 3c0 + c1

c0 + c1F ncz

n. (35)

The right-hand side of Eq. (35) goes to unity as T 0, beingconsistent with Eq. (33). The first term on the right-hand side ofEq. (35) describes the shift in the boundary due to the thermaldepletion of the condensate, while the second term describesthe interaction of the noncondensed component back on thecondensate and acts to reduce the value of pb since F ncz < 0[see Fig. 2(c)].

This result can be understood in terms of two underlyingeffects that compete against each other:

(i) The noncondensate magnetization F ncz (< 0) increasesthe effective linear Zeeman energy [see Eq. (31)], i.e., increasesthe energy difference between the i = 1 and 1 componentsof the condensate [see Eq. (29)]. This causes |1| to increaserelative to |1|, and thus tends to reduce the value of pb atthe phase boundary (where 1 = 0).

(ii) The noncondensate spin coherence plays a nontrivialrole through exchange (Fock) collisions between condensateand noncondensate atoms of the type (i,0) + (j,k) (i,k) +(j,0): Off-diagonal elements of nncij contribute to enhancingthe coupling between 1 and 1 [see Eq. (29)], thus actingto make |1| and |1| more similar, and hence supporting theAFM phase (i.e., this effect tends to increase pb).

To quantify the competition between these two effects weneglect the noncondensate spin coherence by explicitly settingnnc1,1 = 0 in Eq. (28) and calculate pb. In such a case, Eq. (29)has an AFM solution when p = 0, resulting in

pb

c1n= n

c

n+ c0 + 3c1

2c1F ncz

n. (36)

The larger prefactor of the last term demonstrates that whennoncondensate spin coherence is neglected [i.e., only effect(i) contributes] the phase boundary pb is more significantlyreduced.

Figure 3 shows the temperature dependence of pb for aparticular choice of the quadratic Zeeman energy (q = 3c1n)obtained by the full HF calculation (I) and the HF calculationwith the off-diagonal elements of nncij neglected (II),2 whichshow good agreement with Eqs. (35) and (36), respectively.The deviations of the curves I and II from pb/(c1n) = nc/nare the effects of the thermal components (F ncz /n). Notethat the value of pb significantly decreases when we neglect

2Irrespective of whether off-diagonal parts of nncij arise, we includeonly the diagonal parts when evaluating the self-consistent HFHamiltonian [see Eq. (20)].

FIG. 3. (Color online) Temperature dependence of the AFM-FMphase boundary pb at q = 3c1n and c1/c0 = 0.05 obtained by (I)the full HF calculation and (II) the HF calculation but neglectingthe off-diagonal elements of nncij , together with the curves indicatingnc/n, Eq. (35), and Eq. (36).

the spin coherence of the noncondensate. We find that thetemperature dependence of the condensate fraction nc/n andthe noncondensate magnetization F ncz are almost the same for Iand II, so the difference in the phase boundaries arises from thecoefficients of F ncz /n in Eqs. (35) and (36). For the case of thefull HF calculation [Eq. (35)], pb is insensitive to the value ofc1 as long as c1/c0 1. On the other hand, Eq. (36) is stronglydependent on c1/c0, in particular when c1/c0 is small. We havealso numerically calculated pb for the interaction parametersof c1/c0 = 0.005 and 0.5. The results agree with Eqs. (35) and(36).

C. Ferromagnetic interactions1. Numerical results

The numerical results for c1/c0 = 0.05 are summarizedin Fig. 4. Figure 4(a) shows the temperature dependence of theq-p phase diagram. The region of the FM phase is unchanged,whereas the BA-P phase boundary moves to the left-handside as the temperature increases. Figures 4(b) and 4(c) areplots of the longitudinal and transverse magnetizations ofcondensate atoms, respectively, at T/T0 = 0.1, and Figs. 4(d)and 4(e) show the same quantities for the noncondensate. InFig. 4(e), F nc < 0 means that the transverse magnetization ofthe noncondensate is antiparallel to that of the condensate.As in the case of the AFM and FM phases of Fig. 2,the noncondensate magnetization is roughly antiparallel tothat of the condensate, except for the vicinity of the BA-Pphase boundary where the condensate magnetization becomessmall.

2. BA-P phase boundaryWe investigate the temperature dependence of the BA-P

phase boundary qb at p = 0. In the BA phase at p = 0 thecondensate magnetization is purely transverse and vanishes atq = qb. Note that the numerical result [Fig. 4(e)] shows that thenoncondensed component is also magnetized in the transversedirection (see also Ref. [16]), indicating the existence of spincoherence in the noncondensate. This is because the spinrotational symmetry about the z axis is broken in the HF

053611-5


FM

FIG. 4. (Color) Results of the HF calculation for ferromagneticinteractions with c1/c0 = 0.05. (a) Temperature dependence of thephase diagram in (q,p) space, where the FM-BA phase boundary isindependent of temperature. The region of the BA phase shrinks as thetemperature increases. The longitudinal and transverse magnetizationper atom at T/T0 = 0.1 for (b),(c) the condensate and (d),(e) the non-condensate. In (e), F nc < 0 means that the transverse magnetizationof the noncondensate is antiparallel to that of the condensate.

Hamiltonian (11) due to the existence of the transverselymagnetized condensate.

At T = 0, the BA-P phase boundary is given by [1]qb

|c1|n = 2. (37)

At finite temperature, by solving the Gross-Pitaevskii and HFequations self-consistently, we obtain the following relationfor the BA-P boundary (see Appendix C 1 for the derivation):

qb

|c1|n= 2n

c

n 4(3c0 5|c1|)

c0 |c1|dnc

n, (38)

FIG. 5. (Color online) Temperature dependence of the BA-Pphase boundary qb at p = 0 and c1/c0 = 0.05 obtained by (I)the full HF calculation and (II) the HF calculation but neglectingthe off-diagonal elements of nncij , together with the curves indicating2nc/n, Eq. (38), and Eq. (40).

where

dnc = 12(nnc1,1 nnc0,0 + nnc1,1

). (39)

As in the case of antiferromagnetic interactions, thenoncondensate spin coherence has a significant effect on thelocation of the phase boundary. If we neglect the off-diagonalelements of nncij , the phase boundary is changed to

qb

|c1|n = 2nc

n c0 + |c1||c1|

dnc

n, (40)

where dnc is defined in Eq. (39) but with nnc1,1 = 0. Thederivation of Eq. (40) is given in Appendix C 2.

Figure 5 shows the temperature dependence of qb at p = 0obtained by the full HF calculation (I) and the HF calculationwith the off-diagonal elements of nncij neglected (II), whichshow good agreement with Eqs. (38) and (40), respectively.The deviations of the curves I and II from qb/(|c1|n) = 2nc/nare the effects of the noncondensate (dnc/n). As in the caseof Fig. 3, qb is greatly suppressed when we neglect thecoherence of the noncondensate. The difference also comesfrom the coefficients of dnc/n in Eqs. (38) and (40): Eq. (38)is insensitive to the value of c1 as long as |c1|/c0 1; whileEq. (40) is strongly dependent on c1/c0, in particular when|c1|/c0 is small. We have also numerically calculated qb forthe interaction parameters of c1/c0 = 0.005 and 0.5. Theresults agree with Eqs. (38) and (40).

The interpretation of the above results is similar to thecase of antiferromagnetic interactions. In Eq. (39) the maincontribution to dnc comes from the population differencebetween i = 1 and 0 components, nnc1,1 nnc0,0 (= nnc1,1 nnc0,0for p = 0), which induces an energy difference betweencondensed atoms in the i = 0 and 1 components via theexchange (Fock) terms [the last terms in the first and secondlines of Eq. (17)]. Hence, dnc contributes to increasing |0|relative to |1|, and thus tends to reduce the value of qb. On theother hand, the off-diagonal elements of nncij , in particular nnc1,0and nnc0,1, compete against this by coupling condensate atomsin i = 0 and 1 states [see Eq. (17)], which acts to balancethe condensate population in these states and strengthen theBA phase (i.e., this effect tends to increase qb).

053611-6


V. CONCLUSIONS AND OUTLOOK

In this work we have formulated a self-consistent HF theoryto characterize the phase diagram of a spin-1 Bose gas atfinite temperature. Numerical results, presented over a wideparameter regime, show that certain phase boundaries changeappreciably with temperature. We have developed analyticalresults that accurately describe these shifts in phase boundariesas a function of the interaction parameters and the propertiesof the noncondensate.

Our treatment includes spin coherence for the nonconden-sate component of the system, which naturally develops viacoherent collisions with the condensate. Our calculations showthat the noncondensate spin coherence is crucial to stabilizingthe AFM and BA phases, in which the spin rotational symmetryis spontaneously broken. Indeed, neglect of spin coherence inthe thermal cloud leads to significant shifts in the locations ofthe phase boundaries from the full HF calculations.

The effect of the thermal fluctuations on the condensateorder is a key prediction that could be explored in experiments.Early measurements made by Miesner and co-workers [4]mapped out parts of the phase diagram using a 23Na con-densate (with antiferromagnetic interactions). In that workthe temperature of the system was estimated to be about100 nK, sufficiently hot that thermal effects should be relevant;however, the authors measured the P-AFM phase boundarywhich we predict to be temperature insensitive [see Fig. 2(a)].Aided by improvements in techniques for measuring spinorgas properties (e.g., see Refs. [7,27]), it should be feasible toprecisely determine the finite-temperature phase diagram inexperiments and compare to our predictions.

It would also be interesting to experimentally investigatethe role of the noncondensate spin coherence. Our resultsshow that a large change in the phase boundary occurs whenthe noncondensate coherence is removed (see Figs. 3 and 5).Given the large difference in the decoherence times for thecondensate and noncondensate spin coherence [27], it maybe possible to use external fields to reduce (or remove) thespin coherence of the noncondensate, yet leave that of thecondensate intact. In the vicinity of the phase boundary thiscould allow the condensate to exist in a metastable state whichwould transition to a new phase as the noncondensate spincoherence is eventually reestablished.

On the theoretical front many challenges and opportunitiesexist for extending our understanding of spinor gases beyondthe HF approximation. A natural extension is to developa quasiparticle-based mean-field theory such as the HF-Bogoliubov-Popov formalism [12,16]. In Ref. [16] we appliedthis theory to compute the BA-P phase boundary as a functionof temperature for p = 0 and the parameters of 87Rb. Thepredictions of Ref. [16] are quantitatively similar to the HFresults we present here, with the notable exception of theT 0 limit where we have found that the quantum depletion(excluded in the HF theory) acts to increase qb to a value greaterthan 2|c1|nc. An alternative direction is the use of classicalfield techniques [28], which, within their regime of validity,will provide a dynamical description of the finite-temperaturespinor system, and have already seen some initial applicationsto quasi-two-dimensional spinor gases [29]. Another avenuefor consideration is the inclusion of dipole-dipole interactions

between atoms into the finite-temperature description (e.g.,see [30]). These long-range interactions have been predictedto give rise to interesting new features in the ground-state phasediagram [31], and are thought to be important for explainingsome of the observations in the 87Rb spinor gas [8].

ACKNOWLEDGMENTS

Y.K. and N.T.P. were supported by KAKENHI (Grants No.22340114, No. 22740265, and No. 22103005), a Global COEProgram Physical Sciences Frontier, and the Photon FrontierNetwork Program from MEXT of Japan, and by JSPS andFRST under the JapanNew Zealand Research CooperativeProgram. P.B.B. was supported by Marsden Contract No.UOO0924 and FRST IIOF Contract No. UOOX0915.

APPENDIX A: HARTREE-FOCK THEORY ANDTHERMODYNAMIC PARAMETERS

The HF theory can be derived by assuming that the many-body density matrix is given by

D0 = 1Z0

e KHF , (A1)

where Z0 = Tr{e KHF} and KHF =

dr

i,ji Aij

j is theassumed single-particle form for the HF Hamiltonian. Thevariational principle applied to determine KHF (or, equiva-lently, Aij ) is that D0 makes the thermodynamic potential

(D) stationary, where

(D) = Tr{kBTD ln D + D H D N}, (A2)

with N being the number operator. This procedure gives theform of the Gross-Pitaevskii and HF equations used in thispaper [i.e., Eqs. (5), (11), and (12)].

In terms of the self-consistent solution of the HF equationsthermodynamic parameters can be evaluated. The HF energyis given by

EHF = Tr{D0 H} (A3)

=

dr

{j

[j (h0)jjj +

nuj (h0)jjuj

]

+ c02

[(nc + nnc)2 +

ij

nncij(2ncji + nncji

)]

+

c1

2

[(F c + F nc

)2

+ijkl

(f)ij (f)kl nnckj(2ncil + nncil

)]}, (A4)

and by evaluating Eq. (A2), using the self-consistently deter-mined HF density matrix, the free energy of the HF solution

053611-7


(HF) can be determined. Equivalently, it can be evaluatedas

HF = EHF N T SHF, (A5)where the entropy is

SHF = kB

[n ln n (1 + n) ln(1 + n)]. (A6)

APPENDIX B: DERIVATION OF EQ. (35)In this and the following appendixes, we use boldface

quantities to represent matrix quantities for notational effi-ciency, for example, ncij nc, nncij nnc, and ij 1. Wealso introduce dc nc1,1 = nc1,1, dnc nnc1,1, nci ncii , andnnci nncii .

We start from Eq. (29). From the condition that Eq. (29)has a nontrivial solution, is obtained as

=

p2 + C2|dnc|2. (B1)

Choosing the lower chemical potential, the order parameter isgiven by

1 =

nc

2

(1 + p

), (B2a)

1 = ei

nc

2

(1 p

), (B2b)

where

=

p2 + C2|dnc|2, (B3)

= arg(dnc). (B4)Since we have chosen 1 to be positive real numbers, dncis a negative number ( = ). From Eq. (B2), we obtain therelation between the condensate spin density and dnc:

dc

F cz= 11|1|2 |1|2 =

Cdnc

2p. (B5)

Next, by substituting Eqs. (27) and (28) into Eq. (20), weobtain

A = ( + c0n)1 +

p + c1Fz + q 0 00 0 00 0 p c1Fz + q

+

C+n1 + c1n0 0 Cd0 c0n0 + c1(n1 + n1) 0

Cd 0 C+n1 + c1n0

,(B6)

where ni = nci + nnci , Fz = F cz + F ncz , and d = dc + dnc. Theeigenvalue for the i = 0 component is immediately obtainedas

0 = c0n + c0n0 + c1(n1 + n1) . (B7)For the i = 1 components, we need to diagonalize thefollowing 2 2 matrix:

A =[

+ c1nc0 +C+2(nc1 + nc1

)]1

+(p + C+F cz /2 Cd

Cd p C+F cz /2). (B8)

This matrix is almost the same as Eq. (29), and the eigenvaluesand eigenvectors are given by

= + c1nc0 +C+2(nc1 + nc1

) , (B9)(u

()1

u()1

)= 1

2

( (p C+F cz /2)

sgn(d) (p C+F cz /2)), (B10)

where

(

p C+F cz /2)2 + C2d2. (B11)

Since nnc1 and dnc are self-consistently determined so asto satisfy Eq. (22), we obtain the relation between F ncz

and dnc:

dnc

F ncz=

= u

()1 u

()1 Li3/2(e )

=[u

()1 u

()1 u()1 u()1

]Li3/2(e )

= Cd2p C+F cz

. (B12)

Equations (B5) and (B12) are rewritten as a linear equationof dc and dnc:( 2p CF cz

CF ncz 2p C+F cz + CF ncz

)(dc

dnc

)= 0. (B13)

In order for dc and dnc to have a nontrivial solution, p, F cz ,and F ncz have to satisfy

2p(2p C+F cz + CF ncz

) C2F cz F ncz = 0. (B14)Solving Eq. (B14) in terms of p, we obtain Eq. (34), where wehave chosen the sign in front of the square-root term so thatEq. (34) continuously goes to the solution at T = 0.

APPENDIX C: DERIVATION OF EQS. (38) and (40)In the BA phase at p = 0 the condensate is magnetized in

the transverse direction. Without loss of generality, we canchoose the direction of the magnetization in the x direction,i.e., the magnetic field is applied in the z direction and

053611-8


spontaneous magnetization arises in the x direction. We thenmove to the frame of reference which is rotated around the yaxis by /2. In this frame of reference, the magnetic field isapplied in the x direction and the magnetization arises in thez direction. In this appendix all results are given in this frameof reference unless specified otherwise. The magnetic subleveli in the rotated frame corresponds to the eigenvalue of fx inthe laboratory frame.

1. Full HF calculationThe matrices L and A in the rotated frame are given by

L = qf 2x + c0[n1 + (nnc)T]+ c1

[Ff + f(nnc)Tf], (C1)

A = L 1 + c0(nnc)T + c1

f(nc)Tf. (C2)Since the matrix elements of f 2x are given by

f 2x =

1/2 0 1/20 1 0

1/2 0 1/2

, (C3)

we can assume that i = 0 and i = 1 components aredecoupled:

nc =

nc1 0 dc

0 0 0dc 0 nc1

, nnc =

nnc1 0 (dnc)0 nnc0 0dnc 0 nnc1

. (C4)

The order parameter for the BA phase in the laboratory frame isgiven by

nc/2(a,2b,a)T (a,b R,a2 + b2 = 1,0 a

1/

2) [1], which is transformed in the rotated frame asnc/2(a + b,0,a b)T. It follows that dc is always negative

in the BA phase because 0 a 1/

2 b 1. Whenthe system is in the polar phase (a = 0,b = 1), we havedc = nc/2.

From Eq. (C1), the i = 1 components should satisfy(p q/2 + Cdnc

q/2 + C(dnc) p )(

1

1

)= 0, (C5)

where

= (q/2 + c0n + c1nnc0 + C+

nnc1 + nnc12

), (C6)

p = c1F cz +c0 + 3c1

2F ncz , (C7)

and C are defined in Eqs. (32). From the condition thatEq. (C5) has a nontrivial solution, is determined as

=

p2 +q2 + Cdnc

2. (C8)

Choosing the lower chemical potential, the order parameter isgiven by

1 =

nc

2

(1 p

), (C9a)

1 = ei

nc

2

(1 + p

), (C9b)

where

p2 +(q

2+ Cdnc

)2, (C10)

arg(q

2+ Cdnc

). (C11)

Since 1 is assumed to be a negative real number, has to bezero, that is, dnc is real and satisfies q/2 + Cdnc > 0. FromEq. (C9), we obtain the relation between F cz and dc:

dc

F cz= q/2 + Cd

nc

(C+ C)F cz + (2C+ C)F ncz. (C12)

Next, we consider the equation for the noncondensate part.The matrix A is given by

A = ( + c0n)1 +

c1Fz + q/2 0 q/20 q 0

q/2 0 c1Fz + q/2

+

C+n1 + c1n0 0 Cd0 c0n0 + c1(n1 + n1) 0

Cd 0 C+n1 + c1n0

.

(C13)For the i = 1 components, we need to diagonalize the 2 2 matrix:

A =(

+ c1nc0 + C+nc1 + nc1

2

)1 +

(p + C+F cz /2 q/2 + Cdq/2 + Cd p C+F cz /2

). (C14)

The eigenvalues and eigenvectors of A are given by

= + c1nc0 + C+nc1 + nc1

2 , (C15)(

u()1

u()1

)= 1

2

( (p + C+F cz /2)

ei (p + C+F cz /2)), (C16)

where

(

p + C+F cz

2

)2+(q

2+ Cd

)2, (C17)

arg(q

2+ Cd

). (C18)

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Since nnc1 and dnc are self-consistently determined so as tosatisfy Eq. (22), we obtain the relation between F ncz and dnc as

dnc

F ncz=

= u

()1 u

()1 Li3/2(e )

=[u

()1 u

()1 u()1 u()1

]Li3/2(e )

= q/2 + Cd(2C+ C)Fz . (C19)

Equations (C12) and (C19) are rewritten as a linear equationfor F cz and F ncz :(2c1dc Cdnc q/2 (2C+ C)dc

(2C+ C)dnc Cdc 4c1dnc + q/2)

(

F cz

F ncz

)= 0. (C20)

From the condition that F cz and F ncz have a non-trivial solution,we obtain

q = 2(C+ C)dc(

1 + 4C+ CC+

dnc

dc

)(C21)

= 4c1dc(

1 + 3c0 + 5c1c0 + c1

dnc

dc

), (C22)

where we have expanded q to first order in the parameterdnc/dc. Since dc = nc/2 at the BA-P boundary, we obtainthe boundary qb as Eq. (38).

In the laboratory frame, nnc is related to that in the rotatedframe as

nnc(lab) = eify/2nnceify/2, (C23)from which dnc is rewritten in terms of nnc in the laboratoryframe as

dnc nnc1,1 = 12(n

nc(lab)1,1 + nnc(lab)1,1 nnc(lab)0,0

). (C24)

2. Neglecting the off-diagonal partWhen we neglect the off-diagonal part of nnc in the

laboratory frame,

nnc(lab) =

nnc(lab)1 0 00 nnc(lab)0 00 0 nnc(lab)1

, (C25)

the noncondensed component has no transverse magnetization,which means F ncz = 0, i.e., nnc1 = nnc1, in the rotated frame.Hence, the calculation for the condensate part is the sameas that for the full HF calculation if we impose nnc1 = nnc1.Equation (C12) then reduces to

dc

F cz= q/2 + Cd

nc

(C+ C)F cz, (C26)

from which we obtain Eq. (40).

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