Finite temperature phase diagram of spin-1/2 bosons in two-dimensional optical lattice

0.33 MB·English

by  L. de Forges de Parny

Finite temperature phase diagram of spin-1/2 bosons in two-dimensional optical lattice
L. de Forges de Parny1, F. H´ebert1, V.G. Rousseau2, and G.G. Batrouni1,3,4
1INLN, Universit´e de Nice-Sophia Antipolis, CNRS; 1361 route des Lucioles, 06560 Valbonne, France,
2Department of Physics and Astronomy, Louisiana State University, Bˆaton Rouge, Louisiana 70803, USA,
3Institut Universitaire de France and
4Centre for Quantum Technologies, National University of Singapore; 2 Science Drive 3 Singapore 117542.
Westudya two-species bosonic Hubbardmodel on a two-dimensional squarelattice bymeans of
quantumMonteCarlosimulationsandfocusonfinitetemperatureeffects. Weshowintwodifferent
2
cases, ferro- and antiferromagnetic spin-spin interactions, that the phase diagram is composed of
1
a superfluid phase and an unordered phase that can be separated into weakly compressible Mott
0
2 insulators regions and compressible Bose liquid regions. The superfluid-liquid transitions are of
the Berezinsky-Kosterlitz-Thouless type whereas the insulator-liquid passages are crossovers. We
n
analyse the pseudo-spin correlations that are present in the different phases, focusing particularly
a
on the existenceof a polarization in this system.
J

4 PACSnumbers: 05.30.Jp,03.75.Hh,67.40.Kh,75.10.Jm03.75.Mn

]
s
a I. INTRODUCTION simpler spin-1/2 bosonic system takes us a step toward
g understanding the more elaborate, and more difficult to
- simulate, models.
t The study of strongly interacting quantum models by
n
direct realization of an experimental system reproduc- The spin-1/2 model has been extensively studied with
a
u ing the model properties, an idea proposed by Feynman mean-field theory (MFT) at zero or finite temperatures
q [1], was realized in the past ten years with the produc- and, in previous work, we explored its zero temperature
. tion of Bose-Einstein condensates (BEC) and their use behaviorwithquantumMonteCarlo(QMC)simulations
t
a as “quantum simulators” [2]. In particular, BEC in con- in one [17] and two [18] dimensions for on-site repulsive
m
junctionwithopticallatticesareusedtoproducesystems interactions. At zero temperature, the phase diagrams
- reproducing the physics of well known quantum statisti- obtainedinoneandtwodimensions,withMFTorQMC,
d
caldiscretemodelssuchasfermionicorbosonicHubbard aresimilar. Generallyspeaking,atzerotemperature,the
n
models. system can adopt two different kinds of phases: insulat-
o
c Used to study simple models of bosons [3] or fermions ing Mott phases that appear for integer density, ρ, and
[ [4] at low temperature, the flexibility offered by these for large enough repulsion between particles and super-

systems extends the range of interesting models to more fluidphases(SF)otherwise. Thedetailednatureofthese
2
exotic ones which can be treated both experimentally phases depends on the interactions between the differ-
v
5 and theoretically. Examples include systems with long ent kinds of particles. In the case where the repulsion
4 range interactions [5], fermions with imbalanced popula- between identical particles is smaller than between dif-
0 tions [6], mixtures of different kinds of particles [7] and ferent particles (U2 >0 in our previous work [18] and in
3 spin-1bosonswithspin-independentandspin-dependent thefollowing)thesuperfluidisfoundtobepolarized,that
.
9 interactions which allow interplay between superfluid- is, an imbalance develops in the populations of the two
0 ity and magnetism [8–10]. Furthermore, it is possible kinds of particles and one of the species becomes dom-
1 to study systems of bosons with two effective internal inant. The ρ = 1 Mott phase is also polarized whereas
1
degrees of freedom on an optical lattice, the so-called the ρ=2 phase is not(we did notstudy higherdensities
:
v “spin-1/2 bosons”. Such a system, with spin-dependent with QMC). In the opposite case, U2 <0, all the phases
i interactions, could be produced by applying a periodic are unpolarized. Noteworthy is the presence of coherent
X
optical lattice on a bosonic system with two triply de- exchangemovements[19],wheretwoparticlesofdifferent
r
a generateinternalenergylevels. Theopticalpotentialap- types exchange their position in the ρ = 1 Mott phase
plied would localise the atoms at the nodes of a regular in both cases, as well as in the ρ = 2 Mott phase for
network, but would also couple the internal states by Λ U2 < 0. Finally, in one dimension, all zero temperature
orVvirtualprocesses,thusleavingonlytwointernallow phase transitions were found to be continuous whereas
energy degenerated states denoted 0 and Λ and realiz- in two dimensions and when U2 >0 is small enough, the
ing an effective spin-1/2 model [11, 12]. The presence of ρ = 2 Mott-superfluid transition was predicted by MFT
the spin-dependent interaction introduces a term in the to be first order near the tip of the Mott lobe and con-
Hamiltonian which permits the conversion of two parti- tinuous otherwise, whereas for larger U2 the transition
cles of one type into the other type and renders numeri- was predicted to be always continuous. This was con-
calsimulations more difficult. This model is related, but firmed by QMC simulations [18]. Related spin-1 models
not identical, to other models including p-band super- were studied using MFT [20, 21] or QMC in one dimen-
fluid models [13–15] and the bosonic Kondo model [16]. sion [10] and a similar spin 1/2 bosons model was also
Understanding the phase diagram and properties of the recently studied [22].
2
Inthispaper,wewillstudythespin-1/2modelatfinite A. Mean Field Theory
temperature in two dimensions and compare with MFT
predictions. The results for finite system sizes and tem- TheonlytermthatcouplesdifferentsitesintheHamil-
peratures are relevant to experimental efforts to study tonian is the hopping term (1). Introducing the field
tahnidstshyestMemF.TIanndSeQctMioCntIeIc,hwneiqwueilsluinsetdrotdouscteudthyeitm. oSdece-l oψpσe=rathoar†σsr′oin=shitaeσrr′′i,bwyetrheepilracmeetahnecvraelauteisonψ/d,efsotrlluocwtiinong
σ
tionIII andIVwillbedevotedtothepresentationofthe the approach used in [11]. The Hamiltonian on site r is
resultsobtainedfortheU2 >0andU2 <0cases,respec- then decoupled from neighboring sites and can be easily
tively. We will summarize these results and give some
diagonalized numerically in a finite basis. The optimal
final remarks in Section IV.
value of the fields ψ are then chosen by minimizing the
σ
grand canonical potential G = −kTlnQ with respect to
ψ where Q is the grand canonical partition function.
σ
The system is in a superfluid phase when ψ is nonzero,
σ
with superfluid density ρs = |ψ0|2+|ψΛ|2, and is other-
II. SPIN-1/2 MODEL wise in an unordered phase where two cases can be dis-
tinguished: an almost incompressible case, i.e. a Mott
insulator, and a compressible case, i.e. a liquid.
The model we will study is the same we previously
In these two cases there is no broken symmetry and
studied in the low temperature limit in one [17] and two
they cannot be distinguished by symmetry considera-
dimensions[18]andpreviouslyintroducedin[11]. Itisan
tions. If there is a first order transition between the
extended Hubbard model governed by the Hamiltonian
MI and the liquid, characterized by discontinuities in
the density or other thermodynamic functions, the MI
and the liquid would be two distinct phases. If, how-
H=−t X (cid:16)a†σraσr′ +a†σr′aσr(cid:17)−µXnˆσr (1) ever, there is no discontinuity in the evolution from the
σ,hr,r′i σ,r MI to the liquid then they are only two limiting cases
U0 of the same unordered phase and there is only a smooth
+ 2 Xnˆσr(nˆσr−1)+(U0+U2)Xnˆ0rnˆΛr (2) crossover between the MI and the liquid regions of the
σ,r r
phase diagram. We shall see that, indeed, there is a
+U22 X(cid:16)a†0ra†0raΛraΛr+a†Λra†Λra0ra0r(cid:17), (3) crossoverin the system we are considering here.
r One can distinguish between almost-incompressible
and liquid regions by calculating the local density vari-
ance which is a measure of the local compressibility,
twyhpeereσo=pe0r,aΛtoornasσitre(rao†σfra)tdweost-droimysen(csiroenaatelss)quaarbeolsaotnticoef κp˜ar=ticβle(cid:0)shnon2ris−itehnrr.i2κ˜(cid:1)iwshcleorseentro zisertoheinttohtealMnoutmtbpehrasoef
ofsizeL×L. Theoperatorsnˆσr measuresthe numberof and much larger in the liquid phase.
particles of type σ onsite r. The densities of particlesof
While this MFT was shownto reproduce qualitatively
type0andΛarecalledρ0 andρΛ whilethe totaldensity the phase diagram at zero temperature [18], it is rather
is called ρ=ρ0+ρΛ. limited at finite temperature. Indeed, whenever the ψ
σ
The first term (1) of the Hamiltonian is the kinetic are zero the hopping parameter t no longer plays a rˆole
term that lets particles hop from site r to its nearest in the MFT. Then, while the MFT can distinguish be-
neighbours r′. The associated hopping energy t=1 sets tweenSFandunorderedphases,itdoesnotcorrectlydis-
the energy scale. A chemical potential µ is added if one tinguish Mott Insulator (MI) regions from normal Bose
worksinthegrandcanonicalensemble. Thesecondterm liquidsones,asthecrossoverboundarybetweenthosere-
(2)describeson-siterepulsionbetweenidenticalparticles gionswillnotdepend ont andwillbe the same asin the
with a strength U0 or between different particles with a t=0 case.
strength U0 +U2. We will study both the positive and
negative U2 cases but will keep only repulsive interac-
tions, that is |U2| < U0, and a fixed moderate value of B. Quantum Monte Carlo simulations
|U2|/U0 =0.1. Thelastterm(3)providesapossibilityto
change the “spins” of the particles: When two identical To simulate this system, we used the stochastic Green
particles are on the same site, they can be transformed functionalgorithm(SGF)[23],anexactQuantumMonte
intotwoparticlesoftheothertype. Itwasdemonstrated Carlo (QMC) technique that allows canonical or grand
in [11] that the matrix element associatedwith this con- canonical simulations of the system at finite tempera-
versionisU2/2. Weareusingadifferentsignfortheterm tures as well as measurements of many-particle Green
(3) compared to the articles where the model was orig- functions. In particular, this algorithm can simulate ef-
inally introduced [11] but we have shown in a previous ficiently the spin-flip term in the Hamiltonian. We stud-
work [17] that this sign can indeed be chosen freely due ied sizes up to L = 14. The density ρ is conserved in
to a symmetry of the model. canonicalsimulations,butindividualdensitiesρ0 andρΛ
3
fluctuateduetotheconversiontermEq. (3). Thesuper- Sr+ = a†0raΛr, and Sr− = a†Λra0r the Hamiltonian takes
fluid densityis givenbyfluctuations ofthe totalwinding the form
number, (W0 +WΛ), of the world lines of the particles
[24] H = Kˆ + U20 Xnˆr(nˆr−1)+ U42 Xnˆ2r (7)
r r
ρs = h(W04+tβWΛ)2i. (4) + U2X(cid:0)(Srx)2−[(Sry)2+(Srz)2](cid:1) (8)
r
The superfluid density cannot be measured separately
where Kˆ is the hopping term Eq. (1). The terms in (7)
for 0 and Λ particles due to the conversionterm [25]. It,
are invariant under spin rotations. On the other hand,
thereforedependsonthetotaldensitynotontheseparate
term(8)favorspseudo-spincorrelationstodevelopalong
densities of the two species. We also calculate the one
particle Green functions the x axis if U2 < 0 or in the yz plane for U2 > 0. We
note that the total spin S2r = sr(sr+1) on a given site
1 is not fixedbut depends onthe totalnumber ofparticles
G (R)= ha† a +a† a i, (5)
σ 2L2 X σr+R σr σr σr+R on the site (sr =nr/2) and will then fluctuate with this
r
number.
with σ = 0,Λ. G (R) measures the phase coherence of An order along z is measured through the densities
σ
or through density-density correlations of 0 or Λ parti-
individual particles.
cles, i.e. it corresponds to the polarization of the sys-
In a strongly correlated system it is useful to study
tem. An order along the x or y axes is exposed through
correlated movements of particles which can be done,
for example, by studying two-particle Green functions. the behavior of Ga which, in terms of spins, is equal to
Wefound[18]thatanticorrelatedmovementsofparticles Ga(R) = rhSrxSrx+R +SrySry+Ri/L2. Our QMC algo-
governthedynamicsofparticlesinsideMottlobes,asthe rithm allowPs the calculation of Ga(R) but does not give
access to correlations along the x and y axes indepen-
particles of different types exchange their positions. The
dently.
two-particles anti-correlatedGreen function
Weremarkthatthey andz axishavethesamebehav-
1 ior, which means that we expect a spin QLRO in the yz
Ga(R) = 2L2 XDa†Λra†0r+Ra0raΛr+R+H.c.E, (6) plane to appear at low enough temperature for U2 > 0
r
in addition to the expected QLRO of the global phase
measures the phase coherence of such exchange move- of the particles discussed earlier. On our finite size sys-
ments as a function of distance. Due to its defini- tems, this means that we should simultaneously observe
tion, Ga cannot be larger that ρ0ρΛ and is equal to a polarization and a QLRO for Ga. We will call such
Ga(R)=G0(R)GΛ(R)ifthereisnocorrelationbetween a phenomenon a “quasi-polarization” (QP). For U2 > 0
the movements of 0 and Λ particles. and finite T, we expect a QLRO to develop along the x
In two dimensions, at low but finite temperatures, we axisand,consequently,nopolarizationbutstillaplateau
expect,insomecases,toobserveaBerezinsky-Kosterlitz- at long distances in the function Ga.
Thouless(BKT)phasetransitionandthedifferentGreen
functionstoadoptapowerlawbehavioratlargedistance
R, G(R) ∝ R−η, characteristic of the appearance of a III. U2 >0 CASE
quasi long range order (QLRO) in the phase, long range
order(LRO)beingachievedonlyatT =0(η varyingbe- At zero temperature, the results obtained with QMC
tween1/4and0 asT is lowered)[26]. Inother words,at and MFT were in good qualitative agreement [18]. The
finiteT wecanexpectasuperfluidwherethephaseisstiff phase diagram, studied for densities up to two, exhibits
but not orderedand not a BEC whith anorderedphase. three phases. The first two phases are incompressible
At high temperature, the Green functions are of course Mott phases obtained for integer densities ρ = 1 and
expected to decay exponentially. On the finite size sys- ρ=2 forlargeenoughinteractionsU0. Atzerotempera-
temswhichwestudy(L≤14),itisdifficulttodistinguish ture,theentireρ=1Mottphaseispolarized,thatisthe
the QLRO from a true LRO, whereas one can easily dis- system sustains a spontaneous symmetry breaking and
tinguish between the QLRO and exponentially decreas- the density of one type of particles becomes dominant.
ingregimes. ThisdifficultyofdistinguishingQLROfrom This polarizationcanbe understoodinthe frameworkof
LRO is also encountered in experiments where the sizes an effective spin-1/2 model [19] as the coupling in the
of systems that can be studied are typically of the same yz plane is stronger than along the x axis. As expected,
orderasinourQMCsimulations(hundredsofparticles). exchange movements of particles coexist with polariza-
The finite size results are, therefore, directly relevant to tion in this phase and Ga develop a long range phase
experiments. coherence. The ρ = 2 Mott phase is unpolarized and
To elucidate the properties the model, we formulate show no sign of exchange moves. In terms of spin, this
it in terms of spins using a Schwinger bosons approach correspondsto the factthat,neglectingthe kinetic term,
[27]. Defining the spin operators Srz = (nˆ0r−nˆΛr)/2, the ground state for a given site is uniquely determined
4
2 boundary between the MI and the NBL shown in phase
L=8, kT=0.006 U
0 diagrams, we use the following criterion: when the den-
L=10, kT=0.006 U
0 sity deviates by 1% from the total integer density we
L=12, kT=0.006 U
1.5 L=8, kT=0.01 U 0 consider that the system is no longer in the MI region
MIρ=2 L=8, kT=0.02 U0 but in the NBL region.
0
L=8, kT=0.03 U0 Figure 1 shows the limits of the MI regions for differ-
0 L=8, kT=0.04 U
U 1 0 ent temperatures. As expected, the MI regions progres-
/
µ sively disappear as the temperature is increased. The
QP-SF or NBL
limits of the superfluid phase are located by direct mea-
U /U =0.1
2 0 surement of ρ . When the system is compressible and
s
0.5
MI ρ=1 has zero superfluid density, there is a normal Bose liq-
QMC
uid. Fig. 2 (top) shows the QMC phase diagram of the
0
0 0.02 0.04 0.06 0.08 0.1
t/U
0 2
QMC, U /U =0.1, kT/U =0.03
2 0 0
FIG.1: (Coloronline)Thelimitsoftheρ=1andρ=2Mott L=12
regions(MI)fordifferentvaluesofthetemperaturekT andfor 1.5 MIρ=2
differentsizesL. Asthetemperatureincreases,theMottlobes
shrink and totally disappear for kT > 0.04U0. Outside the
0
Mott,thesystemcrosses overtoanormalBose liquid(NBL) U
1 QP-SF
and, eventually, transitions to a quasi-polarized superfluid / NBL
µ
(QP-SF) phase (see below). In the limit of zero temperature
(here represented by the low temperature kT/U0 = 0.006),
there is a direct phase transition between a Mott phase and 0.5 MIρ=1
a superfluid.
0
as the state with Srx = 0. The third possible phase is a 0 0.02 0.04 0.06 0.08
t/U
polarizedsuperfluid(SF) andoccursatanyU0 whenthe 0
density is incommensurate and alsoat small U0 for com-
2
mensurate values. It is notpossible to discuss this phase
NBL MFT, U /U =0.1, kT/U =0.03
in terms of a simple effective spin degree of freedom as 2 0 0
the number of particles on a site is not fixed.
At zero temperature, the transition from the ρ = 1 1.5 MI ρ=2
Mott phase to the SF is continuous, whereas at the tip
of the ρ = 2 Mott lobe, the transition to the SF is first U0 SF polar
order for small values of U2/U0 becoming second order / 1 NBL
for larger values. This was predicted by the MFT [11] µ
and confirmed by QMC simulations [18].
0.5 MI ρ=1
A. Phase diagram at T 6=0
NBL
0
To map the phase diagram at finite temperature with 0 0.02 0.04 0.06 0.08
QMC, we determine the limit of the Mott Insulator re- t/U
0
gions by measuring the density as a function of µ and
determining the boundaries of the plateaux indicating
FIG. 2: (Color online) Phase diagram of the system at
the almost incompressible regions. As explained in Sec.
kT = 0.03U0 for U2/U0 = 0.1. The QMC simulations (top)
IIA, althoughthe compressibilityofthese regionsis very
show that a liquid (NBL) region appears between the quasi
small, they are not strictly incompressible due to ther-
polarizedsuperfluid(QP-SF)phaseandtheMottregion. The
mal fluctuations. The evolution from MI to NBL does
MFT (bottom) is not able to reproduce correctly this result
not show any singularity and is then simply a crossover as it does not take into account the kinetic term in the un-
between two different limiting behaviors, incompressible ordered (Mott or NBL) phase. It predicts first order phase
and compressible, of the same unordered phase. Strictly transition between the SF and the other phase at the tip of
speaking, a truly incompressible Mott phase exists only the ρ = 1 and ρ = 2 lobes: the region of coexistence of the
atT =0but,followingconvention,wewillcontinuetore- twophases islimited bythedashed lineand theredlinewith
fertothisfiniteT regionasaMI.Todefinethecrossover squares.
5
0.2 0.2
L=8 U /U =0.1, ρ=1 L=8 U /U =0.1, ρ=2
2 0 2 0
L=10 L=10
0.15 L=12 0.15 L=12
L=12
0 0
U U
/ 0.1 / 0.1 NBL
T NBL T
k k
Superfluid Superfluid
0.05 Quasi Polar 0.05 Quasi Polar
MI MI BEC
0 0
0 MI polar 0.05 0.1 BEC 0.15 0 0.02 0.04 0.06 0.08 0.1
t/U t/U
0 0
FIG.3: (Coloronline)TheQMCphasediagramforρ=1and FIG. 4: (Color online) The QMC phase diagram for ρ = 2
fordifferentsizesshowingthequantumphasetransitionpoint and for different sizes showing the quantum phase transition
betweentheMottinsulatingphase(MI)andthesuperfluidat point between the MI and the SF occurring only at kT =0.
kT =0. The finiteT superfluid phase does not haveBEC in The superfluid phase is always quasi-polarized whereas the
this two-dimensional system. The system is quasi-polarized NBL and MI are not. The SF-NBL transition is continuous,
throughout the superfluid phase, but it is not in the NBL whereas at T = 0 the MI-SF quantum phase transition was
andMI.TheMottphaseispolarized atT =0andshouldbe found to be discontinuous for U2/U0 small enough [18]. The
quasi-polarizedatextremelysmallT. Thetransitionbetween dash line corresponds to thecase studied in Fig. 5.
theSFandtheNBLiscontinuousandthereisonlyacrossover
separating NBL from MI.
we observe that the SF phase appears to be polarizedat
finite T as it is at T = 0: the histogram of the density
system at a constant finite temperature. As the SF and of one of the species shows two peaks at low and high
MI regionsare destroyeddue to thermal fluctuations, an densities (see Fig. 5). However, due to the continuous
intermediate NBL region appears. The MFT used with symmetry in the yz plane of the pseudo-spin part of the
success at T = 0, where it reproduces qualitatively the Hamiltonian Eq. (8), no LRO exists at finite T. There-
phase diagram, is unable to do so at finite temperature fore, this apparent polarization is due to the finite size
as explained in Sec. IIA. Figure 2 (bottom) shows the of the system, and is, in fact, a quasi-polarization. An
MFT phase diagramwherethe aforementionedproblem intuitive way to understand this quasi-polarization is as
clearlyappears: theboundariesbetweentheMIandNBL follows: In the superfluid phase, the pseudo-spins in the
do not depend on the value of t/U0 and the MFT is un- yz plane are stiff and appear to be mostly alignedin the
able to give correct predictions regarding this crossover. same direction on a finite lattice, such as the case here.
However, the boundaries of the SF are reasonably well But since the symmetry is not broken, this “magnetiza-
reproduced. Asurprisingresultisthatthe transitionbe- tion” direction in the yz plane will drift and point in all
tweentheunorderedphaseandtheSFappearstobefirst directions. As the direction changes, so does the projec-
orderatthetipoftheρ=1andρ=2lobesinthisMFT tionofthispseudo-spinonthez-axis. Inotherwords,the
approach. At zero temperature, only ρ = 2 showed a polarizationdriftstoo,andchangeswithtime,givingthe
first order transition. As will be shown below, for finite doublepeakstructuretothepolarizationhistogram,Fig.
temperatures, this is in total contradiction with the re- 5. Wenotethatexperimentalsystemsaretypicallyofthe
sultsobtainedbyQMCsimulationsthatshowcontinuous same sizes as the ones we study here and, therefore, this
phase transitions for ρ=1 and ρ=2. So this MFT pro- polarization drift will be present in these experiments
vides incorrectdescriptionof the phase transitionandof too: The particle content of the system will change as a
thepositionofthedifferentregionsatfinitetemperature. function of time. This quasi-polarization disappears as
For fixed integer density at zero temperature, we have T increases when the system undergoes a thermal BKT
a quantum phase transition (QPT) between the MI and phasetransitionintotheNBL.Inotherwords,the entire
SF phases. As expected [28], when the temperature is SF phase is quasi-polarizedbut the NBL is not (see Fig.
increased from zero, an intermediate compressible un- 5)
orderedregionappearsbetweenthesuperfluidphaseand Ahistogramofthe polarizationintheρ=1MIregion
the Mott region, namely the normal Bose liquid region shows that as soon as the temperature is increased from
(NBL). This is observed for ρ = 1 (Fig. 3) and ρ = 2 zero,thepolarizationdisappearsandthepopulationsbe-
(Fig. 4). come balanced. According to the effective pseudo-spin
Asforthe possiblepolarizationofthedifferentphases, model[18,19],itispossibletoobservequasi-polarization
6
100 (a) ρ=1, t/U=0.02 101 (b) ρ=2, t/U=0.08
0 0
U/U=0.1, L=12 U/U=0.1, L=12
2 0 2 0
10-1
100
G (R)
a kT/U0=0.0004
kT/U=0.0100
0
10-2 kT/U0=0.0200
kT/U=0.0300
kT/U00=0.0400 10-1 kT/U0=0.02
kT/U0=0.0500 kT/U0=0.06
10-3 kkTT//UU00==00..01700000 kkTT//UU00==00..1126
kkTT//UU00==00..11240000 10-2 kkTT//UU00==00..1280
10-4 kkTT//UU00==00..11680000 kkTT//UU00==00..2340
kT/U0=0.2000 kT/U0=0.35
xxxx10-3 kkTT//UU0==00..4405 GΛ(R)
0
FIG. 5: (Color online) Probability P(ρΛ) as a function of 10-5
0 2 4 6 0 2 4 6
temperature for L=8, U2/U0 =0.1, ρ=2 and t/U0 =0.08. R R
WhenkT/U0 <0.175,thesystemisinthequasi-polarizedSF
phase. For kT/U0 > 0.175, the system is in the unordered FIG. 6: (Color online) Evolution of the Green functions Ga,
unpolarized phase. The temperature at which the quasi-
Eq. (6), in theρ=1 MI and GΛ, Eq. (5),in theSFas func-
polarizationdisappearsisapproximatelythesameasthetem-
tions of distance R for different temperatures, with L = 12
perature where ρs becomes zero (see the vertical dashed line and U2/U0 = 0.1. (a) In the ρ = 1 MI, Ga decays exponen-
in Fig. 4). Only theSF phaseis quasi-polarized.
tiallyatfiniteT,unlikeitsT =0behaviorwhereitreachesa
constant value [18]. A regime of QLRO is not observed and
would occur only at very small T. The exponential decay
as in the SF phase. In this case, the coupling generating persistsintheNBLathigherT. (b)GΛ atρ=2forT values
these spin correlations and the polarization of the MI is taking the system from the SF to the NBL. Same case as in
ofordert2/U0. However,evenfortemperaturesaslowas Fig. 5. In the SF, GΛ decays slowly as a power law, but de-
kT/U0 =0.01,the systemis alreadyin the regimewhere cays exponentially in the NBL. Similar behavior is found for
correlations decay exponentially and we do not observe G0 and Ga.
any QLRO at finite T in the Mott region. This is con-
firmed by the behavior of the Green functions that is
shownin Fig. 6. InFig. 6(a) we showthe anticorrelated range orders that occur: the global U(1) phase QLRO
Green function Ga, Eq. (6), in the ρ=1 MI region as T associated with the superfluid behavior and the pseudo-
isincreased. WeseethatassoonasT becomesfinite, Ga spin one, associated with the quasi-ordering of the spins
decays exponentially with distance, contrary to its con- in the yz plane, i.e. the so-called quasi polarization. As
stantvalue atlongdistance observedatT =0[18]. This explainedpreviously,thesetwoQLROappearsimultane-
exponential decay of course persists in the NBL region. ously. Since the transition is BKT, we first determined
In Fig. 6(b) we show GΛ(R), Eq. (5), for various T the transition temperature using the universal jump of
valuesatρ=2andt/U0 =0.08which,atT =0,putsthe the superfluid density [29], where, at the transition tem-
systemintheSFphase. WeseethatforlowT,GΛdecays perature, Tc, we have ρs(Tc) =kTc/πt. To observe this,
slowly. This decayis expected to be a powerlawbut the we calculated ρs as a function of temperature and deter-
systemsize is too smallto show that unambiguously. As minedTc(ρs)graphicallyastheintersectionofρs(T)with
T isincreased,thesystemtransitionsintotheNBLwhere kT/πt (see Fig. 7). We also calculated the specific heat
GΛ exhibits exponential decay clearly distinguishing the C and determined the transitiontemperatures Tc(Cmax)
NBL and SF phases. A similar behavior is found for G0 as the temperature where C reaches its maximum (Fig.
and Ga, the latter being expected since it accompanies 7). Our QMC simulations were done for L = 8 because
the presence of quasi-polarization. C is extremely difficult to calculate at low temperature
for U2 >0 for larger sizes.
Table I compares the values of T obtained from the
c
B. Nature of the transitions universaljump andfromthe maximumofC. The values
areinagreementconfirmingtheuniversaljumphypothe-
At finite temperature, kT > 0.005U0, the transitions sisandtheBKTnatureofthetransitionanddetermining
betweenthe SFandthe NBLarecontinuousforρ=1as the transition temperature for the studied size. We did
well as for ρ = 2 (and of course for all other densities). simulations for sizes up to L = 14 to examine the effect
Since this transition takes place at constant density and offinitesizeforthistransition(seeFig. 8). Asexpected,
is therefore a phase-only transition, it is expected to be thetransitiongetssharperwithincreasingsize. However,
in the Berezinsky-Kosterlitz-Thouless universality class itistoodifficulttoobtainresultswithsmallenougherror
(BKT) for our two-dimensional system. Actually, below bars for C on these large sizes.
thecriticaltemperature,wehavetwodifferentquasilong Asmentionedearlier,theevolutionfromtheMIregion
xx
7
2 3
Filled symbols: ρs t/U0=0.06 Grand canonical ensemble ρρ
Open symbols: C/Cmax tt//UU0==00..0078 2.5 U2/U0=0.1, t/U0=0.035, L=8, kT/U0=0.02 Asκ~
1.5 U /U =0.1, ρ=2, L=8 t/U0=0.09
2 0 0 2
t/U=0.10
x 0
a
m
C kT/πt 1.5
C/ 1 SF
ρ, s0.5 1 SF NBL MI NBL SF NBL Mρ=I 2 NBL
0.5 ρ=1
0
0 0.5 1 1.5 2
0
0 1 2 3 4 5 µ/U
kT
0
FIG.7: (Coloronline)Superfluiddensityρs andspecificheat FIG. 9: (Color online) ρ, ρs and κ˜ as functions of µ. As one
C as functions of temperature for different values of t/U0. goesfromtheMItotheNBL,thedensityvariescontinuously
The universal jump condition states that ρs(Tc) = kTc/πt. indicating a crossover between the two regions. The two re-
Thetransition temperatures are calculated independentlyby gionsaredistinguishedbytheinteger/non-integervalueofthe
determining themaximum of C(T). density and by the larger valueof thelocal compressibility κ˜
intheliquid. Thesuperfluidphasehasnonzeroρsandamuch
largerlocalcompressibilitythanboththeMIandNBL.κ˜has
t/U0 Tc (ρs) Tc (Cmax) been multiplied by an arbitrary factor A for better visibility.
0.06 1.49±0.10 1.67±0.15
0.07 1.82±0.05 1.85±0.15
0.08 1.99±0.05 2.00±0.15
0.09 2.10±0.05 2.10±0.20
0.10 2.19±0.05 2.20±0.20
to the NBL is a continuous crossover. A plot of the den-
sityasafunctionofµshowsnosignofafirstorderphase
TABLE I: Values of thetransition temperatures for different
transitionintheformofajumpinthedensityasoneap-
valuesoft/U0foran8×8system. Tc(ρs)isobtainedfromthe
proaches the Mott plateaux (see Fig. 9). There is no
universal jump while Tc(Cmax) from the maximum of C(T).
Thevaluesareinagreementandshowthattheuniversalpre- phase transition between MI and NBL since, in addition
diction is verified. The data are taken from Fig. 7. to the absence of a first order transition, no symmetries
are broken. Then, at finite temperature, there is only a
crossoverbetweentheMIandtheNBLandnotthephase
2
transition predicted by MFT.
Transition SF-NBL L=8, t/U=0.06
0
AF, U =0.1U , ρ=2 L=10, t/U0=0.06
2 0 L=12, t/U0=0.06 At zero temperature, we have shown [18] that the
1.5 L=14, t/U0=0.06 Mott-SF transition is always second order for ρ = 1 but
L=8, t/U=0.10
L=10, t/U0=0.10 is first order near the tip of the ρ = 2 Mott lobe when
0
L=12, t/U0=0.10 U2/U0 issmallenough,forexampleU2/U0 =0.1. Hence,
ρs1 L=14, t/U0=0.10 while it is easy to imagine that the continuous NBL-SF
transitions observed at moderate temperatures persists
kT/πt at low temperature for ρ = 1, the ρ = 2 case where the
behavior is different at zero and finite temperatures re-
0.5
quires a separate study for the low temperature regime.
Probing temperatures as low as kT = 0.005U0 (see Fig.
10),thetransitionstillappearscontinuous,whichistobe
0 compared with the temperature at which the MI region
0 1 2 3 4
kT is destroyed kT ≈ 0.05U0 (see Figs. 3 and 4). Although
itcannotbeexcludedthatthetransitionisdiscontinuous
inasmallrangeoftemperatures,thisrangewouldbeex-
FIG. 8: (Color online) Superfluid density ρs as a function of tremely narrow. In order to observe the first order QPT
T for t/U0 = 0.06 and 0.10 and for different sizes L. The inourpreviouswork[18]weusedtemperaturesthatwere
transition becomes sharper as L is varied from L=8 to L= oforderkT/U0 ≃10−3. Thetransitionthenappearsdis-
14.
continuous only at extremely low temperatures.
8
t/U0 Tc (ρs) Tc (Cmax)
0.6 0.06 1.85±0.05 1.75±0.10
kT/U=0.005 0.07 2.00±0.05 2.00±0.10
0
0.5 kT/U=0.006 0.08 2.10±0.05 2.05±0.05
0
kT/U=0.007
0 0.09 2.14±0.05 2.16±0.05
0.4 kT/U=0.008
0 0.10 2.18±0.05 2.15±0.05
kT/U=0.009
s 0
ρ
0.3
TABLEII:Values of theSFtoNBL transition temperatures
fordifferentvaluesoft/U0 andforaL=12sizeforU2/U0 =
0.2 −0.1 and ρ=2. Similar to Table I.
0.1 U /U =0.1, ρ=2, L=12
2 0 2
0 L=8, kT=0.006U0
0.0525 0.05375 0.055 L=10, kT=0.006U
t/U L=12, kT=0.006U0
0 1.5 0
L=8, kT=0.01U
MI ρ=2 L=8, kT=0.02U0
FIG.10: (Color online) Transition from thenormalBose liq- L=8, kT=0.03U0
0 0
uid to the superfluid phase at low temperatures. The tran- U 1 L=8, kT=0.04U0
sition always appears continuous for low temperatures. The /
µ
discontinuity is only found in the zero temperature regime SF ferro or NBL
(kT <10−3U0). U /U =-0.1
2 0
0.5 MIρ=1
IV. U2 <0 CASE
0
0 0.02 0.04 0.06 0.08
It was shown for U2 < 0 [18] that all the phases are t/U
unpolarized and that transitions between the different 0
phases are all continuous in the T =0 limit. The coher-
ent anticorrelated movements are present in the ρ = 1 FIG. 11: (Color online) The limit of the ρ = 1 and ρ = 2
Mott regions (MI) for different valuesof thetemperaturekT
and in the ρ=2 Mott phases.
andfordifferentsizesLintheU2 <0case. Asinthepositive
Proceeding as in the U2 > 0 case, we determine the U2 case, the Mott phases disappear for kT >0.04U0. In the
boundaries of the Mott regions at finite T, Fig. 11, by
T = 0 limit, there is a direct transition between the MI and
calculating the density and the boundary of the super-
SF.
fluid region by measuring ρ . In this case, we chose not
s
topresentresultsfromMFTsinceitshowsthesamelim-
itations as in the U2 > 0 case. We obtain the phase 0.2
diagram for ρ = 1 (shown in Fig. 12) and a similar one L=8 U /U =-0.1, ρ=1
forρ=2(notshownhere). Similarly,weusedhistograms L=10 2 0
of the density similar to Fig. 5 to confirm that all these 0.15 L=12
phases remain unpolarized at finite temperature as they
are at zero temperature. The absence of polarization at
0
T can be understood qualitatively from Eq. (8). For the U NBL
present case, U2 < 0, the last term in Eq. (8) favors T/0.1
k
the alignment of the spins along the x-axis in the low
Superfluid
T phase. Consequently, the polarization, Sz, is always
unpolar
zero. 0.05
As in the U2 > 0 case we observe a slow decay of the
MI
GreenfunctionsG inthesuperfluidphaseandthetran- BEC
σ
sitionisshowntobeoftheBKTtypeusingtheuniversal 0
0 0.05 0.1 0.15
jump argument (see Table II). t/U
0
We also examined the anticorrelated Green function
Gainbothρ=1andρ=2MIregionsatvarioustemper-
FIG. 12: (Color online) The phase diagram for ρ = 1 in the
atures. In the zero temperature limit, both these phases
exhibitnonzerovaluesofG atlongdistances. Thisisex- U2 < 0 case. All phases are unpolarized and the SF-NBL
a transition at finite temperature as well as the MI-SF tran-
pectedinbothcases,consideringthepseudo-spinHamil-
sition at zero temperature are continuous. A similar phase
tonian. Neglectingthekineticenergy,forU2 <0,theen- diagram is observed for ρ=2.
ergyis minimized by maximizing(Sx)2 oneachsite. For
r
9
temperature. On the other hand, in the superfluid, the
100 (a) LU=21/U20, =ρ-=01.1, t/U0=0.02 101 (b) UL=2/1U20,= ρ-=02.1, t/U0=0.02 eisnoebrgvyiosucsallyemasuscohcilaatregderw.iWthhtihleeictoiuspnlointgpoofsspisbeluedtoo-ssppeincs-
100 ifythisscaleaspreciselyasintheMottphases,duetothe
10-1 kT/U=0.0008 itinerantnatureof the particles inthe superfluidregime,
0
kkTT//UU0==00..00120000 10-1 a simple argument shows that it is of order U2: when-
0
10-2 kT/U0=0.0300 ever the particles enter the superfluid phase and adopt
kkTT//UU0==00..00460000 10-2 delocalized states, they overlap; there is an interaction
0
10-3 kT/U0=0.0800 cost which is then of order U0 for identical particles and
kkTT//UU0==00..11020000 10-3 U0 +U2 for different ones. This favors having the par-
0
10-4 Ga(R) kkkkTTTT////UUUU000====0000....1112468000000000 10-4 G (R) thaiacqvleuisnagsmiaopsmotlliayxrtiouzfarettihooenf.psaaFmrotrieclUtey2spa<endf0o,rgtivUhee2si>antse0yrsaatcnetmdionlweaiftadhvsooutrots
0 a
10-5 10-5 any sign of polarization. In both cases, the energy scale
is typically |U2|
0 2 4 6 0 2 4 6
R R The finite temperature phase diagram presented here
is important for the proper interpretation of experimen-
FIG. 13: (Color online) The anticorrelated Green functions tal realization of this and related systems using ultra-
in the ρ = 1 (a) and ρ = 2 (b) Mott regions for negative cold atoms loaded on optical lattices. Such experiments
U2. In both cases, the long range order of the anticorrelated are, of course, always at finite temperature. Similarly
movements that was present at T = 0 is lost even for very to what happens for fermions, we observe that the spin
small temperatures. correlations in the Mott phases will be very difficult to
accessexperimentally as the associatedenergy scalesare
verysmallandasthecorrelationsarealmostimmediately
ρ = 1 and ρ = 2, there are two degenerates states that
wiped out by thermal fluctuations. On the other hand,
achieve this: the Sx = ±1/2 for ρ = 1 and the Sx =±1
r r due to the relatively small sizes of experimental systems
forρ=2. Thesedegenerategroundstatesarecoupledby
andtoitslargerassociatedenergies,thequasi-orderingof
second order hopping processes which lift this degener-
spinsin the superfluidphaseshouldbe immediately visi-
acy through coherent anticorrelated movements [18]. In
bleexperimentallyandindistinguishablefromtruepolar-
Fig. 13, we show Ga in the ρ = 1 and ρ = 2 MI regions ization of the system. In a finite size system, one would
for different temperatures. As expected, the phase co-
expect to observe a slow drift of the polarization as the
herence once again completely disappears rapidly as the
yz symmetry of the system is restored over time.
temperatureisincreasedfromzeroandwedonotobserve
From a more technical point of view, we have eluci-
any sign of quasi long range phase coherence in the MI
dated the limitations of the MFT commonly used in the
region, as well as in the NBL phase.
literature. We found that it is unable to distinguish cor-
rectly the MI and NBL regions. The MFT does predict
reasonably well the NBL-SF boundaries but not the na-
V. CONCLUSION
ture ofthe transitionwhichis sometimespredictedto be
of firstorder whereasdirect QMC simulations,and sym-
Studying a bosonic spin-1/2 Hubbard model at finite
metry considerations,showthat itis in the BKTuniver-
temperature and comparing to the T = 0 case, we find
sality class. Furthermore, we found that MFT predicts
thattheeffectoftemperatureisdramaticallydifferentde-
direct first and second order transitions at finite T be-
pendingonthephaseweconsider. Thesuperfluidphases
tween the MI and SF phases which QMC shows do not
are essentially unchanged by raising the temperature:
exist, (see Figs. 3, 4, 12). Similar incorrectMFT behav-
The long range order present at T = 0 is transformed
ior was found for the spin-1 model and the same caveats
into QLRO.Onthe otherhand, the MI regionsaredras-
should be applied for example in Ref.[20].
ticallymodifiedasthepolarizationthatoccursincertain
cases is almost immediately wiped out by thermal fluc-
tuations: low temperature quasi polarized states are not
found in the regime of temperatures we studied. Acknowledgments
Thisisduetothefactthattheenergyscalesassociated
with the polarization of the system take different values This work was supported by: the CNRS-UC Davis
inthesedifferentphases. IntheMIregions,thepolariza- EPOCALLIA jointresearchgrant;by NSF grantOISE-
tion is due to the coupling between different low energy 0952300; an ARO Award W911NF0710576 with funds
degenerate Mott states, as emphasized by the pseudo- fromtheDARPAOLEProgram. Wewouldliketothank
spin theory [19]. These couplings are of order t2/U0 and Michael Foss-Feig and Ana-Maria Rey for useful input
thepseudo-spinquasi-orderingvanishesveryrapidlywith and discussion.
10
[1] R.P.Feynman,Int.J. Theor. Phys. 21, 467 (1982). [15] C. Wu,Modern Physics Letters B23 1 (2009).
[2] I.Bloch, J. Dalibard, and W. Zwerger, Rev.Mod. Phys. [16] M. Foss-Feig and A.-M.Rey,arXiv:1103.0245v2.
80, 885 (2008). [17] L. de Forges de Parny, M. Traynard, F. H´ebert, V.G.
[3] M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and Rousseau,R.T.Scalettar,andG.G.Batrouni,Phys.Rev.
I.Bloch, Nature415, 39 (2002) A 82, 063602 (2010).
[4] G.-B. Jo, Y.-R. Lee, J.-H. Choi, C.A. Christensen, T.H. [18] L. de Forges de Parny, F. H´ebert, V.G. Rousseau, R.T.
Kim, J.H. Thywissen, D.E. Pritchard, and W. Ketterle, Scalettar, and G.G. Batrouni, Phys. Rev. B 84, 064529
Science 325, 1521 (2009). (2011).
[5] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. [19] A. B. Kuklov and B. V. Svistunov,Phys.Rev.Lett. 90,
Pfau, Phys. Rev.Lett. 94, 160401 (2005). 100401 (2003).
[6] M.W. Zwierlein, A. Schirotzek, C.H. Schunck, and W, [20] R. V. Pai, K. Sheshadri, and R. Pandit, Phys. Rev. B
Ketterle, Science 311, 492 (2006); 77, 014503 (2008).
[7] Y.I.Shin,A.Schirotzek,C.H.Schunck,andW.Ketterle, [21] T. Kimura, S. Tsuchiya, and S. Kurihara,
Phys.Rev.Lett. 101, 070404 (2008). Phys. Rev.Lett. 94, 110403 (2005).
[8] M.Vengalattore,S.R.Leslie,J.Guzman,D.M.Stamper- [22] S. Takayoshi, M. Sato, and S. Furukawa, Phys. Rev. A
Kurn,Phys.Rev.Lett. 100, 170403 (2008); 81, 053606 (2010).
M. Vengalattore, J. Guzman, S. R. Leslie, F. Ser- [23] V.G. Rousseau, Phys. Rev. E 77, 056705 (2008) ;
wane,andD.M.Stamper-Kurn,Phys.Rev.A81,053612 V.G. Rousseau, Phys.Rev. E78, 056707 (2008).
(2010). [24] D.M. Ceperley and E.L. Pollock, Phys. Rev. B39, 2084
[9] D.M. Stamper-Kurn and W. Ketterle, in Coherent (1989).
Atomic Matter Waves, edited by R. Kaiser, C. West- [25] M. Eckholt and T. Roscilde, Phys. Rev. Lett. 105,
brook, and F. David, Springer,p. 137 (2001). 199603 (2010).
[10] G.G. Batrouni, V.G. Rousseau, and R.T. Scalettar, [26] M. Le Bellac, “Quantum and Statistical Field Theory”,
Phys.Rev.Lett. 102, 140402 (2009). Oxford University Press (1992).
[11] K.V.KrutitskyandR.Graham,Phys.Rev.A70,063610 [27] A. Auerbach and D.P. Arovas in “Introduction to Frus-
(2004); K.V. Krutitsky, M. Timmer and R. Graham, tratedMagnetism”,editedbyC.Lacroix,P.Mendels,and
Phys.Rev.A71, 033623 (2005). F. Mila, Springer(2011).
[12] J.LarsonandJ.-P.Martikainen,Phys.Rev.A80,033605 [28] S. Sachdev, “Quantum Phase Transitions”, Cambridge
(2009). University Press (1999).
[13] W. V.Liu and C. Wu,Phys. Rev.A74,13607 (2006). [29] D.R. Nelson and J.M. Kosterlitz, Phys. Rev. Lett. 39,
[14] C.Wu,W.V.Liu,J.E.MooreandS.DasSarma,Phys. 1201 (1977).
Rev.Lett. 97 190406 (2006).