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Phase diagram of ultracold atoms on optical lattice Comparative study to slave fermion and

a r X i v :c o n d -m a t /0404375v 3 [c o n d -m a t .s t r -e l ] 15 M a r 2005

Phase diagram of ultracold atoms in optical lattices:Comparative study of slave

fermion and slave boson approaches to the Bose-Hubbard model

Yue Yu

1,2

and S.T.Chui 2

1.Institute of Theoretical Physics,Chinese Academy of Sciences,P.O.Box 2735,Beijing 100080,China and

2.Bartol Research Institute,University of Delaware,Newark,DE 19716

(Dated:February 2,2008)

We perform a comparative study of the ?nite temperature behavior of ultracold Bose atoms in optical lattices by the slave fermion and the slave boson approaches to the Bose Hubbard model.The phase diagram of the system is presented.Although both approaches are equivalent without approximations,the mean ?eld theory based on the slave fermion technique is quantitatively more appropriate.Conceptually,the slave fermion approach automatically excludes the double occupancy of two identical fermions on the same lattice site.By comparing to known results in limiting cases,we ?nd the slave fermion approach better than the slave boson approach.For example,in the non-interacting limit,the critical temperature of the super?uid-normal liquid transition calculated by the slave fermion approach is closer to the well-known ideal Bose gas result.At zero-temperature limit of the critical interaction strength from the slave fermion approach is also closer to that from the direct calculation using a zero-temperature mean ?eld theory.

PACS numbers:03.75.Lm,67.40.-w,39.25.+k

I.INTRODUCTION

Strongly correlated systems are of longstanding inter-est in studies of condensed matter physics.Ultra-cold atoms in optical lattices o?er new opportunities to study strongly correlated phenomena in a highly controllable environment[1,2,3,4].A quantum phase transition,the super?uid/Mott-insulator transition,was demonstrated using 87Rb atoms in three-[2]and one-dimensional lat-tices [4].Strongly correlated phenomena for boson sys-tems may be studied theoretically by the Bose-Hubbard model [5].Experimental feasibility was established by microscopic calculations of the model parameters for cold boson atoms in optical lattices [6].A review of recent works on the super?uid-insulator quantum phase transi-tion at zero temperature is given in ref.[7].

Strictly speaking,a quantum phase transition can not be observed at any ?nite temperature.The experimen-tal data give only a signal that the system is towards a quantum phase transition if the temperature is extrap-olated to zero.What the experiments really observed was a transition from the super?uid to the normal liquid whose compressibility is very close to zero and the system is practically a Mott insulator.Such a ’classical’phase transition has been investigated recently by Dickerscheid et al [8].Phase diagrams for a given atom density were calculated in the temperature-interaction plane and the chemical potential-interaction plane:For a commensu-rate optical lattice,there are only the super?uid and the Mott insulator phases at zero temperature.At ?nite tem-peratures,starting from the super?uid phase,there is a super?uid/normal liquid phase transition while the Mott insulator phase crossovers to the normal liquid.

In order to extend the ordinary mean ?eld approach for the Bose Hubbard model [9]to include the ?nite tempera-ture e?ects,the slave boson technique [10]was used.The slave particle technique has been widely applied in deal-

ing with the strongly correlated electron systems [11].In principle,the slave boson and slave fermion approaches are equivalent.However,in practical calculations,ap-proximations still have to be used.It was well-known that in the t -J model,the same mean ?eld approximation using the slave boson or slave fermion leads to very di?er-ent phase diagrams [11].It was known that the slave bo-son mean ?eld approximation can qualitatively describe the phase diagram of the cuprates at ?nite doping.How-ever,due to the Bose condensation of the holons,the slave boson mean ?eld approximation does not produce correctly the ferromagnetic Mott insulator phase.One of the purpose of this work is to examine if both slave parti-cle approaches to the Bose Hubbard model give the same physical results under the mean ?eld approximation.In these approaches,there is a constraint that each site can be occupied by only one slave particle.With this exact constraint,the model is very hard to solve.A standard approximation is to relax the constraint on each site to the requirement that the average slave particle per site over the lattice be equal to 1.While both approaches give the same qualitative phase diagram,we shall see that the quantitative behaviors derived from the slave fermion approach are more accurate.The advantage of the slave fermions is that the Fermi statistics automatically ex-cludes two same type slave fermions from occupying the same site even when the constraint is relaxed.We shall see that the con?gurations with the multi-occupations of di?erent types of slave fermion are far away from the mean ?eld state we consider.Thus,these con?gurations will not signi?cantly in?uence our results.However,the statistics of the slave particle a?ect the result remarkably.For repulsive interactions,there are two unsatisfactory features arising from the ?nite temperature mean ?eld theory in the slave boson approache[8].One of them is that the critical on-site repulsive U c ≈5.83from a direct zero temperature mean ?eld calculation and di?ers from

2 the critical U c≈6in T→0from the?nite temperature

mean?eld.This di?erence is much smaller with the slave

fermion approach.The other was that there is a maxi-

mum T c at U=0in the U-T c curve which is obviously

unphysical.We?nd that these de?ciencies are corrected

in the slave fermion approach.Furthermore,for U=0,

the critical temperature of the super?uid-normal liquid

phase transition for the ideal Bose gas was well-known.

We?nd that this critical temperature calculated by the

slave fermion approach is much closer to its exact value

than that by the slave boson approach.

This paper is organized as follows.In Sec.II,we give

an overview of our slave particle approaches.In Sec.III, the perturbation theory is introduced.In Sec.IV,we give our main results according to the mean?eld theory. Section V is our conclusion.

II.SLA VE PARTICLE APPROACH

A boson operator on site i may be expressed by the occupation state|α ,i.e.,

a?i= α√

α+1a?α+1,i aα,i,(2) where aα,i may be either the(slave)boson operator bα,i with[bα,i,b?

β,j

]=δαβδij or the(slave)fermion operator

cα,i with{cα,i,c?

β,j}=δαβδij.As the auxiliary particles, they have to obey the constraint

αnαi= αa?α,i aα,i=1,(3)

on each site,which corresponds to the completeness of the states: α|α α|=1and the original Bose commu-tation relation:[a i,a?j]=δij.

The Bose Hubbard Hamiltonian we will focus on reads H=?t ij a?i a j?μ n i+U

2m?2+V(r)]W(r+a), U=g d r|W(r)|4,(5)

81012141618

FIG.1:The dependence of the Bose Hubbard parameter a s U/6t for the three dimensional optical lattice as a function of the strength of the lattice potential V0.The scattering length a s is in units of a nanometer.V0is in units of the recoil energy E r= 2k2/2m

where g=4πa s 2

α+1

2 i

αα(α?1)nαi,(7)

With the constraint(3),the standard path integral leads to the partition function of the system

Z=T re?βH= DaαDˉaαDλe?S E,

S E[ˉaα,aα,λ]= 1/T0dτ i αˉaα,i[?τ?αμ

α+1

slave boson while being the Grassmann conjugateˉcαi of the fermion?eld cαi for the slave fermion.The integrals over the Lagrange multiplier?eldλi(τ)come from the constraint(3):

iδ( αnαi?1)= Dλexp i β0 iλi

×( αnαi?1)dτ .(9)

In the sense of theδ-function,theλi?elds have to be real to ensure the constraint is correctly taken into account. To decouple the four slave particle term in the Hamil-tonian,we introduce a Hubbard-Stratonovich?eldΦi which is a bosonic?eld and may be identi?ed as the order parameter of the super?uid.The integral

DΦDΦ?exp ? dτt

(Φ?i? α√

α+1ˉaα,j aα+1,j) ,(10)

is obviously a constant.The partition function can be written as

Z= DΦDΦ?DˉaαDaαDλe?S eff[Φ,aα,λ],

S eff[Φ,aα,λ]= dτ i αˉaα,i[?τ?αμ

α+1ˉaα,j aα+1,j

?Φj α√

α+1ˉaα,i aα+1,i = a i .(13)

This means thatΦi indeed serves as an order parameter ?eld.Near the Mott transition,this order parameter is small and one can use perturbation theory to solve the system described by the action(12).The di?culty is that there is no way to exactly solve the problem if the λ?eld varies from site to site.A widely-used approx-imation is to relax the constraint(3)by replacing the local constraint Lagrange multiplierλi(τ)by an imagi-nary time-and site-independent?eldλ.That is,relax-ing the condition of exactly one slave particle per site to one with an average of one particle per site.It im-plies that multi-occupation of the slave particles on the same site is allowed.For slave bosons,this relaxation al-lows the same type of the boson to multi-occupy a single site.However,for the same type of slave fermions,multi-occupation of the same site is automatically forbidden by the Pauli principle.The value ofλwill be variationally determined.

To do the perturbation calculation,it is convenient to make a Fourier transformation for the?elds A i=a i,Φi andλi:

A i=

Lβ k,n Aα,kn e i k·i?iωnτ,(14)

where the Matsubara frequenciesωn=2πnT for bosonic ?elds andωn=(2n+1)πT for fermionic?elds.The approximation of the site-independent ofλi implies that allλk,n=λ0,0=λ

√

β αln(1?e?β?0(α)),(16)

whereβ=1/T and?0(α)=?iλ?αμ+α(α?1)U/2 and?(+)sign corresponds to the slave boson(slave fermion)approximation.The Green’s function is de?ned by

?G?1(k,iωn)=?k+?2k α(α+1)nα?nα+1

.The slave particle occupation num-ber is given by

nα=

IV.MEAN FIELD THEORY

We focus on repulsive interactionis with U>0in this paper.According to Landau theory,the condition G?1(0,0)=0may be used to determine the critical point of the phase transition between the super?uid and the normal liquid[8].

The key di?erence between the slave boson and slave fermion is their quantum statistics,which leads to the sign di?erence?in equation(18).The di?erence ap-pears because we have approximated allλi(τ)by a real constantλ.We discuss the zero temperature and the ?nite temperature cases separately.

A.Zero Temperature

In the zero temperature limit,Dickersheid et al[8]in-vestigated commensurate?llings and assume the num-ber of particles of each well to be?xed at some valueα′: nα′=δα,α′in mean?eld theory.This mean?eld assump-tion works for both kinds of slave particles.In terms of G?1(0,0)=0and the Green’s function(17),it is easy to calculate the phase boundaries in theμ-U plane[8]: The Mott insulator phase is in the regimes whereˉμlies betweenˉμα′±

ˉμα′±=1

β k,n G(k,iωn)?G?1(k,iωn)

the slave boson and slave fermion approaches is shown only on their di?erent statistics.The critical point of the super?uid/normal liquid phase transition,in terms of G?1(0,0)=0,is determined by

α(α+1)nα+1?nα

2 ln(ˉU?24)(ˉU+3)

2 ln(

ˉU+12)(ˉU+3)

dU .It is

seen that the boson statistics of the slave boson sharpens the slope

the T c-U curve in small U.

We next investigate if the?niteαM approximation is good or not.For this purpose,we plot nα(αM=3)

versus U at the critical temperature(Fig.4).It is seen that the occupancy of the0th,1st and2nd types of

the slave fermion is of the order10?1from U=1to 4.However,n3decreases quickly as U increases.For

U=1,n3≈0.05but≈0.009,5×10?4and2×10?5 for U=2,3and4,respectively.This means that for a large enough U,theαM=3cut-o?is a good approx-

imation.In the regime of small U,a largerαM(>3) is required if we would like to have a quantitatively reliable result.ForαM=3,we see that,from Fig.2,ˉT

(U=0,αM=3)≈0.98.It may be expected that

asαM increases,ˉT c(U=0)should be close to, e.g, 1.18in three dimensions,the ideal Bose gas critical temperature.The approximation getting worse for small U means the contribution from largeα(>3)can not be neglected.The maximum of the critical temperature in Fig.2comes from neglecting these degrees of freedom corresponding to largeα.Taking a largerαM,it may be anticipated that the maximum ofˉT c may disappear asˉT c(U=0)tends to1.18.To reveal the quantitative behavior of the system for small U more precisely,we have to work at a largerαM.The numerical work is still in progress which will be present elsewhere.

1.0 1.5

2.0 2.5

3.0 3.5

4.0

U/zt

FIG.4:The slave fermion occupant number n αversus ˉU

at the critical temperature (αM =3).

0246810

/z t U/zt

FIG.5:The phase diagram for the incommensurate ?llings.

The ?lled squares are for n =0.9and the empty circles are for n =0.75.

Incommensurate State

It was also known that in the incommensurate state,there is no Mott insulator phase for any value of U .To con?rm this point,we calculate two incommensurate ?ll-ings with n =0.9and 0.75.As Fig.5shown,only a normal-super?uid transition in the phase diagram is found and there is no Mott insulator phase.

V.CONCLUSION

In summary,we have made a comparative study of the ?nite temperature phase diagrams with the slave boson and the slave fermion mean ?eld theory for the Bose Hub-bard model.We found that both slave particle mean ?eld theories are qualitatively the same but the slave fermion approach is quantitatively more accurate.Many other re-sults obtained in Ref.[8]by the slave boson approach are valuable and may be improved by the slave fermion ap-proach by replacing the bosonic occupation number with the the fermionic occupation number.This approach can be generalized to the two-component Bose Hub-bard model with b ?

i,↑= α↑,α↓ α↓+1c ?α↑,α↓+1c α↑,α↓,while for Bose-Fermi mixture Hubbard model,a slave boson-composite fermion mixture approach,with b ?i = α

√

[14]To eliminate the multi-occupied con?guration beyond the

mean?eld state,one has to deal with the?uctuation corresponding toδλi=λi?λ.However,it is very di?cult to solve this problem.

[15]In the long wave length limit,the critical temperature for

the three-dimensional ideal Bose gas is given byˉT ideal

c =

2π

g3/2(0) 2/3with g i(x)= ∞m=1e mx6≈

1.18.