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arXiv:1001.0021v2 [cond-mat.quant-gas] 16 Jun 2010

Strong-coupling expansion for the two-species Bose-Hubbard model

M. Iskin

Department of Physics, Koc ¸ University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey

(Dated: June 17, 2010)

To analyze the ground-state phase diagram of Bose-Bose mixtures loaded into d-dimensional hypercubic op-

tical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling

analysis) for the two-species Bose-Hubbard model with onsite boson-boson interactions. We consider both

repulsive and attractive interspecies interaction, and obtain an analytical expression for the phase boundary be-

tween the incompressible Mott insulator and the compressible superfluid phase up to third order in the hoppings.

In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite

bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three di-

mensions, when the interspecies interaction is sufficiently large and attractive. We hope that some of our results

could be tested with ultracold atomic systems.

PACS numbers: 03.75.-b, 37.10.Jk, 67.85.-d

I. INTRODUCTION

Single-species Bose-Hubbard (BH) model is the bosonic

generalization of the Hubbard model, and was introduced

originallyto describe4He in porousmediaordisorderedgran-

ular superconductors [1]. For hypercubic lattices in all di-

mensions d, there are only two phases in this model: an in-

compressible Mott insulator at commensurate (integer) fill-

ings and a compressible superfluid phase otherwise. The su-

perfluid phase is well described by weak-coupling theories,

but the insulating phase is a strong-couplingphenomenonthat

only appears when the system is on a lattice. Transition from

the Mott insulator to the superfluid phase occurs as the hop-

ping, particle-particle interaction, or the chemical potential is

varied [1].

It is the recent observation of this transition in effectively

three- [2], one- [3], and two-dimensional [4, 5] optical lat-

tices, which has been considered one of the most remarkable

achievements in the field of ultracold atomic gases, since it

paved the way for studying other strongly correlated phases

in similar setups. Such lattices are created by the intersection

of laser fields, and they are nondissipative periodic potential

energy surfaces for the atoms. Motivated by this success in

experimentally simulating the single-species BH model with

ultracold atomic Bose gases loaded into optical lattices, there

has been recently an intense theoretical activity in analyzing

BH as well as Fermi-Hubbard type models [6].

For instance, in addition to the Mott insulator and single-

species superfluid phases, it has been predicted that the two-

species BH model has at least two additional phases: an in-

compressible super-counter flow and a compressible paired

superfluid phase [7–16]. Our main interest here is in the latter

phase, where a direct transition from the Mott insulator to the

paired superfluid phase (superfluidity of composite bosons,

i.e. Bose-Bose pairs) has been predicted, when both species

have integer fillings and the interspecies interaction is suffi-

ciently large and attractive. Given that the interspecies inter-

actions can be fine tuned in ongoing experiments, e.g. with

41K-87Rb with mixtures [17, 18], via using Feshbach reso-

nances, we hope that some of our results could be tested with

ultracold atomic systems.

Inthispaper,weexaminetheground-statephasediagramof

the two-species BH model with on-site boson-boson interac-

tions in d-dimensional hypercubic lattices, including both the

repulsive and attractive interspecies interaction, via a strong-

coupling perturbation theory in the hopping. We carry the

expansion out to third-order in the hopping, and perform a

scaling analysis using the known critical behavior at the tip

of the insulating lobes, which allows us to accurately predict

the critical point, and the shape of the insulating lobes in the

plane of the chemical potential and the hopping. This tech-

nique was previouslyused to discuss the phase diagram of the

single-species BH model [19–23], extended BH model [24],

and of the hardcore BH model with a superlattice [25], and

its results showed an excellent agreement with Monte Carlo

simulations [23, 25]. Motivated by the success of this tech-

nique with these models, here we apply it to the two-species

BH model, hoping to develop an analytical approach which

could be as accurate as the numerical ones.

The remaining paper is organized as follows. After in-

troducing the model Hamiltonian in Sec. II, we develop the

strong-coupling expansion in Sec. III, where we derive an

analytical expression for the phase boundary between the in-

compressible Mott insulator and the compressible superfluid

phase. Then, in Sec. IV, we propose a chemical-potential ex-

trapolation technique based on scaling theory to extrapolate

our third-order power-series expansion into a functional form

that is appropriate for the Mott lobes, and use it to obtain typ-

ical ground-state phase diagrams. A brief summary of our

conclusions is given in Sec. V.

II.TWO-SPECIES BOSE-HUBBARD MODEL

TodescribeBose-Bosemixturesloadedintoopticallattices,

we consider the following two-species BH Hamiltonian,

H = −

?

i,j,σ

tij,σb†

i,σbj,σ+

?

?

i,σ

Uσσ

2

? ni,σ(? ni,σ− 1)

µσ? ni,σ,

+ U↑↓

?

i

? ni,↑? ni,↓−

i,σ

(1)

Page 2

2

where the pseudo-spin σ ≡ {↑,↓} labels the trapped hyper-

fine states of a given species of bosons, or labels different

types of bosons in a two-species mixture, tij,σ is the tun-

neling (or hopping) matrix between sites i and j, b†

is the boson creation (annihilation) and ? ni,σ = b†

nents, and µσis the chemical potential. In this manuscript,

we consider a d-dimensional hypercubic lattice with M sites,

for which we assume tij,σis a real symmetric matrix with el-

ements tij,σ = tσ ≥ 0 for i and j nearest neighbors and 0

otherwise. The lattice coordination number (or the number of

nearest neighbors) for such lattices is z = 2d.

i,σ(bi,σ)

i,σbi,σ is

the boson number operator at site i, Uσσ′ is the strength of

the onsite boson-boson interaction between σ and σ′compo-

We take the intraspecies interactions to be repulsive

({U↑↑,U↓↓} > 0), but discuss both repulsive and attractive

interspecies interaction U↑↓as long as U↑↑U↓↓> U2

guarantees the stability of the mixture against collapse when

U↑↓≪ 0, and against phase separation when U↑↓≫ 0. How-

ever, when the interspecies interaction is sufficiently largeand

attractive, we note that instead of a direct transition from the

Mott insulator to a single particle superfluid phase, it is possi-

ble to have a transition from the Mott insulator to a paired su-

perfluid phase (superfluidity of composite bosons, i.e. Bose-

Bose pairs) [7–16]. Therefore, one needs to consider both

possibilities, as discussed next.

↑↓. This

III.STRONG-COUPLING EXPANSION

We use the many-body version of Rayleigh-Schr¨ odinger

perturbation theory in the kinetic energy term to perform the

expansion (in powers of t↑and t↓) for the different energies

needed to carry out our analysis. The strong-coupling expan-

sion technique was previously used to discuss the phase di-

agram of the single-species BH model [19–21, 23], extended

BH model[24], andofthehardcoreBH modelwithasuperlat-

tice [25], and its results showed an excellent agreement with

Monte Carlo simulations [23, 25]. Motivated by the success

of this technique with these models, here we apply it to the

two-species BH model.

To determine the phase boundary separating the incom-

pressible Mott phase from the compressible superfluid phase

within the strong-coupling expansion method, one needs the

energyof the Mott phase and of its ‘defect’states (those states

whichhaveexactlyoneextraelementaryparticleorholeabout

the ground state) as a function of t↑ and t↓. At the point

where the energy of the incompressible state becomes equal

to its defect state, the system becomes compressible, assum-

ing that the compressibility approaches zero continuously at

the phase boundary. Here, we remark that this technique can-

notbe usedto calculatethephase boundarybetweentwo com-

pressible phases.

A.Ground-State Wave Functions

The perturbation theory is performed with respect to the

ground state of the system when t↑= t↓= 0, and therefore

we first need zeroth order wave functions of the Mott phase

and of its defect states. To zeroth order in t↑and t↓, the Mott

insulator wave function can be written as,

?

where ?? ni,σ? = nσis an integer number corresponding to the

hand, the wave functions of the defect states are determined

by degenerate perturbation theory. The reason for that lies

in the fact that when exactly one extra elementary particle or

hole is added to the Mott phase, it could go to any of the M

lattice sites, since all of those states share the same energy

when t↑ = t↓ = 0. Therefore, the initial degeneracy of the

defect states is of order M.

Whentheelementaryexcitationsinvolveasingle-σ-particle

(exactly one extra pseudo-spin σ boson) or a single-σ-hole

(exactly one less pseudo-spin σ boson), this degeneracy is

lifted at first order in t↑and t↓. The treatment for this case is

very similar to the single-species BH model [19, 24], and the

wave functions (to zeroth order in t↑and t↓) for the single-σ-

particle and single-σ-hole defect states turn out to be

|Ψins(0)

Mott? =

1

?n↑!n↓!

i

(b†

i,↑)n↑(b†

i,↓)n↓|0?,

(2)

ground-state occupancy of the pseudo-spin σ bosons, ?···? is

the thermal average, and |0? is the vacuum state. On the other

|Ψsσp(0)

def

? =

1

√nσ+ 1

?

fsσh

i

i

fsσp

i

b†

i,σ|Ψins(0)

Mott?,

(3)

|Ψsσh(0)

def

? =

1

√nσ

?

i

bi,σ|Ψins(0)

Mott?,

(4)

where fsσp

tij,σwith the highest eigenvalue (which is ztσwith z = 2d)

such that?

highest eigenvalue of tij,σbecause the hopping matrix enters

the Hamiltonian as −tij,σ, and we ultimately want the lowest-

energy states.

However, when the elementary excitations involve two par-

ticles (exactly one extra boson of each species) or two holes

(exactly one less boson of each species), the degeneracy is

lifted at second order in t↑and t↓. Such elementary excita-

tions occur when U↑↓is sufficiently large and attractive [26],

and the wave functions (to zeroth order in t↑and t↓) for the

two-particle and two-hole defect states can be written as

i

= fsσh

i

is the eigenvector of the hopping matrix

jtij,σfsσp

j

= ztσfsσp

i|fsσp

i

. The normalization condi-

|2= 1. Notice that we choose the

tion requires that?

i

|Ψtp(0)

def? =

1

?(n↑+ 1)(n↓+ 1)

1

√n↑n↓

i

?

i

ftp

ib†

i,↑b†

i,↓|Ψins(0)

Mott?,(5)

|Ψth(0)

def? =

?

turns out to be the eigenvector of the

fth

ibi,↑bi,↓|Ψins(0)

Mott?,

(6)

where ftp

tij,↑tij,↓matrix with the highest eigenvalue (which is zt↑t↓

with z = 2d) such that?

i

= fth

i

jtij,↑tij,↓ftp

j

= zt↑t↓ftp

i. Since the

elementary excitations involve two particles or two holes, the

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3

degenerate defect states cannot be connected by one hopping,

but rather require two hoppings to be connected. Therefore,

one expects the degeneracyto be lifted at least at second order

in t↑and t↓, as discussed next.

B. Ground-State Energies

Next, we employ the many-body version of Rayleigh-

Schr¨ odinger perturbation theory in t↑and t↓with respect to

the ground state of the system when t↑ = t↓ = 0, and cal-

culate the energy of the Mott phase and of its defect states.

The energy of the Mott state is obtained via nondegenerate

perturbation theory, and to third order in t↑and t↓it is given

by

Eins

Mott

M

=

?

?

σ

Uσσ

2

nσ(nσ− 1) + U↑↓n↑n↓−

?

σ

µσnσ

−

σ

nσ(nσ+ 1)zt2

σ

Uσσ

+ O(t4).

(7)

This is an extensive quantity, i.e. Eins

number of lattice sites M. The odd-order terms in t↑and t↓

vanishfor the d-dimensionalhypercubiclattices consideredin

this manuscript, which is simply because the Mott state given

in Eq. (2) cannot be connected to itself by only one hopping,

but rather requires two hoppings to be connected. Notice that

Eq. (7) recovers the known result for the single-species BH

model when one of the pseudo-spin components have vanish-

ing filling, e.g. n↓= 0 [19, 24].

Mottis proportional to the

Thecalculationofthedefect-stateenergiesismoreinvolved

since it requires using degenerate perturbation theory. As

mentioned above, when the elementary excitations involve a

single-σ-particleor a single-σ-hole, the degeneracyis lifted at

first order in t↑and t↓. A lengthy but straightforward calcula-

tion leads to the energyof the single-σ-particle defect state up

to third order in t↑and t↓as

Esσp

def= Eins

Mott+ U↑↓n−σ+ Uσσnσ− µσ− (nσ+ 1)ztσ

?nσ+ 2

− nσ(nσ+ 1)?nσ(z − 1)2+ (nσ+ 1)(z − 1)(z − 4) + (nσ+ 2)(3z/4− 1)? zt3

− 4(nσ+ 1)n−σ(n−σ+ 1)

U2

↑↓

− nσ

2

+ (nσ+ 1)(z − 3)

?zt2

σ

Uσσ

− 2n−σ(n−σ+ 1)

U2

↑↓

U2

−σ−σ− U2

↑↓

zt2

U−σ−σ

−σ

σ

U2

σσ

U2

↑↓

−σ−σ− U2

?

z − 1 −

U2

−σ−σ

U2

−σ−σ− U2

↑↓

?

ztσt2

U2

−σ−σ

−σ

+ O(t4),

(8)

where (− ↑) ≡↓ and vice versa. Here, we assume Uσσ≫ tσand {U−σ−σ,|U−σ−σ± U↑↓|} ≫ t−σ. Equation (8) is valid for

all d-dimensional hypercubic lattices, and it recovers the known result for the single species BH model when n−σ= 0 [19, 24].

Note that this expression also recovers the known result for the single species BH model when U↑↓ = 0, which provides an

independent check of the algebra. To third order in t↑and t↓, we obtain a similar expression for the energy of the single-σ-hole

defect state given by

Esσh

def= Eins

Mott− U↑↓n−σ− Uσσ(nσ− 1) + µσ− nσztσ

?nσ− 1

− nσ(nσ+ 1)?(nσ+ 1)(z − 1)2+ nσ(z − 1)(z − 4) + (nσ− 1)(3z/4 − 1)? zt3

− 4nσn−σ(n−σ+ 1)

U2

↑↓

− (nσ+ 1)

2

+ nσ(z − 3)

?zt2

σ

Uσσ

− 2n−σ(n−σ+ 1)

U2

↑↓

U2

−σ−σ− U2

↑↓

zt2

U−σ−σ

−σ

σ

U2

σσ

U2

↑↓

−σ−σ− U2

?

z − 1 −

U2

−σ−σ

U2

−σ−σ− U2

↑↓

?

ztσt2

U2

−σ−σ

−σ

+ O(t4),

(9)

which is also valid for all d-dimensional hypercubic lattices, and it also recovers the known result for the single-species BH

model when n−σ= 0 or U↑↓= 0 [19, 24]. Here, we again assume Uσσ≫ tσand {U−σ−σ,|U−σ−σ± U↑↓|} ≫ t−σ. We also

checked the accuracy of the second-orderterms in Eqs. (8) and (9) via exact small-cluster (two-site) calculations with one σ and

two −σ particles.

We note that the mean-field phase boundary between the Mott phase and its single-σ-particle and single-σ-hole defect states

can be calculated as

µpar,hol

σ

= Uσσ(nσ− 1/2) + U↑↓n−σ− ztσ/2 ±

?

U2

σσ/4 − Uσσ(nσ+ 1/2)ztσ+ z2t2

σ/4.

(10)

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4

This expression is exact for infinite-dimensional hypercubic lattices, and it recovers the known result for the single species BH

model when n−σ= 0 or U↑↓= 0 [1]. In the d → ∞ limit (while keeping dtσconstant), we checked that our strong-coupling

perturbationresults given in Eqs. (8) and (9) agree with this exact solution when the latter is expandedout to third order in t↑and

t↓, providingan independentcheck ofthe algebra. Equation(10) also shows that, for infinite-dimensionallattices, the Mott lobes

are separated by U↑↓n−σ, but their shapes and critical points (the latter are obtained by setting µpar

U↑↓. This is not the case for finite-dimensional lattices as can be clearly seen from our results. It is also important to mention

here that both the shapes and critical points are independent of the sign of U↑↓in finite dimensions (at the third-order presented

here) as can be seen in Eqs. (8) and (9).

However, when the elementary excitations involve two particles or two holes (which occurs when U↑↓is sufficiently large

and attractive [26]), the degeneracy is lifted at second order in t↑and t↓. A lengthy but straightforward calculation leads to the

energy of the two-particle defect state up to third order in t↑and t↓as

σ

= µhol

σ) are independent of

Etp

def= Eins

Mott+ U↑↓(n↑+ n↓+ 1) +

?

+2nσ(nσ+ 1)

σ

(Uσσnσ− µσ) +2(n↑+ 1)(n↓+ 1)

?

U↑↓

zt↑t↓

+

?

σ

?(nσ+ 1)2

U↑↓

−nσ(nσ+ 2)

2Uσσ+ U↑↓

Uσσ

zt2

σ+ O(t4).

(11)

Here, we assume {Uσσ,|U↑↓|,2Uσσ+ U↑↓} ≫ tσ. Equation (11) is valid for all d-dimensional hypercubic lattices, where the

odd-orderterms in t↑and t↓vanish [27]. To third orderin t↑and t↓, we obtain a similar expressionfor the energyof the two-hole

defect state given by

Eth

def= Eins

Mott− U↑↓(n↑+ n↓− 1) −

?

?

σ

[Uσσ(nσ− 1) − µσ] +2n↑n↓

?

U↑↓

zt↑t↓

+

σ

?n2

σ

U↑↓

−

(n2

2Uσσ+ U↑↓

σ− 1)

+2nσ(nσ+ 1)

Uσσ

zt2

σ+ O(t4),

(12)

which is also valid for all d-dimensional hypercubic lattices,

where the odd-order terms in t↑and t↓vanish [27]. Here,

we again assume {Uσσ,|U↑↓|,2Uσσ+ U↑↓} ≫ tσ. Since

the single-σ-particle and single-σ-hole defect states have cor-

rections to first order in the hopping, while the two-particle

and two-hole defect states have corrections to second order

in the hopping, the slopes of the Mott lobes are finite as

{t↑,t↓} → 0 in the former case, but they vanish in the lat-

ter case. Hence, the shape of the insulating lobes are expected

to be very different for two-particle or two-hole excitations.

In addition, the chemical-potential widths (µσ) of all Mott

lobes are Uσσin the former case, but they [(µ↑+ µ↓)/2] are

U↑↓+ (U↑↑+ U↓↓)/2 in the latter.

We note that in the limit when t↑= t↓= t, U↑↑= U↓↓=

U0, U↑↓ = U′, n↑ = n↓ = n0, µ↑ = µ↓ = µ, and z = 2

(or d = 1), Eq. (12) is in complete agreement with Eq. (3)

of Ref. [11], providing an independent check of the algebra.

However, in the limit when t↑= t↓= J, U↑↑= U↓↓= U,

U↑↓= W ≈ −U, n↑= n↓= m, andµ↑= µ↓= µ, Eqs.(11)

and(12) donot reduceto those givenin Ref. [12] (aftersetting

UNN = 0 there). In their expressions, we believe the terms

that are proportional to t↑t↓are missing. We also checked the

accuracy of Eqs. (11) and (12) via exact small-cluster (two-

site) calculations with one particle of each species.

We would also like to remarkin passing that the energydif-

ferencebetweentheMottphaseandits defectstates determine

the phase boundary of the particle and hole branches. This is

because at the point where the energy of the incompressible

state becomes equal to its defect state, the system becomes

compressible, assuming that the compressibility approaches

zero continuously at the phase boundary. While Eins

its defects Esσp

size M, their difference do not. Therefore, the chemical po-

tentials that determinethe particle and hole branches are inde-

pendent of M at the phase boundaries. This indicates that the

numerical Monte Carlo simulations should not have a strong

dependence on M.

It is known that the third-order strong-coupling expansion

is not very accurate near the tip of the Mott lobes, as t↑and t↓

arenotverysmall there[19, 24]. Forthis reason,an extrapola-

tion technique is highly desirable to determine more accurate

phase diagrams. Therefore, having discussed the third-order

strong-coupling expansion for a general two-species Bose-

Bose mixtures with arbitary hoppings tσ, interactions Uσσ′,

densities nσ, and chemical potentials µσ, next we show how

to develop a scaling theory.

Mottand

def, Esσh

def, Etp

defand Eth

defdepend on the lattice

IV.EXTRAPOLATION TECHNIQUE

In this section, we propose a chemical potential extrapo-

lation technique based on scaling theory to extrapolate our

third-orderpower-series expansion into a functional form that

is appropriate for the entire Mott lobes. It is known that the

critical point at the tip of the lobes has the scaling behavior of

a (d+1)-dimensionalXY model,andthereforethe lobeshave

Kosterlitz-Thouless shapes for d = 1 and power-law shapes

Page 5

5

for d > 1. For illustration purposes, here we analyze only

the latter case, but this technique can be easily adapted to the

d = 1 case [19].

A.Scaling Ansatz

From now on we consider a two-species mixture with t↑=

t↓ = t, U↑↑ = U↓↓ = U, U↑↓ = V , n↑ = n↓ = n, and

µ↑= µ↓= µ. When d > 1, we propose the following ansatz

which includes the known power-law critical behavior of the

tip of the lobes

µ±

U

= A(x) ± B(x)(xc− x)zν,

(13)

whereA(x) = a+bx+cx2+dx3+··· andB(x) = α+βx+

γx2+δx3+··· are regularfunctions of x = 2dt/U, xcis the

critical point which determines the location of the lobes, and

zν is the critical exponent for the (d + 1)-dimensional XY

model which determines the shape of the lobes near xc =

2dtc/U. In Eq. (13), the plus sign corresponds to the particle

branch, and the minus sign corresponds to the hole branch.

The form of the ansatz is taken to be the same for both single-

and two-partice (or single- and two-hole) excitations, but the

parameters are very different.

The parameters a, b, c and d depend on U, V and n, and

they are determined by matching them with the coefficients

given by our third-order expansion such that A(x) = (µpar+

µhol)/(2U). Here, µparand µholare our strong-coupling ex-

pansion results determined from Eqs. (8) and (9) for the

single-particle and single-hole excitations, or from Eqs. (11)

and (12) for the two-particle and two-hole excitations, respec-

tively. Writing our strong-coupling expansion results for the

particle and hole branches in the form µpar= U?3

b = (e+

To determine the U, V and n dependence of the parameters

α, β, γ, δ, xcand zν, we first expand the left hand side of

B(x)(xc− x)zν= (µpar− µhol)/(2U) in powers of x, and

match the coefficients with the coefficients given by our third-

order expansion, leading to

n=0e+

0+ e−

3+ e−

nxn

0)/2,

3)/2.

and µhol= U?3

n=0e−

nxn, leads to a = (e+

1)/2, c = (e+

1+ e−

2+ e−

2)/2, and d = (e+

α =e+

0− e−

2xzν

c

+e+

0

,

(14)

β

α=zν

xc

1− e−

e+

1

0− e−

0

,

(15)

γ

α=zν(zν + 1)

2x2

c

+zν

xc

e+

e+

1− e−

0− e−

+zν(zν + 1)

1

0

+e+

2− e−

e+

2

0− e−

0

1− e−

e+

,

(16)

δ

α=zν(zν + 1)(zν + 2)

e+

e+

6x3

c

2x2

c

e+

0− e−

1

0

+zν

xc

2− e−

0− e−

2

0

+e+

3− e−

e+

3

0− e−

0

.

(17)

We fix zν at its well-known values such that zν ≈ 2/3 for

d = 2 and zν = 1/2 for d > 2. If the exact value of xc

is known via other means, e.g. numerical simulations, α, β,

γ and δ can be calculated accordingly, for which the extrap-

olation technique gives very accurate results [23, 25]. If the

exact value of xcis not known, then we set δ = 0, and solve

Eqs. (14), (15), (16) and the δ = 0 equation to determine

α, β, γ and xcself-consistently, which also leads to accurate

results [19, 24]. Next we present typical ground-state phase

diagrams for (d = 2)- and (d = 3)-dimensional hypercubic

lattices obtained from this extrapolation technique.

0 0

1.5 1.5

3 3

4.5 4.5

0 0 0.09 0.09 0.18 0.18 0.27 0.27

µ/U

x = 2dt/Ux = 2dt/U

(a) Two dimensions (V=0.5U) (a) Two dimensions (V=0.5U)

n=1

n=2

n=3

sp/sh ext

third order third order

µ/U

sp/sh ext

0 0

1.5 1.5

3 3

4.5 4.5

0 0 0.09 0.09 0.18 0.18 0.27 0.27

µ/U

x = 2dt/Ux = 2dt/U

(b) Three dimensions (V=0.5U)(b) Three dimensions (V=0.5U)

n=1

n=2

n=3

sp/sh ext

third orderthird order

µ/U

sp/sh ext

FIG. 1. (Color online) Chemical potential µ (in units of U) versus

x = 2dt/U phase diagram for (a) two- and (b) three-dimensional

hypercubic lattices with t↑ = t↓ = t, U↑↑ = U↓↓ = U, U↑↓ =

V = 0.5U, n↑ = n↓ = n, and µ↑ = µ↓ = µ. The dotted lines

correspond to phase boundary for the Mott insulator to superfluid

state as determined from the third-order strong-coupling expansion,

and the hollow pink-squares to the extrapolation fit for the single-

particle or single-hole excitations discussed in the text.

Page 6

6

B.Numerical Results

In Figs. 1 and 2, the results of the third-order strong-

couplingexpansion(dottedlines) are comparedto those of the

extrapolationtechnique(hollow pink-squaresand solid black-

circles) when V = 0.5U and V = −0.85U, respectively, in

two (d = 2 or z = 4) and three (d = 3 or z = 6) dimensions.

We recall here that t↑= t↓= t, U↑↑= U↓↓= U, U↑↓= V ,

n↑= n↓= n, and µ↑= µ↓= µ.

In Fig. 1, we show the chemical potential µ (in units of U)

versus x = 2dt/U phase diagramfor (a)two-dimensionaland

(b) three-dimensional hypercubic lattices, where we choose

the interspecies interaction to be repulsive V = 0.5U. The

dotted lines correspond to phase boundary for the Mott in-

sulator to superfluid state as determined from the third-order

strong-coupling expansion, and the hollow pink-squares cor-

respond to the extrapolation fits for the single-particle and

single-hole excitations discussed in the text.

TABLE I. List of the critical points (location of the tips) xc =

2dtc/U for the first two Mott insulator lobes that are found from

the chemical potential extrapolation technique described in the text.

Here, t↑= t↓ = t, U↑↑ = U↓↓ = U, U↑↓= V , n↑ = n↓ = n, and

µ↑= µ↓= µ. These critical points for the single-particle or single-

hole excitations are determined from Eqs. (8) and (9), and they tend

to move in as V increases, and are independent of the sign of V .

d = 2

d = 3

V/U n = 1 n = 2 n = 1 n = 2

0.00.234 0.138 0.196 0.116

0.1

0.234 0.138 0.196 0.115

0.2

0.233 0.137 0.195 0.115

0.3

0.230 0.136 0.194 0.114

0.4

0.227 0.134 0.193 0.113

0.5

0.223 0.131 0.190 0.112

0.6

0.217 0.128 0.187 0.110

0.7

0.208 0.123 0.182 0.107

0.8

0.197 0.116 0.174 0.102

0.9

0.193 0.113 0.163 0.095

At t = 0, the chemical potential width of all Mott lobes

are U (similar to the single-species BH model), but they are

separated from each other by V as a function of µ. As t in-

creases from zero, the rangeof µ aboutwhich the groundstate

is a Mott insulator decreases, and the Mott insulator phase

disappears at a critical value of t, beyond which the system

becomes a superfluid. In addition, similar to what was found

for the single-species BH model [19, 24], the strong-coupling

expansion overestimates the phase boundaries, and it leads to

unphysical pointed tips for all Mott lobes, which is expected

since a finite-order expansion cannot describe the physics of

the critical point correctly. A short list of V/U versus the crit-

ical points xc = 2dtc/U is presented for the first two Mott

insulator lobes in Table I, where it is shown that the criti-

cal points tend to move in as V increases. This is because

presence of a second species (say −σ ones) screens the on-

site intraspecies repulsion Uσσbetween σ-species, and hence

increases the superfluid region.

-0.45-0.45

-0.3-0.3

-0.15-0.15

0 0

0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4

µ/U

x = 2dt/Ux = 2dt/U

(a) Two dimensions (V=-0.85U)(a) Two dimensions (V=-0.85U)

n=1n=1

n=2n=2

n=3n=3

tp/th ext

sp/sh ext

third orderthird order

µ/U

tp/th ext

sp/sh ext

-0.45 -0.45

-0.3-0.3

-0.15-0.15

0 0

0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4

µ/U

x = 2dt/Ux = 2dt/U

(b) Three dimensions (V=-0.85U)(b) Three dimensions (V=-0.85U)

n=1n=1

n=2 n=2

n=3n=3

tp/th ext

sp/sh ext

third orderthird order

µ/U

tp/th ext

sp/sh ext

FIG. 2. (Color online) Chemical potential µ (in units of U) versus

x = 2dt/U phase diagram for (a) two- and (b) three-dimensional

hypercubic lattices with t↑ = t↓ = t, U↑↑ = U↓↓ = U, U↑↓ =

V = −0.85U, n↑ = n↓ = n, and µ↑ = µ↓ = µ. The dotted lines

correspond to phase boundary for the Mott insulator to superfluid

state determined from the third-order strong-coupling expansion, the

hollow pink-squares to the extrapolation fit for the single-particle or

single-hole excitations (shown only for illustration purposes), and

the solid black-circles to the extrapolation fit for the two-particle or

two-hole excitations (the expected transition) discussed in the text.

In Fig. 2, we show the chemical potential µ (in units of

U) versus x = 2dt/U phase diagram for (a) two-dimensional

and (b) three-dimensional hypercubic lattices, where in these

Page 7

7

figures we choose the interspecies interaction to be attractive

V = −0.85U. The dotted lines correspondto phase boundary

for the Mott insulator to superfluid state as determined from

the third-order strong-coupling expansion, the hollow pink-

squares correspond to the extrapolation fits for the single-

particle and single-hole excitations (shown only for illustra-

tion purposes), and the solid black-circles correspond to the

extrapolationfits for the two-particle and two-hole excitations

(this is the expected transition) discussed in the text.

At t = 0, the chemical potential width of all Mott lobes

are V + U = 0.15U, which is in contrast with the single-

species BH model. As t increases from zero, the range of µ

aboutwhichthegroundstate isaMottinsulatordecreaseshere

as well, and the Mott insulator phase disappears at a critical

value of t, beyond which the system becomes a paired super-

fluid. The strong-coupling expansion again overestimates the

phase boundaries,andit againleads to unphysicalpointedtips

for all Mott lobes. In addition, a short list of V/U versus the

critical points xc = 2dtc/U are presented for the first two

Mott insulator lobes in Table I. Our results are consistent with

the expectation that, for small V , the locations of the tips in-

crease as a function of V , because the presence of a nonzero

V is what allowed these states to form in the first place. How-

ever, when V is larger than some critical value (∼ 0.6U), the

locationsofthetipsdecrease,andtheyeventuallyvanishwhen

V = −U. This may indicate an instability towards a collapse

since at this point U↑↑U↓↓is exactly equal to U2

↑↓.

Compared to the V > 0 case shown in Fig. 1, note that

shape of the Mott insulator to paired superfluid phase bound-

ary is very different, showing a re-entrant behavior in all di-

mensions from paired superfluid to Mott insulator and again

to a paired superfluid phase, as a function of t. Our results

are consistent with an early numerical time-evolving block

decimation (TEBD) calculation [11], where such a re-entrant

quantum phase transition in one dimension was predicted.

The re-entrant quantum phase transition occurs when co-

efficient of the hopping term in Eq. (12) is negative [so

that the two-hole excitation branch has a negative slope in

(µ↑+ µ↓)/2 versus tσ phase diagram when tσ → 0], i.e.

−(2n↑n↓/U↑↓)zt↑t↓−?

few Mott lobes beyond a critical U↑↓. When this coefficient

is negative, its value is most negative for the first Mott lobe,

and therefore the effect is strongest there. However, the coef-

ficientincreases andeventuallybecomespositiveas a function

of filling, and thus the re-entrant behavior becomes weaker as

filling increases, and it eventually disappears beyonda critical

filling. For the parameters used in Fig. 2, this occurs only for

the first lobe, as can be seen in the figures. We also note that

the sign of this coefficient is independent of the dimensional-

ity of the lattice, since z = 2d enters into the coefficient only

as an overall factor.

σ[n2

σterm, which occurs for the first

σ/U↑↓− (n2

σ− 1)/(2Uσσ+

U↑↓)+2nσ(nσ+1)/Uσσ]zt2

TABLE II. List of the critical points (location of the tips) xc =

2dtc/U that are found from the chemical potential extrapolation

technique described in the text. Here, t↑= t↓= t, U↑↑= U↓↓= U,

U↑↓ = V , n↑ = n↓ = n, and µ↑ = µ↓ = µ. These critical

points for the two-particle or two-hole excitations are determined

from Eqs. (11) and (12) when V < 0. Note that, for small V , xc’s

tend to increase as a function of V , since the presence of a nonzero

V is what allowed these states to form in the first place. However,

xc’s decrease beyond a critical V , and they eventually vanish when

V = −U, which may indicate an instability towards a collapse.

d = 2

d = 3

V/U n = 1 n = 2 n = 1 n = 2

-0.01 0.0543 0.0337 0.0611 0.0379

-0.03 0.0937 0.0582 0.105 0.0655

-0.05 0.121 0.0749 0.136 0.0843

-0.07 0.142 0.0883 0.160 0.0994

-0.1

0.1690.105

-0.2

0.2330.145

-0.3

0.2770.173

-0.4

0.3070.193

-0.5

0.3250.205

-0.6

0.3310.209

-0.7

0.3210.203

-0.8

0.2910.183

-0.9

0.2250.141

-0.93 0.193

0.121

-0.95 0.166

0.103

-0.97 0.1304 0.0812 0.147 0.0913

-0.99 0.0764 0.0474 0.0860 0.0534

0.190

0.262

0.311

0.345

0.366

0.372

0.362

0.327

0.253

0.217

0.187

0.118

0.164

0.195

0.217

0.230

0.235

0.228

0.206

0.159

0.136

0.116

V.CONCLUSIONS

We analyzed the zero temperature phase diagram of the

two-species Bose-Hubbard (BH) model with on-site boson-

boson interactions in d-dimensional hypercubic lattices, in-

cluding both the repulsive and attractive interspecies in-

teraction.We used the many-body version of Rayleigh-

Schr¨ odinger perturbation theory in the kinetic energy term

with respect to the ground state of the system when the ki-

netic energy term is absent, and calculate ground state ener-

gies needed to carry out our analysis. This technique was

previously used to discuss the phase diagram of the single-

species BH model [19–21, 23], extended BH model [24], and

of the hardcore BH model with a superlattice [25], and its

results showed an excellent agreement with Monte Carlo sim-

ulations [23, 25]. Motivated by the success of this technique

with these models, here we generalized it to the two-species

BH model, hoping to develop an analytical approach which

could be as accurate as the numerical ones.

We derived analytical expressions for the phase boundary

between the incompressible Mott insulator and the compress-

iblesuperfluidphaseuptothirdorderinthehoppings. Wealso

proposed a chemical potential extrapolation technique based

on the scaling theory to extrapolate our third-order power se-

Page 8

8

ries expansionintoa functionalformthat is appropriateforthe

Mott lobes. In particular, when the interspecies interaction is

sufficiently large and attractive, we found a re-entrant quan-

tum phase transition from paired superfluid (superfluidity of

composite bosons, i.e. Bose-Bose pairs) to Mott insulator and

again to a paired superfluid in all one, two and three dimen-

sions. Since the available Monte Carlo calculations [9, 10] do

not provide the Mott insulator to superfluid transition phase

boundary in the experimentally more relevant chemical po-

tentialversushoppingplane, wecouldnotcompareourresults

with them. This comparison is highly desirable to judge the

accuracy of our strong-couplingexpansion results.

A possible direction to extend this work is to consider the

limit where hopping of one-species is much larger than the

other. In this limit, the two-species BH model reduces to

the Bose-Bose version of the Falicov-Kimball model [28], the

Fermi-Fermi version of which has been widely discussed in

the condensed-matter literature and the Fermi-Bose version

has just been studied [29]. It is known for such models that

there is a tendency towards both phase separation and density

wave order [30], which requires a new calculation partially

similar to that of Ref. [24]. One can also examine how the

momentum distribution changes with the hopping in the insu-

lating phases [23, 31], which has direct relevance to ultracold

atomic experiments.

VI. ACKNOWLEDGMENTS

The author thanks Anzi Hu, L. Mathey and J. K. Freer-

icks fordiscussions, andThe Scientific and TechnologicalRe-

search Council of Turkey (T¨UB˙ITAK) for financial support.

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[26] Recall that U2

erwise the mixture would be unstable against collapse. In addi-

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↑↓cannot be greater than or equal to U↑↑U↓↓, oth-

σmust be kept con-