1001.0021.txt

arXiv:1001.0021v3  [cond-mat.quant-gas]  8 Oct 2010Strong-coupling expansionforthe two-species Bose-Hubba rd model
M. Iskin
Department of Physics, Koc ¸ University, Rumelifeneri Yolu , 34450 Sariyer, Istanbul, Turkey
(Dated: August 28, 2018)
Toanalyze the ground-state phase diagram ofBose-Bose mixt ures loadedinto 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 onsi te boson-boson interactions. We consider both
repulsive and attractive interspecies interaction, and ob tain an analytical expression for the phase boundary be-
tweentheincompressibleMottinsulatorandthecompressib lesuperfluidphaseuptothirdorderinthehoppings.
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, whentheinterspecies interactionissufficientl ylargeandattractive. Wehope thatsome ofourresults
couldbe testedwithultracoldatomic 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
originallytodescribe4Heinporousmediaordisorderedgran-
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,
buttheinsulatingphaseisastrong-couplingphenomenonth at
only appearswhen the system is on a lattice. Transition from
the Mott insulator to the superfluid phase occurs as the hop-
ping, particle-particleinteraction,or the chemical pote ntial is
varied[1].
It is the recent observation of this transition in effective ly
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 intersec tion
of laser fields, and they are nondissipative periodic potent ial
energy surfaces for the atoms. Motivated by this success in
experimentally simulating the single-species BH model wit h
ultracoldatomic Bose gasesloaded into optical lattices, t here
has been recently an intense theoretical activity in analyz ing
BH aswell asFermi-Hubbardtypemodels[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
superfluidphase[7–16]. Ourmaininteresthereisinthelatt er
phase,wherea directtransitionfromtheMott insulatorto t he
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 su ffi-
ciently large and attractive. Given that the interspecies i nter-
actions can be fine tuned in ongoing experiments, e.g. with
41K-87Rb with mixtures [17, 18], via using Feshbach reso-
nances,we hopethat someof ourresults couldbe tested with
ultracoldatomicsystems.Inthispaper,weexaminetheground-statephasediagramof
the two-species BH model with on-site boson-boson interac-
tionsind-dimensionalhypercubiclattices, includingboth the
repulsive and attractive interspecies interaction, via a s trong-
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 ti p
of the insulating lobes, which allows us to accurately predi ct
the critical point, and the shape of the insulating lobes in t he
plane of the chemical potential and the hopping. This tech-
niquewaspreviouslyusedtodiscussthephasediagramofthe
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
couldbeasaccurateasthenumericalones.
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 superflui d
phase. Then, in Sec. IV, we proposea chemical-potentialex-
trapolation technique based on scaling theory to extrapola te
ourthird-orderpower-seriesexpansioninto a functionalf orm
thatisappropriatefortheMottlobes,anduse ittoobtainty p-
ical ground-state phase diagrams. A brief summary of our
conclusionsisgiveninSec.V.
II. TWO-SPECIESBOSE-HUBBARDMODEL
TodescribeBose-Bosemixturesloadedintoopticallattice s,
weconsiderthe followingtwo-speciesBH Hamiltonian,
H=−/summationdisplay
i,j,σtij,σb†
i,σbj,σ+/summationdisplay
i,σUσσ
2/hatwideni,σ(/hatwideni,σ−1)
+U↑↓/summationdisplay
i/hatwideni,↑/hatwideni,↓−/summationdisplay
i,σµσ/hatwideni,σ, (1)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,σis the tun-
neling (or hopping) matrix between sites iandj,b†
i,σ(bi,σ)
is the boson creation (annihilation) and /hatwideni,σ=b†
i,σbi,σis
the boson number operator at site i,Uσσ′is the strength of
the onsite boson-bosoninteraction between σandσ′compo-
nents, and µσis the chemical potential. In this manuscript,
we considera d-dimensionalhypercubiclattice with Msites,
forwhich we assume tij,σis a real symmetricmatrix with el-
ementstij,σ=tσ≥0foriandjnearest neighbors and 0
otherwise. Thelattice coordinationnumber(orthe numbero f
nearestneighbors)forsuchlatticesis z= 2d.
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
↑↓. This
guarantees the stability of the mixture against collapse wh en
U↑↓≪0,andagainstphaseseparationwhen U↑↓≫0. How-
ever,whentheinterspeciesinteractionissufficientlylar geand
attractive, we note that instead of a direct transition from the
Mottinsulatortoasingleparticlesuperfluidphase,itispo ssi-
bletohaveatransitionfromtheMottinsulatortoa pairedsu -
perfluid phase (superfluidity of composite bosons, i.e. Bose -
Bose pairs) [7–16]. Therefore, one needs to consider both
possibilities,asdiscussednext.
III. STRONG-COUPLINGEXPANSION
We use the many-body version of Rayleigh-Schr¨ odinger
perturbation theory in the kinetic energy term to perform th e
expansion (in powers of t↑andt↓) for the different energies
needed to carryout our analysis. The strong-couplingexpan -
sion technique was previously used to discuss the phase di-
agram of the single-species BH model [19–21, 23], extended
BHmodel[24],andofthehardcoreBHmodelwithasuperlat-
tice [25], and its results showed an excellent agreement wit h
Monte Carlo simulations [23, 25]. Motivated by the success
of this technique with these models, here we apply it to the
two-speciesBH 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
energyoftheMottphaseandofits‘defect’states(thosesta tes
whichhaveexactlyoneextraelementaryparticleorholeabo ut
the ground state) as a function of t↑andt↓. 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 phaseboundary. Here,we remarkthat thistechniquecan-
notbeusedtocalculatethephaseboundarybetweentwocom-
pressiblephases.A. Ground-StateWave 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 zerothorderin t↑andt↓, the Mott
insulatorwavefunctioncanbewrittenas,
|Ψins(0)
Mott/an}bracketri}ht=1/radicalbig
n↑!n↓!/productdisplay
i(b†
i,↑)n↑(b†
i,↓)n↓|0/an}bracketri}ht,(2)
where/an}bracketle{t/hatwideni,σ/an}bracketri}ht=nσis anintegernumbercorrespondingto the
ground-stateoccupancyofthe pseudo-spin σbosons,/an}bracketle{t···/an}bracketri}htis
thethermalaverage,and |0/an}bracketri}htisthevacuumstate. Ontheother
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 o r
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 energ y
whent↑=t↓= 0. Therefore, the initial degeneracy of the
defectstates isoforder M.
Whentheelementaryexcitationsinvolveasingle- σ-particle
(exactly one extra pseudo-spin σboson) or a single- σ-hole
(exactly one less pseudo-spin σboson), this degeneracy is
lifted at first order in t↑andt↓. The treatment for this case is
very similar to the single-species BH model [19, 24], and the
wave functions(to zerothorderin t↑andt↓) forthe single- σ-
particleandsingle- σ-holedefectstates turnouttobe
|Ψsσp(0)
def/an}bracketri}ht=1√nσ+1/summationdisplay
ifsσp
ib†
i,σ|Ψins(0)
Mott/an}bracketri}ht,(3)
|Ψsσh(0)
def/an}bracketri}ht=1√nσ/summationdisplay
ifsσh
ibi,σ|Ψins(0)
Mott/an}bracketri}ht, (4)
wherefsσp
i=fsσh
iis the eigenvector of the hopping matrix
tij,σwith the highest eigenvalue (which is ztσwithz= 2d)
such that/summationtext
jtij,σfsσp
j=ztσfsσp
i.The normalizationcondi-
tion requires that/summationtext
i|fsσp
i|2= 1. Notice that we choose the
highest eigenvalue of tij,σbecause the hoppingmatrix enters
theHamiltonianas −tij,σ,andweultimatelywantthelowest-
energystates.
However,whentheelementaryexcitationsinvolvetwopar-
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↑andt↓. Such elementary excita-
tions occur when U↑↓is sufficiently large and attractive [26],
and the wave functions (to zeroth order in t↑andt↓) for the
two-particleandtwo-holedefectstatescanbewrittenas
|Ψtp(0)
def/an}bracketri}ht=1/radicalbig
(n↑+1)(n↓+1)/summationdisplay
iftp
ib†
i,↑b†
i,↓|Ψins(0)
Mott/an}bracketri}ht,(5)
|Ψth(0)
def/an}bracketri}ht=1√n↑n↓/summationdisplay
ifth
ibi,↑bi,↓|Ψins(0)
Mott/an}bracketri}ht, (6)
whereftp
i=fth
iturns out to be the eigenvector of the
tij,↑tij,↓matrix with the highest eigenvalue (which is zt↑t↓
withz= 2d)suchthat/summationtext
jtij,↑tij,↓ftp
j=zt↑t↓ftp
i.Sincethe
elementaryexcitationsinvolvetwo particlesor two holes, the3
degeneratedefectstatescannotbeconnectedbyonehopping ,
but rather require two hoppings to be connected. Therefore,
oneexpectsthedegeneracytobeliftedatleastatsecondord er
int↑andt↓, asdiscussednext.
B. Ground-StateEnergies
Next, we employ the many-body version of Rayleigh-
Schr¨ odinger perturbation theory in t↑andt↓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↑andt↓it is given
by
Eins
Mott
M=/summationdisplay
σUσσ
2nσ(nσ−1)+U↑↓n↑n↓−/summationdisplay
σµσnσ
−/summationdisplay
σnσ(nσ+1)zt2
σ
Uσσ+O(t4). (7)Thisis anextensivequantity,i.e. Eins
Mottis proportionalto the
number of lattice sites M. The odd-order terms in t↑andt↓
vanishforthe d-dimensionalhypercubiclatticesconsideredin
thismanuscript,whichissimplybecausetheMott state give n
in Eq. (2) cannot be connected to itself by only one hopping,
but ratherrequirestwo hoppingsto be connected. Notice tha t
Eq. (7) recovers the known result for the single-species BH
modelwhenoneofthepseudo-spincomponentshavevanish-
ingfilling,e.g. n↓= 0[19,24].
Thecalculationofthedefect-stateenergiesismoreinvolv ed
since it requires using degenerate perturbation theory. As
mentioned above, when the elementary excitations involve a
single-σ-particleorasingle- σ-hole,thedegeneracyisliftedat
firstorderin t↑andt↓. Alengthybutstraightforwardcalcula-
tionleadstotheenergyofthesingle- σ-particledefectstateup
tothirdorderin t↑andt↓as
Esσp
def=Eins
Mott+U↑↓n−σ+Uσσnσ−µσ−(nσ+1)ztσ
−nσ/bracketleftbiggnσ+2
2+(nσ+1)(z−3)/bracketrightbiggzt2
σ
Uσσ−2n−σ(n−σ+1)U2
↑↓
U2
−σ−σ−U2
↑↓zt2
−σ
U−σ−σ
−nσ(nσ+1)/bracketleftbig
nσ(z−1)2+(nσ+1)(z−1)(z−4)+(nσ+2)(3z/4−1)/bracketrightbigzt3
σ
U2σσ
−4(nσ+1)n−σ(n−σ+1)U2
↑↓
U2
−σ−σ−U2
↑↓/parenleftBigg
z−1−U2
−σ−σ
U2
−σ−σ−U2
↑↓/parenrightBigg
ztσt2
−σ
U2
−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
alld-dimensionalhypercubiclattices,andit recoversthe know nresult forthesinglespeciesBH modelwhen 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
independentcheckofthe algebra. To thirdorderin t↑andt↓, we obtaina similarexpressionfortheenergyofthe single- σ-hole
defectstate givenby
Esσh
def=Eins
Mott−U↑↓n−σ−Uσσ(nσ−1)+µσ−nσztσ
−(nσ+1)/bracketleftbiggnσ−1
2+nσ(z−3)/bracketrightbiggzt2
σ
Uσσ−2n−σ(n−σ+1)U2
↑↓
U2
−σ−σ−U2
↑↓zt2
−σ
U−σ−σ
−nσ(nσ+1)/bracketleftbig
(nσ+1)(z−1)2+nσ(z−1)(z−4)+(nσ−1)(3z/4−1)/bracketrightbigzt3
σ
U2σσ
−4nσn−σ(n−σ+1)U2
↑↓
U2
−σ−σ−U2
↑↓/parenleftBigg
z−1−U2
−σ−σ
U2
−σ−σ−U2
↑↓/parenrightBigg
ztσt2
−σ
U2
−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
modelwhen n−σ= 0orU↑↓= 0[19, 24]. Here, we againassume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. We also
checkedtheaccuracyofthesecond-ordertermsinEqs.(8)an d(9)viaexactsmall-cluster(two-site)calculationswith oneσand
two−σparticles.
We note that the mean-field phase boundarybetween the Mott ph ase and its single- σ-particle and single- σ-holedefect states
canbecalculatedas
µpar,hol
σ=Uσσ(nσ−1/2)+U↑↓n−σ−ztσ/2±/radicalbig
U2σσ/4−Uσσ(nσ+1/2)ztσ+z2t2σ/4. (10)4
This expression is exact for infinite-dimensionalhypercub iclattices, and it recoversthe knownresult for the single s pecies BH
model when n−σ= 0orU↑↓= 0[1]. In the d→ ∞limit (while keeping dtσconstant), we checked that our strong-coupling
perturbationresultsgiveninEqs.(8)and(9)agreewiththi sexactsolutionwhenthelatterisexpandedouttothirdorde rint↑and
t↓,providinganindependentcheckofthealgebra. Equation(1 0)alsoshowsthat,forinfinite-dimensionallattices,theM ottlobes
are separatedby U↑↓n−σ, but theirshapesandcritical points(thelatter are obtain edbysetting µpar
σ=µhol
σ) are independentof
U↑↓. This is not the case for finite-dimensional lattices as can b e clearly seen from our results. It is also important to menti on
herethat boththe shapesandcritical pointsare independen tofthe sign of U↑↓in finite dimensions(at the third-orderpresented
here)ascanbeseen inEqs.(8) and(9).
However, when the elementary excitations involve two parti cles or two holes (which occurs when U↑↓is sufficiently large
and attractive [26]), the degeneracyis lifted at second ord erint↑andt↓. A lengthybut straightforwardcalculationleads to the
energyofthetwo-particledefectstate uptothirdorderin t↑andt↓as
Etp
def=Eins
Mott+U↑↓(n↑+n↓+1)+/summationdisplay
σ(Uσσnσ−µσ)+2(n↑+1)(n↓+1)
U↑↓zt↑t↓
+/summationdisplay
σ/bracketleftbigg(nσ+1)2
U↑↓−nσ(nσ+2)
2Uσσ+U↑↓+2nσ(nσ+1)
Uσσ/bracketrightbigg
zt2
σ+O(t4). (11)
Here, we assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Equation (11) is valid for all d-dimensional hypercubiclattices, where the
odd-ordertermsin t↑andt↓vanish[27]. Tothirdorderin t↑andt↓,weobtainasimilarexpressionfortheenergyofthetwo-hol e
defectstate givenby
Eth
def=Eins
Mott−U↑↓(n↑+n↓−1)−/summationdisplay
σ[Uσσ(nσ−1)−µσ]+2n↑n↓
U↑↓zt↑t↓
+/summationdisplay
σ/bracketleftbiggn2
σ
U↑↓−(n2
σ−1)
2Uσσ+U↑↓+2nσ(nσ+1)
Uσσ/bracketrightbigg
zt2
σ+O(t4), (12)
which is also valid for all d-dimensional hypercubic lattices,
where the odd-order terms in t↑andt↓vanish [27]. Here,
we again assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Since
the single- σ-particleandsingle- σ-holedefectstateshavecor-
rections to first order in the hopping, while the two-particl e
and two-hole defect states have corrections to second order
in the hopping, the slopes of the Mott lobes are finite as
{t↑,t↓} →0in the former case, but they vanish in the lat-
tercase. Hence,theshapeoftheinsulatinglobesareexpect ed
to be very different for two-particle or two-hole excitatio ns.
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↓↓)/2inthelatter.
We note that in the limit when t↑=t↓=t,U↑↑=U↓↓=
U0,U↑↓=U′,n↑=n↓=n0,µ↑=µ↓=µ, andz= 2
(ord= 1), Eq. (12) is in complete agreementwith Eq. (3) of
Ref. [11], providing an independent check of the algebra. In
addition, in the limit when t↑=t↓=J,U↑↑=U↓↓=U,
U↑↓=W≈ −U,n↑=n↓=m, andµ↑=µ↓=µ,
Eqs. (11) and (12) reduce to those given in Ref. [12] (after
settingUNN= 0there). However, the terms that are propor-
tional tot↑t↓are not included in their definitions of the two-
particle and two-hole excitation gaps. We also checked the
accuracy of Eqs. (11) and (12) via exact small-cluster (two-
site) calculationswithoneparticleofeachspecies.
Wewouldalsoliketoremarkinpassingthattheenergydif-
ferencebetweentheMottphaseanditsdefectstatesdetermi ne
the phase boundaryof the particle and hole branches. This is
because at the point where the energy of the incompressiblestate becomes equal to its defect state, the system becomes
compressible, assuming that the compressibility approach es
zero continuously at the phase boundary. While Eins
Mottand
its defects Esσp
def,Esσh
def,Etp
defandEth
defdepend on the lattice
sizeM, their difference do not. Therefore, the chemical po-
tentialsthatdeterminetheparticleandholebranchesarei nde-
pendentof Mat thephaseboundaries. Thisindicatesthat the
numerical Monte Carlo simulations should not have a strong
dependenceon M.
It is known that the third-order strong-coupling expansion
isnotveryaccuratenearthetipoftheMottlobes,as t↑andt↓
arenotverysmallthere[19,24]. Forthisreason,anextrapo la-
tion technique is highly desirable to determine more accura te
phase diagrams. Therefore, having discussed the third-ord er
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
todevelopa scalingtheory.
IV. EXTRAPOLATIONTECHNIQUE
In this section, we propose a chemical potential extrapo-
lation technique based on scaling theory to extrapolate our
third-orderpower-seriesexpansionintoafunctionalform that
is appropriate for the entire Mott lobes. It is known that the
criticalpointatthetipofthelobeshasthescalingbehavio rof
a(d+1)-dimensional XYmodel,andthereforethelobeshave5
Kosterlitz-Thouless shapes for d= 1and power-law shapes
ford >1. For illustration purposes, here we analyze only
the latter case, but this techniquecan be easily adapted to t he
d= 1case [19].
A. ScalingAnsatz
Fromnowonwe considera two-speciesmixturewith t↑=
t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
µ↑=µ↓=µ. Whend >1, we proposethe followingansatz
which includes the known power-law critical behavior of the
tipofthe lobes
µ±
U=A(x)±B(x)(xc−x)zν, (13)
whereA(x) =a+bx+cx2+dx3+···andB(x) =α+βx+
γx2+δx3+···areregularfunctionsof x= 2dt/U,xcisthe
critical point which determines the location of the lobes, a nd
zνis 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 correspondsto the particle
branch, and the minus sign corresponds to the hole branch.
Theformoftheansatzistakentobethesameforbothsingle-
and two-partice (or single- and two-hole) excitations, but the
parametersareverydifferent.
The parameters a,b,candddepend on U,Vandn, and
they are determined by matching them with the coefficients
givenbyourthird-orderexpansionsuchthat A(x) = (µpar+
µhol)/(2U).Here,µparandµholare our strong-couplingex-
pansion results determined from Eqs. (8) and (9) for the
single-particle and single-hole excitations, or from Eqs. (11)
and(12)forthetwo-particleandtwo-holeexcitations,res pec-
tively. Writing our strong-coupling expansion results for the
particleandhole branchesin the form µpar=U/summationtext3
n=0e+
nxn
andµhol=U/summationtext3
n=0e−
nxn, leads to a= (e+
0+e−
0)/2,
b= (e+
1+e−
1)/2,c= (e+
2+e−
2)/2, andd= (e+
3+e−
3)/2.
To determine the U,Vandndependence of the parameters
α,β,γ,δ,xcandzν, we first expand the left hand side of
B(x)(xc−x)zν= (µpar−µhol)/(2U)in powers of x, and
matchthecoefficientswiththecoefficientsgivenbyourthir d-
orderexpansion,leadingto
α=e+
0−e−
0
2xzνc, (14)
β
α=zν
xc+e+
1−e−
1
e+
0−e−
0, (15)
γ
α=zν(zν+1)
2x2c+zν
xce+
1−e−
1
e+
0−e−
0+e+
2−e−
2
e+
0−e−
0,(16)
δ
α=zν(zν+1)(zν+2)
6x3c+zν(zν+1)
2x2ce+
1−e−
1
e+
0−e−
0
+zν
xce+
2−e−
2
e+
0−e−
0+e+
3−e−
3
e+
0−e−
0. (17)
We fixzνat its well-known values such that zν≈2/3for
d= 2andzν= 1/2ford >2. If the exact value of xcis known via other means, e.g. numerical simulations, α,β,
γandδcan be calculated accordingly, for which the extrap-
olation technique gives very accurate results [23, 25]. If t he
exact value of xcis not known, then we set δ= 0, and solve
Eqs. (14), (15), (16) and the δ= 0equation to determine
α,β,γandxcself-consistently, which also leads to accurate
results [19, 24]. Next we present typical ground-state phas e
diagrams for (d= 2)- and (d= 3)-dimensional hypercubic
latticesobtainedfromthisextrapolationtechnique.
B. Numerical Results
In Figs. 1 and 2, the results of the third-order strong-
couplingexpansion(dottedlines)arecomparedtothoseoft he
extrapolationtechnique(hollowpink-squaresandsolidbl ack-
circles) when V= 0.5UandV=−0.85U, respectively, in
two (d= 2orz= 4) andthree ( d= 3orz= 6) dimensions.
We recall here that t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,
n↑=n↓=n, andµ↑=µ↓=µ.
In Fig. 1, we show the chemical potential µ(in units of U)
versusx= 2dt/Uphasediagramfor(a)two-dimensionaland
(b) three-dimensional hypercubic lattices, where we choos e
the interspecies interaction to be repulsive V= 0.5U. Com-
paring Eqs. (8) and (9) with Eqs. (11) and (12), we expect
that the excited state of the system to be the usual superfluid
for allV >0for allt. The dotted lines correspond to phase
boundary for the Mott insulator to superfluid state as deter-
mined from the third-order strong-coupling expansion, and
the hollow pink-squares correspond to the extrapolation fit s
forthesingle-particleandsingle-holeexcitationsdiscu ssedin
the text. We recall here that an incompressible super-count er
flow phase [7–9, 13] also exists outside of the Mott insulator
lobes, but our current formalism cannot be used to locate its
phaseboundary.
TABLE I. List of the critical points (location of the tips) xc=
2dtc/Ufor 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 th ey tend
tomove inas Vincreases, andare independent of the signof V.
d= 2 d= 3
V/Un= 1n= 2n= 1n= 2
0.00.234 0.138 0.196 0.116
0.10.234 0.138 0.196 0.115
0.20.233 0.137 0.195 0.115
0.30.230 0.136 0.194 0.114
0.40.227 0.134 0.193 0.113
0.50.223 0.131 0.190 0.112
0.60.217 0.128 0.187 0.110
0.70.208 0.123 0.182 0.107
0.80.197 0.116 0.174 0.102
0.90.193 0.113 0.163 0.0956
 0 1.5 3 4.5
 0  0.09  0.18  0.27µ/U
x = 2dt/U(a) Two dimensions (V=0.5U)
n=1n=2n=3sp/sh ext
third order
 0 1.5 3 4.5
 0  0.09  0.18  0.27µ/U
x = 2dt/U(a) Two dimensions (V=0.5U)
sp/sh ext
third order
 0 1.5 3 4.5
 0  0.09  0.18  0.27µ/U
x = 2dt/U(b) Three dimensions (V=0.5U)
n=1n=2n=3sp/sh ext
third order
 0 1.5 3 4.5
 0  0.09  0.18  0.27µ/U
x = 2dt/U(b) Three dimensions (V=0.5U)
sp/sh ext
third order
FIG. 1. (Color online) Chemical potential µ(in units of U) versus
x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
V= 0.5U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
correspond to phase boundary for the Mott insulator to super fluid
state as determined from the third-order strong-coupling e xpansion,
and the hollow pink-squares to the extrapolation fit for the s ingle-
particle or single-hole excitations discussed in the text. Recall that
anincompressiblesuper-counterflowphasealsoexistsouts ideofthe
Mott insulator lobes.
Att= 0, the chemical potential width of all Mott lobes
areU(similar to the single-species BH model), but they are
separated from each other by Vas a function of µ. Astin-
creasesfromzero,therangeof µaboutwhichthegroundstate
is a Mott insulator decreases, and the Mott insulator phasedisappears at a critical value of t, beyond which the system
becomes a superfluid. In addition, similar to what was found
forthesingle-speciesBH model[19,24],thestrong-coupli ng
expansionoverestimatesthe phase boundaries,and it leads to
unphysical pointed tips for all Mott lobes, which is expecte d
since a finite-order expansion cannot describe the physics o f
thecriticalpointcorrectly. Ashortlistof V/Uversusthecrit-
ical points xc= 2dtc/Uis 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 Vincreases. This is because
presence of a second species (say −σones) screens the on-
site intraspeciesrepulsion Uσσbetweenσ-species, and hence
increasesthesuperfluidregion.
In Fig. 2, we show the chemical potential µ(in units of
U)versusx= 2dt/Uphasediagramfor(a) two-dimensional
and (b) three-dimensionalhypercubiclattices, where in th ese
figures we choose the interspecies interaction to be attract ive
V=−0.85U. Comparing Eqs. (8) and (9) with Eqs. (11)
and (12), we expect that the excited state of the system to
be a paired superfluid for all V <0whent→0. This is
clearlyseen inthefigurewherethedottedlinescorrespondt o
phaseboundaryfortheMottinsulatortosuperfluidstateasd e-
termined from the third-orderstrong-couplingexpansion, the
hollow pink-squares correspond to the extrapolation fits fo r
thesingle-particleandsingle-holeexcitations(shownon lyfor
illustration purposes), and the solid black-circles corre spond
to the extrapolation fits for the two-particle and two-hole e x-
citations(thisisthe expectedtransition)discussedin th etext.
Att= 0, the chemical potential width of all Mott lobes
areV+U= 0.15U, which is in contrast with the single-
species BH model. As tincreases from zero, the range of µ
aboutwhichthegroundstateisaMottinsulatordecreaseshe re
as well, and the Mott insulator phase disappears at a critica l
value oft, beyondwhich the system becomesa paired super-
fluid. The strong-couplingexpansionagain overestimatest he
phaseboundaries,anditagainleadstounphysicalpointedt ips
for all Mott lobes. In addition, a short list of V/Uversus the
critical points xc= 2dtc/Uare presented for the first two
MottinsulatorlobesinTableI. Ourresultsareconsistentw ith
the expectation that, for small V, the locations of the tips in-
crease as a function of V, because the presence of a nonzero
Viswhatallowedthesestatestoforminthefirstplace. How-
ever, when Vis largerthan some critical value ( ∼0.6U), the
locationsofthetipsdecrease,andtheyeventuallyvanishw hen
V=−U. Thismay indicatean instabilitytowardsa collapse
sinceat thispoint U↑↑U↓↓is exactlyequalto U2
↑↓.
Compared to the V >0case shown in Fig. 1, note that
shape of the Mott insulator to paired superfluidphase bound-
ary is very different, showing a re-entrant behavior in all d i-
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-entran t
quantumphasetransitionin onedimensionwaspredicted.
The re-entrant quantum phase transition occurs when co-
efficient of the hopping term in Eq. (12) is negative [so7
-0.45-0.3-0.15 0
 0 0.1 0.2 0.3 0.4µ/U
x = 2dt/U(a) Two dimensions (V=-0.85U)
n=1n=2n=3tp/th ext
sp/sh ext
third order
-0.45-0.3-0.15 0
 0 0.1 0.2 0.3 0.4µ/U
x = 2dt/U(a) Two dimensions (V=-0.85U)
n=1n=2n=3tp/th ext
sp/sh ext
third order
-0.45-0.3-0.15 0
 0 0.1 0.2 0.3 0.4µ/U
x = 2dt/U(b) Three dimensions (V=-0.85U)
n=1n=2n=3tp/th ext
sp/sh ext
third order
-0.45-0.3-0.15 0
 0 0.1 0.2 0.3 0.4µ/U
x = 2dt/U(b) Three dimensions (V=-0.85U)
n=1n=2n=3tp/th ext
sp/sh ext
third order
FIG. 2. (Color online) Chemical potential µ(in units of U) versus
x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
V=−0.85U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
correspond to phase boundary for the Mott insulator to super fluid
statedeterminedfromthethird-order strong-coupling exp ansion, the
hollow pink-squares to the extrapolation fit for the single- particle or
single-hole excitations (shown only for illustration purp oses), and
the solid black-circles to the extrapolation fit for the two- particle or
two-hole excitations (the expected transition) discussed inthe text.
that the two-hole excitation branch has a negative slope in
(µ↑+µ↓)/2versustσphase diagram when tσ→0], i.e.
−(2n↑n↓/U↑↓)zt↑t↓−/summationtext
σ[n2
σ/U↑↓−(n2
σ−1)/(2Uσσ+
U↑↓)+2nσ(nσ+1)/Uσσ]zt2
σterm,whichoccursforthefirst
few Mott lobes beyond a critical U↑↓. When this coefficient
is negative, its value is most negative for the first Mott lobe ,TABLE II. List of the critical points (location of the tips) xc=
2dtc/Uthat are found from the chemical potential extrapolation
techniquedescribedinthetext. Here, t↑=t↓=t,U↑↑=U↓↓=U,
U↑↓=V,n↑=n↓=n, andµ↑=µ↓=µ. These critical
points for the two-particle or two-hole excitations are det ermined
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
Vis what allowed these states to form in the first place. Howeve r,
xc’s decrease beyond a critical V, and they eventually vanish when
V=−U,which mayindicate an instabilitytowards a collapse.
d= 2 d= 3
V/Un= 1n= 2n= 1n= 2
-0.010.0543 0.0337 0.0611 0.0379
-0.030.0937 0.0582 0.105 0.0655
-0.050.121 0.0749 0.136 0.0843
-0.070.142 0.0883 0.160 0.0994
-0.10.169 0.105 0.190 0.118
-0.20.233 0.145 0.262 0.164
-0.30.277 0.173 0.311 0.195
-0.40.307 0.193 0.345 0.217
-0.50.325 0.205 0.366 0.230
-0.60.331 0.209 0.372 0.235
-0.70.321 0.203 0.362 0.228
-0.80.291 0.183 0.327 0.206
-0.90.225 0.141 0.253 0.159
-0.930.193 0.121 0.217 0.136
-0.950.166 0.103 0.187 0.116
-0.970.1304 0.0812 0.147 0.0913
-0.990.0764 0.0474 0.0860 0.0534
and thereforethe effect is strongest there. However,the co ef-
ficientincreasesandeventuallybecomespositiveasafunct ion
offilling,andthusthere-entrantbehaviorbecomesweakera s
fillingincreases,anditeventuallydisappearsbeyondacri tical
filling. For the parametersused in Fig. 2, this occursonlyfo r
the first lobe, as can be seen in the figures. We also note that
the sign of this coefficientis independentof the dimensiona l-
ity of the lattice, since z= 2dentersinto the coefficient only
asanoverallfactor.
What happenswhen t↑/ne}ationslash=t↓and/orU↑↑/ne}ationslash=U↓↓? We donot
expectany qualitativechangefor attractiveinterspecies inter-
actions. However, for repulsive interspecies interaction s, this
lifts the degeneracyof the single-particle or single-hole exci-
tation energies. While the transition is from a double Mott
insulator to a double superfluid of both species in the degen-
erate case, it is from a double-Mott insulator of both specie s
toaMottinsulatorofonespeciesandasuperfluidoftheother
inthenondegeneratecase.
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-8
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 ter m
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-
speciesBH model[19–21, 23], extendedBH model[24],and
of the hardcore BH model with a superlattice [25], and its
resultsshowedanexcellentagreementwithMonteCarlosim-
ulations [23, 25]. Motivated by the success of this techniqu e
with these models, here we generalized it to the two-species
BH model, hoping to develop an analytical approach which
couldbeasaccurateasthe numericalones.
We derived analytical expressions for the phase boundary
betweentheincompressibleMottinsulatorandthecompress -
iblesuperfluidphaseuptothirdorderinthehoppings. Weals o
proposed a chemical potential extrapolation technique bas ed
on the scaling theory to extrapolateour third-orderpower s e-
riesexpansionintoafunctionalformthatisappropriatefo rthe
Mott lobes. In particular, when the interspecies interacti on is
sufficiently large and attractive, we found a re-entrant qua n-
tum phase transition from paired superfluid (superfluidity o f
compositebosons,i.e. Bose-Bosepairs)toMottinsulatora nd
again to a paired superfluid in all one, two and three dimen-sions. SincetheavailableMonteCarlocalculations[9,10] do
not provide the Mott insulator to superfluid transition phas e
boundary in the experimentally more relevant chemical po-
tentialversushoppingplane,wecouldnotcompareourresul ts
with them. This comparison is highly desirable to judge the
accuracyofourstrong-couplingexpansionresults.
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
theBose-BoseversionoftheFalicov-Kimballmodel[28],th e
Fermi-Fermi version of which has been widely discussed in
the condensed-matter literature and the Fermi-Bose versio n
has just been studied [29]. It is known for such models that
thereisa tendencytowardsbothphaseseparationanddensit y
wave order [30], which requires a new calculation partially
similar to that of Ref. [24]. One can also examine how the
momentumdistributionchangeswiththehoppingintheinsu-
latingphases[23, 31], whichhasdirect relevanceto ultrac old
atomicexperiments.
VI. ACKNOWLEDGMENTS
The author thanks Anzi Hu, L. Mathey and J. K. Freer-
icksfordiscussions,andTheScientificandTechnologicalR e-
searchCouncilofTurkey(T ¨UB˙ITAK)forfinancialsupport.
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