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. [1] M.P.A.Fisher,P.B.Weichman,G.Grinstein,andD.S.Fis her, Phys.Rev. B 40, 546(1989). [2] M. 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