Superfluid-Insulator Transitions in Attractive Bose-Hubbard Model with Three-Body Constraint
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Superfluid-Insulator Transitions in Attractive Bose-Hubbard Model with Three-BodyConstraint
Yu-Wen Lee ∗ and Min-Fong Yang Department of Physics, Tunghai University, Taichung 40704, Taiwan (Dated: March 27, 2018)By means of the method of the effective potential, the phase transitions from the Mott insulatingstate to either the atomic or the dimer superfluid state in the three-body constrained attractive Boselattice gas are analyzed. Due to the appearance of the Feshbach resonance coupling between the twokinds of order parameters in the derived effective potential function, it is found that the continuousMott insulator-to-superfluid transitions can be preempted by first-order ones. Since the employedapproach can provide accurate predictions of phase boundaries in the strong coupling limit, wherethe dimer superfluid phase can emerge, our work hence sheds light on the search of this novel phasein real ultracold Bose gases in optical lattices.
PACS numbers: 67.85.Hj, 11.15.Me, 64.70.Tg
The impressive developments on the manipulation ofultracold gases in optical lattices provides one of the bestenvironments in the search for exotic quantum phases [1].Because unprecedented control over microscopic modelparameters can be achieved, it is possible to explore pa-rameter regimes which are not available in other analo-gous condensed matter systems. The remarkable experi-mental demonstration of the superfluid to Mott insulatortransition in ultracold lattice bosons [2] has paved theway for investigating other strongly correlated phases insimilar setups. For instance, the search for novel andunconventional quantum phases in the mixtures of ultra-cold atoms obeying either the same or different statisticshas attracted considerable attention [1].It was recently suggested that intriguing quantum crit-ical behaviors can occur in attractive bosonic lattice gaseswith three-body on-site constraint [3, 4]. The system isdescribed by the Bose-Hubbard model with a three-bodyconstraint a † i ≡ H = − t X h i,j i a † i a j + U X i n i ( n i − − µ X i n i , (1)Here, a i ( a † i ) is the bosonic annihilation (creation) opera-tor at site i , t is the hopping matrix element, U < µ the chemical poten-tial. The convention h i, j i signifies a sum over nearest-neighbor sites i and j . The on-site constraint can arisenaturally due to large three-body loss processes [5, 6],and it stabilizes the attractive bosonic system againstcollapse. Therefore, besides the conventional atomic su-perfluid state (ASF) with non-vanishing order parame-ters h a i 6 = 0 and (cid:10) a (cid:11) = 0 appearing in the weakly-interacting limit, a dimer superfluid phase (DSF) withvanishing atomic order parameter ( h a i = 0) but nonzeropairing correlation ( (cid:10) a (cid:11) = 0) can be realized for suffi-ciently strong attraction [5]. It was shown in Refs. [3, 4] ∗ Electronic address: [email protected] that this model provides a simple realization of thephysics of Ising quantum transition together with theColeman-Weinberg mechanism [7] without resorting tothe Feshbach-resonant mechanism [8–10]. While the na-ture around the ASF-DSF transition has been discussed,the detailed physics of the Mott insulator (MI) to super-fluid (either ASF or DSF) transitions is not addressed inRefs. [3, 4].In the present work, we focus our attention on theMI-ASF and the MI-DSF transitions in this three-bodyconstrained attractive Bose lattice gas. Since there is nohopping term for dimers in the model of Eq. (1), usualstrong-coupling theory based on simple mean-field decou-pling is not appropriate for our purpose, because it failsto describe the DSF phase. Instead, the method of the ef-fective potential developed in Refs. [11, 12] is employed,which has been applied successfully to the usual Bose-Hubbard model with repulsive interaction. Accurate ana-lytical results for the phase boundaries of Mott insulator-to-superfluid transitions have been obtained when thecontributions from higher orders in the hopping term areincluded systematically [13, 14]. Here, we generalize theirapproach to the case with two order parameters . For theattractive bosonic lattice gases under consideration, weshow that the effective potential function has the sameform as the mean-field Ginzburg-Landau theory of reso-nant Bose gases [8–10]. Following similar discussions inRef. [10], it is found that the MI-DSF transitions are al-ways second-order, while the MI-ASF transitions can beeither second-order or first-order. The tricritical pointson the MI-ASF transition lines and the critical end pointsof the MI-DSF phase boundaries are determined withinthe present approach. From our results, it is shown thatthe DSF phase exists only in a narrow region of chemi-cal potential µ/ | U | for small hopping parameters t/ | U | .Since the approach used in the present work can be con-sidered as a kind of strong-coupling expansion, it is ex-pected to provide accurate phase boundaries of the MIphases for the transitions occurring at small values of t/ | U | . Our results hence provides a useful guide to theexperimental search of the DSF phase and the associ-ated quantum phase transitions in ultracold Bose gasesin optical lattices.To derive the effective potential with two order pa-rameters for the model in Eq. (1), we begin by addingtwo symmetry breaking source terms to the Hamiltonian,which are spatially and temporally global: ˜ H = H + V with H being the single-site zero-hopping contributionin Eq. (1) and V = − t P h i,j i a † i a j + P i ( χ ∗ a i + η ∗ a i +H . c . ). It is easy to see that the exact ground state forthe on-site part H is the n = 0 MI state (i.e., the emptystate) for the chemical potential µ < −| U | /
2, while itbecomes the n = 2 MI state (i.e., the completely filledstate) for µ > −| U | /
2. By treating V as a perturbationand following the specific adaption of high-order many-body perturbation theory proposed in Refs. [11, 12], thefree energy (or the ground state energy at the T = 0 limitunder consideration) in the MI phases for the modifiedHamiltonian ˜ H can be calculated as a double power se-ries in both the hopping parameter t and the source fields χ , χ ∗ and η , η ∗ . Up to the forth power in source fields,the general expression of free energy per site f takes thefollowing form f ≃ f + r | χ | + u | χ | + r | η | + u | η | − λ η ∗ χ + c . c . ) + u | χ | | η | , (2)where f corresponds to the ground state energy densityin the absence of the source fields. We point out thatour expression of f does respect the symmetry for theU(1) phase transformation χ → χ e iθ and η → η e iθ ,which can be understood from the form of the modifiedHamiltonian ˜ H .The coefficients in the expansion of free energy density f in Eq. (2) can be determined perturbatively in hoppingparameter t . In the following, energy unit is set to be | U | and we define the dimensionless parameters ¯ t = t/ | U | and ¯ µ = µ/ | U | for convenience. Up to leading order in ¯ t ,we find that, for the n = 0 MI state with ¯ µ < − / r = 1¯ µ (cid:18) − z ¯ t ¯ µ + z ¯ t ¯ µ (cid:19) ,u = − µ (2¯ µ + 1) (cid:18) − z ¯ t ¯ µ (cid:19) ,r = 22¯ µ + 1 (cid:20) z ¯ t ¯ µ (2¯ µ + 1) (cid:21) , (3) u = − µ + 1) (cid:2) O (¯ t ) (cid:3) ,λ = − µ (2¯ µ + 1) (cid:18) z ¯ t ¯ µ (cid:19) ,u = − µ + 1)¯ µ (2¯ µ + 1) (cid:20) − µ + 6¯ µ − µ (3¯ µ + 1) z ¯ t (cid:21) ; while, for the n = 2 MI state with ¯ µ > − / r = − µ + 1 (cid:20) z ¯ t ¯ µ + 1 + 4 z ¯ t (¯ µ + 1) (cid:21) ,u = 4(3¯ µ + 1)(¯ µ + 1) (2¯ µ + 1) (cid:18) z ¯ t µ + 1 (cid:19) ,r = − µ + 1 (cid:20) z ¯ t (¯ µ + 1)(2¯ µ + 1) (cid:21) , (4) u = 8(2¯ µ + 1) (cid:2) O (¯ t ) (cid:3) ,λ = − µ + 1)(2¯ µ + 1) (cid:18) − z ¯ t ¯ µ + 1 (cid:19) ,u = 4(3¯ µ + 2)(¯ µ + 1) (2¯ µ + 1) (cid:20) − µ + 96¯ µ + 32(¯ µ + 1)(3¯ µ + 2) z ¯ t (cid:21) . Here z is the coordination number of the underlying lat-tices. Form the free energy density f , the order parame-ters of the atomic condensate φ a ≡ h a i and the molecular(or dimer) condensate φ m ≡ h a i can be obtained by thefirst derivative of f with respect to corresponding exter-nal sources. That is, φ a = ∂f /∂χ ∗ and φ m = ∂f /∂η ∗ ,respectively. We note that, due to the mixing terms in f with coefficients λ and u , there exists a nontrivialrelation between the two order parameters φ a and φ m .The effective potential in terms of these order parame-ters is then derived by performing the Legendre transfor-mation on the free energy density f , Γ( φ a , φ ∗ a , φ m , φ ∗ m ) = f − χ ∗ φ a − χφ ∗ a − η ∗ φ m − ηφ ∗ m , which can be used todetermine the phase boundaries of the Mott insulator-to-superfluid transition and their nature as shown below.From Eq. (2), the Ginzburg-Landau expansion of the ef-fective potential as a power series of the order parametervariables can be obtained:Γ( φ a , φ ∗ a , φ m , φ ∗ m ) ≃ f + m | φ a | + g | φ a | + m | φ m | + g | φ m | − α φ ∗ m φ + c . c . ) + g | φ a | | φ m | , (5)with the coefficients given by m = − /r , g = u /r +3 λ / r r , m = − /r , g = u /r , α = λ/r r , and g = 3 λ /r r − u /r r . An interesting feature of thetwo-component Landau theory in Eq. (5) is the appear-ance of the Feshbach resonance coupling with coefficient α . The existence of such a term has important conse-quences on the Ising quantum phase transition in theboson Feshbach resonance problems [8, 9]. Starting froma specific mean-field ansatz for the DSF state, the anal-ogy between the present problem and the usual bosonFeshbach resonance model was also noticed and exploredin some detail in Ref. [4]. Here we approach this issuefrom the MI states, and focus our attention on its impli-cation on the MI-ASF and the MI-DSF transitions.As usual discussions for Ginzburg-Landau theory, the continuous MI-DSF and MI-ASF transitions occur whenthe coefficients of the quadratic terms for the correspond-ing order parameters vanish. That is, m = 0 and m = 0 give continuous MI-DSF and MI-ASF transi-tions, respectively [15]. From the relation between m and r derived above and the perturbative results for r in Eqs. (3) and (4), the critical value of hopping pa-rameter for the continuous n = 0 MI-DSF transition isgiven by z ¯ t c, ≃ p z ¯ µ (2¯ µ + 1) /
2, while that forthe continuous n = 2 MI-DSF transition is z ¯ t c, ≃ p z (¯ µ + 1)(2¯ µ + 1) /
2. Similar reasoning applies also forthe continuous MI-ASF transition. Our calculations givethe critical values of ¯ t for the continuous n = 0 MI-ASFtransition: z ¯ t c, ≃ − ¯ µ , and that for the continu-ous n = 2 MI-ASF transition: z ¯ t c, ≃ (¯ µ + 1) / z = 4 aredepicted in Fig. 1(a) as the red and the blue solid lines,respectively.As mentioned before, the presence of the Feshbach res-onance term plays an important role in characterizingthe nature of the ASF state, and it may even modify thefeature of the phase transitions. In general, the phaseboundaries of the MI phases are determined by minimiz-ing the effective potential Γ in Eq. (5), which leads to thefollowing extremum equations: φ a ( m + g | φ a | − αφ m + g | φ m | ) = 0 , (6a) − α φ + φ m ( m + g | φ m | + g | φ a | ) = 0 . (6b)From Eq. (6b), it is clear that, once φ a becomes nonzero,the molecular condensate φ m will acquire a non-vanishingvalue also. Thus the fact that the atomic condensateand the molecular condensate are both finite in the ASFphase can be explained straightforwardly as long as theFeshbach resonance term exists in Γ. Nevertheless, since φ a = 0 in both the DSF and the MI phases, one canrealize again from Eq. (6b) that the condition for thecontinuous MI-DSF transition is not affected by the pres-ence of this Feshbach-resonance term. Similarly, far awayfrom the ASF-DSF phase boundary, φ m ≃ φ m = 0 in the MI phase, thus the effect of theFeshbach resonance coupling on the aforementioned con-tinuous MI-ASF transition should be negligible, whichcan be understood from Eq. (6a).However, around the parameter region such that both m and m are zero, the continuous MI-to-superfluidtransitions mentioned above will be preempted by first-order MI-ASF transitions caused by the Feshbach res-onance term. Therefore, crossing this phase boundary,the atomic and the molecular condensates as well as theparticle density will no longer change continuously. Theoccurrence of this first-order MI-ASF transition seems tobe overlooked in Refs. [3, 4]. As explained below, thisconclusion can be reached by carefully analyzing the ex-tremum equations in Eq. (6a) and (6b), which has beendiscussed in the context of the boson Feshbach resonanceproblem [10].Without loss of generality, we may take both φ a and φ m to be real. We remind that, on the first-order MI- −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 000.10.20.30.40.50.60.70.80.91 µ / |U| z t / | U | −0.48 −0.46 −0.44 −0.420.20.250.3 ASF DSF n = 2 MIn = 0 MI (a) −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 000.10.20.30.40.50.60.70.80.91 µ / |U| z t / | U | −0.49 −0.47 −0.450.20.250.3 ASF DSF n = 2 MI n = 0 MI(b)
FIG. 1: (Color online) Phase diagram of the three-bodyconstrained attractive Bose-Hubbard model on (a) two-dimensional square lattices (b) three dimensional cubic lat-tices. The red solid lines indicate the continuous MI-DSFtransitions, and the blue solid (dashed) lines show the continu-ous (first-order) MI-ASF transitions. The black dots (crosses)denote the tricritical (critical end) points. The schematicphase boundary between the ASF and the DSF phases isadded as the green dashed line for clarity. The insets showthe details of the first-order n = 2 MI-ASF transition lines. ASF phase boundary, there should exist a discrete jumpfor φ a to a finite value. Therefore, to determine thisfirst-order phase boundary, we need to seek for non-trivial solutions of φ a . From Eq. (6a), this is given by | φ a | = ( αφ m − g | φ m | − m ) /g . Substituting this con-dition into Eqs. (6b), one leads to a saddle-point equa-tion, which is expressed solely in terms of φ m , m α g + (cid:18) m − m g g − α g (cid:19) φ m + 3 αg g φ + (cid:18) g − g g (cid:19) φ = 0 . (7)Besides, the corresponding energy density difference be-tween the ASF and the MI phases is given by∆Γ = − m g + m αg φ m + (cid:18) m − m g g − α g (cid:19) φ + αg g φ + (cid:18) g − g g (cid:19) φ . (8)The first-order MI-ASF phase boundaries are determinedby the simultaneous solutions of Eq. (7) and ∆Γ = 0 inEq. (8). Moreover, the tricritical points separating thecontinuous and the first-order MI-ASF transitions canbe found by satisfying the conditions for both kinds oftransitions. That is, by requiring nontrivial solution ofEq. (7) with m = 0, one leads to the condition for thetricritical points: m − α / g = 0. From the perturba-tive results for the coefficients of the free energy density f in Eqs. (3) and (4) as well as the relations betweenthe coefficients in f and those in the effective potentialΓ, these first-order transition lines and the correspond-ing tricritical points can be calculated numerically. Ourresults of the first-order MI-ASF phase boundaries forthe models defined on 2D square lattices with z = 4are depicted as the blue dashed lines in Fig. 1(a). Thevalue of the tricritical point on the n = 0 MI-ASF tran-sition line is found to be (¯ t T , ¯ µ T ) ≃ (0 . , − . t T , ¯ µ T ) ≃ (0 . , − . n = 2MI-ASF transition line. On the other hand, the valueof the critical end point of the n = 0 MI-DSF transi-tion line is given by (¯ t E , ¯ µ E ) ≃ (0 . , − . t E , ¯ µ E ) ≃ (0 . , − . n = 2 MI-DSFtransition line. These tricritical (critical end) points aredenoted by black dots (crosses) in Fig. 1(a). Within the present approach, the dependence ofthe lattice structure and the dimensionality enter onlythrough the coordination number z . As an illustrationfor its influence, the results for the models on three-dimensional cubic lattice with z = 6 are presented inFig. 1(b). We find that the ranges in ¯ µ (and in ¯ t also) forboth the first-order transition lines and the DSF phaseare narrower than those in the 2D case [16]. This impliesthat the role played by the Feshbach resonance term be-comes less important as z increases.To summarize, the phase boundaries of the MI-ASFand the MI-DSF transitions in the three-body con-strained attractive Bose lattice gas are determined bygeneralizing the effective potential approach developedin Refs. [11, 12]. While the coefficients of the free energydensity f are calculated only up to leading order in t/ | U | ,our results should be quantitatively accurate at least inthe strong-coupling region of the phase diagram. We findthat, due to the presence of the Feshbach resonance termin the Ginzburg-Landau expansion, the continuous MI-to-superfluid transitions can be driven to be first-order ones. From our results, it is found that the DSF phaseexists only in a narrow region of chemical potential µ/ | U | for small hopping parameters t/ | U | . Therefore, carefullytuning system parameters into the suggested parameterregime are necessary to uncover experimentally this novelphase in real ultracold Bose gas in optical lattices.We are grateful to K.-K. Ng for for many enlighteningdiscussions. Y.-W. Lee and M.-F.Yang thank the supportfrom the National Science Council of Taiwan under grantNSC 96-2112-M-029-006-MY3 and NSC 96-2112-M-029-004-MY3, respectively. [1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A.Sen, and U. Sen, Adv. Phys. , 243 (2007); I. Bloch,J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885(2008).[2] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. 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