Supersolidity in a Bose-Holstein model
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Supersolidity in a Bose-Holstein model
Sanjoy Datta and Sudhakar Yarlagadda
CAMCS and TCMP Div., Saha Institute of Nuclear Physics, 1/AF Salt Lake, Kolkata-64, India (Dated: November 17, 2018)We derive an effective d-dimensional Hamiltonian for a system of hard-core-bosons coupled tooptical phonons in a lattice. At non-half-fillings, a superfluid-supersolid transition occurs at inter-mediate boson-phonon couplings, while at strong-couplings the system phase separates. We demon-strate explicitly that the presence of next-nearest-neighbor hopping and nearest-neighbor repulsionleads to supersolidity. Thus we present a microscopic mechanism for the homogeneous coexistenceof charge-density-wave and superfluid orders.
PACS numbers: 67.80.kb, 64.70.Tg, 71.38.-k, 71.45.Lr
Introduction:
Coexistence of charge-density-wave(CDW) and superconductivity is manifested in a varietyof systems such as the perovskite type bismuthates (i.e.,BaBiO doped with K or Pb) [1], quasi-two-dimensionallayered dichalcogenides (e.g., 2H − TaSe , 2H − TaS ,and 2H − NbSe ) [2], etc. What is special about theabove systems is that they defy the usual expectationthat the competing CDW and superconductivity ordersoccur mutually exclusively. CDW and superconductiv-ity are examples of diagonal long range order (DLRO)and off-diagonal long range order (ODLRO) respectively.Here, DLRO breaks a continuous translational invarianceinto a discrete translational symmetry, whereas ODLRObreaks a global U(1) phase rotational invariance [3].Another interesting example of DLRO-ODLRO con-currence is a supersolid (SS). A SS state is characterizedby the homogeneous coexistence of two seemingly mu-tually contradictory phases, namely, a crystalline solidand a superfluid (SF). Here all particles simultaneouslyparticipate in both types of long range order. It hasbeen conjectured long ago that solid helium-4 may ex-hibit supersolidity due to Bose condensation of vacanciespresent [4, 5]. Only recently, Chan and Kim (using atorsional oscillator) observed a decoupling of a small per-centage of the helium solid from a container’s walls [6].They interpreted this as supersolidity. This discovery ledto studies of bosonic models in different kinds of latticestructures [7] and with various types of particle interac-tions [8]. Phenomenologically, Ginzburg-Landau theory[9] and a quantum solid based on Gross-Pitaevskii equa-tion [10] have been used to study supersolidity.In this paper, we address the above ongoing puzzlesby studying the quantum phase transitions exhibited byhard-core-bosons (HCB) coupled to optical phonons. Tothis end, in contrast to the above treatments, we em-ploy a microscopic approach involving a minimal Bose-Holstein (BH) lattice model. Examples of real systemsdescribable by our BH model are as follows. In the bis-muthate systems, the observed valence skipping of thebismuth ion is explained by invoking non-linear screen-ing which is said to produce a large attractive interactionresulting in the formation of local pairs or HCB [11, 12]. Such HCB couple to the cooperative breathing mode ofthe oxygen octahedra surrounding the Bismuth ion. Ina helium-4 crystal, vacancies produce a local distortionand can be treated as HCB coupled to finite frequencyphonons. Furthermore, the concentration of vacancies isvery small [5] and hence direct interactions among theseHCB is negligible. Lastly, for dichalcogenides such asNbSe , where homogeneous coexistence of the two longrange orders has been unambiguously established [13],our results on the BH model should be quite relevant.Starting with our BH model, we derive an effective d-dimensional Hamiltonian for HCB by using a transparentnon-perturbative technique. The region of validity of oureffective Hamiltonian is governed by the small parame-ter ratio of the adiabaticity t/ω and the boson-phonon(b-p) coupling g . The most interesting feature of thiseffective Hamiltonian is that it contains an additionalnext-nearest-neighbor (NNN) hopping compared to theHeisenberg xxz-model involving only nearest-neighbor(NN) hopping and NN repulsion [14, 15]. We employed amodified Lanczos algorithm [16] (on lattices of sizes 4 × √ × √ √ × √
20, and 4 ×
6) and found that (ex-cept for the extreme anti-adiabatic limit) the BH modelshows supersolidity at intermediate b-p coupling strengths whereas the xxz-model produces only a phase separated ** * *
SF SSPS CD W g Np (in a 4 x 4 lattice) (a) CD W *** SF PS g Np (in a 4 x 4 lattice) (b) * FIG. 1: Quantum phase diagram at various particle numbers N p , t/ω = 1 . J > J = 0 (xxz-model). Su-persolidity occurs only in (a). The continuous lines in (a) and(b) are guides for the eye. (PS) state. Here we present the calculations for only a4 × Effective Bose-Holstein Hamiltonian:
We startwith a system of spinless HCB coupled with opticalphonons on a square lattice. This system is describedby a BH Hamiltonian [17] H = − t X j,δ b † j b j + δ + ω X j a † j a j + gω X j n j ( a j + a † j ) , (1)where δ corresponds to nearest-neighbors, ω is the op-tical phonon frequency, b j ( a j ) is the destruction op-erator for HCB (phonons), and n j ≡ b † j b j . Then weperform the Lang-Firsov (LF) transformation [18, 19]on this Hamiltonian and this produces displaced simpleharmonic oscillators and dresses the hopping particleswith phonons. Under the LF transformation given by e S He − S = H + H ′ with S = − g P i n i ( a i − a † i ), b j and a j transform like fermions and phonons in the Holsteinmodel. This is due to the unique (anti-) commutationproperties of HCB given by[ b i , b j ] = [ b i , b † j ] = 0 , for i = j, { b i , b † i } = 1 . (2)Next, we take the unperturbed Hamiltonian H to be [18] H = − J X j,δ b † j b j + δ + ω X j a † j a j − g ω X j n j , (3)and the perturbation H ′ to be H ′ = − J X j,δ b † j b j + δ {S j † + S j − − } , (4)where S j ± = exp[ ± g ( a j − a j + δ )] , J = t exp( − g ), and g ω is the polaronic binding energy. We then follow thesame steps as in Ref. [18] to get the following effectiveHamiltonian in d-dimensions for our BH model H e = − g ω X j n j − J X j,δ b † j b j + δ − J X j,δ,δ ′ = δ b † j + δ ′ b j + δ − . J z X j,δ n j (1 − n j + δ ) , (5)where J z ≡ ( J /ω )[4 f ( g ) + 2 f ( g )] and J ≡ ( J /ω ) f ( g ) with f ( g ) ≡ P ∞ n =1 g n / ( n ! n ) and f ( g ) ≡ P ∞ n =1 P ∞ m =1 g n + m ) / [ n ! m !( n + m )] . Here we would liketo point out that, as shown in Ref. 20, the small param-eter for our perturbation theory is t/ ( gω ). Long range orders:
Diagonal long range order(DLRO) can be characterized by the structure factor de-fined in terms of the particle density operators as follows: S ( q ) = 1 N X i,j e q · ( R i − R j ) ( h n i n j i − h n i i h n j i ) . (6) Off-diagonal long range order (ODLRO) in a Bose-Einstein condensate (BEC), as introduced by Penroseand Onsager [3], is characterized by the order parame-ter h b i = p h n i e iθ where n is the occupation numberfor the k = 0 momentum state or the BEC. It is usefulto define the general one-particle density matrix˜ ρ ( i, j ) = D b † i b j E = 1 N X k , q e ( k · R i − q · R j ) D b † k b q E , (7)where hi denotes ensemble average. Eq. (7) gives theBEC fraction as n b = h n i N p = X i,j ˜ ρ ( i, j ) N N p . (8)In general, to find n b , one constructs the generalized one-particle density matrix ˜ ρ and then diagonalizes it to findout the largest eigenvalue. To characterize a SF, an im-portant quantity is the SF fraction n s which is calculatedas follows. Spatial variation in the phase of the SF orderparameter will increase the free energy of the system. Weuse a linear phase variation θ ( x ) = θ ( x/L ) with θ beinga small angle and L the linear dimension in x-direction.This is done by imposing twisted boundary conditions(TBC) on the many-particle wave function. At T = 0 K,we can write the change in energy to be E [ θ ] − E [0] = 12 mN p n s (cid:12)(cid:12)(cid:12)(cid:12) ~ m ~ ∇ θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . (9)Then the SF fraction is given by [21, 22] n s = (cid:18) NN p t eff (cid:19) E [ θ ] − E [0] θ , (10)where t eff = ~ / m . For our Hamiltonian in Eq. (5), wefind t eff = J + 8 J . The phase variation (taken to bethe same for BEC and SF order parameters) is introducedin our calculations by modifying the hopping terms with b j → b j exp[ i ˆ x · R j θ /L ] which is gauge-equivalent toTBC. Results and discussion:
We employ the mean fieldanalysis (MFA) of Robaszkiewicz et al. [14] to study thephase transitions dictated by the effective Hamiltonian ofEq. (5). We obtain the following expression for the SF-PS (SF-CDW) phase boundary at non-half-filling (half-filling): J z J − J J = 1 + (2 n − − (2 n − . (11)Eq.(11) leads to the same phase diagram as that (for thexxz-model) in Ref. 14 but with J effz = J z / − J asthe y-coordinate instead of J effz = J z /
2. Thus, withinmean field, NNN hopping does not change the qualitativefeatures of the phase diagram; it only increases the criti-cal value of J z /J at which the transition from SF state -14-12-10-8-6-4-2 0 3 4 5 6 7 8 F r ee E n e r g y / J N p t/ ω =0.1J > 0 g = 2.1 2.2 2.3 2.4 2.5 3.0 (a) -30-25-20-15-10-5 0 3 4 5 6 7 8 F r ee E n e r g y / J N p J = 0 g ≥ 1.3 g = 1.2 (b) FIG. 2: Free energy at different fillings and different valuesof g for (a) J > J = 0, and t/ω = 0 .
1; and (b) J = 0, J = 0, and t/ω = 1 . to PS or CDW state occurs. However, as demonstratedby the numerics below at non-half-fillings, MFA fails tocapture the supersolid phase.We studied the stability of the phases by examining thenature of the free energy versus N p curves at different b-pcouplings g (see Figs. 2 and 3) and by using an analysisequivalent to the Maxwell construction. For the systemat a given N p , g , and t/ω , if the free energy point P lies above the straight line joining the nearest two stablepoints Q and R (lying on either side of P ) on the samefree energy curve, then the system at P breaks up intotwo phases corresponding to points Q and R .For the range of parameters that we considered (i.e.,0 . ≤ t/ω ≤ g > t/ ( gω ) <
1. The behavior of the system for t/ω = 0 . J = 0 and 0 = J /J [= ( J /ω ) f ( g )]because J /J is negligible in the latter case. Further-more, for J = 0, the behavior of the system as a func-tion of g is qualitatively the same for all values of t/ω asthere is only one dimensionless parameter J z /J involvedin Eq. (5).We will now analyze together, in one plot, thequantities n b , n s , and the normalized structure factor S ∗ ( π, π ) = S ( π, π ) /S max ( π, π ) where S max ( π, π ) corre-sponds to all particles occupying only one sub-lattice. -18-16-14-12-10-8-6-4-2 0 3 4 5 6 7 8 F r ee E n e r g y / J N p t/ ω =1.0 g = 3.0 = 2.8 = 2.6 = 2.4 = 2.3 = 2.2 = 2.1 = 2.0 = 1.9 J > 0 FIG. 3: Plot of free energy for different number of particlesand various values of g when J = 0, J = 0, and t/ω = 1 . g N p = 8, t/ ω = 0.1 n b n s S * ( π,π) J > 0(a) g n b n s S * ( π,π) N p =8, t/ ω =1.0J > 0(b) g n b n s S * ( π,π) N p =5, t/ ω =0.1 J > 0(c) g n b n s S * ( π,π) N p =5, t/ ω =1.0J > 0(d) FIG. 4: Comparative plots of S ∗ ( π, π ), n b , and n s when J =0, J = 0, and (a) t/ω = 0 . N p = 8; (b) t/ω = 1 . N p = 8;(c) t/ω = 0 . N p = 5; and (d) t/ω = 1 . N p = 5. At half-filling, we first observe that the system is eithera pure CDW or a pure SF. For a half-filled system (i.e., N p = 8) at J = 0 and t/ω = 0 . t/ω = 1 . g c ≈ .
15 ( g c ≈ . g = g c , whilethere is a sharp rise in S ∗ ( π, π ), there is also a concomi-tant sharp drop in both the condensation fraction n b andthe SF fraction n s . Furthermore, while n s actually goesto zero, n b remains finite [as follows from Eq. (8)] at avalue 1 /N = 1 /
16 which is an artifact of the finitenessof the system. Larger values of t/ω for a half-filled sys-tem leads to lower values of g c . This is in accordancewith the MFA phase boundary Eq. (11) and the factthat ( J z /J ) × ( ω /t ) [( J /J ) × ( ω /t )] is monotonicallyincreasing (decreasing) function of g for g > N p ≤ × J = 0 and J = 0).In Fig. 4 (c) drawn for N p = 5, although S ∗ ( π, π ) dis-plays a CDW transition at a critical value g c = 2 . n s does not go to zero even at large values of g considered.Furthermore, we see clearly from Fig. 2 (a) that, abovethis critical value of g , the curvature of the free energycurves suggests that the system at N p = 5 is an inhomo-geneous mixture of CDW-state and SF-state. Thus awayfrom half-filling, at small values of the adiabaticity t/ω ,our HCB-system undergoes a transition from a SF-stateto a PS-state at a critical b-p coupling strength [similarto the xxz-model in Fig. 1 (b)]. However for t/ω not too small, when NNN hopping ispresent, the system shows a strikingly new behavior for a certain region of the g -parameter space . Let us considerthe system at N p = 5, t/ω = 1 . J = 0, and J = 0.Fig. 4 (d) shows that, above g ≈ .
85, the system entersa CDW state (as can be seen from the structure factor);however, it continues to have a SF character as reflectedby the finite value of n s . Furthermore, Fig. 3 reveals thatthe curve is concave, i.e., the system is PS, only above g = 2 .
0. This simultaneous presence of CDW and SFstates, without any inhomogeneity (for 1 . < g < . supersolid . Similarly, for 6and 7 particles as well, we find that the system undergoestransition from a SF-state to a SS-state and then to a PS-state. This behavior is displayed in Fig. 1 (a).Finally, we shall present the interesting case of J = 0and J = 0 as a means of understanding the SS phase inthe phase diagram of Fig. 1 (a). The physical scenario,when J can be negligibly small compared to J , has beenaddressed in Ref. [20] for cooperative electron-phonon in-teraction in an one-dimensional system. When J = 0,for large values of nearest-neighbor repulsion, it is quitenatural that all the particles will occupy a single sub-lattice. However, the dramatic jump (at a critical valueof g ), from an equal occupation of both sub-lattices toa single sub-lattice occupation, is quite unexpected (seeFig. 5). For a half-filled system, above a critical point, allthe particles get localized which results in an insulatingstate. This can be seen from Fig. 5 (a). One can see that(at g ≈ .
23) the structure factor dramatically jumps toits maximum value, while n s drops to zero and n b takesthe limiting value of 1 /
16 for reasons discussed earlier.This shows that above g ≈ .
23, the system is in an in-sulating state with one sub-lattice being completely full.However, away from half-filling, the system conducts per-fectly while occupying a single sub-lattice because of thepresence of holes in the sub-lattice. For instance, fromFig. 5 (b) drawn for N p = 5, we see that the structurefactor jumps to its maximum value at g ≈ .
26, whereas n s drops to a finite value which remains constant above g = 1 .
26. We see from Fig. 2 (b), based on the curvatureof the free energy curves, that the 5-particle system doesnot phase separate both above and below the CDW tran-sition. In fact, this single-phase-stability is true for anyfilling. This means that the system, at any non-half fill-ing and at J = 0, exhibits supersolidity above a critical g n b n s S * ( π,π) N p = 8 J = 0, t/ ω = 1.0(a) g n b n s S * ( π,π )N p = 5, t/ ω = 1.0 J = 0(b) FIG. 5: Comparison of S ∗ ( π, π ), n b , and n s when t/ω = 1 . J = 0 but J = 0 and for (a) N p = 8; and (b) N p = 5. value of g ! Conclusions:
We demonstrated that our BH modeldisplays supersolidity in two-dimensions (2D). Our effec-tive Hamiltonian of Eq. (5) should be realizable in a2D optical lattice. Furthermore, our BH model too canbe mimicked by designing HCB to move as extra parti-cles in a 2D array of trapped molecules with HCB cou-pled to the breathing mode of the trapped molecules [23].In three-dimensions, supersolidity is more achievable (i)in general, due to an increase in the ratio of NNN andNN coordination-numbers; and (ii) in particular for bis-muthates, because cooperative breathing mode enhancesthe ratio of NNN and NN hoppings.S. Datta thanks Arnab Das for very useful discussionson implementation of Lanczos algorithm. S. Yarlagaddathanks K. Sengupta, S. Sinha, A. V. Balatsky, R. J. Cava,I. Mazin, and M. Randeria for valuable discussions. Thisresearch was supported in part by the National ScienceFoundation under Grant No. PHY05-51164 at KITP. [1] S. H. Blanton et al. , Phys. Rev. B , 996 (1993).[2] For a review, see R. L. Withers and J. A. Wilson, J. Phys.C , 4809 (1986).[3] O. Penrose and L. Onsager, Phys. Rev. , 576 (1956).[4] A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP ,1107 (1969); G. V. Chester, Phys. Rev. A , 256 (1970).[5] A. J. Leggett, Phys. Rev. Lett. , 1543 (1970).[6] E. Kim and M. H. W. Chan, Nature , 225 (2004).[7] D. Heidarian and K. Damle, Phys. Rev. Lett. , 127206(2005); R. G. Melko et al. , ibid. , 127207 (2005); S.Wessel and M. Troyer, ibid. et al. , Phys. Rev. Lett. , 207202 (2005).[9] Jinwu Ye, Phys. Rev. Lett. , 125302 (2006).[10] C. Josserand, Y. Pomeau, and S. Rica, Phys. Rev. Lett. , 195301 (2007).[11] C. M. Varma, Phys. Rev. Lett. , 2713 (1988).[12] A. Taraphder, H. R. Krishnamurthy, Rahul Pandit, andT. V. Ramakrishnan, Phys. Rev. B , 1368 (1995).[13] H. Suderow et al. , Phys. Rev. Lett. , 117006 (2005).[14] S. Robaszkiewicz, R. Micnas, and K. A. Chao, Phys. Rev.B , 1447 (1981).[15] A. S. Alexandrov and J. Ranninger, Phys. Rev. B ,1796 (1981); A. S. Alexandrov, J. Ranninger, and S.Robaszkiewicz , Phys. Rev. B , 4526 (1986).[16] E. R. Gagliano et al. , Phys. Rev. B , 1677 (1986).[17] G. Jackeli and J. Ranninger, Phys. Rev. B , 184512(2001).[18] S. Datta, A. Das, and S. Yarlagadda, Phys. Rev. B ,235118 (2005).[19] I.G. Lang and Yu.A. Firsov, Zh. Eksp. Teor. Fiz. ,1843 (1962) [Sov. Phys. JETP , 1301 (1962)].[20] S. Yarlagadda, arXiv:0712.0366v2.[21] M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev.A , 1111 (1973).[22] R. Roth and K. Burnett, Phys. Rev. A , 023604 (2003).[23] For a similar system, see G.Pupillo et al., Phy. Rev. Lett100