Mott Insulator-like Bose-Einstein Condensation in a Tight-Binding System of Interacting Bosons with a Flat Band
Hosho Katsura, Naoki Kawashima, Satoshi Morita, Akinori Tanaka, Hal Tasaki
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Mott-Insulating Bose-Einstein Condensationin a Tight-Binding System of Interacting Bosons with a Flat Band
Hosho Katsura
1, 2, 3 and Hal Tasaki Department of Physics, Graduate School of Science,The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan Trans-scale Quantum Science Institute, The University of Tokyo, 7-3-1, Hongo, Tokyo 113-0033, Japan Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan (Dated: February 23, 2021)We propose a new class of tight-binding systems of interacting bosons with a flat band, which isexactly solvable in the sense that one can explicitly write down the unique ground state. We arguethat the ground state describes a Mott insulator, and conjecture that in dimensions three or higherit may exhibit Bose-Einstein condensation, i.e., may have nonzero off-diagonal long-range order. Itis notable that a Mott insulator can exhibit a condensation phenomenon. (There is a 17 minutesvideo in which the main results of the paper are described. See https://youtu.be/_Fo4rQFBlhE ) PACS numbers: 03.75.Nt,67.90.+z,67.85.-d
Introduction .—Tight-binding models of interactingparticles with a flat band, i.e., a set of highly degener-ate single-particle energy eigenstates, have been studiedintensively over the decades. Flat band systems do notonly serve as idealized models of materials with a nar-row band, but also provide a theoretical playground forinvestigating various collective phenomena arising fromthe interplay between particle motion and interactionsin quantum many-body systems. This is because the ef-fect of interactions is magnified due to the flatness of theband. Such an approach was fruitful in the study of theorigin of ferrimagnetism [1] and ferromagnetism [2–5] inthe Hubbard model. See [6] for a comprehensive review.For the formation of a Wigner crystal and the effect ofthe change in the density in bosonic systems with a flatband, see [7–11]. The recent proposal that the nearly flatband in twisted bilayer graphene supports superconduc-tivity is quite intriguing [12].In this Letter we propose a new class of tight-bindingsystems of interacting bosons with a flat lowest band.The model is based on the construction in [13] of flatband Hubbard models, and can be regarded as a spinlessversion of the models studied in [14]. We write down theground state of the model explicitly and prove that it isthe unique ground state. The ground state shows verysmall density fluctuation, indicating that it describes aMott insulator. We conjecture that in dimensions threeor higher the ground state for suitable parameters ex-hibits Bose-Einstein condensation (BEC) in the sensethat it has nonzero off-diagonal long-range order. Asfar as we know this is the first example of the notablephenomenon of coexistence of Mott insulator and BEC.We remark that it was found in a two-component systemof bosons that one component may exhibit BEC whilethe other is in the Mott insulating state [15]. We believethat this is different from our situation, where the densityfluctuation on any part of the lattice is small. u x ˆ b u ˆ d x (a) (b) FIG. 1. (a) The lattice Λ =
E ∪ I in the case ( E , B ) isthe square lattice. The black and white dots denote the sitesin E and I , respectively. The states corresponding to the ˆ b and ˆ d operators are also shown. (b) The lattice obtained fromthe non-standard square lattice in which neighboring sites areconnected by two bonds. This corresponds to the case with d = 2 and p = 2. We also note that a class of states (without parentHamiltonians) very similar to ours was proposed and ex-amined in [16]. It was found that these states do notexhibit off-diagonal (quasi) long-range order.
The model and the exact ground state .—Let ( E , B ) bea finite lattice, where E is the set of sites and B is theset of bonds { x, y } = { y, x } with x = y and x, y ∈ E .Although ( E , B ) is arbitrary, one may imagine that itis the standard square or the cubic lattice with periodicboundary conditions. At the center of each bond { x, y } ∈B , we take a new site and denote it as u ( x, y ). We let I be the set of all sites u ( x, y ) with { x, y } ∈ B , and considerthe decorated lattice Λ = E ∪ I . See Fig. 1.We shall define a tight-binding model of boson on Λ.We denote by ˆ a † r and ˆ a r the creation and annihilationoperators, respectively, of a boson at site r ∈ Λ. Theysatisfy the canonical commutation relations [ˆ a r , ˆ a s ] = 0and [ˆ a r , ˆ a † s ] = δ r,s for any r, s ∈ Λ. The number oper-ator is defined as ˆ n r = ˆ a † r ˆ a r . We dente by | Φ vac i thestate without any bosons, i.e., the unique normalizedstate such that ˆ a r | Φ vac i = 0 for any r ∈ Λ. For each x ∈ E , we define ˆ d x = ζ ˆ a x + P u ∈N ( x ) ˆ a u , where ζ > N ( x ) is the set of sites in I thatare written as u ( x, y ) with some y ∈ E .We consider the Hamiltonian ˆ H = ˆ H hop + ˆ H int withˆ H hop = t X x ∈E ˆ d † x ˆ d x , ˆ H int = U X u ∈I ˆ n u (ˆ n u − , (1)where t > U >
0. Our model is character-ized by the three parameters ζ , t , and U . Note thatˆ H hop is rewritten in the standard form as ˆ H hop = P r,s ∈ Λ t r,s ˆ a † r ˆ a s , where the amplitude t r,s describes hop-ping between the nearest and some of the next nearestneighbor sites, and on-site potentials. We remark thatthe hopping amplitude t r,s has “wrong” signs comparedwith standard tight-binding Hamiltonians realizable, e.g.,in a cold atom system. Note also that ˆ H int describes on-site repulsive interaction only on sites in I .The hopping Hamiltonian ˆ H hop describes a tight-binding model with a flat band. To see this we defineˆ b u = ( ζ ˆ a u − ˆ a x − ˆ a y ) / p ζ for u ∈ I where x, y ∈ E is the unique pair such that u = u ( x, y ). One can easilyverify the orthogonality [ ˆ d x , ˆ b † u ] = 0 for any x ∈ E and u ∈ I . This means that ˆ H hop ˆ b † u | Φ vac i = 0 for any u ∈ I .Since the states ˆ b † u | Φ vac i with u ∈ I are linearly indepen-dent and ˆ H hop ≥
0, we see that ˆ H hop has |I| independentsingle-particle ground states (with zero energy).Let us state our first theorem which characterizes theground state of the model. Theorem 1 .— Consider the above model with particlenumber N = |I| . For any ζ > t >
0, and
U > H is unique and written as | Φ GS i =( Q u ∈I ˆ b † u ) | Φ vac i . The ground state energy is zero.Since ˆ b † u creates a particle state localized around thesite u , the particle density in the ground state | Φ GS i isalmost constant. More precisely the fluctuation of theparticle number in any region is strictly bounded by thesize of the boundary of the region. This observation in-dicates that our ground state is a kind of Mott insulator,where the density fluctuation is suppressed by the repul-sive interaction.The proof of the theorem requires the model to havea perfectly dispersionless band. However the Mott-insulating character of the ground state suggests that,when the repulsive interaction U is sufficiently large, thenature of the ground state for N = |I| remains un-changed when ˆ H hop is modified so that the lowest bandis no longer flat. In fact, such robustness of flat-bandground states is proved in some classes of Hubbard mod-els. See [6].The proof of Theorem 1 is an easy application of thetechnique developed in [4, 5, 13] for the (fermionic) Hub-bard model. Proof of Theorem 1 .— It is easily verified thatˆ H | Φ GS i = 0. Since ˆ H ≥
0, this proves that | Φ GS i is a ground state. We only need to prove that it is theunique ground state.We first note that any single-particle state can be ex-pressed by a linear combination of the operators ˆ b † u with u ∈ I and ˆ d † x with x ∈ E . This follows by observing thatˆ b † u and ˆ d † x are all linearly independent, and there are ex-actly |I| + |E| = | Λ | operators. We thus see that any N particle state is written as a linear combination of thebasis states | Φ n i = { Q u ∈I (ˆ b † u ) n u }{ Q x ∈E ( ˆ d † x ) n x }| Φ vac i ,where the “occupation number” n = ( n r ) r ∈ Λ satisfies P r ∈ Λ n r = N .Our proof is based on the standard argument for frus-tration free Hamiltonians. Let | Ψ i be an arbitrary statewith N = |I| particles such that ˆ H | Ψ i = 0, and ex-pand it as | Ψ i = P n α n | Φ n i . Noting that ˆ d † x ˆ d x ≥ n u (ˆ n u −
1) = (ˆ a † u ) (ˆ a u ) ≥
0, we see that ˆ d † x ˆ d x | Ψ i = 0and (ˆ a † u ) (ˆ a u ) | Ψ i = 0 for all x ∈ E and u ∈ I . Theserelations further imply that ˆ d x | Ψ i = 0 for all x ∈ E and(ˆ a u ) | Ψ i = 0 for all u ∈ I . The first condition impliesthat α n = 0 whenever n x = 0 for some x ∈ E , i.e., theground state contains only the ˆ b -states. Then the secondcondition implies that α n = 0 whenever n u ≥ u ∈ I . This means that | Ψ i is a constant multiplicationof | Φ GS i . Correlation function and Bose-Einstein Condensa-tion .—Although the ground state | Φ GS i is representedin a simple form in terms of the ˆ b -operators, its proper-ties are far from trivial.Note first that these operators satisfy the commutationrelation [ˆ b u , ˆ b † v ] = u = vβ u ∼ v , (2)where β = (2 + ζ ) − , and u ∼ v means that u = v andthere is x ∈ E such that u, v ∈ N ( x ). By repeatedly using(2), one finds that the normalization factor h Φ GS | Φ GS i has a compact stochastic geometric representation [16,17] h Φ GS | Φ GS i = X L β |L| , (3)where the sum is over all possible sets L = { ℓ , . . . , ℓ n } with n = 0 , , , . . . of oriented loops. See Figure 2. Byan oriented loop of length m , we mean a sequence ℓ =( u , . . . , u m ) of m distinct sites in I such that u j ∼ u j +1 for j = 1 , . . . , m , where we set u m +1 = u . Here weidentify the new sequence obtained by the shift u j → u j +1 for j = 1 , . . . , m with the original sequence. Thismeans that, for u , u ∈ I such that u ∼ u , thereis a unique loop, ℓ = ( u , u ), of length two, while for u , u , u ∈ I such that u ∼ u , u ∼ u , and u ∼ u ,there are two loops, ℓ = ( u , u , u ) and ¯ ℓ = ( u , u , u ), FIG. 2. A typical configuration of loops on I correspondingto the lattice in Fig. 1 (a). By an arrow from site u to v weindicate the commutator [ˆ b u ˆ b † v ]. of length three. We also assume that, in any set L = { ℓ , . . . , ℓ n } , no loops share a common site. Finally wewrote |L| = P nj =1 | ℓ j | , where | ℓ | denotes the length of aloop ℓ . (In short, we here consider an oriented loop modelon the line graph of ( E , B ) with site-avoiding constraint.)We can derive similar stochastic geometric represen-tation for correlation functions. Noting that ˆ a u | Φ GS i = ζ √ β ( Q w = u ˆ b † w ) | Φ vac i for u ∈ I , we find that h Φ GS | ˆ a † v ˆ a u | Φ GS i = ζ β X ′L , ω : u → v β |L| + | ω | , (4)where L is again summed over sets of oriented loops, and ω is summed over all self-avoiding walks connecting u to v , i.e., a sequence ω = ( u , . . . , u m ) of m + 1 distinctsites such that u = u , u m = v , and u j ∼ u j +1 for j = 0 , . . . , m −
1. The length of the walk is defined as | ω | = m . The prime in the sum indicates that ω andloops in L do not share common sites. By combining (3)and (4), we arrive at the following stochastic geometricrepresentation of the off-diagonal correlation function h ˆ a † v ˆ a u i GS = h Φ GS | ˆ a † v ˆ a u | Φ GS ih Φ GS | Φ GS i = ζ β P ′L , ω : u → v β |L| + | ω | P L β |L| . (5)We note that the general correlation function h ˆ a † r ˆ a s i GS with r, s ∈ Λ has a similar (but slightly more compli-cated) representation, and should behave almost simi-larly as h ˆ a † v ˆ a u i GS , especially when the distance between r and s is large.The summations over loops and walks in these repre-sentations are in general nontrivial and hardly evaluatedexplicitly. When ( E , B ) is a chain with bonds connectingnearest neighbor sites, one can evaluate the summationsby using, e.g., the transfer matrix method to show for along enough chain that h ˆ a † v ˆ a u i GS = C (cid:16) β p β (cid:17) | u − v | ≤ C −| u − v | , (6)where C = ζ β/ p β . The final bound follows byrecalling that 0 < β ≤ /
2. In this case the off-diagonalcorrelation always decays exponentially, and the groundstate does not exhibit BEC. Let us turn to the models in higher dimensions. When β is small, we see from the representation (5) thatlarge loops are suppressed, and the stochastic geomet-ric system should be in the disordered phase where con-figurations with only small loops are dominant. Thismeans that the off-diagonal correlation decays exponen-tially. Such a result indicating the absence of BEC, al-though not very interesting, can be easily proved. Notethat by relaxing the constraint L ∩ ω = ∅ in (4), wesee that h Φ GS | ˆ a † v ˆ a u | Φ GS i ≤ ζ β P L , ω : u → v β |L| + | ω | = ζ β h Φ GS | Φ GS i P ω : u → v β | ω | , which leads to the simpleupper bound h ˆ a † v ˆ a u i GS ≤ ζ β X ω : u → v β | ω | = ζ β ∞ X n =dist( u,v ) Ω u,v ( n ) β n , (7)where Ω u,v ( n ) is the total number of self-avoiding walksof length n that connect u and v , and dist( u, v ) is theminimum length of such walks. Suppose that, for any u ∈I , the number of v ∈ I such that v ∼ u is bounded fromabove by a constant ν + 1. (If ( E , B ) is the standard d -dimensional hypercubic lattice, we have ν +1 = 2(2 d − u,v ( n ) ≤ ( ν + 1) ν n − . ( Proof:
Thereare at most ( ν + 1) choices for the first step, and at most ν choices for each of the following n − v .)This upper bound, with (7), implies the following. Theorem 2 .— Let ζ be such that νβ = ν/ (2 + ζ ) < u, v ∈ I that h ˆ a † v ˆ a u i GS ≤ ζ β ( ν + 1) ν (1 − νβ ) ( νβ ) dist( u,v ) . (8)When the dimension is larger than one and β is suffi-ciently large, on the other hand, it is expected that thestochastic geometric system of loops and a walk is in thepercolating phase where a macroscopically large loop (ora walk) appears. If the critical value of β for the percola-tion transition is less than 1 /
2, which is the upper boundfor β , the ground state undergoes a phase transition.This existence of a phase transition is easily seen inthe standard large- d approximation, in which one fixes˜ β = νβ and lets d ↑ ∞ (and hence ν ↑ ∞ ). It is well-known that, for ˜ β <
1, one can neglect contributions fromloops in this limit [17]. Then the sum of the off-diagonalcorrelation is evaluated for any ˜ β < X v ∈I h ˆ a † v ˆ a u i GS ≃ ζ β X ω : u →· β | ω | ≃ ζ β ∞ X n =0 ν n β n = ζ β − ˜ β , (9)where ω is summed over all walks (with an arbitrarylength) that starts from u . The result is essentially amean-field calculation, and indicates that the groundstate undergoes a phase transition at β MF = ν − , andexhibits BEC for β > β MF . The mean-field critical pointis given by β MF = (4 d − − when ( E , B ) is the d -dimensional hypercubic lattice.The existence of a phase transition is also suggested bya more careful examination of the stochastic geometricrepresentation (5). Recall that, in (3) or (4), every loopof length three or more is summed exactly twice withdifferent orientations. This is equivalent to consideringunoriented loops, but with an extra factor 2 for eachloop. We thus see that our representation resembles thatobtained from the high-temperature expansion in an O(2)symmetric classical ferromagnetic spin system at a finitetemperature. The O(2) symmetry corresponds to theU(1) phase symmetry in the original quantum system.From this analogy we conjecture that in dimensions threeor higher the ground state | Φ GS i exhibits BEC (where theoff-diagonal correlation h ˆ a † v ˆ a u i GS converges to a nonzeroconstant when the distance between u and v are large)while in two dimensions it exhibits a quasi long-rangeorder (characterized by a power law decay of h ˆ a † v ˆ a u i GS ),both at sufficiently large β .We recall that our ground state exhibits a phase tran-sition if and only if the critical value β c is less than 1/2,and that β c should decrease when we increase the param-eter ν . (Recall that ν + 1 is the number of neighboringsites in I .) An advantage of our construction is that ν can be made as large as one wishes by choosing a suitablelattice structure. An easy way to get a lattice with large ν is to consider the hyper cubic lattice in which neighbor-ing sites are connected by p bonds [19]. See Figure 1 (b).We have ν = (4 d − p − β c < / d ≥ p .Unfortunately it is likely that the proof of the existenceof BEC is extremely difficult (if not impossible) since weneed to show the breakdown of a continuous symmetry.See [18] and references therein for rare cases where theexistence of BEC can be established rigorously by usingthe method of reflection positivity. (A readable accountof the reflection positivity method can be found in [6].)In the present model, we have a rigorous result only forthe system defined on a tree. In this case we can applythe standard techniques (see, e.g., [20, 21]) to show thatthe ground state exhibits BEC for sufficiently small ζ [17]. Other particle numbers .—It is easy to characterize theground states of the model when the number of particles, N , is less than |I| . Extending the proof of Theorem 1,we see that the space of the ground states is spanned by | Φ S i = ( Q u ∈ S ˆ b † u ) | Φ vac i where S is an arbitrary subsetof I such that | S | = N . The ground states show macro-scopic degeneracy, which should be immediately liftedwhen a generic infinitesimal perturbation is added to theHamiltonian.The case with N > |I| is difficult, and we only havea conjecture for a class of models. Assume that the lat-tice ( E , B ) is connected and bipartite, i.e., there is a de-composition E = E + ∪ E − such that { x, y } ∈ B only if x ∈ E + , y ∈ E − or x ∈ E − , y ∈ E + . Let us de-fine ˆ D = P x ∈E + ˆ a x − P x ∈E − ˆ a x . It is easily shownthat | Φ n i = ( ˆ D † ) n ( Q u ∈I ˆ b † u ) | Φ vac i is an energy eigen-state with eigenvalue E n = ntζ [17]. We conjecturethat | Φ n i is the unique ground state of the model with N = |I| + n when U is sufficiently large, but the proofseems to be difficult (even in the limiting case U = ∞ ). Discussion .—We proposed a new class of exactly solv-able models of interacting bosons with a flat band, andargued that the Mott-insulating ground states may ex-hibit BEC. We are able to prove the existence of conden-sation only for the model on a tree, but the conjecture isplausible and supported by several arguments.It may be counterintuitive that our exact ground state | Φ GS i , which consists of bosons localized at each u ∈ I ,exhibits off diagonal long-range order. One should notehowever that the operator ˆ b † u ( x,y ) creates a coherent su-perposition of the three states in which a particle is at x , y , and u ( x, y ). Because all the bosons are identi-cal the coherence “propagates” in the system generatingoff-diagonal correlation, which may be short-ranged orlong-ranged depending on the nature of the propagation.At least mathematically, the situation is parallel to thatfor the long-range N´eel order in the exact valence-bondground states of the Affleck-Kennedy-Lieb-Tasaki modelin high dimensions [21, 22].Indeed this point is related to the fact that the statesproposed in [16] only have short-ranged off-diagonalcorrelation while our ground state on a suitable two-dimensional lattice (with sufficiently large ν ) is expectedto show off-diagonal quasi long-range order. The ba-sic difference is not of qualitative but of quantitativenature, namely, the commutator [ˆ b u , ˆ b † v ] of neighboringsites, which give the basic parameter β , and the coordi-nation number ν +1 can be larger in our models comparedwith [16].It is also interesting to investigate the possibility ofsimilar models of electrons, which should exhibit super-conductivity.It is a pleasure to thank Akinori Tanaka for valuablediscussion, constructive proposals, and useful commentson the manuscript, and Kensuke Tamura for valuablecomments. H.K. was supported in part by JSPS Grant-in-Aid for Scientific Research on Innovative Areas No.JP20H04630, JSPS Grant-in-Aid for Scientific ResearchNo. JP18K03445, and the Inamori Foundation, and H.T.was supported by JSPS Grant-in-Aid for Scientific Re-search No. 16H02211. [1] E.H. Lieb, Two theorems on the Hubbard model , Phys.Rev. Lett. , 1201–1204 (1989).[2] A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations , J. Phys.
A24 , 3311(1991).[3] A. Mielke,
Exact ground states for the Hubbard model onthe Kagome lattice , J. Phys.
A25 , 4335 (1992).[4] H. Tasaki,
Ferromagnetism in the Hubbard models withdegenerate single-electron ground states , Phys. Rev. Lett. , 1608 (1992).[5] A. Mielke and H. Tasaki, Ferromagnetism in the Hub-bard model. Examples from models with degenerate single-electron ground states , Comm. Math. Phys. , 341–371(1993). https://projecteuclid.org/euclid.cmp/1104254245 .[6] H. Tasaki,
Physics and Mathematics of Quantum Many-Body Systems , Graduate Texts in Physics (Springer,2020).[7] S.D. Huber and E. Altman,
Bose condensation in flatbands , Phys. Rev. B , 184502 (2010). https://arxiv.org/abs/1007.4640 [8] S. Takayoshi, H. Katsura, N. Watanabe, and H. Aoki, Phase diagram and pair Tomonaga-Luttinger liquid in aBose-Hubbard model with flat bands , Phys. Rev. A ,063613 (2013) https://arxiv.org/abs/1309.6329 [9] M. Tovmasyan, E. van Nieuwenburg, and S. Huber, Ge-ometry induced pair condensation , Phys. Rev. B ,220510R (2013) https://arxiv.org/abs/1310.2589 [10] A. Mielke, Pair formation of hard core bosons in flat bandsystems , J. Stat. Phys. , 679–695 (2018). https://arxiv.org/abs/1708.02508 [11] J. Fronk and A. Mielke,
Localised pair formation inbosonic flat-band Hubbard models , preprint (2020). https://arxiv.org/abs/2008.01756 [12] Y. Cao, V. Fatemi, S. Fang, K. Watanabe., T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero,
Unconventional super-conductivity in magic-angle graphene superlattices , Na-ture , 43–50 (2018).[13] H. Tasaki,
From Nagaoka’s ferromagnetism to flat-bandferromagnetism and beyond — An introduction to ferro-magnetism in the Hubbard model , Prog. Theor. Phys. ,489–548 (1998). https://arxiv.org/abs/cond-mat/9712219 .[14] H. Yang, H. Nakano, and H. Katsura, Symmetry-protected Topological Phases in Spinful Bosons with aFlat Band , preprint (2020). https://arxiv.org/abs/2003.01705 [15] G.-H. Chen and Y.-S. Wu,
Quantum phase transition ina multi-component Bose-Einstein condensate in opticallattices , Phys. Rev. A , 013606 (2003). https://arxiv.org/abs/cond-mat/0205440v1 [16] I. Kimchi, S.A. Parameswaran, A.M. Turner, F. Wang,and A. Vishwanath, Featureless and non-fractionalizedMott insulators on the honeycomb lattice at / site fill-ing , Porc. Nat. Acad. Sc. (41) 16378–16383, (2013). https://arxiv.org/abs/1207.0498 [17] See Supplemental Material at (URL will be inserted).[18] M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, andJ. Yngvason, Bose-Einstein Quantum Phase Transitionin an Optical Lattice Model , Phys. Rev. A , 023612(2004). https://arxiv.org/abs/cond-mat/0403240v1 [19] A. Tanaka, private communication.[20] See, e.g., C.J. Thompson, Local properties of an Isingmodel on a Cayley tree , J. Stat. Phys. , 441–456 (1982).[21] I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Va-lence bond ground states in isotropic quantum antiferro-magnets , Comm. Math. Phys. , 477–528 (1988). https://projecteuclid.org/euclid.cmp/1104161001 [22] D.P. Arovas, A. Auerbach, and F.D.M. Haldane,
Ex-tended Heisenberg models of antiferromagnetism: Analo-gies to the fractional quantum Hall effect , Phys. Rev.Lett. , 531 (1988). Supplemental Material for “Mott-Insulating Bose-Einstein Condensation in aTight-Binding System of Interacting Bosons with a Flat Band”
Hosho Katsura and Hal Tasaki
A. Stochastic geometric representations
Let us describe in some detail the derivations of the stochastic geometric representations (3) and (4).From ˆ b u | Φ vac i = 0, we have the standard relation h Φ GS | Φ GS i = h Φ vac | (cid:0) Y u ∈I ˆ b u (cid:1)(cid:0) Y u ∈I ˆ b † u (cid:1) | Φ vac i = X π Y u ∈I [ˆ b u , ˆ b † π ( u ) ] , (A.1)where π is summed over all permutations of the elements of I . To see that (A.1) leads to the claimed representation(3), it is best to first examine a simple example. Suppose that I = { u , u , u } , and it holds that u ∼ u , u ∼ u ,and u ∼ u . Then we see explicitly from (A.1) that h Φ GS | Φ GS i = 1 + [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] + [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] + [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] + [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] + [ˆ b , ˆ b † ] [ˆ b , ˆ b † ] [ˆ b , ˆ b † ]= 1 + 3 β + 2 β , (A.2)where we abbreviated ˆ b u j as ˆ b j .To treat general cases, fix π and let I ′ = { u ∈ I | π ( u ) = u } . It is well-known (and easily proved) that I ′ isdecomposed into a disjoint union as I ′ = S nk =1 c k , where each c k is a cycle. A cycle c is a set of more than one sitethat can be rearranged into an ordered sequnce ( u , . . . , u m ) such that π ( u j ) = u j +1 for j = 1 , . . . , m , where we wrote u m +1 = u . Defining the weight for the cycle c as W ( c ) = Q mj =1 [ˆ b u j , ˆ b † u j +1 ], we see that the summand in the right-handside of (A.1) factorizes as Q u ∈I [ˆ b u , ˆ b † π ( u ) ] = Q nk =1 W ( c k ). By recalling (2), we get the desired representation (3).To derive (4), we take arbitrary v = u , and evaluate h Φ GS | ˆ a † v ˆ a u | Φ GS i = ζ h Φ vac | (cid:0) Y w = v ˆ b w (cid:1)(cid:0) Y w = u ˆ b † w (cid:1) | Φ vac i . (A.3)Since there is ˆ b u but no ˆ b † u , we must have [ˆ b u , ˆ b † u ] with u ∼ u to have a nonzero contribution. This implies that wealso need [ˆ b u , ˆ b † u ] with u ∼ u , and so on. This is terminated only when we have [ˆ b u m − , ˆ b † v ] with u m − ∼ v . Wehave obtained the contribution from the random walk. The contribution from loops can be derived as in the above. B. Large- d approximation In the large- d approximation, we can neglect the possibility that a random walk accidentally intersect its trajectory.This means that the number of length n walks is given by ν n . We used this estimate in (9). Note that we here donot make distinction between ν and ν + 1.Let us see why we can neglect the contribution from loops. Let Ω( n ) be the number of loops that contain a givensite in I . Since there are at most ( ν + 1) choices for the first step, at most n choices for each of the following n − n ) ≤ ( ν + 1) ν n − . (B.1)Thus the total contribution from all the loops containing a site is bounded from above by ∞ X n =2 Ω( n ) β n . ∞ X n =2 ν n − β n = 1 ν ∞ X n =2 ˜ β n . (B.2)This vanishes as ν ↑ ∞ provided that the sum converges. ( Remark:
To be precise, to fix a site and sum over all theloops containing it is not a proper way of evaluating the summations in (5). But it gives a correct order estimate interms of ν .) ( E , B ) ( E , B ) ( E , B ) o o o FIG. C.1. The first three generations of the tree with branching number three. The site at the root is denoted as o I I o ′ o ′ FIG. C.2. The lattices I and I . The bonds denote the connection u ∼ u ′ . The gray circles are the sites in the boundary ∂ I n , and the root of the lattice is denoted as o ′ . C. Bose-Einstein condensation in the model on a tree
We shall study the model defined on a tree, and show that the ground state exhibits Bose-Einstein condensationfor sufficiently small ζ .Let ( E n , B n ) be the regular n -generation tree with branching number three, as depicted in Figure C.1. As in themain text, we define the corresponding set of sites I n , and consider the model of interacting boson on E n ∪ I n . Ourgoal is to show that the ground state exhibits spontaneous symmetry breaking associated with BEC. We note thatthe model on the tree with branching number two does not exhibit a phase transition.A standard method to test for the existence of spontaneous symmetry breaking is to impose boundary conditionsthat explicitly favor certain order, and see if the effect of the boundary conditions survives in the infinite volume limit.In the case of ferromagnetic spin systems, this is done by enforcing spins at the boundary to point in a certain fixeddirection. In the case of BEC, the corresponding procedure is to replace ˆ b † u at the boundary by α + γ ˆ b † u with somenonzero α, γ ∈ C and then to examine the expectation value of the annihilation operator deep inside the tree. In theferromagnetic Ising model, for example, it is known that the same procedure exactly recovers the result of the Betheapproximation [20]. See also [21] for a treatment of a quantum spin state.To be precise, let ∂ I n be the set of sites at the boundary in I n . See Figure C.2. We then consider the ground statewith plus boundary conditions defined as | Φ +GS i = (cid:16) Y u ∈I n \ ∂ I n ˆ b † u (cid:17)(cid:16) Y u ∈ ∂ I n (1 + ˆ b † u ) (cid:17) | Φ vac i . (C.1)Note that we have chosen α = γ = 1 for simplicity. We are interested in the expectation value h ˆ a o i +GS = h Φ +GS | ˆ a o | Φ +GS ih Φ +GS | Φ +GS i , (C.2)especially in its limiting value as n ↑ ∞ , where o denotes the site at the root of the tree ( E n , B n ).As in the main text, we develop stochastic geometric representations for h Φ +GS | Φ +GS i and h Φ +GS | ˆ a o | Φ +GS i . Reflectingthe special geometry of I n , the representations contain loops of length two, three, or four, but not larger. Apart fromthese loops the representations contain random walks that starts from a site in ∂ I n and ends in another site in ∂ I n .Note that the symmetry breaking boundary terms play the roles of sources and sinks of the walks. Of course theloops and the walks should satisfy the site-avoiding conditions. See Figure C.3 (a). These are all contributions for (a) (b) FIG. C.3. Allowed configurations of loops and walks for stochastic geometric representations of (a) h Φ +GS | Φ +GS i and(b) h Φ +GS | ˆ a o | Φ +GS i . X + YX + Y X X X X + Y X XX X XX X X Z + Z − Z + Z − β X ( X + Y ) β X ( X + Y ) β X β X β ( X + Y ) Z + Z − β XZ + Z − X + Y Z + Z − X Z + Z − β XZ + Z − β XZ + Z − Y ′ Z + Z − Z + Z − X + YX + YX + Y X + Y X X X X X X + Y X ( X + Y ) β X ( X + Y ) β X β ( X + Y ) Z + Z − β XZ + Z − X ′ Z + Z + Z + Z + Z + Z + Z − X + YX + Y X + Y X X X X X β ( X + Y ) Z + β X ( X + Y ) Z + β X Z + β X Z + ( Z + ) ′ β ( Z + ) Z − FIG. C.4. The diagrammatic derivation of the recursion relations (C.7), (C.8), and (C.9). I n +1 is decomposed into the squarecontaining the root o ′ and three copies of I n . By specifying a configuration (of loops and walks) on the square, possible types ofconfigurations on each branch is determined separately. By counting the number of similar configurations, we get the recursionrelations. the representation for h Φ +GS | Φ +GS i , and we have h Φ +GS | Φ +GS i = ∞ X m =0 X ℓ ,...,ℓ m ∞ X k =0 X ω ,...,ω k : ∂ I→ ∂ I β P | ℓ j | + P | ω j | . (C.3)The representation of h Φ +GS | ˆ a o | Φ +GS i must contain a random walk that starts from the site o ′ , the root of I n , andends at a site in ∂ I n . See Figure C.3 (b). Thus the representation is given by h Φ +GS | ˆ a o | Φ +GS i = p β X ω : o ′ → ∂ I ∞ X m =0 X ℓ ,...,ℓ m ∞ X k =0 X ω ,...,ω k : ∂ I→ ∂ I β P | ℓ j | + P | ω j | . (C.4)Following the standard procedure for models on a tree [20, 21], we shall evaluate these sums by using exact recursionrelations. We define four sums X n , Y n , Z + n , and Z − n of loops and walks as in (C.3) and (C.4) with different conditionon the site o ′ at the root of I n . In X n we sum over all configurations where no loop or walk touching o ′ . In Y n wesum over all configurations where there are two segments (which are part of a loop or a walk) touching o ′ . In Z + n ( resp. Z − n ) we sum over all configurations where there is exactly one segment (which is a part of a walk) cominginto ( resp. going out of) o ′ . Note that we have X = 1, Y = 0, and Z ± = 1. We see that h Φ +GS | Φ +GS i = X n + Y n , h Φ +GS | ˆ a o | Φ +GS i = √ β Z − n , and hence h ˆ a o i +GS = p β Z − n X n + Y n = p β z n y n , (C.5)where we defined y n = Y n X n , z n = Z − n X n . (C.6)Now it is straightforward (although tedious) to see that X n , Y n , Z + n , and Z − n satisfy the exact recursion relations X ′ = ( X + Y ) + 3 β X ( X + Y ) + 2 β X + 6 β ( X + Y ) Z + Z − + 6 β XZ + Z − , (C.7) Y ′ = 3 β X ( X + Y ) + 6 β X ( X + Y ) + 6 β X + 3 β X + 6 β ( X + Y ) Z + Z − + 18 β XZ + Z − , (C.8)( Z + ) ′ = 3 β ( X + Y ) Z + + 6 β X ( X + Y ) Z + + 9 β X Z + + 6 β ( Z + ) Z − , (C.9)( Z − ) ′ = 3 β ( X + Y ) Z − + 6 β X ( X + Y ) Z − + 9 β X Z − + 6 β Z + ( Z − ) , (C.10)where we wrote X n − , Y n − , and Z ± n − as X , Y , and Z ± , and X n , Y n , and Z ± n as X ′ , Y ′ , and ( Z ± ) ′ . See Figure C.4.Since (C.9) and (C.10) are symmetric under the exchange of Z + and Z − and we have Z +0 = Z − , we see that Z + n = Z − n for all n .Then the relations (C.7), (C.8), (C.9), and (C.10) lead to the following recursion relations for y n and z n = Z ± n /X n . y ′ = 3 β (1 + y ) + 6 β (1 + y ) + 9 β + 6 β (1 + y ) z + 18 β z (1 + y ) + 3 β (1 + y ) + 2 β + 6 β (1 + y ) z + 6 β z , (C.11) z ′ = 3 β (1 + y ) z + 6 β (1 + y ) z + 9 β z + 6 β z (1 + y ) + 3 β (1 + y ) + 2 β + 6 β (1 + y ) z + 6 β z , (C.12)where we wrote y n − and z n − as y and z , and y n and z n as y ′ and z ′ . Our task is to start from the initial values( y , z ) = (0 , y n , z n ) in thelimit n ↑ ∞ . We get a reliable conclusion from a simple numerical calculation. We see that there is a critical value of β , which is estimated to be β c ≃ . β ∈ (0 , β c ), we see that y n → y ∗ ( β ) >
0, and z n →
0. This means thatthe order parameter h ˆ a o i +GS tends to zero as n ↑ ∞ , indicating that there is no BEC. For β ∈ ( β c , / y n → y ∗ ( β ) >
0, and z n → z ∗ ( β ) >
0. Thus the order parameter h ˆ a o i +GS converges to a nonzero value as n ↑ ∞ . Thismeans that the ground state exhibits spontaneous symmetry breaking of the U(1) symmetry, which corresponds toBEC. D. On the operator ˆ D Here we assume that the lattice ( E , B ) is connected and bipartite, i.e., there is a decomposition E = E + ∪ E − suchthat { x, y } ∈ B only if x ∈ E + , y ∈ E − or x ∈ E − , y ∈ E + . Let us denote by E ( x ) the set of sites y ∈ E such that { x, y } ∈ B . The number of elements in E ( x ), which is the coordination number, is denoted as z x .From the definition ˆ d x = ζ ˆ a x + X u ∈N ( x ) ˆ a u , (D.1)we see that [ ˆ d x , ˆ d † y ] = ζ + z x x = y ;1 { x, y } ∈ B ;0 otherwise . (D.2)The commutation relation, along with the definition (1) of ˆ H hop , implies[ ˆ H hop , ˆ d † x ] = t ( ζ + z x ) ˆ d † x + t X y ∈E ( x ) ˆ d † y . (D.3)Since E is bipartite, we can express ˆ D in terms of the ˆ d -operators asˆ D = X x ∈E + ˆ a x − X x ∈E − ˆ a x = 1 ζ (cid:16) X x ∈E + ˆ d x − X x ∈E − ˆ d x (cid:17) . (D.4)0Then, by using (D.3), we find that[ ˆ H hop , ˆ D † ] = 1 ζ (cid:16) X x ∈E + n t ( ζ + z x ) ˆ d † x + t X y ∈E ( x ) ˆ d † y o − X x ∈E − n t ( ζ + z x ) ˆ d † x + t X y ∈E ( x ) ˆ d † y o(cid:17) = 1 ζ (cid:16) X x ∈E + tζ ˆ d † x − X x ∈E − tζ ˆ d † x (cid:17) = tζ ˆ D † . (D.5)We shall show below that tζ is the minimum energy among single-particle energy eigenstates that are orthogonal tothe flat band (i.e., the states generated by the ˆ b † operators), and that the state generated by ˆ D † is the unique energyeigenstate with energy tζ .Note that by construction the operator ˆ D † does not contain ˆ a † u for any u ∈ I . This immediately implies that[ ˆ H int , ˆ D † ] = 0 . (D.6)As in the main text we let | Φ n i = ( ˆ D † ) n ( Y u ∈I ˆ b † u ) | Φ vac i . (D.7)It is clear from (D.5) and (D.6) that ˆ H hop | Φ n i = ntζ | Φ n i and ˆ H int | Φ n i = 0. We thus find that | Φ n i is an eigenstateof ˆ H = ˆ H hop + ˆ H int with eigenvalue E n = ntζ .The state | Φ n i clearly minimizes the interaction Hamiltonian ˆ H int by avoding any double occupancy on sites in I . Although | Φ n i does not minimize ˆ H hop , we recall that ˆ D † creates the state with the lowest energy among thoseorthogonal to the ˆ b † states. Since an addition of any ˆ b † state to the state ( Q u ∈I ˆ b † u ) | Φ vac i generates doubly occupiedsites in I (which costs extra repulsive energy), it is very likely that | Φ n i is a ground state provided that the interaction U is large enough. Unfortunately, we are still not able to make this heuristic argument into a proof, even in the limitingcase with U = ∞ .It remains to justify our claim about the minimum energy among states orthogonal to the flat band. Note that anysingle-particle state orthogonal to the flat band is written as | ϕ i = X x ∈E ϕ x ˆ d † x | Φ vac i (D.8)with arbitrary coefficients ϕ x ∈ C . By using the commutation relation (D.3), we see thatˆ H hop | ϕ i = X x ∈E ϕ x n t ( ζ + z x ) ˆ d † x + t X y ∈E ( x ) ˆ d † y o | Φ vac i = X x ∈E n t ( ζ + z x ) ϕ x + t X y ∈E ( x ) ϕ y o ˆ d † x | Φ vac i . (D.9)Thus the Schr¨odinger equation ˆ H hop | ϕ i = ǫ | ϕ i reduces to t ( ζ + z x ) ϕ x + t X y ∈E ( x ) ϕ y = ǫϕ x for any x ∈ E . (D.10)Multiplying this by ϕ ∗ x and summing over x , we find tζ X x ∈E | ϕ x | + t X { x,y }∈B | ϕ x + ϕ y | = ǫ X x ∈E | ϕ x | , (D.11)which implies that ǫ ≥ tζ . We also see that the minimum eigenvalue tζ is attained only when ϕ x + ϕ y = 0 for all { x, y } ∈ B , i.e., ϕ x = c for all x ∈ E + and ϕ x = − c for all x ∈ E − with a nonzero constant cc