Two-body problem in a multiband lattice and the role of quantum geometry
TTwo-body problem in a multi-band lattice and the role of quantum geometry
M. Iskin
Department of Physics, Ko¸c University, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey (Dated: February 9, 2021)We consider the two-body problem in a periodic potential, and study the bound-state dispersion ofa spin- ↑ fermion that is interacting with a spin- ↓ fermion through a short-range attractive interaction.Based on a variational approach, we obtain the exact solution of the dispersion in the form of a setof self-consistency equations, and apply it to tight-binding Hamiltonians with onsite interactions.We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversalsymmetry, and show that the lowest-energy bound states disperse quadratically with momentum,whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlyingBloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study thespecial role played by the interband processes in producing a finite effective mass for the boundstates in a non-isolated flat band. I. INTRODUCTION
A flat band refers to a featureless Bloch band in whichthe energy of a single particle does not change when thecrystal momentum is varied across the 1st Brillouin zone.Because of their peculiar properties [1–5], there is a grow-ing demand in designing and studying physical systemsthat exhibit flat bands in their spectrum [6–11]. For in-stance, such a dispersionless band indicates that not onlythe effective mass of the particle is literally infinite butalso its group velocity is zero. This further suggests thatthe particle remains localized in real space. Then, up un-til very recently [12], one of the puzzling questions waswhether the diverging effective mass is good or bad newsfor the fate of superconductivity in a material that is to alarge extent characterized by a flat band, given that su-perconductivity, by definition, requires a finite effectivemass for its superfluid carriers.Despite such a complicacy that prevents the motion ofparticles through the intraband processes in a flat band,it turns out that the superfluidity of many-body boundstates is still possible through the interaction-induced in-terband transitions in the presence of other flat and/ordispersive bands [12]. Furthermore, in the case of an iso-lated flat band, i.e., a flat band that is separated by someenergy gaps from the other bands, it has been shown thatthe effective mass of the two-body bound states becomesfinite as soon as the attractive interaction between theparticles is turned on, independently of its strength [13].Moreover, assuming that the interaction is weak, theeffective-mass tensor is characterized by the summationof the so-called quantum-metric tensor [14–16] of the flatband in the 1st Brillouin zone. There is no doubt thatsuch few-body problems offer a bottom-up approach forthe analysis of the many-body problem, e.g., it may bepossible to use the two-body problem as a universal pre-cursor of superconductivity in a flat band [13].Motivated also by related proposals in other con-texts [17, 18], here we construct a variational approachto study the two-body bound-state problem in a genericmulti-band lattice, and give a detailed account of bipar-tite lattices with a two-point basis and an onsite interac- tion that manifest time-reversal symmetry. For this case,we show that the lowest-energy bound states dispersequadratically with momentum, whose effective-mass ten-sor has two physically distinct contributions coming from(i) the intraband processes that depend only on the one-body dispersion and (ii) the interband processes that alsodepend on the quantum-metric tensor of the underlyingBloch states. In particular we apply our theory to theMielke checkerboard lattice for its simplicity [19], andreveal how the interband processes help produce a finiteeffective mass for the bound states in a non-isolated flatband, i.e., a flat band that is in touch with others. Re-cent realizations of non-isolated flat bands include theKagome and Lieb lattices [6–11], but they both involvea relatively complicated three-point basis.The remaining parts of this paper are organized as fol-lows. In Sec. II we introduce the two-body Hamiltonianfor a general multi-band lattice, and present its bound-state solutions through a variational approach. In Sec. IIIwe focus on the tight-binding lattices with a two-pointbasis, and derive their self-consistency equations in thepresence of a time-reversal symmetry. In Sec. IV we ana-lyze the bound-state problem in a non-isolated flat band,and discuss the role of quantum metric. In Sec. V we endthe paper with a brief summary of our conclusions.
II. VARIATIONAL APPROACH
In this paper we are interested in the dispersion ofthe two-body bound-state in a periodic potential when aspin- ↑ fermion interacts with a spin- ↓ fermion through ashort-range attractive interaction [13, 20]. Our startingHamiltonian can be written as H = H + H ↑↓ , where theone-body contributions H = (cid:80) σ H σ are governed by H σ = (cid:90) d x ψ † σ ( x ) (cid:20) − ∇ m σ + V σ ( x ) (cid:21) ψ σ ( x ) . (1)Here the operator ψ σ ( x ) annihilates a spin- σ fermion atposition x , the Planck constant (cid:126) is set to unity, and V σ ( x ) is the periodic one-body potential. Without loss a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b of generality, the one-body problem can be expressed as H σ | n k σ (cid:105) = ε n k σ | n k σ (cid:105) , (2)where | n k σ (cid:105) represents a particle in the Bloch statethat is labeled by the band index n and crystal momen-tum k in the 1st Brillouin zone, and ε n k σ is the cor-responding one-body dispersion. The Bloch wave func-tion can be conveniently chosen as φ n k σ ( x ) = (cid:104) x | n k σ (cid:105) = e i k · x n k σ ( x ) / √ N c , where n k σ ( x ) is a periodic functionin space and N c is the number of unit cells in the sys-tem. We note that if N b is the number of basis sites ina unit cell, i.e., the number of sublattices in the system,then the total number of lattice sites is N = N b N c , and (cid:82) d x = N c (cid:82) unitcell d x . The two-body contribution to the Hamiltonian can bewritten in general as H ↑↓ = (cid:90) d x d x ψ †↑ ( x ) ψ †↓ ( x ) U ( x ) ψ ↓ ( x ) ψ ↑ ( x ) , (3)where the two-body potential U ( x ) depends on the rel-ative position x = x − x of the particles and hasthe same periodicity as the one-body potentials. It isconvenient to express H ↑↓ in terms of the Bloch wavefunctions. For this purpose, we combine the Fourier ex-pansions of the Bloch state | n k σ (cid:105) = √ N c (cid:80) j e i k · x j | njσ (cid:105) , where x j is the position of the lattice site j , and the Wan-nier function W nσ ( x − x j ) = √ N c (cid:80) k e − i k · x j φ n k σ ( x ) , where W nσ ( x − x j ) = (cid:104) x | njσ (cid:105) is the usual definition inthe tight-binding approximation. This leads to | x σ (cid:105) = (cid:80) nj W ∗ nσ ( x − x j ) | njσ (cid:105) , suggesting that ψ σ ( x ) = (cid:88) n k φ n k σ ( x ) c n k σ . (4)Here the operator c n k σ annihilates a spin- σ fermion inthe n th Bloch band with momentum k .The two-body dispersion E q is determined by theSchr¨odinger equation H | Ψ q (cid:105) = E q | Ψ q (cid:105) , (5)where q is the total momentum of the particles and | Ψ q (cid:105) represents the two-body bound state for a given q . Here the conservation of q is due to the discretetranslational invariance of H . The exact solutions of E q can be achieved by the functional minimization of (cid:104) Ψ q | H − E q | Ψ q (cid:105) [13, 20], where | Ψ q (cid:105) = (cid:88) nm k α q nm k c † n, k + q , ↑ c † m, − k + q , ↓ | (cid:105) (6)is the most general variational ansatz (i.e., for a given q ) with complex parameters α q nm k . Here | (cid:105) representsthe vacuum of particles and the normalization of | Ψ q (cid:105) requires (cid:80) nm k | α q nm k | = 1 . Unlike the continuum modelof uniform systems where the bound-state wave function involves pairs of particles with k + q and − k + q withina single parabolic band, here we also allow n (cid:54) = m termsto take the interband couplings that are induced by theperiodic lattice potential into account. They correspondto pairs of particles whose center-of-mass momenta areshifted by reciprocal-lattice vectors in the extended-zonescheme [20]. By plugging Eq. (4) in Eq. (3), a compactway to present the functional is (cid:104) H − E q (cid:105) = (cid:88) nm k ( ε n, k + q , ↑ + ε m, − k + q , ↓ − E q ) | α q nm k | + 1 N c (cid:88) nmn (cid:48) m (cid:48) ; kk (cid:48) U nm k n (cid:48) m (cid:48) k (cid:48) ( q ) α q ∗ n (cid:48) m (cid:48) k (cid:48) α q nm k , (7)where the non-interacting terms are simply determinedby Eq. (2) and the most general interaction-dependentmatrix elements are given by a complicated integral U nm k n (cid:48) m (cid:48) k (cid:48) ( q ) = 1 N c (cid:90) d x d x n (cid:48)∗ k (cid:48) + q , ↑ ( x ) m (cid:48)∗− k (cid:48) + q , ↓ ( x ) × U ( x ) e i ( k − k (cid:48) ) · x m − k + q , ↓ ( x ) n k + q , ↑ ( x ) . (8)Then we set ∂ (cid:104) H − E q (cid:105) /∂α q ∗ nm k = 0 , and obtain an inte-gral equation that must be self-consistently satisfied byboth α q nm k and E q as α q nm k = − N c (cid:80) n (cid:48) m (cid:48) k (cid:48) U n (cid:48) m (cid:48) k (cid:48) nm k α q n (cid:48) m (cid:48) k (cid:48) ε n, k + q , ↑ + ε m, − k + q , ↓ − E q . (9)To simplify Eqs. (8) and (9), next we restrict our analysisto the zero-ranged contact interactions where U ( x ) = U ( x ) δ ( x ) with δ ( x ) the Dirac-delta function. Suchlocal two-body potentials are known to be well-suited formost of the cold-atom systems.For instance, in the case of Hubbard-type tight-bindingHamiltonians with onsite interactions, Eq. (8) can bewritten as U nm k n (cid:48) m (cid:48) k (cid:48) ( q ) = (cid:88) S U S n (cid:48)∗ k (cid:48) + q , ↑ S m (cid:48)∗− k (cid:48) + q , ↓ S × m − k + q , ↓ S n k + q , ↑ S , (10)where S labels the basis sites in a unit cell, i.e., sub-lattices in the system, U S is the onsite interaction withthe possibility of a sublattice dependence, and n k σS = (cid:104) S | n k σ (cid:105) is the projection of the Bloch function onto the S th sublattice. Thus Eq. (9) reduces to α q nm k = − (cid:80) S U S n ∗ k + q , ↑ S m ∗− k + q , ↓ S ε n, k + q , ↑ + ε m, − k + q , ↓ − E q × N c (cid:88) n (cid:48) m (cid:48) k (cid:48) m (cid:48)− k (cid:48) + q , ↓ S n (cid:48) k (cid:48) + q , ↑ S α q n (cid:48) m (cid:48) k (cid:48) . (11)This integral equation suggests that one can determineall possible E q solutions by representing Eq. (11) as aneigenvalue problem in the nm k basis, i.e., the two-bodyproblem reduces to finding the eigenvalues of an N × N matrix for each q . Alternatively, one can introduce a newparameter set β S q = (cid:80) nm k n k + q , ↑ S m − k + q , ↓ S α q nm k , andreduce the integral Eq. (11) to a self-consistency relation β S q = − N c (cid:88) nm k S (cid:48) U S (cid:48) n ∗ K ↑ S (cid:48) m ∗− K (cid:48) ↓ S (cid:48) m − K (cid:48) ↓ S n K ↑ S ε n, k + q , ↑ + ε m, − k + q , ↓ − E q β S (cid:48) q , (12)where K = k + q and K (cid:48) = k − q are used as a shorthandnotation. Thus, for a given q , the two-body problem re-duces to finding the roots of a nonlinear equation thatis determined by setting the determinant of an N b × N b matrix to 0. We illustrate these two approaches in thenext section, where we focus on the experimentally morerelevant case of a sublattice-independent onsite interac-tions, and set U S = − U with U ≥ III. BIPARTITE LATTICES
For the sake of simplicity, below we consider a genericbipartite lattice with a two-point basis as a nontrivial il-lustration of our results, and denote its sublattices with S = { A, B } . In this case, the self-consistency equationscan be combined to give (cid:18) M q AA M q AB M q BA M q BB (cid:19) (cid:18) β A q β B q (cid:19) = 0 , where the matrix elements are M q SS = 1 − UN c (cid:88) nm k | n k + q , ↑ S | | m − k + q , ↓ S | ε n, k + q , ↑ + ε m, − k + q , ↓ − E q , (13) M q AB = − UN c (cid:88) nm k n ∗ K ↑ B m ∗− K (cid:48) ↓ B n K ↑ A m − K (cid:48) ↓ A ε n, k + q , ↑ + ε m, − k + q , ↓ − E q , (14)with M q BA = M q ∗ AB . Thus the nontrivial bound-statesolutions require the condition det M q = M q AA M q BB −| M q AB | = 0 to be satisfied. In this paper we are in-terested in the time-reversal symmetric systems where n k ↑ S = n ∗− k ↓ S ≡ n k S = (cid:104) S | n k (cid:105) . In the presence of two sublattices, the one-body con-tributions to the Hamiltonian can be written as H = (cid:88) k σ (cid:0) c † A k σ c † B k σ (cid:1) ( d k τ + d k · τ ) (cid:18) c A k σ c B k σ (cid:19) , (15)where c S k σ annihilates a spin- σ fermion in the S th sub-lattice with momentum k , and d k and d k = ( d x k , d y k , d z k )parametrize the most general Hamiltonian matrix in thesublattice basis. Here τ is an identity matrix and τ = ( τ x , τ y , τ z ) is a vector of Pauli spin matrices. Theone-body dispersions ε s k ↑ = ε s, − k , ↓ = ε s k are given by ε s k = d k + sd k , (16)where s = ± denotes the upper and lower bands, and d k = (cid:112) ( d x k ) + ( d y k ) + ( d z k ) is the magnitude of d k . The sublattice projections of the Bloch functions can bewritten as (cid:104) A | s k (cid:105) = − d x k + id y k (cid:112) d k ( d k − sd z k ) , (17) (cid:104) B | s k (cid:105) = d z k − sd k (cid:112) d k ( d k − sd z k ) . (18)By plugging these expressions into Eqs. (13) and (14),we find M q AA = 1 − U N c (cid:88) ss (cid:48) k (cid:16) s d z K d K (cid:17) (cid:16) s (cid:48) d z K (cid:48) d K (cid:48) (cid:17) ε s, k + q + ε s (cid:48) , k − q − E q , (19) M q BB = 1 − U N c (cid:88) ss (cid:48) k (cid:16) − s d z K d K (cid:17) (cid:16) − s (cid:48) d z K (cid:48) d K (cid:48) (cid:17) ε s, k + q + ε s (cid:48) , k − q − E q , (20) M q AB = − U N c (cid:88) ss (cid:48) k s d x K − id y K d K s (cid:48) d x K (cid:48) + id y K (cid:48) d K (cid:48) ε s, k + q + ε s (cid:48) , k − q − E q . (21)Before proceeding with the numerical applications, nextwe show that these exact expressions are in perfect agree-ment with those of the Gaussian-fluctuation theory thatis presented in Ref. [21].To reveal a direct link between the variational ap-proach to the two-body bound-state problem and theeffective-action approach to the many-body pairing prob-lem in the Gaussian approximation, first we considerthe normal state with a vanishing saddle-point order pa-rameters in the system, i.e., ∆ A = ∆ B = 0 for thesublattices. Then we substitute ω + 2 µ = E q afterthe analytical continuation of the Matsubara frequency iν (cid:96) = ω + i + of the pairs, and take the zero-temperaturelimit. Within the Gaussian approximation, the fluctua-tion contribution to the thermodynamic potential can bewritten as Ω G = (cid:80) q (cid:0) Λ ∗ T q Λ ∗ R q (cid:1) (cid:18) F q T T F q T R F q RT F q RR (cid:19) (cid:18) Λ T q Λ R q (cid:19) , where Λ T q = (Λ A q + Λ B q ) / R q = (Λ A q − Λ B q ) / Sq is defined as the fluctu-ations of the complex Hubbard-Stratonovich field ∆ Sq around the saddle-point order parameter ∆ S for the S thsublattice, i.e., ∆ Sq = ∆ S + Λ Sq . The matrix elementsare reported as [21] F q T T = 1 U − N (cid:88) ss (cid:48) k ss (cid:48) d x K d x K (cid:48) + d y K d y K (cid:48) + d z K d z K (cid:48) d k + q d k − q ε s, k + q + ε s (cid:48) , k − q − E q , (22) F q RR = 1 U − N (cid:88) ss (cid:48) k − ss (cid:48) d x K d x K (cid:48) + d y K d y K (cid:48) − d z K d z K (cid:48) d k + q d k − q ε s, k + q + ε s (cid:48) , k − q − E q , (23) F q T R = − N (cid:88) ss (cid:48) k s d z K d K + s (cid:48) d z K (cid:48) d K (cid:48) − iss (cid:48) d x K d y K (cid:48) − d y K d x K (cid:48) d k + q d k − q ε s, k + q + ε s (cid:48) , k − q − E q , (24)where F q RT = F q ∗ T R . Here N = 2 N c is the number oflattice sites in the system, i.e., N b = 2. We note thatsince the elements of F q and M q are related to eachother through a unitary transformation, the conditiondet F q = F q T T F q RR − | F q T R | = 0 coincides precisely withdet M q = 0. IV. NUMERICAL APPLICATION
As a specific illustration of the theory, next we applyour generic results to study the two-body problem in anon-isolated flat band, i.e., a flat band that is in touchwith others. In this context the Mielke checkerboardlattice in two dimensions is one of the simplest one tostudy since it exhibits a single flat band that is in touchwith a single dispersive band at some k points. Such alattice can be described by d k = − t cos( k x a ) cos( k y a ) ,d x k = − t cos( k x a ) − t cos( k y a ) , d y k = 0 , and d z k =2 t sin( k x a ) sin( k y a ) [19]. Here a is the lattice spac-ing between the nearest-neighbor sites of a square lat-tice, and the primitive vectors b = ( π/a, − π/a ) and b = ( π/a, π/a ) determine the reciprocal lattice. Inthis paper we let t → −| t | because it is advantageousto have the flat band as the lower one. This is be-cause, no matter how weak U is, the low-energy boundstates that are most relevant to the presence of a flatband appear just below it, i.e., they do not overlapwith the one-body states. Thus the dispersive band ε + , k = 2 | t | + 4 | t | cos( k x a ) cos( k y a ) touches quadraticallyto the flat band ε − , k = − | t | at the four corners of the 1stBrillouin zone k ≡ { ( ± π/a, , (0 , ± π/a ) } . A portion ofthe band structure is shown in Fig. 1(a) for an extendedzone.For the two-body problem of interest in this paper, firstwe find all possible E q values by solving the eigenvalueproblem that is governed by Eq. (11). The exact solutionsare shown in Fig. 1(b) for U = 5 | t | when q y a = 0. Notethat all of the high-energy bound states have an instabil-ity towards a one-body decay in the − | t | ≤ E q ≤ | t | region. For this reason we focus only on the low-energystates with E q < − | t | . In Fig. 1(b) there are two dis-tinct bound-state branches appearing in the two-bodyproblem. In contrast to the upper branch that appearsnearly featureless in the shown scale, the lower one dis-perses quadratically with momentum in the small- q limit.Given that our quadratic expansion E q = E b + q / (2 m b )is an excellent fit around q = 0, next we analyze boththe offset E b < − | t | of the lower branch and its effectivemass m b > F q RR = 0 and F q T T = 0 are in perfect agree-ment with the upper and lower branches, respectively.This is because the coupling term F q T R integrates to 0when q x = 0 and/or q y = 0. Then, in contrast toEq. (11), we note that Eqs. (22) and (23) offer an ana-lytically tractable approach. For instance one can deter-mine both E b and m b of the lower branch by substituting E q = E b + (cid:80) ij q i ( m − ) ij q j / F q T T = 0 up to second order in q . Here FIG. 1: (a) One-body dispersion ε s k is shown for the Mielkecheckerboard lattice when the lower band is flat. The bandstouch at the four corners of the 1st Brillouin zone. (b) Two-body dispersion E q is shown for U = 5 | t | as a function of q x a when q y a = 0. The conditions F q RR = 0 and F q TT = 0are in perfect agreement with the upper and lower branches,respectively. The quadratic expansion E q = E b + q / (2 m b ) isan excellent fit for the lower branch in the small- q limit. ( m − ) ij corresponds to the ij th element of the inverse ofthe effective-mass tensor m b of the lower branch. Thusthe condition F T T = 0 for the zeroth-order term leads toa closed-form expression1 = UN (cid:88) s k ε s k − E b (25)for the E b of the lower branch. Note that the familiarone-band result is recovered by Eq. (25), after setting d k = 0 in the one-body dispersion shown in Eq. (16).Similarly the condition F RR = 0 gives an expression forthe E b of the upper branch. In Fig. 2(a) we show E b forboth the upper and lower branches as a function of U .For the lower branch of main interest here, we find that E b = − | t | − U/ U limitbut it approaches to E b = − | t | − U in the large- U limit. FIG. 2: (a) Lowest energy E b = E q = of the bound stateis shown for the upper and lower branches as a function of U . (b) Inverse of the effective mass m b of the bound stateis shown for the lower branch as a function of U togetherwith its intraband and interband contributions, where 1 /m b =1 /m intra b + 1 /m inter b . Here m b = 5 π/ [ Ua ln(64 | t | /U )] fits verywell in the small- U limit. While the condition ∂F q T T /∂q i | q = = 0 for thefirst-order term is always satisfied, the condition ∂ F q T T / ( ∂q i ∂q j ) | q = = 0 for the second-order term leadsto a closed-form expression ( m − ) ij = ( m − ) intra ij +( m − ) inter ij for the effective-mass tensor, where( m − ) intra ij = 12 (cid:80) s k ∂ ε s k / ( ∂k i ∂k j )(2 ε s k − E b ) (cid:80) s k ε s k − E b ) , (26)( m − ) inter ij = − (cid:80) s k sd k g ij k (2 d k + E b )(2 ε s k − E b ) (cid:80) s k ε s k − E b ) , (27)are the so-called intraband and interband contributions,respectively. Here 2 g ij k = ∂ ( d k /d k ) ∂k i · ∂ ( d k /d k ) ∂k j is precisely the quantum-metric tensor of the Blochstates [19, 21, 22]. It is truly delightful to note thatthe expressions Eqs. (26) and (27) are formally equiva-lent to the ones reported in the recent literature in anentirely different but a related context, i.e., the effective-mass tensor of the Cooper pairs in the presence of helicitybands that is induced by spin-orbit coupling [17]. In par-ticular they suggest that while the intraband processesdepend only on the one-body band structure, the inter-band ones are controlled by the quantum geometry of theBloch states. In addition the familiar one-band result isrecovered merely by Eq. (26), after setting d k = 0 in theone-body dispersion shown in Eq. (16). This leads notonly to ( m − ) inter ij = 0 but also to ( m − ) intra ij = δ ij / (2 m )for the one-body dispersion that is quadratic in k , e.g., d k = ε + k / (2 m ), where δ ij is the Kronecker-delta. For the specific case of a Mielke checkerboard lattice, m b turns out to be a diagonal matrix with isotropic ele-ments, leading to 1 /m b = 1 /m intra b + 1 /m inter b , and theyare shown in Fig. 2(b) as a function of U . By the trial anderror approach, we find that m b = 5 π/ [ U a ln(64 | t | /U )]fits very well in the small- U limit. Since the effectiveintraband mass of the one-body dispersion diverges forthe flat band to begin with, we note that U (cid:54) = 0 isresponsible for m b (cid:54) = ∞ through the interband pro-cesses with the dispersive band, e.g., it can be shownthat ( m − ) inter ij ≈ UN (cid:80) k g ij k [1 − U/ (4 ε + , k − E b )] in the U → + limit. Here N c (cid:80) k g ij k diverges by itself due tothe touching points, and the second term is crucial forproducing a finite effective mass in the Mielke flat band,i.e., it cancels precisely those diverging points. Thusour calculation reveals the quantum-geometric mecha-nism that gives rise to a finite m b in the U → + limitas long as U is nonzero. However, away from the small- U limit, Fig. 2(b) shows that the intraband processeswithin the dispersive band also give a similar contribu-tion. The physical mechanism is known to be very differ-ent in the large- U limit [20, 23, 24], where the tunnelingof the bound state is possible only through virtual disso-ciation of the pair, and this leads to m b ∝ U/ ( a t ) asshown in Fig. 2(b).In particular to the small- U limit, we would like toemphasize that our generic result m b ∝ A / [ U ln( B /U )]for the non-isolated flat bands is in distinct contrast withthat m b ∝ A /U of the isolated ones [13], where A and B are real constants depending on the lattice structure.To be more precise, it was found that the quadraticexpansion of E q works very well for some isolated flatbands with an offset E b = − U/N b defined from the flatband and an effective-mass tensor ( m − ) ij = UN (cid:80) k g ij k in the small- U limit [13]. Here g ij k is the correspond-ing quantum-metric tensor of the Bloch states in theflat band in the presence of other flat and/or dispersivebands. In comparison to the intraband contribution ofEq. (26) for a non-isolated flat band, there is no suchcontribution for an isolated flat band in the small- U limitdue to the presence of a band gap between the flat bandand others. However, we again note that U (cid:54) = 0 is fullyresponsible for m b (cid:54) = ∞ through merely the interbandprocesses with the rest of the Bloch states in the system. V. CONCLUSION
In summary, above we constructed a variational ap-proach to study the two-body bound-state problem in ageneric multi-band lattice, and gave a detailed account ofbipartite lattices with an onsite interaction that manifesttime-reversal symmetry. For this case we showed that thelowest-energy bound states disperse quadratically withmomentum, whose effective-mass tensor has two physi-cally distinct contributions coming from (i) the intrabandprocesses that depend only on the one-body dispersionand (ii) the interband processes that also depend on thequantum-metric tensor of the underlying Bloch states. Inparticular we applied our theory to the Mielke checker-board lattice for its simplicity, and revealed how the in-terband processes help produce a finite effective mass forthe bound states in a non-isolated flat band. As an out-look, our theory can be extended to the non-isolated flatbands of Kagome and Lieb lattices that have recently been realized in a number of physical systems [6–11].
Acknowledgments
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