Construction of interacting flat-band models by molecular-orbital representation: Correlation functions, energy gap, and entanglement

We calculate correlation functions of exactly-solvable one-dimensional flat-band models by utilizing the"molecular-orbital"representation. The models considered in this paper have a gapped ground state with flat-band being fully occupied, even in the presence of the interaction. In this class of models, the space spanned by the"molecular-orbitals"is the co-space of that spanned by the flat bands. Thanks to this property, the correlation functions are calculated by using the information of the molecular-orbitals rather than the explicit forms of the flat-band wave functions, which simplifies the calculations. As a demonstration, several one-dimensional models and their correlation functions are presented. We also calculate the entanglement entropy by using the correlation function.

In flat-band models, complete quench of kinetic energy may allow us to obtain the exact ground state by simply finding a state that minimizes the interaction term. The ferromagnetic state in the flat-band Hubbard model is a representative example. Recently, this protocol was also applied to the Wigner crystal [33] and the quantum scar [34][35][36]. In these examples, the interaction has a natural form (e.g., the Hubbard interaction or the nearest-neighbor density-density interaction), but the flat-band wave functions have a suitable real-space profile so that the interaction terms do not act on the many-body states composed of the flat-bands.
On the other hand, it is also possible to tune the interaction such that it is vanishing when acting on the many-body states with the flat-band states being occupied. In this paper, we argue this type of interacting flat-band models. For the construction of the models, we rely on the molecular-orbital (MO) representation, which we have developed to describe flat-band models [32,[37][38][39][40][41]. The key idea of the MO representation is to write down the single-particle Hamiltonian by using the unnormalized and non-orthogonal wave functions, i.e., the "MOs", whose number is smaller than the number of atomic sites. Based on this, we consider the interaction term written down by the MOs. The form of the interaction in the atomic-site basis is, in general, not in a natural form as the conventional interactions, but is highly fine-tuned. Nevertheless, the models possess several intriguing features. Namely, we not only obtain the exact ground state, but also calculate the correlation function for the ground state. Specifically the correlation function can be obtained without explicitly deriving the flat-band wave functions. Rather, the forms of the MO have all the information needed to calculate the correlation function, which is a unique feature of this type of models and has not yet been pointed out to our knowledge.
The rest of this paper is structured as follows. In Sec. 2, we review the MO representation of the flat-band models. In Sec. 3, we describe the details of the single-particle eigenvalues and eigenfunctions of the models. In Sec. 4, we introduce the interaction term to construct the exactly-solvable flat-band models. The form of the ground state is also shown. In Sec. 5, we present the formal expression of the correlation function for the ground state. In Sec. 6, we discuss two concrete examples, namely, the saw-tooth lattice model and the diamond chain, and show the explicit forms of the correlation functions. For the saw-tooth lattice model, we also show the entanglement entropy computed by using the correlation function. Section 7 is devoted to a brief summary of this paper.

Molecular-orbital representation of flat-band models
We first review the MO representation of generic flat-band models [32,[37][38][39][40][41]. This model construction method can be viewed as an extension of the "cell construction", which has been developed in the context of the ferromagnetism in the Hubbard-type models [3,5,8,9]. Let us consider a spinless-fermion lattice model of N sites: whereĈ i is an annihilation operator of a MO i, (i = 1, · · · , M ) aŝ and For later use, we define which is an N × M matrix. Note thatĈ i satisfies the anti-commutation relation: where O := Ψ † Ψ is the M × M matrix and is referred to as an overlap matrix. The single-particle Hamiltonian of Eq. (1) can be written by using c aŝ where Using the following formula for N × M and M × N matrices Then if N > M , there are at least N − M (> 0) zero modes 1 . When applying this argument to the momentum space representation, one obtains flat-bands. One may apply it to the random case in a real space as well [32,39]. We also note that additional zero modes appear when Equation (

Single-particle eigenvalues and eigenstates
In what follows, we assume that detO = 0. 1 The lower bound of the number of the zero modes is straightforwardly obtained from Eq. (8), because the dimension of the kernel of Ψ † as a linear map is equal to or greater than N − M . It can also be found from Eq. (8) that the number of the zero modes is equal to the dimension of the kernel of hΨ † .

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Using the MOs, we can also derive the eigenstates other than the zero modes. Let Ψ ′ be the eigenmultiplet: with E = diag (ε 1 , · · · , ε M ) and Φ being the M × M matrix. Note that Φ is not unitary. For later use, we assume ε 1 ≤ ε 2 · · · ≤ ε M . From Eqs. (8) and (10), we have By further operating Ψ † from the left, we obtain Defining h ψ := O 1/2 hO 1/2 , Eq. (12) can be further deformed as where Φ ′ := O 1/2 Φ. Importantly, we can choose Φ ′ as a unitary matrix since h ψ is an Hermitian matrix. This leads to and thus From Eqs. (13) (14), and (15), we have Substituting Eq. (16) into Eq. (8), we can write down the single-particle Hamiltonian aŝ whereĈ ′ := Ψ ′ †ĉ = Φ † Ψ †ĉ is a set of orthogonal MOs. We note that holds, and thatĈ ′ satisfies the anti-commutation relation: To obtain (18) and (19), we have used (14). Let us here note that Ψ ′ is orthonormalized as where we have used (15). However, it is not complete generically, since Here we define P MO which is the projector to the vector space spanned by the orthonormalized MOs Ψ ′ . Note that the relation P 2 MO = P MO holds, which is the generic requirement of the projector.

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The co-space of the space spanned by the MOs corresponds to the zero-energy flat-band. To be concrete, the projector of the zero-energy flat bands is given as One can see that the followings hold: and The rank of P MO is M and that of P 0 is Z = N − M due to the assumption that detO = 0. 2 . This means the eigenspace of P 0 is spanned by Z orthonormalized vectors ϕ 1 , · · · , ϕ Z . Defining by which P 0 can be written as we have and Equation (27) leads to This implies for i = 1, · · · , Z and j = 1, · · · , M ; here we have introduced ψ ′ j such that Now, let us define a set of zero-mode fermions as 3 They satisfy the anti-commutation relation: We note that ẑ i ,Ĉ ′ † j = 0 due to (30); this also implies ẑ i ,Ĉ † j = 0 because of (18). 2 We emphasize that this does not mean the number of zero modes is Z. Indeed, additional zero modes can appear when ε n = 0 for some n. In that case, deth = 0.
3 Note that z i zero-mode fermions are not necessarily compactly supported localized states; as a concrete example, see Ref. [16]. Generally the zero mode has an extended tail. Also, in this sense, the zero-mode states are not necessarily compact localized states [42].

Interaction and many-body ground state
In the following, we consider the case where ε n > 0 for all n. In other words, the number of zero modes forĤ 0 is Z and they are given as z † i (i = 1, · · · , Z). Let |t be an arbitrary many-body state. Noting we see thatĤ 0 is positive semi-definite.
To construct the exactly-solvable model with interactions, we consider the interaction of the form:Ĥ which is the interaction among MOs. This interaction Hamiltonian is positive semi-definite as well, since The Z-particle many-body ground state of the HamiltonianĤ =Ĥ 0 +Ĥ int is given by where |0 is a vacuum ofĉ i :ĉ for all i = 1, · · · , N . The state |z is the zero-energy eigenstate ofĤ because of the anticommutation relation betweenẑ's and (Ĉ)'s. More precisely, |z satisfiesĤ 0 |z = 0 and H int |z = 0 simultaneously. Note that similar exact many-body eigenstates with vanishing interaction energy have been discussed in several contexts [5,33,35,36,43]. AsĤ is positive semi-definite, |z is a ground state forĤ. Also, |z is the unique ground state for the Z-particle system, since |z is the only state that satisfies z|Ĥ 0 |z = 0. It is also worth mentioning that, even when the conditions ε n > 0 and V i,j ≥ 0 are relaxed, |z is an exact eigenstate ofĤ 0 +Ĥ int . In fact, as we will show in Sec. 6, |z has short-range correlations and the entanglement entropy for |z obeys an exact area law for a typical model, which implies that |z becomes a quantum scar when it is embedded in the middle of the many-body spectrum.

Correlation function
The correlation function of the fermions at i and j with respect to the ground state is Noting and thusĉ we have Using Eqs. (21) and (22), we obtain Note that the expression of Eq. (44) was obtained by Mielke in the study of flat-band ferromagnetism [44,45]. In this regard, the merit of using Eq. (45) is that one can obtain the correlation function without deriving the explicit form of P 0 , i.e., without diagonalizing the single-particle HamiltonianĤ 0 . This is advantageous whenĤ 0 is not translationally invariant due to disorders [32]. We also emphasize that the result does not depend on the interaction V i,j , and that the higher-order correlation function can be obtained by using the Wick's theorem. 7/17

Analytic calculation in one-dimensional systems
In this section, we study concrete examples. Specifically, we focus on the one-dimensional models, namely the saw-tooth-lattice model and the diamond-chain model, where the analytic form of the correlation function can be obtained. We first study the saw-tooth lattice [ Fig. 1(a)], whose single-particle Hamiltonian iŝ

Saw-tooth lattice
Here, a 1 , a 2 , a 3 ∈ C are the parameters. Note that the periodic boundary condition is imposed asĉ 2L+j =ĉ j . Due to the translational invariance, we can perform the Fourier transformation, and derive the band structure. In Fig. 1(b), we plot the band structure for a representative set of parameters. We see that the first band is the zero-energy flat-band, and there is a finite band gap between the first and the second bands. In this model, the MO is set aŝ by which the single-particle Hamiltonian can be written aŝ Corresponding Ψ † and O have the forms, and respectively.

8/17
Here we focus on a many-body ground state at half-filling. To obtain the correlation function of Eq. (45), it is helpful to utilize the momentum-space representation. Namely, we define a L × L unitary matrix U L : and a 2L × 2L unitary matrix U 2L : for n, m = 1, · · · , L, where q n = 2π L · (n − 1). Then, we have with Ψ † qn being the 1 × 2 matrix: For later use, we define A qn = a 1 + a 3 e −iqn , B qn = a 2 . From Eq. (53), we have a blockdiagonal form of the overlap matrix: where Combining Eqs. (53) and (55), we have with For the calculation of the correlation function g i,j , we focus on the case of i = 2m − 1 and j = 2m ′ − 1; calculations for the other cases can be carried out in the same manner. In this case, we have To obtain the third line of Eq. (59), we have used L n=1 e iqn(m−m ′ ) = Lδ m,m ′ . The above expression of the correlation function is exact, but we perform a numerical demonstration employing the exact diagonalization to check it. As for the interaction term, 9/17  we considerĤ int = V mĈ † mĈ † m+1Ĉ m+1Ĉm . To obtain the site-basis representation of the interaction term of Eq. (35), we use a numerical code in Dirac-Q [46]. For the exact diagonalization, we use the quantum lattice-model solver HΦ [47]. In Fig. 2, the correlation function for the system with L = 10 (the number of sites is 20) is shown for several values of V . We see that the correlation function is indeed independent of V . We also see that the numerical results show a good agreement with the analytic one. We further evaluate the correlation length in the limit of L → ∞. For L → ∞, we can replace 1 L L n=1 with 2π 0 dq 2π . Then, we have where we have introduced x := m − m ′ , T := |a 1 | 2 + |a 2 | 2 + |a 3 | 2 , and ǫ := (a 1 a * 3 )/T . Up to the coefficient |a2| 2 T , the right-hand side of Eq. (60) can be calculated by replacing the variable as z = e i(q+argǫ) : where z <,> are the solutions of the quadratic equation z 2 + |ǫ| −1 z + 1 = 0, i.e., and satisfying |z < | < 1 and |z > | > 1. Therefore, the correlation function is given as This implies the exponential decay of the correlation function. The correlation length ξ, by which the correlation function is written as g (2m−1),(2m ′ −1) ∼ e −|m−m ′ |/ξ , is In Fig. 3, we plot the energy gap ∆ and the correlation length, fixing a 1 = a 3 = 1 and varying a 2 . In this case, the energy gap is given as ∆ = (a 2 ) 2 . We see that ξ is diverging for ∆ → 0 and that ξ decreases upon increasing ∆. We note that the correlation function for ∆ = 0 is vanishing due to the factor |a 2 | 2 in Eq. (64). This is accounted for as follows. For a 2 = 0, the even-numbered sites are completely decoupled from the odd-numbered sites, and the atomic states at the even-numbered sites form the flat-band. Then, the odd-numbered sites are empty for the ground state, resulting in the vanishing of the correlation function. Furthermore, we show another useful application of the correlation function analytically obtained by Eq. (45). That is, the entanglement entropy can be obtained directly from the correlation function. The reduced density matrix for a certain subsystem (A-subsystem) denoted by ρ A can be extracted from the correlation function [48][49][50] Here, the eigenvalues of ρ A are the entanglement spectra denoted by λ n (n = 1, 2, · · · , N A , where N A is the number of lattice sites of the A-subsystem). Then, from the values of λ n , 11/17 the entanglement entropy is given by [49], We can extract the entanglement property of the many-body ground state at half-filling in the saw-tooth lattice. By substituting Eq. (49) and Eq. (50) into Eq. (45) and setting the entanglement cut where the A-subsystem includes all even-numbered sites as shown in Fig. 4, we calculated S A . Here the eigenvalues of the correlation matrix, λ n , are computed numerically. We note, however, that this bipartition is translationally symmetric, and thus S A can be computed by using the momentum space representation; see Appendix A for details. The result of S A versus the energy gap ∆(= |a 2 | 2 ) for a 1 = a 3 = 1 is shown in Fig. 4(b). The entanglement entropy between even (2m) sites and odd (2m + 1) sites depends on the size of the gap ∆. The entanglement entropy increases for small ∆ to a peak at S A /N A ∼ ln 2, and then decays as increasing ∆. Actually, this behavior is related to the form of the Wannier function of the flat band. From the Hamiltonian H 0 of Eq. (46), the Wannier function can be calculated. At the saturation point of S A , ∆ ∼ 1.8, the Wannier function is mainly localized on three sites as shown in the inset of Fig. 4(b). Here, the two odd-numbered sites and the single even-numbered site are highly entangled. On the other hand, for larger ∆, the Wannier function is mainly localized on the two adjacent odd sites as shown in the inset of Fig. 4(b). This means that the odd-numbered sites and even-numbered sites are weakly entangled. And also, the result in Fig. 4(b) shows almost no system-size dependence. This indicates that the entanglement of the many-body ground state shows the area-law, which is characteristic in generic unique gapped ground states. The second example is the diamond chain [ Fig. 5(a)]. The single-particle Hamiltonian readŝ where a 1 -a 6 ∈ C are the parameters and the periodic boundary condition is imposed aŝ c 3L+j =ĉ j . The band structure for the representative parameters is shown in Fig. 5(b), where we see the zero-energy flat-band with a finite energy gap to the other bands. For this model, we define two species of MOs: and then the Hamiltonian can be written aŝ The momentum-space representation of Ψ † and O can be obtained by the same procedure as the previous subsection, and we have and 13/17 where X 1 = |a 1 | 2 + |a 2 | 2 + |a 3 | 2 , X 2 = |a 4 | 2 + |a 5 | 2 + |a 6 | 2 , and Y qn = a 1 a * 6 e iqn + a 2 a * 4 + a 3 a * 5 . Then we find where and We focus on a many-body ground state at 1/3-filling. Then the remaining calculation of the correlation function follows that of the previous subsection, and we obtain the correlation function of the limit of L → ∞ as where γ = β/α ′ andz We thus obtain the correlation length,

Summary
We have formulated how to calculate the correlation function in the flat-band models, whose single-particle Hamiltonian and the interaction term are both constructed from the MOs. Our main result is represented by Eq. (45), which implies the correlation function depends only on the structure of MOs, irrespective of the details of V i,j (as far as the interaction term is positive semi-definite). This formulation of calculating the correlation function has some useful applications such as obtaining the correlation length and the entanglement entropy of the many-body ground state.
As concrete examples, we study the saw-tooth lattice and the diamond chain, where we presented the analytic forms of the correlation functions and the correlation length. In these examples, the MOs are composed of the atomic sites in a range of neighboring unit cells. Thus, the analytic calculation of the correlation function can be performed by using the changing the variable as z ∝ e iq and using the residue theorem. It is noteworthy that we can perform the analytic calculation in the same way for the models with the MOs are composed of the atomic sites with farther unit cells. In such cases, the correlation length is determined by the pole of the integrand with the absolute value being the closest to 1. 14/17 In this appendix, we derive the analytic form of the entanglement entropy for the saw-tooth lattice model shown in Fig. 4, derived by using the momentum-space representation. As we have seen, the correlation function does not depend on the interaction, so it is sufficient to consider the single-particle Hamiltonian.
To begin with, we introduce the Bloch Hamiltonian by performing the unitary transformation of Eq. (52) to H 0 . By doing so, we have with where A qn and B qn are those defined in Sec. 6.1. The flat-band eigenstate is obtained as the zero-energy eigenvector of H 0,qn , ϕ qn , whose explicit form is where the subscripts o and e stand for the odd-numbered and even-numbered sites, respectively, we can write the creation operator of the flat band state, z † qn , as Then, the many-body ground state is written as Now, the correlation function for the even-numbered sites are g 2m,2m ′ = Ψ 0 | c † 2m c 2m ′ |Ψ 0 = n,n ′ e i(qnm−q n ′ m ′ ) Ψ 0 | c † qn,e c q n ′ ,e |Ψ 0 . |A qn | 2 |A qn | 2 + |B qn | 2 . (A10) Having this at hand, we can compute the entanglement entropy S A by using the eigenvalues of the L × L matrix g 2m,2m ′ (m, m ′ = 1, · · · , L). After some algebras, we find that the eigenvalues are given as  ∆ dependence of S A /N A for L = 64. The blue circles and orange triangles denote the result obtained by the method described in the main text and that of analytical expression derived in the appendix, respectively.
We now focus on the case of a 1 = a 3 = 1. Figure A1 shows the ∆ (= a 2 2 ) dependence of S A /N A for L = 64, comparing the result obtained in the main text with that of the analytical expression derived in this appendix. Clearly, they show a good agreement within the numerical accuracy.