Building Flat-Band Lattice Models from Gram Matrices

Tight-binding lattice models that support flat bands Leykam et al. (2018), i.e., with single-particle energy dispersion $E(\mathbf{k})$ independent of momentum $\mathbf{k}$, are of great importance. The quenched kinetic energy and the associated macroscopic degeneracy in a flat band makes the system extremely sensitive to perturbations. In particular, in a many-body setting, interaction between particles in the flat band, no matter how weak it is, can result in strong correlations and exotic quantum phases. This is exactly what happens in, for example, fractional quantum Hall systems Laughlin (1983); Jain (1989); Stormer et al. (1999), where the underlying single-particle spectrum features flat Landau levels. It is therefore important to understand what model Hamiltonians can support flat bands and, conversely, how one can systematically design models supporting flat bands.

At the first glance, it seems quite simple to construct a flat-band lattice model in momentum space. Given dispersion relations $\left\{E_{1}\left(\mathbf{k}\right),E_{2}\left(\mathbf{k}\right),...,E_{n}\left(\mathbf{k}\right)\right\}$ in which one or a few of them are set to be constant, e.g., $E_{1}\left(\mathbf{k}\right)=0$, we can construct a $n$-band flat-band model from an arbitrary unitary matrix $U\left(\mathbf{k}\right)$: $h\left(\mathbf{k}\right):=U\left(\mathbf{k}\right)E\left(\mathbf{k}\right)U^{{\dagger}}\left(\mathbf{k}\right)$, where $E\left(\mathbf{k}\right)$ is the diagonal matrix with elements $\left\{E_{1}\left(\mathbf{k}\right),E_{2}\left(\mathbf{k}\right),...,E_{n}\left(\mathbf{k}\right)\right\}$. However, this approach is too general to be practical. Specifically, if we want the flat-band model to possess certain properties (e.g., finite-range hopping, certain symmetry properties, etc.), it is not straightforward to put appropriate constraints on the matrices $U\left(\mathbf{k}\right)$ and $E\left(\mathbf{k}\right)$ to make $h\left(\mathbf{k}\right)$ possess the desired properties. To solve this problem, we turn our attention from the momentum space to the real position space.

Suppose the Hilbert space is spanned by the basis $\left|\mathbf{R};i=1,2,\dots,n\right\rangle$, where $\mathbf{R}=\sum_{j=1}^{d}x_{j}\mathbf{e}_{j}$ ($x_{j}\in\mathbb{\mathbb{Z}}$) represents the position of cells in a $d$-dimension lattice and $i$’s distinguish the $n$ sites in a cell. In general, as long as a Hamiltonian $H$ defined on this basis satisfies the translational symmetry $U_{\mathbf{R}}HU_{\mathbf{R}}^{{\dagger}}=H$ for arbitrary $\mathbf{R}$, where $U_{\mathbf{R}}\left|\mathbf{R}^{\prime};i\right\rangle\equiv\left|\mathbf{R+R}^{\prime};i\right\rangle$, it depicts a $n$-band system. Now it is not difficult to make the hopping terms in $H$ finite ranged. This usually comes, however, at the cost of losing control on the band dispersions.

Although it is difficult to manipulate the energy of all the $n$ bands simultaneously in the position representation, it is still possible to set some of the bands flat. It can be done by putting the bands into the null space of the Hamiltonian. For example, the famous Lieb’s lattice, whose flat bands are crucial in proving magnetization when repulsive Hubbard interaction is turned on Lieb (1989), consists of two sublattice $A$ and $B$ with unequal numbers of sites and zero hopping magnitude between any two sites in the same sublattice. This special bipartite connectivity results in $\left|n_{A}-n_{B}\right|$ flat bands that reside in the null space of the Hamiltonian, where $n_{A}$ and $n_{B}$ are the number of sites in the two sublattice respectively. The flatness originates from the fact that the Hamiltonian is a direct sum of $h_{AB}$ and $h_{BA}$, consisting of hopping terms starting from sites in $A$ and $B$ respectively, and the rank of $h_{AB}$ ($h_{BA}$) cannot be greater than the dimension of $h_{BA}$ ($h_{AB}$). Sutherland Sutherland (1986) studies the band structure of general bipartite systems and find that these connectivity-induced flat bands occur at the middle of the spectrum, while there exist dispersive upper and lower bands that are reflectively symmetric about the average on-site energy of the two sublattices. It is remarkable that the flat bands of a bipartite lattice will not be lifted by an external magnetic field because the flatness is not due to fine-tuning of the hopping strengths Aoki et al. (1996). However, a general flat-band lattice model is not necessarily bipartite.

An alternative method to put a band into the null space of the Hamiltonian is based on the following feature of any flat bands: One can superimpose Bloch states on the flat band to make localized states which remain as eigenstates on the same band. Such localized states that occupy fewest cells are called compact localized states (CLSs). CLSs can be annihilated by their parent Hamiltonian if the band energy is shifted to zero. Conversely, as long as we find one of the CLSs, we can generate a whole family of them by translation, each centering at a different cell. Altogether, this family of CLSs form a flat band. Thus, using CLSs as generators, one can devise a systematic way of constructing flat-band models. However, it is tricky to find appropriate CLSs. For an arbitrary localized state, its parent Hamiltonian may not exist, because the Hamiltonian is found by solving an inverse eigenvalue problem that may not have a solution. Another drawback of the CLS method is that information about the band spectrum cannot be obtained readily. In particular, one cannot know a priori whether the flat band is a ground or an excited band. Moreover, the inverse problem is in general computationally cumbersome, particularly for spatial dimensions larger than one. Only very recently, the complete flat band generators on one dimensional (1D) lattice was found Maimaiti et al. (2019), but its generalization to higher dimensions is not straightforward. It is thus highly desirable to develop a more powerful and convenient method of generating flat-band models. Aoki et al. (1996); Maimaiti et al. (2017, 2019); Vicencio et al. (2015); Zong et al. (2016).

In this paper, we will present such a new method based on the simple mathematical properties of Gram matrices. The Gram matrix method put the ground states into the null space of the Hamiltonian. It guarantees the flatness of the lowest band by simple dimension-counting procedures without invoking complicated inverse problems. Also, generating flat-band models in high-dimensional lattices is straightforward as the method is insensitive to spatial dimensions. The remaining of the paper is organized as follows. We will describe the general principle of our method in Sec. II. In the next two sections, we will present some specific examples to demonstrate its usage. In Sec. III, we demonstrate the construction of several flat-band models with finite-range hopping. In Sec. IV, we construct the long-range Kapit-Mueller model in two-dimensional square lattice and present its generalization to arbitrary lattice geometry. Finally, we conclude in Sec. V.

A Gram matrix $G$ defines a semi-inner product in a linear vector space $V$. Given a linear transformation $T:V\rightarrow V^{\prime}$, $G$ is the pullback of the inner product defined in $V^{\prime}$, that is, ${G}\equiv{T}^{{\dagger}}{T}$. Obviously, $G$ is Hermitian and semi-positive definite. Under a basis $\{|v_{i}\rangle\}_{i=1}^{N}$ of the $N$-dimensional space $V$, the $i^{\mathrm{th}}$ column of the $T$ matrix is the image of $|v_{i}\rangle$, $\left|v_{i}^{\prime}\right\rangle:=T\left|v_{i}\right\rangle$, so the matrix element of $G$ is the inner product $\langle v_{i}|G|v_{j}\rangle=\langle v_{i}^{\prime}|v_{j}^{\prime}\rangle$. If the set of vectors $\{|v_{i}^{\prime}\rangle\}_{i=1}^{N}$ are linearly independent, then ${G}$ is positive definite; otherwise, $G$ would be singular and possess zero eigenvalues, and the number of zero eigenvalues equals the dimension of the kernel of $T$. If $N^{\prime}<N$, where $N^{\prime}$ is the dimension of $V^{\prime}$, then the set $\{|v_{i}^{\prime}\rangle\}_{i=1}^{N}$ is necessarily linearly dependent, and the number of zero eigenvalues that ${G}$ possesses is at least $N-N^{\prime}$.

This simple property serves as the basic principle underlying our method. Up to a shift of energy making its ground state energy zero, a Hamiltonian can always be written as $H\equiv T^{{\dagger}}T$ and can thus be interpreted as a Gram matrix. Now $T$ is a linear transformation from the Hilbert space $V$ to an auxiliary space $V^{\prime}$. We take the states $\left|\mathbf{R};i=1,2,\dots,n\right\rangle$’s as the basis of $V$. If the auxiliary space $V^{\prime}$ is spanned by sites $\left|\mathbf{R};i^{\prime}=1,2,\dots,n^{\prime}\right\rangle_{\text{aux}}$ in an auxiliary lattice, where $n^{\prime}<n$, then we claim that any $n$-band model with $n-n^{\prime}$ flat lowest bands is associated with a matrix $T$ whose elements are

Thus we can interpret the $T$ matrix graphically as hopping terms that connects a single cell in the real lattice to sites in the auxiliary lattice. Because $H\equiv T^{{\dagger}}T$, as long as $T$ is finite-ranged, so will the Hamiltonian $H$. We will construct some specific finite-range flat-band models in Section III.

The proof our statement is straightforward: Any desired flat-band Hamiltonian can be decomposed as $H=\sum_{\mathbf{k}}T_{\mathbf{k}}^{{\dagger}}T_{\mathbf{k}}$, where $T_{\mathbf{k}}$’s are $n^{\prime}$-by-$n$ matrices acting on $\mathbf{k}$-Bloch states with the usual definition $\left|\mathbf{k};i\right\rangle\equiv\sum\frac{e^{i\mathbf{k\cdot R}}}{\sqrt{N}}\left|\mathbf{R};i\right\rangle$ in momentum space. Without loss of generality, we can identify the image of $T_{\mathbf{k}}$ with the $\mathbf{k}$-Bloch states in the auxiliary space $\left|\mathbf{k};i^{\prime}\right\rangle_{\text{ aux}}\equiv\sum\frac{e^{i\mathbf{k\cdot R}}}{\sqrt{N^{\prime}}}\left|\mathbf{R};i^{\prime}\right\rangle_{\text{aux}}$. Thus $T=\sum_{\mathbf{k}}T_{\mathbf{k}}$ and the matrix elements of $T$ in Eq. (1) are related by a simple Fourier transformation. We can say that the translational symmetry of the Hamiltonian is inherited by the $T$ matrix.

To summarize our Gram matrix method, here is the protocol to generate an $n$-band model in $d$-dimensional space whose lowest few bands are degenerate:

1. 1.

   The Hilbert space is spanned by $\left|\mathbf{R};i=1,2,\dots,n\right\rangle$, where $\mathbf{R}$’s are positions of the unit cells and each unit cell contains $n$ internal sites.

2. 2.

   Construct an auxiliary space spanned by $\left|\mathbf{R};i^{\prime}=1,2,\dots,n^{\prime}\right\rangle_{\text{aux}}$, where $n^{\prime}<n$.

3. 3.

   A mapping $T$, whose matrix elements are given by Eq. (1), is constructed to map between the Hilbert and the auxiliary spaces. Note that $T$ is completely arbitrary unless some special properties are desired for the resulting Hamiltonian.

4. 4.

   The Hamiltonian can the constructed as $H=T^{{\dagger}}T$. The lowest $n-n^{\prime}$ bands of $H$ are guaranteed to be flat with energy 0.

The above is the most general protocol to generate models with flat lowest bands. The choice of the auxiliary space $V^{\prime}$ is flexible in the sense that the space is not necessarily spanned by the lattice $\left|\mathbf{R};i^{\prime}=1,2,\dots,n^{\prime}\right\rangle_{\text{aux}}$. Especially, when the restriction on finite-range hopping is lifted, then an alternative choice of $V^{\prime}$ may better serve our purpose. In Section IV, we will see that if we select the auxiliary space to be spanned by a subset of coherent states, then we can elegantly reproduce the Kapit-Mueller model and its generalizations, revealing the nature of the massive degeneracy in such models, which turns out to be a universal feature as required by the (over-)completeness of the coherent states.

Let us now demonstrate the usage of the Gram matrix method by constructing several specific models. In this section, we show the simplest models constructed by the Gram matrix method. By simplest we mean that the model Hamiltonian contains the fewest number of hopping terms in the $T$ matrix. Considering a $d$-dimensional lattice, the simplest choice of the $T$ matrix is $T\left|\mathbf{R};i\right\rangle=\left|\mathbf{R};i\right\rangle_{\text{aux}}$. However, it is trivial because the resulting Hamiltonian does not contain hopping terms between sites in the real lattice. To obtain a non-trivial Hamiltonian, the $T$ matrix has to map the real lattice cell at $\mathbf{R}$ to the auxiliary lattice cell at $\mathbf{R}$ and to at least $d$ of its nearest neighbor cells. For example,

where $a_{j},\,\,b_{j}\in\mathbb{\mathbb{C}}$. The corresponding Hamiltonian is given by

The underlying lattice is the $d$-dimensional Tasaki’s lattice (examples in 1D and 2D are presented in Fig. 1(a) and (b), respectively), and the hopping amplitudes in the original Tasaki’s Hamiltonian Tasaki (1992) represents a special case of Eq. (2) with $a_{i}=b_{i}=1/\lambda$. This Hamiltonian has $n=d+1$ bands. Since each unit cell in the auxiliary space has $n^{\prime}=d$ internal sites, $H$ possesses $n-n^{\prime}=1$ flat band. This flat band has zero energy and represents the ground band of the system. The CLSs of the flat band can be found as

If we group the sites $\left|\mathbf{R};1\right\rangle$ as sublattice $A$ and the remaining sites $\left|\mathbf{R};i=2,3,...,d+1\right\rangle$ as sublattice $B$, we find that the particle on a $B$ site can only hop to an $A$ site. Following the same argument about the origin of the flat bands in bipartite lattices, we conclude that there must be ($d-1$) additional flat bands at energy one, whose CLSs are

Finally, there exists a dispersive top band with energy $E_{\mathbf{k}}=1+\sum_{i=1}^{d}\left|\alpha_{\mathbf{k}}^{i}\right|^{2}$ where $\alpha_{\mathbf{k}}^{i}:=a_{i}+b_{i}\exp\left(-i\mathbf{k\cdot e}_{i}\right)$. The corresponding eigenstate is

The above example of constructing the $d$-dimensional Tasaki’s model serves a pedagogical purpose, from which we see the power of the Gram matrix method that it takes no extra effort to generalize the model to high dimensions, whereas the conventional CLS method involves solving more and more complicated equation when $d$ increases. By modifying the $T$ matrix, any model with a desired number of lowest few flat bands can be readily constructed. We can also artificially put constraints on the $T$ matrix so that the resulting model possesses desired properties. To this end, we now turn to construct a finite-range hopping lattice model which possesses a special feature: all of its bands are flat.

The basic idea rooted in the reflection symmetry of the upper and lower bands in a bipartite model. From Sutherland (1986) we know the middle bands in a bipartite model are flat, and if there is only one lower band (i.e., the ground band) whose flatness is guaranteed by the Gram matrix, then the only upper band must also be flat because of the reflection symmetry. So in order to build a model in which all bands are flat, we are going to design a bipartite structure which guarantees $n-2$ middle flat bands and, by choosing $n^{\prime}=n-1$, there is a zero-energy flat ground band. Such a $T$ matrix can be chosen as follows:

where $c_{j}\in\mathbb{\mathbb{C}}$ and $u_{i,j}$ is an arbitrary unitary matrix. The unitary matrix guarantees the bipartite structure: Different rows of $u_{i,j}$ are orthogonal to each other, so the hopping amplitudes between sites $\left|\mathbf{R};i=2,3,\dots,d+2\right\rangle$ are zero, and the onsite energy of these sites is uniform because each row of the unitary matrix has the same norm. The resulting Hamiltonian is given by:

The lattice connectivity in 1D and 2D are illustrated in Fig. 1(c) and (d), respectively. The bipartite nature can be easily seen if we group $\left|\mathbf{R};1\right\rangle$ as sublattice $A$ with onsite energy $E_{A}=\sum_{i}^{d+1}\left|c_{i}\right|^{2}$ , and $\left|\mathbf{R};i=2,..,d+2\right\rangle$ as sublattice $B$ with onsite energy $E_{B}=1$. This $\left(d+2\right)$-band model consists of a flat ground band with energy zero, $d$ flat bands with energy $E_{B}=1$, and a flat top band with energy $E_{A}+E_{B}=1+\sum_{i}^{d+1}\left|c_{i}\right|^{2}$, reflecting the reflection symmetry of bipartite lattices.

We have seen two examples of finite-range flat-band models. The flat bands in a finite-range Hamiltonian are necessarily non-topological with zero Chern number. This is due to a theorem Chen et al. (2014) which states that the following three conditions concerning a band cannot be simultaneously satisfied: (1) Being flat; (2) Having a non-zero Chern number; (3) The Hamiltonian is finite-range. To create a flat band with finite Chern number, it is then necessary to construct a model with infinite-range hopping amplitudes. In general, as we have demonstrated in the Introduction, a flat topological ground band can always be obtained by writing down a topologically non-trivial unitary matrix $U\left(\mathbf{k}\right)$ to construct the Hamiltonian in the momentum space: $h\left(\mathbf{k}\right)=U\left(\mathbf{k}\right)E\left(\mathbf{k}\right)U^{{\dagger}}\left(\mathbf{k}\right)$, where $E\left(\mathbf{k}\right)=$ diag$\left(0,E_{2}\left(\mathbf{k}\right),E_{3}\left(\mathbf{k}\right),\dots,E_{n}\left(\mathbf{k}\right)\right)$. Usually, we want the constructed Hamiltonian to be short ranged, in the sense that the hopping strength decays fast when the hopping distance increases. It is, however, inconvenient to work in the momentum space to restrain the hopping distance.

In Kapit and Mueller (2010), Kapit and Mueller, working in the real space, found such a topological flat band in a 2D square lattice, and attributed the massive degeneracies in the flat band to some unrevealed symmetries. It was realized that the degenerate ground states can be regarded as discrete lowest Landau levels (LLLs), and that the degeneracy of the LLLs give birth to the flatness. Here we find an alternative way to understand the origin of this topological flat band by reproducing the model with Gram matrices. More specifically, we reproduce the Kapit-Mueller model by a Gram matrix built upon a subset of coherent states. From this construction, the massive degeneracy of the ground band can be straightforwardly understood as a result of the linear dependency of the coherent states. And we find the generalizations of the model, which are beyond the LLL descriptions, can also share the ground band degeneracy.

We have proposed a powerful method of constructing lattice models supporting flat bands. The method is based on the mathematical properties of Gram matrices. Any lattice model with flat lowest band can be constructed through the method. The method does not require any elaborate calculations such as solving the inverse eigenvalue problem, works for arbitrary spatial dimensions, and guarantees to produce a flat ground band. We have presented a variety of examples, including both finite- and infinite-range hopping, topologically trivial and nontrivial flat bands. Specifically, we have constructed the $d$-dim Tasaki lattice, a $d$-dim bipartite lattice whose bands are all flat, and the generalized Kapit-Mueller lattice whose flat ground band features universal (i.e., geometry-independent) degeneracy. We study the generalized Kapit-Mueller in detail and, especially, conclude that the (over-)completeness of the coherent states is the origin of the universal degeneracy.

Finally, we want to comment on realizing flat-band models in laboratory. Over the past few years, we have witnessed rapid progress in realizing lattice models in synthetic materials, particularly synthetic dimensions, in both atomic Celi et al. (2014); Mancini et al. (2015); Stuhl et al. (2015); Zeng et al. (2015); Barbarino et al. (2016); Sundar et al. (2019, 2018); An et al. (2018); Lohse et al. (2018); Sugawa et al. (2018) and photonic Lustig et al. (2019); Jukić and Buljan (2013); Luo et al. (2015); Yuan et al. (2016); Ozawa et al. (2016); Yuan et al. (2018); Zhou et al. (2017); Ma et al. (2017); Wang et al. (2019); Anderson et al. (2016); Schine et al. (2016) systems, where the lattice sites are represented by different atomic states or photonic modes, respectively. Nearly arbitrary hopping amplitudes can be realized in such systems. Realizing the flat band models constructed using our method with synthetic materials should therefore pose no essential difficulties.

We acknowledge the support from the NSF and the Welch Foundation (Grant No. C-1669).