Symmetry-protected flatband conditions for Hamiltonians with local symmetry

Hamiltonians with discrete translational symmetry can have dispersionless bands known as flatbands, with energy spectra $E(\bm{k})$ independent of momentum $\bm{k}$ [1, 2, 3, 4, 5]. The flatness of a band implies zero group velocity, infinite effective mass of electrons, and suppressed electron and wave transport. The origin of flatbands is destructive interference due to a fine-tuning of the hopping or lattice symmetry. An important property of flatbands in short-range Hamiltonians is compact localized states (CLSs)—flatband eigenstates that are perfectly localized on a finite number of lattice sites. This is in contrast to Anderson localization where eigenstates are localized exponentially over the entire lattice [6]. Since the first report of a flatband in a dice lattice [7], a variety of flatband models have been identified, e.g., Lieb [8, 9, 10, 11, 12, 13], kagome [14, 15, 16, 17, 18, 19, 20], and honeycomb [21, 22, 23, 24] lattices. Despite their fine-tuned character and strong sensitivity to perturbations, flatbands have been realized in multiple experiments in different settings: superconducting networks [25, 26], photonic flatbands [27, 28, 29, 30, 31], optical lattices for cold atoms [32, 33, 34, 35, 36], and engineered atomic lattices [37, 38, 39].

Symmetry, one of the fundamental principles of physics, allows us to predict certain properties of a system without solving the often complicated underlying equations. In quantum mechanics, symmetry is associated with an operator that commutes with a Hamiltonian. A symmetry is global if the respective operator is independent of lattice sites, and a symmetry is local if the operator is dependent on lattice sites. It has been discovered that certain classes of local symmetries can indeed be systematically linked to CLSs and flatbands [3, 40, 41]. Although it is well known that flatbands and CLSs result from destructive interference caused by fine-tuning [42] or by specific symmetries [7, 8], a relation between fine-tuning and symmetry has not been fully established yet [43, 44]. In this work, we derive the exact conditions for CLSs occupying a single unit cell and the corresponding flatbands from the global and local symmetries of the system. We then propose a method to design lattice Hamiltonians with flatbands in terms of given symmetries and corresponding unitary operators. As a result, we demonstrate that if a Hamiltonian possesses a local symmetry for which the associated unitary operators are also operators of a global symmetry of the Hamiltonian, such Hamiltonian should have at least one compact localized state and corresponding flatband.

This paper is structured as follows. In Section II, we derive the conditions for CLSs and corresponding flatbands in a Hamiltonian with discrete translational symmetry in terms of both global and local symmetries. In Section III, we illustrate the method with several simple examples, including already-known flatband models. Section IV introduces generalizations of our method to longer-range hopping and higher dimensions. In Section V, we summarize and discuss our results.

Consider a one-dimensional (1D) tight-binding model with nearest-neighbor unit cell hopping. The Hamiltonian in the second quantized form is given by

where $\alpha=1,2,\cdots,m$ labels the basis in the unit cell, $\hat{c}_{i\alpha}$ ($\hat{c}_{i\alpha}^{\dagger}$) is an annihilation (creation) operator at unit cell $i$ and basis site $\alpha$, and $N$ is the total number of cells. $H_{0;\alpha\beta}$ are on-site energies ($\alpha=\beta$) and hopping constants ($\alpha\neq\beta$) within the unit cell, and $H_{1;\alpha\beta}$ are inter-cell hoppings. The matrix notation with respect to the basis $\alpha$ is introduced in the second equation. Since we are interested in single-particle models, the statistics of $\hat{c}_{i\alpha}$ (fermionic or bosonic) is irrelevant.

Let us assume that a given matrix $H_{0}$ has a nontrivial finite symmetry group $\mathcal{G}$ of order larger than one, e.g., with more than one element. Then all matrices of a representation $U_{i}$ of group $\mathcal{G}$ commute with the intra-cell Hamiltonian

Here the subscript $i$ indicates the lattice site. Our goal is to systematically find $H_{1}$ consistent with the symmetries $\mathcal{G}$, such that the full Hamiltonian, Eq. (1), has flatbands. This consistency condition has important implications when the group $\mathcal{G}$ has more than one element. The symmetry operation on the total system can be represented as a product of these operators, $U_{\mathrm{tot}}(g)=\otimes_{i=1}^{N}U_{i}(g)$. For a global symmetry, the operator acting on site $i$ $U_{i}(g)=U(g)$ is the same for all sites. Requesting that the Hamiltonian $\mathcal{H}$ is invariant under global symmetry, one obtains the condition

in addition to the condition in Eq. (2).

Now, consider two different symmetries of $H_{0}$, $g_{1}$ and $g_{2}$, both elements of $\mathcal{G}$. Requesting that the total Hamiltonian is invariant under $g_{1}$ and $g_{2}$ globally, we have the following commutation relations,

where $\sigma_{1,2}=U(g_{1,2})$. However, the presence of two distinct elements allows to construct more complex symmetries. In particular, to impose a CLS, we consider a kink operation, namely a symmetry $U_{\mathrm{tot}}=\otimes_{i=1}^{N}\sigma_{i}$ acting as $\sigma_{1}$ for sites $i\leq l$ and $\sigma_{2}$ for $i\geq l+1$, as shown in Fig. 1. We now require that the total Hamiltonian is invariant under the kink operation. Then at the location of the kink, we have the condition

To show that this condition implies the existence of a CLS, we rewrite the condition as

Acting with this operator on eigenstates $\phi_{n}$ of $H_{0}$, we find

If $\phi_{n}^{\prime}=0$ for all $n$, then $(1-\sigma_{2}^{-1}\sigma_{1})=0$ or $\sigma_{1}=\sigma_{2}$, due to the completeness of the eigenstates $\{\phi_{n}\}$. This contradicts the assumption of two different symmetries $g_{1}$ and $g_{2}$. There is therefore at least one nontrivial state $\phi_{n}^{\prime}$ that is a zero mode of $H_{1}$ and that is also an eigenstate of $H_{0}$, since $\sigma_{1,2}$ commute with $H_{0}$ [Eq. (4)]. Accordingly, the state $\phi_{n}^{\prime}$ is an eigenstate of the total Hamiltonian with $H_{1}\phi_{n}^{\prime}=0$, and so it is a CLS.

We note that the operator $\sigma_{i}$ is determined up to phase; see Eq. (2). The number of allowed phases is finite for a finite symmetry group. Consider an element $g$ of order $N_{g}$,

where $e$ is the identity element. Let $U(g)$ be the operator corresponding to $g$. Then the operator $e^{i\theta}U(g)$, $\theta\in\mathbb{R}$ also satisfies Eq. (2). However, the condition $g^{N_{g}}=e$ restricts the values of $\theta$, $e^{iN_{g}\theta}=1$, or $\theta_{n}=2\pi\frac{n}{N_{g}}$, $n=0,1,\cdots,N_{g}-1$. Thus there can be $N$ operators

corresponding to the element $g$. Due to the unitarity and homeomorphic property of $U(g)$, the phase $e^{i\theta}$ is the same as the eigenvalues of $\sigma_{g}\equiv U(g)$. Therefore, we can rewrite the flatband condition in Eq. (5) as

where $\lambda_{g}^{i}$ are eigenvalues of $\sigma_{g}$ with $i=1,2,3,\cdots,N_{g}$. This is the condition that is mainly used in the following sections. From Eqs. (4) and (10), one can obtain the total Hamiltonian $\mathcal{H}$ with flatbands.

We illustrate the generic method outlined above by considering a given intra-cell hopping matrix $H_{0}$. First, we determine the symmetry group $\mathcal{G}$ of $H_{0}$, e.g., the unitary operators commuting with $H_{0}$. Then we identify inter-cell hopping matrices $H_{1}$ from the given $H_{0}$ and the associated unitary operators using the flatband condition derived in the previous section, Eq. (10). We note that setting $H_{1}=0$ (or equivalent) is also a solution; however, this corresponds to the trivial case of disconnected unit cells and we do not consider such cases in the derivations below.

Similarly as in the previous section, one can extend the analysis to the case of an arbitrary number of bands. Our approach can be extended to longer-range hopping and higher dimensions, as we demonstrate below.

We derived the conditions for lattice Hamiltonians with flatbands to have compact localized eigenstates that localize perfectly in a single unit cell. The conditions are based on unitary operators commuting with the Hamiltonian and are associated with local symmetries that guarantee compact localized states and flatbands. Beyond flatbands in lattice models, we can also apply our results to perturbed Hamiltonians where some internal states are not affected by additional perturbations (see Appendix E). We expect that the conditions derived here can be extended to design extraordinary states robust against local perturbations or environmental changes in a variety of coupled systems.

The authors thank Emil Yuzbashyan for helpful discussions. We acknowledge financial support from the Institute for Basic Science in the Republic of Korea through the project IBS-R024-D1.