Compact localized states and flat bands from local symmetry partitioning

TL;DR

This paper introduces a graph-theoretic framework linking local symmetries in discrete Hamiltonians to the design of compact localized states and flat bands, enabling efficient analysis and tailored state creation.

Contribution

It presents a novel method using recent graph theory theorems to partition Hamiltonians based on local symmetries, simplifying eigenproblem solutions and aiding in the design of localized states.

Findings

Reduces Hamiltonian diagonalization to smaller matrices.

Automatically classifies eigenstates into localized and extended.

Provides a unified approach for designing localized states at specific energies.

Abstract

We propose a framework for the connection between local symmetries of discrete Hamiltonians and the design of compact localized states. Such compact localized states are used for the creation of tunable, local symmetry-induced bound states in an energy continuum and flat energy bands for periodically repeated local symmetries in one- and two-dimensional lattices. The framework is based on very recent theorems in graph theory which are here employed to obtain a block partitioning of the Hamiltonian induced by the symmetry of a given system under local site permutations. The diagonalization of the Hamiltonian is thereby reduced to finding the eigenspectra of smaller matrices, with eigenvectors automatically divided into compact localized and extended states. We distinguish between local symmetry operations which commute with the Hamiltonian, and those which do not commute due to an asymmetric coupling to the surrounding sites. While valuable as a computational tool for versatile discrete systems with locally symmetric structures, the approach provides in particular a unified, intuitive, and efficient route to the flexible design of compact localized states at desired energies.