Non-Hermitian lattices with a flat band and polynomial power increase

TL;DR

This paper explores methods to construct non-Hermitian flat bands with real dispersionless parts, identifies a special flat band with exceptional points, and analyzes power dynamics of localized excitations in such systems.

Contribution

It systematically presents three approaches to create non-Hermitian flat bands and introduces a unique flat band with exceptional points of order 3, analyzing its power increase behaviors.

Findings

Three methods to construct non-Hermitian flat bands.

Identification of an EP3 flat band with unique properties.

Localized excitations can show conserved, quadratic, or quartic power increase.

Abstract

In this work we first discuss systematically three general approaches to construct a non-Hermitian flat band, defined by its dispersionless real part. They resort to, respectively, spontaneous restoration of non-Hermitian particle-hole symmetry, a persisting flat band from the underlying Hermitian system, and a compact Wannier function that is an eigenstate of the entire system. For the last approach in particular, we show the simplest lattice structure where it can be applied, and we further identify a special case of such a flat band where every point in the Brillouin zone is an exceptional point of order 3. A localized excitation in this "EP3 flat band" can display either a conserved power, quadratic power increase, or even quartic power increase, depending on whether the localized eigenstate or one of the two generalized eigenvectors is initially excited. Nevertheless, the asymptotic wave function in the long time limit is always given by the eigenstate, and in this case, the compact Wannier function or its superposition in two or more unit cells.