Generalized Lieb's theorem for noninteracting non-Hermitian $n$-partite tight-binding lattices

TL;DR

This paper extends Lieb's theorem to non-Hermitian, multi-sublattice tight-binding models with unidirectional, cyclical connections, revealing how zero-energy flat bands relate to sublattice imbalances and generalized chiral symmetry.

Contribution

It introduces a generalized Lieb's theorem for non-Hermitian multi-sublattice models with unidirectional coupling, expanding understanding of zero-energy flat bands beyond bipartite systems.

Findings

Derived a formula for zero-energy flat bands based on sublattice imbalances.

Showed models obey a generalized chiral symmetry similar to clock or parafermionic systems.

Discussed potential physical realizations of the models.

Abstract

Hermitian bipartite models are characterized by the presence of chiral symmetry and by Lieb's theorem, which derives the number of zero-energy flat bands of the model from the imbalance of sites between its two sublattices. Here, we introduce a class of non-Hermitian models with an arbitrary number of sublattices connected in a unidirectional and cyclical way and show that the number of zero-energy flat bands of these models can be found from a generalized version of Lieb's theorem, in what regards its application to noninteracting tight-binding models, involving the imbalance between each sublattice and the sublattice of lowest dimension. Furthermore, these models are also shown to obey a generalized chiral symmetry, of the type found in the context of certain clock or parafermionic systems. The main results are illustrated with a simple toy model, and possible realizations in different platforms of the models introduced here are discussed.