Generating function for Hermitian and non-Hermitian models

TL;DR

This paper introduces a unified generating-function approach for Hermitian and non-Hermitian lattice models, linking non-Hermitian physics with recurrence relations and discrete mathematics.

Contribution

It develops a general framework that maps finite lattice models to generating functions, simplifying eigenstate analysis and revealing boundary effects and topological transitions.

Findings

Boundary conditions influence zero locations in the generating function.

The method uncovers the boundary sensitivity of non-Hermitian systems.

Identifies topological phase transition in the non-Hermitian SSH model.

Abstract

It is well known that Hermitian and non-Hermitian models exhibit distinct physics and require different theoretical tools. In this work, we propose a unified generating-function framework for both classes with generic boundary conditions and local impurities. Within this framework, any finite lattice model can be mapped to a generating function of the form G(z)=P(z)/Q(z), where Q(z) and P(z) denote the bulk recurrence relation and boundary terms or impurities, respectively. The problem of solving for eigenstates reduces to a simple criterion based on the cancellation of zeros of Q(z) and P(z). Applying this method to the Hatano-Nelson (HN) model, we show how boundary conditions and impurities determine the location of the zeros, thereby demonstrating the boundary sensitivity of non-Hermitian systems. We further investigate topological edge states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model and identify its topological phase transition. Inspired by generating-function techniques widely used in discrete mathematics, particularly in the study of the Fibonacci sequence, our results establish a direct connection between non-Hermitian physics and recurrence relations, providing a new perspective for analyzing non-Hermitian systems and exploring their connections with discrete mathematical structures.