Analytical topological invariants for 2D non-Hermitian phases using Morse theory

TL;DR

This paper develops analytical methods using Morse theory to understand topological invariants and edge states in 2D non-Hermitian systems, addressing the complexity introduced by gain and loss phenomena.

Contribution

It introduces an analytical framework for calculating topological invariants in 2D non-Hermitian systems, including explicit eigenstates and phase diagram divisions.

Findings

Derived closed-form eigenstates and phase diagrams for non-Hermitian SSH models

Used Morse theory to analyze topology of exceptional points in momentum space

Established a phase-based topological invariant that remains well-defined at exceptional points

Abstract

As energy dissipation and gain are ubiquitous in the real world, such phenomena demand the generalization of Hermitian methods such as the analysis of topological properties for non-Hermitian systems. However, as non-Hermitian systems typically contain more degrees of freedom, this poses a challenge for analytical approaches to understand their topology and invariants. In this work, we analytically calculate the 2D Zak phase for a 2D non-Hermitian SSH-type Hamiltonian that supports a rich structure and edge currents. Closed-form expressions for eigenstates and divisions of the phase diagram are obtained, including for regions in the phase diagram where different types of exceptional points exist. We use Morse theory to determine the topology of exceptional points in momentum space. Although the band structure breaks down at exceptional points, we show that a specific phase-based topological invariant remains well-defined. Furthermore, our work yields an analytic derivation for counting edge states in the Hermitian limit. These results provide new conceptual and analytical tools for the study of complex topological systems.