# Non-Hermitian systems and topology: A transfer-matrix perspective

## TL;DR

This paper introduces a comprehensive transfer-matrix framework to analyze non-Hermitian topological systems, elucidating phenomena like the skin effect and boundary modes through analytical methods and topological invariants.

## Contribution

It provides a unifying analytical approach using transfer matrices to understand non-Hermitian topological phases and their boundary phenomena, connecting spectral properties with topological invariants.

## Key findings

- Transfer matrix determinant indicates non-Hermitian skin effect.
- Real-space exceptional points scale with system size.
- Topological invariants are derived for non-Hermitian systems.

## Abstract

Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulk-boundary correspondence. These features can change drastically for their non-Hermitian generalizations, as exemplified by a general breakdown of bulk-boundary correspondence and a localization of all states at the boundary, termed the non-Hermitian skin effect. In this article, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices -- a real-space approach suitable for systems with periodic as well as open boundary conditions. We show that various qualitative properties of these systems can be easily deduced from the transfer matrix. For instance, the connection between the breakdown of the conventional bulk-boundary correspondence and the existence of a non-Hermitian skin effect, previously observed numerically, is traced back to the transfer matrix having a determinant not equal to unity. The vanishing of this determinant signals real-space exceptional points, whose order scales with the system size. We also derive previously proposed topological invariants such as the biorthogonal polarization and the Chern number computed on a complexified Brillouin zone. Finally, we define an invariant for and thereby clarify the meaning of topologically protected boundary modes for non-Hermitian systems.