# Topology and Observables of the Non-Hermitian Chern Insulator

## TL;DR

This paper investigates the relationship between topological invariants and observable quantization in non-Hermitian Chern insulators, revealing that the non-Hermitian Chern number does not correspond to quantized Hall conductivity.

## Contribution

It demonstrates that the non-Hermitian Chern number is not linked to quantized Hall conductivity, challenging its use as a topological classification in open, lossy systems.

## Key findings

- Non-Hermitian Chern number does not imply quantized Hall conductivity.
- Explicit calculations show non-quantized Hall response with finite Chern number.
- The Chern number is not a physically meaningful topological invariant in non-Hermitian systems.

## Abstract

Topology plays a central role in nearly all disciplines of physics, yet its applications have so far been restricted to closed, lossless systems in thermodynamic equilibrium. Given that many physical systems are open and may include gain and loss mechanisms, there is an eminent need to reexamine topology within the context of non-Hermitian theories that describe open, lossy systems. The recent generalization of the Chern number to non-Hermitian Hamiltonians initiated this reexamination; however, there is so far no established connection between a non-Hermitian topological invariant and the quantization of an observable. In this work, we show that no such relationship exists between the Chern number of non-Hermitian bands and the quantization of the Hall conductivity. Using field theoretical techniques, we calculate the longitudinal and Hall conductivities of a non-Hermitian Hamiltonian with a finite Chern number to explicitly demonstrate the physics of a non-quantized Hall conductivity despite an invariable Chern number. These results demonstrate that the Chern number does not provide a physically meaningful classification of non-Hermitian Hamiltonians.