# Metal-insulator phase transition in a non-Hermitian Aubry-Andr\'e-Harper Model

## TL;DR

This paper provides rigorous analytical results on the energy spectrum, phase transition, and localization length in a non-Hermitian, PT-symmetric Aubry-André-Harper model, advancing understanding of localization and topological transitions in disordered non-Hermitian systems.

## Contribution

It offers the first analytical derivation of localization length and phase transition properties in a non-Hermitian Aubry-André-Harper model, extending classical results to the non-Hermitian realm.

## Key findings

- Analytical form of localization length in the insulating phase.
- Localization length is energy-independent, similar to Hermitian case.
- Identification of a topological metal-insulator phase transition.

## Abstract

Non-Hermitian extensions of the Anderson and Aubry-Andr\'e-Harper models are attracting a considerable interest as platforms to study localization phenomena, metal-insulator and topological phase transitions in disordered non-Hermitian systems. Most of available studies, however, resort to numerical results, while few analytical and rigorous results are available owing to the extraordinary complexity of the underlying problem. Here we consider a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andr\'e-Harper model, undergoing a topological metal-insulator phase transition, and provide rigorous analytical results of energy spectrum, symmetry breaking phase transition and localization length. In particular, by extending to the non-Hermitian realm the Thouless$^{\prime}$s result relating localization length and density of states, we derive an analytical form of the localization length in the insulating phase, showing that -- like in the Hermitian Aubry-Andr\'e-Harper model-- the localization length is independent of energy.