# Anyonic $\mathcal{PT}$ symmetry, drifting potentials and non-Hermitian delocalization

## TL;DR

This paper explores wave dynamics in non-Hermitian Schrödinger equations with anyonic PT symmetry, revealing how drifting potentials and symmetry phases influence spectral properties, scattering, and localization, including reflectionless barriers and delocalization transitions.

## Contribution

It introduces the concept of anyonic PT symmetry in non-Hermitian systems and analyzes how drifting potentials affect spectral deformation, scattering, and localization phenomena.

## Key findings

- Drifting potentials can make barriers reflectionless in unbroken PT phase.
- The number of bound states decreases with increasing drift velocity.
- Non-Hermitian delocalization transition occurs due to drift effects.

## Abstract

We consider wave dynamics for a Schr\"odinger equation with a non-Hermitian Hamiltonian $\mathcal{H}$ satisfying the generalized (anyonic) parity-time symmetry $\mathcal{PT H}= \exp(2 i \varphi) \mathcal{HPT}$, where $\mathcal{P}$ and $ \mathcal{T}$ are the parity and time-reversal operators. For a stationary potential, the anyonic phase $\varphi$ just rotates the energy spectrum of $\mathcal{H}$ in complex plane, however for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when $\varphi \neq 0$. In particular, in the unbroken $\mathcal{PT}$ phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.