Effects of disorder on non-unitary $\mathcal{PT}$ symmetric quantum walks

TL;DR

This paper investigates how spatial disorder affects non-unitary quantum walks with generalized $ ext{PT}$ symmetry, revealing cases where real quasi-energies persist despite broken symmetry.

Contribution

It introduces non-unitary quantum walks with generalized $ ext{PT}$ symmetry and demonstrates their robustness to disorder through numerical analysis.

Findings

Real quasi-energies can exist despite $ ext{PT}$ symmetry breaking.

Disorder can preserve real eigenvalues in non-unitary quantum walks.

Generalized $ ext{PT}$ symmetry can be resilient to spatial randomness.

Abstract

$\mathcal{PT}$ symmetry, namely, a combined parity and time-reversal symmetry can make non-unitary quantum walks exhibit entirely real eigenenergy. However, it is known that the concept of $\mathcal{PT}$ symmetry can be generalized and an arbitrary anti-unitary symmetry has a possibility to substitute $\mathcal{PT}$ symmetry. The aim of the present work is to seek such non-unitary quantum walks with generalized $\mathcal{PT}$ symmetry by focusing on effects of spatially random disorder which breaks $\mathcal{PT}$ symmetry. We numerically find non-unitary quantum walks whose quasi-energy is entirely real despite $\mathcal{PT}$ symmetry is broken.