Spread complexity and localization in $\mathcal{PT}$-symmetric systems

TL;DR

This paper introduces a framework using spread complexity and entropy to analyze wave function localization and the non-Hermitian skin effect in $ ext{PT}$-symmetric quantum systems, revealing phase-dependent delocalization and localization behaviors.

Contribution

It presents a novel approach employing spread measures in Krylov space to characterize the $ ext{PT}$-symmetry breaking transition and the associated localization phenomena.

Findings

Wave functions are delocalized in the $ ext{PT}$-unbroken phase.

Wave functions become localized at edges in the $ ext{PT}$-broken phase.

Krylov basis measures effectively characterize the skin effect and phase transition.

Abstract

We present a framework for investigating wave function spreading in $\mathcal{PT}$-symmetric quantum systems using spread complexity and spread entropy. We consider a tight-binding chain with complex on-site potentials at the boundary sites. In the $\mathcal{PT}$-unbroken phase, the wave function is delocalized. We find that in the $\mathcal{PT}$-broken phase, it becomes localized on one edge of the tight-binding lattice. This localization is a realization of the non-Hermitian skin effect. Localization in the $\mathcal{PT}$-broken phase is observed both in the lattice chain basis and the Krylov basis. Spread entropy, entropic complexity, and a further measure that we term the Krylov inverse participation ratio probe the dynamics of wave function spreading and quantify the strength of localization probed in the Krylov basis. The number of Krylov basis vectors required to store the information of the state reduces with the strength of localization. Our results demonstrate how measures in Krylov space can be used to characterize the non-hermitian skin effect and its localization phase transition.