Nonequilibrium dynamics of the localization-delocalization transition in the non-Hermitian Aubry-Andr\'{e} model

TL;DR

This paper studies the non-Hermitian Aubry-Andre9 model's localization transition, revealing universal critical exponents and demonstrating that driven dynamics follow Kibble-Zurek scaling, applicable to both ground and excited states.

Contribution

It provides a detailed analysis of critical exponents and demonstrates the applicability of Kibble-Zurek scaling to driven localization transitions in a non-Hermitian system.

Findings

Localization length exponent =1 is universal.

Dynamic exponent z=2 from finite-size scaling.

Kibble-Zurek scaling describes driven dynamics across the transition.

Abstract

In this paper, we investigate the driven dynamics of the localization transition in the non-Hermitian Aubry-Andr\'{e} model with the periodic boundary condition. Depending on the strength of the quasi-periodic potential $\lambda$, this model undergoes a localization-delocalization phase transition. We find that the localization length $\xi$ satisfies $\xi\sim \varepsilon^{-\nu}$ with $\varepsilon$ being the distance from the critical point and $\nu=1$ being a universal critical exponent independent of the non-Hermitian parameter. In addition, from the finite-size scaling of the energy gap between the ground state and the first excited state, we determine the dynamic exponent $z$ as $z=2$. The critical exponent of the inverse participation ratio (IPR) for the $n$th eigenstate is also determined as $s=0.1197$. By changing $\varepsilon$ linearly to cross the critical point, we find that the driven dynamics can be described by the Kibble-Zurek scaling (KZS). Moreover, we show that the KZS with the same set of the exponents can be generalized to the localization phase transitions in the excited states.