Nonequilibrium dynamics of the localization-delocalization transition in the non-Hermitian Aubry-André model

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 quasiperiodic potential $\ensuremath{\lambda}$, this model undergoes a localization-delocalization phase transition. We find that the localization length $\ensuremath{\xi}$ satisfies $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\varepsilon}$ being the distance from the critical point and $\ensuremath{\nu}=1$ 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 for the $n\mathrm{th}$ eigenstate is also determined as $s=0.1197$. By changing $\ensuremath{\varepsilon}$ 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.