Equilibrium and nonequlibrium scaling behaviors of localization transition in a non-Hermitian Aubry-Andr\'{e} model with onsite gain and loss

Abstract

The interplay between non-Hermiticity and localization has attracted considerable interest, yet the driven dynamics of localization transitions in non-Hermitian systems with on-site gain and loss remains largely unexplored. Here we investigate the critical scaling behavior and driven dynamics of the non-Hermitian Aubry-Andr\'e (AA) model with on-site gain and loss. Through finite-size scaling analyses of the localization length, the inverse participation ratio (IPR), and the energy gap, we extract the critical exponents $\nu = 1.00(2)$, $s = 0.7965(2)$, and $z = 1.999(2)$. These exponents are different from those of both the Hermitian AA model and the nonreciprocal hopping AA model, particularly the IPR exponent $s$, demonstrating that the gain-loss mechanism belongs to a distinct universality class. For the driven dynamics, we focus on the case where the system is initially prepared in a gapless extended state and linearly driven across the critical point. We verify that the finite-time scaling (FTS) framework remains applicable provided that the criterion $z' < r$ is satisfied, where $z' = 1.999(2)$ characterizes the gap closure in the extended phase and $r = z + 1/\nu \approx 2.999$. The predicted FTS scaling forms for the IPR are numerically validated across a wide range of system sizes and driving rates, demonstrating that the unified scaling description can be successfully generalized to the gain-loss type non-Hermitian AA model. Our work not only establishes the gain-loss AA model as a new universality class of localization transitions but also extends the applicability of the FTS framework to non-Hermitian systems with gapless initial states.