Scaling behavior of dissipative systems with imaginary gap closing

Point-gap topology, characterized by spectral winding numbers, is crucial to non-Hermitian topological phases and dramatically alters real-time dynamics. In this paper, we study the evolution of quantum particles in dissipative systems with imaginary gap closing, using the saddle-point approximation method. For trivial point-gap systems, imaginary gap-closing points can also be saddle points. This leads to a single power-law decay of the local Green's function, with the asymptotic scaling behavior determined by the order of these saddle points. In contrast, for nontrivial point-gap systems, imaginary gap-closing points do not coincide with saddle points in general. This results in a dynamical behavior characterized by two different scaling laws for distinct time regimes. In the short-time regime, the local Green's function is governed by the dominant saddle points and exhibits an asymptotic exponential decay. In the long-time regime, however, the dynamics is controlled by imaginary gap-closing points, leading to a power-law decay envelope. Our findings advance the understanding of quantum dynamics in dissipative systems and provide predictions testable in future experiments.