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.

💡 Research Summary

In this work the authors investigate the long‑time dynamics of quantum particles evolving under non‑Hermitian (dissipative) Hamiltonians that feature an “imaginary gap closing,” i.e., a point in the Brillouin zone where the imaginary part of the complex energy spectrum touches zero. The central theme is how the point‑gap topology—characterized by a winding number of the spectrum around a reference energy—determines the scaling behavior of the local Green’s function (G(x_0,t)=\langle x_0|e^{-iHt}|x_0\rangle).

The paper begins with a concise review of non‑Hermitian physics, emphasizing the non‑Hermitian skin effect (NHSE) and the role of point‑gap topology. It then introduces a general multi‑band tight‑binding formulation, writes the time‑domain Green’s function as a contour integral over the complex Bloch phase (\beta=e^{ik}), and shows how the eigenvalues (E_n(\beta)) and bi‑orthogonal eigenvectors enter the integrand. The point‑gap winding number (W(E_b)=\oint d\beta/(2\pi i),\partial_\beta\ln\det