Scaling behavior of dissipative systems with imaginary gap closing

TL;DR

This work investigates real-time quantum dynamics in dissipative, non-Hermitian systems with imaginary gap closing, introducing a non-Hermitian multi-band Green's function framework linked to point-gap topology. Using saddle-point methods, it reveals that trivial point-gap systems exhibit a single power-law decay in the long-time Green's function, with $G(t) o t^{-1/n}$ where $n$ is the saddle-order of the imaginary-gap point, while nontrivial point-gap systems display two dynamical regimes: an initial saddle-point–driven exponential decay followed by a long-time power-law envelope $G(t) o t^{-1/n'}$, with $n'$ determined by the order of the saddle point at the imaginary-gap closing. The findings, supported by analytic and numerical results on concrete two-band models, illuminate how boundary conditions and spectral topology shape dissipative quantum dynamics and suggest feasible experimental tests in various platforms. The work provides a theoretical framework for observing saddle-point dynamics and the interplay between imaginary-gap closing and point-gap topology in non-Hermitian systems.

Abstract

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.

Figures (4)

• Figure 1: (a) Schematic of a one-dimensional lossy ladder lattice with trivial point-gap topology, which corresponds to the Hamiltonian in Eq. (\ref{['III(1)']}). Each unit cell, labeled by the spatial coordinate $x$, comprises two sites, denoted $A$ and $B$. The particle is initially localized on the $A$ sublattice of the unit cell at $x_0$. The energy spectra and corresponding numerical results for the local Green's function $G_{A}\left( x_{0};t\right)=\left\langle x_{0},A\left\vert e^{-iHt}\right\vert x_{0},A\right\rangle$ are as follows: (b)-(c) OBC (red points) and PBC (blue points) energy spectra $E_{\pm}$, with saddle points $S^{\pm}_i$ explicitly marked by cross symbols. The saddle points $S^{+}_{1}$ and $S^{+}_{2}$ in (b), as well as the threefold saddle point $S^{+}_{1}$ in (c), are also imaginary gap-closing points. (d)-(e) Numerical results for the scaling of $|G_{A}\left( x_{0};t\right)|$ with bulk $x_0=75$ under OBC (red solid lines) and PBC (blue dashed lines); these agree well with the theoretical results (black dashed lines). In (b) and (d), the parameters are $t_0=0.5$,$t_1=0.5$, and $\gamma=0.8$; in (c) and (e), $t_0=1.0$, $t_1=0.5$, and $\gamma=0.8$. The number of unit cells is $L=150$ for the numerical calculation.
• Figure 2: Schematic of a one-dimensional lossy ladder lattice with nontrivial point-gap topology, which corresponds to the Hamiltonian in Eq. (\ref{['Eq(17)']}). A nonzero Peierls phase $\phi$ is introduced via the hopping term $t_p$, which breaks the time-reversal symmetry of the system.
• Figure 3: Energy spectra and corresponding numerical results of $G_{A}\left( x_{0};t\right)$ in an imaginary gap-closing system with nontrivial point-gap topology. (a)-(b) OBC (red points) and PBC (blue points) energy spectra $E_{\pm}$. Saddle points $S^{\pm}_i$ are explicitly marked by cross symbols. With the exception of $\operatorname{Im} S^{+}_1=0.01 >0$ in (a), the imaginary parts of all other saddle points are negative. (c)-(d) Numerical results for short-time evolution of $|G_{A}\left( x_{0};t\right)|$. The fitted values $\lambda$ are in agreement with the theoretical saddle-point predictions. (e)-(f) Numerical results for long-time evolution of $|G_{A}\left( x_{0};t\right)|$ exhibit a power law envelope, fitted by black dashed line. The local peaks within this envelope recur with a period $T$. In (a), (c), and (e), the parameters are $t_0=0.5$, $t_1=0.5$, $t_p=0.3$, and $\gamma=0.8$; in (b), (d), and (f), the parameters are $t_0=1.0$, $t_1=0.5$, $t_p=0.7$, and $\gamma=0.8$. For all panels, the Peierls phase is $\phi =\pi/2$ and the number of unit cells is $L=150$.
• Figure 4: Numerical results for the amplitude of world-line Green's function $G_{AA}\left( m,x_{0};t\right)$ under PBC, where $m= x_0+v_{n,g}(k_0)t$. Here, $v_{n,g}(k_0)$ is the group velocity at the imaginary gap-closing point for the $E_n$ band. The parameter values are: $t_0=0.5$, $t_1=0.5$, $t_p=0.3$, $\gamma=0.8$, and $v_{+,g}(k_0)=v_{-,g}(k_0)=-0.3$ in (a); $t_0=1.0$, $t_1=0.5$, $t_p=0.7$, $\gamma=0.8$, and $v_{+,g}(k_0)=v_{-,g}(k_0)=-1.4$ in (b). For both panels, the initial conditions are $\psi_{x}^{A}(0)=\delta_{x,x_{0}}$ and $\psi_{x}^{B}(0)=0$, with the number of unit cells is $L$=150.