Characterization of Chaotic Evolution in Quantum Systems Induced by Random Hermitian Matrices

How can the early-stage chaotic evolution and information scrambling of complex quantum systems be quantitatively characterized without relying on large-scale simulations or systems with a classical limit? The authors address this by applying a recently developed semiclassical formalism to finite-dimensional random Hermitian matrices. They define an energy-dependent quantum Poincar’e map whose squared matrix elements generate a classical bistochastic Markov process on a discrete phase space of directed edges. The early-stage evolution toward an ergodic state is quantified by the mean Lyapunov exponent (LE), whereas the late-stage relaxation rate is governed by the spectral gap of the Markov matrix.

The study investigates five distinct random-matrix ensembles: adjacency matrices of d-regular graphs, the Gaussian orthogonal ensemble (GOE), the Gaussian unitary ensemble (GUE), and tridiagonal G’E ensembles with parameters beta equals 1 and 2. To put these models on an equal footing, the Hamiltonians are normalized such that their asymptotic spectral distributions span the interval from minus one to one. Using ergodic theory and perturbation techniques for non-Hermitian operators, the authors derive analytical expressions for the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy and temperature.

Numerical simulations show excellent agreement with the analytical predictions across all examined ensembles. The mean Lyapunov exponent is found to be a self-averaging quantity that depends primarily on local matrix structure rather than total dimension. Furthermore, the variance of the LE is decomposed into F-component and G-component contributions, revealing how exceptional points in the non-Hermitian spectrum affect fluctuations without causing divergences. The spectral gaps complement this picture by characterizing the asymptotic rate of convergence to equilibrium.