Integrating Random Matrix Theory Predictions with Short-Time Dynamical Effects in Chaotic Systems

TL;DR

This paper introduces a method that enhances Random Matrix Theory predictions by incorporating short-time dynamical effects in chaotic systems, improving accuracy without requiring Hamiltonian diagonalization.

Contribution

The authors develop a systematic approach to include non-universal short-time dynamics into RMT eigenstate statistics, bridging the gap between RMT and actual chaotic system behavior.

Findings

Significant accuracy improvement over standard RMT in simple chaotic systems

Method works with classical approximations of short-time dynamics

Combining with correlation function bootstrapping further accelerates convergence

Abstract

We discuss a modification to Random Matrix Theory eigenstate statistics, that systematically takes into account the non-universal short-time behavior of chaotic systems. The method avoids diagonalization of the Hamiltonian, instead requiring only a knowledge of short-time dynamics for a chaotic system or ensemble of similar systems. Standard Random Matrix Theory and semiclassical predictions are recovered in the limits of zero Ehrenfest time and infinite Heisenberg time, respectively. As examples, we discuss wave function autocorrelations and cross-correlations, and show that significant improvement in accuracy is obtained for simple chaotic systems where comparison can be made with brute-force diagonalization. The accuracy of the method persists even when the short-time dynamics of the system or ensemble is known only in a classical approximation. Further improvement in the rate of convergence is obtained when the method is combined with the correlation function bootstrapping approach introduced previously.