Decay of a Thermofield-Double State in Chaotic Quantum Systems

TL;DR

This paper investigates how initially localized information in chaotic quantum systems, modeled by random matrices and disordered spin chains, decays over time, revealing universal features and differences near localization.

Contribution

It provides exact analytical results for the survival probability in GUE random matrices and compares these with numerical results from disordered spin chains, highlighting universal and non-universal behaviors.

Findings

Exact finite-N expressions for survival probability in GUE.

Numerical validation with GOE matrices and disordered spin chains.

Identification of features common to chaotic regimes and deviations near localization.

Abstract

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite $N$ for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.