Entanglement Dynamics of Random GUE Hamiltonians

TL;DR

This paper investigates how entanglement evolves in quantum systems governed by random non-integrable Hamiltonians, comparing with integrable cases, and derives universal formulas for average dynamics using random matrix theory.

Contribution

It introduces a universal approach to analyze entanglement dynamics in non-integrable quantum systems using random matrix ensembles and Weingarten functions.

Findings

Universal average time evolution formulas for reduced density matrices.

Comparison of entanglement dynamics between non-integrable and integrable Hamiltonians.

Effective description of complex models like SYK and Spin Glass using random matrix theory.

Abstract

In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential $n$-point correlation functions in the gas of GUE eigenvalues.