Onset of universality in the dynamical mixing of a pure state

TL;DR

This paper investigates how the spectral statistics of a pure quantum state evolve under random Hamiltonian dynamics, revealing a universal crossover from GOE to GUE statistics that depends on the Hilbert space dimension.

Contribution

It demonstrates the onset of universal spectral statistics in the dynamical mixing of pure states and characterizes the timescale of this crossover using a semi-analytical approach.

Findings

Spectral statistics follow RMT predictions during evolution.

Crossover time scales inversely with Hilbert space dimension.

GUE statistics emerge at large times in chaotic many-body systems.

Abstract

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the GOE to the Gaussian unitary ensemble (GUE) for short and large times respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the Hilbert space dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace.