Relaxation of non-integrable systems and correlation functions

TL;DR

This paper studies how quickly quantum many-body systems reach equilibrium by comparing observable and correlation function relaxation rates, finding they often coincide for generic initial states, supported by analytical and numerical evidence.

Contribution

It demonstrates the universality of relaxation timescales in non-integrable quantum systems and links these timescales to effective dimensions and Haar-random Hamiltonian dynamics.

Findings

Relaxation rates of observables and correlation functions often coincide.

Effective dimensions quantify participation of energy levels in dynamics.

Haar-random Hamiltonians confirm the universality of these timescales.

Abstract

We investigate early-time equilibration rates of observables in closed many-body quantum systems and compare them to those of two correlation functions, first introduced by Kubo and Srednicki. We explore whether these different rates coincide at a universal value that sets the timescales of processes at a finite energy density. We find evidence for this coincidence when the initial conditions are sufficiently generic, or typical. We quantify this with the effective dimension of the state and with a state-observable effective dimension, which estimate the number of energy levels that participate in the dynamics. Our findings are confirmed by proving that these different timescales coincide for dynamics generated by Haar-random Hamiltonians. This also allows to quantitatively understand the scope of previous theoretical results on equilibration timescales and on random matrix formalisms. We approach this problem with exact, full spectrum diagonalization. The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian, and the dynamics are investigated for several pairs of observables and states.