Timescale for macroscopic equilibration in isolated quantum systems: a rigorous derivation for free fermions

TL;DR

This paper rigorously proves that certain free-fermion quantum systems on a lattice equilibrate in a timescale proportional to the system size, confirming theoretical expectations for macroscopic equilibration.

Contribution

It provides a rigorous derivation of the equilibration timescale for translation-invariant free-fermion systems, establishing the optimal $O(L)$ scaling.

Findings

Equilibration occurs within a timescale proportional to system size $L$.

The $O(L)$ timescale is proven to be optimal for these systems.

Results apply to a broad class of free-fermion models with uniform hopping.

Abstract

For a class of translation-invariant free-fermion systems (including those with uniform nearest neighbor hopping) on a $d$-dimensional $L \times \cdots \times L$ hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order $L$. This scaling is optimal, since there exist initial states whose equilibration requires time of order $L$. Our result establishes $O(L)$ as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.