Thermalization in the mixed-field Ising model: An occupation number perspective

TL;DR

This paper investigates thermalization in the mixed-field Ising model by analyzing occupation numbers in both quantum and classical regimes, revealing how ergodicity and thermalization emerge with increasing system size.

Contribution

It introduces a classical ergodicity criterion using spin dynamics and random matrix theory, linking classical and quantum thermalization behaviors.

Findings

Classical ergodicity deviations decay algebraically with system size.

Quantum thermalization approaches the microcanonical prediction as system size increases.

Classical model enables larger system size analysis than quantum model.

Abstract

The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi--Dirac, Bose--Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the $z$-direction. We use this quantity to probe the onset of thermalization in the isolated quantum and classical one-dimensional spin-1 Ising model with transverse and longitudinal fields. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length $L$ increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with $L$. To overcome this limitation, we turn to the corresponding classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.