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

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 an isolated one-dimensional quantum spin-1 Ising model with transverse and longitudinal fields and in its classical counterpart. 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 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.