Thermalization in the mixed-field Ising model: An occupation number perspective
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Vallejo-Fabila, Isaías Borgonovi, Fausto Izrailev, Felix M. Santos, Lea F. |
| author_facet | Vallejo-Fabila, Isaías Borgonovi, Fausto Izrailev, Felix M. Santos, Lea F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02497 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Thermalization in the mixed-field Ising model: An occupation number perspective Vallejo-Fabila, Isaías Borgonovi, Fausto Izrailev, Felix M. Santos, Lea F. Statistical Mechanics 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. |
| title | Thermalization in the mixed-field Ising model: An occupation number perspective |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2601.02497 |