Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910603893800960 |
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| author | Tasaki, Hal |
| author_facet | Tasaki, Hal |
| contents | To demonstrate the implication of the recent important theorem by Roos, Teufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study thermalization in the two-dimensional Ising model in the low-temperature phase. We consider the Hamiltonian $\hat{H}_L$ of the standard ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator $λ\hat{V}$ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. Suppose that the system is initially in a classical spin configuration with a specified energy that may be very far from thermal equilibrium. It is proved that, for most choices of the random perturbation, the unitary time evolution $e^{-i(\hat{H}_L+λ\hat{V})t}$ brings the initial state into thermal equilibrium after a sufficiently long and typical time $t$, in the sense that the measurement result of the magnetization density at time $t$ almost certainly coincides with the spontaneous magnetization expected in the corresponding equilibrium. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09395 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem Tasaki, Hal Statistical Mechanics Mathematical Physics Quantum Physics To demonstrate the implication of the recent important theorem by Roos, Teufel, Tumulka, and Vogel [1] in a simple but nontrivial example, we study thermalization in the two-dimensional Ising model in the low-temperature phase. We consider the Hamiltonian $\hat{H}_L$ of the standard ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator $λ\hat{V}$ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. Suppose that the system is initially in a classical spin configuration with a specified energy that may be very far from thermal equilibrium. It is proved that, for most choices of the random perturbation, the unitary time evolution $e^{-i(\hat{H}_L+λ\hat{V})t}$ brings the initial state into thermal equilibrium after a sufficiently long and typical time $t$, in the sense that the measurement result of the magnetization density at time $t$ almost certainly coincides with the spontaneous magnetization expected in the corresponding equilibrium. |
| title | Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem |
| topic | Statistical Mechanics Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2409.09395 |