Macroscopic thermalization by unitary time-evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Teufel-Tumulka-Vogel theorem

TL;DR

This paper demonstrates that in a two-dimensional Ising model with small random perturbations, unitary evolution leads to thermalization, aligning measurement outcomes with equilibrium magnetization after sufficient time, illustrating the theorem's implications.

Contribution

It applies the Roos-Teufel-Tumulka-Vogel theorem to show thermalization in a nontrivial quantum spin system with random perturbations.

Findings

Unitary evolution causes the system to reach thermal equilibrium.

Measurement results match spontaneous magnetization in equilibrium.

Thermalization occurs for most random perturbations.

Abstract

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 $\lambda\hat{V}$ 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 $e^{-i(\hat{H}_L+\lambda\hat{V})t}$ 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.