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.
Categories: cond-mat.stat-mech math-ph math.MP quant-ph