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

Abstract

By applying the theory recently developed by Roos, Sugimoto, Teufel, Tumulka, and Vogel [1], we construct a rigorous example of an isolated macroscopic quantum system in which every initial state chosen from a microcanonical energy shell exhibits thermalization solely by the unitary time evolution determined by the Hamiltonian. We consider the Hamiltonian $\hat{H}_L$ of the standard two-dimensional ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator $\lambda_L\hat{V}_L$ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. The Roos-Sugimoto-Teufel-Tumulka-Vogel theory then guarantees that, for all but an exponentially small fraction of the random perturbations, the model exhibits strong ETH (energy eigenstate thermalization hypothesis), i.e., every energy eigenstate in a microcanonical energy shell is in thermal equilibrium in the sense of large deviation, as precisely stated in the main text. It is then standard that the strong ETH implies the following result for thermalization. Suppose that the initial state $\Psi(0)\rangle$ is chosen from the microcanonical energy shell and may be very far from thermal equilibrium. Then the time-evolved state $|\Psi(t)\rangle=e^{-i(\hat{H}_L+\lambda_L\hat{V}_L)t}|\Psi(0)\rangle$ is in thermal equilibrium at sufficiently long and typical times $t$, in the sense that the measurement result of the magnetization density almost certainly coincides with the corresponding spontaneous magnetization. When we state ``typical'' and ``almost certainly'' above, we mean that the corresponding exceptional fractions are exponentially small in the system size $L$.