Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

TL;DR

This paper extends the understanding of thermalization in quantum systems, demonstrating that even degenerate Hamiltonians thermalize under ETH, and that small perturbations generally induce ETH and thermalization.

Contribution

It proves that degenerate Hamiltonians also satisfy ETH if all eigenbases are in MATE, and shows most eigenbases are in MATE if one is, implying generic perturbations lead to thermalization.

Findings

Degenerate Hamiltonians can satisfy ETH and thermalize.

Existence of one eigenbasis in MATE implies most are in MATE.

Small generic perturbations induce ETH and thermalization.

Abstract

We say of an isolated macroscopic quantum system in a pure state $\psi$ that it is in macroscopic thermal equilibrium (MATE) if $\psi$ lies in or close to a suitable subspace $\mathcal{H}_{eq}$ of Hilbert space. It is known that every initial state $\psi_0$ will eventually reach and stay there most of the time (``thermalize'') if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation $H_\theta^{fF}$ of the Hamiltonian $H_0^{fF}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{fF}$. Here, we first point out that also for degenerate Hamiltonians all $\psi_0$ thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $H_0^{fF}$. Inspired by the fact that there is one eigenbasis of $H_0^{fF}$ for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given $H_0$ that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $H_0$ lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $H=H_0+\lambda V$ with $\lambda\ll 1$, for most perturbations $V$ the perturbed Hamiltonian $H$ satisfies ETH and all states thermalize.