Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

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 $H_{\text{eq}}$ 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 $H_{\theta }^{\text{fF}}$ of the Hamiltonian $H_0^{\text{fF}}$ of $N\gg 1$ free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $H_0^{\text{fF}}$ . 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 $H_0^{\text{fF}}$ . Inspired by the fact that there is one eigenbasis of $H_0^{\text{fF}}$ 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.