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 $\mathcal{H}_{eq}$ 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.