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