Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

We say of an isolated macroscopic quantum system in a pure state ψ that it is in macroscopic thermal equilibrium (MATE) if ψ lies in or close to a suitable subspace Heq of Hilbert space. It is known that every initial state ψ0 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 HθfF of the Hamiltonian H0fF of N≫1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of H0fF. Here, we first point out that also for degenerate Hamiltonians all ψ0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for H0fF. Inspired by the fact that there is one eigenbasis of H0fF 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 H0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of H0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, H=H0+λV with λ≪1, for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.