Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

Abstract

Abstract We say of an isolated macroscopic quantum system in a pure state $$\psi $$ ψ that it is in macroscopic thermal equilibrium (MATE) if $$\psi $$ ψ lies in or close to a suitable subspace $$\mathcal {H}_\textrm{eq}$$ H eq of Hilbert space. It is known that every initial state $$\psi _0$$ ψ 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 $$H_\theta ^\textrm{fF}$$ H θ fF of the Hamiltonian $$H_0^\textrm{fF}$$ H 0 fF of $$N\gg 1$$ N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of $$H_0^\textrm{fF}$$ H 0 fF . Here, we first point out that also for degenerate Hamiltonians all $$\psi _0$$ ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for $$H_0^\textrm{fF}$$ H 0 fF . Inspired by the fact that there is one eigenbasis of $$H_0^\textrm{fF}$$ 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 $$H_0$$ H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of $$H_0$$ H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, $$H=H_0+\lambda V$$ H = H 0 + λ V with $$\lambda \ll 1$$ λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.