Temperature and conditions for thermalization after canonical quenches

Abstract

We consider quenches of a quantum system that is prepared in a canonical equilibrium state of one Hamiltonian and then evolves unitarily in time under a different Hamiltonian. Technically, our main result is a systematic expansion of the pre- and post-quench canonical ensembles in the quench strength. We first demonstrate how this can be used to predict the system's temperature after the quench from equilibrium properties at the pre-quench temperature. For a thermalizing post-quench system, it furthermore allows us to calculate equilibrium observable expectation values. Finally, in the presence of additional conserved quantities besides the Hamiltonian, we obtain a hierarchy of necessary conditions for thermalization towards the (post-quench) canonical ensemble. At first order, these thermalization conditions have a nice geometric interpretation in operator space with the canonical covariance as a semi-inner product: The quench operator (difference between post- and pre-quench Hamiltonians) and the conserved quantity must be orthogonal in the orthogonal complement of the post-quench Hamiltonian. We illustrate the results numerically for a variety of setups involving integrable and nonintegrable models.