Temperature and conditions for thermalization after canonical quenches

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 postquench 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 prequench temperature. For a thermalizing postquench 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 (postquench) 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 prequench Hamiltonians) and the conserved quantity must be orthogonal in the orthogonal complement of the postquench Hamiltonian. We illustrate the results numerically for a variety of setups involving integrable and nonintegrable models.