Frictional work and entropy production in integrable and non-integrable spin chains

The maximum work extractable from a quantum system is achieved when the system is driven adiabatically. Frictional work then quantifies the difference in work output between adiabatic and non-adiabatic driving. Here we show that frictional work in a non-integrable spin chain is well-described by the diagonal entropy production associated with the build up of quantum coherence. The relationship is characterized by an effective temperature of the final adiabatic state and holds for slow to moderate driving protocols. For fast protocols, the frictional work is instead described by the quantum relative entropy between the final non-adiabatic and adiabatic states. We compare our results to those obtained from an integrable spin chain, in which case the adiabatic state is no longer described by a single temperature. In this case, the frictional work is described by a sum of terms for each independent subspace of the spin chain, which are at different effective temperatures. We show how integrability breaking can enhance work extraction in the adiabatic limit, but degrade work extraction in sufficiently non-adiabatic regimes.