What is the relationship between the integrability of a quantum many-body system and thermalization? If a quantum many-body system is integrable, does it imply the system would always thermalized or many-body localized?
 A: *

*First of all, I only discuss closed quantum system here.

*Usually integrable systems do not contain disorders (but 1D Kondo model has impurity while being integrable), hence generally not many-body localised.

*Integrable systems do not thermalise in a conventional way (I mean it does not thermalise to a Gibbs ensemble). Be careful about the definition of thermalisation here. Because for any closed quantum system, the dynamics should be unitary, i.e. if one starts with a pure state, it will stay as a pure state. But "thermalisation" in this context means the expectation value of a local operator can be expressed as statistical expectation value of a Gibbs ensemble. (Tracing out the rest of the system, this is possible, similar to what happens to entanglement entropy.)

*Integrable systems will thermalise into a "generalised Gibbs ensemble" (GGE) due to the existence of (at least) extensive many local/quasi-local conserved charges. This is well understood for an integrable system relaxing after a quantum quench. See review such as 1604.03990. A complete description of the GGE in integrable systems is explained here:1603.00440.

