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For a thermalised system, at equilibrium, the usual quantum statistical ensembles are valid and we can define temperature at an equilibrium between system and its bath. But for integrable systems (non-thermalisable) we don't have those ensembles, so how do we define a temperature?
PS- I had this doubt specific to many body localized state. And my intution is that the ensemble for a many body localised state would be similar to any other integrable system. Please correct me if I am wrong.
To the best of my knowledge, to describe thermalization in integrable models one should to pass to the so-called "generalized Gibbs ensemble". For a prototypical integrable model, the XXX spin chain, the former was constructed by Ilievski et al arXiv:1507.02993; see also arXiv:1610.06911. (For the dynamics, away from thermal equilibrium, the key concept is "generalized hydrodynamics", see e.g. J. Dubail's Viewpoint in Physics)
