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While temperature is well understood as an intensive quantity in standardthermodynamics, it is less clear whether the same holds in the presence ofstrong correlations, especially in the case of quantum systems, which may evendisplay correlations with no classical analogue. The problem lies in the factthat, under the presence of strong correlations, subsystems of a system inthermal equilibrium are, in general, not described by a thermal state at thesame temperature as the global system and thus one cannot simply assign a localtemperature to them. However, there have been identified situations in whichcorrelations in thermal states decay sufficiently fast so that the state oftheir subsystems can be very well approximated by the reduced states ofequilibrium systems that are only slightly bigger than the subsystemsthemselves, hence allowing for a valid local definition of temperature. In thiswork, we address the question of whether temperature is locally well definedfor a bosonic system with local interactions that undergoes a phase transitionat non-zero temperature. We consider a three-dimensional bosonic model in thegrand canonical state and verify that a certain form of locality of temperatureholds regardless of the temperature, and despite the presence of infinite-rangecorrelations at and below the critical temperature of the phase transition.

Locality of temperature and correlations in the presence of non-zero-temperature phase transitions