  While temperature is well understood as an intensive quantity in standard
thermodynamics, it is less clear whether the same holds in the presence of
strong correlations, especially in the case of quantum systems, which may even
display correlations with no classical analogue. The problem lies in the fact
that, under the presence of strong correlations, subsystems of a system in
thermal equilibrium are, in general, not described by a thermal state at the
same temperature as the global system and thus one cannot simply assign a local
temperature to them. However, there have been identified situations in which
correlations in thermal states decay sufficiently fast so that the state of
their subsystems can be very well approximated by the reduced states of
equilibrium systems that are only slightly bigger than the subsystems
themselves, hence allowing for a valid local definition of temperature. In this
work, we address the question of whether temperature is locally well defined
for a bosonic system with local interactions that undergoes a phase transition
at non-zero temperature. We consider a three-dimensional bosonic model in the
grand canonical state and verify that a certain form of locality of temperature
holds regardless of the temperature, and despite the presence of infinite-range
correlations at and below the critical temperature of the phase transition.
