From thermal to excited-state quantum phase transitions ---the Dicke model

We study the thermodynamics of the full version of the Dicke model, including all the possible values of the total angular momentum $j$, with both microcanonical and canonical ensembles. We focus on how the excited-state quantum phase transition, which only appears in the microcanonical description of the maximum angular momentum sector, $j=N/2$, change to a standard thermal phase transition when all the sectors are taken into account. We show that both the thermal and the excited-state quantum phase transitions have the same origin; in other words, that both are two faces of the same phenomenon. Despite all the logarithmic singularities which characterize the excited-state quantum phase transition are ruled out when all the $j$-sectors are considered, the critical energy (or temperature) still divides the spectrum in two regions: one in which the parity symmetry can be broken, and another in which this symmetry is always well defined.