Thermal and quantum phase transitions in atom-field systems: a microcanonical analysis

The thermodynamical properties of a generalized Dicke model are calculated and related with the critical properties of its energy spectrum, namely the quantum phase transitions (QPT) and excited state quantum phase transitions (ESQPT). The thermal properties are calculated both in the canonical and the microcanonical ensembles. The latter deduction allows for an explicit description of the relation between thermal and energy spectrum properties. While in an isolated system the subspaces with different pseudospin are disconnected, and the whole energy spectrum is accessible, in the statistical ensemble the situation is radically different. The multiplicity of the lowest energy states for each pseudospin completely dominates the thermal behavior, making the set of degenerate states with the smallest pseudospin at a given energy the only ones playing a role in the thermal properties. As a result, the states in the region with positive thermal energy cannot be thermally populated because their negligible probability, making that energy region thermally unreachable at finite temperatures. The quantum phase transitions of the lowest energy states, from a normal to a superradiant phase, produce the thermal transition. The other critical phenomena, the ESQPTs occurring at excited energies, have no manifestation in the thermodynamics, although their effects could be seen in finite size corrections. A new superradiant phase is found, which only exists in the generalized model, and can be relevant in finite size systems.