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 pseudo spin are disconnected, and the whole energy spectrum is accesible, in the thermal ensamble the situation is radically different. The multiplicity of the lowest energy states for each pseudo spin completely dominates the thermal behavior, making the set of degenerate states with the smallest pseudo spin at a given energy the only ones playing a role in the thermal properties, making the positive energy states thermally inaccesible. Their quantum phase transitions, from a normal to a superradiant phase, are closely associated with 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 sizes corrections. A new superradiant phase is found, which only exists in the generalized model, and can be relevant in finite size systems. 
Miguel A. Bastarrachea-Magnani, Sergio Lerma-Hern\'andez, Jorge G.   Hirsch
quant-ph