Phase transitions of the Dicke model

We revisit the three main phase transitions of the Dicke model: quantum phase transition (QPT), thermal phase transition (TPT) and excited state quantum phase transition (ESQPT). It is known that in the QPT the system goes from the normal phase (NP) to the super-radiant phase (SP). We show here that in the SP, the ground state participation ratio scales as square root of the total Hilbert space dimension, indicating multifractal behavior of the system. The TPT is studied with the aid of the mutual information between two spins - this yields a striking signature in excellent agreement with the analytical expression for the critical temperature Tc, above which the system comes back from the SP to the NP for a sufficiently large coupling strength. The ESQPT which is traditionally understood as a property of the spectrum is here studied also with the aid of eigenstate properties. We find that not only is there a lower cut-off energy but also there exists an upper cutoff energy, which is signalled by a number of eigenstate properties: von-Neumann entanglement entropy between spins and bosons, the boson particle number, concurrence between two spins and the participation ratio.