Excited-state quantum phase transition in light-matter systems with discrete and continuous symmetries

We systematically investigate the excited-state quantum phase transition (ESQPT) in the anisotropic quantum Rabi model, which interpolates between the quantum Rabi model with Z2 symmetry and the Jaynes-Cummings model with U (1) symmetry. We calculate the model energy spectra and density of states (DOS) with the cumulants in both analytical and numerical ways to describe the ESQPT by the singularities. In the Jaynes-Cummings limit, its continuous U (1) symmetry presents different nonanalytic behaviors from the quantum Rabi model with the discrete Z2 symmetry, and there exists a finite discontinuous jump at the critical energy. For the general anisotropy case with Z2 symmetry, there are two types of ESQPTs characterized by the finite discontinuous jump and the logarithmic divergence in the DOS, respectively. Different from the ground-state quantum phase transition, the ESQPT of the anisotropic quantum Rabi model strongly depends on the anisotropy which leads to a discontinuity even though the Z2 symmetry is still preserved.