In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counter-rotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the orig
inal model. We derive the corresponding $G$-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities and are doubly degenerate. We find analytically that the ground-state and the first excited state can cross several times, indicating multiple first-order phase transitions as function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model which are caused by a hidden symmetry.
In this paper, we analyze the quantum criticality of the Rabi-Stark model at finite ratios of the qubit and cavity frequencies in terms of the energy gap, the order parameter, as well as the fidelity, if the Stark coupling strength is the same as the
cavity frequency. The critical exponents are derived analytically. The energy gap and the length critical exponents are different from those in the quantum Rabi model and the Dicke model. The finite size scaling analysis for the order parameter and the fidelity susceptibility is also performed. The universal scaling behaviors are demonstrated and several finite size exponents can be then extracted. Furthermore, universal critical behavior can be also established in terms of the bosonic Hilbert space truncation number, and the corresponding critical scaling exponents are found. Interestingly, the critical correlation length exponents in terms of the photonic truncation number as well as the equivalently effective length scales are different in the Rabi-Stark model and the quantum Rabi model, suggesting they belong to different universality classes. The second-order quantum phase transition is convincingly corroborated in the Rabi-Stark model at finite frequency ratios, by contrast, it only emerges at the infinite frequency ratio in the original quantum Rabi model without the Stark coupling.
Various quantum phase transitions in the anisotropic Rabi-Stark model with both the nonlinear Stark coupling and the linear dipole coupling between a two-level system and a single-mode cavity are studied in this work. The first-order quantum phase tr
ansitions are detected by the level crossing of the ground-state and the first-excited state with the help of the pole structure of the transcendental functions derived by the Bogoliubov operators approach. As the nonlinear Stark coupling is the same as the cavity frequency, this model can be solved by mapping to an effective quantum oscillator. All energy levels close at the critical coupling in this case, indicating continuous quantum phase transitions. The critical gap exponent is independent of the anisotropy as long as the counter-rotating wave coupling is present, but essentially changed if the counter-rotating wave coupling disappears completely. It is suggested that the gapless Goldstone mode excitations could appear above a critical coupling in the present model in the rotating-wave approximation.
