Ground-state phase diagram,symmetries,excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

We investigate the two-mode quantum Rabi model(QRM)describing the interaction between a two-level atom and a two-mode cavity field.The quantum phase transitions are found when the ratioηof transition frequency of atom to frequency of cavity field approaches infinity.We apply the Schrieffer–Wolff(SW)transformation to derive the low-energy effective Hamiltonian of the two-mode QRM,thus yielding the critical point and rich phase diagram of quantum phase transitions.The phase diagram consists of four regions:a normal phase,an electric superradiant phase,a magnetic superradiant phase and an electromagnetic superradiant phase.The quantum phase transition between the normal phase and the electric(magnetic)superradiant phase is of second order and associates with the breaking of the discrete Z_(2) symmetry.On the other hand,the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1)symmetry.Several important physical quantities,for example the excitation energy and average photon number in the four phases,are derived.We find that the excitation spectra exhibit the Nambu–Goldstone mode.We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities.To confirm the validity of the low-energy effective Hamiltonians analytically derived by us,the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.