Critical Property of the Geometric Phase in the Dicke Model with the Dipole-dipole Interactions

We obtained the ground-state energy level and associated geometric phase in the Dicke model with the dipole-dipole interactions analytically by the Holstein-Primakoff transformation and the boson expansion approach in the thermodynamic limit.The nonadiabatic geometric phase induced by the photon field was derived with the time-dependent unitary transformation.It is shown that dipole-dipole interactions have a deep influence on scaled behavior of the geometric phase at the critical point.