Stability of polarizable materials against superradiant phase transition

The possibility of the superradiant phase transition in polarizable materials described by the minimal-coupling Hamiltonian with the longitudinal dipole-dipole interaction is examined. We try to reduce the Hamiltonian into the Dicke one in the homogeneous and infinite case, and discuss the stability of normal ground state by the formalism of the Green function in the spatially inhomogeneous case. The presence of the longitudinal dipole-dipole interaction does not enable the superradiant phase transition, if the transverse and longitudinal fields are decoupled. Although the full dipole-dipole interaction can be eliminated in the electric-dipole gauge in the absence of overlap between individual atomic dipoles, we cannot reduce the Hamiltonian to the Dicke one, because the elimination is justified only if all the transverse and longitudinal fields remain. Even if the transverse and longitudinal fields are mixed in spatially inhomogeneous systems, the normal ground state is still stable if the system does not show the superradiant phase transition in the homogeneous case.