Dipolar phase transition in the Dicke model with infinitely coordinated frustrating interaction

We consider the Dicke Hamiltonian of a system of N half spins with infinitely coordinated antiferromagnetic interaction. This Hamiltonian arises when one considers a single-mode microwave cavity coupled to low-capacitance Josephson junctions via the gauge-invariant Josephson phases. We found analytically a critical coupling strength causing a first order quantum phase transition of the system into dipolar phase with symmetry breaking coherent electromagnetic field emerging in the cavity. A new analytic tool: self-consistently ’rotating’ Holstein-Primakoff representation for the Cartesian components of the total spin is proposed.