Quantum phase transition in a pseudo-Hermitian Dicke model

We show that a Dicke-type non-Hermitian Hamiltonian admits entirely real spectra by mapping it to the 'dressed Dicke model' through a similarity transformation. We find a positive-definite metric in the Hilbert space of the non-Hermitian Hamiltonian so that the time evolution is unitary and allows a consistent quantum description. We then show that this non-Hermitian Hamiltonian describing nondissipative quantum processes undergoes quantum phase transition. The exactly solvable limit of the non-Hermitian Hamiltonian has also been discussed.