  We investigate a generalized Dicke model by introducing two interacting spin
ensembles coupled with a single-mode bosonic field. Apart from the normal to
superradiant phase transition induced by the strong spin-boson coupling,
interactions between the two spin ensembles enrich the phase diagram by
introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The
mean-field approach reveals a phase diagram comprising three phases:
paramagnetic-normal phase, ferromagnetic-superradiant phase, and
antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can
significantly reduce the required spin-boson coupling strength to observe the
superradiant phase, where the macroscopic excitation of the bosonic field
occurs. Conversely, antiferromagnetic spin-spin interaction can strongly
suppress the superradiant phase. To examine higher-order quantum effects beyond
the mean-field contribution, we utilize the Holstein-Primakoff transformation,
which converts the generalized Dicke model into three coupled harmonic
oscillators in the thermodynamic limit. Near the critical point, we observe the
close of the energy gap between the ground and the first excited states, the
divergence of entanglement entropy and quantum fluctuation in certain
quadrature. These observations further confirm the quantum phase transition and
offer additional insights into critical behaviors.
