  Circuit QED on a chip has become a powerful platform for simulating complex
many-body physics. In this report, we realize a Dicke-Ising model with an
antiferromagnetic nearest-neighbor spin-spin interaction in circuit QED with a
superconducting qubit array. We show that this system exhibits a competition
between the collective spin-photon interaction and the antiferromagnetic
nearest-neighbor spin-spin interaction, and then predict four quantum phases,
including: a paramagnetic normal phase, an antiferromagnetic normal phase, a
paramagnetic superradiant phase, and an antiferromagnetic superradiant phase.
The antiferromagnetic normal phase and the antiferromagnetic superradiant phase
are new phases in many-body quantum optics. In the antiferromagnetic
superradiant phase, both the antiferromagnetic and superradiant orders can
coexist, and thus the system possesses $Z_{2}^{z}\otimes Z_{2}$\ symmetry.
Moreover, we find an unconventional photon signature in this phase. In future
experiments, these predicted quantum phases could be distinguished by detecting
both the mean-photon number and the magnetization.
