  We propose a quantum Rabi square model where both the nearest-neighbor and
the next-nearest-neighbor photon hopping are allowed among four quantum Rabi
systems located at the vertices of a square. By tuning the next-nearest hopping
strength, we realize a first-order phase transition between the
antiferromagnetic superradiant phase and the frustrated superradiant phase, as
well as a second-order phase transition between the normal and the superradiant
phases. To understand the emergence of such phases, we show analytically that
the effect induced by next-nearest hopping is equivalent to that of an
artificial gauge phase. Our findings suggest that the next-nearest-neighbor
hopping can serve as an alternative for the gauge phase to realize quantum
control in applications of quantum simulation and quantum materials, and that
our model represents a basic building block for the frustrated $J_1$-$J_2$
quantum spin model on square lattices.
