  Frustration occurs when a system cannot find a lowest-energy configuration
due to conflicting constraints. We show that a frustrated superradiant phase
transition occurs when the ground-state superradiance of cavity fields due to
local light-matter interactions cannot simultaneously minimize the positive
photon hopping energies. We solve the Dicke trimer model on a triangle motif
with both negative and positive hopping energies and show that the latter
results in a six-fold degenerate ground-state manifold in which the
translational symmetry is spontaneously broken. In the frustrated superradiant
phase, we find that two sets of diverging time and fluctuation scales coexist,
one governed by the mean-field critical exponent and another by a novel
critical exponent. The latter is associated with the fluctuation in the
difference of local order parameters and gives rise to site-dependent photon
number critical exponents, which may serve as an experimental probe for the
frustrated superradiant phase. We provide a qualitative explanation for the
emergence of unconventional critical scalings and demonstrate that they are
generic properties of the frustrated superradiant phase at the hand of a
one-dimensional Dicke lattice with an odd number of sites. The mechanism for
the frustrated superradiant phase transition discovered here applies to any
lattice geometries where the anti-ferromagnetic ordering of neighboring sites
are incompatible and therefore our work paves the way towards the exploration
of frustrated phases of coupled light and matter.
