Higher symmetry breaking and non-reciprocity in a driven-dissipative Dicke model

Higher symmetries in interacting many-body systems often give rise to new phases and unexpected dynamical behavior. The canonical Dicke model, which describes identical coupling of many two-level systems to a single bosonic mode, exhibits a Z₂ symmetry-breaking superradiant phase. Here, we theoretically investigate a variant of the Dicke model with higher-order discrete symmetry, resulting from complex-valued coupling coefficients between quantum emitters and a bosonic mode. We propose a driven-dissipative realization of this model focusing on optomechanical response of a driven atom tweezer array comprised of n sub-ensembles and placed within an optical cavity, with the phase of the driving field advancing stepwise between sub-ensembles.  Examining stationary points and their dynamical stability, we identify a phase diagram for n≥3 with three distinctive features: a Z_n (Z_2n) symmetry-breaking superradiant phase for even (odd) n, a normal unbroken-symmetry phase that is dynamically unstable due to non-reciprocal forces between emitters, and a first-order phase transition separating these phases. This n-phase Dicke model may be equivalently realized in a variety of optomechanical or opto-magnonic settings, where it can serve as a testbed for studying high-order symmetry breaking and non-reciprocal interactions in open systems.