Exotic frustrated phases of coupled light and matter on the periodic lattice

Symmetry breaking leads to exotic phases and critical phenomena in quantum many-body systems. In this work, we investigate this phenomenon in large arrays of coupled Dicke models on a periodic lattice, where strong light-matter coupling drives the system into a superradiant phase. Unlike the conventional Dicke model alone, competition between order parameters of neighboring sites leads to spontaneous translational symmetry breaking. We show that the broken translational symmetry in the superradiant phase can be predicted from the excitation spectrum of the normal phase, significantly simplifying energy minimization in numerical calculations. Furthermore, we identify an abnormal excitation mode that evolves into a zero-energy mode as the system size approaches infinity, which we interpret as an asymptotic Nambu-Goldstone mode. Our study also reveals the emergence of quasi-periodic ordering in the superradiant phase, akin to incommensurate charge-density-wave phases and quasicrystals. Introducing a magnetic flux allows for the engineering of translational symmetry and commensurate ordering. These findings provide new insights into symmetry breaking, quantum criticality, and collective excitations, with potential applications to topological phases and localized states in cavity quantum electrodynamics and quantum optics.