Time crystal in a static cavity QED Hamiltonian

In this work, we study how localization in the presence of a long-range coupling can hosts new symmetry breaking and topological phases with non-trivial correlations. The long-range coupling can be mediated by photons in cavity QED or other global modes, such as a central spin-$S$. One of the interesting emergent phases is a time crystal, which exists in thermal equilibrium for this cavity QED Hamiltonian. We investigate the stability of time crystal to dissipation, which naturally occurs in real experimental systems. We also discuss analytical methods to determine stability, including a high-frequency expansion that allows adiabatic elimination of the photon degree of freedom.