Interplay of quantum phase transition and flat band in hybrid lattices

We establish a connection between quantum phase transitions (QPTs) and energy band theory in an extended Dicke-Hubbard lattice, where the periodical critical curves modulated by wave number k leads to rich equilibrium dynamics. Interestingly, the chiral-symmetry-protected flat band and the localization that it engenders exclusively occurs in the normal phase and disappears in the superradiant phase. This originates from the QPT induced simultaneous breaking up of the on-site resonance condition and off-site chiral symmetry of the system, which prohibits the destructive interference for obtaining a flat band. Our work offers an approach to identify different phases of the lattice via detecting the flat bands or simply the related localizations in a single cell, and, in turn, to control the appearance of flat bands by QPT.