Emergent superfluid crystals, frustration, and topologically defected states in multimode cavity QED

We propose that condensed matter phenomena involving the spontaneous emergence and dynamics of crystal lattices can be realized in the setting of ultracold Bose-condensed atoms coupled to multimode cavities. Previously, it was shown that in the case of a transversely pumped single-mode cavity, the atoms self-organize at either the even or the odd antinodes of the cavity mode, given sufficient pump intensity, and hence spontaneously break a discrete translational symmetry. Here, we demonstrate that in multimode cavities the self-organization brings the additional feature of continuous translational symmetry breaking, via a variant of Brazovskii's transition, thus paving the way for realizations of compliant lattices and associated phenomena, e.g., quantum melting, topological defects, frustration, glassiness, and even supersolidity; such phenomena are absent in ultracold atomic systems when the optical lattices are externally imposed. We apply a functional integral approach to this many-body cavity QED system, which enables us, inter alia, to calculate transition thresholds, explore fluctuations near this transition, and determine how such fluctuations are manifest in the light emitted from the cavity.