  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.
