Intertwined and vestigial order with ultracold atoms in multiple cavity modes

Atoms in transversely pumped optical cavities "self-organize" by forming a density wave and emitting superradiantly into the cavity mode(s). For a single-mode cavity, the properties of this self-organization transition are well characterized both theoretically and experimentally. Here, we explore the self-organization of a Bose-Einstein condensate in the presence of two cavity modes---a system that was recently experimentally realized [Leonard \emph{et al.}, \emph{Nature} {\bf 543}, 87 (2017)]. We argue that this system can exhibit a "vestigially ordered" phase in which neither cavity mode exhibits superradiance but the cavity modes are mutually phase-locked by the atoms. We argue that this vestigially ordered phase should generically be present in multimode cavity geometries.