Spontaneous breaking of multipole symmetries

Multipole symmetries are of interest both as a window on fracton physics and crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question whether when multipole can be spontaneously broken, thermal equilibrium at zero temperature. We derive generalized Mermin-Wagner arguments total or partial breaking multipolar symmetry groups Imry-Ma robustness such to disorder. present general results explicit examples. Our should directly applicable dynamics with also provide useful stepping stone understanding phases fluctuations,