Spontaneous breaking of multipole symmetries

TL;DR

This work establishes generalized Mermin-Wagner and Imry-Ma bounds for spontaneous breaking of multipole symmetries, covering both full and partial symmetry breaking in clean and disordered systems. By constructing invariant multipole field theories with derivative operators that annihilate the symmetry polynomials, it derives dispersion-dependent constraints on long-range order and identifies how disorder and compressibility affect the spectrum of massless modes. The authors provide explicit lattice realizations that illustrate how dipolar and monopolar components can independently undergo SSB with distinct dynamical exponents, and they extend the analysis to anisotropic and non-maximal multipole groups. These results offer a framework to assess robustness of fracton-like phases to thermal fluctuations, quantum fluctuations, and disorder, with implications for quantum dynamics and emergent universality classes. The work also outlines future directions, including extensions to higher-form, subsystem, and non-Abelian multipole structures and the role of topological defects in non-ordered regimes.

Abstract

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