Mermin-Wagner theorems for quantum systems with multipole symmetries
Timo Feistl, Severin Schraven, Simone Warzel
TL;DR
This work generalizes the Mermin-Wagner theorem to quantum lattice systems with multipole symmetries, showing that higher-order symmetries can elevate the dimensional threshold for spontaneous symmetry breaking of lower-order charges. The authors establish a robust, operator-algebraic framework on a countable metric lattice with effective dimension $\gamma$, introduce multipole automorphisms $\tau^{(a)}_s$, and impose $k$-symmetric interactions with suitable decay. Under the condition $|a|\le k$ and $\gamma\le 2(k-|a|+1)$, they prove that $\tau^{(a)}_s$ is preserved on all $\beta$-KMS states, recovering the classic result for $k=0$ and demonstrating protection in scenarios with dipole and higher multipole symmetries. The proof combines existence of infinite-volume multipole symmetries, a relative entropy bound via smooth truncations, and an entropy-based Mermin-Wagner criterion, with extensions to slabs and discussions of practical decay assumptions. The results have implications for systems with fracton-like behavior and broaden the scope of symmetry-protection mechanisms in quantum many-body physics.
Abstract
We prove Mermin-Wagner-type theorems for quantum lattice systems in the presence of multipole symmetries. These theorems show that the presence of higher-order symmetries protects against the breaking of lower-order ones. In particular, we prove that the critical dimension in which the charge symmetry can be broken increases if the system admits higher multipole symmetries, e.g. $ d = 4 $ on the regular lattice $ \mathbb{Z}^d $ in the presence of dipole symmetry.