Ferromagnets, a New Anomaly, Instantons, and (noninvertible) Continuous Translations

TL;DR

This paper explores how classical field theories with continuous translation symmetry exhibit quantum anomalies that break this symmetry to a discrete one, with implications for ferromagnets and Landau level lattices.

Contribution

It introduces a unified framework describing anomalies, discrete and noninvertible symmetries, and their relation to lattice models in translation-invariant field theories.

Findings

Quantum anomalies break continuous translation symmetry to a discrete one.

Discrete translation symmetry extends with a $d-2$-form global symmetry.

Broken continuous symmetry can be restored as a noninvertible symmetry.

Abstract

We discuss a large class of classical field theories with continuous translation symmetry. In the quantum theory, a new anomaly explicitly breaks this translation symmetry to a discrete symmetry. Furthermore, this discrete translation symmetry is extended by a $d-2$-form global symmetry. All these theories can be described as $U(1)$ gauge theories where Gauss law states that the system has nonzero charge density. Special cases of such systems can be phrased as theories with a compact phase space. Examples are ferromagnets and lattices in the lowest Landau level. In some cases, the broken continuous translation symmetry can be resurrected as a noninvertible symmetry. We clarify the relation between the discrete translation symmetry of the continuum theory and the discrete translation symmetry of an underlying lattice model. Our treatment unifies, clarifies, and extends earlier works on the same subject.