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

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.

arXiv comment54 pages, minor changes

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Cited by in corpus (7)

• Symmetry Transmutation and Anomaly Matching
• Anomalous Continuous Translations
• LSM and CPT
• IR side of bounds on Theories with Spontaneously Broken Lorentz Symmetry
• Topological dipoles of quantum skyrmions
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
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