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.

54 pages, minor changes

References in corpus (19)

• Generalized Global Symmetries
• Higher Gauging and Non-invertible Condensation Defects
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
• $(k,n)$-fractonic Maxwell theory
• 2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Classification of fractional quantum Hall states with spatial symmetries
• A Comment On Berry Connections
• Is Schwinger Model at Finite Density a Crystal?
• Conserved momenta of ferromagnetic solitons through the prism of differential geometry
• A Duality Between U(1) Haah Code and 3D Smectic A Phase
• Gappability Index for Quantum Many-Body Systems
• Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum

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
• Non-invertible translation from Lieb-Schultz-Mattis anomaly