A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases

Abstract

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases.

arXiv comment31 pages, 4 figures. The mathematical sections have been removed and now appear in 2405.04649

References in corpus (29)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Topological orders and Edge excitations in FQH states
• Braiding statistics approach to symmetry-protected topological phases
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Theta, Time Reversal, and Temperature
• Fermion Path Integrals And Topological Phases
• Classification of interacting electronic topological insulators in three dimensions
• Symmetry protected topological phases from decorated domain walls
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• Reflection positivity and invertible topological phases
• The "Parity" Anomaly On An Unorientable Manifold
• Dai-Freed anomalies in particle physics
• Comments on Abelian Higgs Models and Persistent Order
• Symmetry Protected Topological phases and Generalized Cohomology
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Candidate Phases for SU(2) Adjoint QCD${}_4$ with Two Flavors from $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Duality and bosonization of (2+1)d Majorana fermions
• Higher-dimensional generalizations of the Berry curvature
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• Anomalous Symmetries End at the Boundary
• Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models
• The algebra of anomaly interplay
• Toric 2-group anomalies via cobordism
• Bordism for the 2-group symmetries of the heterotic and CHL strings

Cited by in corpus (3)

• The Smith Fiber Sequence and Invertible Field Theories
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation