Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation

Abstract

We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\omega$ of the cohomology group $H^3(G,U_T(1))$. This element $\omega \in H^3(G,U_T(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$-symbol associated with domain walls in a symmetry broken edge theory; this domain wall $F$-symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.

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• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
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Cited by in corpus (23)

• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Pauli stabilizer models of twisted quantum doubles
• Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• (SPT-)LSM theorems from projective non-invertible symmetries
• Gauging non-invertible symmetries on the lattice
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Average-exact mixed anomalies and compatible phases
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• LSM and CPT
• Self-$G$-ality in 1+1 dimensions
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Disentangling anomaly-free symmetries of quantum spin chains
• $N$-ality symmetry and SPT phases in (1+1)d
• SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Non-onsite symmetry breaking: topological phase coexistence and criticality