Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge

Abstract

It is well known that (1+1)-D bosonic symmetry-protected topological (SPT) phases with symmetry group $G$ can be identified by the projective representation of the symmetry at the edge. Here, we generalize this result to higher dimensions. We assume that the representation of the symmetry on the spatial edge of a ($d+1$)-D SPT is /local/ but not necessarily /on-site/, such that there is an obstruction to its implementation on a region with boundary. We show that such obstructions are classified by the cohomology group $H^{d+1}(G,U(1))$, in agreement with the classification of bosonic SPT phases proposed in [Chen et al, Science 338, 1604 (2012)]. Our analysis allows for a straightforward calculation of the element of $H^{d+1}(G,U(1))$ corresponding to physically meaningful models such as non-linear sigma models with a theta term in the action. SPT phases outside the classification of Chen et al are those in which the symmetry cannot be represented locally on the edge. With some modifications, our framework can also be applied to fermionic systems in (2+1)-D.

23 pages + 11 pages of appendices; v2 added references; v3 final published version

References in corpus (12)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• A classification of symmetry enriched topological phases with exactly solvable models
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Interaction effects on 1D fermionic symmetry protected topological phases

Cited by in corpus (7)

• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Non-Abelian Hybrid Fracton Orders
• Rokhsar-Kivelson Models of Bosonic Symmetry-Protected Topological States
• Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions
• From orbifolding conformal field theories to gauging topological phases
• Field theory of symmetry-protected valence bond solid states in (2+1) dimensions
• Perturbative and global anomalies in bosonic analogues of integer quantum Hall and topological insulator phases