Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

By A. Kapustin. 2014

225 citations recorded by the source corpus.

We propose that Symmetry Protected Topological Phases with a finite symmetry group G are classified by cobordism groups of the classifying space of G. This provides an explanation for the recent discovery of bosonic SPT phases which do not fit into the group cohomology classification. We discuss the connection of the cobordism classification of SPT phases to gauge and gravitational anomalies in various dimensions.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/7f7f3abc15b065375f46e229ed4d6b5532d881af

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Gapless Topological Phases and Symmetry-Enriched Quantum Criticality (2019, background): “[6] Anton Kapustin, “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,” arXiv e-prints , arXiv:1403.”
Topological phases with generalized global symmetries (2015, background): “Recently, generalizations of SPT phases with higher-form symmetry have been discussed [22–26].”
Classification of topological quantum matter with symmetries (2015, methodology): “While the group cohomology approach is one of the most systematic and general methods to classify SPT phases, it was shown that it does not describe all possible phases (Kapustin, 2014b; Vishwanath and Senthil, 2013; Wang and Senthil, 2013).”
On 2-group global symmetries and their anomalies (2018, methodology): “This reproduces the result in [27].”
Average Symmetry-Protected Topological Phases (2022, background): “For an invertible phase (such as SPT) in a clean setup, the global properties are included in a topological quantum field theory (TQFT) as the imaginary phase of the Euclidean partition function[36],”
Pre-thermal phases of matter protected by time-translation symmetry (2016, unknown): “[14] Anton Kapustin, “Symmetry Protected Topological”
Generalized global symmetries (2014, methodology): “If one allows dependence on the topology of the space-time manifold (which corresponds to taking into account discrete gravitational and mixed gauge-gravitational anomalies), then ’t Hooft anomalies are classified by the degree d+1 cobordism group of BG with U(1) coefficients [28].”
Spin TQFTs and fermionic phases of matter (2015, background): “It was proposed in [11] that fermionic Short Range Entangled phases in d space-time dimensions with symmetry G and vanishing thermal Hall conductivity are classified by the Pontryagin dual of the torsion part of Ω d (BG).”
Classification of Interacting Topological Floquet Phases in One Dimension (2016, background): “[37] Anton Kapustin, “Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology,” arXiv preprint arXiv:1403.”