Fermion path integrals and topological phases

2015/08/19 by Edward Witten · 1 voice · 15 citations
Materials Science · Physics and Astronomy · #Atiyah–Singer index theorem #Cohomology #Fermion #Graphene research and applications #Invariant (physics) #Mathematical physics #Path integral formulation #Physics #Pure mathematics #Quantum #Quantum many-body systems #Quantum mechanics #Symmetry (geometry) #Symmetry group #Theoretical physics #Topological Materials and Phenomena #Topological insulator #cond-mat.mes-hall #cond-mat.other #hep-th

paper · pdf · doi:10.1103/revmodphys.88.035001

published as Rev. Mod. Phys. 88, 35001 (2016) · 67 pp plus appendices, minor corrections in v. 2

arxiv published 2015/08/19 · arxiv created 2015/12/19 · openalex publication_date 2016/07/13 · arxiv updated 2016/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Symmetry-protected phases of matter have been at the forefront of condensed matter physics in recent years. Bosonic symmetry-protected phases have been interpreted in terms of anomalies and group cohomology. The present article aims to develop an analogous description of fermionic symmetry-protected phases, such as the topological insulators that have been seen experimentally in 2 or 3 space dimensions. The relevant mathematical concepts include the Atiyah-Singer index theorem and the Atiyah-Patodi-Singer eta invariant.

Citations

• Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets
• A PCAC puzzle: π0→γγ in the σ-model
• Topological insulators and superconductors
• Heterotic and Type I string dynamics from eleven dimensions
• Quantal phase factors accompanying adiabatic changes
• Topological Insulators
• Short-range entanglement and invertible field theories
• E₈ gauge theory, and a derivation of K-theory from M-theory
• The index of elliptic operators on compact manifolds
• The Index of Elliptic Operators: I
• Axial-Vector Vertex in Spinor Electrodynamics
• The Index of Elliptic Operators: V

Cited by

• Surface Variables Description of Axion Topological Materials
• Theory of the Matchgate Commutant
• Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
• Exploring two dimensional ℤ₂ invariant phases with time reversal symmetry and their transitions with topological operations
• Boundary Witten effect in multi-axion insulators
• Intrinsically gapless topological phases
• Three-dimensional topological insulators and bosonization
• Bosonization and mirror symmetry
• Master 3d bosonization duality with boundaries
• Quantum annealed criticality: A scaling description
• Catalysis of dynamical chiral symmetry breaking by chiral chemical potential in Dirac semimetals
• Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries
• High-temperature domain walls of QCD with imaginary chemical potentials
• Fermionic matrix product states and one-dimensional topological phases
• Unquantized anomalies in topological semimetals
• Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Discussions

• I have a question about the symmetry protecting a 3+1d topological insulator (TI). In the ten-fold way classification I am considering class AII, which is protected by time reversal alone. This gives [stackexchange, 2 points]

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• Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective
• Classification and Construction of Topological Phases of Quantum Matter
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Symmetry protected topological orders and the group cohomology of their symmetry group