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Symmetry Protected Topological phases and Generalized Cohomology

Davide Gaiotto, Theo Johnson-Freyd

TL;DR

The paper develops a homotopy-theoretic framework to classify SPT phases via a generalized cohomology theory valued in the spectrum GP^× of invertible gapped phases. It shows that SPT phases arise from decorating symmetry domain walls and their junctions, encoded by k-invariants and stable cohomology operations, and provides concrete computations for bosonic and fermionic cases including group cohomology, Gu-Wen restricted supercohomology, extended supercohomology, and Majorana/E8 layers. It also integrates time-reversal, categorical actions, and anomaly considerations, formulating a twisted cohomology perspective and connecting to higher-dimensional anomaly theories. Overall, the framework unifies several known classifications and clarifies the role of E8, Majorana, and extended fermionic phases within a spectrum-based, cohomological paradigm suitable for comparing with cobordism and topological-field-theory approaches.

Abstract

We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.

Symmetry Protected Topological phases and Generalized Cohomology

TL;DR

The paper develops a homotopy-theoretic framework to classify SPT phases via a generalized cohomology theory valued in the spectrum GP^× of invertible gapped phases. It shows that SPT phases arise from decorating symmetry domain walls and their junctions, encoded by k-invariants and stable cohomology operations, and provides concrete computations for bosonic and fermionic cases including group cohomology, Gu-Wen restricted supercohomology, extended supercohomology, and Majorana/E8 layers. It also integrates time-reversal, categorical actions, and anomaly considerations, formulating a twisted cohomology perspective and connecting to higher-dimensional anomaly theories. Overall, the framework unifies several known classifications and clarifies the role of E8, Majorana, and extended fermionic phases within a spectrum-based, cohomological paradigm suitable for comparing with cobordism and topological-field-theory approaches.

Abstract

We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.

Paper Structure

This paper contains 30 sections, 26 equations, 8 figures.

Table of Contents

  1. Introduction and Conclusions
  2. Outline of the paper
  3. Generalities
  4. Stacking and invertibility
  5. SREs, invertible phases of matter, and invertible TFTs
  6. Defects
  7. Invertible defects
  8. The spectrum of invertible phases
  9. Definition of the spectrum
  10. Cohomology with coefficients in $\mathrm{GP}^\times$ and internal symmetries in lattice models
  11. SPT phases and spectra
  12. Symmetries and defects
  13. Stable cohomology operations
  14. SPT phases and reduced cohomology
  15. Phases versus choices of realization
  16. ...and 15 more sections

Figures (8)

  • Figure 1: A system (red) defines an invertible phase of matter if we can find another system (blue) such that the combination of the two system is in a trivial phase
  • Figure 2: Top: A collection of defects of various codimension in an invertible system can be mapped to a lower dimensional stand-alone system by stacking with an inverse system and deforming the bulk to a trivial system. Bottom: Lower dimensional systems can be stacked on top of a system to create simple defects. The two operations are essentially inverse of each other, up to important ambiguities in the choice of how to deform the bulk to a trivial system.
  • Figure 3: Composition of defects
  • Figure 4: A continuous interpolation between two systems can be identified with an invertible interface at a sufficiently large scale.
  • Figure 5: A mesoscopic lattice of invertible interfaces between two systems realizes a continuous interpolation between them.
  • ...and 3 more figures