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

Dominic V. Else, Chetan Nayak

TL;DR

This work reframes the bosonic SPT classification as arising from obstructions to a local but non-on-site edge symmetry, proving these obstructions are captured by the cohomology group $H^{d+1}(G, U(1))$ across dimensions. Through a structured edge-reduction procedure, it shows how 2-cocycles and 3-cocycles emerge in (1+1)-D and (2+1)-D respectively, and extends to higher dimensions under locality assumptions, with nonlinear sigma models and theta terms providing concrete computation routes. The paper demonstrates concrete NL$\sigma$M examples, lattice-model realizations, and a fermionic generalization, including a Z2×Z2^f case yielding a $Z_4$ classification, while also discussing beyond-cohomology phases and the limitations of the approach. Overall, the framework connects edge anomalies to bulk SPT data, offering a practical, wavefunction-based diagnostic for identifying SPT phases from ground-state information and guiding future extensions to SETs and interacting fermionic systems.

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.

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

TL;DR

This work reframes the bosonic SPT classification as arising from obstructions to a local but non-on-site edge symmetry, proving these obstructions are captured by the cohomology group across dimensions. Through a structured edge-reduction procedure, it shows how 2-cocycles and 3-cocycles emerge in (1+1)-D and (2+1)-D respectively, and extends to higher dimensions under locality assumptions, with nonlinear sigma models and theta terms providing concrete computation routes. The paper demonstrates concrete NLM examples, lattice-model realizations, and a fermionic generalization, including a Z2×Z2^f case yielding a classification, while also discussing beyond-cohomology phases and the limitations of the approach. Overall, the framework connects edge anomalies to bulk SPT data, offering a practical, wavefunction-based diagnostic for identifying SPT phases from ground-state information and guiding future extensions to SETs and interacting fermionic systems.

Abstract

It is well known that (1+1)-D bosonic symmetry-protected topological (SPT) phases with symmetry group 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 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 , 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 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.

Paper Structure

This paper contains 21 sections, 4 theorems, 83 equations, 4 figures, 1 table.

Table of Contents

  1. The general formalism
  2. (1+1)-D SPT's
  3. (2+1)-D SPT's
  4. Higher dimensions
  5. Example: "Chiral" symmetry on the edge of a (2+1)-D SPT
  6. Proof of separation of phases in (2+1)-D.
  7. Non-linear sigma models
  8. Examples
  9. $Z_2^T$ in (1+1)-D
  10. $Z_2$ in (2+1)-D
  11. Lattice models of SPT phases
  12. Beyond the cohomological classification
  13. Fermionic systems
  14. Example: Fermionic SPT with $Z_2$ symmetry
  15. Conclusions
  16. ...and 6 more sections

Key Result

Lemma 1

A $\mathrm{U}(1)$$k$-cochain $\omega$ on a manifold $T$ is exact if and only if $\omega(C) = 0$ for all closed (i.e. boundaryless) $k$-chains $C$.

Figures (4)

  • Figure 1: Obtaining a 2-cocycle on the (0+1)-D edge of a (1+1)-D SPT.
  • Figure 2: Obtaining a 3-cocycle on the (1+1)-D edge of a (2+1)-D system.
  • Figure 3: The reduction process to obtain a 4-cocycle $\omega$ on the (2+1)-D edge of a (3+1)-D system, assuming a symmetry representation on the edge of the form Eq. (\ref{['simple_symmetry']}).
  • Figure 4: The regions $A$ and $A^{\prime}$ on which we can prove that the anomalous symmetry on the boundary is classified by the same element of $H^3(G, \mathrm{U}(1))$. The orientations of the boundaries $\partial A$ and $\partial A^{\prime}$ are depicted with arrows.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof