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Classification of topological phases with finite internal symmetries in all dimensions

Liang Kong, Tian Lan, Xiao-Gang Wen, Zhi-Hao Zhang, Hao Zheng

TL;DR

The paper presents a unified, two-pronged framework to classify symmetry-protected trivial (SPT) and anomaly-free symmetry-enriched topological (SET) orders across all spatial dimensions with finite internal symmetries. It develops both: (i) gauging the symmetry in the same dimension via minimal modular extensions and (ii) a boundary–bulk approach that uses the center and condensation completion to encode intrinsic data. The main contributions include a complete description for 1d and 2d cases, explicit group identifications (e.g., $H^3(G,U(1))$ for 2d bosonic SPT, $\mathbb{Z}_{16}$ for fermionic 2d cases with $G=\mathbb{Z}_2$, etc.), and a general higher-dimensional classification framework leveraging $E_k$-monoidal categories and delooping. The results establish the equivalence of the two approaches and provide a rigorous mathematical blueprint for predicting and classifying SPT/SET orders via higher-categorical centers, with condensation completion formalizing the inclusion of all defect descendants.

Abstract

We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

Classification of topological phases with finite internal symmetries in all dimensions

TL;DR

The paper presents a unified, two-pronged framework to classify symmetry-protected trivial (SPT) and anomaly-free symmetry-enriched topological (SET) orders across all spatial dimensions with finite internal symmetries. It develops both: (i) gauging the symmetry in the same dimension via minimal modular extensions and (ii) a boundary–bulk approach that uses the center and condensation completion to encode intrinsic data. The main contributions include a complete description for 1d and 2d cases, explicit group identifications (e.g., for 2d bosonic SPT, for fermionic 2d cases with , etc.), and a general higher-dimensional classification framework leveraging -monoidal categories and delooping. The results establish the equivalence of the two approaches and provide a rigorous mathematical blueprint for predicting and classifying SPT/SET orders via higher-categorical centers, with condensation completion formalizing the inclusion of all defect descendants.

Abstract

We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

Paper Structure

This paper contains 14 sections, 4 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. SPT/SET orders via gauging the symmetry
  3. 2d SPT/SET orders
  4. 1d SPT/SET orders
  5. Mathematical classification
  6. Physical classification
  7. SPT/SET orders in higher dimensions
  8. SPT/SET orders via boundary-bulk relation
  9. 0d SPT/SET orders
  10. 1d SPT/SET orders revisit
  11. Condensation completion
  12. 2d SPT/SET orders
  13. nd SPT/SET orders
  14. Appendix

Key Result

Lemma 2.6

If $\mathrm{M}_{\mathrm{ex}}(\EuScript{C},\eta_\EuScript{C})$ and $\mathrm{M}_{\mathrm{ex}}(\EuScript{D},\eta_\EuScript{D})$ are not empty, then is a well-defined map. Here, the notation $(-)_{L_\EuScript{E}}^0$ denotes the category of local $L_\EuScript{E}$-modules in the braided category $-$, and the functor $\iota_\EuScript{M}\star\iota_\EuScript{N}$ is the one induced from $\iota_\EuScript{M}

Figures (6)

  • Figure 1: Picture (a) illustrates the relation among $\EuScript{E},\EuScript{A},\EuScript{X},\EuScript{M}$ in a physical way and provides a proof of the canonical isomorphism in (\ref{['eq:pic-aut']}); Picture (b) illustrates the compatibility between the multiplications in $\mathrm{Pic}(\EuScript{E})$ and $\mathop{\mathrm{Aut}}\nolimits^{br}(\mathfrak{Z}_1(\EuScript{E}),\iota_0)$, where $\EuScript{Y}_{\phi_i}$ denotes the invertible 1d domain wall associated to $\phi_i$ for $i=1,2$ and is itself a 1d SPT order.
  • Figure 2: This picture illustrates some 1d, 0d domain walls in an anomaly-free 2d topological order $\EuScript{M}$.
  • Figure 3: These pictures illustrate the physical meaning of the proof of Theorem$^{\mathrm{ph}}$\ref{['pthm:cc-P']}.
  • Figure 4: Based on the boundary-bulk relation kwz1kz, these pictures illustrate the physical meanings of the functor $\Phi$ and its inverse in the proof of Theorem$^{\mathrm{ph}}$\ref{['thm:fun-PP']}.
  • Figure 5: Picture (a) depicts a physical configuration that illustrates the relation among $\Sigma\EuScript{E},\Sigma\EuScript{C}$. $\EuScript{Y}_\phi$ denotes the 2d invertible domain wall between two 3d bulks associated to the braiding equivalence $\phi:\mathfrak{Z}_1(\Sigma\EuScript{E})\to \mathfrak{Z}_1(\Sigma\EuScript{C})$ and $\iota\simeq\phi\circ\iota_0$. Picture (b) depicts the result of the closing-fan process indicated by the dotted lines in Picture (a) (see (\ref{['eq:sigma-M']})).
  • ...and 1 more figures

Theorems & Definitions (77)

  • Remark 1.2
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6: LW160205936
  • ...and 67 more