Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

Peng Ye, Zheng-Cheng Gu

TL;DR

The paper develops a 3D TQFT description for bosonic SPT phases with unitary Abelian symmetry, showing that bulk excitations remain trivial while symmetry acts nontrivially on the partition function. Starting from a GL action for $G=\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}$, it identifies a novel topological term with quantized coefficient $p$ that classifies distinct SPTs in accord with group cohomology; dualization yields a BF-type TQFT with a higher-order interaction $a^1\wedge a^2\wedge d a^2$, and the framework clarifies how to implement symmetry, gauge structure, and domain-wall physics. The work extends to more general Abelian symmetries, including multiple $\mathbb{Z}_N$ factors and $\mathrm{U}(1)$ components, reproducing the full group-cohomology classifications via TQFTs and connecting to decorated-domain-wall constructions and (conjecturally) 3-loop statistics upon gauging. Together, these results provide a continuum, gauge-theoretic description of 3D SPTs that complements lattice group-cohomology approaches and informs boundary constructions and SET extensions.

Abstract

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state(or atomic insulator) as long as certain global symmetry $G$ is unbroken. At low energies, most of two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of multi-component Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\cdots$). In addition to the usual multi-component $BF$ topological term at level-$1$, we find that there are new topological terms with quantized coefficients (e.g., $a^1\wedge a^2\wedge d a^2$ and $a^1\wedge a^2\wedge a^3\wedge a^4$) in TQFT actions, where $a^{1},a^2,\cdots$ are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as $G$ is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.

Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

TL;DR

The paper develops a 3D TQFT description for bosonic SPT phases with unitary Abelian symmetry, showing that bulk excitations remain trivial while symmetry acts nontrivially on the partition function. Starting from a GL action for , it identifies a novel topological term with quantized coefficient that classifies distinct SPTs in accord with group cohomology; dualization yields a BF-type TQFT with a higher-order interaction , and the framework clarifies how to implement symmetry, gauge structure, and domain-wall physics. The work extends to more general Abelian symmetries, including multiple factors and components, reproducing the full group-cohomology classifications via TQFTs and connecting to decorated-domain-wall constructions and (conjecturally) 3-loop statistics upon gauging. Together, these results provide a continuum, gauge-theoretic description of 3D SPTs that complements lattice group-cohomology approaches and informs boundary constructions and SET extensions.

Abstract

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state(or atomic insulator) as long as certain global symmetry is unbroken. At low energies, most of two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of multi-component Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ). In addition to the usual multi-component topological term at level-, we find that there are new topological terms with quantized coefficients (e.g., and ) in TQFT actions, where are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.

Paper Structure

This paper contains 23 sections, 53 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results and outline
  4. SPT as a quantum disordered phase: Ginzburg-Landau action approach
  5. A review of quantum disordered phase
  6. Ginzburg-Landau action description for a $3+1$D bosonic SPT phase with $G=\mathbb{Z}_{N_1}\times \mathbb{Z}_{N_2}$ symmetry
  7. Topological Quantum Field Theory of Symmetry-Protected Topological Phases
  8. Duality between GL theory and TQFT
  9. Exotic gauge transformations
  10. Global symmetry $G= \mathbb{Z}_{N_1}\times \mathbb{Z}_{N_2}$ in TQFT
  11. Symmetry-protected quantization
  12. Other symmetries and applications
  13. Other Abelian symmetries
  14. Connection to "decorated domain walls" picture
  15. Connection to 3-loop statistics
  16. ...and 8 more sections

Figures (2)

  • Figure 1: (Color online) Bosonic wormhole effect as a physical mechanism of SPT phases with $\prod^2_I \mathbb{Z}_{N_I}$ symmetry. (a) A $\mathbb{Z}_{N_1}$ symmetry domain wall "$\mathcal{D}(\mathbb{Z}_{N_1})$" and a $\mathbb{Z}_{N_2}$ symmetry domain wall "$\mathcal{D}(\mathbb{Z}_{N_2})$" intersect along a closed loop "$\mathcal{C}$" (only a segment is shown due to the limited space). (b) shows the bosonic wormhole effect induced by intersections of symmetry domain walls. Each intercepting line forms a closed loop in the bulk, but some of them may form open strings and end at two 2D boundaries between which the bulk is sandwiched. Along such an open string $\mathcal{C}$ (the line with arrow), $\mathbb{Z}_{N_2}$ symmetry charge $\mathcal{Q}$ is pumped from the endpoint on the lower boundary to the endpoint on the upper boundary. See main text for more detailed explanation.
  • Figure 2: (Color online) Illustration of gauge-invariant operators introduced in Sec. \ref{['sec:gauge']}. (a) The Wilson loop operator $e^{i\int_{\mathcal{M}^1} a^I}$. (b) The operator formed by $\exp\{i\int_{\mathcal{M}^2} b^I-i2\pi p\int_{\mathcal{V}^3} \epsilon^{IJ3}a^J\wedge \,d a^2\}$ where $\mathcal{M}^2=\partial \mathcal{V}^3$. In (b), the cube represents $\mathcal{V}^3$ and its surface represents $\mathcal{M}^2$. The star symbols in (b) represent nonzero contributions of the "Chern-Simons density" $\int_{\mathcal{V}^3} \epsilon^{IJ3}a^J\wedge \,d a^2$ in $\mathcal{V}^3$.