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Classification of fractional quantum Hall states with spatial symmetries

Naren Manjunath, Maissam Barkeshli

TL;DR

This work develops a comprehensive framework to classify fractional quantum Hall states with spatial symmetries by employing G-crossed braided tensor categories, capturing both intrinsic topological order and symmetry-protected invariants. It systematically computes symmetry fractionalization and defect invariants for a range of groups including continuous and discrete translations and rotations, and derives their physical consequences as fractionally quantized responses (e.g., Hall conductivity, shift, angular momentum) and defect quantum numbers. The authors provide explicit invariant formulas, connect them to crystalline gauge theory via effective actions, and demonstrate applications to lattice FQH analogs and fractional Chern insulators, including counting distinct SET phases for bosonic Moore-Read and Read-Rezayi orders. The framework yields new categorical definitions of Hall conductivity, extends LSM-type relations beyond Galilean invariance, and offers a path to extracting symmetry-enriched topological data from ground-state wavefunctions, with clear experimental relevance in FCIs and photonic/optical lattice platforms.

Abstract

Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs) for SETs. We consider systems with $U(1)$ charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all $5$ orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains $9$ terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the $G\times$BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the $G\times$BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.

Classification of fractional quantum Hall states with spatial symmetries

TL;DR

This work develops a comprehensive framework to classify fractional quantum Hall states with spatial symmetries by employing G-crossed braided tensor categories, capturing both intrinsic topological order and symmetry-protected invariants. It systematically computes symmetry fractionalization and defect invariants for a range of groups including continuous and discrete translations and rotations, and derives their physical consequences as fractionally quantized responses (e.g., Hall conductivity, shift, angular momentum) and defect quantum numbers. The authors provide explicit invariant formulas, connect them to crystalline gauge theory via effective actions, and demonstrate applications to lattice FQH analogs and fractional Chern insulators, including counting distinct SET phases for bosonic Moore-Read and Read-Rezayi orders. The framework yields new categorical definitions of Hall conductivity, extends LSM-type relations beyond Galilean invariance, and offers a path to extracting symmetry-enriched topological data from ground-state wavefunctions, with clear experimental relevance in FCIs and photonic/optical lattice platforms.

Abstract

Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of -crossed braided tensor categories (BTCs) for SETs. We consider systems with charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.

Paper Structure

This paper contains 102 sections, 8 theorems, 417 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Classification method
  3. Applicability of $G$-crossed BTC to spatial symmetries
  4. Continuity and finiteness of $G$
  5. Relation to other classification approaches
  6. Summary of main results
  7. Symmetries under consideration
  8. General results on $G$-crossed invariants
  9. $G = U(1)$, charge conservation
  10. $G = U(1) \leftthreetimes \mathbb{R}^2$, continuous magnetic translational symmetry
  11. $G = U(1) \leftthreetimes \mathbb{E}^2$, continuous magnetic translational and rotational symmetry
  12. $G = U(1) \leftthreetimes_\phi \mathbb{Z}^2$, discrete magnetic translational symmetry
  13. $G = \mathbb{Z}_M$, discrete rotational symmetry
  14. $G = U(1)\leftthreetimes_{\phi} [\mathbb{Z}^2\rtimes \mathbb{Z}_M]$, discrete magnetic translational and rotational symmetry
  15. Possible experimental applications
  16. ...and 87 more sections

Key Result

Theorem J.1

For $M$ discrete and $G$ finite-dimensional, locally compact, $\sigma$-compact, (see page 522 of Stasheff1978)

Figures (3)

  • Figure 1: Diagrammatic representations of the actions of the $U$ and $\eta$ symbols. Anyon lines are black and symmetry defect sheets are orange.
  • Figure 2: The $E_{2}$ page of the LHSS. Here we only draw the first quadrant: the other three quadrants are trivial. The domain and range of the differentials $d_2$ are indicated through directed arrows.
  • Figure 3: We can use the $E_{\infty}$ page to retrieve the information of $\mathcal{H}^{4}(G,M)$. The group $A_{4}$ is $E_{\infty}^{4,0}$. The group $A_{3}$ is $E_{\infty}^{3,1}$ extended by $A_{4}$. The group $A_{2}$ is $E_{\infty}^{2,2}$ extended by $A_{3}$. The group $A_{1}$ is $E_{\infty}^{1,3}$ extended by $A_{2}$. Finally, the group $A_{0}=\mathcal{H}^{4}(G,M)$ is $E_{\infty}^{0,4}$ extended by $A_{1}$.

Theorems & Definitions (8)

  • Theorem J.1
  • Theorem J.2
  • Theorem J.3: Cyclic groups
  • Theorem J.4: Long-Exact Sequence
  • Theorem J.5: Universal Coefficient Theorem
  • Theorem J.6: Künneth decomposition
  • Theorem J.7: Cohomology of noncompact Lie groups
  • Theorem K.1: Lyndon-Hochschild-Serre spectral sequence