Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-invertible SPT, gauging and symmetry fractionalization

Yabo Li, Mikhail Litvinov

TL;DR

The work provides explicit UV lattice realizations of Rep$(G)$ SPT states by gauging anomalous abelian subgroups of bulk 2+1d SPTs and employing partial electric-magnetic (PEM) duality to connect to non-abelian and non-invertible boundary symmetries. It develops concrete lattice models for Rep$(D_8)$, Rep$(Q_8)$, Rep$(G_1)$, and Rep$(G_{4,4})$, and shows how the same framework extends to general Rep$(G)$ for class-2 nilpotent groups, with the bulk symmetry fractionalization data explaining the boundary fusion and symmetry structure. The paper demonstrates how gauging abelian anomalous symmetries yields dual Abelian/non-Abelian and invertible/non-invertible boundary symmetries, organized through a web of dualities and bulk SET fractionalization patterns, including TY-category connections. These results provide a versatile, tensor-product-Hilbert-space platform for realizing and analyzing non-invertible SPT phases, with potential applications to measurement-based quantum computation and beyond.

Abstract

We explicitly realize the Rep($Q_8$) non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep($Q_8$) duality web. We further show that we can construct a class of lattice models with Rep($G$) symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.

Non-invertible SPT, gauging and symmetry fractionalization

TL;DR

The work provides explicit UV lattice realizations of Rep SPT states by gauging anomalous abelian subgroups of bulk 2+1d SPTs and employing partial electric-magnetic (PEM) duality to connect to non-abelian and non-invertible boundary symmetries. It develops concrete lattice models for Rep, Rep, Rep, and Rep, and shows how the same framework extends to general Rep for class-2 nilpotent groups, with the bulk symmetry fractionalization data explaining the boundary fusion and symmetry structure. The paper demonstrates how gauging abelian anomalous symmetries yields dual Abelian/non-Abelian and invertible/non-invertible boundary symmetries, organized through a web of dualities and bulk SET fractionalization patterns, including TY-category connections. These results provide a versatile, tensor-product-Hilbert-space platform for realizing and analyzing non-invertible SPT phases, with potential applications to measurement-based quantum computation and beyond.

Abstract

We explicitly realize the Rep() non-invertible symmetry-protected topological (SPT) state as a 1+1d cluster state on a tensor product Hilbert space of qubits. Using the Kramers-Wannier operator, we construct the lattice models for the phases of all the symmetries in the Rep() duality web. We further show that we can construct a class of lattice models with Rep() symmetry including non-invertible SPT phases if they have a dual anomalous abelian symmetry. Upon dualizing, there is a rich interplay between onsite symmetries, non-onsite symmetries, non-abelian symmetries, and non-invertible symmetries. We show that these interplay can be explained using the symmetry fractionalization in the 2+1d bulk SET.
Paper Structure (26 sections, 109 equations, 4 figures)

This paper contains 26 sections, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rep($G$) SPT phases
  3. Rep($D_8$) symmetry
  4. Rep($D_8$) SPT models
  5. Form Rep($D_8$) to $D_8$ in one shot
  6. Rep($Q_8$) symmetry
  7. ${\mathbb Z}_2^3$ type II+II+III anomaly
  8. Rep($Q_8$) SPT model
  9. Gauging Rep($Q_8$) symmetry
  10. Rep($G_1$) symmetry
  11. Rep($G_{4,4}$) symmetry
  12. Rep($G_{4,4}$) SPT models
  13. 1. Rep($G_{4,4}$) SPT from ${\mathbb Z}_2^4/\hat{{\mathbb Z}}_2^{v}$ SSB.
  14. 2. Rep($G_{4,4}$) SPT from ${\mathbb Z}_2^4/\mathrm{diag}(\hat{{\mathbb Z}}_2^{o}\times \hat{{\mathbb Z}}_2^{v})$ SSB.
  15. 3. Rep($G_{4,4}$) SPT from ${\mathbb Z}_2^4/\mathrm{diag}(\hat{{\mathbb Z}}_2^{o'}\times \hat{{\mathbb Z}}_2^{v})$ SSB.
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Some $G$ twisted quantum double and $G'$ twisted quantum double can be identical according to the PEM duality (more details in Sec. \ref{['app:PEM']}). When $\alpha=1$, different gapped boundaries of $G$ quantum double and $G'$ twisted quantum double are all related to each other by gauging. For example, the non-invertible Rep($G$) symmetry is dual to an anomalous $G'$ symmetry as a result.
  • Figure 2: Bulk symmetry fractionalization describes the fusion of the bulk symmetry defects. It captures the fusion rules of symmetry operators on the boundary, which could become non-abelian or non-invertible.
  • Figure 3: The 3-colorable lattice and the type-III SPT stabilizers,
  • Figure 4: Stabilizers for type II+II+III SPT given by topological action $\frac{1}{2}A_e\cup A_e\cup A_v+\frac{1}{2}A_o\cup A_o\cup A_v+\frac{1}{2}A_e\cup A_o\cup A_v$.