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Cluster state as a non-invertible symmetry protected topological phase

Sahand Seifnashri, Shu-Heng Shao

TL;DR

The paper demonstrates that the 1+1d cluster model possesses a non-invertible symmetry described by the fusion category Rep(D8), rendering the cluster state a non-invertible SPT phase in addition to its ${\mathbb{Z}}_2^e\times{\mathbb{Z}}_2^o$ protection. Through a two-step gauging procedure (KT/TST), it identifies three distinct Rep(D8) SPT phases realized by commuting Pauli Hamiltonians H_cluster, H_odd, and H_even, and analyzes their edge modes via a local projective algebra at interfaces. It shows that these phases are not connected by any Rep(D8)-symmetric entangler and discusses the absence of a stacking operation for non-invertible SPTs, highlighting fundamental differences from invertible SPTs. The work also elucidates how gauging converts Rep(D8) to an anomalous invertible symmetry and connects the lattice constructions to continuum field theories and fusion-category language. Overall, it provides a concrete, minimal-qubit realization of non-invertible SPT phases and outlines routes for generalizing to other non-invertible symmetry categories.

Abstract

We show that the standard 1+1d $\mathbb{Z}_2\times \mathbb{Z}_2$ cluster model has a non-invertible global symmetry, described by the fusion category Rep(D$_8$). Therefore, the cluster state is not only a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry protected topological (SPT) phase, but also a non-invertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D$_8$) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these non-invertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

Cluster state as a non-invertible symmetry protected topological phase

TL;DR

The paper demonstrates that the 1+1d cluster model possesses a non-invertible symmetry described by the fusion category Rep(D8), rendering the cluster state a non-invertible SPT phase in addition to its protection. Through a two-step gauging procedure (KT/TST), it identifies three distinct Rep(D8) SPT phases realized by commuting Pauli Hamiltonians H_cluster, H_odd, and H_even, and analyzes their edge modes via a local projective algebra at interfaces. It shows that these phases are not connected by any Rep(D8)-symmetric entangler and discusses the absence of a stacking operation for non-invertible SPTs, highlighting fundamental differences from invertible SPTs. The work also elucidates how gauging converts Rep(D8) to an anomalous invertible symmetry and connects the lattice constructions to continuum field theories and fusion-category language. Overall, it provides a concrete, minimal-qubit realization of non-invertible SPT phases and outlines routes for generalizing to other non-invertible symmetry categories.

Abstract

We show that the standard 1+1d cluster model has a non-invertible global symmetry, described by the fusion category Rep(D). Therefore, the cluster state is not only a symmetry protected topological (SPT) phase, but also a non-invertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these non-invertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.
Paper Structure (28 sections, 121 equations, 1 figure, 3 tables)

This paper contains 28 sections, 121 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Non-invertible symmetry of the cluster model
  3. Non-invertible SPT phases
  4. Gaugings and the KT transformation
  5. Three Rep(D$_8$) SPT phases
  6. $\hat{{\mathbb{Z}}}_2^\mathsf{V}$ preserving phase:
  7. $\text{diag}(\hat{{\mathbb{Z}}}_2^\mathsf{V} \times \hat{{\mathbb{Z}}}_2^\mathrm{e})$ preserving phase:
  8. $\text{diag}(\hat{{\mathbb{Z}}}_2^\mathsf{V} \times \hat{{\mathbb{Z}}}_2^\mathrm{o})$ preserving phase:
  9. Edge modes
  10. No symmetric entangler for non-invertible SPT phases
  11. Conclusions and outlook
  12. Lattice gauging
  13. KW$^{\otimes2}$: untwisted lattice gauging $(\mathbf{S})$
  14. KT: twisted lattice gauging $(\mathbf{TST})$
  15. Non-invertible symmetry under gauging
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Localization of the non-invertible operator $\mathsf{D}$ at the interfaces between SPT phases. The interface system \ref{['interface']} is locally in the cluster state in the region $j=1,2,\cdots, \ell$, and locally in the $\ket{\text{odd}}$ state in the complement region. The operator $\mathsf{D}$ factorizes into local factors $\mathsf{D}_\text{L}^{(1)}\mathsf{D}_\text{R}^{(1)}+\mathsf{D}_\text{L}^{(2)}\mathsf{D}_\text{R}^{(2)}$ (multiplied by $(-1)^{(L-\ell)/ 4}$) on the ground space .