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Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations

Linhao Li, Masaki Oshikawa, Yunqin Zheng

TL;DR

This work introduces a unified KT-based framework to generate gapless symmetry protected topological phases from decoupled spin models, clarifying how non-invertible dualities can produce both intrinsically gapless and purely gapless topologies. By applying the Kennedy-Tasaki transformation to combinations of SSB, Ising CFT, and XX/XXZ chains, the authors realize gSPT, igSPT, pgSPT, and ipgSPT, and derive their field-theoretic descriptions, twist-sector mappings, and edge/phase-diagram characteristics. The results demonstrate analytical control over the stability of gapless SPTs under symmetric perturbations and connect decorated-domain-wall pictures to a non-invertible gauging viewpoint, with explicit constructions for both conventional and intrinsically gapless variants. The framework can be generalized to broader symmetry groups and higher dimensions, offering a systematic route to classify and analyze gapless topological phases and their transitions in lattice models and field theories.

Abstract

The Kennedy-Tasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work [L. Li et al. arXiv:2301.07899] on a ring by sacrificing the unitarity, and should be understood as a non-invertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations.

Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations

TL;DR

This work introduces a unified KT-based framework to generate gapless symmetry protected topological phases from decoupled spin models, clarifying how non-invertible dualities can produce both intrinsically gapless and purely gapless topologies. By applying the Kennedy-Tasaki transformation to combinations of SSB, Ising CFT, and XX/XXZ chains, the authors realize gSPT, igSPT, pgSPT, and ipgSPT, and derive their field-theoretic descriptions, twist-sector mappings, and edge/phase-diagram characteristics. The results demonstrate analytical control over the stability of gapless SPTs under symmetric perturbations and connect decorated-domain-wall pictures to a non-invertible gauging viewpoint, with explicit constructions for both conventional and intrinsically gapless variants. The framework can be generalized to broader symmetry groups and higher dimensions, offering a systematic route to classify and analyze gapless topological phases and their transitions in lattice models and field theories.

Abstract

The Kennedy-Tasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work [L. Li et al. arXiv:2301.07899] on a ring by sacrificing the unitarity, and should be understood as a non-invertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations.
Paper Structure (53 sections, 149 equations, 4 figures, 1 table)

This paper contains 53 sections, 149 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Overview of gapless SPT:
  3. Kennedy Tasaki transformation:
  4. Constructing (gapped and gapless) SPTs from Kennedy Tasaki transformation:
  5. Advantages of the present construction of gapless SPT phases using the KT transformation:
  6. Outlook and the structure of the present paper:
  7. Review of Kennedy-Tasaki transformation
  8. Definition of the KT transformation for spin-$\frac{1}{2}$ chains
  9. Properties of KT trnasformation
  10. Mapping between symmetry-twist sectors:
  11. Fusion rules:
  12. Gapped SPT from KT transformation
  13. Gapped SPT from KT transformation:
  14. Ground state charge under TBC:
  15. String order parameter:
  16. ...and 38 more sections

Figures (4)

  • Figure 1: Phase diagram of the gapless SPT under perturbation.
  • Figure 2: The phase diagram of \ref{['eq:igSPTpert']}.
  • Figure 3: The phase diagram of ipgSPT.
  • Figure 4: The phase diagram of pgSPT.