[Paper Review] Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations

Linhao Li, Masaki Oshikawa+1 more|arXiv (Cornell University)|Jul 10, 2023

Physics of Superconductivity and Magnetism22 citations

TL;DR

The paper uses Kennedy-Tasaki transformations to systematically construct and analyze gapless symmetry protected topological (SPT) phases, including intrinsically gapless and purely gapless variants, from decoupled models.

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.

Motivation & Objective

• Motivate and classify gapless SPTs and their intrinsic vs. non-intrinsic nature.
• Show how Kennedy-Tasaki transformations map decoupled models to gapless SPTs, including purely gapless cases.
• Provide field-theory descriptions and phase diagrams for gapless SPTs.
• Demonstrate stability analysis of gapless SPTs under symmetric perturbations using KT-derived mappings.

Proposed method

• Review and apply the Kennedy-Tasaki (KT) transformation to spin-1/2 chains with Z2 x Z2 symmetry.
• Map decoupled Z2 SSB or gapless sectors to gapped SPT, gapless SPT, and intrinsically gapless SPT via STS (gauge, stack) steps.
• Use KT transformation to relate symmetry-twist sectors and derive energy spectra and edge-state features.
• Derive field-theory descriptions for gapless SPTs from the Ising CFT and free boson CFT inputs.
• Analyze phase diagrams and topological features under symmetry twists and boundary conditions.

Experimental results

Research questions

• RQ1 Can the Kennedy-Tasaki transformation generate gapless SPTs from decoupled theories with simple symmetries?
• RQ2 What are the topological and boundary-condition signatures of gSPT, igSPT, pgSPT, and ipgSPT generated by KT?
• RQ3 How does KT-induced mapping connect edge-state degeneracies and ground-state charges under twisted boundary conditions to gapless SPTs?
• RQ4 What field theories describe the gapless SPTs obtained via KT, and how do they capture topological features?
• RQ5 How stable are these gapless SPTs under symmetric perturbations and what does the phase diagram look like?

Key findings

• KT transformation can generate known gSPT and igSPT from decoupled Z2 sectors, reproducing decorated defect structures when gapped sectors exist.
• The KT construction yields intrinsically gapless SPT and intrinsically purely gapless SPT, extending beyond decorated defect pictures.
• Field theories for gSPT and gapless SPT can be obtained from Ising CFT and free boson CFT inputs via KT, clarifying their topological features.
• Ground-state charges under twisted boundary conditions and edge-state quasi-degeneracies are linked through KT mappings, aiding edge-state identification.
• Phase diagrams and stability analyses under symmetric perturbations can be performed analytically by undoing KT to decoupled theories and then transforming back.