Contributor info: Mr Li Linhao

Linhao Li, Masaki Oshikawa, Yunqin Zheng

SciPost Phys. 18, 153 (2025) · published 13 May 2025 | Toggle abstract · pdf

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 [Phys. Rev. B 108, 214429 (2023)] 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.

Weiguang Cao, Linhao Li, Masahito Yamazaki

SciPost Phys. 17, 104 (2024) · published 4 October 2024 | Toggle abstract · pdf

Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami $\mathbb Z_N×\mathbb Z_N$ type, we obtain a new non-invertible symmetry in models with $\mathbb Z_N$ dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.

Linhao Li, Masaki Oshikawa, Yunqin Zheng

SciPost Phys. 17, 013 (2024) · published 15 July 2024 | Toggle abstract · pdf

Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected topological states. We construct several simple gapless-SPT models using the decorated defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary (quasi)-degeneracy under open boundary conditions. We also comment on the stability of the gapless-SPT models under symmetric perturbations, and apply small-scale exact diagonalization when direct analytic understanding is not available.

Weiguang Cao, Linhao Li, Masahito Yamazaki, Yunqin Zheng

SciPost Phys. 15, 155 (2023) · published 12 October 2023 | Toggle abstract · pdf

We explore non-invertible symmetries in two-dimensional lattice models with subsystem $\mathbb Z_2$ symmetry. We introduce a subsystem $\mathbb Z_2$-gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.