Non-Invertible Duality Transformation Between SPT and SSB Phases

Abstract

In 1992, Kennedy and Tasaki constructed a non-local unitary transformation that maps between a Z 2 × Z 2 spontaneously symmetry breaking phase and the Haldane gap phase, which is a prototypical Symmetry-Protected Topological phase in modern framework, on an open spin chain. In this work, we propose a way to define it on a closed chain, by sacrificing unitarity. The operator realizing such a non-unitary transformation satisfies non-invertible fusion rule, and implements a generalized gauging of the Z 2 × Z 2 global symmetry. These findings connect the Kennedy-Tasaki transformation to numerous other concepts developed for SPT phases, and opens a way to construct SPT phases systematically using the duality mapping.

Subsystem non-invertible symmetry operators and defects

“…In this work, we will study both symmetry operators and defects on the lattice. Our discussion of the symmetry operators directly generalizes our previous work on (1 + 1)d KW duality operators [154]. The discussion of symmetry defects is relatively new, and is consistent with earlier results for the (1 + 1)d KW duality defect in the Ising model on the lattice [157].…”

“…In the literature, most of the discussions of non-invertible symmetries are either in continuum field theories or in the statistical models with infinite system sizes [52]. The authors of [154] systematically discussed the properties of the KW-duality operator acting on a finitesize Hilbert space of a quantum system, and carefully discussed the effects of boundary conditions for closed spin chains. However, it is also important to study symmetry defects to fully uncover the properties of symmetry.…”

“…However, on a closed chain, one has to specify how the boundary conditions transform under gauging [167][168][169][170]. In this subsection, we mainly follow the discussion in [154], where the fusion rule, exchange of symmetry and twist sectors, and mapping of operators are all discussed on closed chains. We only briefly summarize the results, and refer the interested reader to [154] for further details.…”

“…The KW transformation is realized by an operator N acting on the Hilbert space. It is sufficient to specify how it acts on the basis states [154] N…”

Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence

“…He also thanks the organizers and speakers at the conference "Bootstat 2021: Conformal bootstrap and statistical at Orsay, France, for the introduction of a modern understanding of the operator formalism of CFT. We thank Yuji Tachikawa for notifying us [104], and Yunqin Zheng for sharing the manuscript in [160], which contains related topics. This work is supported by the NTU grant for Nanyang Assistant Professorship and the National Research Foundation, Singapore under the NRF fellowship award (NRF-NRFF12-2020-005), and a Nanyang Technological University start-up grant (NTU-SUG).…”