[論文レビュー] Non-Invertible Duality Transformation Between SPT and SSB Phases

Linhao Li, Masaki Oshikawa+1 more|arXiv (Cornell University)|Jan 19, 2023

Quantum many-body systems参考文献 74被引用数 23

ひとこと要約

本論文は閉鎖チェーンにおいて non-unitary Kennedy-Tasaki duality を定義し、非可逆的な融合則と twisted Z2 gauging によって Z2×Z2 SSB および SPT 相を関連づけ、リング構成との同値性を示す。

ABSTRACT

In 1992, Kennedy and Tasaki constructed a non-local unitary transformation that maps between a $\mathbb{Z}_2 imes \mathbb{Z}_2$ 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 $\mathbb{Z}_2 imes \mathbb{Z}_2$ 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.

研究の動機と目的

• Revisit the Kennedy-Tasaki transformation from a modern perspective and extend it to closed chains.
• Show that the KT transformation on closed chains is non-unitary and non-invertible, with twisted sectors essential for definitions.
• Connect KT duality to Kramers-Wannier duality and Z2 gauging concepts.
• Provide a framework to systematically construct SPT phases using duality mappings.

提案手法

• Define a non-unitary Kennedy-Tasaki transformation on a ring with spin-1 per unit cell and on a ring with two spin-1/2 per unit cell.
• Express the KT transformation as a non-invertible operator with fusion rules that generalize Kramers-Wannier duality.
• Analyze symmetry-twist sector mappings under KT to relate Z2×Z2 SSB and SPT phases.
• Relate KT transformation to twisted gauging of Z2 symmetries in a field-theory formulation.
• Show unitary behavior on intervals in certain spin-1/2 constructions and establish equivalences between spin-1 and spin-1/2 KT mappings.

実験結果

リサーチクエスチョン

• RQ1 How can the Kennedy-Tasaki transformation be defined on closed chains without violating noninvertibility?
• RQ2 What are the fusion rules and sector mappings that characterize KT duality on rings?
• RQ3 How does KT duality realize twisted Z2 gauging and relate SSB and SPT phases?
• RQ4 What is the relationship and equivalence between KT transformations in spin-1 and spin-1/2 systems?
• RQ5 Can KT duality be used to systematically construct SPT phases from dual models?

主な発見

• The KT transformation on a ring is non-unitary and obeys non-invertible fusion rules, extending Kramers-Wannier duality.
• KT duality maps Z2×Z2 SSB phases to Z2×Z2 SPT phases on closed and open chains.
• The transformation implements a twisted gauging of Z2 symmetries, connecting KT to broader SPT constructions.
• The ring construction with two spin-1/2 per unit cell decouples Z2 charges, facilitating twisted gauging interpretations.
• KT transformations for spin-1 and two spin-1/2 systems are shown to be equivalent in the appropriate settings.