[论文解读] Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry

Chao Meng, Xinping Yang+2 more|arXiv (Cornell University)|Dec 29, 2024

Atomic and Subatomic Physics Research被引用 5

一句话总结

本文提出一个框架，用于对由非可互逆融合范畴对称性保护的(1+1)d SPT相进行分类和构建，并给出 Rep†(D8) 的现场晶格实现的显式方案，包含三个通过 S3 对偶性相关的不同 SPT相。

ABSTRACT

We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three $\mathsf{Rep}^\dagger(D_8)$ SPT phases, including a trivial phase, and further demonstrate the $S_3$-duality among these three SPT phases.

研究动机与目标

• Define anomaly-free fusion category symmetry via a fiber functor, ensuring a local onsite action and a trivial symmetric phase.
• Develop an onsite matrix-product-operator (MPO) realization of Rep†(D8) symmetry operators on a lattice.
• Classify and construct (1+1)d SPT phases as Q-systems in the charge category and relate them to matrix algebras under forgetful functors.
• Explicitly realize all three Rep†(D8) SPT phases on a lattice and demonstrate their S3-duality.
• Show how dual monoidal equivalences between fiber functors map to duality transformations between lattice phases.

提出的方法

• Introduce the anomaly-free pair (C, f) with C a unitary fusion category and f: C → Hilb as the fiber functor.
• Construct onsite Rep†(D8) MPOs using dualizable MPOs whose virtual bond dimension equals quantum dimension (on-site criterion).
• Decompose Rep†(D8) MPOs into projective charges and analyze phase realizations as W(a,c), W(a,b), and W(c,b,a) sectors.
• Use Q-systems in the charge category to model fixed-point SPT tensors and relate them to matrix algebras via forgetful functors.
• Establish S3-duality between the three Rep†(D8) SPT phases through a duality construction with defect-enlarged Hilbert spaces and Gauss-law projection.
• Discuss connection to edge modes and ground-state structure on open chains.

实验结果

研究问题

• RQ1 如何在1D晶格模型中一致地实现非可互逆融合范畴对称性？
• RQ2 Rep†(D8) 融合范畴对称性保护的(1+1)d SPT相的分类是什么？
• RQ3 纤维函子如何组织不同的 SPT 相，以及在晶格上如何实现纤维函子之间的对偶？
• RQ4 能否在微观上实现全部三个 Rep†(D8) SPT 相，以及边缘模如何表征？
• RQ5 在电荷范畴中 Q-系统对于识别固定点 SPT 模型的作用是什么？

主要发现

• 一个无异常的融合范畴对称性由带有纤维函子的对 (C, f) 给出，确保局部性和一个平凡的对称相。
• 存在一个现场 Rep†(D8) MPO 表示，其键结维度等于量子维度，能够在张量积希尔伯特空间上实现现场对称作用。
• 晶格上实现了三个 Rep†(D8) SPT 相，分别对应投射电荷 W(a,c)、W(a,b) 和 W(c,b,a)，S3 对偶性将它们联系起来。
• 开放链上的边缘模和基态结构显示可简并的边缘态，与 SPT 行为一致。
• 电荷范畴中的 Q-系统模型再现 SPT 相；纤维函子之间的单合成等价性会在晶格相之间诱导对偶性。
• S3 对偶性通过对 Defects 的双重构造与高斯定律投影实现，对相的作用是置换 S3/K 的陪集。