[论文解读] Gapped boundaries of (3+1)d topological orders

Zhu-Xi Luo|arXiv (Cornell University)|Dec 19, 2022

Theoretical and Computational Physics被引用 5

一句话总结

本文通过商去不仅包括解耦的(2+1)d拓扑序，还包括如分形子系统等奇异表面相，重新定义了(3+1)d拓扑序中隙边界类。研究表明，不同的边界类由哪些体激发的弦状激发可以在边界上终结决定，而与其末端无关——展示了对于(3+1)d玻色子$β$-拓扑码，这些类对应于底层群$G$的正规子群，从而涌现出超越已知的粗糙、平滑及扭曲平滑类型的新型边界。

ABSTRACT

Given a gapped boundary of a (3+1)d topological order (TO), one can stack on it a decoupled (2+1)d TO to get another boundary theory. Should one view these two boundaries as "different"? A natural choice would be no. Different classes of gapped boundaries of (3+1)d TO should be defined modulo these decoupled (2+1)d TOs. But is this enough? We examine the possibility of coupling the boundary of a (3+1)d TO to additional (2+1)d TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk point-like excitations, when touching the boundary, become excitations in the added (2+1)d phase, while the string-like excitations in the bulk may end on the boundary but with endpoints dressed by some other excitations in the (2+1)d phase. For a good definition of "class" for gapped boundaries of (3+1)d TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the string-like excitations that can end on the boundary, whatever their endpoints are. A concrete example is the (3+1)d bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the $m$-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a (3+1)d TO characterized by a finite group $G$, different classes correspond to different subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model and field theory.

研究动机与目标

• 通过去除解耦的(2+1)d拓扑序带来的冗余，解决对(3+1)d拓扑序中隙边界分类的模糊性。
• 研究仅商去解耦的(2+1)d相是否足以定义不同的边界类。
• 探讨与奇异2维相（如分形子系统或任意Anyonic拓扑序）耦合在生成新的、物理上不同的中隙边界理论中的作用。
• 建立一个统一的分类方案，基于哪些体弦激发可以终结于边界，且独立于其末端结构。
• 将分类推广至有限群模型，表明边界类对应于表征体的群$G$的正规子群。

提出的方法

• 使用层堆叠构造实现中隙边界，将(3+1)d体与(2+1)d拓扑序或分形子系统堆叠。
• 应用Walker-Wang模型实现(3+1)d体拓扑序，边界使用弦网模型，通过任意Anyon凝聚实现耦合。
• 利用场论与异常流入机制从边界数据重建体，特别分析表面Anyon凝聚与单连通性条件。
• 利用范畴论与群上同调对边界类型进行分类，通过拉格朗日子代数及体Anyon与表面Anyon的复合。
• 引入一个基于体弦算符是否能终结于边界（无论其末端激发如何）的判据，作为边界类的定义不变量。
• 证明对于(3+1)d$\mathbb{Z}_2$拓扑码，边界类与$\mathbb{Z}_2$的正规子群一一对应，即$\{e\}$与$\mathbb{Z}_2$。

实验结果

研究问题

• RQ1 能否在商去解耦的(2+1)d拓扑序之外，对(3+1)d拓扑序的中隙边界进行有意义的分类？
• RQ2 当(3+1)d拓扑序与奇异2维相（如分形子系统或任意Anyonic拓扑序）耦合时会发生什么？
• RQ3 体弦激发能否终结于边界是否为一个稳健的不变量，可用于分类不同的边界类型？
• RQ4 边界束缚弦的末端如何影响中隙边界的分类？这种依赖关系能否被商去？
• RQ5 对于由有限群$G$表征的(3+1)d拓扑序，$G$中何种代数结构对应于不同的边界类？

主要发现

• 本文根据$m$-弦是否能终结于边界，识别出(3+1)d玻色子$\mathbb{Z}_2$拓扑码的两类不同中隙边界，分别对应于$\mathbb{Z}_2$的平凡子群与全子群。
• 此前已知的三种边界——粗糙、平滑与扭曲平滑——被归为两类：{粗糙}与{平滑, 扭曲平滑}，依据$m$-弦的可终结性。
• 通过将(3+1)d体与额外的(2+1)d拓扑序或分形子系统耦合，构造出新的中隙边界，产生在体中不存在的异常表面激发。
• 边界类的分类被证明等价于表征(3+1)d拓扑序的有限群$G$的正规子群的分类。
• 该框架允许构造具有非平凡表面Anyon与移动性约束的边界，即使体中无此类激发。
• 该方法提供了一种物理上合理且数学上一致的分类，通过体弦的可终结性作为关键不变量，同时商去了解耦的(2+1)d拓扑序与奇异表面相。