Entanglement entropy for (3+1)-dimensional topological order with excitations
Entanglement entropy for (3+1)-dimensional topological order with excitations
复制标题
DOI：
10.1103/physrevb.97.085147
复制
发表时间：
2017-10
影响因子：
3.7
通讯作者：
X. Wen;Huan He;A. Tiwari;Yunqin Zheng;P. Ye
中科院分区：
文献类型：
--
作者：
X. Wen;Huan He;A. Tiwari;Yunqin Zheng;P. Ye
(3+1)维拓扑有序相中的激发态具有非常丰富的结构。(3+1)维拓扑相既支持点状激发，也支持弦激发，尤其是环(闭弦)激发可能存在打结和连接结构。在这项工作中，我们提出了这样一个问题：不同类型的拓扑激发如何对纠缠熵做出贡献，或者，我们是否可以利用纠缠熵来检测激发态的结构，并进一步获得潜在的拓扑序信息？我们主要研究在Dijkgraaf-Witten规范理论中可以实现的(3+1)维拓扑序，它被数学上的{H}^4[G；U(1)]$上的有限群$G及其4-上循环$\omega所标记，直到群自同构。我们发现，每个拓扑激发对纠缠熵贡献一个普适常数$lndi$，其中$di$是依赖于激发结构和数据$(G，\，\omega)$的量子维度。连接/非连接拓扑的激发纠缠熵可以捕捉到DW理论$(G，\，\omega)$的不同信息。特别是，由Hopf-link环激发引入的纠缠熵可以将某些群4-余环区分开来。
Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.