Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy
Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy
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有间隙的磁畴壁、有间隙的边界和拓扑简并
DOI：
10.1103/physrevlett.114.076402
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发表时间：
2015-02-18
影响因子：
8.6
通讯作者：
Wen, Xiao-Gang
中科院分区：
文献类型：
--
作者：
Lan, Tian;Wang, Juven C.;Wen, Xiao-Gang
检查作为 (2 + 1)D 拓扑有序状态之间的拓扑线缺陷的带隙畴壁。我们提供了简单的标准来确定带隙畴壁的存在，这适用于阿贝尔和非阿贝尔拓扑顺序。我们的标准还确定哪些 (2 + 1)D 拓扑序必须具有无间隙边缘模式，即哪些 (1 + 1)D 全局引力异常确保无间隙。此外，我们引入了一个新的数学对象，隧道矩阵 W，其条目是融合空间维度 W-ia，来标记不同类型的带隙畴壁。通过研究许多例子，我们发现证据表明隧道矩阵是对不同类型的带隙畴壁进行分类的强大数量。由于带隙边界是体拓扑序与真空之间的带隙畴壁，被视为平凡拓扑序，因此我们的带隙畴壁理论包含了带隙边界理论。此外，我们推导了拓扑基态简并公式，应用于具有间隙域壁的任意可定向空间2流形，包括封闭2流形和具有间隙边界的开放2流形。
Gapped domain walls, as topological line defects between (2 + 1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2 + 1)D topological orders must have gapless edge modes, namely, which (1 + 1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W-ia, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.