拓扑绝缘体与超导体：十重途径与维层次
Topological insulators and superconductors: ten-fold way and dimensional hierarchy
摘要：
最近的研究表明，在每一个空间维度上，精确地存在五种不同的拓扑绝缘体或超导体。在一个给定的类中，不同的拓扑扇区可以根据情况通过Z或Z_2拓扑不变量来区分。这是一个详尽的分类。在这里，我们构造的代表拓扑绝缘体和超导体的所有五个类别和任意空间维d，在狄拉克哈密顿。使用这些代表，我们展示了如何拓扑绝缘体（超导体）在不同的维度和不同的类可以通过压缩一个或多个空间维度（在卡鲁扎-克莱因的方式）通过降维相关。对于Z-拓扑绝缘体（超导体），这是通过每次下降一维到不同的类别来进行的。另一方面，Z_2-拓扑绝缘体（超导体）被证明是同一类中父Z-拓扑绝缘体的低维后代，它们继承了它们的拓扑性质。对于具有满足至少一个现实条件（由时间反演或电荷共轭/粒子空穴对称性引起）的哈密顿量的拓扑绝缘体（超导体），在d维中存在的8倍周期性是正交群SO（N）的旋量表示的8倍周期性（波特周期性的一种形式）的反映。我们推导了表征具有手征对称性的拓扑绝缘体/超导体的拓扑不变量与Chern-Simons不变量之间的关系：它将不变量与电极化的强度（d=1）或磁电极化率（d=3）联系起来。最后，我们讨论拓扑场论描述的线性响应的时空理论，并研究如何反演对称性的存在修改的分类。
英文摘要：
It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a Z or a Z_2 topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via dimensional reduction by compactifying one or more spatial dimensions (in Kaluza-Klein-like fashion). For Z-topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z_2-topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Z-topological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). We derive a relation between the topological invariant that characterizes topological insulators/superconductors with chiral symmetry and the Chern-Simons invariant: it relates the invariant to the electric polarization (d=1), or to the magnetoelectric polarizability (d=3). Finally, we discuss topological field theories describing the space time theory of linear responses, and study how the presence of inversion symmetry modifies the classification.
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