Topological insulators and superconductors: tenfold way and dimensional hierarchy
Topological insulators and superconductors: tenfold way and dimensional hierarchy
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DOI：
10.1088/1367-2630/12/6/065010
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发表时间：
2010-06-17
影响因子：
3.3
通讯作者：
Ludwig, Andreas W. W.
中科院分区：
文献类型：
--
作者：
Ryu, Shinsei;Schnyder, Andreas P.;Ludwig, Andreas W. W.
最近的研究表明，在每一个空间维度上，都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，我们构建代表拓扑绝缘体和超导体的所有五个类和任意空间维度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“降维”相关的紧致一个或多个空间维度（在“卡鲁扎-克莱因”样的时尚）。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。拓扑绝缘体（超导体）在d维上存在的八重周期性，其哈密顿量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），是正交群SO（N）的旋量表示的八重周期性的反映（一种博特周期性的形式）。此外，我们推导出一般空间维度的拓扑不变量之间的关系，表征拓扑绝缘体和超导体与手征对称性（即，缠绕数）和Chern-Simons不变量。对于低维情况，该公式将绕组数与电极化（d = 1个空间维度）或磁电极化率（d = 3个空间维度）相关联。最后，我们还讨论了拓扑场理论描述的拓扑绝缘体（超导体）的线性响应的时空理论，并研究如何反转对称的存在修改拓扑绝缘体（超导体）的分类。
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 eightfold 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 eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern-Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d = 1 spatial dimensions) or to the magnetoelectric polarizability (d = 3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).