Topological insulators and superconductors: tenfold way and dimensional hierarchy

By S. Ryu, A. Schnyder, A. Furusaki, A. Ludwig. 2009

2,001 citations recorded by the source corpus.

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 or a 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 -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -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).

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/f1421e7b6a3e55a2854cb2e38917e04459279f54

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Citations in the corpus

Roadmap on topological photonics (2022, background): “Being freely engineered, they are also involved in defining the effective dimensionality and symmetries of a system, provide the topologically distinct configurations as well as the defects and interfaces that pin topological modes, and can act on internal degrees of freedom provided by polarization”
Topology by dissipation in atomic quantum wires (2011, background): “, the constraint is provided by the chiral symmetr”
Topological Phases Protected by Point Group Symmetry (2016, background): “systems with internal symmetry [4,5], and much attention”