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Published May 14, 2009 | Published
Book Section - Chapter Open

Periodic table for topological insulators and superconductors

Abstract

Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.

Additional Information

© 2009 American Institute of Physics. Issue Date: 14 May 2009. I am grateful to Andreas Ludwig and Shinsey Ryu for teaching me about ^(3)He-5 and (p_x + ip_y)↑ + (_Px - iP_y)↓ and helping to fit these phases into the periodic table. I also thank John Preskill, Michael Freedman, John Roe, Charles Kane, and Grigori Volovik for stimulating discussions. This research is supported in part by NSF under grant No. PHY-0456720.

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August 20, 2023
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