Bott–Kitaev periodic table and the diagonal map. (25th August 2015)

• Record Type: Journal Article
• Authors: Kennedy, R; Zirnbauer, M R
• Is Part Of: Physica scripta. Volume T164(2015:Dec.)
• Publication Date: 2015-08-25
• Subjects: topological insulators and superconductors -- free-fermion ground states -- symmetry-protected topological phases -- equivariant vector bundles -- Bott periodicity -- homotopy theory; Physics -- Periodicals; 530.05
• DOI: 10.1088/0031-8949/2015/T164/014010
• Languages: English
• ISSNs: 0031-8949

Abstract

Abstract: Building on the ten-way symmetry classification of disordered fermions, the authors have recently given a homotopy-theoretic proof of Kitaev's 'periodic table' for topological insulators and superconductors. The present paper offers an introduction to the physical setting and the mathematical model used. Basic to the proof is the so-called diagonal map, a natural transformation akin to the Bott map of algebraic topology, which increases by one unit both the momentum-space dimension and the symmetry index of translation-invariant ground states of gapped free-fermion systems. This mapping is illustrated here with a few examples of interest. (Based on a talk delivered by the senior author at the Nobel Symposium on 'New Forms of Matter: Topological Insulators and Superconductors'; Stockholm, 13–15 June, 2014.)