Bott-Kitaev Periodic Table and the Diagonal Map

TL;DR

This paper introduces the Diagonal Map, a mathematical tool akin to the Bott map, which helps classify topological phases of free-fermion systems by increasing their dimension and symmetry index, building on the periodic table of topological insulators.

Contribution

It provides an introduction to the Diagonal Map and its role in the homotopy-theoretic proof of Kitaev's periodic table for topological insulators and superconductors.

Findings

The Diagonal Map increases dimension and symmetry index of ground states.

Examples illustrate the application of the Diagonal Map.

The approach connects physical models with algebraic topology concepts.

Abstract

Building on the 10-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.