Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

TL;DR

This paper reviews and systematically classifies topological phases of matter in low dimensions using Fermi projections, providing explicit algorithms and numerical indices for symmetry classes, and illustrating the method with simple 0- and 1-dimensional systems.

Contribution

It introduces a systematic, explicit, and constructive classification method for topological phases, including algorithms to deform projections with matching indices.

Findings

Identifies numerical indices for all symmetry classes.

Provides algorithms for deformation of projection families.

Recovers known topological invariants for low-dimensional systems.

Abstract

We review Kitaev's celebrated "periodic table" for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections which depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for 0- and 1-dimensional systems, and recover the (weak and strong) topological invariants proposed by Kitaev and others.