Topological classification of crystalline insulators through band structure combinatorics

TL;DR

This paper introduces a combinatorial method to classify all topologically distinct electronic band structures in crystals, applicable in various dimensions and symmetry classes, linking mathematical topology with crystal band theory.

Contribution

It provides a simple counting algorithm for classifying topological phases in 2D and 3D crystals, connecting combinatorics with K-theory-based topological classification.

Findings

Classifies all topological phases in 2D class A crystals.

Extends classification to 3D space groups.

Identifies possible phase transitions and Weyl semimetal phases.

Abstract

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of K-theory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band inversions at high symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure, and identify whether or not they give rise to intermediate Weyl semimetallic phases.