Topological classification of crystalline insulators through band structure combinatorics

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented 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 all allowed topological phases of spinless particles in crystals in class A. Employing this classification, 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 to identify whether or not they give rise to intermediate Weyl semimetallic phases.