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.
Highlights
Over the past two decades, topological order has been established as an organizing principle in the classification of matter, alongside the traditional symmetry-based approach
We study transitions between topological phases within class A that are driven by band inversions at high-symmetry points in the first Brillouin zone
It is well known that, within time-reversal symmetric topological insulators, the discrete translational symmetries surviving within the atomic lattice lead to the definition of weak invariants in three dimensions, which need to be used in addition to the tenfold periodic table to get a full classification of the topological state [10,11]
Summary
Over the past two decades, topological order has been established as an organizing principle in the classification of matter, alongside the traditional symmetry-based approach. It is well known that, within time-reversal symmetric topological insulators, the discrete translational symmetries surviving within the atomic lattice lead to the definition of weak invariants in three dimensions, which need to be used in addition to the tenfold periodic table to get a full classification of the topological state [10,11] This procedure can be generalized to include any space group symmetry in two and three spatial dimensions [12] and is expected to become experimentally accessible and relevant in the presence of lattice defects [13,14,15,16,17,18].
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