Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem

TL;DR

This paper introduces a graph theory approach to analyze and classify global band structures in topological quantum chemistry, enabling the identification of topologically distinct insulating phases.

Contribution

It formulates the problem of band connectivity across the Brillouin zone using graph theory, providing a systematic way to classify topological phases.

Findings

Graph connectivity constrains band structures.

Enumeration of all allowed band connectivities.

Identification of topologically distinct insulating phases.

Abstract

The conventional theory of solids is well suited to describing band structures locally near isolated points in momentum space, but struggles to capture the full, global picture necessary for understanding topological phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298 (2017)], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local "k-dot-p" band structures across the Brillouin zone in terms of graph theory. In the current manuscript we give the details of our full theoretical construction. We show that crystal symmetries strongly constrain the allowed connectivities of energy bands, and we employ graph-theoretic techniques such as graph connectivity to enumerate all the solutions to these constraints. The tools of graph theory allow us to identify disconnected groups of bands in these solutions, and so identify topologically distinct insulating phases.