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

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.

19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529 v2: Accepted version, minor typos corrected and references added. Now 19+epsilon pages

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• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• The $8Pmmn$ borophene sheet: A solid-state platform for space-time engineering
• Purely rotational symmetry-protected topological crystalline insulator $\alpha$-Bi4Br4
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Topological insulators in the NaCaBi family with large SOC gaps
• The interplay between structural, magnetic and electronic states in the pyrochlore iridate Eu2Ir2O7
• Efficient Design of Helical Higher-Order Topological Insulators in 3D Elastic Medium