Band connectivity for topological quantum chemistry: Band structures as a graph theory problem

Barry Bradlyn, L. Elcoro, M. G. Vergniory, Jennifer Cano, Zhijun Wang, C. Felser, M. I. Aroyo, B. Andrei Bernevig

Research output: Contribution to journalArticle

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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, Nature (London) 547, 298 (2017)NATUAS0028-083610.1038/nature23268], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local k·p band structures across the Brillouin zone in terms of graph theory. In this paper, 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.

Original languageEnglish (US)
Article number035138
JournalPhysical Review B
Volume97
Issue number3
DOIs
StatePublished - Jan 16 2018

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics

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    Bradlyn, B., Elcoro, L., Vergniory, M. G., Cano, J., Wang, Z., Felser, C., Aroyo, M. I., & Bernevig, B. A. (2018). Band connectivity for topological quantum chemistry: Band structures as a graph theory problem. Physical Review B, 97(3), [035138]. https://doi.org/10.1103/PhysRevB.97.035138