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Graph theory data for topological quantum chemistry.

M G Vergniory1,2,3, L Elcoro4, Zhijun Wang5

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This study bridges global topological band structures with local group theory using graph construction. Algorithms are developed to identify all topologically distinct band structures, orbital types, and lattices.

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Area of Science:

  • Condensed Matter Physics
  • Materials Science
  • Quantum Mechanics

Background:

  • Topological phases in noninteracting particles are defined by global band structure properties.
  • Conventional group theory in solid-state physics analyzes local properties within the Brillouin zone.
  • A gap exists between global topological characterization and local symmetry analysis.

Purpose of the Study:

  • To connect global topological band structures with local symmetry properties.
  • To develop algorithms for constructing and classifying topologically distinct band structures.
  • To tabulate orbital types and lattices supporting topologically disconnected band structures.

Main Methods:

  • Mapping band structure construction to a graph problem.
  • Formulating algorithms to generate all topologically distinct graphs.
  • Applying algorithms to elementary band structures.

Main Results:

  • Explicit data and algorithms for producing topologically distinct graphs.
  • Identification and tabulation of orbital types and lattices for topologically disconnected band structures.
  • Demonstration of the bandrep program for accessing computational results.

Conclusions:

  • The developed methods and algorithms enable a comprehensive classification of topological band structures.
  • This work provides a systematic way to identify materials with non-trivial topological properties.
  • The bandrep program offers a practical tool for researchers in topological materials science.