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  • Open Access

Topological Classification of Crystalline Insulators through Band Structure Combinatorics

Jorrit Kruthoff1, Jan de Boer1, Jasper van Wezel1, Charles L. Kane2, and Robert-Jan Slager3

  • 1Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
  • 2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
  • 3Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany

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Phys. Rev. X 7, 041069 – Published 22 December, 2017

DOI: https://doi.org/10.1103/PhysRevX.7.041069

Abstract

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented 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 all allowed topological phases of spinless particles in crystals in class A. Employing this classification, 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 to identify whether or not they give rise to intermediate Weyl semimetallic phases.

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Popular Summary

Topological insulators are exotic materials that are electrical insulators in their interior but can conduct electricity on their surface, and their discovery has fundamentally changed our understanding of how phases of matter may be organized. Most phases of matter are categorized by the symmetries that they break. Crystals break translational symmetry, magnets break rotational symmetry, and so on. Topological insulators, however, show that some phases can be distinct even though their symmetries are equal. Researchers have come up with a classification scheme—called the tenfold way—which allows for the categorization of topological phases depending on some general properties, such as whether or not they have time-reversal or particle-hole symmetry. We complement this categorization by providing a method for listing all possible topologically distinct phases of matter that do not have external symmetries but do have the types of internal (or lattice) symmetries that appear in the atomic arrangements of real solid materials.

We explicitly list all possible phases in two-dimensional materials and provide an intuitive and easily applicable method for identifying the phases possible within a given lattice type in any dimension. Our method matches the known results based on the mathematically involved predictions of K-theory, which is known to be a rigorous way of identifying all possible topological phases. It thus provides insight into this mathematical arena based on a physical understanding of topological band structures. We also show how our method can be used to study the transitions between topological phases and predict whether they will result in topologically protected Weyl semimetals.

This new classification can now be used to guide the search for new types of topological materials and related edge modes or exotic intermediate phases.

Article Text

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Phys. Rev. X 7, 041069– Published 22 December, 2017
  • Received 31 January 2017
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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