Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -

Abstract

We formulate topological crystalline materials on the basis of the twisted equivariant $K$-theory. Basic ideas of the twisted equivariant $K$-theory is explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry.

72 pages

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• Braiding Majorana corner modes in a second-order topological superconductor
• Modern semiclassical theory of magnetic transport and breakdown
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
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• ${\mathbb{Z}}_2$ Topological Invariant for Magnon Spin Hall Systems
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• Glide-symmetric magnetic topological crystalline insulators with inversion symmetry
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• Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling
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• Boundary edge networks induced by bulk topology
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• Effective field theories of topological crystalline insulators and topological crystals
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
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• Topological classification of intrinsic 3D superconductors using anomalous surface construction
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