Classification of topological crystalline insulators based on representation theory

Abstract

Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the $p4m$, $p31m$ and $p6m$ groups, and topological nonsymmorphic crystalline insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we also obtain the following new results: (1) there are two integer mirror Chern numbers (${\mathbb{Z}}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the $cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$ ($cmm$) groups, there is no topological classification in the spinless case but ${\mathbb{Z}}^4$ (${\mathbb{Z}}^2$) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the $pg$ group is related to that in the $pm$ group; (4) we identify topological classification of the $p4m$, $p31m$, and $p6m$ for the spinful case; (5) we find topological non-symmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups, which exhibit new features compared to those in $pg$ and $pmg$ groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.

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• Topological Materials Discovery from Crystal Symmetry
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• Line nodes, Dirac points and Lifshitz transition in 2D nonsymmorphic photonic crystals
• Hourglass Fermion Surface States in Stacked Topological Insulators with Nonsymmorphic Symmetry
• Coupled-layer description of topological crystalline insulators
• Valley-dependent Corner States in Honeycomb Photonic Crystal without Inversion Symmetry
• Hourglass semimetals with nonsymmorphic symmetries in three dimensions
• Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors
• Piezoelectricity and Topological Quantum Phase Transitions in Two-Dimensional Spin-Orbit Coupled Crystals with Time-Reversal Symmetry
• Floquet engineering of individual band gaps in an optical lattice using a two-tone drive
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Electronic properties of SnTe-class topological crystalline insulator materials
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• Mirror protected Dirac fermions on a Weyl semimetal NbP surface
• Topological Gapless Phases in Non-Symmorphic Antiferromagnets
• Glide-symmetric topological crystalline insulator phase in a nonprimitive lattice
• Rigidity of topological invariants to symmetry breakings
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• Current Noise in Tunnel Junctions
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