Mirror Chern numbers in the hybrid Wannier representation

Abstract

The topology of electronic states in band insulators with mirror symmetry can be classified in two different ways. One is in terms of the mirror Chern number, an integer that counts the number of protected Dirac cones in the Brillouin zone of high-symmetry surfaces. The other is via a ${\mathbb{Z}}_2$ index that distinguishes between systems that have a nonzero quantized orbital magnetoelectric coupling ("axion-odd"), and those that do not ("axion-even"); this classification can also be induced by other symmetries in the magnetic point group, including time reversal and inversion. A systematic characterization of the axion ${\mathbb{Z}}_2$ topology has previously been obtained by representing the valence states in terms of hybrid Wannier functions localized along one chosen crystallographic direction, and inspecting the associated Wannier band structure. Here we focus on mirror symmetry, and extend that characterization to the mirror Chern number. We choose the direction orthogonal to the mirror plane as the Wannierization direction, and show that the mirror Chern number can be determined from the winding numbers of the touching points between Wannier bands on mirror-invariant planes, and from the Chern numbers of flat bands pinned to those planes. In this representation, the relation between the mirror Chern number and the axion ${\mathbb{Z}}_2$ index is readily established. The formalism is illustrated by means of $\text{ab initio}$ calculations for SnTe in the monolayer and bulk forms, complemented by tight-binding calculations for a toy model.

References in corpus (29)

• Maximally-localized generalized Wannier functions for composite energy bands
• Topological Insulators with Inversion Symmetry
• Topological Field Theory of Time-Reversal Invariant Insulators
• Maximally-localized Wannier functions for entangled energy bands
• Topological Crystalline Insulators in the SnTe Material Class
• The Computational 2D Materials Database: High-Throughput Modeling and Discovery of Atomically Thin Crystals
• Prediction and observation of the first antiferromagnetic topological insulator
• Experimental realization of a topological crystalline insulator in SnTe
• Magnetoelectric polarizability and axion electrodynamics in crystalline insulators
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• Antiferromagnetic topological insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator
• Quantized Response and Topology of Insulators with Inversion Symmetry
• Axion Electrodynamics in Topological Materials
• Chiral Flat Bands: Existence, Engineering and Stability
• Crystal Field Effect Induced Topological Crystalline Insulators In Monolayer IV-VI Semiconductors
• Topological insulator and the Dirac equation
• Berry Phase and Pseudospin Winding Number in Bilayer Graphene
• Discovering new two-dimensional topological insulators from computational screening
• The Axion Insulator as a Pump of Fragile Topology
• Designing in-plane heterostructures of quantum spin hall insulators from first principles: 1T'-MoS2 with adsorbates
• On the matter of topological insulators as magnetoelectrics
• Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries
• Adiabatic Pumping of Chern-Simons Axion Coupling
• Coupled-layer description of topological crystalline insulators
• Axion coupling in the hybrid Wannier representation
• Layered Topological Crystalline Insulators

Cited by in corpus (3)

• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
• Topological Insulators with Hybrid-order Boundary States
• Evolution of the Chern Gap in Chern Magnet HoMn${}_6$Sn${}_{6-x}$Ge${}_x$