Topological Crystalline Insulators

TL;DR

The paper introduces topological crystalline insulators, extending 3D band-structure topology to include crystal point-group symmetry, notably $C_4$ (and $C_6$) in spinless systems. Using a tight-binding model on a tetragonal lattice with two inequivalent atoms per unit cell and a $p_x/p_y$-orbital basis, it derives a Bloch Hamiltonian $H({\bf k})$ and demonstrates gapless (001) surface states protected by time-reversal and $C_4$ symmetry, with surface states exhibiting quadratic dispersion. A $Z_2$ invariant $\nu_0 \in \{0,1\}$ is defined for 3D TR-invariant crystals with $C_4$ symmetry from occupied doublet wavefunctions at high-symmetry momenta, with $\nu_0=1$ signaling a nontrivial phase and a protected surface. The framework generalizes to $C_6$ symmetry and suggests a broader class of symmetry-protected topological phases, potentially realizable in layered materials and photonic crystals; the work emphasizes the interplay between symmetry representations and band topology and expands the landscape of topological phases with practical implications for material search.

Abstract

The recent discovery of topological insulators has revived interest in the topological properties of insulating band structures. In this work, we extend the topological classification of insulating band structures to include certain point group symmetry of crystals. We find a class of three-dimensional "topological crystalline insulators" which have metallic surface states on certain high symmetry crystal surfaces. These topological crystalline insulators can be viewed as the counterpart of topological insulators in materials without spin-orbit coupling. Their surface states have quadratic band degeneracy instead of linear Dirac dispersion. Their band structures are characterized by new Z2 invariants. We hope this work will enlarge the family of topological phases in band insulators and stimulate the search for them in real materials.

Figures (2)

• Figure 1: (a) tetragonal lattice with two atoms $A$ and $B$ along the $c$-axis in the unit cell; (b) the Brillouin zone and four high symmetry points.
• Figure 2: (a) bulk band structure of the tight binding model along high symmetry lines; (b) surface states with quadratic band touching exist on (001) face. The tight-binding parameters are shown in the text below Eq.(\ref{['Hk']}).