New classes of three-dimensional topological crystalline insulators: Nonsymmorphic and magnetic

We theoretically predict two new classes of three-dimensional topological crystalline insulators (TCIs), which have an odd number of unpinned surface Dirac cones protected by crystal symmetries. The first class is protected by a single nonsymmorphic glide plane symmetry; the second class is protected by a composition of a twofold rotation and time-reversal symmetry (a magnetic group symmetry). Both classes of TCIs are characterized by a quantized $\pi$-Berry phase associated with surface states and a $Z_2$ topological invariant associated with the bulk bands. In the presence of disorder, these TCI surface states are protected against localization by the average crystal symmetries, and exhibit critical conductivity in the universality class of the quantum Hall plateau transition. These new TCIs exist in time-reversal-breaking systems with or without spin-orbital coupling, and their material realizations are discussed.

Figure 1

(a) The SBZ of the $xz$ surface of a 3D (spinless) system with glide plane defined in Eq. (1). (b)–(d) are schematics of the process where two identical Dirac points on a single mirror symmetric line annihilate each other through finite surface perturbation. Here in (b) we set $k_{1,2}$ slightly away from each other only to indicate that there are two instead of one Dirac points. The color bar on the left of (b) shows the color code for the phase of the glide plane eigenvalues.

Figure 2

(a)–(c) show the process of gapping the spectral flow when there are two band crossings, on ${\overline{X}}^{\prime }\overline{\Gamma }\overline{X}$ and ${\overline{M}}^{\prime }\overline{\Gamma }\overline{M}$, respectively. (d)–(f) show the continuous flow when there is only one band crossing, and how the crossing moves from ${\overline{X}}^{\prime }\overline{\Gamma }\overline{X}$ to ${\overline{M}}^{\prime }\overline{\Gamma }\overline{M}$.