Topology of nonsymmorphic crystalline insulators and superconductors

Abstract

Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$ theory, we complete the classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space-group symmetries. The order-two nonsymmorphic space groups include half-lattice translation with ${\mathbit{Z}}_{2}$ flip, glide, twofold screw, and their magnetic space groups. We find that the topological periodic table shows modulo-2 periodicity in the number of flipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow ${\mathbb{Z}}_{2}$ topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with time-reversal and/or particle-hole symmetries provides novel ${\mathbb{Z}}_{4}$ topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and analytic expressions of the ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{4}$ topological invariants. The half-lattice translation with ${\mathbit{Z}}_{2}$ spin flip and glide symmetry are compatible with the existence of boundaries, leading to topological surface gapless modes protected by the order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.