Topology of nonsymmorphic crystalline insulators and superconductors

Abstract

Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B ${\bf 90}$, 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 $Z_2$ flip, glide, two-fold 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 the 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 the analytic expression of the $\mathbb{Z}_2$ and $\mathbb{Z}_4$ topological invariants. The half lattice translation with $Z_2$ spin flip and glide symmetry are compatible with the existence of the boundary, leading to topological surface gapless modes protected by such order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.Comment: 38 pages, 13 figures, final versio

Topological Insulators from Group Cohomology

“…The advantage of this analogy is that the Wilson bands may be interpolated [35,38] [39], and all topological insulators with robust edge states protected by space-time symmetries. Here, we refer to topological insulators with either symmorphic [1,2,8] or nonsymmorphic spatial symmetries [4,7,19,40], the time-reversal-invariant quantum spin Hall phase [25], and magnetic topological insulators [18,[41][42][43]. These case studies are characterized by extensions of G ∘ by TABLE I.…”

“…In particular, D 4 6h falls outside the K-theoretic classification of nonsymmorphic topological insulators in Ref. [7].…”

“…The point group (D 6h ) of KHgX, defined as the quotient of its space group by translations, is generated by four spatial transformations-this typifies the complexity of most space groups. This work describes a systematic method to topologically classify space groups with similar complexity; in contrast, previous classifications [1,2,[4][5][6][7]14] (with one exception by us [8]) have expanded the Altland-Zirnbauer symmetry classes [22,23] to include only a single point-group generator. For point groups with multiple generators, different submanifolds of the Brillouin torus are invariant under different symmetries; e.g., mirror and glide planes are respectively mapped to themselves by a symmorphic reflection and a glide reflection, as illustrated in Fig.…”

“…Spatial symmetries have enriched the topological classification of insulators and superconductors [1][2][3][4][5][6][7][8][9][10]. A basic geometric property that distinguishes spatial symmetries regards their transformation of the spatial origin: symmorphic symmetries preserve the origin, while nonsymmorphic symmetries unavoidably translate the origin by a fraction of the lattice period [11].…”

“…1(a) for D 4 6h . Wave functions in each submanifold are characterized by a lower-dimensional topological invariant that depends on the symmetries of that submanifold; e.g., mirror planes are characterized by a mirror-Chern number [24] and glide planes by a glidesymmetric analog [7,21] of the quantum spin Hall (QSH) effect [25] [in short, a quantum glide Hall effect (QGHE)]. The various invariants are dependent because wave functions must be continuous where the submanifolds overlap; e.g., the intersection of planes in Fig.…”