Classification of topological insulators and superconductors in the presence of reflection symmetry

Abstract

We discuss a topological classification of insulators and superconductors in the presence of both (nonspatial) discrete symmetries in the Altland-Zirnbauer classification and spatial reflection symmetry in any spatial dimensions. By using the structure of bulk Dirac Hamiltonians of minimal matrix dimensions and explicit constructions of topological invariants, we provide the complete classification, which still has the same dimensional periodicities with the original Altland-Zirnbauer classification. The classification of reflectionsymmetry-protected topological insulators and superconductors depends crucially on the way reflection symmetry operation is realized. When a boundary is introduced, which is reflected into itself, these nontrivial topological insulators and superconductors support gapless modes localized at the boundary.

Topological Insulators from Group Cohomology

“…By our cohomological classification of quasimomentum submanifolds through Eq. (1), we provide a unifying framework to classify chiral topological insulators [39] and topological insulators with robust edge states protected by space-time symmetries [1,2,4,7,8,18,19,25,[40][41][42][43]. Our framework is also useful in classifying some topological insulators without edge states [26,55,56]; one counterexample that eludes this framework may nevertheless by classified by bent Wilson loops [46] rather than the straight Wilson loops of this work.…”

“…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 our companion work [21], we have identified a criterion on the surface group that characterizes all robust surface states that are protected by space-time symmetries [1,2,4,7,8,18,19,25,[40][41][42][43]54]. Our criterion introduces the notion of connectivity within a submanifold (M) of the surface-Brillouin torus and generalizes the theory of elementary band representations [57,62].…”

“…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.…”

Controlled Topological Phases and Bulk-edge Correspondence

“…In [FM13], Freed and Moore studied quantum systems with symmetries given by finite extensions of translation groups such as topological crystalline insulators [Fu11,CYR13]. A remarkable feature of their approach is that they developed the theory in an axiomatic way starting from a small number of general principles.…”