Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Abstract

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-two and four period-eight cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme the classification in arbitrary dimensions $d\ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)]. The use of a scattering-matrix inspired method allows us to address the second descendant $\bZ_2$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

18 pages, 7 figures

References in corpus (11)

• Topological Insulators with Inversion Symmetry
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• The space group classification of topological band insulators
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Surface density of states and topological edge states in non-centrosymmetric superconductors
• Nodal Structure of Superconductors with Time-Reversal Invariance and Z2 Topological Number
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Möbius topological superconductivity in UPt${}_3$

Cited by in corpus (12)

• Higher-Order topological insulator in a dodecagonal quasicrystal
• Higher-order topological band structures
• Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors
• $N$-band Hopf insulator
• Topology invisible to eigenvalues in obstructed atomic insulators
• Boundary edge networks induced by bulk topology
• Rational boundary charge in one-dimensional systems with interaction and disorder
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Higher-order topological superconductor phases in a multilayer system
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Scattering theory of higher order topological phases
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry