Homotopy Theory of Strong and Weak Topological Insulators

Abstract

We use homotopy theory to extend the notion of strong and weak topological insulators to the non-stable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly d-dimensional", i.e. not realizable by stacking lower-dimensional insulators, a more restrictive definition of "strong" is required. However, this does not exclude weak topological insulators from being "truly d-dimensional", which we demonstrate by an example. Additionally, we prove some useful technical results, including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime, as well as the rigorous justification of replacing $T^d$ by $S^d$ and $T^{d_k}\times S^{d_x}$ by $S^{d_k+d_x}$ as is common in the current literature.

11 pages, 3 figures

References in corpus (9)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Crystalline Insulators
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological surface states in three-dimensional magnetic insulators

Cited by in corpus (9)

• Geometric approach to fragile topology beyond symmetry indicators
• Compactly-supported Wannier functions and algebraic $K$-theory
• Floquet Engineering Ultracold Polar Molecules to Simulate Topological Insulators
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Three dimensional two-band Floquet topological insulator with $Z_2$ index
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• The cohomology invariant for class DIII topological insulators
• Neutrino zeromodes on electroweak strings in light of topological insulators
• Topology vs localization in synthetic dimensions