Homotopy Theory of Strong and Weak Topological Insulators

TL;DR

This paper applies homotopy theory to extend the classification of strong and weak topological insulators into the non-stable regime, clarifying their dimensionality and providing rigorous mathematical justifications for existing methods.

Contribution

It introduces a refined definition of 'strong' topological insulators in low-band regimes and proves technical results on invariant factorization and manifold replacements.

Findings

Strong topological insulators require a stricter 'strong' definition in low-band regimes.

Weak topological insulators can still be 'truly d-dimensional' in the non-stable regime.

Homotopy theory justifies the factorization of invariants and the replacement of tori by spheres.

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.