Quantitative K-Theory Related to Spin Chern Numbers

Abstract

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant.

References in corpus (2)

• Computing a logarithm of a unitary matrix with general spectrum
• Almost commuting matrices with respect to normalized Hilbert-Schmidt norm

Cited by in corpus (7)

• K-Theory and Pseudospectra for Topological Insulators
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Real-space representation of winding number for one-dimensional chiral-symmetric topological insulator
• On the Bott index of unitary matrices on a finite torus
• Time-dependent topological systems: A study of the Bott index
• Topological charge pumping in quasiperiodic systems characterized by Bott index
• Higher dimensional Bott classes and the stability of rotation relations