Gap-labelling conjecture with nonzero magnetic field

Abstract

Given a constant magnetic field on Euclidean space ${\mathbb R}^p$ determined by a skew-symmetric $(p\times p)$ matrix $\Theta$, and a ${\mathbb Z}^p$-invariant probability measure $\mu$ on the disorder set $\Sigma$ which is by hypothesis a Cantor set, where the action is assumed to be minimal, the corresponding Integrated Density of States of any self-adjoint operator affiliated to the twisted crossed product algebra $C(\Sigma) \rtimes_\sigma {\mathbb Z}^p$, where $\sigma$ is the multiplier on ${\mathbb Z}^p$ associated to $\Theta$, takes on values on spectral gaps in the magnetic gap-labelling group. The magnetic frequency group is defined as an explicit countable subgroup of $\mathbb R$ involving Pfaffians of $\Theta$ and its sub-matrices. We conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group. We give evidence for the validity of our conjecture in 2D, 3D, the Jordan block diagonal case and the periodic case in all dimensions.