Gap-labelling conjecture with nonzero magnetic field

TL;DR

This paper investigates the magnetic gap-labelling conjecture for operators influenced by a constant magnetic field on Euclidean space, providing evidence for the conjecture's validity in various dimensions and special cases.

Contribution

It introduces the magnetic gap-labelling group and conjectures its inclusion in the magnetic frequency group, extending the gap-labelling theory to magnetic fields.

Findings

The magnetic gap-labelling group is a subgroup of the magnetic frequency group in 2D and 3D.

Evidence supports the conjecture in the Jordan block diagonal case.

The conjecture holds in the periodic case across all dimensions.

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.