A Non-Commutative Formula for the Isotropic Magneto-Electric Response

TL;DR

This paper derives a non-commutative mathematical formula for the isotropic magneto-electric response in disordered insulators, linking it to topological invariants and Chern numbers, with implications for topological insulators.

Contribution

It introduces a novel non-commutative formula for magneto-electric response, connecting it to topological invariants in disordered systems under magnetic fields.

Findings

The formula captures the quantization and topological invariance of the response.

Connection established between the magneto-electric response and a second Chern number.

Applicability to 3D disordered topological insulators with symmetry considerations.

Abstract

A non-commutative formula for the isotropic magneto-electric response of disordered insulators under magnetic fields is derived using the methods of non-commutative geometry. Our result follows from an explicit evaluation of the Ito derivative with respect to the magnetic field of the non-commutative formula for the electric polarization reported in Ref. 1. The quantization, topological invariance and connection to a second Chern number of the magneto-electric response are discussed in the context of 3-dimensional, disordered, time-reversal or inversion symmetric topological insulators.