The Faraday effect revisited: General theory

TL;DR

This paper provides a rigorous theoretical analysis of the Faraday effect, deriving formulas for conductivity in magnetic fields, and clarifies previous ambiguities by expressing results in terms of zero-field quantities.

Contribution

It offers a comprehensive, rigorous framework for understanding the electronic response in magnetic fields, including explicit formulas and removal of ambiguities in the theory.

Findings

Proved the Widom-Streda formula at zero temperature and frequency.

Explicitly computed transverse conductivity for free electrons.

Derived magnetic perturbation expansions for conductivity tensor.

Abstract

This paper is the first in a series revisiting the Faraday effect, or more generally, the theory of electronic quantum transport/optical response in bulk media in the presence of a constant magnetic field. The independent electron approximation is assumed. At zero temperature and zero frequency, if the Fermi energy lies in a spectral gap, we rigorously prove the Widom-Streda formula. For free electrons, the transverse conductivity can be explicitly computed and coincides with the classical result. In the general case, using magnetic perturbation theory, the conductivity tensor is expanded in powers of the strength of the magnetic field $B$. Then the linear term in $B$ of this expansion is written down in terms of the zero magnetic field Green function and the zero field current operator. In the periodic case, the linear term in $B$ of the conductivity tensor is expressed in terms of zero magnetic field Bloch functions and energies. No derivatives with respect to the quasi-momentum appear and thereby all ambiguities are removed, in contrast to earlier work.