Horia D. Cornean, G. Nenciu, Thomas G. Pedersen
The Faraday effect revisited: General theory
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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. 
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 quasimomentum appear and 
thereby all ambiguities are removed, in contrast to earlier work.