The Faraday effect revisited: Thermodynamic limit

Abstract

This paper is the second in a series revisiting the (effect of) Faraday rotation. We formulate and prove the thermodynamic limit for the transverse electric conductivity of Bloch electrons, as well as for the Verdet constant. The main mathematical tool is a regularized magnetic and geometric perturbation theory combined with elliptic regularity and Agmon-Combes-Thomas uniform exponential decay estimates.Comment: 35 pages, accepted for publication in Journal of Functional Analysi

On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schrödinger Operators

“…The resolvent (H(0)−z) −1 is an integral operator given by an integral kernel Q 0 (x, x ′ ; z). The singularities of Q 0 (x, x ′ ; z) are the same as in the case of the free Laplacean and there exists some δ > 0 and C M < ∞ such that uniformly in x = x ′ [12,10]:…”

“…Note that the operator K ± (0) has an integral kernel given by: 10) where the local singularity at x = x ′ dissapears due to the integral with respect to z. Now using (4.8), (4.3) and (4.5) we have:…”

Středa formula for charge and spin currents

“…where Γ is a positively-oriented contour in the complex energy plane surrounding all the bands below E F and contained in the resolvent set of H B1,B2 . Notice that the terms in the Hamiltonian which are proportional to S z produce only bounded perturbations, therefore we can apply general results from the theory of magnetic Schrödinger operators [28,9] and conclude that the resolvent operator has an integral kernel that decays away from the diagonal and has a logarithmic singularity on the diagonal [6,3], that is,…”

“…The previous formula can now be treated in the same way as in the charge current case, see [10,9]. Due to the magnetic covariance of the operators in the trace, we first obtain that the spin conductivity admits a thermodynamic limit in the form of a trace per unit volume: After that, we can perform the limit of zero temperature and zero frequency, where in particular the equilibrium state converges to the Fermi projection, obtaining a closed formula for the transverse spin conductivity which reads…”

On the transfer matrix of a MIMO system

“…Our main mathematical tool is the geometric perturbation theory for closed but non‐selfadjoint operators. These methods have been developed for apparently unrelated quantum scattering problems in 13–16 and they can be used for describing the propagation of electromagnetic waves in frequency domain.…”