Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; to a large class of symmetries for a non-magnetic system, one can associate "magnetic" symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bundle, a rank m hermitian vector bundle over the Brillouin zone, we prove that magnetic time-reversal symmetry is sufficient to ensure the triviality of the Bloch bundle in spatial dimension d=1,2,3. For d=4, an exponentially localized Wannier basis exists provided that the trace per unit volume of a suitable function of the Fermi projection vanishes. For d>4 and d \leq 2m (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of [Kuc09]. Finally, for d>4 and d>2m (unstable rank regime) we show that the mere analysis of Chern classes does not suffice in order to prove trivility and thus exponential localization.Comment: 48 pages, updated introduction and bibliograph

The Schr\"odinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous

This paper systematically develops the Schr\"odinger formalism that is valid also for gyrotropic media where the material weights $W = \left ( \begin{smallmatrix} \varepsilon & \chi \chi^* & \mu \end{smallmatrix} \right ) \neq \overline{W}$ are complex. This is a non-trivial extension of the Schr\"odinger formalism for non-gyrotropic media (where $W = \overline{W}$) that has been known since at least the 1960s. Here, Maxwell's equations are rewritten in the form $\mathrm{i} \partial_t \Psi = M \Psi$ where the selfadjoint (hermitian) Maxwell operator $M = W^{-1} \, \mathrm{Rot} \, \big |_{\omega \geq 0} = M^*$ takes the place of the Hamiltonian and $\Psi$ is a complex wave representing the physical field $(\mathbf{E},\mathbf{H}) = 2 \mathrm{Re} \, \Psi$. Writing Maxwell's equations in Schr\"odinger form gives us access to the rich toolbox of techniques initially developed for quantum mechanics and allows us to apply them to classical waves. To show its utility, we explain how to identify conserved quantities in this formalism. Moreover, we sketch how to extend our ideas to other classical waves.Comment: 58 pages, updated version incorporates suggestions from the communit

The Perturbed Maxwell Operator as Pseudodifferential Operator

As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M_0. In particular, we characterize the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |\omega|. Among other things, we prove that generically the band spectrum is symmetric with respect to inversions at k = 0 and that there are exactly 4 ground state bands with approximately linear dispersion near k = 0.Comment: 41 pages, rewritten introduction, generalized results to include electric permittivity and magnetic permeability tensor