Peierls substitution and magnetic pseudo-differential calculus
Mathematical Physics
2015-07-23 v1 Other Condensed Matter
Analysis of PDEs
math.MP
Spectral Theory
Abstract
We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudo-differential calculus for weak magnetic fields with no spatial decay conditions, when the non-magnetic symbols have a certain spatial periodicity. We show in great generality that the symbol of the magnetic band Hamiltonian admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian admits a localized composite Wannier basis, we show that the magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In addition, if the magnetic field perturbation is slowly variable, then the spectrum of this matrix is close to the spectrum of a Weyl quantized, minimally coupled symbol.
Cite
@article{arxiv.1507.06114,
title = {Peierls substitution and magnetic pseudo-differential calculus},
author = {Horia D. Cornean and Viorel Iftimie and Radu Purice},
journal= {arXiv preprint arXiv:1507.06114},
year = {2015}
}