Peierls substitution and magnetic pseudo-differential calculus

TL;DR

This paper revisits the Peierls-Onsager substitution using magnetic pseudo-differential calculus, demonstrating a convergent expansion for magnetic band Hamiltonians and their spectral properties under weak magnetic fields.

Contribution

It generalizes the Peierls substitution by employing magnetic pseudo-differential calculus without decay conditions and links magnetic Hamiltonians to Hofstadter-like matrices.

Findings

Magnetic band Hamiltonian admits a convergent symbol expansion.

Magnetic Hamiltonian is unitarily equivalent to a Hofstadter-like matrix.

Spectrum of the magnetic matrix approximates a Weyl quantized symbol under slow magnetic field variation.

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.