A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fields

TL;DR

This paper develops a rigorous mathematical framework for the effective dynamics of semimetals under long-range magnetic fields, extending previous models to more complex band structures without trivial Bloch bundle assumptions.

Contribution

It introduces a general Peierls-Onsager substitution for semimetals with overlapping Bloch bands, using localized tight-frames and magnetic matrices, and proves approximate time evolution with error bounds.

Findings

Established a rigorous effective dynamics model for semimetals in magnetic fields.

Extended previous analysis to non-trivial Bloch bundles with overlapping bands.

Provided error estimates for the approximate time evolution.

Abstract

We consider periodic (pseudo)differential {elliptic operators of Schr\"odinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.