Energy of electrons on a two-dimensional lattice in a magnetic field: perturbation theory versus ?Peierls substitution?

1991 ◽  
Vol 83 (2) ◽  
pp. 237-244 ◽  
Author(s):  
A. Alexandrov ◽  
H. Capellmann
Keyword(s):  
Magnetic Field ◽  
Perturbation Theory ◽  
Two Dimensional ◽  
Field Perturbation ◽  
Dimensional Lattice ◽  
Theory Versus ◽  
Magnetic Field Perturbation

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

2019 ◽  
Vol 31 (03) ◽  
pp. 1950008
Author(s):  
Horia D. Cornean ◽  
Viorel Iftimie ◽  
Radu Purice
Keyword(s):  
Magnetic Field ◽  
Differential Calculus ◽  
Spectral Band ◽  
Spatial Decay ◽  
Minimal Coupling ◽  
Field Perturbation ◽  
Wannier Functions ◽  
Weak Magnetic Fields ◽  
The Magnetic Field ◽  
Magnetic Field Perturbation

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic [Formula: see text]-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by [Formula: see text] exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in [Formula: see text]. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on [Formula: see text] and is [Formula: see text]-periodic; if [Formula: see text], the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

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