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

Decebal Horia Cornean, Viorel Iftimie, Radu Purice

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We revisit the celebrated Peierls-Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic τ-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by N 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 [ℓ2(τ)]N. 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 ξ and is τ- -periodic; if N = 1, the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

Original languageEnglish
Article number1950008
JournalReviews in Mathematical Physics
Issue number3
Publication statusPublished - 1 Apr 2019



  • Peierls-Onsager substitution
  • periodic Hamiltonian
  • magnetic field
  • pseudo-differential calculus

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