### Abstract

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 language | English |
---|---|

Article number | 1950008 |

Journal | Reviews in Mathematical Physics |

Volume | 31 |

Issue number | 3 |

ISSN | 0129-055X |

DOIs | |

Publication status | Published - 1 Apr 2019 |

### Fingerprint

### Keywords

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

### Cite this

*Reviews in Mathematical Physics*,

*31*(3), [1950008]. https://doi.org/10.1142/S0129055X19500089