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A fresh look at the Peierls-Onsager substitution

Horia D. Cornean, Bernard Helffer, Radu Purice

TL;DR

The paper develops a rigorous generalization of the Peierls-Onsager substitution for finite sets of Bloch bands under a local spectral gap, using strongly localized Parseval frames and magnetic matrix symbols to handle long-range regular magnetic perturbations. It constructs a magnetic Parseval frame for the isolated Bloch family, derives a reduced, near-invariant subspace under an external magnetic field, and shows that the dynamics on this subspace is governed by a magnetic Toeplitz-quantized matrix that matches the matrix-valued symbol of the Bloch data, up to controllable ε-dependent errors. A Schur-complement analysis yields precise spectral and time-evolution estimates, with the reduced dynamics converging to an effective Hamiltonian in the presence of non-constant but regular magnetic fields. The framework applies to a broad class of elliptic periodic pseudo-differential operators in any dimension d ≥ 2 and accommodates nonzero constant magnetic components, thereby extending PO-type approximations beyond slowly varying or Wannier-dependent regimes and offering a robust tool for solid-state models in magnetic fields.

Abstract

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. 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.

A fresh look at the Peierls-Onsager substitution

TL;DR

The paper develops a rigorous generalization of the Peierls-Onsager substitution for finite sets of Bloch bands under a local spectral gap, using strongly localized Parseval frames and magnetic matrix symbols to handle long-range regular magnetic perturbations. It constructs a magnetic Parseval frame for the isolated Bloch family, derives a reduced, near-invariant subspace under an external magnetic field, and shows that the dynamics on this subspace is governed by a magnetic Toeplitz-quantized matrix that matches the matrix-valued symbol of the Bloch data, up to controllable ε-dependent errors. A Schur-complement analysis yields precise spectral and time-evolution estimates, with the reduced dynamics converging to an effective Hamiltonian in the presence of non-constant but regular magnetic fields. The framework applies to a broad class of elliptic periodic pseudo-differential operators in any dimension d ≥ 2 and accommodates nonzero constant magnetic components, thereby extending PO-type approximations beyond slowly varying or Wannier-dependent regimes and offering a robust tool for solid-state models in magnetic fields.

Abstract

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. 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.
Paper Structure (59 sections, 54 theorems, 277 equations, 2 figures)

This paper contains 59 sections, 54 theorems, 277 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Peierls-Onsager approximation procedure
  3. Very brief report on the existing results
  4. Significance of our results
  5. Brief overview of the paper
  6. Precise formulation of the problem and the main result
  7. Notations
  8. The framework
  9. The phase space.
  10. Some classes of functions.
  11. The Hörmander classes of symbols.
  12. The magnetic field.
  13. The $d$-dimensional torus.
  14. The unperturbed Hamiltonian.
  15. The isolated Bloch family.
  16. ...and 44 more sections

Key Result

Proposition 2.15

Suppose given a symbol $F\in{S}^p_1(\Xi)$ for some $p\in\mathbb{R}$ and two magnetic fields $B=dA$ and $B^\prime=dA^\prime$ in $\mathlarger{\mathlarger{\mathbf{\Lambda}}}_{\text{\tt bd}}^2(\mathcal{X})$ with vector potentials $A\in\mathlarger{\mathlarger{\mathbf{\Lambda}}}_{\text{\tt pol}}^1(\mathca with:

Figures (2)

  • Figure 1: Here $k_0=2$ and $N=1$.
  • Figure 2: Here $k_0=2$ and $N=0$.

Theorems & Definitions (110)

  • Definition 1.2
  • Definition 2.2
  • Definition 2.3
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.11
  • Definition 2.12
  • Definition 2.14
  • Proposition 2.15
  • ...and 100 more