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

Horia D. Cornean, Bernard Helffer, Radu Purice

TL;DR

The paper addresses rigorous derivation of Peierls-Onsager-type effective dynamics for semimetals under long-range magnetic perturbations without assuming a spectral gap or trivial Bloch bundle. It develops a magnetic Weyl calculus with a direct reduction to a quasi-invariant subspace associated with an isolated Bloch family, even when bands overlap, and constructs strongly localized Parseval frames to realize a matrix representation of the effective dynamics via magnetic matrices and Zak translations. The main contributions are (i) a Schur-complement-based reduction yielding an effective Hamiltonian 𝔥^{ε,c}_𝔅 that approximates the perturbed spectrum and unitary evolution up to O(ε^2), (ii) a robust Parseval-frame framework enabling a tight-binding type matrix model even for non-trivial Bloch bundles, and (iii) explicit spectral and dynamical error bounds for the perturbed dynamics, including cases with non-zero Chern numbers where Wannier bases may not exist. The work extends prior gap-based results to semimetallic regimes and provides a rigorous path to magnetic spectral modifications and Hofstadter-like matrices in a broad periodic setting. Overall, it delivers a mathematically solid connection between Peierls-Onsager substitutions, magnetic pseudodifferential calculus, and tight-frame representations, with potential implications for modeling semimetal transport in complex magnetic fields.

Abstract

We consider periodic (pseudo)differential {elliptic operators of Schrödinger 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.

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

TL;DR

The paper addresses rigorous derivation of Peierls-Onsager-type effective dynamics for semimetals under long-range magnetic perturbations without assuming a spectral gap or trivial Bloch bundle. It develops a magnetic Weyl calculus with a direct reduction to a quasi-invariant subspace associated with an isolated Bloch family, even when bands overlap, and constructs strongly localized Parseval frames to realize a matrix representation of the effective dynamics via magnetic matrices and Zak translations. The main contributions are (i) a Schur-complement-based reduction yielding an effective Hamiltonian 𝔥^{ε,c}_𝔅 that approximates the perturbed spectrum and unitary evolution up to O(ε^2), (ii) a robust Parseval-frame framework enabling a tight-binding type matrix model even for non-trivial Bloch bundles, and (iii) explicit spectral and dynamical error bounds for the perturbed dynamics, including cases with non-zero Chern numbers where Wannier bases may not exist. The work extends prior gap-based results to semimetallic regimes and provides a rigorous path to magnetic spectral modifications and Hofstadter-like matrices in a broad periodic setting. Overall, it delivers a mathematically solid connection between Peierls-Onsager substitutions, magnetic pseudodifferential calculus, and tight-frame representations, with potential implications for modeling semimetal transport in complex magnetic fields.

Abstract

We consider periodic (pseudo)differential {elliptic operators of Schrödinger 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.

Paper Structure

This paper contains 44 sections, 45 theorems, 247 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Informal brief overview.
  3. The basic framework
  4. The magnetic Weyl calculus.
  5. The magnetic fields.
  6. The problem
  7. The unperturbed Hamiltonian
  8. The Hörmander classes of symbols.
  9. The unperturbed Hamiltonian.
  10. The $d$-dimensional torus.
  11. The Bloch-Floquet structure of the unperturbed Hamiltonian.
  12. The Bloch eigenprojections.
  13. The isolated Bloch family hypothesis.
  14. The perturbed magnetic Hamiltonian.
  15. The main result.
  16. ...and 29 more sections

Key Result

Theorem 2.5

If $H^{\circ}$ is a lower semi-bounded self-adjoint operator in $L^2(\mathcal{X})$ with domain $\mathscr{H}^p(\mathcal{X})$ ($p>0$), which commutes with all the translations $\{U_\gamma,\,\gamma\in\Gamma\}$ as in DF-h-circ, then:

Figures (2)

  • Figure 1: Here $k_0=2$ and $N=1$. The isolated band consists of two crossing eigenvalues in red and green. Formally, the green colour should always be on top of the red colour because $\lambda_{k_0}\leq \lambda_{k_0+N}$, but we will never treat them individually, only as a well-defined isolated family. The energy interval we are interested in is $(E_-,E_+)$ where $E_-$ is the maximum of the blue eigenvalue $\lambda_{k_0-1}$ and $E_+$ is the minimum of the the black one $\lambda_{k_0+N+1}$. The Hamiltonian $H^\circ$ does not have a spectral gap.
  • Figure 2: Here $k_0=2$ and $N=0$. The isolated band consists of just one eigenvalue and it forms a spectral island for $H^\circ$, separated from the rest of the spectrum by two gaps.

Theorems & Definitions (89)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Theorem 2.5: Bloch-Floquet decomposition
  • Remark 2.6
  • Theorem 2.7
  • Definition 2.9
  • Proposition 2.12
  • Theorem 2.14
  • ...and 79 more