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Magnetic Dirac operator in strips submitted to strong magnetic fields

Loïc Le Treust, Julien Royer, Nicolas Raymond

TL;DR

This work analyzes the spectral properties of magnetic Dirac operators on curved 2D strips with infinite mass boundary in the strong-field limit. By reducing to a straight-strip model via unitary equivalence and employing a fibered operator framework, the authors derive precise asymptotics for the essential spectrum thresholds and establish the existence of discrete spectrum for small h. They develop a robust min-max and Hardy-space–based approach (incl. Hardy-Taylor expansions) to obtain sharp asymptotics for the first few positive eigenvalues, linking them to effective quantities that capture the interplay between geometry and magnetic field. The results extend prior bounded-domain analyses to unbounded curved waveguides, providing detailed quantitative bounds with explicit constant data and revealing the spectral confinement mechanisms relevant to graphene-like nanoribbon systems in strong magnetic fields.

Abstract

We consider the magnetic Dirac operator on a curved strip whose boundary carries the infinite mass boundary condition. When the magnetic field is large, we provide the reader with accurate estimates of the essential and discrete spectra. In particular, we give sufficient conditions ensuring that the discrete spectrum is non-empty.

Magnetic Dirac operator in strips submitted to strong magnetic fields

TL;DR

This work analyzes the spectral properties of magnetic Dirac operators on curved 2D strips with infinite mass boundary in the strong-field limit. By reducing to a straight-strip model via unitary equivalence and employing a fibered operator framework, the authors derive precise asymptotics for the essential spectrum thresholds and establish the existence of discrete spectrum for small h. They develop a robust min-max and Hardy-space–based approach (incl. Hardy-Taylor expansions) to obtain sharp asymptotics for the first few positive eigenvalues, linking them to effective quantities that capture the interplay between geometry and magnetic field. The results extend prior bounded-domain analyses to unbounded curved waveguides, providing detailed quantitative bounds with explicit constant data and revealing the spectral confinement mechanisms relevant to graphene-like nanoribbon systems in strong magnetic fields.

Abstract

We consider the magnetic Dirac operator on a curved strip whose boundary carries the infinite mass boundary condition. When the magnetic field is large, we provide the reader with accurate estimates of the essential and discrete spectra. In particular, we give sufficient conditions ensuring that the discrete spectrum is non-empty.
Paper Structure (34 sections, 39 theorems, 188 equations, 2 figures)

This paper contains 34 sections, 39 theorems, 188 equations, 2 figures.

Table of Contents

  1. Motivations and main results
  2. The magnetic Dirac operator on a strip
  3. The strips
  4. The choice of magnetic gauge
  5. The magnetic Dirac operators
  6. Main results
  7. Organization and strategy
  8. Estimate of the essential spectrum
  9. The operators on the straight and curved waveguides share the same essential spectrum
  10. A fibered family of Dirac operators
  11. Proof of the symmetry of the dispersion curves
  12. Proof of the invertibility of the operators
  13. A characterization of the eigenvalues
  14. Study of the first positive dispersion curve
  15. Proof of Point \ref{['pt.propdisp1']} and \ref{['pt.propdisp2']} of Proposition \ref{['prop.curves+']}
  16. ...and 19 more sections

Key Result

Proposition 1.1

Let $\hat{\phi}_0 = \phi_0 \circ \Theta^{-1} \in \mathscr C^\infty(\overline \Omega)$. There exists a unique $\phi\in\mathscr{C}^\infty(\overline{\Omega})$ such that $\Delta\phi=1$, $\phi_{|\partial\Omega}=0$, and $\phi-\hat{\phi}_0\in\mathscr{S}(\overline{\Omega})$. Moreover, there exists $c_0>0$ s

Figures (2)

  • Figure 1: Typical Waveguide Profile
  • Figure 2: Dispersion curves for $h = 0.05$

Theorems & Definitions (75)

  • Proposition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • Lemma 2.1
  • ...and 65 more