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Magnetic Dirac systems: Violation of bulk-edge correspondence in the zigzag limit

Loïc Le Treust, Jean-Marie Barbaroux, Horia D. Cornean, Edgardo Stockmeyer, Nicolas Raymond

Abstract

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.

Magnetic Dirac systems: Violation of bulk-edge correspondence in the zigzag limit

Abstract

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.
Paper Structure (14 sections, 17 theorems, 113 equations, 6 figures)

This paper contains 14 sections, 17 theorems, 113 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. The energy dispersion curves
  3. Setting
  4. Main results for the energy dispersion curves
  5. Edge conductance formula
  6. Energy dispersion curves
  7. Preliminaries
  8. A characterization of the eigenvalues for the non-zigzag case
  9. The auxiliary quadratic forms
  10. Study of $\alpha\mapsto \nu_n(\alpha,\xi)$
  11. Study of $\xi\mapsto \nu_n^{\pm}(\alpha,\xi)$
  12. Proofs for the energy dispersion curves
  13. Proof of Theorems \ref{['thm.main']} and \ref{['prop.limitgamma']}
  14. Lemma on trace class operators

Key Result

Theorem 1.2

Let $b>0$ and let $\chi=\mathds{1}_{(0,1)}$ be the indicator function of the interval $(0,1)$. Let $\gamma\in\mathbb{R}\cup\{+\infty\}$. Let $F\in C_0^2(\mathbb{R})$ be such that it equals $1$ near $n\geqslant 1$ Landau levels, and $0$ near the others. Then, the operator $\chi(X_1)J_1 F'(\mathscr{D}

Figures (6)

  • Figure 1: Dispersion curves for the zigzag boundary conditions
  • Figure 2: The dispersion curves of $\mathscr{D}_{\gamma,\xi}$ for $\gamma = b= 1$
  • Figure 3: The value of the maximal negative energy of the full operator as a function of $\gamma$ for $b=1$.
  • Figure 4: The location of the critical point of the first negative dispersion curve for $b= 1$. The red bullet refers to $\gamma=1$.
  • Figure 5: The dispersion curves for $b=1$ and various $\gamma$
  • ...and 1 more figures

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3: On the boundary condition
  • Proposition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9: Symmetries
  • Proposition 2.1
  • ...and 27 more