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Absence of small magic angles for disordered tunneling potentials in twisted bilayer graphene

Simon Becker, Izak Oltman, Martin Vogel

Abstract

We consider small random perturbations of the standard high-symmetry tunneling potentials in the Bistritzer-MacDonald Hamiltonian describing twisted bilayer graphene. Using methods developed by Sjöstrand for studying the spectral asymptotics of non-selfadjoint pseudo-differential operators, we prove that for sufficiently small twisting angles the Hamiltonian will not exhibit a flat band with overwhelming probability, and hence the absence of the so-called \textit{magic angels}. Moreover, we prove a probabilistic Weyl law for the eigenvalues of the non-selfadjoint tunneling operator, subject to small random perturbations, of the Bistritzer-MacDonald Hamiltonian in the chiral limit.

Absence of small magic angles for disordered tunneling potentials in twisted bilayer graphene

Abstract

We consider small random perturbations of the standard high-symmetry tunneling potentials in the Bistritzer-MacDonald Hamiltonian describing twisted bilayer graphene. Using methods developed by Sjöstrand for studying the spectral asymptotics of non-selfadjoint pseudo-differential operators, we prove that for sufficiently small twisting angles the Hamiltonian will not exhibit a flat band with overwhelming probability, and hence the absence of the so-called \textit{magic angels}. Moreover, we prove a probabilistic Weyl law for the eigenvalues of the non-selfadjoint tunneling operator, subject to small random perturbations, of the Bistritzer-MacDonald Hamiltonian in the chiral limit.
Paper Structure (23 sections, 21 theorems, 262 equations, 1 figure)

This paper contains 23 sections, 21 theorems, 262 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Random tunneling perturbation
  4. Absence of small magic angels with overwhelming probability
  5. Eigenvalue asymptotics -- a probabilistic Weyl law
  6. Outline of the paper
  7. Notation and conventions
  8. Review of Grushin problems and singular values
  9. Singular values
  10. Singular values for matrix-valued differential operators
  11. Singular values and vectors of $D_h$
  12. Grushin problem
  13. Pseudo-differential calculus and Sobolev class perturbations
  14. The unperturbed operator
  15. Sobolev perturbations
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $0 < \varepsilon < \exp( -C_1 h^{-2} \varepsilon_0(h))$, $C_1>1$ sufficiently large, let $N=N(\tau_0^2)$, and let with $C>0$ large enough. Then, for $h>0$ small enough

Figures (1)

  • Figure 1: Top left: Away from magic angles $h$, $\operatorname{Spec}(D_h)=h\Gamma^*$ with two-fold multiplicity. Top right: Spectrum of randomly perturbed operator $D_h$ away from magic angles $h$. Bottom left: Spectrum of $D_h$ numerically computed at largest magic angle $h$. In reality the spectrum is the entire complex plane. Bottom right: Spectrum of random perturbation of $D_h$ for $h$ the largest magic angle.

Theorems & Definitions (31)

  • Remark 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • proof
  • ...and 21 more