Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral theory of twisted bilayer graphene in a magnetic field

Simon Becker, Xiaowen Zhu

Abstract

In this article we study the Bistritzer-MacDonald (BM) model with external magnetic field. We study the spectral properties of the Hamiltonian in an external magnetic field with a particular emphasis on the flat band of the chiral model at magic angles. Our analysis includes different types of interlayer tunneling potentials, the so-called chiral and anti-chiral limits. One novelty of our article is that we show that using a magnetic field one can discriminate between flat bands of different multiplicities, as they lead to different Chern numbers in the presence of magnetic fields, while for zero magnetic field their Chern numbers always coincide.

Spectral theory of twisted bilayer graphene in a magnetic field

Abstract

In this article we study the Bistritzer-MacDonald (BM) model with external magnetic field. We study the spectral properties of the Hamiltonian in an external magnetic field with a particular emphasis on the flat band of the chiral model at magic angles. Our analysis includes different types of interlayer tunneling potentials, the so-called chiral and anti-chiral limits. One novelty of our article is that we show that using a magnetic field one can discriminate between flat bands of different multiplicities, as they lead to different Chern numbers in the presence of magnetic fields, while for zero magnetic field their Chern numbers always coincide.
Paper Structure (17 sections, 16 theorems, 129 equations, 8 figures)

This paper contains 17 sections, 16 theorems, 129 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Introduction of magnetic BM model
  3. Moiré lattices and TBG
  4. Chiral and anti-chiral limits
  5. Spectral properties of chiral and anti-chiral model
  6. Characterization of magic angles
  7. Magnetic Bloch-Floquet theory
  8. The chiral limit
  9. Flat bands away from magic angles
  10. Flat bands at magic angles
  11. Preliminaries
  12. Spectral properties
  13. Hörmander condition and exponential localization of bands
  14. Exponential squeezing in chiral model
  15. Anti-chiral model
  16. ...and 2 more sections

Key Result

Lemma 3.1

Let $A$ satisfy Assumption assump: magnetic potential. Let $T_{{\mathbf n},\operatorname{mag}}$ be defined as in eq: def_of_Tn. Then the following families of translations form Abelian groups and where ${\mathbf n}\cdot\lambda := n_1\lambda_1 + n_2 \lambda_2$, $\omega = \exp(\tfrac{2\pi i}{3})$.

Figures (8)

  • Figure 1: Left: Visible moiré pattern at $\theta = 5^{\circ}$. Right: Single moiré hexagon, with (A=red, B=blue) and (A'=green, B'=black) denote vertices of two sheets of graphene respectively.
  • Figure 2: The tunneling potentials for different coupling types on unit-size honeycomb lattice.
  • Figure 3: Figure showing modulus of two flat bands at $\alpha=0.2$ (non-magic) away from magic angles with magnetic flux $\Phi=2\pi.$
  • Figure 4: Figure showing modulus of two flat bands at $\alpha_1=0.5865$, the first magic angle, with $\Phi=2\pi.$
  • Figure 5: Spectrum of $T_{\mathbf k}$ for $\mathbf k \notin \Gamma^*$ with periodic magnetic field. The magic $\alpha_1$ do not depend on the magnetic field strength (left). Number of magic angles within radius $r$ (as in figure on the left) showing quadratic dependence (right), cf. Theorem \ref{['theo:magic_angles']}.
  • ...and 3 more figures

Theorems & Definitions (36)

  • Lemma 3.1
  • proof
  • Theorem 1: Magnetic Bloch-Floquet theory
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Corollary 3.2
  • proof
  • ...and 26 more