Degenerate flat bands in twisted bilayer graphene

TL;DR

This work analyzes the chiral limit of the Bistritzer–MacDonald Hamiltonian for twisted bilayer graphene and shows that magic angles can yield flat bands of multiplicity four, with the zero-energy sector carrying a Chern number of $-1$. The authors develop a theta-function framework and Birman–Schwinger trace techniques to prove the existence of infinitely many degenerate magic angles for generic tunneling potentials (including BM’s), and establish that generically flat bands are limited to twofold or fourfold degeneracies. They further show that for degenerate bands the Bloch bundle decomposes into a trivial rank $m(\alpha)-1$ component plus a line bundle, yielding $c_1(E)=-1$ in the twofold case, and they provide extensive numerical evidence about complex magic angles and curvature behavior. A generic-simplicity program shows that, within a broad class of potentials, higher degeneracies are non-generic and that simple or twofold degeneracies are typical across representations. Together, these results deepen the topological and spectral understanding of moiré flat bands and their degeneracies, with potential implications for superconductivity and correlated physics in twisted graphene systems.

Abstract

We prove that in the chiral limit of the Bistritzer--MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be $ -1 $, and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer--MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials flat bands have only twofold or fourfold multiplicities.

Figures (11)

• Figure 1: Magic angles $\alpha$ derived from potentials $U_{\pm}=U_1(\pm \bullet)$ (left) and $U_{\pm}= U_2 (\pm \bullet)$ (right) in \ref{['eq:Hamiltonian']}. The multiplicity of the flat bands $u$ of $(D(\alpha)+k)u_{k}=0$ is illustrated by the numbers (no number $\rightarrow$ simple magic angle, 2 $\rightarrow$ two-fold degenerate magic angle) in the figure. The movie https://math.berkeley.edu/ zworski/Interpolation.mp4 shows the magic angles for interpolation between these potentials: $U ( z ) = ( \cos \theta -\sin \theta ) U_1 ( z ) + \sin \theta U_2 ( z )$; multiplicity one magic angles are coded by $\textcolor{red}{*}$ and multiplicity two by $\textcolor{blue}{*}$.
• Figure 2: Let $\alpha \approx 0.853799$ as in Table \ref{['table:double']}, lowest two Bloch band with positive energy close to the first magic angle with $U_{\pm}= U_2 (\pm \bullet)$. We plot $E_1(k)$ (left) and $E_2(k)$ (right).
• Figure 3: Let $\alpha \approx 0.9628 + 0.9873i$ the first complex magic angle for $U_{\pm}= U_1 (\pm \bullet)$, lowest two Bloch band with positive energy close to the first degenerate magic angle. We plot $E_1(k)$ (left) and $E_2(k)$ (right).
• Figure 4: Modulus of flat band wavefunctions of $\ker_{X}(D(\alpha))$ at first magic angle $\alpha=0.853799$ with $X=L^2_{i,j}$ with $i=K$(top), $i=-K$ (bottom), $j=0$ (left), $j=1$ (right) for potential $U_{\pm}= U_2 (\pm \bullet)$ in \ref{['eq:potential']}.
• Figure 5: Flat band wavefunctions of $\ker_{X}(D(\alpha))$ at first magic angle $\alpha=0.853799$ with $X=L^2_{0,0}$ (left) and $X=L^2_{0,1}$ (right) for potential $U_{\pm}= U_2 (\pm \bullet)$ in \ref{['eq:potential']} upper component, top and lower component, bottom.

Theorems & Definitions (28)

• Theorem 1: Zeros and multiplicities
• Theorem 2: Rigidity
• Theorem 3: Degenerate magic angles
• Theorem 4: Generic simplicity
• Theorem 5: Flat band topology
• Proposition 2.1: beta,bhz2
• Proposition 2.2
• proof
• Proposition 3.1
• proof