Higher-order continuum models for twisted bilayer graphene

TL;DR

This work develops and rigorously justifies higher-order continuum models for twisted bilayer graphene at small twist angles by extending the Bistritzer–MacDonald framework through a systematic multiscale expansion. It introduces an effective Hamiltonian $H^{\rm eff}=H^{(1)}+\zeta(\varepsilon)H^{({\rm NNN})}+\xi(\varepsilon)H^{(\nabla,{\rm NN})}+\varepsilon H^{(2)}$, and proves two complementary results: a four-term multiscale construction yielding $O(\varepsilon^{2+\eta_-})$ accuracy for wavepacket dynamics localized near monolayer Dirac points, and a direct second-order PDE formulation with the same convergence order. The analysis relaxes prior assumptions and demonstrates that higher-order corrections capture qualitative features (e.g., spiral patterns) and break emergent particle-hole symmetry present in first-order BM models, with numerical experiments corroborating the theoretical gains. The results provide a more accurate and robust continuum description of TBG dynamics, facilitating analysis and simulations of moiré-scale phenomena in graphene and related layered materials. Overall, the paper advances rigorous moiré continuum modeling by quantifying convergence, illustrating symmetry relations, and validating the higher-order model against tight-binding dynamics.

Abstract

The first-order continuum PDE model proposed by Bistritzer and MacDonald in \cite{bistritzer2011moire} accurately describes the single-particle electronic properties of twisted bilayer graphene (TBG) at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer-MacDonald model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis from \cite{watson2023bistritzer}, which rigorously established the validity of the (first-order) BM model.

Figures (8)

• Figure 1.1: The modulus of the wave-function for the tight-binding model and the first and second order continuum models. The initial data is concentrated on a flat band. The group velocity is approximately zero. The tight-binding model display a spiral pattern, which is also present in the second order BM model. The axes have units Å, and one unit of time is $\hbar \cdot \mathrm{eV}^{-1} \approx 6.6 \times 10^{-16} \text{ s}$.
• Figure 2.1: Monolayer graphene lattice with the blue and red dots respectively corresponding to the $A$ and $B$ sites. If the two right-most dots are indexed by $R = 0$ (so that their positions are $\tau^A$ and $\tau^B$), then the point $o := \frac{1}{2} (\tau^A + \tau^B) - (\frac{a}{2}, 0)$ marks the center of the right hexagon.
• Figure 2.2: Geometry of the monolayer graphene reciprocal lattice.
• Figure 4.1: TBG lattice at twist angle $\theta = 5^\circ$ with the red and blue dots respectively corresponding to layers $1$ and $2$. The point "$\times$" marks the center of rotation. Left: parameters are given by \ref{['eq:particular']}. Right: same value of $\tau^A$, but now $\mathfrak{d} := (0, a/10)$.
• Figure 5.1: Left: Illustration of monolayer Brillouin zone (BZ) in red and blue for twisted bilayer graphene, high symmetry point $K_1$ and $K_2$, moiré Brillouin zone (mBZ) in green and moiré high symmetry points $K := K_1$, $\Gamma$ and $M$. Right: The band structure showing the dispersion relation along a specific path in mBZ for first and second order BM continuum models, with Slater-Koster hopping function. The second order corrections breaks the particle-hole symmetry of the first order model.

Theorems & Definitions (60)

• Definition 2.1
• Lemma 2.1
• proof
• Theorem 2.1
• Corollary 2.1
• Remark 2.1
• Example 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4