Modeling of electronic dynamics in twisted bilayer graphene (Q6554470)

The authors propose numerical computations to simulate the quantum dynamics of an electron in twisted bilayer graphene, especially comparing results obtained through tight-binding dynamics and the Bistritzer-MacDonald model. They first describe the 2D lattice structure of the twisted bilayer graphene and the moiré lattice vectors.\ They present the tight-binding Hamiltonian for an electron in twisted bilayer graphene, as a linear self-adjoint operator \(H:\mathcal{H}\rightarrow \mathcal{H}\) that acts on the wave functions according to: \((H\psi )_{\mathbf{R}_{i}\sigma }=\sum_{ \mathbf{R}_{j}\sigma ^{\prime }\in \Omega }H_{\mathbf{R}_{i}\sigma ,\mathbf{R }_{j}^{\prime }\sigma ^{\prime }}\psi _{\mathbf{R}_{j}^{\prime }\sigma ^{\prime }}\), with \(H_{\mathbf{R}_{i}\sigma ,\mathbf{R}_{j}^{\prime }\sigma ^{\prime }}=\overline{H_{\mathbf{R}_{j}^{\prime }\sigma ^{\prime },\mathbf{R} _{i}\sigma }}\), where \(\mathcal{H}=\ell ^{2}(\Omega )\), \(\Omega =(\mathcal{R} _{1}\times \mathcal{A}_{1})\cup (\mathcal{R}_{2}\times \mathcal{A}_{2})\) being the index set describing the full degree of freedom space of twisted bilayer graphene and \(\mathbf{R}_{i}\sigma \) is the index of an atom in \( \Omega \). They assume an exponential decay on the operator \(H_{\mathbf{R} _{i}\sigma ,\mathbf{R}_{j}^{\prime }\sigma ^{\prime }}\) and they prove that \( H\) is bounded and self-adjoint, and an exponential decay of the resolvent associated with \(H\). \N\NDefining the finite dimensional injection map \(P_{R}: \mathcal{H}\rightarrow \ell ^{2}(\Omega _{R})\), with \(\Omega _{R}=\{R_{i}\sigma \in \Omega :\left\vert R_{i}+\tau _{i}^{\sigma }\right\vert \leq R\}\) through \(P_{R}\psi =(\psi R_{i}\sigma )_{R_{i}\sigma \in \Omega _{R}}\) and its dual, they prove a truncation estimate. They define the Bistritzer-MacDonald Hamiltonian \(H_{BM}\) as an unbounded self-adjoint operator on the space \(L^{2}(\mathbb{R}^{2};\mathbb{C}^{4})\) with domain \(H^{1}(\mathbb{R}^{2};\mathbb{C}^{4})\) defined through: \( H_{BM}=\left( \begin{array}{cc} v\sigma \cdot (-i\nabla _{r}) & w\sum_{n=1}^{3}T_{n}e^{-is_{n}\cdot r} \\ w\sum_{n=1}^{3}T_{n}^{\dag }e^{is_{n}\cdot r} & v\sigma \cdot (-i\nabla _{r}) \end{array} \right) \), where \(\sigma \) denotes the vector of Pauli matrices, and the parameters \(v\) and \(w\) control the strength of intralayer and interlayer hopping. They show how the Bistritzer-MacDonald Hamiltonian can be used to approximate the dynamics of wave-packets in twisted bilayer graphene, in a specific parameter regime. The main parts of the paper discuss numerical simulations obtained through both methods. The authors finally analyze the sensitivity of parameters for the Bistritzer-MacDonald approximation.