Stop using Landau gauge for Tight-binding Models

Abstract

To analyze the electronic band structure of a two-dimensional (2D) crystal under a commensurate perpendicular magnetic field, tight-binding (TB) Hamiltonians are typically constructed using a magnetic unit cell (MUC), which is composed of several unit cells (UC) to satisfy flux quantization. However, when the vector potential is constrained to the Landau gauge, an additional constraint is imposed on the hopping trajectories, further enlarging the TB Hamiltonian and preventing incommensurate atomic rearrangements. In this paper, we demonstrate that this constraint persists, albeit in a weaker form, for any linear vector potential ($\mathbf{A}(\mathbf{r})$ linear in $\mathbf{r}$). This restriction can only be fully lifted by using a nonlinear vector potential. With a general nonlinear vector potential, a TB Hamiltonian can be constructed that matches the minimal size dictated by flux quantization, even when incommensurate atomic rearrangements occur within the MUC, such as moiré reconstructions. For example, as the twist angle $θ$ of twisted bilayer graphene (TBG) approaches zero, the size of the TB Hamiltonian scales as $1/θ^4$ when using linear vector potentials (including the Landau gauge). In contrast, with a nonlinear vector potential, the size scales more favorably, as $1/θ^2$, making small-angle TBG models more tractable with TB.

Figures (3)

• Figure 1: Example of how parameters define the lattice and the magnetic lattice. Crystal in this figure has primitive vectors $\vb{v_1},\vb{v_2}$ and has three atoms ($n=1,2,3$) in its UC (marked with purple boundaries). Indices $j_1$ and $j_2$ are marking its lattice points. Among four hoppings of the crystal ($m=1,2,3,4$), second hopping was described as $(2,3,( 10),t_2)$. Magnetic lattice of the crystal was constructed with $T=(5112)$. Magnetic lattice has primitive vectors $\vb{u_1},\vb{u_2}$ and its MUC (shaded in light blue) is composed of 9 UCs, each corresponding to a point in $J$ (marked with green dots). Hopping is inherited from the original lattice and two of them are described in the text boxes with blue boundary lines.
• Figure 2: Even textbook-lattices need an application of non-Landau-like linear or nonlinear vector potential to minimize the size of their MUC (shaded in blue). For Honeycomb lattice with $\Phi=\frac{1}{2}$, (a) Landau-like vector potential with $A= (00\frac{1}{2}0),~T=(61)$ needed MUC three times bigger than (b) linear vector potential with $A= (10\frac{1}{2}0),~T=(21)$. For Kagome lattice with $\Phi=\frac{1}{2}$, (c) Landau-like/linear vector potential with $A= (00\frac{1}{2}0),~T=(41)$ needed MUC two times bigger than (d) nonlinear vector potential with $A= (00\frac{1}{2}0),~T=(21)$. Each figure shows the hopping network whose hopping phase (assumed $\vb{k}=0$) of the edges are depicted with a color scale and the direction of the arrow. (If the arrow is unidirectional with phase $\phi \in (0,\pi)$ depicted as the color of the arrow, the hopping in the reverse direction has the phase $2\pi - \phi$.)
• Figure 3: Size of the TB Hamiltonian of the moiré-TBG with uniform magnetic flux $\Phi=\frac{5}{7}$. As the twisted angle $\theta$ decreases to 0, the size of the TB Hamiltonian is $q_{\Phi}=7$ times size of the UC in the case of nonlinear vector potential, while linear vector potentials such as Landau gauge requires quadratically larger TB Hamiltonian. Y-axis on the right shows the computation time for retrieving 10 eigenpairs near the Fermi energy (E=0). Using a laptop with i7-8550U CPU and 16GB RAM, the computation time was measured for the Python scipy.sparse.linalg.eigs function applied to moiré-TBG Hamiltonians, where all hoppings with distances less than $\frac{4}{\sqrt{3}} |\vb{e}_1|$ were consideredKoshino. The transparent area on the right Y-axis indicates the region where the computation time is estimated through extrapolation. The graph shows that moiré-TBG near MATBG is only tractable with the use of nonlinear vector potentials.

Theorems & Definitions (4)

• Remark
• Remark
• Theorem 8.1
• proof