Stop using Landau gauge for Tight-binding Models

TL;DR

This paper shows that using nonlinear vector potentials in tight-binding models for 2D materials under magnetic fields allows for more accurate and smaller Hamiltonians, especially in twisted bilayer graphene at small angles.

Contribution

It demonstrates that the Landau gauge imposes constraints that can be lifted by nonlinear vector potentials, enabling minimal TB Hamiltonians that accommodate incommensurate atomic arrangements.

Findings

Linear vector potentials impose size constraints on TB Hamiltonians.

Nonlinear vector potentials reduce Hamiltonian size scaling from 1/θ^4 to 1/θ^2.

Enhanced tractability for small-angle twisted bilayer graphene 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 moir\'e reconstructions. For example, as the twist angle $\theta$ of twisted bilayer graphene (TBG) approaches zero, the size of the TB Hamiltonian scales as $1/\theta^4$ when using linear vector potentials (including the Landau gauge). In contrast, with a nonlinear vector potential, the size scales more favorably, as $1/\theta^2$, making small-angle TBG models more tractable with TB.