Finite difference method for two-dimensional Dirac billiards with arbitrary shape and potentials

The emergence of relativistic pseudoparticles described by the Dirac equation in condensed-matter systems, combined with the ability to exert external potentials, highlights the importance of solving the Dirac equation in confined domains with general potentials. In this work, we introduce a finite-difference scheme to solve the Dirac equation in general two-dimensional (2D) domains of arbitrary shapes and potential configurations. These domains may be simply connected or multiply connected (with holes), and the potential can be of electrostatic type, magnetostatic type, or mass type, or their combinations. The method incorporates particle states, hole states, or Klein tunneling cases automatically, and accommodates diverse boundary conditions, such as positive infinite mass, negative infinite mass, or mixed configurations. By transforming the discretized Dirac equation into a sparse generalized eigenvalue problem, our approach efficiently addresses challenges posed by underdetermination or overdetermination, which arise from domain topology, while avoiding nonphysical solutions. This robust numerical framework offers a general and efficient tool for investigating a broad range of relativistic quantum phenomena in 2D Dirac billiards featuring complex potentials and boundaries.