Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Abstract

Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. The strategy based on pseudodifferential operators allows for efficient computations of these convolution products by using an ordinary fast Fourier transform algorithm. The resulting numerical methods are efficient and have spectral convergence. Simultaneously, wave absorption at the boundary can be easily derived using absorbing layers to cope with some potential negative effects of periodic conditions inherent to spectral methods. The numerical schemes are first tested on simple systems to verify the convergence and are then applied to the dynamics of charge carriers in strained graphene. 12], tremendous efforts have been put these past two decades on the development of numerical methods for the computation of the time-dependent Dirac equation in flat Minkowski space. Real space methods such as Quantum Lattice Boltzman techniques [13,14,15,16,17], Galerkin methods [18,19,20], and pseudospectral or spectral methods [21,22,23,24,25,26,27,28] were developed to efficiently and accurately solve the Dirac equation. Semi-classical regimes were considered in [29] using Gaussian beams or Frozen Gaussian Approximations [30,31], while the non-relativistic limit has been studied in several recent papers [32,33]. The computational difficulties for solving the time-dependent Dirac equation include the fermion doubling problem, related to numerical dispersion [16], and the Zitterbewegung [1], resulting in highly oscillating solutions whose origin can be traced back to the presence of the mass term βmc 2 . Finally, drastic stability conditions can lead to numerical diffusion related to the finite wave propagation speed, the speed of light c.Another numerical challenge for the Dirac equation shared by any other wave equation in real space solved on a truncated domain is the need of imposing special boundary conditions in order to avoid spurious wave reflections at the computational domain boundary. Therefore, the computational methods on truncated domains require non-reflective boundary conditions [15,34,35,36,37], absorbing or perfectly matched layers (PML) [35,38,39,40,41], or the introduction of an artificial potential [38]. On the other hand, Fourier-based methods applied on bounded domains naturally induce periodic boundary conditions, which can be problematic when dealing with delocalized wave functions. In this respect, the technique developed in [42,43] for the Dirac equation in flat space, where a spectral method is combined with PML, is an interesting alternative. One of the goals of this article is to extend this numerical scheme to the Dirac equation in curved space.Recently, the Dirac equation in curved space-time has gained important intere...

Perfectly matched layers for the Dirac equation in general electromagnetic texture

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Spin-1/2 Particles under the Influence of a Uniform Magnetic Field in the Interior Schwarzschild Solution

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