Single-cone real-space finite difference scheme for the time-dependent Dirac equation

Abstract

A finite difference scheme for the numerical treatment of the (3+1)D Dirac equation is presented. Its staggered-grid intertwined discretization treats space and time coordinates on equal footing, thereby avoiding the notorious fermion doubling problem. This explicit scheme operates entirely in real space and leads to optimal linear scaling behavior for the computational effort per space-time grid-point. It allows for an easy and efficient parallelization. A functional for a norm on the grid is identified. It can be interpreted as probability density and is proved to be conserved by the scheme. The single-cone dispersion relation is shown and exact stability conditions are derived. Finally, a single-cone scheme for the two-component (2+1)D Dirac equation, its properties, and a simulation of scattering at a Klein step are presented. 1 E.g., spin-1/2 particle in a homogeneous magnetic field [6], the Dirac oscillator [7], an electromagnetic plane wave [8], or the Coulomb potential [9].2 The electron is accelerated to relativistic speeds during one laser cycle.

A spectral method for half-integer spin fields based on spin-weighted spherical harmonics

“…This will allow for a test of the spin-weighted spectral transformations we present in the context of evolution equations. Common techniques for treating the numerical problem of the Dirac equation consist of: FD schemes formulated on a flat-lattice in configuration space [12,13], on a grid within a finite-volume in momentumspace [22] and using methods based on the split-step operator technique [7,8,20,21]. Particular to the FD approach special care must be taken so as to avoid the Fermion-doubling problem [24].…”

“…Particular to the FD approach special care must be taken so as to avoid the Fermion-doubling problem [24]. Elimination of spurious modes introduced to the solution may be accomplished by means of nonlocal approximation for the spatial derivative operator [32,38] or by staggered-grid schemes [12,13]. As remarked upon in [13] the issue of spurious modes is not particular to the Dirac equation but can occur whenever a symmetric FD approximant is used for a first derivative on a uniform grid.…”

Staggered grid leap-frog scheme for the Dirac equation

“…Here it should be recalled that by using one-sided difference operators with alternating direction for for u and v, fermion doubling can be avoided for the (1+1)D Dirac equation. For the (1+1)D case the latter is equivalent to the present spatial staggering of the grid [23,24]. We use staggering in time to further improve the dispersion relation, which will be shown below.…”

“…Applications to TI surface state dynamics based on this algorithm have and will be presented elsewhere [20,21]. A related numerical treatment of the (1+1)D two-spinor-component Dirac equation including exact absorbing boundary conditions, displaying a single Dirac cone, has been presented by us recently [24]. Furthermore, we have been able to develop a scheme, respectively, for the to the (2+1)D two-spinorcomponent Dirac equation and the (3+1)D four-spinorcomponent Dirac equation with a single cone only [42].…”

Simulation of Acoustic Wave Propagation in Aluminium Coatings for Material Characterization

“…The field variables, velocity and stress, are placed in a staggered manner onto the computational grid, which greatly improves the dispersion relation of the scheme. Interestingly, this property is also exploited in electrodynamics, where the finite difference time domain (FDTD) scheme is unsurpassed for inhomogeneous domains [37], and recently also for the relativistic wave equation [38]. The elastic wave discretization using a staggered grid and the EFIT are related, with the difference being that the material properties are placed differently on the grid, allowing the EFIT to more accurately describe material inhomogeneities [35].…”