Corrigendum to “Single-cone real-space finite difference scheme for the time-dependent Dirac equation” [J. Comput. Phys. 265 (2014) 50–70]

Abstract

We would like to correct some typos which are contained in the original paper. 1. Equations (9)–(12) should read (They are obtained from the originally published equations by the substitution [Formula presented].): For B one has on the [Formula presented] sub-grid [Formula presented] and on the [Formula presented] sub-grid one uses [Formula presented] The update for the C-components living on [Formula presented] writes [Formula presented] and for the C-component on [Formula presented] [Formula presented] 2. In Eq. (19) [Formula presented] should be replaced by [Formula presented], giving [Formula presented] 3. [Formula presented] in the denominator on the lhs. of Eqs. (40)–(43) should be replaced by [Formula presented]. 4. Eq. (49) should read [Formula presented] 5. “that is has” in the line following Eq. (58) should read “that it has”. Finally, we would like to clarify the use of the terms “single-cone real-space finite difference scheme” and “the avoidance of fermion doubling”. Under constant coefficients (mass and scalar potential) the presented finite-difference scheme (FDS) supports a single-cone dispersion only. There are no doublers. In the presence of electromagnetic potentials, which should be introduced in gauge-invariant fashion, the situation is more subtle [1]. In the long-wave-length limit of the potentials, with grid spacing [Formula presented], our FDS locally supports a single-cone energy-momentum dispersion: there is no (counter-propagating) doubler. This limit is the one relevant for a meaningful numerical simulation of physical processes [2]. However, on the grid one can construct (periodic) scalar potentials V with [Formula presented] where the scheme supports more than one energy-momentum dispersion [3]. For a potential periodic with [Formula presented], there are as many “cones” (i.e. particle-antiparticle-symmetric pairs of bands) as there are grid points in the unit cell of the time sheets. However, in the long-wavelength limit (i.e. keeping λ constant, with [Formula presented]) this family of cones merges into a family of single-cone dispersions, e.g. Eq. (49) for the (2+1)D case and constant mass m, adiabatically modulated by the potential V. If the limit [Formula presented] is taken with [Formula presented] held constant the limiting potential is not defined (as a smooth function of space) and a comparison with the continuum Dirac equation regarding fermion doubling cannot be made. Finally, we note that the energy-momentum dispersion supported by the FDS depends on the time step chosen within the CFL condition. The authors would like to apologise for any inconvenience or confusion the original presentation may have caused.