Real-space finite difference scheme for the von Neumann equation with the Dirac Hamiltonian

A finite difference scheme for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian is presented. It is based on a sequential left–right (ket–bra) application of a staggered space–time scheme for the pure-state Dirac equation and offers a numerical treatment of the general mixed-state dynamics of an isolated quantum system within the von Neumann equation. Thereby this direct scheme inherits all the favorable features of the finite-difference scheme for the pure-state Dirac equation, such as the single-cone energy–momentum dispersion, convergence conditions, and scaling behavior. A conserved functional is identified. Moreover this scheme is shown to conserve both Hermiticity and positivity. Numerical tests comprise a numerical analysis of stability, as well as the simulation of a mixed-state time-evolution of Gaussian wave functions, illustrating Zitterbewegung and transverse current oscillations. Imaginary-potential absorbing boundary conditions and parameters which pertain to topological insulator surface states were used in the numerical simulations.