# Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation

## Abstract

An explicit finite difference scheme is presented for the von Neumann equation for (2+1)D Dirac fermions. It is founded upon a staggered space–time grid which ensures a single-cone energy dispersion and performs the time-derivative in one sweep using a three-step leap-frog procedure. It enables a space–time-resolved numerical treatment of the mixed-state dynamics of Dirac fermions within the effective single-particle density matrix formalism. Energy-momentum dispersion, stability and convergence properties are derived. Elementary numerical tests to demonstrate stability properties use parameters which pertain to topological insulator surface states. A method for the simulation of charge injection from an electric contact is presented and tested numerically. Potential extensions of the scheme to a Dirac–Lindblad equation, real-space–time Green's function formulations, and higher-order finite-difference schemes are discussed.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22701624

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 348; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY MATRIX; DIRAC EQUATION; ELECTRIC CONTACTS; FERMIONS; MIXED STATES; SPACE-TIME; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Pötz, Walter, and Schreilechner, Magdalena.
```*Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.07.037.

```
Pötz, Walter, & Schreilechner, Magdalena.
```*Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation*. United States. doi:10.1016/J.JCP.2017.07.037.

```
Pötz, Walter, and Schreilechner, Magdalena. Wed .
"Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation". United States. doi:10.1016/J.JCP.2017.07.037.
```

```
@article{osti_22701624,
```

title = {Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation},

author = {Pötz, Walter and Schreilechner, Magdalena},

abstractNote = {An explicit finite difference scheme is presented for the von Neumann equation for (2+1)D Dirac fermions. It is founded upon a staggered space–time grid which ensures a single-cone energy dispersion and performs the time-derivative in one sweep using a three-step leap-frog procedure. It enables a space–time-resolved numerical treatment of the mixed-state dynamics of Dirac fermions within the effective single-particle density matrix formalism. Energy-momentum dispersion, stability and convergence properties are derived. Elementary numerical tests to demonstrate stability properties use parameters which pertain to topological insulator surface states. A method for the simulation of charge injection from an electric contact is presented and tested numerically. Potential extensions of the scheme to a Dirac–Lindblad equation, real-space–time Green's function formulations, and higher-order finite-difference schemes are discussed.},

doi = {10.1016/J.JCP.2017.07.037},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 348,

place = {United States},

year = {2017},

month = {11}

}