[Paper Review] Single-cone real-space finite difference schemes for the Dirac von Neumann equation

Magdalena Schreilechner, W. Pötz|arXiv (Cornell University)|Mar 6, 2015

Topological Materials and Phenomena10 references3 citations

TL;DR

This paper presents two single-cone real-space finite difference schemes for solving the von Neumann equation of the (2+1)D Dirac Hamiltonian, using a staggered grid to eliminate fermion doubling. The bra-ket scheme preserves positivity by applying the Hamiltonian separately to kets and bras, while the direct scheme computes the time derivative in one step; both ensure single-cone dispersion and are stable, with applications to topological insulator surface states and extensions to Lindblad and Green’s function formulations.

ABSTRACT

Two finite difference schemes for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian are presented. Both utilize a single-cone staggered space-time grid which ensures a single-cone energy dispersion to formulate a numerical treatment of the mixed-state dynamics within the von Neumann equation. The first scheme executes the time-derivative according to the product rule for "bra" and "ket" indices of the density operator. It therefore directly inherits all the favorable properties of the difference scheme for the pure-state Dirac equation and conserves positivity. The second scheme proposed here performs the time-derivative in one sweep. This direct scheme is investigated regarding stability and convergence. Both schemes are tested numerically for elementary simulations using parameters which pertain to topological insulator surface states. Application of the schemes to a Dirac Lindblad equation and real-space-time Green's function formulations are discussed.

Motivation & Objective

• Address the numerical challenge of simulating mixed-state dynamics in (2+1)D Dirac systems, particularly for topological insulator surface states.
• Overcome the fermion doubling problem inherent in standard finite-difference schemes for first-order space-time derivatives.
• Develop efficient, stable, and positivity-preserving numerical schemes for the von Neumann equation that scale linearly with grid size.
• Enable gauge-invariant treatment of electromagnetic potentials via Peierls substitution on the staggered lattice.
• Extend the framework to open quantum systems via the Lindblad equation and to real-space-time Green’s function formulations.

Proposed method

• Employ a single-cone staggered space-time grid to ensure a single Dirac cone in the energy-momentum dispersion relation.
• Implement the bra-ket scheme by applying the Hamiltonian separately to the left (bra) and right (ket) indices of the density operator, using the product rule for the commutator.
• Develop a direct scheme that computes the time derivative of the density matrix in a single sweep, avoiding separate bra/ket propagation.
• Use centered finite differences over one lattice spacing on the staggered grid to maintain accuracy and avoid fermion doubling.
• Apply Peierls substitution to incorporate electromagnetic vector potentials in a gauge-invariant manner.
• Validate schemes numerically using parameters relevant to topological insulator surface states and analyze stability and convergence.

Experimental results

Research questions

• RQ1 Can a finite difference scheme for the von Neumann equation preserve positivity and avoid fermion doubling in (2+1)D Dirac systems?
• RQ2 How does the bra-ket scheme’s structure inherit stability and positivity from the pure-state Dirac equation?
• RQ3 What are the stability and convergence properties of the direct finite difference scheme for the density matrix evolution?
• RQ4 Can the proposed schemes be extended to include open quantum system dynamics via the Lindblad equation?
• RQ5 How can the framework be adapted to real-space-time Green’s function formulations for non-equilibrium transport?

Key findings

• The bra-ket scheme inherits positivity and stability from the pure-state Dirac equation due to its product rule implementation.
• The direct scheme achieves computational efficiency with fewer matrix operations per time step but exhibits a reduced radius of convergence compared to the bra-ket scheme.
• Both schemes utilize a staggered grid to eliminate fermion doubling and ensure a single Dirac cone in the energy dispersion.
• The schemes are stable and converge under numerical testing with parameters relevant to topological insulator surface states.
• The Peierls substitution can be applied to the staggered grid in a gauge-invariant way, preserving the stability of the schemes under electromagnetic potentials.
• Extensions to the Lindblad equation and Green’s function formulations are feasible, enabling modeling of dissipative effects and non-equilibrium transport.