Single-cone finite-difference schemes for the (2+1)-dimensional Dirac equation in general electromagnetic textures

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Computational Physics

Background:

• The Dirac equation describes relativistic fermions, crucial for understanding particle physics and condensed matter phenomena.
• Simulating quantum systems on a lattice is essential for studying complex behaviors not accessible through analytical methods.
• Accurate lattice formulations are needed to capture phenomena like topological insulator surface states.

Purpose of the Study:

• To develop a robust finite-difference lattice scheme for the (2+1)-dimensional Dirac equation.
• To incorporate general electromagnetic textures and magnetization effects into the lattice model.
• To ensure the scheme preserves fundamental physical properties like energy dispersion and gauge invariance.

Main Methods:

• A single-cone finite-difference lattice scheme is developed using a staggered grid.
• A Peierls-Schwinger substitution is employed to introduce electromagnetic potentials.
• Conservation laws and stability are analyzed by comparing with a zero-potential scheme.
• Magnetization terms are incorporated based on consistency with electromagnetic terms.

Main Results:

• The developed lattice scheme successfully models the (2+1)-dimensional Dirac equation with electromagnetic textures.
• Single-cone energy dispersion and gauge invariance are preserved from the continuum to the lattice.
• The scheme demonstrates stability and conservation laws.
• Numerical simulations show single-fermion transport in the presence of in-plane magnetization.

Conclusions:

• The proposed finite-difference lattice scheme provides an accurate and stable method for simulating relativistic fermions in electromagnetic fields.
• This approach is suitable for studying phenomena in materials like topological insulators.
• The scheme lays the groundwork for more complex simulations involving electromagnetic textures and magnetism.