Two-dimensional Dirac matter in the semiclassical regime

TL;DR

This paper derives a semiclassical framework for 2D Dirac materials in curved space-time, linking geometry to particle dynamics and demonstrating the effects of curvature on trajectories.

Contribution

It introduces a method to analyze 2D Dirac matter in curved space using isothermal coordinates and derives the relativistic Lorentz force with curvature-induced fictitious forces.

Findings

Relativistic Lorentz force emerges in semiclassical approximation

Effective refractive index linked to space-time metric

Numerical trajectories show curvature effects on particle motion

Abstract

The semiclassical regime of 2D static Dirac matter is obtained from the Dirac equation in curved space-time. To simplify the formulation, the Cartesian space-time geometry parametrization is transformed to isothermal coordinates using quasi-conformal transformations. Using this framework, it is demonstrated that to first order, the semiclassical approximation yields the relativistic Lorentz force equation with additional fictitious forces related to the space curvature and to the mass gradient. An effective graded index of refraction is defined and is directly linked to the metric component in isothermal coordinates. Semiclassical trajectories are evaluated in a simple example by solving the equation of motion and the Beltrami equation numerically.