Gaugeless Formulation and Finite-Difference Modeling of Superconducting Thin Films (Part II)

Rather than using gauge-dependent quantities such as the vector potential and superconducting phase, we model a thin-film London superconductor entirely in terms of a gauge-independent scalar magnetization field Μ(x,y) obeying a Poisson-like equation. All observable quantities, including current and magnetic field, can be directly obtained from Μ. This formulation provides a versatile framework for treating inhomogeneous superfluid density, complex boundaries, and holes and can be applied in both analytical and numerical calculations. We show that, to treat Pearl vortices correctly within this framework, the vortex free energy cost scales as F_v∝ln(R_e/ξ_e), where R_e is a long-range cutoff (determined by geometry or screening) and ξ_e≈0.966ξ represents the effective vortex-core radius. In finite-difference simulations, the grid spacing should be a_FD≈4.87ξ, ensuring that the discrete vortex energy matches its continuum counterpart. This establishes a universal correspondence between discrete and continuum models, enabling accurate finite-difference simulations of vortex free energy and magnetization in superconducting films.