Minimal model for vortex nucleation and reversal in spherical magnetic nanoparticles

Magnetic nanoparticles beyond the single-domain limit often develop vortex-like magnetization textures arising from the competition between exchange and magnetostatic energies. While such states are routinely studied using micromagnetic simulations, transparent analytical descriptions of vortex-mediated hysteresis and nucleation remain scarce. Here, we develop a semi-analytical minimal framework for vortex states in spherical magnetic nanoparticles. Guided by micromagnetic simulations, we introduce a parametrized vortex magnetization Ansatz based on hyperbolic functions that continuously interpolates between uniform and vortex states. In this way, we achieve a complexity reduction leading to a minimal Hamiltonian, which enables the efficient computation of magnetization curves and provides insight into vortex-mediated magnetization reversal. As an application, we derive analytical estimates for the critical vortex nucleation radius and field, recovering the functional form of Brown's classic result and extending it within a variational framework.