Structural Permeability of Quantum Vortices in Bose-Einstein Condensates and Screening of Tkachenko Waves

This paper presents a mathematical formalism for the generation of emergent electrodynamic properties from the internal structure of quantum vortices in Bose-Einstein condensates (BECs) and the screening of transverse Tkachenko waves. From the local energy balance of the condensate, the physical boundary of the vortex core is determined as R = σ, where the kinetic energy of the flow per particle reaches the background chemical potential. Analytical integration of the Gross-Pitaevskii density profile yields the structural redistribution of the vortex mass deficit between the core (47.30%) and the outer halo (52.70%). By analyzing the local displacement of the vortex, wherein the Magnus force is balanced by the Ginzburg-Landau elastic restorative force, we derive the screening of the effective mass density of the Bose-condensate and the emergent permeability coefficient κ_BEC. Sensitivity analysis conducted for various variational coefficients σ_core = γξ demonstrates that the microscopic permeability of the vortex core lies within the interval 4.20 ≤ κ_individual ≤ 7.01, while the macroscopic permeability lies within 1.25 ≤ κ_BEC ≤ 1.48. Utilizing coarse-grained wave equations and the Lagrangian formalism, we derive the frequency screening formula for Tkachenko waves. The theoretical results are compared with JILA experimental data and Simula's numerical solutions, demonstrating high quantitative agreement.