Coupled Vortex Flux Tube Modes and Pinning Induced Relaxation in Neutron Star Interior

Neutron star interiors provide an extreme condensed matter environment where neutron superfluid vortices coexist and interact with proton superconducting flux tubes. We introduce a nonlinear, frequency-dependent coupling law between neutron vortices and proton flux tubes and demonstrate its consequences for magnetic-field decay and glitch recovery using numerical mode calculations and three-component relaxation modeling. We develop a phenomenological framework to examine the coupled dynamics of vortices and flux tubes, focusing on three central ingredients: pinning interactions, mutual friction, and collective oscillatory modes. Our model incorporates variable pinning strengths and lattice geometries to assess how micro-physical parameters shape large-scale relaxation processes. We find that strong pinning produces metastable configurations with extended relaxation times, while weaker pinning enables creep-like motion and enhanced dissipation. Crucially, the coupled system supports slow collective oscillation modes whose frequencies and damping rates depend sensitively on defect density and coupling strength. These modes provide a natural explanation for the protracted relaxation observed in some glitch recoveries and may imprint signatures on timing noise and quasi-periodic variability in magnetars. Our results highlight universal aspects of interacting vortex-flux-tube systems by connecting microscopic defect physics with macroscopic relaxation dynamics, this work offers new predictions for neutron star evolution and motivates deeper links between condensed matter theory and astrophysical observation.