Strong pinning transition with arbitrary defect potentials

Dissipation-free current transport in type II superconductors requires vortices to be pinned by defects in the underlying material. The pinning capacity of a defect is quantified by the Labusch parameter $\kappa \sim f_p/\xi\bar{C}$, measuring the pinning force $f_p$ relative to the elasticity $\bar{C}$ of the vortex lattice, with $\xi$ denoting the coherence length (or vortex core size) of the superconductor. The critical value $\kappa = 1$ separates weak from strong pinning, with a strong defect at $\kappa > 1$ able to pin a vortex on its own. So far, this weak-to-strong pinning transition has been studied for isotropic defect potentials, resulting in a critical exponent $\mu = 2$ for the onset of the strong pinning force density $F_\mathrm{pin} \sim n_p f_p (\xi/a_0)^2(\kappa-1)^\mu$, with $n_ p$ denoting the density of defects and $a_0$ the intervortex distance. The behavior changes dramatically when studying anisotropic defects with no special symmetries: the strong pinning then originates out of isolated points with length scales growing as $\xi (\kappa - 1)^{1/2}$, resulting in a different force exponent $\mu = 5/2$. Our analysis of the strong pinning onset for arbitrary defect potentials $e_p(\mathbf{R})$, with $\mathbf{R}$ a planar coordinate, makes heavy use of the Hessian matrix describing its curvature and leads us to interesting geometrical structures. Both, onset and merger points are defined by local differential properties of the Hessian's determinant $D(\mathbf{R})$, specifically, its minima and saddle points. Extending our analysis to the case of a random two-dimensional pinning landscape, we discuss the topological properties of unstable and bistable regions as expressed through the Euler characteristic, with the latter related to the local differential properties of $D(\mathbf{R})$ through Morse theory.