Abstract
Supercurrent transport requires vortices in superconductors to be pinned by material defects. Vortices are trapped in minima of the pinning potential landscape, and one has to determine what fraction of this landscape comes with a positive curvature. Such a curvature analysis is done by studying the Hessian matrix of the landscape. The authors find the stable area fraction to be unexpectedly low (yellow color in the figure). In particular, for Gaussian random landscapes it is $(3\ensuremath{-}\sqrt{3})/6\ensuremath{\approx}21%$. The results offer new hints for optimizing pinning landscapes. Their transdisciplinary application to natural landscapes exhibits a surprising universality.
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