Hessian characterization of a vortex in a maze

TL;DR

This paper introduces the Hessian character as a new method to analyze vortex pinning landscapes, providing insights into stability regions across different potential types, including disordered landscapes.

Contribution

The paper presents the Hessian stability map and character as novel tools for characterizing vortex pinning landscapes, extending analysis to various potential shapes and disordered systems.

Findings

The stable area fraction varies with potential type, with a universal value of approximately 21% for disordered landscapes.

Hessian character effectively distinguishes different pinning potential geometries.

The approach offers a new perspective on vortex dynamics in complex energy landscapes.

Abstract

Recent advances in vortex imaging allow for tracing the position of individual vortices with high resolution. Pushing an isolated vortex through the sample with the help of a controlled $dc$ transport current and measuring its local $ac$ response, the pinning energy landscape could be reconstructed along the vortex trajectory [$\text{L. Embon } et\ al.$, $\text{Scientific Reports}$ $\mathbf{5}$, $7598$ $(2015)$]. This setup with linear tilts of the potential landscape reminds about the dexterity game where a ball is balanced through a maze. The controlled motion of objects through such tilted energy landscapes is fundamentally limited to those areas of the landscape developing local minima under appropriate tilt. We introduce the Hessian stability map and the Hessian character of a pinning landscape as new quantities to characterize a pinning landscape. We determine the Hessian character, the area fraction admitting stable vortex positions, for various types of pinning potentials: assemblies of cut parabolas, Lorentzian- and Gaussian-shaped traps, as well as a Gaussian random disordered energy landscape, with the latter providing a universal result of $(3-\sqrt{3})/6 \approx 21\%$ of stable area. Furthermore, we discuss various aspects of the vortex-in-a-maze experiment.