Potential energy landscape networks of confined vortex matter

Advances in nanoscale patterning of superconducting films [Y. L. Wang et al ., Nat. Nanotechnol. 13 , 560–565 (2018)] and the availability of submicron-sized superconducting single crystals with controlled geometry allow for the investigation of superconducting vortex matter under extreme confinement [A. K. Geim et al ., Nature 390 , 259–262 (1997)]. It has been observed in these systems that the interplay between sample geometry and vortex number results in the emergence of so-called “magic number” configurations which show unusually high stability against thermal fluctuations. However, a detailed, comprehensive theoretical study of the ground state and metastable states of confined vortex matter has not yet been developed. Here we show that the intricate properties of a system of vortices under extreme confinement can be characterized in detail by mapping the potential energy landscape of the system onto a network [J. P. K. Doye, Phys. Rev. Lett. 88 , 238701 (2002)] with nodes corresponding to local potential energy minima and edges to transition states between local minima. We apply these techniques to systems featuring repelling particles/vortices confined in small containers of various geometric shapes. We demonstrate that the ground state almost always belongs to a subset of system states within the deepest network core. We find that the emergence of magic number configurations in the ground state results in a drastic decrease in the complexity of the entire network. We use large-scale molecular dynamics simulations to study the size dependence of melting in different configurations. We find that as confinement effects increase, magic number configurations transition to states with lower symmetry.