A superconducting vortex resembles a whirlpool in a quantum fluid. When a magnetic field pierces a superconductor, it gathers into thin filaments, vortices, around which persistent currents circulate. In the core of such a vortex, superconductivity is suppressed, and special electronic states, discovered back in the sixties, accumulate there. The authors of the new work took a simple model, a step-like gap profile, and showed that the normalization of these states is almost independent of angular momentum. This allowed them to collapse a cumbersome sum into a single compact formula.
Inside the vortex, where the superconducting order parameter twists, one component drops out of the full set of cylindrical waves. As a result, the charge density vanishes right at the vortex axis and begins to oscillate with a wave vector 2k_F (twice the electron momentum at the Fermi surface). Outside, these oscillations decay as 1/r over the coherence length, leaving behind a faint ripple — much like the rings from a thrown stone that gradually fade away.
Why is this alternating ripple so noticeable? It's all about the inefficiency of screening at large momentum transfers. When electrons try to screen charge on scales of the interatomic distance, they don't do a good job. As a result, the modulation amplitude decreases only by a factor of one-half to three-quarters — for any metal. This is not a fine-tuning but a direct consequence of conservation laws and quantum mechanics. The vortex doesn't just expel charge: it transforms it into a standing wave that follows the laws of interference.
The wave vector of oscillations, 2k_F, is roughly the inverse of the interatomic distance: these are the shortest waves that can exist in an electron liquid.