How a Neutron Vortex Gathers a Bouquet of Magnetic Filaments ✛

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Deep inside a neutron star, where matter is compressed to the density of an atomic nucleus, neutrons form a superfluid and protons form a superconductor. A neutron vortex is a quantized vortex in the superfluid, akin to a miniature whirlpool that cannot stop spinning. A proton flux tube is a quantized thread of magnetic flux, a thin tube through which the magnetic field penetrates the superconductor. Previously, these defects were studied separately, but now scientists have for the first time placed them in the same computational cell and let them interact.

The authors constructed a two-dimensional Ginzburg–Landau model, a mathematical description of two coupled condensates, the neutron and proton ones. They minimized the free energy numerically, fixing the total number of vortices and flux tubes via quasiperiodic boundary conditions. Calculations on grids with a spacing of about 5 points per unit length were performed using the Adam optimizer in PyTorch.

The results showed that with attractive local couplings (parameters α<0 or g<0), the overlap of a vortex and a flux tube is energetically favorable. The binding energy per unit length for typical parameters was on the order of 0.13 MeV/fm, a tiny quantity, but on stellar scales it accumulates. In a configuration with one vortex and nine flux tubes, a 'bouquet' formed: between 6 and 9 flux tubes gathered around the vortex. In a dense bouquet, individual flux tubes are indistinguishable by condensate amplitude or magnetic field, but their nine phase singularities remain topologically independent, like nine stems in a single bouquet that can still be counted.

These structures could influence dissipative processes in neutron stars, such as mutual friction between the superfluid and charged components. Understanding how vortices latch onto flux tubes is important for explaining glitches, sudden spin-ups in pulsars. Previously, it was unclear whether vortices and flux tubes could form stable bound states without additional phase entrainment. Now it has been shown that they can, and the binding energy is known quantitatively. In the future, the scientists plan to include phase entrainment and move to three-dimensional simulations to test the stability of bouquets in volume and connect them with observations.