Supercurrent coupling destabilizes knot solitons

Abstract

In an influential paper of 2002, Babaev, Faddeev and Niemi conjectured that two-component Ginzburg-Landau (TCGL) theory in three dimensions should support knot solitons, where the projective equivalence class of the pair of complex condensate fields [psi_1,psi_2]:R^3 -> CP^1 has non-zero Hopf degree. The conjecture was motivated by a certain truncation of the TCGL model which reduced it to the Faddeev-Skyrme model, long known to support knot solitons. Physically, the truncation amounts to ignoring the coupling between [psi_1,psi_2] and the supercurrent of the condensates. The current paper presents a direct test of the validity of this truncation by numerically tracking the knot solitons as the supercurrent coupling is turned back on. It is found that the knot solitons shrink and disappear as the true TCGL model is reached. This undermines the reasoning underlying the conjecture and, when combined with other negative numerical studies, suggests the conjecture, in its original form, is very unlikely to be true.Comment: replaced with the published version, with added PACS numbes and removed a footnote; 12 page

Isospinning hopfions

“…The numerical algorithm employed was similar to that used in [9]. A simple forward differencing scheme was used on a cubic lattice with a lattice spacing ∆x = 0.1 on a grid typically containing (240) 3 latice points.…”

Skyrmions with low binding energies

“…In a finite box, large enough that boundary pressure is negligible, D(ε) should be small and positive [14]. If D(ε) becomes negative, or large and positive, this indicates that our numerical solution is not reliable, and we discard it.…”

“…As well as monitoring the accuracy of B, we also keep track of the Derrick scaling constraint, as in [14]. That is, any critical point of E ε should, by the Derrick scaling argument [15], satisfy the virial constraint…”

Vortex states in mesoscopic three-band superconductors

“…Other parts of the program and support routines are an adaption of previous work done in Ref. 48 . More details on the discretisation can be found in Refs.…”