Scattering of knotted vortices (Hopfions) in the Faddeev–Skyrme model

Abstract

Several materials, such as ferromagnets, spinor Bose-Einstein condensates and some topological insulators, are now believed to support knotted structures. One of the most successful base-models having stable knots is the Faddeev-Skyrme model and it is expected to be contained in some of these experimentally relevant models. The taxonomy of knotted topological solitons (Hopfions) of this model is known. In this paper, we describe some aspects of the dynamics of Hopfions and show that they indeed behave like particles: during scattering the Hopf charge is conserved and bound states are formed when the dynamics allows it. We have also investigated the dynamical stability of a pair of Hopfions in stacked or side-by-side configurations, whose theoretical stability has recently been discussed by Ward.

Three-dimensional crystals of adaptive knots

“…Hopf indices of a heliknoton can also be evaluated numerically by integrating in the configuration space ℝaccording to Eq. (2) in the main text (40)(41)(42)(43) (44,45). In addition to the chiral nematics, orthorhombic biaxial nematics also have a similar target space ] -/E´.…”

Coherence simplices

“…Physically, we can view B(x, y) as a continuous background deformation, upon which sits the screw-type phase dislocation given by the term proportional to m. The amplitude | f (x 0 , y 0 )| must vanish at the location of the phase singularity to ensure single valuedness of the complex function. In three-dimensional (3D) systems such zero-amplitude phase singularities are nodal lines or line vortices that may close on themselves, forming loops or vortex rings [1,5,9,[34][35][36] or even vortex knots [11][12][13][14][15]. In superfluids, the gradient of the phase function ∇ arg[ψ(r)] of the complex order parameter field ψ(r) describing the system can be associated with the velocity field v s (r) of the superfluid.…”

“…, x 2 p ) with an associated nonzero coherence circulation κ p . Note that the propagator, (15), being a correlation function itself, may also be vortical. Figure 3(b) also illustrates the conservation of the coherence circulation.…”

“…Such coherence vortex pairs appear as a manifestation of coherence vortex space-time loops, such as N 2 , along which the coherence vanishes. Coherence-vortex nodal-line knots (not shown) are also possible; such knotted vortices have been studied both theoretically and experimentally, for the case of coherent fields [11][12][13][14][15].…”

“…The study of such optical vortices has now blossomed into the area of singular optics, with several key reviews including those of Berry [8], Nye [9], Soskin and Vasnetsov [10] and Dennis et al [6]. The nodal lines that thread vortex cores can form closed loops or extend to infinity [5]; they can terminate at points where the potential is discontinuous; and they can even form knots [11][12][13][14][15]. Now, partially coherent optical fields or mixed-state wavefunctions are often studied via correlation functions such as the mutual intensity or the cross-spectral density (for partially coherent optical fields) [16,17] and the density matrix (for mixed-state wavefunctions) [18].…”