Supercurrent coupling in the Faddeev–Skyrme model

Speight, J.M.

doi:10.1016/j.geomphys.2009.12.007

Cited by 15 publications

(17 citation statements)

References 19 publications

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“…Again, these solitons disappear as the true TCGL theory is approached. It also complements the exact results of [16] where it was shown analytically that supercurrent coupling destabilizes the Q = 1 Hopf soliton on physical space S 3 . In the absence of evidence to the contrary, one should wield Occam's razor and conclude that, in all likelihood, the basic TCGL theory does not possess knot solitons.…”

Section: Introductionsupporting

confidence: 80%

“…The failure of the direct approach to find knot solitons is not surprising given the results of [16]. Even if one imposes that ρ is non-vanishing, so that Q is well-defined, in every degree class the infimum of E(ψ 1 , ψ 2 , A) is 0 because, for suitably chosen A, any spatially localized configuration ψ 1 , ψ 2 is unstable against Derrick scaling.…”

Section: Introductionmentioning

confidence: 99%

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Supercurrent coupling destabilizes knot solitons

Jäykkä

Speight

2011

Phys. Rev. D

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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

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Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Supercurrent coupling destabilizes knot solitons

Jäykkä

Speight

2011

Phys. Rev. D

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“…Finally, for the models (43) (the DDI model) and (44), the condition (45) is sufficient, so both models have the inverse stereographic projection as a solution saturating the bound. For the DDI model, this solution was already found in [38].…”

Section: Saturating the Boundsmentioning

confidence: 99%

Topological energy bounds in generalized Skyrme models

Adam

Wereszczyński

2014

Phys. Rev. D

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The Skyrme model has a natural generalization amenable to a standard hamiltonian treatment, consisting of the standard sigma model and the Skyrme terms, a potential, and a certain term sextic in first derivatives. Here we demonstrate that, in this theory, each pair of terms in the static energy functional which may support topological solitons according to the Derrick criterion (i.e., each pair of terms with opposite Derrick scaling) separately posesses a topological energy bound. As a consequence, there exists a four-parameter family of topological bounds for the full generalized Skyrme model. The optimal bounds, i.e., the optimal values of the parameters, depend both on the form of the potential and on the relative strength of the different terms. It also follows that various submodels of the generalized Skyrme model have one-parameter families of topological energy bounds. We also consider the case of topological bounds for the generalized Skyrme model on a compact base space as well as generalizations to higher dimensions.

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“…Let us recall that for a map ϕ defined on a 3-manifold (M 3 , g) and taking values in a surface (N 2 , J, h) with fundamental 2-form Ω(X, Y ) = h(JX, Y ), the symplectic Dirichlet energy is defined [41,42] as F (ϕ) = 1 2 ϕ * Ω 2 L 2 . Assuming M is a compact, connected, oriented 3-manifold with H 2 (M, Z) = 0, the following topological lower bound can be established [40,42]:…”

Section: Remarkmentioning

confidence: 99%

Energy Minimizing Beltrami Fields on Sasakian 3-Manifolds

Peralta‐Salas

Slobodeanu

2019

International Mathematics Research Notices

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We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magnetic relaxation and the hydrodynamic stability of steady Euler flows. We characterize the Sasakian manifolds where the Reeb field is a minimizer in terms of the first positive eigenvalue of the curl operator and show that for a > a 0 (a constant that depends on the Sasakian structure) the Reeb field of the D-homothetic deformation of the manifold with constant a (which is still Sasakian) is an unstable critical point of the energy, and hence not even a local minimizer. We also provide some examples of Sasakian manifolds where the Reeb field is a minimizer, highlighting the case of the weighted 3-spheres, on which another minimization problem (for the quartic Skyrme-Faddeev energy) is shown to admit exact solutions.

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Supercurrent coupling in the Faddeev–Skyrme model

Cited by 15 publications

References 19 publications

Supercurrent coupling destabilizes knot solitons

Supercurrent coupling destabilizes knot solitons

Topological energy bounds in generalized Skyrme models

Energy Minimizing Beltrami Fields on Sasakian 3-Manifolds

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