Gauged Hopfions

Abstract

We discuss the U (1) gauged version of the 3+1 dimensional Faddeev-Skyrme model supplemented by the Maxwell term. We show that there exist axially symmetric static solutions coupled to the non-integer toroidal flux of magnetic field, which revert to the usual Hopfions Am,n of lower degrees Q = mn in the limit of the gauge coupling constant vanishing. The masses of the static gauged Hopfions are found to be less than the corresponding masses of the usual ungauged solitons A1,1 and A2,1 respectively, they become lighter as gauge coupling increases. The dependence of the solutions on the gauge coupling is investigated. We find that in the strong coupling regime the gauged Hopfion carries two magnetic fluxes, which are quantized in units of 2π, carrying n and m quanta respectively. The first flux encircles the position curve and the second one is directed along the symmetry axis. Effective quantization of the field in the gauge sector may allow us to reconsider the usual arguments concerning the lower topological bound in the Faddeev-Skyrme-Maxwell model.

Magnetothermodynamics of BPS baby skyrmions

“…Let us remark that some first numerical results for the Skyrme model minimally coupled to the electromagnetic field (but without the anomalous or Wess-Zumino-Witten term contribution) have been found in [8,9]. Further, very recently some knotted soliton solutions have been obtained for the S 2 restriction of the minimally gauged Skyrme model i.e., the gauged Faddeev-Skyrme-Niemi model, however within the toroidal ansatz which limits the solutions to the charge Q = 1, 2 sectors [10]. As has been mentioned already, a precise derivation of the Skyrme model (or in fact any effective low energy model) from QCD is one of the most urgent, however still unsolved, tasks in modern theoretical physics.…”

Magnetic Hopfions in the Faddeev-Skyrme-Maxwell model

“…The smooth function Aðϕ 3 Þ must satisfy the restrictions AðAE1Þ ¼ 0 and Að0Þ ≃ 1, and it agrees with the parametrization used previously to construct axially symmetric solutions of the gauged Faddeev-Skyrme model [33]. Thus, we can take…”

“…Note that under a spatial rescaling x → λx, this term in the action scales as λ −1 ; i.e., it has the same scaling properties as the quartic in the derivative Skyrme term. The flat metric is g μν ¼ diagð1; −1; −1; −1Þ, and the coupling of the scalar triplet to the gauge field is given by the covariant derivative [27,28,33,34] …”

“…The consideration above explains the effective quantization of the magnetic fluxes of the gauged axially symmetric Hopfions of degrees Q ¼ 1, 2 in the strong coupling limit [33]. It was observed that the configuration is associated with two magnetic fluxes, one of which represents a circular vortex, and the second one is orthogonal to the position curve [37].…”

“…However, it is known that the Hopfions of lowest degrees Q ¼ 1, 2 are axially symmetric [30][31][32]; thus, in Ref. [33], the consideration was restricted to the case of the static axially symmetric gauged unlinked Hopfions A 1;1 and A 2;1 . An assumption of axial symmetry simplifies the consideration significantly, since the problem then can be reduced to the numerical solution of the system of coupled ordinary differential equations.…”