Static Hopfions in the extended Skyrme-Faddeev model

TL;DR

This paper constructs static soliton solutions with non-zero Hopf charges in an extended Skyrme-Faddeev model, revealing unique behaviors of energies and sizes near special coupling values and exploring integrable sectors with conserved currents.

Contribution

It introduces new static Hopfion solutions in an extended Skyrme-Faddeev model and analyzes their properties, including energy bounds and integrable sectors, advancing understanding of topological solitons in gauge theories.

Findings

Solutions with Hopf charge up to four were numerically constructed.

Energy and size tend to zero near a specific coupling constant value.

An integrable sector with infinite conserved currents was identified.

Abstract

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is an extension of the Skyrme-Faddeev model by the addition of a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled non-linear partial differential equations in two variables by a successive over-relaxation (SOR) method. We construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms. Their energies and sizes tend to zero as that combination approaches a particular special value. We calculate the equivalent of the Vakulenko and Kapitanskii energy bound for the theory and find that it vanishes at that same special value of the coupling constants. In addition, the model presents an integrable sector with an infinite number of local conserved currents which apparently are not related to symmetries of the action. In the intersection of those two special sectors the theory possesses exact vortex solutions (static and time dependent) which were constructed in a previous paper by one of the authors. It is believed that such model describes some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and our results may be important in identifying important structures in that strong coupling regime.