SU(3) knot solitons: Hopfions in the F2 Skyrme–Faddeev–Niemi model

We discuss the existence of knot solitons (Hopfions) in a Skyrme–Faddeev–Niemi-type model on the target space $SU\left(3\right)/U{\left(1\right)}^2$, which can be viewed as an effective theory of both the $SU\left(3\right)$ Yang–Mills theory and the $SU\left(3\right)$ anti-ferromagnetic Heisenberg model. We derive the knot solitons with two different types of ansatz: the first is a trivial embedding configuration of $SU\left(2\right)$ into $SU\left(3\right)$, and the second is a non-embedding configuration that can be generated through the Bäcklund transformation. The resulting Euler–Lagrange equations for both ansatz reduce exactly to those of the $C{P}^1$ Skyrme–Faddeev–Niemi model. We also examine some quantum aspects of the solutions using the collective coordinate zero-mode quantization method.