Bound fermion states in the field of a soliton of the nonlinear $O(3)$ $\sigma$ model

The $(2 + 1)$-dimensional nonlinear $O(3)$ $\sigma$ model whose n field is coupled to the fermion field by the Yukawa interaction has been examined. The cases of the isosinglet and isodoublet fermion fields with respect to the internal symmetry group have been considered. It has been shown that bound states of the fermion in the n field of a soliton of the nonlinear $O(3)$ $\sigma$ model exist for some variants of the Yukawa interaction. The absence of zeroth fermion modes in the $\mathbf n$ field of the soliton has been established. The properties of the ground state of the fermion have been numerically studied. In particular, it has been shown that an increase in the spatial size of the soliton results in a decrease in the energy of the ground state. This leads to the instability of the soliton in a certain region of the parameters of the model.