Exact solutions of a fermion-soliton system in two dimensions

We investigate a coupled system of a Dirac particle and a pseudoscalar field in the form of a soliton in (1 + 1) dimensions and find some of its exact solutions numerically. We solve the coupled set of equations self-consistently and non-perturbatively by the use of a numerical method and obtain the bound states of the fermion and the shape of the soliton. That is the shape of the static soliton in this problem is not prescribed and is determined by the equations themselves. This work goes beyond the perturbation theory in which the back-reaction of the fermion on soliton is its first order correction. We compare our results to those of an exactly solvable model in which the soliton is prescribed. We show that, as expected, the total energy of our system is lower than the prescribed one. We also compute non-perturbatively the vacuum polarization of the fermion induced by the presence of the soliton and display the results. Moreover, we compute the soliton mass as a function of the boson and fermion masses and find that the results are consistent with Skyrme's phenomenological conjecture. Finally, we show that for fixed values of the parameters, the shape of the soliton obtained from our exact solutions depends slightly on the fermionic state to which it is coupled. However, the exact shape of the soliton is always very close to the isolated kink.