  The magnetization process of a superconductor is determined by the potential
barrier for vortex nucleation and escape. In multicomponent superconductors,
fractional vortices with a winding in the phase of only one of the components
can be stable topological solitons that carry a fraction of the flux quantum.
While the formation of such objects in the bulk costs logarithmically or
linearly divergent energy, these objects were shown to be stable near samples'
boundaries in the two-component London model. Therefore, the conventional
Bean-Livingston picture of magnetic flux entry does not apply to these
superconductors, since the entry process can involve fractionalization of a
vortex. In this paper, we address the nonlinear problem of determining the
potential barrier for fluxoid penetration in a multicomponent superconductor,
including the effects of various intercomponent couplings, by using the
recently developed gauged string method. The method allows numerically exact
(i.e., convergent) calculation of a sphaleron configuration in a gauge theory
and thus the height of the nucleation barrier. We show how the fractionalized
nucleation processes result in multiple sphalerons and intermediate states due
to the complex shape of the energy landscape of multicomponent superconductors.
