  This work deals with the behavior of fermions in the background of kinklike
structures in the two-dimensional spacetime. The kinklike structures appear
from bosonic scalar field models that engender distinct profiles and interact
with the fermion fields via the standard Yukawa coupling. We first consider two
models that engender parity symmetry, one leading to the exclusion of fermion
bound states, and the other to the inclusion of bound states, when the
parameter that controls the bosonic structure varies from zero to unity. We
then go on and investigate another model where the kinklike solution explicitly
breaks parity symmetry, leading to fermion bound states that are spatially
asymmetric.
