  This work deals with models described by a single real scalar field in
two-dimensional spacetime. The aim is to propose potentials that support
massless minima and investigate the presence of kinklike structures that
engender polynomial tails. The results unveil the presence of families of
asymmetric solutions with energy density and linear stability that behave
adequately, enhancing the importance of the analytical study. We stress that
the novel topological structures which we find in this work engender long range
interactions that are of current interest to statistical mechanics, dipolar
quantum gases and the study of quantum information with Rydberg atoms.
