  The importance of models with an exact solution for the study of materials
with non-trivial topological properties has been extensively demonstrated.
Among these, the Kitaev model of a one-dimensional $p$-wave superconductor
plays a guiding role in the search for Majorana modes in condensed matter
systems. Also, the $sp$ chain, with an anti-symmetric mixing among the $s$ and
$p$ bands provides a paradigmatic example of a topological insulator with well
understood properties. There is an intimate relation between these two models
and in particular their topological quantum phase transitions share the same
universality class. Here we consider a two-band $sp$ model of spinless fermions
with an attractive (inter-band) interaction. Both the interaction and
hybridization between the $s$ and $p$ fermions are anti-symmetric. The zero
temperature phase diagram of the model presents a variety of phases including a
Weyl superconductor, topological insulator and trivial phases. The quantum
phase transitions between these phases can be either continuous or
discontinuous. We show that the transition from the topological superconducting
phase to the trivial one has critical exponents different from those of an
equivalent transition in Kitaev's model.
