  Mixed order phase transitions (MOT), which display discontinuous order
parameter and diverging correlation length, appear in several seemingly
unrelated settings ranging from equilibrium models with long-range interactions
to models far from thermal equilibrium. In a recent paper [1] an exactly
soluble spin model with long-range interactions that exhibits MOT was
introduced and analyzed both by a grand canonical calculation and a
renormalization group analysis. The model was shown to lay a bridge between two
classes of one dimensional models exhibiting MOT, namely between spin models
with inverse distance square interactions and surface depinning models. In this
paper we elaborate on the calculations done in [1]. We also analyze the model
in the canonical ensemble, which yields a better insight into the mechanism of
MOT. In addition, we generalize the model to include Potts and general Ising
spins, and also consider a broader class of interactions which decay with
distance with a power law different from 2.
