  We consider Majorana lattices with two-site interactions consisting of a
general function of the fermion bilinear. The models are exactly solvable in
the limit of a large number of on-site fermions. The four-site chain exhibits a
quantum phase transition controlled by the hopping parameters and manifests
itself in a discontinuous entanglement entropy, obtained by constraining the
one-sided modular Hamiltonian. Inspired by recent work within the AdS/CFT
correspondence, we identify transitions between types of von Neumann operator
algebras throughout the phase diagram. We find transitions of the form
II$_1\leftrightarrow\,$III$\,\,\leftrightarrow\,\,$I$_\infty$ that reduce to
II$_1\leftrightarrow\,\,$I$_\infty$ in the strongly interacting limit, where
they connect non-factorized and factorized ground states. Our results provide
novel realizations of such transitions in a controlled many-body model.
