  Several condensed-matter platforms have been proposed recently to realize the
Sachdev-Ye-Kitaev (SYK) model in their low-energy limit. In these proposed
realizations, the characteristic SYK behavior is expected to occur under
certain assumptions about the underlying physical system that (i) render all
bilinear terms small compared to four-fermion interactions and (ii) ensure that
the coupling constants are approximately all-to-all and independent random
variables. In this work we explore, both analytically and numerically, the
family of models that arises when we relax these assumptions in ways motivated
by real physical systems. By relaxing (i) and allowing large bilinear terms, we
obtain a novel, exactly-solvable cousin of the SYK model. It exhibits two
distinct phases separated by a quantum phase transition characterized by a
power-law, $\sim |\omega|^{-1/3}$ scaling of the low-energy spectral density,
despite being a non-interacting model. By relaxing (ii), we obtain close
relatives of the SYK model which exhibit interesting behaviors, including a
chaotic non-Fermi liquid phase with continuously varying fermion scaling
dimension, and a phase transition to a disordered Fermi liquid as a function of
interaction range and disorder length scale.
