  The Sachdev-Ye-Kitaev (SYK) model is a rare example of a strongly-interacting
system that is analytically tractable. Tractability arises because the model is
largely structureless by design and therefore artificial: while the interaction
is restricted to two-body terms, interaction matrix elements are "randomized"
and therefore the corresponding interaction operator does not commute with the
local density. Unlike conventional density-density-type interactions, the
SYK-interaction is, in this sense, not integrable. We here investigate a
variant of the (complex) SYK model, which restores this integrability. It
features a randomized single-body term and a density-density-type interaction.
We present numerical investigations suggesting that the model exhibits two
integrable phases separated by several intermediate phases including a chaotic
one. The chaotic phase carries several characteristic SYK-signatures including
in the spectral statistics and the frequency scaling of the Green's function
and therefore should be adiabatically connected to the non-Fermi liquid phase
of the original SYK model. Thus, our model Hamiltonian provides a bridge from
the SYK-model towards microscopic realism.
