  We present a toy model for interacting matter and geometry that explores
quantum dynamics in a spin system as a precursor to a quantum theory of
gravity. The model has no a priori geometric properties, instead, locality is
inferred from the more fundamental notion of interaction between the matter
degrees of freedom. The interaction terms are themselves quantum degrees of
freedom so that the structure of interactions and hence the resulting local and
causal structures are dynamical. The system is a Hubbard model where the graph
of the interactions is a set of quantum evolving variables. We show
entanglement between spatial and matter degrees of freedom. We study
numerically the quantum system and analyze its entanglement dynamics. We
analyze the asymptotic behavior of the classical model. Finally, we discuss
analogues of trapped surfaces and gravitational attraction in this simple
model.
