  We show that atoms in tilted optical superlattices provide a platform for
exploring coupled spin chains of forms that are not present in other systems.
In particular, using a period-2 superlattice in 1D, we show that coupled Ising
spin chains with XZ and ZZ spin coupling terms can be engineered. We use
optimized tensor network techniques to explore the criticality and
non-equilibrium dynamics in these models, finding a tricritical Ising point in
regimes that are accessible in current experiments. These setups are ideal for
studying low-entropy physics, as initial entropy is "frozen-out" in realizing
the spin models, and provide an example of the complex critical behaviour that
can arise from interaction-projected models.
