  We study the influence of atomic interactions on quantum simulations in
momentum-space lattices (MSLs), where driven atomic transitions between
discrete momentum states mimic transport between sites of a synthetic lattice.
Low energy atomic collisions, which are short ranged in real space, relate to
nearly infinite-ranged interactions in momentum space. However, the
distinguishability of the discrete momentum states coupled in MSLs gives rise
to an added exchange energy between condensate atoms in different momentum
orders, relating to an effectively attractive, finite-ranged interaction in
momentum space. We explore the types of phenomena that can result from this
interaction, including the formation of chiral self-bound states in topological
MSLs. We also discuss the prospects for creating squeezed states in
momentum-space double wells.
