  We study two many-body systems of bosons interacting via an infinite
three-body contact repulsion in a lattice: a pairs quasi-condensate induced by
correlated hopping and the discrete version of the Pfaffian wavefunction. We
propose to experimentally realise systems characterized by such interaction by
means of a proper spin-1 lattice Hamiltonian: spin degrees of freedom are
locally mapped into occupation numbers of emerging bosons, in a fashion similar
to spin-1/2 and hardcore bosons. Such a system can be realized with ultracold
spin-1 atoms in a Mott Insulator with filling-factor one. The high versatility
of these setups allows us to engineer spin-hopping operators breaking the SU(2)
symmetry, as needed to approximate interesting bosonic Hamiltonians with
three-body hardcore constraint. For this purpose we combine bichromatic
spin-independent superlattices and Raman transitions to induce a different
hopping rate for each spin orientation. Finally, we illustrate how our setup
could be used to experimentally realize the first setup, i.e. the transition to
a pairs quasi-condensed phase of the emerging bosons. We also report on a route
towards the realization of a discrete bosonic Pfaffian wavefunction and list
some open problems to reach this goal.
