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The combination of optical tweezer arrays with strong interactions -- via
dipole-exchange of molecules and van-der-Waals interactions of Rydberg atoms --
has opened the door for the exploration of a wide variety of quantum spin
models. A next significant step will be the combination of such settings with
mobile dopants: This will enable to simulate the physics believed to underlie
many strongly correlated quantum materials. Here we propose an experimental
scheme to realize bosonic $t$-$J$ models via encoding the local Hilbert space
in a set of three internal atomic or molecular states. By engineering
antiferromagnetic (AFM) couplings between spins, competition between charge
motion and magnetic order similar to that in high-$T_c$ cuprates can be
realized. Since the bosonic AFM version of the 2D $t$-$J$ model we propose has
not been studied previously, we start by analyzing the case of two dopants --
the simplest instance in which their bosonic statistics plays a role, and
contrast our results to the fermionic case. We perform large-scale density
matrix renormalization group (DMRG) calculations on six-legged cylinders, and
find a strong tendency for bosonic holes to form stripes. This demonstrates
that bosonic, AFM $t$-$J$ models may contain similar physics as the collective
phases in strongly correlated electrons.
