We investigate the groundstate properties and excitations of Rydbergdressed bosons in both three and two dimensions, using the hypernettedchain EulerLagrange approximation, which accounts for correlations and thus goes beyond the mean field approximation. The shortrange behavior of the pair distribution function signals the instability of the homogeneous system towards the formation of droplet crystals at strong couplings and large softcore radius. This tendency to spatial density modulation coexists with offdiagonal longrange order. The contribution of the correlation energy to the groundstate energy is significant at large coupling strengths and intermediate values of the softcore radius while for a larger softcore radius the groundstate energy is dominated by the meanfield (Hartree) energy. We have also performed path integral Monte Carlo simulations to verify the performance of our hypernettedchain EulerLagrange results in three dimensions. In the homogeneous phase, the two approaches are in very good agreement. Moreover, Monte Carlo simulations predict a firstorder quantum phase transition from a homogeneous superfluid phase to the quantum droplet phase with facecentered cubic symmetry for Rydbergdressed bosons in three dimensions.
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