\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We investigate the ground-state properties and excitations of Rydberg-dressed bosons in both three and two dimensions, using the hypernetted-chain Euler-Lagrange approximation, which accounts for correlations and thus goes beyond the mean field approximation. The short-range behavior of the pair distribution function signals the instability of the homogeneous system towards the formation of droplet crystals at strong couplings and large soft-core radius.
This tendency to spatial density modulation coexists with off-diagonal long-range order.
The contribution of the correlation energy to the ground-state energy is significant at large coupling strengths and intermediate values of the soft-core radius while for a larger soft-core radius the ground-state energy is dominated by the mean-field (Hartree) energy.
We have also performed path integral Monte Carlo simulations to verify the performance of our hypernetted-chain Euler-Lagrange results in three dimensions.
In the homogeneous phase, the two approaches are in very good agreement. Moreover, Monte Carlo simulations predict a first-order quantum phase transition from a homogeneous superfluid phase to the quantum droplet phase with face-centered cubic symmetry for Rydberg-dressed bosons in three dimensions.
\end{document}