  We report on the ground state phase diagram of interacting Rydberg atoms in
the unfrustrated square lattice array. Using new tensor network algorithms, we
scale to large systems in two dimensions while including all long-range
interactions, revealing the phases in the bulk and their analogs in accessible
finite arrays. We find a greatly altered phase diagram from earlier numerical
and experimental studies, and in particular, we uncover an emergent entangled
quantum nematic phase that appears in the absence of frustration. Broadly our
results yield a conceptual guide for future experiments, while our techniques
provide a blueprint for converging numerical studies in other lattices.
