  Spontaneous symmetry breaking underlies much of our classification of phases
of matter and their associated transitions. The nature of the underlying
symmetry being broken determines many of the qualitative properties of the
phase; this is illustrated by the case of discrete versus continuous symmetry
breaking. Indeed, in contrast to the discrete case, the breaking of a
continuous symmetry leads to the emergence of gapless Goldstone modes
controlling, for instance, the thermodynamic stability of the ordered phase.
Here, we realize a two-dimensional dipolar XY model -- which exhibits a
continuous spin-rotational symmetry -- utilizing a programmable Rydberg quantum
simulator. We demonstrate the adiabatic preparation of correlated
low-temperature states of both the XY ferromagnet and the XY antiferromagnet.
In the ferromagnetic case, we characterize the presence of long-range XY order,
a feature prohibited in the absence of long-range dipolar interaction. Our
exploration of the many-body physics of XY interactions complements recent
works utilizing the Rydberg-blockade mechanism to realize Ising-type
interactions exhibiting discrete spin rotation symmetry.
