  Synthetic dimension platforms offer unique pathways for engineering quantum
matter. We compute the phase diagram of a many-body system of ultracold atoms
(or polar molecules) with a set of Rydberg states (or rotational states) as a
synthetic dimension, where the particles are arranged in real space in optical
microtrap arrays and interact via dipole-dipole exchange interaction. Using
mean-field theory, we find three ordered phases - two are localized in the
synthetic dimension, predicted as zero-temperature ground states in Refs. [Sci.
Rep., 8, 1 (2018) and Phys. Rev. A 99, 013624 (2019)], and a delocalized phase.
We characterize them by identifying the spontaneously broken discrete
symmetries of the Hamiltonian. We also compute the phase diagram as a function
of temperature and interaction strength, for both signs of the interaction. For
system sizes with more than six synthetic sites and attractive interactions, we
find that the thermal phase transitions can be first or second order, which
leads to a tri-critical point on the phase boundary. By examining the
dependence of the tri-critical point and other special points of the phase
boundary on the synthetic dimension size, we shed light on the physics for
thermodynamically large synthetic dimension.
