  We introduce a variational manifold of simple tensor network states for the
study of a family of constrained models that describe spin-1/2 systems as
realized by Rydberg atom arrays. Our manifold permits analytical calculation
via perturbative expansion of one- and two-point functions in arbitrary spatial
dimensions and allows for efficient computation of the matrix elements required
for variational energy minimization and variational time evolution in up to
three dimensions. We apply this framework to the PXP model on the hypercubic
lattice in 1D, 2D, and 3D, and show that, in each case, it exhibits quantum
phase transitions breaking the sub-lattice symmetry in equilibrium, and hosts
quantum many body scars out of equilibrium. We demonstrate that our variational
ansatz qualitatively captures all these phenomena and predicts key quantities
with an accuracy that increases with the dimensionality of the lattice, and
conclude that our method can be interpreted as a generalization of mean-field
theory to constrained spin models.
