  Arrays of Rydberg atoms constitute a highly tunable, strongly interacting
venue for the pursuit of exotic states of matter. We develop a new strategy for
accessing a family of fractionalized phases known as quantum spin liquids in
two-dimensional Rydberg arrays. We specifically use effective field theory
methods to study arrays assembled from Rydberg chains tuned to an Ising phase
transition that famously hosts emergent fermions propagating within each chain.
This highly entangled starting point allows us to naturally access spin liquids
familiar from Kitaev's honeycomb model, albeit from an entirely different
framework. In particular, we argue that finite-range repulsive Rydberg
interactions, which frustrate nearby symmetry-breaking orders, can enable
coherent propagation of emergent fermions between the chains in which they were
born. Delocalization of emergent fermions across the full two-dimensional
Rydberg array yields a gapless Z2 spin liquid with a single massless Dirac
cone. Here, the Rydberg occupation numbers exhibit universal power-law
correlations that provide a straightforward experimental diagnostic of this
phase. We further show that explicitly breaking symmetries perturbs the gapless
spin liquid into gapped, topologically ordered descendants: Breaking lattice
symmetries generates toric-code topological order, whereas introducing
chirality generates non-Abelian Ising topological order. In the toric-code
phase, we analytically construct microscopic incarnations of non-Abelian
defects, which can be created and transported by dynamically controlling the
atom positions in the array. Our work suggests that appropriately tuned Rydberg
arrays provide a cold-atoms counterpart of solid-state 'Kitaev materials' and,
more generally, spotlights a new angle for pursuing experimental platforms for
Abelian and non-Abelian fractionalization.
