  A deterministic and scalable array of single photon nonlinearities in the
solid state holds great potential for both fundamental physics and
technological applications, but its realization has proved extremely
challenging. Despite significant advances, leading candidates such as quantum
dots and group III-V quantum wells have yet to overcome their respective
bottlenecks in random positioning and weak nonlinearity. Here we consider a
hybrid light-matter platform, marrying an atomically thin two-dimensional
material to a photonic crystal cavity, and analyze its second-order coherence
function. We identify several mechanisms for photon antibunching under
different system parameters, including one characterized by large dissipation
and weak nonlinearity. Finally, we show that by patterning the two-dimensional
material into different sizes, we can drive our system dynamics from a coherent
state into a regime of strong antibunching with $g^{(2)}(0) \sim 10^{-3}$,
opening a possible route to building scalable, on-chip quantum simulators.
