  Harnessing the enhanced light-matter coupling and quantum vacuum fluctuations
resulting from mode volume compression in optical cavities is a promising route
towards functionalizing quantum materials and realizing exotic states of
matter. Here, we extend cavity quantum electrodynamical materials engineering
to correlated magnetic systems, by demonstrating that a Fabry-P\'erot cavity
can be used to control the magnetic state of the proximate quantum spin liquid
$\alpha$-RuCl$_3$. Depending on specific cavity properties such as the mode
frequency, photon occupation, and strength of the light-matter coupling, any of
the magnetic phases supported by the extended Kitaev model can be stabilized.
In particular, in the THz regime, we show that the cavity vacuum fluctuations
alone are sufficient to bring $\alpha$-RuCl$_3$ from a zigzag antiferromagnetic
to a ferromagnetic state. By external pumping of the cavity in the few photon
limit, it is further possible to push the system into the antiferromagnetic
Kitaev quantum spin liquid state.
