  We develop a theory for optical Faraday rotation noise in two-dimensional
Dirac materials. In contrast to spin noise in conventional semiconductors, we
find that the Faraday rotation fluctuations are influenced not only by spins
but also the valley degrees of freedom attributed to intervalley scattering
processes. We illustrate our theory with two-dimensional transition metal
dichalcogenides and discuss signatures of spin and valley noise in the Faraday
noise power spectrum. We propose optical Faraday noise spectroscopy as a
technique for probing both spin and valley relaxation dynamics in
two-dimensional Dirac materials.
