Disorder and non-Hermiticity in Kitaev spin liquids with a Majorana Fermi surface

We study the effect of disorder on Z${}_2$ quantum spin liquids with a Majorana Fermi line (respectively surface in three dimensions) and show that depending on the symmetries that are preserved \emph{on average} qualitatively different scenarios will occur. In particular, we identify the relevant non-Hermitian symmetries for which disorder will effectively split the Fermi line into two exceptional lines, with $\mathrm{\Re }(E)=0$ states filling the area in between. We demonstrate the different scenarios using both toy models as well as large-scale numerical simulations.