  The bond-disordered Kitaev model attracts much attention due to the
experimental relevance in $\alpha$-RuCl$_3$ and $A_3$LiIr$_2$O$_6$ ($A=$ H, D,
Ag, $\textit{etc.}$). Applying a magnetic field to break the time-reversal
symmetry leads to a strong modulation in mass terms for Dirac cones. Because of
the smallness of the flux gap of the Kitaev model, a small bond disorder can
have large influence on itinerant Majorana fermions, and Majorana fermions will
be in the Anderson localization state immediately. We call this immobile liquid
state Anderson-Kitaev liquid state with two localized Majorana fermions, one
frozen by gauge fluctuations and the other localized by disordered mass terms.
The quantization of the thermal Hall conductivity $\kappa/T$ disappears by a
quantum Hall transition induced by a small disorder, and $\kappa/T$ shows a
rapid crossover into the Anderson-Kitaev liquid with a negligible Hall current.
Especially, the critical disorder strength $\delta J_{c1} \sim 0.05$ in the
unit of the Kitaev interaction would have many implications for the stability
of Kitaev spin liquids.
