  In recent years, several magnetic Mott insulators with strong spin-orbit
couplings were suggested to be proximate to the Kitaev quantum spin liquid
(QSL) whose one of the most exciting features is the fractionalization of spin
excitations into itinerant Majorana fermions and static $Z_2$ fluxes. Motivated
by the emergence of this plethora of 4d and 5d transition metal Kitaev
materials and by the fact that some level of disorder is inevitable in real
materials, here we study how the Kitaev QSL responds to various forms of
disorder, such as vacancies, impurities, and bond randomness. First, we argue
that the presence of the quenched disorder in the Kitaev QSL can lead to the
Anderson localization of Majorana fermions and the appearance of Lifshitz
tails. We point out that the Anderson localization of low-energy states is
particularly strong in the extended Kitaev model with the time reversal
symmetry breaking term. Second, we show that the disorder effects on the
low-energy Majorana fermion modes can be detected in thermal transport. Third,
we show that at finite temperatures the $Z_2$ fluxes become thermally excited
and give rise to an additional disorder for the Majorana fermions. This source
of the disorder dominates at high temperatures. Fourth, we demonstrate that
both the structure of the energy spectrum and the thermal transport properties
of the disordered Kitaev QSL depend strongly on the character of disorder.
While we find that both the site disorder and the bond randomness suppress the
longitudinal thermal conductivity, the low-energy localization is stronger in
the case of the site disorder.
