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We study the effect of adding disorder to the exactly solvable Kitaev modelon the hyper-honeycomb lattice, which hosts both gapped and gapless spin liquidphases with an emergent $\mathbb{Z}_2$ gauge field. The latter has an unusualgapless spectrum of Majorana fermion excitations, with a co-dimension-two Fermiring. We thus address the question of the interplay of topological physics anddisorder by considering the properties of isolated single and pair ofvacancies. We show that near the vacancies, the local magnetic response to afield $h_z$ is parametrically enhanced in comparison to the pristine bulk.Unlike the previously studied case of the 2D honeycomb Kitaev model, thevacancies do not bind a flux of the $\mathbb{Z}_2$ gauge field. In the gappedphase, an isolated vacancy gives rise to effectively free spin-half momentswith a non-universal coupling to an external field. In the gapless phase, thelow-field magnetization is suppressed parametrically, to $(-\ln h_z)^{-1/2}$because of interactions with the surrounding spin-liquid. We also show that apair of vacancies is subject to a sublattice-dependent interaction on accountof coupling through the bulk spin liquid, which is spatially anisotropic evenwhen all Kitaev couplings have equal strength. This coupling is thusexponentially suppressed with distance in the gapped phase. In the gaplessphase, two vacancies on the same (opposite) sublattice exhibit an enhanced(suppressed) low-field response, amounting to an effectively(anti-)ferromagnetic interaction.

Vacancies in the Kitaev quantum spin liquids on the 3D hyper-honeycomb lattice