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Chiral spin liquids (CSLs) are time-reversal symmetry breaking ground statesof frustrated quantum magnets that show no long-range magnetic ordering, butinstead exhibit topological order and fractional excitations. Their realizationin simple and tractable microscopic models has, however, remained an openchallenge for almost two decades until it was realized that Kitaev models onlattices with odd-length loops are natural hosts for such states, even in theabsence of a time-reversal symmetry breaking magnetic field. Here we report onthe formation of CSLs in a three-dimensional Kitaev model, which differ fromtheir widely studied two-dimensional counterparts, namely, they exhibit acrystalline ordering of the $\mathbb{Z}_2$ gauge fluxes and thereby break someof the underlying lattice symmetries. We study the formation of theseunconventional CSLs via extensive quantum Monte Carlo simulations anddemonstrate that they are separated from the featureless paramagnet at hightemperatures by a single first-order transition at which both time-reversal andlattice symmetries are simultaneously broken. Using variational approaches forthe ground state, we explore the effect of varying the Kitaev couplings andfind at least five distinct CSL phases, all of which possess crystallineordering of the $\mathbb{Z}_2$ gauge fluxes. For some of these phases, thecomplementary itinerant Majorana fermions exhibit gapless band structures withtopological features such as Weyl nodes or nodal lines in the bulk and Fermiarc or drumhead surface states.

Chiral spin liquids with crystalline $\mathbb{Z}_2$ gauge order in a three-dimensional Kitaev model