Chiral spin liquids (CSLs) are exceptional magnetic states that break time-reversal symmetry without long-range magnetic ordering. Here, the authors showcase a family of CSLs in a three-dimensional extension of the Kitaev model on the hypernonagon lattice, using quantum Monte Carlo simulations supplemented by variational calculations. Notably, all the CSLs are found to not only break time-reversal symmetry but concurrently also a number of lattice symmetries by forming a crystalline ordering of the underlying \ensuremath{\mathbb{Z}}
2 gauge structure. It is further shown that some of the CSLs exhibit topological features with gapless Majorana modes.
Chiral spin liquids (CSLs) in three dimensions and thermal phase transitions to paramagnet are studied by unbiased Monte Carlo simulations. For an extension of the Kitaev model to a three-dimensional tricoordinate network dubbed the hypernonagon lattice, we derive low-energy effective models in two different anisotropic limits. We show that the effective interactions between the emergent
Z2 degrees of freedom called fluxes are unfrustrated in one limit, while highly frustrated in the other. In both cases, we find a first-order phase transition to the CSL, where both time-reversal and parity symmetries are spontaneously broken. In the frustrated case, however, the CSL state is highly exotic---the flux configuration is subextensively degenerate while showing a directional order with broken
C3 rotational symmetry. Our results provide two contrasting archetypes of CSLs in three dimensions, both of which allow approximation-free simulation for investigating the thermodynamics.