Spin liquid crystals are magnetic analogs of liquid crystals, possessing properties of both liquids and solids, a typical example of which are spin nematics. Spin nematics share many features with spin liquids, and the interplay between them is a promising, but little explored, route to uncovering new phases of matter. Here, we address this question in the context of a spin-1 magnet on the honeycomb lattice, by considering a model with both biquadratic interactions, favoring spin-nematic states, and Kitaev-like interactions, supporting spin liquids. Accompanying these, where dipole and quadrupole moments compete, we find a plethora of exotic phases, including multiple-$q$ states with nonzero scalar spin chirality, a quasi-one-dimensional coplanar phase, a twisted conical phase, and a noncoplanar order state which gives way to a chiral spin liquid at finite temperature. The implication of these results for experiment is discussed.

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• Frustrated magnetism
• Nematic order
• Spin liquid
• Honeycomb lattice
• Magnetic insulators
• Strongly correlated systems
• Kitaev model
• Monte Carlo methods
• Numerical techniques