The hydrodynamic stresses created by active particles can destabilize orientational order present in the system. This is manifested, for example, by the appearance of a bend instability in active nematics or in quasi-two-dimensional living liquid crystals consisting of swimming bacteria in thin nematic films. Using large-scale hydrodynamics simulations, we study a system consisting of spherical microswimmers within a three-dimensional nematic liquid crystal. We observe a spontaneous chiral symmetry breaking, where the uniform nematic state is kneaded into a continuously twisting state, corresponding to a helical director configuration akin to a cholesteric liquid crystal. The transition arises from the hydrodynamic coupling between the liquid crystalline elasticity and the swimmer flow fields, leading to a twist-bend instability of the nematic order. It is observed for both pusher (extensile) and puller (contractile) swimmers. Further, we show that the liquid crystal director and particle trajectories are connected: in the cholesteric state the particle trajectories become helicoidal.
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