Chiral Broken Symmetry Descendants of the Kagom\'e Lattice Chiral Spin Liquid

The breaking of chiral and time-reversal symmetries provides a pathway to exotic quantum phenomena and topological phases. In particular, the breaking of chiral (mirror) symmetry in quantum materials has been shown to have important technological applications. Recent work has extensively explored the resulting emergence of chiral charge orders and chiral spin liquids on the kagom\'e lattice. Such chiral spin liquids are closely tied to bosonic fractional quantum Hall states and host anyonic quasiparticles; however, their connection to nearby magnetically ordered states has remained a mystery. Here, we show that two distinct non-coplanar magnetic orders with uniform spin chirality, the XYZ umbrella state and the Octahedral spin crystal, emerge as competing orders in close proximity to the kagom\'e chiral spin liquid. Our results highlight the intimate link between a many-body topologically ordered liquid and broken symmetry states with nontrivial real-space topology.Comment: Main text : 9 pages, 4 figures; Supplementary : 12 pages, 11 figure