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**Quantum Loop States in Spin-Orbital Models on the Honeycomb and Hyperhoneycomb Lattices
**

In the quest for quantum spin liquids, the challenges are many: neither is it clear how to look for nor how to describe them, and definitive experimental examples of quantum spin liquids are still missing. In this talk I will show how to devise a realistic model on the honeycomb lattice whose ground state realizes Haldane chains whose physical supports fluctuate, hence naturally providing the hallmark “fractional excitations” of quantum spin liquids. When taken to the three-dimensional hyperhoneycomb lattice, the ground state becomes a full-fledged symmetry-enriched U(1) quantum spin-orbital liquid, “disordered” both in the spin and orbital channels. The phase diagram also contains an interacting bosonic topological insulator phase. Crucially, this model is expected to describe actual materials, and I will provide a detailed set of material-specific constraints which may be readily used for an experimental realization.