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Exactly solvable models play a special role in Condensed Matter physics,serving as secure theoretical starting points for investigation of newphenomena. Changlani et al. [Phys. Rev. Lett. 120, 117202 (2018)] havediscovered a limit of the XXZ model for $S=1/2$ spins on the kagome lattice,which is not only exactly solvable, but features a huge degeneracy of exactground states corresponding to solutions of a three-coloring problem. Thisspecial point of the model was proposed as a parent for multiple phases in thewider phase diagram, including quantum spin liquids. Here, we show that theconstruction of Changlani et al. can be extended to more general forms ofanisotropic exchange interaction, finding a line of parameter space in an XYZmodel which maintains both the macroscopic degeneracy and the three-coloringstructure of solutions. We show that the ground states along this line arepartially ordered, in the sense that infinite-range correlations of some spincomponents coexist with a macroscopic number of undetermined degrees offreedom. We therefore propose the exactly solvable limit of the XYZ model oncorner-sharing triangle-based lattices as a tractable starting point fordiscovery of quantum spin systems which mix ordered and spin liquid-likeproperties.

Exactly solvable spin-1/2 XYZ models with highly-degenerate, partially ordered, ground states