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Spin-orbital liquids are quantum disordered states in systems with entangledspin and orbital degrees of freedom. We study exactly solvable spin-orbitalmodels in two dimensions with selected Heisenberg-, Kitaev-, and $\Gamma$-typeinteractions, as well as external magnetic fields. These models realize avariety of spin-orbital-liquid phases featuring dispersing Majorana fermionswith Fermi surfaces, nodal Dirac or quadratic band touching points, or fullgaps. In particular, we show that Zeeman magnetic fields can stabilizenontrivial flux patterns and induce metamagnetic transitions between stateswith different topological character. Solvable nearest-neighbor biquadraticspin-orbital perturbations can be tuned to stabilize zero-energy flat bands. Wediscuss in detail the examples of $\mathrm{SO}(2)$- and$\mathrm{SO}(3)$-symmetric spin-orbital models on the square and honeycomblattices, and use group-theoretical arguments to generalize to$\mathrm{SO}(\nu)$-symmetric models with arbitrary integer $\nu > 1$. Theseresults extend the list of exactly solvable models with spin-orbital-liquidground states and highlight the intriguing general features of such exoticphases. Our models are thus excellent starting points for more realisticmodellings of candidate materials.

Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids