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Motivated by the recent synthesis of $\beta$-Li$_2$IrO$_3$ -- a spin-orbitentangled $j=1/2$ Mott insulator with a three-dimensional lattice structure ofthe Ir$^{4+}$ ions -- we consider generalizations of the Kitaev model believedto capture some of the microscopic interactions between the Iridium moments onvarious trivalent lattice structures in three spatial dimensions. Of particularinterest is the so-called hyperoctagon lattice -- the premedial lattice of thehyperkagome lattice, for which the ground state is a gapless quantum spinliquid where the gapless Majorana modes form an extended two-dimensionalMajorana Fermi surface. We demonstrate that this Majorana Fermi surface isinherently protected by lattice symmetries and discuss possible instabilities.We thus provide the first example of an analytically tractable microscopicmodel of interacting SU(2) spin-1/2 degrees of freedom in three spatialdimensions that harbors a spin liquid with a two-dimensional spinon Fermisurface.

Quantum spin liquid with a Majorana Fermi surface on the three-dimensional hyperoctagon lattice