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