  Motivated by systems where a high temperature non-Fermi liquid gives way to
low temperature $\mathbb{Z}_3$ Potts nematic order, we studied a three-orbital
Sachdev-Ye-Kitaev (SYK) model in the large-$N$ limit. In the single-site limit,
this model exhibits a spontaneous orbital-selective transition which preserves
average particle-hole symmetry, with two orbitals becoming insulators while the
third orbital remains a non-Fermi liquid down to zero temperature. We extend
this study to lattice models of three-orbital SYK dots, exploring uniform
symmetry broken states on the triangular and cubic lattices. At high
temperature, these lattice models exhibit an isotropic non-Fermi liquid metal
phase. On the three-dimensional (3D) cubic lattice, the low temperature uniform
$\mathbb{Z}_3$ nematic state corresponds to an orbital selective layered state
which preserves particle-hole symmetry at small hopping and spontaneously
breaks the particle-hole symmetry at large hopping. Over a wide range of
temperature, the transport in this layered state shows metallic in-plane
resistivity but insulating out-of-plane resistivity. On the 2D triangular
lattice, the low temperature state with uniform orbital order is also a
correlated $\mathbb{Z}_3$ nematic with orbital-selective transport but it
remains metallic in both principal directions. We discuss a Landau theory with
$\mathbb{Z}_3$ clock terms which captures salient features of the phase diagram
and nematic order in all these models. We also present results on the
approximate wavevector dependent orbital susceptibility of the isotropic
non-Fermi liquid states.
