  We study a tight-binding model on the two-dimensional ruby lattice. This
lattice supports several types of first and second neighbor spin-dependent
hopping parameters in an $s$-band model that preserves time-reversal symmetry.
We discuss the phase diagram of this model for various values of the hopping
parameters and filling fractions, and note an interesting competition between
spin-orbit terms that individually would drive the system to a $Z_2$
topological insulating phase. We also discuss a closely related spin-polarized
model with only first and second neighbor hoppings and show that extremely flat
bands with finite Chern numbers result, with a ratio of the band gap to the
band width approximately 70. Such flat bands are an ideal platform to realize a
fractional quantum Hall effect at appropriate filling fractions. The ruby
lattice can be possibly engineered in optical lattices, and may open the door
to studies of transitions between quantum spin liquids, topological insulators,
and integer and fractional quantum Hall states.
