  We describe a variational theory of multi-band two-dimensional electron gases
that captures the interplay between electrostatic confining potentials,
orbital-dependent interlayer electronic hopping and electron-electron
interactions, and apply it to the d-band two-dimensional electron gases that
form near perovskite oxide surfaces and heterojunctions. These multi-band
two-dimensional electron gases are prone to the formation of
Coulomb-interaction-driven orbitally-ordered nematic ground-states. We find
that as the electron density is lowered and interaction effects strengthen,
spontaneous orbital order occurs first, followed by spin order. We compare our
results with known properties of single-component two-dimensional electron gas
systems and comment on closely related physics in semiconductor quantum wells
and van der Waals heterostructures.
