  The spin-orbit interaction of holes in zinc-blende semiconductors is much
stronger than that of electrons. This makes the hole systems very attractive
for possible spintronics applications. In three dimensions (3D) dynamics of
holes is described by well known Luttinger Hamiltonian. However, most of recent
spintronics applications are related to two dimensional heterostructures where
dynamics in one direction is frozen due to quantum confinement. The confinement
results in dimensional reduction of the Luttinger Hamiltonian, 3D ->2D. Due to
interplay of the spin-orbit interaction, the external magnetic field, and the
lateral gate potential imposed on the heterostructure the reduction is highly
nontrivial and not known. In the present work we perform the reduction and
hence derive the general effective Hamiltonian which describes spintronics
effects in symmetric two-dimensional (2D) heterostructures. In particular, we
do the following, (i) derive the spin-orbit interaction and the Darwin
interaction related to the lateral gate potential, (ii) determine the momentum
dependent out-of-plane g-factor, (iii) point out that there are two independent
in-plane g-factors, (iv) determine momentum dependencies of the in-plane
g-factors.
