  We study theoretically cubic anisotropy of Zeeman splitting of a hole
localized in semiconductor nanocrystal. This anisotropy originates from three
contributions: crystallographic cubically-symmetric spin and kinetic energy
terms in the bulk Luttinger Hamiltonian and the spatial wave function
distribution in a cube-shaped nanocrystal. From symmetry considerations, an
effective Zeeman Hamiltonian for the hole lowest even state is introduced,
containing a spherically symmetric and a cubically symmetric term. The values
of these terms are calculated numerically for spherical and cube-shaped
nanocrystals as functions of the Luttinger Hamiltonian parameters. We
demonstrate that the cubic shape of the nanocrystal and the cubic anisotropy of
hole kinetic energy (so called valence band warping) significantly affect
effective $g$ factors of hole states. In both cases, the effect comes from the
cubic symmetry of the hole wave functions in zero magnetic field. Estimations
for the effective $g$ factor values in several semiconductors with zinc-blende
crystal lattices are made. Possible experimental manifestations and potential
methods of measurement of the cubic anisotropy of the hole Zeeman splitting are
suggested.
