Spin density matrix of spin-3/2 hole systems

For hole systems with an effective spin j=3/2, we present an invariant decomposition of the spin density matrix that can be interpreted as a multipole expansion. The charge density corresponds to the monopole moment and the spin polarization due to a magnetic field corresponds to a dipole moment while heavy hole-light hole splitting can be interpreted as a quadrupole moment. For quasi-two-dimensional hole systems in the presence of an in-plane magnetic field B-parallel to the spin polarization is a higher-order effect that is typically much smaller than one even if the minority spin subband is completely depopulated. On the other hand, the field B-parallel to can induce a substantial octupole moment, which is a unique feature of j=3/2 hole systems.