  The contribution of bulk inversion asymmetry to the total spin-orbit coupling
is commonly neglected for group III-V nanowires grown in the generic [111]
direction. We have solved the complete Hamiltonian of the circular nanowire
accounting for bulk inversion asymmetry via exact numerical diagonalization.
Three different symmetry classes of angular momentum states exist, which
reflects the threefold rotation symmetry of the crystal lattice about the [111]
axis. A particular group of angular momentum states contains degenerate modes
which are strongly coupled via the Dresselhaus Hamiltonian, which results in a
significant energy splitting with increasing momentum. Hence, under certain
conditions Dresselhaus spin-orbit coupling is relevant for [111] InAs and [111]
InSb nanowires. We demonstrate momentum-dependent energy splittings and the
impact of Dresselhaus spin-orbit coupling on the dispersion relation. In view
of possible spintronics applications relying on bulk inversion asymmetry we
calculate the spin expectation values and the spin texture as a function of the
Fermi energy. Finally, we investigate the effect of an axial magnetic field on
the energy spectrum and on the corresponding spin polarization.
