  We study the effects of a Zeeman magnetic field on the electron transport of
one-dimensional quantum rings which are marked by electronic states with
$d-$orbital symmetry in the presence of spin-orbit and orbital Rashba
couplings. By considering phase-coherent propagation, we analyse the geometric
Aharonov-Anandan (AA) phase of the channels which is acquired in a closed path,
by demonstrating that the orbital polarization can influence the electronic
transport when amplitude and magnetic field directions are varied. We explore
all the possible cases for the injection of electrons at various energies in
the regime of low electron filling. The magnetic field can allow the selection
of only one channel where the transmission is uniquely affected by the AA
phase. Conversely, when more orbital channels are involved there is also a
dynamical contribution that lead to oscillations in the transmission as the
magnetic field is varied. In particular, the transmission is chiral when the
energy states are close to the absolute minimum of the energy bands. Instead,
when an interference between the channels occurs the orbital and spin
contributions tend to balance each other with the increasing of the magnetic
field amplitude resulting in a trivial AA phase. This saturation effect does
not occur in the high magnetic field regime when orbital and spin properties of
the channels exhibit sharp variations with direct consequences on the
transport.
