  One-dimensional quantum rings with Rashba and Dresselhaus spin-orbit
couplings are studied analytically and are in perfect agreement with the
numerical results. The topological charge of the spin field defined by the
winding number along the ring is also studied analytically and numerically in
the presence of the spin-orbit interactions. We also demonstrate the cases
where the one-dimensional model is invalid for a relatively large radius.
However, the numerical results of the two-dimensional model always remain
reliable. Just as many physical properties of the quantum rings are influenced
by the Aharonov-Bohm effect, the topological charge is also found to vary
periodically due to the step-like change of the angular momentum with an
increase of the magnetic field. This is significantly different from the cases
of quantum dots. We also study how the current is induced by the magnetic field
and spin-orbit couplings, which is strong enough that it could to be detected.
The magnetic induction lines induced by the spin field and the current are also
analyzed which can be observed and could perhaps help identifying the
topological features of the spin fields in a quantum ring.
