Aharonov-Bohm effect in Luttinger liquid and beyond

In systems with the spin-charge separation, the period of the Aharonov-Bohm (AB) oscillation becomes half of the flux quantum. This effect is at least related to the fact that for the creation of the holons (spinons) are needed two electrons. The effect is illustrated on the example of the Hubbard Hamiltonian with the aid of the bosonization including topological numbers and exists also in the Luttinger liquid on two chains. The relation to a fractional 1/N- AB effect, which can be associated with a modified Luttinger liquid, is discussed.