Magnetic responses in 1D mesoscopic rings and cylinders

I investigated a detailed study of persistent current and low-field magnetic susceptibility in one-dimensional mesoscopic rings and cylinders threaded by slowly varying magnetic flux φ in the tight-binding model. In perfect rings described by constant number of electrons N_e, current shows only saw-tooth variation with φ, while for those rings described by constant chemical potential μ, current varies saw-tooth like for some special choices of μ, but in all other cases it shows kink-like structures. On the other hand, in perfect cylinders I get both saw-tooth and kink-like structures in persistent current whether these cylinders are described by constant N_e or μ. In presence of impurity, current gets a continuous variation with φ only for the rings described by constant N_e, while in all other cases it depends on the choice of μ. My exact calculation predicts that the diamagnetic and paramagnetic sign of the low-field currents can be determined exactly for the rings described by constant N_e. In perfect rings, I get only diamagnetic currents both for odd and even N_e, while in presence of impurity current always shows diamagnetic sign for the rings with odd N_e and paramagnetic sign for the rings with even N_e. Both for the perfect and disordered rings described by constant μ the sign of the current cannot be mentioned exactly since it depends on the choice of μ and disordered configurations. Similar arguments are also true for the cylinders those are described either by constant N_e or by constant μ since the sign of the current in these systems depends on N_e, μ and disordered configurations.