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 φφ in the tight-binding model. In perfect rings described by constant number of electrons NeNe, current shows only saw-tooth variation with φφ, while for those rings described by constant chemical potential μμ, current varies saw-tooth like for some special choices of μμ, 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 NeNe or μμ. In presence of impurity, current gets a continuous variation with φφ only for the rings described by constant NeNe, while in all other cases it depends on the choice of μμ. 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 NeNe. In perfect rings, I get only diamagnetic currents both for odd and even NeNe, while in presence of impurity current always shows diamagnetic sign for the rings with odd NeNe and paramagnetic sign for the rings with even NeNe. Both for the perfect and disordered rings described by constant μμ the sign of the current cannot be mentioned exactly since it depends on the choice of μμ and disordered configurations. Similar arguments are also true for the cylinders those are described either by constant NeNe or by constant μμ since the sign of the current in these systems depends on NeNe, μμ and disordered configurations.