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tent current. For illustrative purposes, in Fig. 8 we show the current-flux characteristics for some typi- cal moebius strips ( N = 50) considering v$_{⊥}$ = − 1, where (a) and (b) correspond to the results for N$_{e}$ = 40 and 35, respectively. The solid and dotted lines represent the similar meaning as in Fig. 7. The saw-tooth like behavior is still preserved in perfect case. But, some additional kinks may appear in I - φ spectrum at different values of φ depending on the choices of N$_{e}$ . For instance, the current shows a kink across φ = 0 when N$_{e}$ = 40. These kinks are associated with the crossing of energy levels for the cylindrical systems. While, for disordered cases, current varies continuously with much reduced am- plitude, as expected, and no kink will appear. 

## 4 Low-field magnetic suscep- tibility 

Next, we focus our attention on the determination of magnetic susceptibility χ ( φ ) in the limit φ → 0. Evaluating the sign of χ ( φ ) one can able to predict whether the current is paramagnetic or diamagnetic in nature. It was a long standing problem over the past few many years, and, here we try to emphasize about it both for one-channel rings and moebius strips described with fixed number of electrons N$_{e}$ . The general expression of magnetic susceptibility at any flux φ is expressed in the form, 

where, N corresponds to the total number of atomic sites in the system. Here we will determine χ ( φ ) only in the limit φ → 0 i.e., we are interested only on the low-field magnetic susceptibility. 

As representative examples, in Fig. 9 we display the variation of low-field magnetic susceptibility as a function of N$_{e}$ for strictly one-channel mesoscopic rings, where (a) and (b) correspond to the perfect ( W = 0) and disordered ( W = 1) rings, respec- tively. The system size N is fixed at 200. Our re- 

<img src='content_image/412824.jpg'>

sults predict that in absence of any impurity χ ( φ ) is always negative irrespective of the total number of electrons N$_{e}$ i.e., whether N$_{e}$ is odd or even. It re- veals that for perfect rings low-field current provides only the diamagnetic response. This diamagnetic behavior of low-field currents can be clearly under- stood by noting the slope of I - φ characteristics plot- 

<img src='content_image/412822.jpg'>

ted by the solid lines in Figs. 3(a) and (b). The response drastically changes as long as impurities are introduced in the system (9(b)). For odd and even N$_{e}$ , it shows completely opposite behavior. The low-field current gives paramagnetic response for the rings with even N$_{e}$ (solid line), while for the rings with odd N$_{e}$ it shows diamagnetic na- ture (dotted line). These diamagnetic and para- magnetic natures of persistent current for the dis- ordered case can be easily understood by observing the slopes of I - φ curves presented by the dotted 

