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This provides φ 0 / 2 flux-quantum periodicity and it is the speciality of such a twisted geometry. Now, as long as electrons are allowed to hop between the two strands i.e., v$_{⊥}$ ̸ = 0, energy levels get back φ 0 flux- quantum periodicity. This is quite obvious since for the case when v$_{⊥}$ ̸ = 0 one can treat the moebius strip as a conventional cylinder. For illustrations, 

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see the results plotted in Fig. 6, where the solid and dotted lines correspond to the identical meaning as in Fig. 5. 

With the above energy-flux characteristics now we concentrate our study on the current-flux spec- tra. Let us begin with the results plotted in Fig. 7, where the currents are evaluated for some moebius strips with v$_{⊥}$ = 0. The system size N is fixed at 50, where (a) and (b) correspond to N$_{e}$ = 40 (even) and 35 (odd), respectively. The solid and dotted curves represent the currents for ordered and disor- dered moebius strips, respectively, where the disor- 

der strength W is set to 1. Like single channel meso- scopic ring, persistent current reveals saw-tooth like behavior as a function of φ both for odd and even number of electrons. Similarly, in presence of disor- der current varies continuously, as studied earlier, 

<img src='content_image/407122.jpg'>

and gets much reduced value than the perfect case. These features are clearly understood from our pre- vious discussion. The main point is that, for this particular case where v$_{⊥}$ = 0, current varies peri- odically with φ showing φ 0 / 2 periodicity, instead of conventional φ 0 periodicity. This unconventional period halving behavior of persistent current may be clearly visible from our energy-flux spectrum, for instance see Fig. 5. 

The situation becomes quite different when elec- trons can able to hop along the transverse direction i.e., for v$_{⊥}$ ̸ = 0. In this case a moebius can be regarded as an ordinary mesoscopic cylinder, and therefore, we get the conventional features of persis- 

