GLYPH<Minus> 1.0 1.0 Φ GLYPH<Minus> 0.2 0.2 I GLYPH<LParen1> Φ GLYPH<RParen1> GLYPH<LParen1> b GLYPH<RParen1> GLYPH<Minus> 1.0 1.0 Φ GLYPH<Minus> 0.2 0.2 I GLYPH<LParen1> Φ GLYPH<RParen1> GLYPH<LParen1> a GLYPH<RParen1> GLYPH<Minus> 1.0 1.0 Φ GLYPH<Minus> 0.3 0.3 I GLYPH<LParen1> Φ GLYPH<RParen1> GLYPH<LParen1> b GLYPH<RParen1> GLYPH<Minus> 1.0 1.0 Φ GLYPH<Minus> 0.2 0.2 I GLYPH<LParen1> Φ GLYPH<RParen1> GLYPH<LParen1> a GLYPH<RParen1> Figure 8: I - φ characteristics for ordered (solid curve) and disordered (dotted curve) moebius strips ( N = 50 ) with v$_{⊥}$ = − 1 . (a) N$_{e}$ = 40 and (b) N$_{e}$ = 35 . For the disordered case, W is fixed at 1 . Figure 7: I - φ characteristics for ordered (solid curve) and disordered (dotted curve) moebius strips ( N = 50 ) with v$_{⊥}$ = 0 . (a) N$_{e}$ = 40 and (b) N$_{e}$ = 35 . For the disordered case, W = 1 . 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, 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, see the results plotted in Fig. 6, where the solid and dotted lines correspond to the identical meaning as in Fig. 5. 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. 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- 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- 6