there are N atomic sites) can be written within the non-interacting electron picture as, to this, it is also observed that the energy levels vary periodically with flux φ providing φ 0 / 2 flux- H = 2 N ∑ i =1 ϵ$_{i}$c † $_{i}$c$_{i}$ + v 2 N ∑ i $_{=1}$( e $^{iθ}$c † $_{i}$c$_{i}$ +1 + e − $^{iθ}$c † i $_{+1}$c$_{i}$ ) + v$_{⊥}$ 2 N ∑ i =1 c † $_{i}$c$_{i}$ + N (5) GLYPH<Minus> 1 GLYPH<Minus> 0.5 0.5 1 Φ GLYPH<Minus> 2 GLYPH<Minus> 1 1 2 E GLYPH<LParen1> Φ GLYPH<RParen1> Figure 5: Energy-flux characteristics for ordered (solid curve) and disordered (dotted curve) moebius strips ( N = 5 ) with v$_{⊥}$ = 0 . where, ϵ$_{i}$ represents the on-site energy of an electron in site i and θ = 2 πφ/N is the phase factor associ- ated with flux φ . v gives the nearest-neighbor hop- ping integral in each strand and v$_{⊥}$ corresponds to the hopping strength between the nearest-neighbor sites in the two strands, the so-called intra-chain interaction strength. It ( v$_{⊥}$ ) significantly controls the periodicity of persistent current in such a spe- quantum periodicity instead of simple φ 0 period- icity, as observed in traditional one-channel meso- scopic rings or multi-channel cylinders. This phe- nomenon of half flux-quantum periodicity can be explained as follow. In the typical case where cial type of geometric model which we will describe in forthcoming sub-section. Now we start our discussion with the energy-flux characteristics of a moebius strip. For illustrative examples, in Fig. 5 we plot the energy levels as a function of φ for a moebius strip considering N = 5 in the typical case where v$_{⊥}$ = 0. The solid and dotted lines correspond to the perfect and disor- dered moebius strips, respectively. The presence (disordered strip) or absence (ordered strip) of W provides exactly the similar behavior i.e., gap or gap less spectrum as we observe in the case of one- channel mesoscopic rings. Most importantly we no- tice that the energy levels have extremas at inte- ger multiples of φ = ± nφ 0 / 4 or ± nφ 0 / 2. Addition v$_{⊥}$ = 0, electrons cannot able to hop along the ver- tical direction. Therefore, in a moebius strip if an electron starts to move from any point, it will come back at its initial position after traversing twice the path length, and accordingly, it encloses 2 φ flux. 5 Figure 6: Energy-flux characteristics for ordered (solid line) and disordered (dotted line) moebius strips ( N = 5 ) with v$_{⊥}$ = − 1 . Figure 4: Schematic representation of a moebius strip penetrated by a magnetic flux φ . GLYPH<Minus> 1 GLYPH<Minus> 0.5 0.5 1 Φ GLYPH<Minus> 3 GLYPH<Minus> 1 1 3 E GLYPH<LParen1> Φ GLYPH<RParen1>