content_image/388732.jpg content_image/388744.jpg content_image/388733.jpg content_image/388743.jpg
there are N atomic sites) can be written within the non-interacting electron picture as, 

<img src='content_image/388732.jpg'>

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- 

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 

<img src='content_image/388744.jpg'>

to this, it is also observed that the energy levels vary periodically with flux φ providing φ 0 / 2 flux- 

<img src='content_image/388733.jpg'>

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 

<img src='content_image/388743.jpg'>

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. 

