E$_{n}$ ( φ ) = 2 v cos [ 2 π N ( n + φ φ 0 )] (2) trates the behavior of low-field magnetic suscep- tibility both for one-channel mesoscopic rings and moebius strips, and finally, we summarize our study in Section 5. In order to study current-flux ( I - φ ) characteris- tics first it is necessary to know the variation of energy levels with φ . For an ordered ring, we cal- culate the energy eigenvalues analytically. Mathe- matically, the energy eigenvalue for n -th eigenstate can be written in the form, Here we explore the behavior of persistent current in strictly one-channel mesoscopic rings. Let us start by referring to Fig. 1 where a mesoscopic ring is penetrated by a magnetic flux φ . To describe the where, n is an integer bounded in the range − N/ 2 ≤ n < N/ 2. But as long as impurities are intro- duced in the ring i.e., the ring becomes a disordered one, energy eigenvalues cannot be obtained analyt- ically. In that case we do exact numerical diagonal- ization of the tight-binding Hamiltonian to achieve the eigenvalues. As illustrative examples, in Fig. 2 2 Persistent current in one- channel mesoscopic rings 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 1: Schematic view of a 1 D mesoscopic ring penetrated by a magnetic flux φ . The filled black circles correspond to the position of atomic sites. A persistent current I is established in the ring. Figure 2: Energy-flux characteristics for ordered (solid line) and disordered (dotted line) mesoscopic rings with N = 10 . For the disordered case, we choose W = 1 . system we use a tight-binding framework. Within a non-interacting electron picture, the tight-binding Hamiltonian for a N -site ring, enclosing a magnetic flux φ (in units of the elementary flux quantum φ 0 = ch/e ), can be expressed in Wannier basis as, we plot the energy-flux ( E - φ ) characteristics for a mesoscopic ring considering N = 10. The solid and dotted curves correspond to the perfect and disor- dered rings, respectively. For the disordered ring, we set W = 1. In absence of any impurity ( W = 0), energy levels intersect with each other at integer or half-integer multiples of φ 0 / 2, while gaps open at these points in presence of impurity. In both the two cases energy levels vary periodically as a function of φ showing φ 0 (= 1, in our chosen unit where, ϵ$_{i}$ ’s are the on-site energies, v is the hop- ping strength between nearest-neighbor atomic sites and θ = 2 πφ/N is the phase factor due to the flux threaded by the ring. As the magnetic field associ- ated with flux φ does not penetrate anywhere of the circumference of the ring, we neglect Zeeman term in the above Hamiltonian (Eq. 1). Throughout our calculations we set the hopping strength v = − 1 and use the units where c = e = h = 1. H = ∑ i ϵ$_{i}$c † $_{i}$c$_{i}$ + ∑ <ij> v ( e $^{iθ}$c † $_{i}$c$_{j}$ + e − $^{iθ}$c † $_{j}$c$_{i}$ ) (1) 3