(AB) fluxes φ$_{1}$ and φ$_{2}$ , respectively. The schematic view of the bridge system is depicted in Fig. 1. A simple tight-binding model is used to describe the system and all the calculations are done nu- merically, which illustrate conductance-energy and current-voltage characteristics as functions of the interferometer-to-electrode coupling strength, mag- netic fluxes and the difference of these two fluxes. Several exotic features are observed from this study. These are: (i) semiconducting or metallic nature depending on the coupling strength of the inter- ferometer to the side attached electrodes, (ii) ap- pearance of anti-resonant states [7, 8, 9] and (iii) unconventional periodic behavior of typical conduc- tance/current as a function of the difference of two AB fluxes. where, T becomes the transmission probability of an electron across the interferometer. It ( T ) can be expressed in terms of the Green’s function of the interferometer and its coupling to the side-attached electrodes by the relation [46, 47], T = Tr [Γ$_{S}$ G r $_{I}$Γ$_{D}$ G a $_{I}$] (2) where, G r I and G a I are respectively the retarded and advanced Green’s functions of the interferometer in- cluding the effects of the electrodes. Here Γ$_{S}$ and Γ$_{D}$ describe the coupling of the interferometer to The scheme of the paper is as follows. Follow- ing the introduction (Section 1), in Section 2, we describe the model and theoretical formulation for the calculation. Section 3 explores the results, and finally, we conclude our study in Section 4. 2 1 2 3 4 1 Source Drain Quantum Interferometer Figure 1: (Color online). Schematic view of a quan- tum interferometer with four atomic sites ( N = 4) attached to two semi-infinite one-dimensional metallic electrodes. 2 Model and synopsis of the theoretical background the source and drain, respectively. For the complete system i.e., the interferometer, source and drain, Green’s function is defined as, G = ( E − H ) − 1 (3) Let us begin with the model presented in Fig. 1. A quantum interferometer with four atomic sites ( N = 4, where N gives the total number of atomic sites in the interferometer) is attached symmet- rically to two semi-infinite one-dimensional (1D) metallic electrodes, namely, source and drain. The atomic sites 2 and 3 of the interferometer are di- rectly coupled to each other, and accordingly, two sub-rings, left and right, are formed. These two sub-rings are subject to the AB fluxes φ$_{1}$ and φ$_{2}$ , respectively. where, E is the injecting energy of the source elec- tron. To Evaluate this Green’s function, inversion of an infinite matrix is needed since the complete system consists of the finite size interferometer and two semi-infinite electrodes. However, the entire system can be partitioned into sub-matrices cor- responding to the individual sub-systems and the Green’s function for the interferometer can be ef- fectively written as, Considering linear transport regime, conductance g of the interferometer can be obtained using the Landauer conductance formula [43, 44, 45, 46, 47], where, H$_{I}$ is the Hamiltonian of the interferometer that can be expressed within the non-interacting G$_{I}$ = ( E − H$_{I}$ − Σ$_{S}$ − Σ$_{D}$) − 1 (4) g = 2 e 2 h T (1) 3