TUD

We examine transmission zeros and phases in transport through an one-dimensional ring where inversion-symmetry about a transmission axis is broken by different length of branches. It is found that zeros at eigenstates lead to \pi-jump in the Friedel phase, leading discrete set of delta functions in the density of states (DOS), while those due to destructive interference gaurantee continuity of DOS. It is also found that when the ratio of the upper-branch length to the lower on is given by \gamma= p/q satisfying p+q=0 (mod 4), there exist double zero points in transmittance, and further they are accompanied by 2\pi-jump in a transmittance phase, making the transmittance phase identical to the Friedel phase.