Electron localization in linear chains of identical loop scatterers

We show that electron localization is generic in a linear chain of identical simple quantum wire loops with equal arm lengths in the presence of either a perpendicular magnetic field or the spin-orbit interaction, and has less to do with the shapes of the loops. We calculate the transfer matrices for a general simple loop scatterer in the presence of these effects. Based on the knowledge of the transfer matrices, we thus provide a criterion for the occurrence of the localization and present a simple formalism to integrate the transmission probability over the injection wave vector of electron.