Charge counting statistics and weak localization in a quantum chain

Abstract

We study transport properties of a quantum chain formed by an array of chaotic quantum dots coupled to each other and to electron reservoirs via barriers of arbitrary transparencies. We introduce two exact representations of the generating function (GF) of charge counting statistics: a transfer matrix model for numerical simulations and a supersymmetric $\ensuremath{\sigma}$ model for analytical calculations. Using the $\ensuremath{\sigma}$ model, we calculate analytically the semiclassical term and the leading quantum correction (weak localization) of GF as a function of barriers' transparencies and the number of quantum dots. We observe that the density of transmission eigenvalues, obtained from the semiclassical term of the GF, exhibits a quantum transition, associated with the emergence of Fabry-Perot modes, in a region of the parameter space of barriers' transparencies that expands by increasing the number of quantum dots in the chain. Simple analytical expressions for the transition lines are derived, which may be used to tune the different regimes. We demonstrate that the presence of these Fabry-Perot (FP) modes plays a decisive role in controlling the ballistic-diffusive crossover in the quantum chain. We also find interesting nonmonotonic behaviors in the leading semiclassical terms and changes of sign in the weak-localization corrections of high-order charge transfer cumulants, as a function of the number of dots. Our results agree well with independent analytical calculations using a diagrammatic technique for integration over the unitary group and also with numerical simulations using the transfer matrix model.