Renormalization group study of the conductances of interacting quantum wire systems with different geometries

We examine the effect of interactions between the electrons on the Landauer-Buttiker conductances of some systems of quantum wires with different geometries. The systems include a long wire with a stub in the middle, a long wire containing a ring which can enclose a magnetic flux, and a system of four long wires which are connected in the middle through a fifth wire. Each of the wires is taken to be a weakly interacting Tomonaga-Luttinger liquid, and scattering matrices are introduced at all the junctions present in the systems. Using a renormalization group method developed recently for studying the flow of scattering matrices for interacting systems in one dimension, we compute the conductances of these systems as functions of the temperature and the wire lengths. We present results for all three regimes of interest, namely, high, intermediate, and low temperature. These correspond, respectively, to the thermal coherence length being smaller than, comparable to, and larger than the smallest wire length in the different systems, i.e., the lengths of the stub or each arm of the ring or the fifth wire. The renormalization group procedure and the formulas used to compute the conductances are different in the three regimes. In particular, the dimensionality of the scattering matrix effectively changes when the thermal length becomes larger than the smallest wire length. We also present a phenomenologically motivated formalism for studying the conductances in the intermediate regime where there is only partial coherence. At low temperatures, we study the line shapes of the conductances versus the energy of the electrons near some of the resonances; the widths of the resonances are found to go to zero with decreasing temperature. Our results show that the Landauer-Buttiker conductances of various systems of experimental interest depend on the temperature and lengths in a non-trivial way when interactions are taken into account.