Model of resistances in systems of Tomonaga-Luttinger liquid wires

In a recent paper, we combined the technique of bosonization with the concept of a Rayleigh dissipation function to develop a model for resistances in one-dimensional systems of interacting spinless electrons Europhys. Lett. 93, 57007 (2011)]. We also studied the conductance of a system of three wires by using a current splitting matrix M at the junction. In this paper, we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of M and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the M matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. We study the conductance of a three-wire system of interacting spin-1/2 electrons, and show that the charge and spin conductances can generally be different from each other. Finally, we consider a system of two wires that are coupled by a dissipation function, and we show that this leads to a current in one wire when a voltage bias is applied across the other wire.