We study numerically the universal conductance of Luttinger-liquid wire with a single impurity via the multiscale entanglement renormalization ansatz (MERA). The scale-invariant MERA provides an efficient way to extract scaling operators and scaling dimensions for both the bulk and the boundary conformal field theories. By utilizing the key relationship between the conductance tensor and ground-state correlation function, the universal conductance can be evaluated within the framework of the boundary MERA. We construct the boundary MERA to compute the correlation functions and scaling dimensions for the Kane-Fisher fixed points by modeling the single impurity as a junction (weak link) of two interacting wires. We show that the universal behavior of the junction can be easily identified within the MERA and argue that the boundary MERA framework has tremendous potential to classify the fixed points in general multiwire junctions.

We study the electronic transport in a molecular junction, in which each unit is coupled to a local phonon bath, using the nonequilibrium Green's function method. We observe that the conductance oscillates with the molecular chain length and that the oscillation period in odd-numbered chains depends strongly on the applied bias. This oscillatory behavior is smeared out at the bias voltage near the phonon energy. For the phonon-free case, we find a crossover from tunneling to thermally activated transport as the length of the molecule increases. In the presence of the electron-phonon interaction, the transport is thermally driven and a crossover from thermally suppressed conduction to assisted conduction is observed.