  We study numerically the universal conductance of Luttinger liquids wire with
a single impurity via the Muti-scale 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
multi-wire junctions.
