Scaling behavior of impurities in mesoscopic Luttinger liquids

Using a functional renormalization group we compute the flow of the renormalized impurity potential for a single impurity in a Luttinger liquid over the entire energy range - from the microscopic scale of a lattice-fermion model down to the low-energy limit. The non-perturbative method provides a complete real-space picture of the effective impurity potential. We confirm the universality of the open chain fixed point, but it turns out that very large systems (10^4-10^5 sites) are required to reach the fixed point for realistic choices of the impurity and interaction parameters.