Functional renormalization group approach to correlated electron systems

The functional renormalization group is an ideal tool for dealing with the diversity of energy scales and competition of correlations in interacting Fermi systems. An exact hierarchy of flow equations yields the gradual evolution from a microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff. Suitable truncations of the hierarchy have recently led to powerful new approximation schemes. I derive and discuss the structure of the flow equations for several versions of the functional renormalization group. I then review applications of truncated flow equations to the two-dimensional Hubbard model, focussing in particular on magnetic correlations and d-wave superconductivity, and to one-dimensional Luttinger liquids with impurities, where a strikingly simple truncation captures a surprising amount of nontrivial correlation effects.