We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in form of the Polchinski equation. We apply this method to the Hubbard model on a square lattice using both zero-and finite-temperature methods. Truncating the effective action at the sixth term in fermionic variables we obtain the one-loop functional renormalization equations for the effective interaction.We find the temperature of the instability T RG c as function of doping. We calculate furthermore the renormalization of the angle-resolved correlation functions for the superconductivity (SC) and for the antiferromagnetism (AF). The dominant component of the SC correlations is of the type d x 2 −y 2 while the AF fluctuations are of the type s. Following the strength of both SC and AF fluctuation along the instability line we obtain the phase diagram. The temperature T RG c can be identified with the crossover temperature Tco found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime T RG c corresponds to the superconducting critical temperature. 74.20.Mn, 74.25.Dw, 75.30.Fv

The influence of an external magnetic field on a quasi one-dimensional system with a charge density wave (CDW) instability is treated within the random phase approximation (RPA) which includes both CDW and spin density wave (SDW) correlations. We show that the CDW is sensitive to both orbital and Pauli effects of the field. In the case of perfect nesting, the critical temperature decreases monotonously with the field, and the wave vector of the instability starts to shift above some critical value of magnetic field. Depending on the ratio between the spin and charge coupling constants and on the direction of the applied magnetic field, the wave vector shift is either parallel (CDW x order) or perpendicular (CDW y order) to the most conducting direction. The CDW x order is a field dependent linear combination of the charge and spin density waves and is sensible only to the Pauli effect. The wave vector shift in CDW y depends on the interchain coupling, but the critical temperature does not. This order is affected by the confinement of the electronic orbits. By increasing the relative strength of the orbital effect with respect to the Pauli effect, one can destroy the CDW y , establishing either a CDW x , or a CDW 0 (corresponding to perfect nesting wave vector). We also show that by increasing the imperfect nesting parameter, one passes from the regime where the critical temperature decreases with the field to the regime where it is initially enhanced by the orbital effect and eventually suppressed by the Pauli effect. For a bad nesting, the quantized phases of the field-induced CDW appear.

The problem of weakly correlated electrons on a square lattice is formulated in terms of one-loop renormalization group. Starting from the action for the entire Brillouin zone (and not with a low-energy effective action) we reduce successively the cutoff Λ about the Fermi surface and follow the renormalization of the coupling U as a function of three energy-momenta. We calculate the intrinsic scale T co where the renormalization group flow crosses over from the regime (Λ > T co ) where the electron-electron (e-e) and electron-hole (eh) terms are equally important to the regime (Λ < T co ) where only the e-e term plays a role. In the low energy regime only the pairing interaction V is marginally relevant, containing contributions from all renormalization group steps of the regime Λ > T co . After diagonalization of V Λ=Tco , we identify its most attractive eigenvalue λ min . At low filling, λ min corresponds to the B 2 representation (d xy symmetry), while near half filling the strongest attraction occurs in the B 1 representation (d x 2 −y 2 symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. Using the form of pairing and the structure of the renormalization group equations in the low energy regime, we give our interpretation of ARPES experiments trying to determine the symmetry of the order parameter in the Bi2212 high-T c compound.