Weakly correlated electrons on a square lattice: Renormalization-group theory

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 the form of the Polchinski equation. The advantage of this renormalization scheme is that the flow itself has a physical interpretation, i.e., the cutoff has the meaning of the temperature. 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 and neglecting self-energy renormalization, we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability $T_c^{\mathrm{RG}}$ as a function of doping. Furthermore we calculate 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 fluctuations along the instability line, we obtain the phase diagram. The temperature $T_c^{\mathrm{RG}}$ can be identified with the crossover temperature $T_{\mathrm{co}}$ found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime $T_c^{\mathrm{RG}}$ corresponds to the superconducting critical temperature.