Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution

We study the phase diagram of the Hubbard model in the limit where U, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for T_c, the superconducting transition temperature, in terms of the correlation functions of the noninteracting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing T_c involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff Ω₀. Then, the renormalization group (RG) flows of the resulting effective action are computed and T_c is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that T_c is independent of Ω₀. Using this method, we find a variety of unconventional superconducting ground states in two- and three-dimensional lattice systems, and present explicit results for T_c and pairing symmetries as a function of the electron concentration.