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 non-interacting 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 $\Omega_0$. 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 $\Omega_0$.
Prof. Srinivas Raghu (Rice University/Stanford)