N-chain Hubbard model in weak coupling

We present a systematic weak-coupling renormalization group (RG) technique for studying a collection of N coupled one-dimensional interacting electron systems, focusing on the example of N-leg Hubbard ladders. For N=2,3, we recover previous results, and find that also more generally broad regions of the phase space of these models are unstable to pairing, usually with approximate d-wave symmetry. We show how these instabilities can be understood in terms of a fairly conventional “gap” function Δ at the discretized Fermi surface, and describe how this function is calculated. The dimensional crossovers as N→∞ and as many such ladders are weakly coupled together are also discussed.