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 $N\to \infty$ and as many such ladders are weakly coupled together are also discussed.