content_image/372221.jpg content_image/372220.jpg content_image/372219.jpg
the LSR holds if we consider the sign changes the Green’s function undergoes not at ω = 0 but at ω = µ . 

2. Superconducting Instability 

The residual interactions between the Fermi pockets and the Cooperons will lead to instabilities in the RPA solutions as temperature goes to zero. Provided a finite chemical potential is present the leading instability will be to a superconducting state. While gapless quasi-particles only exist in the A-bands, both A and B bands will go superconducting simultaneously. The general form of the Cooperon-quasiparticle interaction is 

<img src='content_image/372221.jpg'>

Here L is the length of the ladders, a is the interladder spacing, and N is the number of ladders in the array. Φ$_{A,B}$ are the Cooperon fields whose bare propagators are defined as 

<img src='content_image/372220.jpg'>

We see that g has the dimensionality of energy × length 2 and Γ$_{i}$ has the dimensionality of energy × length 1 / $^{2}$. 

The different terms in Eqn. (3.11) have different origins. The strongest interactions are presumably Γ$_{i}$ as this term already exists for uncoupled ladders. Inter-ladder interactions, such as interladder Coulomb repulsion, also contribute to Γ$_{i}$. However interladder hoping does not – this contribution is suppressed due to a mismatch between the Fermi momenta of the A$_{i}$ and B$_{i}$ bands. The coupling g is smaller than Γ$_{i}$: it arises only in second or- der perturbation theory from intraladder interactions and from presumed weak inter-ladder Coulomb interactions. 

The pair susceptibility for the quasiparticles Δ$_{QPA}$ in an RPA approximation is given by 

<img src='content_image/372219.jpg'>

