content_image/371041.jpg content_image/371040.jpg content_image/371042.jpg content_image/371038.jpg content_image/371039.jpg
In the second scenario, we assume Δ$_{A}$ ≪ t$_{⊥}$ ≪ Δ$_{B}$. In this case t$_{⊥}$ wipes out the effects of interactions on the A-bands. Coupling them together then gives us an anisotropic two dimensional Fermi liquid. But as t$_{⊥}$ is much smaller than Δ$_{B}$, the Cooperons on the outer bands at zeroth order remain unperturbed. The coupling then between the anisotropic Fermi liquid and the B-Cooperons induces superconductivity in the system as a whole. This superconductivity is d-wave in nature. 

We now elaborate on these two scenarios. 

## A. Scenario I 

We treat the interladder hopping through a random phase approximation (RPA) analysis of the interladder hopping. The form of the hopping is taken to be long range 

<img src='content_image/371041.jpg'>

where a, b = 1 , ..., 4 run over the legs of an individual ladder and n and m mark the n ’th and m ’th ladders. By particle-hole symmetry the hopping is assumed to have peaks both near k$_{⊥}$ = 0 and k = G/ 2 where G = (0 , π/ 2) is the inverse lattice vector perpendicular to the ladders. In particular the hopping takes the form 

<img src='content_image/371040.jpg'>

where f$_{ab}$ ( m − n ) = f$_{ba}$ ( n − m ) and f$_{ab}$ (0) = 0 (i.e. no (additional) hopping within a ladder). 

By treating H$_{interladder}$ in an RPA approach, we find that the single particle Green’s function takes the form 

<img src='content_image/371042.jpg'>

where 

<img src='content_image/371038.jpg'>

<img src='content_image/371039.jpg'>

