4,j=1,...,4 n−1 n n+1 1 2 3 4 1 2 3 4 1 2 3 4 n+1, n t FIG. 2: An array of four leg ladders. As an example of the hopping assumed in the RPA analysis (hopping between every second leg), we show how electrons can hop between the fourth chain of the n-th 4-leg ladder and the chains on the n + 1-th ladders. t eff 2 ( k$_{y}$ ) = 2 ∑ n> 0 cos(4 k$_{y}$ ) $_{×}$( 2( s 2 1 + s 2 $_{2}$) t n, 0 1 , 1 − (2 s$_{1}$s$_{2}$ − s 2 $_{1}$)( t n, 0 1 , 2 + t n, 0 2 , $_{1}$) − 2 s$_{1}$s$_{2}$ ( t n, 0 3 , 1 + t n, 0 1 , $_{3}$) + s 2 $_{2}$( t n, 0 1 , 4 + t n, 0 4 , $_{1}$) ) ; Γ$_{1}$( k$_{y}$ ) = 2( s 2 1 + s 2 $_{2}$) + 2(2 s$_{1}$s$_{2}$ − s 2 $_{1}$) cos( k$_{y}$ ) − 4 s$_{1}$s$_{2}$ cos(2 k$_{y}$ ) − 2 s 2 $_{2}$cos(3 k$_{y}$ ); Γ$_{2}$( k$_{y}$ ) = 2( s 2 1 + s 2 $_{2}$) − 2(2 s$_{1}$s$_{2}$ − s 2 $_{1}$) cos( k$_{y}$ ) − 4 s$_{1}$s$_{2}$ cos(2 k$_{y}$ ) + 2 s 2 $_{2}$cos(3 k$_{y}$ ) , (3.4) and s$_{1}$ = sin( π/ 5) and s$_{2}$ = sin(2 π/ 5). We have assumed the hopping is real and that the low energy contribution to G 2D RPA comes from the A -bands as Δ$_{A}$ ≪ Δ$_{B}$. Thus G$_{A}$$_{1}$ ( ω, k$_{x}$ ) /G$_{A}$$_{2}$ ( ω, k$_{x}$ ) are the Green’s functions of the A -band electrons on a given 4- leg ladder. As we have discussed in the previous section G$_{A}$$_{1}$/G$_{A}$$_{2}$ are no more than the bonding/anti-bonding electron Greens functions for a 2-leg ladder. The RPA does not mix G$_{A}$$_{1}$ and G$_{A}$$_{2}$ as the weights of the two are found near differing Fermi wavevectors (i.e. we can take G$_{A}$$_{1}$ ( k ) G$_{A}$$_{2}$ ( k ) ∼ 0 safely for all k ). The presence of Γ$_{1}$( k$_{y}$ ) and Γ$_{2}$( k$_{y}$ ) act as structure factors which cause the quasiparticle weight at various k$_{y}$ to be negligible. While the denominator of G 2D RPA has the periodicity of the reduced Brillouin zone i.e. k$_{y}$ and k$_{y}$ + π/ 2 are identified) these structure functions merely have the periodicity of the original zone i.e. k$_{y}$ and k$_{y}$ + 2 π are identified). 8