where κ$_{0}$ is the small parameter guaranteeing that the RPA is a good approximation. The dispersion relations of the quasi-particles near the hole pockets are In Figure 3 are plotted the expected Fermi pockets. On the l.h.s. of Figure 3 are plotted the pockets found at zero chemical potential while on the r.h.s. are plotted the pockets for a chemical potential such that 2Δ$_{A}$ + 2 µ > t$_{0}$ > 2Δ$_{A}$ − 2 µ . For such a condition one obtains only hole pockets. We see that hole pockets occur in the vicinity of ( ± π/ 2 , ± π/ 2). The Luttinger sum rule (LSR) for the single particle Green’s functions at the particle hole symmetric point takes the form where n is the electron density. The corresponding Luttinger surface of G ( ω, k ) is defined as the loci of points in k -space where G ( ω = 0 , k ) changes sign. These sign changes occur both at the poles and the zeros of G . In order to apply the Luttinger sum rule, we must take G ( ω, k ) to be one of G 2 D A 1 / $_{2}$, i.e. we must apply the LSR to each band separately (see Eqn. (3.3) for the definition of G 2 D A 1 / $_{2}$). (We only apply the LSR to the electrons in the A-bands – the LSR also holds separately for electrons in the B-bands.) At the particle-hole symmetric point, zeros are present in G 2 D A 1 / $_{2}$(0 , k ) along the lines k$_{y}$ = ± K$_{F}$$_{A}$$_{i}$ . In the absence of pockets the LSR is satisfied G ( ω = 0 , k$_{x}$ = K$_{F}$$_{A}$$_{1}$$_{,}$$_{2}$, k$_{y}$ ) because of these zeros. And when t eff i becomes strong enough so that pockets form, the appearance of equally size electron and hole pockets on either side of ± K$_{FA}$$_{i}$ ensure that the Luttinger sum rule continues to hold. Introducing a finite chemical potential (with µ < Δ$_{A}$ / 2) leaves the LSR violated as expressed in Eqn. (3.10). However it continues to hold in a modified form. Because in a finite chemical potential, the ladder Greens functions are given by G$_{A}$$_{i}$ ( ω, µ, k ) = G$_{A}$$_{i}$ ( ω − µ, 0 , k ), 10 n = 2 (2 π ) 2 ∫$_{G}$$_{(}$$_{ω}$$_{=0}$$_{,k}$$_{)}$$_{>}$$_{0}$ d $^{d}$k, (3.10) where p$_{i}$ = ± K$_{FA}$$_{i}$ ∓ t eff i (0) 2 v$_{Fi}$ , ϵ$_{Fi}$ = ( γ$_{i}$t eff i (0)) 2 8 m$_{|}$$_{|}$$_{i}$v 2 Fi , γ$_{i}$ = (1 − 4 ( t eff i (0)) $_{2}$(Δ 2 − µ 2 + µt eff i (0))) 1 / $^{2}$, m$_{⊥}$$_{i}$ = κ 2 0 2 t eff i $_{(0)}$, and m$_{|}$$_{|}$$_{i}$ = t eff i (0) − 2 µ 2 v 2 Fi . E$_{i}$ ( k$_{x}$, k$_{y}$ ) = ( k$_{x}$ − p$_{i}$ ) 2 2 m$_{|}$$_{|}$$_{i}$ + ( k$_{y}$ − K$_{y}$ ) 2 2 m$_{⊥}$$_{i}$ − ϵ$_{Fi}$ (3.9) 1. Luttinger sum rule