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Double fluctuations on the attractive Hubbard model: ladder approximation

We explore, for the first time, the effect of double fluctuations on both the diagonal and off-diagonal self-energy. We use the T-Matrix equations below $T_c$, developed recently by the Zürich group (M.H. Pedersen et al) for the local pair attraction Hamiltonian. Here, we include as well the effect of fluctuations on the order parameter (beyond the BCS solution) up to second order in $U/t$. This is equivalent to approximating the effective interaction by $U$ in the off-diagonal self-energy. For $U/t = -6.0$, $T/t = 0.05$, $μ/t = - 5.5$ and $Δ/t = 1.5$, we find four peaks both for the diagonal, $A(n(π/16,π/16),ω)$, and off-diagonal, $B(n(π/16,π/16),ω)$, spectral functions. These peaks are not symmetric in pairs as previously found. In addition: (a) in $A(n(π/16,π/16),ω)$, the far left peak has a vanishing small weight; (b) in $B(n(π/16,π/16),ω)$ the far left and far right peaks have very small weights. The physical picture is, then, that the pair physics in the normal phase ($T > T_c$) is still valid below $T_c$. However, the condensation of the e-h pairs produces an additional gap around the chemical potential as in BCS, in other words, superconductivity opens a gap in the lower branch of a Hubbard-type-I solution.