Double fluctuations on the attractive Hubbard model: ladder approximation

TL;DR

This paper investigates the impact of double fluctuations on the Hubbard model's spectral functions below the critical temperature, revealing complex peak structures and the persistence of pair physics into the superconducting phase.

Contribution

It introduces a novel approach to include double fluctuations in both diagonal and off-diagonal self-energies using T-Matrix equations, extending beyond traditional BCS theory.

Findings

Four peaks observed in spectral functions with asymmetric weights.

Additional gap forms around the chemical potential due to pair condensation.

Pair physics remains valid below the critical temperature.

Abstract

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 Z\"urich 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 $U/t = -6.0$, $T/t = 0.05$, $\mu/t = - 5.5$ and $\Delta/t = 1.5$, we find four peaks both for the diagonal, $A(n(\pi/16,\pi/16),\omega)$, and off-diagonal, $B(n(\pi/16,\pi/16),\omega)$, spectral functions. These peaks are not symmetric in pairs as previously found. In addition: (a) in $A(n(\pi/16,\pi/16),\omega)$, the far left peak has a vanishing small weight; (b) in $B(n(\pi/16,\pi/16),\omega)$ 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.