The attractive Fermi-Hubbard model stands out as a simple model for studying the pairing and superconductivity of fermions on a lattice. In this article, we apply several many-body theories in the three-dimensional attractive Hubbard model. Specifically, we compare the results of various $GW$ methods with determinant quantum Monte Carlo simulations, and we observe that they provide reliable results in the weak- to intermediate-coupling regime. The critical exponents also agree well with the accurate results obtained from the three-dimensional $XY$ model. Additionally, we propose a method to determine the temperature at which the pseudogap appears. In the superconducting phase, we implement the covariance method within the $GW$ framework to calculate correlation functions. This method satisfies both the Ward-Takahashi identity (WTI) and the fluctuation-dissipation theorem, whereas the random phase approximation and the Bethe-Salpeter equation within $GW$ violate these principles. For systems with continuous spontaneous symmetry breaking, the WTI rigorously ensures the existence of massless Goldstone modes. Moreover, we numerically verify the appearance of Goldstone modes in the superconducting phase within the covariant-$GW$ approach. Based on the covariance method, we further incorporate corrections beyond the $GW$ approximation—a technique termed the post-$GW$ method which gives quite accurate results for the Green's function in the superconducting phase.

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• Superconductivity
• Diffusion quantum Monte Carlo
• GW method
• Green's function methods
• Hubbard model