The $GW$ approximation is a widely used framework for studying correlated materials, but it struggles with certain limitations, such as its inability to explain pseudogap phenomena. To overcome these problems, we propose a systematic theoretical framework for Green's function corrections and apply it specifically to the $GW$ approximation. In this new theory, the screened potential is reconnected to the physical response function, i.e., the covariant response function proposed by Li et al. [Phys. Rev. B 107, 085106 (2023)], rather than using the random phase approximation formula. We apply our scheme to calculate Green's function, the spectral function, and the charge compressibility in the two-dimensional Hubbard model. Our scheme yields significant qualitative and quantitative improvements over the standard $GW$ method and successfully captures the pseudogap behavior.

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• Electronic structure
• Magnetism
• Pseudogap
• GW method
• Hubbard model
• Many-body techniques