This review provides a comprehensive survey of weak-coupling theoretical approaches applied to understand the emergent superconductivity in the pressurized nickelate bilayer system La₃Ni₂O₇. Following the landmark discovery of its high-T_c superconductivity under pressure, this material has rapidly become a new paradigmatic platform in the field of unconventional superconductivity, joining cuprates and iron-based systems. We focus on three pivotal theoretical frameworks: the random phase approximation (RPA), the fluctuation-exchange approximation (FLEX), and the functional renormalization group (FRG). These methods are deployed to analyze the effective pairing interactions and emergent instabilities arising from the low-energy electronic structure, which is commonly modeled by a bilayer two-orbital Hubbard Hamiltonian incorporating the Ni \mathrmd_x^2-y^2 and \mathrmd_3z^2-r^2 orbitals.

A central theme consolidating the weak-coupling perspective is the crucial role of Fermi surface topology and nesting. Theoretical studies consistently identify a multi-pocket Fermi surface under pressure, featuring an electron-like α pocket and two hole-like β and γ pockets. The γ pocket, predominantly derived from the \mathrmd_3z^2-r^2 orbital, exhibits strong nesting with the other pockets. This nesting significantly enhances antiferromagnetic spin fluctuations, which in turn mediate attractive pairing interactions. Consequently, a dominant \mathrms_\pm-wave pairing symmetry is widely predicted across different methodologies. In this state, the superconducting gap function maintains the same sign on the γ and α pockets but reverses sign on the β pocket, a structure intimately linked to the interlayer pairing channel dominated by the \mathrmd_3z^2-r^2 orbital.

We further elaborate on the methodological distinctions and complementary strengths of the three weak-coupling approaches surveyed. The random phase approximation (RPA) provides an efficient description of spin and charge susceptibilities by summing an infinite series of bubble and ladder diagrams, offering a transparent link between Fermi surface nesting and the emergence of leading pairing instabilities. The fluctuation-exchange approximation (FLEX) improves upon RPA by incorporating self-consistency, thereby capturing the mutual renormalization between quasiparticle properties and collective spin fluctuations, which is essential for a more accurate determination of the pairing interaction in the intermediate coupling regime. The functional renormalization group (FRG) goes beyond static susceptibility calculations by integrating out high-energy degrees of freedom progressively, allowing for an unbiased treatment of competing instabilities—such as superconductivity, spin-density wave, and charge-density wave orders—within a unified framework. These complementary techniques collectively reinforce the conclusion that spin-fluctuation-mediated pairing, driven by the interlayer nesting between the \mathrmd_3z^2−r^2 -dominated γ pocket and other Fermi surface sheets, is the primary mechanism underlying superconductivity in the bilayer nickelate system.

The review systematically compares predictions from these approaches regarding the pairing strength, competition between different symmetries (e.g., s-wave vs. d-wave), and the influence of key material parameters such as interaction strength U, Hund’s coupling J_H, crystal field splitting, and interlayer hybridization. Notably, the theoretical understanding is extended to ambient-pressure thin films, exploring the effects of substrate strain and the ongoing debate concerning the presence of the γ pocket. We also discuss how the pressure dependence of T_c and the material’s response to disorder are addressed within the weak-coupling paradigm.

By synthesizing results from these advanced many-body techniques, this review highlights the success of weak-coupling, itinerant-based theories in capturing the essential physics of La₃Ni₂O₇, particularly in linking its high-T_\mathrmc superconductivity to spin-fluctuation-mediated pairing driven by specific Fermi surface geometries. Finally, we outline outstanding challenges and future directions, emphasizing the need for closer integration with strong-coupling pictures, more precise ab initio-derived model parameters, and definitive experimental tests to distinguish between competing theoretical scenarios and fully elucidate the superconducting mechanism in this fascinating nickelate system.