Recent advances and prospects in theoretical study of bilayer nickelate superconductor La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>

The high-temperature superconductivity in bilayer nickelate La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> under high pressures, which was discovered in 2023, has spurred intensive theoretical and numerical investigations. These studies aim to unravel physical properties of La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> from various aspects, with particular emphasis on its pairing symmetry and underlying superconducting mechanism. Moreover, significant effort has also been made to explore and predict novel nickel-based superconductors related to La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>. This article reviews these recent advancements aimed at elucidating the physical properties and superconducting mechanism of La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>, whose multi-orbital characteristics and intricate electronic correlations have spawned diverse theories for its pairing mechanism. In this article, the recent findings on La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> are summarized regarding its macroscopic models, pairing symmetry, normal state characteristics, and the structure of spin and charge density waves. Particular attention is paid to the debate surrounding the role of <i>σ</i>-bonding band metallization in superconductivity. Finally, this article also presents an outlook on future studies crucial for advancing our understanding of La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> superconductivity.