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Motivated by the recently discovered high-temperature superconductorLa$_3$Ni$_2$O$_7$ under pressure, we comprehensively study this system usingdensity functional theory and the random phase approximation (RPA), focusing onpressure effects. At low pressures, the Amam phase is stable, containing theY$^{2-}$ mode distortion from the Fmmm phase, while the Fmmm phase is unstable.Because of small differences in enthalpy and a considerable Y$^{2-}$ modeamplitude, the two phases may coexist in the range between 10.6 and 14 Gpa,beyond which the Fmmm phase dominates. In addition, our density functionaltheory and RPA calculations found that the magnetic stripe-type spin order withwavevector ($\pi$, 0) becomes stable at intermediate Hubbard and Hundcouplings. Pairing is induced in the $s_{\pm}$-wave channel due to nestingbetween the {\bf M}=$(\pi,\pi)$ and the {\bf X}=$(\pi,0)$ and {\bf Y}=$(0,\pi)$points in the Brillouin zone. This order resembles results for iron-basedsuperconductors, but for the nickelate of our focus the orbital $d_{3z^2-r^2}$dominates, establishing a fundamental difference with iron pnictides andselenides. Within RPA, we predict that the critical temperature willsubstantially increase if the Fmmm phase is stabilized at lower pressures thancurrently achieved, via strain or growing on appropriate substrates, inagreement with tendencies in recent experimental results.

Structural phase transition, $s_{\pm}$-wave pairing and magnetic stripe  order in the bilayered nickelate superconductor La$_3$Ni$_2$O$_7$ under  pressure