Recently, ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ thin film on the ${\mathrm{LaAlO}}_3$ substrate is shown to be superconducting, while the bulk ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ with the same in-plane lattice constant under pressure does not superconduct. This difference suggests the interlayer distance $d_{\mathrm{Ni}-\mathrm{Ni}}$ is crucial to control superconductivity, and its variation under pressure may tune the ground state sensitively. We investigate systematically the ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7/{\mathrm{LaAlO}}_3$ thin films in a reasonable range of $d_{\mathrm{Ni}-\mathrm{Ni}}$, by a combination of the first-principle calculations and the singular-mode functional renormalization group. For smaller (larger) $d_{\mathrm{Ni}-\mathrm{Ni}}$, the ground state is a C-type (G-type) spin density wave with spins coupled ferromagnetically (antiferromagnetically) across the two layers. Between the two phases, $s_{\pm }$-wave superconductivity emerges with dominant pairings between nickel $3d_{3z^2-r^2}$ orbitals. The results explain the experimental superconductivity in the thin film under ambient pressure, and predicts that the applied pressure will decrease the superconducting transition temperature, until the system enters the C-type spin density wave. Experimental verification would provide profound insights into the nature of electron correlations in this system, since the C-type spin density wave is achieved most naturally in the itinerant picture, while it would be hard in the local moment picture where spins are always coupled antiferromagnetically across the layers.

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• Pressure effects
• Spin density waves
• Superconductivity
• Nickelates
• Thin films
• Unconventional superconductors
• Density functional calculations
• Renormalization group