Electronic structure and two-orbital model of the quadlayer La${}_5$Ni${}_4$O${}_{13}$

The discovery of pressure-induced superconductivity in Ruddlesden–Popper (RP) nickelates has stimulated extensive interest in high-T${}_c$ superconductors. Here, we systematically study the electronic properties of the quadlayer RP nickelate La${}_5$Ni${}_4$O${}_{13}$ under ambient pressure, 5% isotropic compressive strain, and 4% $c$-axis uniaxial strain using density functional theory (DFT) and random phase approximation (RPA) calculations. DFT calculations show that isotropic strain broadens the Ni-$e_g$ bands and induces charge transfer from O-$p$ to Ni-$d$ orbitals, whereas $c$-axis uniaxial strain selectively shifts the $d_{z^2}$-derived bonding1 band upward while leaving the $d_{x^2-y^2}$ dispersion nearly unchanged. From Wannier downfolding, we construct a quadlayer two-orbital model that reproduces the low-energy Ni-$e_g$ bands. Our model reveals that under ambient pressure and 5% isotropic strain, the Fermi surface consists of two electron pockets ($\alpha$ and $\delta$) and three hole pockets ($\beta$, ${\beta }^{\mathrm{\prime }}$, and ${\beta }^{\mathrm{\prime }\mathrm{\prime }}$), while under uniaxial strain, a $\gamma$ hole pocket with $d_{z^2}$ orbital character emerges. RPA calculations reveal that the leading spin response shifts from $\mathbf{q}\approx (2\pi \mathrm{/}3,2\pi \mathrm{/}3)$ at ambient pressure to $\mathbf{q}\approx (\pi ,\pi )$ under both strain conditions and is enhanced under $c$-axis compression. These results suggest that $c$-axis compression may provide a favorable route to superconductivity in the quadlayer nickelate analogous to that in bilayer and trilayer nickelates.