The recently discovered high-$T_c$ superconductivity in ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ with $T_c\approx 80$ K provides another intriguing platform to explore the microscopic mechanism of unconventional superconductivity. In this work, we study a previously proposed bilayer two-orbital model Hamiltonian for ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ [Y. Shen et al., Chin. Phys. Lett. 40, 127401 (2023)] on a plaquette ladder, which is a minimum setup with two-dimensional characteristic. We employ large-scale density matrix renormalization group calculations to accurately determine the ground state of the model. We determine the density, magnetic structure, and the pairing property of the model. We find that with large effective interlayer antiferromagnetic exchange for the $3d_{z^2}$ orbital, the spin, charge, and pairing correlation display quasi-long-range behavior, which could be viewed as a precursor of possible true long-range order in the two-dimensional limit. Interestingly, sign oscillation for the pairing correlation is observed for both the $3d_{x^2-y^2}$ and $3d_{z^2}$ orbitals, indicating the presence of a possible pair density wave in the system. Even though we only study the model on a quasi-one-dimensional plaquette ladder geometry due to the computational difficulty, the results for the spin, charge, and pairing correlation provide valuable insight towards the clarification of the properties of ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$.

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• Charge density waves
• Exchange interaction
• Superconductivity
• 1-dimensional systems
• Nickelates
• Density matrix renormalization group
• Lattice models in condensed matter