Magnetism and superconductivity in bilayer nickelate

The discovery of high-temperature superconductivity in bilayer nickelate La${3}$Ni${2}$O${7}$ necessitates a minimal theoretical model that unifies the superconducting phase with the spin-density-wave (SDW) phase without external pressure or strain. We propose a model where half-filled $d{z^{2}}$ local moments interact with itinerant $d_{x^{2}-y^{2}}$ electrons via strong Hund’s coupling $J_H$, which reduces to a bilayer type-II t-J model in the large $J_H$ limit. Using iDMRG calculations on an $L_y=4, L_z=2$ cylinder, we demonstrate that the competition between double-exchange ferromagnetism and in-plane superexchange generates period-4 stripe-like SDW order-a feature absent in one-orbital t-J model with only $d_{x^2-y^2}$ orbital. Furthermore, increasing the interlayer exchange coupling suppresses magnetic order and stabilizes interlayer s-wave superconductivity. These results identify the type-II t-J model as a minimal framework for capturing the interplay of magnetism and superconductivity in bilayer nickelates.

💡 Research Summary

The paper addresses the pressing need for a unified theoretical framework that can simultaneously describe the spin‑density‑wave (SDW) phase observed at ambient pressure and the high‑temperature superconducting (SC) phase that emerges under pressure or strain in the bilayer nickelate La₃Ni₂O₇. The authors argue that a minimal model must retain both the itinerant d_{x²‑y²} electrons and the localized d_{z²} moments, because the latter are essential for reproducing the experimentally observed magnetic order. Starting from a bilayer Kondo‑lattice description, they take the strong Hund’s coupling limit (J_H → ∞) and project onto the low‑energy subspace. This yields a “type‑II t‑J” model with five local states per site: two spin‑½ singlons (Ni³⁺) and three spin‑1 doublons (Ni²⁺). The Hamiltonian contains intra‑layer hopping t, intra‑layer super‑exchange J_∥ (acting on the d_{x²‑y²} spins), and inter‑layer exchange J_⊥ originating from the d_{z²} orbitals. Importantly, the inter‑layer couplings satisfy J_ss⊥ = 2 J_sd⊥ = 4 J_dd⊥ = J_⊥, while intra‑layer spin‑spin and spin‑doublon couplings are set by J_∥.

Using infinite‑density‑matrix‑renormalization‑group (iDMRG) on cylinders with width L_y = 4 and two layers (L_z = 2), the authors explore the phase diagram as a function of J_⊥ and J_∥. At small J_⊥ (≤ 0.5 t) the system exhibits magnetic order. When J_∥ = 0 the ground state is ferromagnetic due to double‑exchange; a finite J_∥ introduces antiferromagnetic super‑exchange that competes with double‑exchange, leading to a compromise: a period‑4 stripe‑like SDW with ordering vector Q ≈ (π/2, 0). This SDW is absent in a one‑orbital t‑J model that contains only the d_{x²‑y²} orbital, underscoring the necessity of the d_{z²} moments.

Increasing J_⊥ suppresses the SDW. Finite‑size DMRG calculations on L_y = 2 cylinders reveal that for J_⊥ ≤ 0.1 t the spin gap Δ_S is essentially zero, indicating a gapless Luttinger‑liquid phase with central charge c ≈ 3. For J_⊥ ≥ 0.3 t a finite spin gap opens, signalling a Luther‑Emery liquid—the one‑dimensional analogue of a superconducting state. Pair‑pair correlation functions for inter‑layer Cooper pairs Δ_y(x) = ε_{σσ′} c_{t,σ}(x) c_{b,σ′}(x) decay as a power law with exponent α < 2 when J_⊥ ≥ 0.5 t (x = 0.1) or J_⊥ ≥ 1 t (x = 0.25), confirming robust inter‑layer s‑wave pairing. For higher doping (x = 0.5) the system becomes insulating at large J_⊥, with one Cooper pair per unit cell, consistent with a fully paired state.

On wider cylinders (L_y = 4) the same trend persists: at J_⊥ ≥ 1 t the inter‑layer pair‑pair correlations exhibit clear power‑law decay, while the spin correlations are short‑ranged, indicating a dominant superconducting tendency. Correlation‑length analysis shows the ratio ξ_{pair}/ξ_{single} grows with bond dimension for large J_⊥, further supporting pair formation. The SDW‑SC transition occurs in the interval J_⊥,c ∈ (0.5, 1), although the precise nature of the transition (continuous vs first‑order, possible coexistence) remains unresolved due to computational limitations.

The authors connect their findings to experiment by proposing that pressure or epitaxial strain primarily enhances J_⊥ by straightening the buckled Ni‑O bonds along the c‑axis. This explains why SDW order is observed at ambient conditions (small J_⊥) and superconductivity appears under pressure (large J_⊥). They also compute the itinerant‑electron spin susceptibility χ(q) and find its maximum at Q ≈ (0.63π, 0.63π), close to the experimentally reported Q = (π/2, π/2). They suggest that small symmetry‑breaking perturbations—such as anisotropic inter‑layer hopping terms or structural orthorhombicity—could shift the ordering vector to the diagonal direction observed experimentally.

In summary, the work establishes the bilayer type‑II t‑J model as a minimal yet realistic description of La₃Ni₂O₇, capturing both the period‑4 SDW at weak inter‑layer exchange and the inter‑layer s‑wave superconductivity at strong inter‑layer exchange. The study highlights the crucial role of the localized d_{z²} moments, the competition between double‑exchange ferromagnetism and super‑exchange antiferromagnetism, and the tunability of magnetic versus superconducting order via the inter‑layer exchange coupling. This framework provides a solid foundation for future theoretical and experimental investigations of multiorbital high‑Tc nickelates.