Bilayer ${t\text{−}J\text{−}J}_{\perp }$ Model and Magnetically Mediated Pairing in the Pressurized Nickelate ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$

The recently discovered nickelate superconductor ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ has a high transition temperature near 80 K under pressure, providing an additional avenue for exploring unconventional superconductivity. Here, with state-of-the-art tensor-network methods, we study a bilayer ${t\text{−}J\text{−}J}_{\perp }$ model for ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ and find a robust $s$-wave superconductive (SC) order mediated by interlayer magnetic couplings. Large-scale density matrix renormalization group calculations find algebraic pairing correlations with Luttinger parameter $K_{\mathrm{SC}}\lesssim 1$. Infinite projected entangled-pair state method obtains a nonzero SC order directly in the thermodynamic limit, and estimates a strong pairing strength ${\overline{\mathrm{\Delta }}}_z\sim \mathcal{O}(0.1)$. Tangent-space tensor renormalization group simulations elucidate the temperature evolution of SC pairing and further determine a high SC temperature $T_c^{*}/J\sim \mathcal{O}(0.1)$. Because of the intriguing orbital selective behaviors and strong Hund’s rule coupling in the compound, ${t\text{−}J\text{−}J}_{\perp }$ model has strong interlayer spin exchange (while negligible interlayer hopping), which greatly enhances the SC pairing in the bilayer system. Such a magnetically mediated pairing has also been observed recently in the optical lattice of ultracold atoms. Our accurate and comprehensive tensor-network calculations reveal a robust SC order in the bilayer ${t\text{−}J\text{−}J}_{\perp }$ model and shed light on the pairing mechanism of the high-$T_c$ nickelate superconductor.

Figure 1

(a) Two $e_g$ orbitals in the bilayer structure of the nickelate ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$. The quarter-filled $d_{x^2-y^2}$ orbitals form an effective $t\text{−}J$ model with intralayer hopping $t$ and spin exchange $J$. The $d_{z^2}$ orbital is localized and has an interlayer AF exchange through the $\sigma$ bonding. The spins of $d_{x^2-y^2}$ and $d_{z^2}$ orbitals are coupled through an on-site FM Hund’s rule coupling $J_{\mathrm{H}}$. In the large $J_{\mathrm{H}}$ limit, we arrive at (b) bilayer ${t\text{−}J\text{−}J}_{\perp }$ model, where the interlayer AF coupling $J_{\perp }$ is strong while the interlayer hopping is absent. The SC pairing correlation ${\mathrm{\Phi }}_{zz}(r)$ is between two interlayer pairing ${\mathrm{\Delta }}^{(†)}$ along the vertical $z$ direction and separated by a distance $r$ along the $x$ direction.

Figure 2

(a) Pairing correlation ${\mathrm{\Phi }}_{zz}$ on the $2\times W\times L$ bilayer lattices with widths $1\le W\le 3$ and long length $L$, namely, $2\times 1\times 128$ ($W=1$, $n_e=0.5$), $2\times 2\times 64$ ($W=2$, $n_e\approx 0.54$), and $2\times 3\times 48$ ($W=3$, $n_e=0.5$). The SC correlations exhibit algebraic behaviors as ${\mathrm{\Phi }}_{zz}(r)\sim r^{-K_{\mathrm{SC}}}$, enhanced with interlayer coupling $J_{\perp }$. The $W=2$ data fall into algebraic scaling with oscillations [31], leading to inaccurate extraction of the Luttinger parameters. (b) Spin-spin correlation $F(r)$ and the single-particle Green’s function $G(r)$ decay exponentially (see definitions in the main text) in the SC phase [44].

Figure 3

(a) The Luttinger parameter $(2-K_{\mathrm{SC}})$ vs $J_{\perp }$ calculated on various bilayer systems. The binding energy for $W=1$ system is also shown. (b) The SC order parameter ${\overline{\mathrm{\Delta }}}_z$ obtained with iPEPS (dashed horizontal lines) and DMRG (solid lines) for $J_{\perp }=2$, 4. In the latter, ${\overline{\mathrm{\Delta }}}_z$ is estimated within central columns, and the error bars represent the difference between the maximal and minimal values by varying the number of columns involved. We set $n_e=0.5$ for the $W=1$, 3 cases, while for the $W=2$ case it is shifted slightly to $n_e\simeq 0.54$. (c) Pairing correlation $|{\mathrm{\Phi }}_{zz}|$ with various interlayer hopping $t_{\perp }$. For the $t_{\perp }=0$ case, we show the pairing correlations in excellent data convergence as computed with $L_x=64$ and 128. (d) The electron density profiles $n_e(i)$, $1\le i\le L$ for various $J_{\perp }$ computed on the $2\times 1\times 64$ system.

Figure 4

Finite-temperature results for $W=1$ systems with length up to $L=128$. In panels (a)–(e), the data for $J_{\perp }=2$ (red lines) and 4 (green) are shown. In (a),(c)–(e) the hole density is fixed as $n_h\equiv 1-n_e=0.5$, and in (b) the data are calculated with fine-tuned chemical potential that leads to $n_h\simeq 0.5$. (a) Shows the specific heat $c_e$ with $T_l$ the lower characteristic temperature. (b) Shows the pairing susceptibility ${\chi }_{\mathrm{SC}}$, which diverges with a power-law scaling $T^{-\alpha }$ (the dashed lines) for $T\le T_c^{*}$. The interlayer pairing and AF correlations are illustrated in the inset. (c) Shows the interlayer hole correlations on the rungs, with $T_l$ and $T_c^{*}$ determined from (a) and (b), respectively. (d) Shows the magnetic susceptibility ${\chi }_m$ with a hump at $T_m$. (e) Shows the rung spin correlations. (f) The contour plot of the pairing correlation ${\mathrm{\Phi }}_{zz}(r=L/4)$ for various hole densities $n_h$ computed with $J_{\perp }=2$. The vertical red line denotes the $n_h=0.5$ case relevant for ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ (LNO).