Magnetism and superconductivity in bilayer nickelate

TL;DR

We address the question of unifying spin-density-wave magnetism with high-temperature superconductivity in bilayer nickelates without external stimuli such as pressure or strain. We propose the bilayer type-II $t$-$J$ model, derived from a double-Kondo description in the large $J_H$ limit, which retains a five-state local Hilbert space per site corresponding to two spin-1/2 and three spin-1 degrees of freedom. Using iDMRG on cylinders with $L_y=2$ and $L_y=4$ ($L_z=2$), we find a competition between double-exchange ferromagnetism and in-plane superexchange that yields period-4 SDW order at weak interlayer coupling, and a transition to interlayer $s$-wave superconductivity as $J_ $ increases. The interlayer pairing shows power-law decay and a spin gap opening in the high-$J_ $ regime, with the type-II model giving a more conservative estimate of pairing strength than a one-orbital model, underscoring the essential role of $d_{z^2}$ local moments. Overall, the bilayer type-II $t$-$J$ model provides a minimal, unified framework for magnetism and superconductivity in bilayer nickelates and suggests pressure/strain enhances interlayer coupling by straightening buckled Ni–O bonds.

Abstract

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.

Figures (13)

• Figure 1: (a) Illustration of the double Kondo model and type-II t-J model in the limit $J_H\rightarrow\infty$. The single arrow and double arrow correspond to the spin-$1/2$ and spin-$1$ moment, respectively. (b), (c) schematic phase diagrams of the bilayer type-II t-J model for $J_\parallel=0$ (b) and finite $J_\parallel>0$ (c). At small $J_\perp$, there is a magnetic ordered phase. As the interlayer coupling $J_\perp$ increases, the magnetism is suppressed and an interlayer s-wave superconductor emerges.
• Figure 2: iDMRG results for type II t-J model with $L_y=4, L_z=2$ and hole doping $x=0.5$. (a) spin structure factor $\langle\vec{S}(q_x,q_y=0)\cdot \vec{S}(-q_x,q_y=0)\rangle$ for single layer model with $J_\parallel=0$; (b) real space spin-spin correlation function for single layer type-II t-J model with $J_\parallel=0$; (c) the spin-spin correlation function $\langle \vec{S}(q_x,q_y=0)\cdot\vec{S}(-q_x,q_y=0)\rangle$ with $J_{\parallel}=0.5$ for single layer (red line) and bilayer (blue line) type-II t-J model with $J_\perp=0.2$; (d) the spin-spin correlation in real space for $J_{\parallel}=0.5, J_{\perp}=0.2$. In real-space spin-spin correlation plot, the size of the circle indicates the value of the correlation function $\langle \vec{S}(\mathbf r)\cdot \vec{S}(0)\rangle$, and the arrow indicates the direction of the spin configuration. $q_x$ is in units of $\pi$. In the plot of (c), we scale the data for single layer correlation (red line) to $4\langle\vec{S}({\bf q})\cdot\vec{S}(-{\bf q})\rangle$. The bond dimension is $m=7000$ for single layer model and $m=10000$ for bilayer model.
• Figure 3: The log-log plot of pair-pair correlation function for $L_y=2$, with $t=1$, $J_\parallel=0.5$ and hole doping (a) $x=0.1$, (b) $x=0.25$. The bond dimension is $m=10000$.
• Figure 4: iDMRG results for bilayer type II t-J model with $L_y=4$ and $J_\perp=0.5$, it is achieved with bond dimension $m=10000$. (a) the log-log plot of interlayer pair-pair correlation function for $L_y=4$, with $J_\parallel=0.5$ and hole doping $x=0.25$ at large $J_\perp$. (b) the ratio of correlation length $\xi_{2e}/\xi_e$ at $x=0.25$. (c) and (d) same plot as (a) and (b) at doping $x=0.5$. The interlayer pairing operator is defined as $\Delta(\mathbf r)=\epsilon_{\sigma\sigma^\prime}c_{t,\sigma}(\mathbf r)c_{b,\sigma^\prime}(\mathbf r)$.
• Figure 5: The spin susceptibility $\chi^{tt}(q,\omega=0)$ from itinerant electrons. The peak is around ${\bf Q}=(0.632\pi,0.632\pi)$. The red line correspond to two Fermi surfaces from the dispersion. To fit the $\alpha$ and $\beta$ Fermi pockets, we add the interlayer hopping term as $t^\perp_{{\bf k}}=4t^\perp(\cos(k_x)-\cos(k_y)+\eta(\cos(2k_x)-\cos(2k_y)))^2$ and use the parameters from Ref. PhysRevB.110.024514. The nearest, next-nearest neighbor, second-next-nearest neighbor intralayer hoppings are $t=0.29eV$, $t_2/t=-0.18$, $t_3/t=0.05$, and we use $t^\perp/t=-0.14$ and $\eta=0.15$.