High spin, low spin or gapped spins: magnetism in the bilayer nickelates

Abstract

Inspired by the recent discovery of high-temperature superconductivity in bilayer nickelates, we investigate the role of magnetism emerging from a hypothetical insulating $d^8$ parent state. We demonstrate that due to the interplay of superexchange and Hund's coupling, the system can be in a high-spin, low-spin or spin-gapped state. The low-spin state has singlets across the bilayer in the $d_{z^2}$ orbital, with charge carriers in the $d_{x^2-y^2}$ orbital. Thus, at low energy scales, it behaves as an effective one band system when hole doped. By contrast, the high-spin state is a more robust, spin-1 antiferromagnet. Using Hartree-Fock methods, we find that for fixed interaction strength and doping, high-spin magnetism remains more robust than the low-spin counterpart. Whether this implies that the high spin state provides a stronger pairing glue, or more strongly competes with superconductivity remains an open question. Our analysis therefore underscores the importance of identifying the spin state for understanding superconductivity in nickelates.

Figures (10)

• Figure 1: (a) Illustration of the bilayer two-orbital Hubbard model. The system consists of a bilayer square lattice with in-plane hopping $t_\parallel$ for $d_{x^2 - y^2}$ orbital and interlayer hopping $t_\perp$ for $d_{z^2}$ orbital. A ferromagnetic Hund’s coupling $J_H$ and an onsite Hubbard $U$ are included. (b) Schematic phase diagram across hole doping from the hypothetical $3d^8$ parent state. La$_3$Ni$_2$O$_7$, with an average Ni valence of $d^{7.5}$, can be viewed as a 50% hole-doped state from a $d^8$ Mott insulator. (c) Electronic configuration of Ni at $d^8$ and $d^{7}$.
• Figure 2: (a,b) Schematic phase diagram and magnetic phases of $J_\parallel-J_\perp-J_H$ model, Eq. \ref{['eq:two_heisenberg']}. (a) The phase diagram exhibits a second-order phase transition between a Néel-ordered phase at small $J_\perp$ and a interlayer singlet phase at large $J_\perp$. Within the Néel phase, a crossover occurs from a low-spin to a high-spin AFM as $J_H$ increases. (b) Illustrations of the three phases: (I) interlayer singlet, (II) low-spin AFM, and (III) high-spin AFM. Each panel shows four horizontal layers representing two orbitals on each bilayer plane. Black ellipses with red lines in (I) and (II) indicate interlayer singlet bonds.
• Figure 3: (a) Phase diagram from bond-operator mean-field theory (BOMFT). Blue, yellow region denotes interlayer singlet and Néel order, respectively. The black line indicates the second order phase transition boundary between these two phases. (b) $J_H$ dependence of orbital-summed Néel order, $N=N_{x^2-y^2}+N_{z^2}$ (purple line) from Holstein-Primakov theory at $J_\perp = 5$. Orbital-resolved Néel orders, $N_{x^2-y^2} = (-1)^{i + l} \langle S^{d;z}_{i,l} \rangle$ and $N_{z^2} = (-1)^{i + l} \langle S^{f;z}_{i,l} \rangle$, are plotted in blue, red dashed line, respectively. The system transitions from low-spin (spin-1/2) AFM on the $X$-orbital at small $J_H$ to high-spin (spin-1) AFM from both orbitals at large $J_H$. Inset plots $\partial N / \partial J_H$ which highlights that $N_{x^2-y^2}$ decreases while $N_{z^2}$ increases with increasing $J_H$ at large $J_H$ regime.
• Figure 4: Hartree-Fock (HF) results for Néel order ($N$) under hole doping $x$ of (a) low-spin and (b) high-spin configurations for values of $U/t_\parallel$. Here, $N=(-1)^{i+\ell}\langle S^{d;z}_{i,\ell}+S^{f;z}_{i,\ell}\rangle$ is an orbital-summed Néel order. We used the hopping parameters $t_\parallel = 0.483$, $t_\perp = 0.635$, and onsite energy difference $\Delta = 0.367$ from DFT calculations PhysRevLett.131.126001. In both cases, we observe first-order or weakly first-order transitions. The AFM order in the high-spin case is more robust than in the low-spin case, as indicated by a larger critical doping $x_c$
• Figure S1: Phase diagram from Schwinger-boson mean-field theory and bond-operator mean-field theory. Red and blue lines denote the critical coupling $J_\perp^{c}$ separating the Néel and disordered phases, obtained from Schwinger-boson mean-field theory (SBMFT) and bond-operator mean-field theory (BOMFT), respectively. The red star indicates the QMC result of $J_\perp^c=7.15$ for the spin-1 bilayer Heisenberg model PhysRevB.84.214412.