[Paper Review] Hund's Rule, Interorbital Hybridization, and High-$T_c$ Superconductivity in the Bilayer Nickelate

Xing-Zhou Qu, Dai-Wei Qu+3 more|arXiv (Cornell University)|Nov 21, 2023

Physics of Superconductivity and Magnetism8 citations

TL;DR

The paper studies a two-orbital bilayer $t$-$J$ model for La3Ni2O7 using DMRG and finite-$T$ tensor-network methods, showing robust interorbital superconductivity dominated by the $d_{x^{2}-y^{2}}$ orbital and clarifying the roles of Hund’s rule coupling and interorbital hybridization.

ABSTRACT

Understanding the pairing mechanism in bilayer nickelate superconductors constitutes a fascinating quest. Here we investigate the intriguing interplay between Hund's rule coupling and interorbital hybridization in a two-orbital model for bilayer nickelates, using a comprehensive tensor network approach: density matrix renormalization group for finite-size systems, infinite projected entangled-pair states in the thermodynamic limit, and thermal tensor networks for finite-temperature properties. We explain the pressure-dependent high-$T_c$ superconductivity observed in experiment, by identifying three distinct superconductive (SC) regimes: hybridization dominant, Hund's rule dominant, and the hybrid-Hund synergistic SC regimes. In these SC regimes, both $d_{x^2-y^2}$ and $d_{z^2}$ orbitals exhibit algebraic pairing correlations with similar Luttinger parameters $K_{\mathrm{SC}}$. However, the former exhibits a much stronger amplitude than the latter, with a distinctly higher SC characteristic temperature $T_c^*$, below which the pairing susceptibility diverges as $χ_{\mathrm{SC}}(T) \sim 1/T^{2-K_{\mathrm{SC}}}$. With realistic model parameters, we find the pressurized La$_3$Ni$_2$O$_7$ falls into the Hund's rule dominated SC regime. As hybridization further enhances with pressure, it leads to significant interorbital frustration and in turn suppresses the SC correlations, explaining the rise and fall of high-$T_c$ superconductivity under high pressure. Our results offer a comprehensive understanding of the interlayer pairing in superconducting La$_3$Ni$_2$O$_7$.

Motivation & Objective

• Investigate how the two $e_g$ orbitals ($d_{x^{2}-y^{2}}$ and $d_{z^{2}}$) contribute to superconductivity in La3Ni2O7 under pressure.
• Determine the impact of interorbital hybridization and Hund’s rule coupling on superconducting order.
• Quantify ground-state pairing correlations and finite-temperature pairing susceptibilities for both orbitals.
• Identify conditions under which each mechanism (Hund’s rule vs hybridization) stabilizes superconductivity.
• Clarify the comparative roles of itinerancy vs localization of the two orbitals in mediating pairing.

Proposed method

• Study a bilayer $t$-$J$ model with intralayer $t_c$, $J_c$ and interlayer $t_ ext{perp}$, $J_ ext{perp}$ for $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals.
• Include inter-orbital hybridization $V$ and on-site Hund’s rule coupling $J_H$ with rotationally invariant form.
• Use zero-temperature DMRG on $2 imes L_x$ ladders (up to $L_x=64$) with up to $D^*=4{,}500$ multiplets and finite-temperature tangent-space TRG on $L_x=24$.
• Compute orbital-resolved pairing correlations $\
• :Phi_{zz}(r)\
• and pairing susceptibility $SC$ under a small pairing field to identify divergences.
• Tune model parameters to reflect realistic values from DFT (e.g., $t_c$, $t_ ext{perp}$, $t_d$, $V$, $ abla \, $) and explore parameter space for Hund’s vs hybridization dominance.

Experimental results

Research questions

• RQ1 What is the relative strength of superconducting correlations in the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals?
• RQ2 How do inter-orbital hybridization $V$ and Hund’s rule coupling $J_H$ influence the emergence and robustness of SC order?
• RQ3 Can Hund’s rule coupling transfer interlayer AF correlations from $d_{z^{2}}$ to $d_{x^{2}-y^{2}}$ to mediate pairing?
• RQ4 Are there distinct SC phases dominated by Hund’s rule versus hybridization, and is there a non-SC regime between them?
• RQ5 What is the estimated high-temperature SC scale $T_c^*$ from orbital-resolved pairing susceptibilities?

Key findings

• Both $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals show algebraic SC pairing in the ground state, with $d_{x^{2}-y^{2}}$ stronger.
• Finite-$T$ pairing susceptibilities diverge as a power law, with $T_c^* \,\approx\, 0.03 t_c$ for the $d_{x^{2}-y^{2}}$ orbital and weaker divergence for $d_{z^{2}}$.
• Sufficient Hund’s rule coupling ($J_H \gtrsim 1$ eV) enables interlayer AF correlations to transfer to the $d_{x^{2}-y^{2}}$ orbital, promoting robust SC.
• Hybridization can also stabilize SC; with strong $V$, SC can emerge even without large $J_H$, but an intermediate non-SC regime appears when $J_H$ and $V$ compete.
• Two SC regimes are identified: SC dominated by Hund’s rule and SC dominated by hybridization, separated by a non-SC region.
• The $d_{x^{2}-y^{2}}$ orbital is itinerant within layers and primarily responsible for high-$T_c$ SC, while $d_{z^{2}}$ is localized and nearly half-filled.