Pairing Symmetry Crossover from $d$-wave to $s_{\pm}$-wave in a Bilayer Nickelate Driven by Hund's Coupling and Crystal Field Splitting

Abstract

The pairing symmetry of the recently discovered bilayer nickelate superconductor La$_3$Ni$_2$O$_7$ is a subject of intense debate in condensed matter physics, with the two leading theoretical candidates being a sign-reversing $s_{\pm}$-wave and a $d$-wave state. To investigate its ground-state properties in the intermediate coupling regime which is critical for real materials, we construct a two-orbital bilayer Hubbard model and employ the constrained-path quantum Monte Carlo method for large-scale simulations. By systematically calculating ground-state pairing correlation functions across parameter spaces, we map its pairing symmetry phase diagram. We find that an increasing Hund's coupling selectively enhances the interlayer $s_{\pm}$-wave pairing while suppressing the intralayer $d$-wave pairing. Similarly, a larger crystal field splitting drives a transition from $d$-wave- to $s_{\pm}$-wave-dominant states. Further analysis reveals that the strength of the intralayer $d$-wave pairing is strongly correlated with the $(π, π)$ antiferromagnetic spin fluctuations, which are in turn effectively suppressed by a large crystal field splitting, thereby weakening the $d$-wave pairing channel. Additionally, the dominant pairing symmetry transition region roughly overlaps with the inversion of orbital occupancy response to Hubbard $U$, suggesting an intrinsic link between pairing competition and orbital physics. Our results indicate that, within the parameter regime relevant to the actual material, the $s_{\pm}$-wave is the most probable pairing symmetry.

Figures (6)

• Figure 1: (a) Crystal structure of $\text{La}_3\text{Ni}_2\text{O}_7$ under high pressure. (b) Schematic of the hopping processes in the model. (c) Band structure and (d)Fermi surface calculated using the tight-binding parameters from first-principles calculations for the material at 29.5 GPa (see Table \ref{['tab:model_parameters_combined']}). Near the Fermi level, there are three bands $\alpha, \beta, \gamma$ and one unoccupied $\delta$ band. The splitting between the bonding and antibonding bands, arising from the interlayer coupling, is approximately $2t_\perp/t$ . The resulting Fermi surface consists of one electron-like pocket $\alpha$ at the Brillouin zone center ($\Gamma$ point) and two hole-like pockets $\beta, \gamma$ at the Brillouin zone corner ($M$ point). Here, red represents the $d_{x^2-y^2}$ orbital character and blue represents the $d_{3z^2-r^2}$ orbital character.
• Figure 2: (a)(b) Evolution of electron occupancies $n_z$ and $n_x$ for the $z$ and $x$ orbitals as a function of the on-site Coulomb interaction $U/t$, at fixed interlayer coupling $t_{\perp}/t = 1.315$ and crystal field splitting $\Delta E/t = 0.76$. Curves of different colors correspond to different values of $J_H/U$. (c)(d) Orbital occupancies as a function of crystal field splitting $\Delta E/t$ at fixed $U/t=4$ and $J_H/U=0.20$. (e)(f) Intra-orbital double occupancies $D_z$ and $D_x$ for the two orbitals as a function of $1/U$. The total electron filling is fixed at $\langle n \rangle = 0.75$.
• Figure 3: (a) $s_{\pm}$-wave pairing strength $V_{s_{\pm}}$ as a function of $U/t$ (parameters are $t_{\perp}/t = 1.315$, $\Delta E/t = 0.76$; different colors correspond to different $J_H/U$, same for below). (b) $d$-wave pairing strength $V_d$ as a function of $U/t$. (c) Evolution of the difference between the two pairing channel strengths, $V_{s_{\pm}}-V_d$, as a function of $U/t$. The inset shows the finite-size scaling analysis. (d) Phase diagram of the dominant pairing symmetry in the $U/t$–$J_H/U$ parameter plane: the pink region is $s_{\pm}$-wave dominant, the blue region is $d$-wave dominant, and the dashed line indicates the phase boundary ($V_{s_{\pm}} = V_d$).
• Figure 4: (a) $V_{s_{\pm}}$ as a function of $\Delta E/t$ (parameters are $U/t=4$, $J_H/U=0.20$; different curves correspond to different $t_{\perp}/t$, same for below). (b) $V_d$ as a function of $\Delta E/t$. (c) Evolution of the pairing potential difference $V_{s_{\pm}}-V_d$ as a function of $\Delta E/t$. (d) Phase diagram of the dominant pairing symmetry in the $\Delta E/t$–$t_{\perp}/t$ parameter plane: the pink region is $s_{\pm}$-wave dominant, the blue region is $d$-wave dominant, and the dashed line indicates the phase boundary.
• Figure 5: (a) $(\pi,\pi)$ static spin structure factor $S_{zz}(M)$ as a function of $U/t$ (parameters are $t_{\perp}/t=1.315$, $\Delta E/t = 0.76$; different colors correspond to different $J_H/U$, same for below). (b) Nearest-neighbor spin correlation $C_{nnspin}$ as a function of $U/t$. (c) $S_{zz}(M)$ as a function of $\Delta E/t$ (parameters are $U/t=4$, $J_H/U=0.20$; different colors correspond to different $t_{\perp}/t$, same for below). (d) $C_{nnspin}$ as a function of $\Delta E/t$.