Extended $s$-wave pairing from an emergent Feshbach resonance in bilayer nickelate superconductors

TL;DR

This work analyzes a mixed-dimensional bilayer $t$-$J+V$ model motivated by La$_3$Ni$_2$O$_7$ superconductivity, where holes form spinon-chargon ($sc$) and chargon-chargon ($cc$) quasi-particles. The authors derive an effective two-channel Hamiltonian via Schrieffer-Wolff in the regime $t_{\parallel}\ll J_{\perp},V$ and solve it with a mean-field ansatz combining $sc$-BCS and $cc$-BEC sectors, capturing the BCS–BEC crossover and a Feshbach-type coupling between channels. Benchmarking against DMRG in 1D ladders and quasi-2D systems shows good quantitative agreement, validating the MF approach for both densities and energies. In 2D, the MF solution predicts an extended $s$-wave pairing gap $\Delta_k$ for the $(sc)^2$ pairs, with the crossover location tunable by doping and $J_{\perp}/V$. The results offer a microscopic mechanism for unconventional pairing in bilayer nickelates and suggest experimental tests in ultracold-atom setups.

Abstract

Since the discovery of unconventional superconductivity in cuprates, unraveling the pairing mechanism of charge carriers in doped antiferromagnets has been a long-standing challenge. Motivated by the discovery of high-T$_c$ superconductivity in nickelate bilayer La$_3$Ni$_2$O$_7$ (LNO), we study a minimal mixed dimensional (MixD) $t-J$ model supplemented with a repulsive Coulomb interaction $V$. When hole-doped, previous numerical simulations revealed that the system exhibits strong binding energies, with a phenomenology resembling a BCS-to-BEC crossover accompanied by a Feshbach resonance between two distinct types of charge carriers. Here, we perform a mean-field analysis that enables a direct observation of the BCS-to-BEC crossover as well as microscopic insights into the crossover region and the pairing symmetry for two-dimensional bilayers. We benchmark our mean-field description by comparing it to density-matrix renormalization group (DMRG) simulations in quasi-one dimensional settings and find remarkably good agreement. For the two-dimensional system relevant to LNO our mean-field calculations predict a BCS pairing gap with an extended $s$-wave symmetry, directly resulting from the pairing mechanism's Feshbach-origin. Our analysis hence gives insights into pairing in unconventional superconductors and, further, can be tested in currently available ultracold atom experiments.

Figures (8)

• Figure 1: Schematic depiction of the 2D MixD+V bilayer model (right) and its quasi-particle constituents (left). We highlight the emergent constituents that characterize the two sides of the BCS-to-BEC crossover existing in this system: on the BCS side the system is dominated by spatially extended pairs ($sc^2$) of spinon-chargons ($sc$'s), involving one extended singlet (highlighted in the figure). The BEC side features tightly bound chargon-chargons ($cc$'s) characterized by the Coulomb repulsion $V$ (yellow shadow).
• Figure 2: In the crossover regime, the spinon-chargon pairs are described by the BCS wavefunction. Here we consider a 2D MixD+V model on a $16 \times 16$ square lattice with $\frac{t_{\parallel}}{V}=0.1$, $J_{\perp}=V$ and $\delta = \frac{128}{256}$ holes in the system. The figure shows the value of the BCS order parameter $\Delta_k$ realizing an extended $s$-wave pairing symmetry.
• Figure 3: Quasi-particle densities and ground state energy for the doped 1D ladder. (a) We compare the $sc$ and $cc$ densities for a 1D ladder as the ratio $\frac{J_{\perp}}{V}$ and the doping $\delta$ are tuned. We compute $(N_{sc}-2N_{cc})/N_h$ where $N_{sc}$ and $N_{cc}$ are respectively the number of $sc$'s and $cc$'s in the system as defined in Sec. \ref{['Sec:Benchmark']}; $N_h$ is the total number of holes. This allows to distinguish the two sides of the crossover. The inset shows the section at constant doping $\delta=\frac{8}{30}$ with the corresponding colors. We find good agreement between DMRG (crosses) and mean-field (colorplot/solid line) for a MixD+V ladder of size $L=30$ where we have fixed $\frac{t_\parallel}{V}=0.1$. (b) The ground state energy $E$ (left) of the microscopic model for a 1D ladder system from DMRG (markers) is reproduced well by the mean-field prediction for the effective Hamiltonian (solid lines). The agreement is good also for the ground state energies without considering the contribution from the background distortion $E - E_{bg} = E-J_{\perp}N_{sc}-(V+ J_{\perp})N_{cc}$ (right).
• Figure 4: Quasi-particle densities and ground state energy for the doped two-ladder system.(a) We compare the number of $sc$ and $cc$ pairs for a two‑ladder system with $L_x=30$ and $L_y=2$ as the number of holes is tuned. We show the DMRG (markers) and mean‑field (solid lines) for $J_\perp=V$ and $J_\perp=1.5V$, where $t_\parallel = 0.1V$ is fixed. (b) The ground state energy $E$ (left) of the microscopic model for a two‑ladder system from DMRG (markers) is compared to the mean‑field prediction for the effective Hamiltonian (solid lines). We also show the results for the ground state energies without considering the contribution from the background distortion $E - E_{bg} = E - J_{\perp}N_{sc} - (V+J_{\perp})N_{cc}$ (right).
• Figure 5: The binding energies $E_B$ computed for a 1D ladder system via DMRG (markers) is compared to mean-field theory (solid line) as $J_{\perp}$ is tuned. Here, we also show the value of the order parameter $\Delta_k$ at the minimum of $E_k$, i.e. where the band gap is minimal (dashed line). We consider a ladder system of size $L=30$ with fixed doping $\delta=\frac{14}{30}$ and $\frac{t_\parallel}{V}=0.1$.