Simulating superconductivity in mixed-dimensional $t_\parallel$-${J}_\parallel$-${J}_\perp$ bilayers with neural quantum states

Abstract

Motivated by the recent discovery of superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ (LNO) under pressure, we study a mixed-dimensional (mixD) bilayer $t_\parallel$-$J_\parallel$-$J_\perp$ model, which has been proposed as an effective low-energy description of LNO. Using neural quantum states (NQS), and in particular Gutzwiller-projected Hidden Fermion Pfaffian State, we access the ground-state properties on large lattices up to $8\times 8\times 2$ sites. We show that this model exhibits superconductivity across a wide range of dopings and couplings, and analyze the pairing behavior in detail. We identify a crossover from tightly bound, Bose-Einstein-condensed interlayer pairs at strong interlayer exchange to more spatially extended Bardeen-Cooper-Schrieffer-like pairs as the interlayer exchange is decreased. Furthermore, upon tuning the intralayer exchange, we observe a sharp transition from interlayer $s$-wave pairing to intralayer $d$-wave pairing, consistent with a first-order change in the pairing symmetry. We verify that our simulations are accurate by comparing with matrix product state simulations on coupled ladders. Our results represent the first simulation of a fermionic multi-orbital system with NQS, and provide the first evidence for superconductivity in two-dimensonal bilayers using high-precision numerics. These findings provide insight into superconductivity in bilayer nickelates and cold atom quantum simulation platforms.

Figures (22)

• Figure 1: a. We consider the mixed-dimensional (mixD) $t$-$J$ bilayer model (right), which can be derived from the multiband model of LNO, including $d_{z^2}$ and $d_{x^2-y^2}$ orbitals. b. Energies per site $E/N$ obtained with the NQS for coupled ladders, where comparisons to MPS are possible (left) and for bilayers up to size $8\times 8\times 2$. We consider two sets of parameters, with $J_\perp/t_\parallel=3.0$ and $J_\parallel/t_\parallel=0.0$ (blue) as well as $J_\perp/t_\parallel=0.6$ and $J_\parallel/t_\parallel=0.4$ (orange). For NQS (MPS), we consider periodic (open) boundaries in the long direction.
• Figure 2: BEC-to-BCS crossover when changing $t_\parallel/J_\perp$. a. Sketch of the phase diagram, with tightly bound BEC-like pairs for $t_\parallel/J_\perp \ll 1$ and more spatially extended BCS-like pairs when $t_\parallel/J_\perp$ is increased. b. We show exemplary maps of the momentum-resolved pairing order parameter estimated by the mixed-estimator $\langle \hat{P}^{s_\perp}(\mathbf{k})\rangle$ (left) and its Fourier transform $\langle \hat{P}^{s_\perp}(\mathbf{r})\rangle$ (right) (see Eq. \ref{['eq:MixedEstimator']}). c. We show cuts of the BEC (BCS) condition given by $W_\delta =\sum_{\vert \mathbf{r}\vert <l_\delta}\vert \langle \hat{P}^{s_\perp}(\mathbf{r})\rangle \vert /\sum_{\vert \mathbf{r}\vert \geq l_\delta}\vert \langle \hat{P}^{s_\perp}(\mathbf{r})\rangle\vert \geq 1 (<1)$ through the phase diagram at different doping levels $\delta$ as a function of $t_\parallel/J_\perp$ and at $J_\parallel/t_\parallel=0.4$.
• Figure 3: Inter- and intralayer pairing symmetries when changing $J_\parallel/t_\parallel$ for fixed $J_\perp/t_\parallel=0.6$ and $\delta=0.5$. a. Long-range pairing correlations from the center site at $(4,4)$ and averaged over all four edges $\langle\! \langle (\hat{\Delta}^\Gamma_\mathrm{center})^\dagger \hat{\Delta}_\mathrm{edge}^\Gamma\rangle \!\rangle$ for $\Gamma=s_\perp$ (orange) and $\Gamma=d_\parallel$ (green). b. Momentum-resolved mixed-estimator $\langle \!\langle \hat{p}^\Gamma(\mathbf{k})\rangle \!\rangle$ for $\Gamma=s_\perp$ (top row) and $\Gamma=d_\parallel$ (bottom row).
• Figure S1: Ground state energies per site $E/N$ for $4\times 4\times 2$ mixD bilayers obtained with the HFPS for different dopings $\delta=0.62,0.5,0.25$ (left to right) and different translational symmetry sectors indicated by the respective $k$-point in the reciprocal lattice.
• Figure S2: Final ground state energies (left) and variance (right) per site $N$ for $8\times 8\times 2$ mixD bilayers obtained with the HFPS for different sublattice geometries. We consider hole doping $\delta=0.5$ with $J_\parallel/J_\perp=0.4, 2.0$ and different mean field pairing fields $\Delta^\mathrm{MF}_\Gamma$ ($\Gamma=s_\perp, d_\parallel)$ respectively, corresponding to states with $s_\perp$- and $d_\parallel$-wave pairing.