Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states

TL;DR

The study addresses the question of superconductivity in the 2D square-lattice Hubbard model using hidden fermion Pfaffian states (HFPS), a Pfaffian-based neural quantum-state variational approach that captures intertwined spin, charge, and pairing orders. It demonstrates that at strong coupling with negative $t'$, partially-filled stripes coexist with $d$-wave superconductivity, while at intermediate coupling with doping $\delta=1/6$ the overdoped regime hosts a uniform $d$-wave superconductor whose momentum-space pairing signature is robust and size-invariant. Momentum-space diagnostics, including the estimator $p(\mathbf{k})$ derived from $M_{\mathbf{k}\mathbf{k'}}$, circumvent finite-size limitations that plague real-space pair correlations, providing compelling evidence for superconductivity in regimes where stripes are suppressed. The results corroborate and extend recent multidisciplinary numerical efforts, showcasing HFPS as a powerful tool for exploring correlated electron phases with implications for cuprates and ultracold atoms.

Abstract

Whether the ground state of the square lattice Hubbard model exhibits superconductivity remains a major open question, central to understanding high temperature cuprate superconductors and ultra-cold fermions in optical lattices. Numerical studies have found evidence for stripe-ordered states and superconductivity at strong coupling but the phase diagram remains controversial. Here, we show that one can resolve the subtle energetics of metallic, superconducting, and stripe phases using a new class of neural quantum state (NQS) wavefunctions that extend hidden fermion determinant states to Pfaffians. We simulate several hundred electrons using fast Pfaffian algorithms allowing us to measure off-diagonal long range order. At strong coupling and low hole-doping, we find that a non-superconducting filled stripe phase prevails, while superconductivity coexisting with partially-filled stripes is stabilized by a negative next neighbor hopping t-prime, with |t-prime| > 0.1. At larger doping levels, we introduce momentum-space correlation functions to mitigate finite size effects that arise from weakly-bound pairs. These provide evidence for uniform d-wave superconductivity at U = 4, even when t-prime = 0. Our results highlight the potential of NQS approaches, and provide a fresh perspective on superconductivity in the square lattice Hubbard model.

Figures (13)

• Figure 1: Depiction of the phases described in this work. The top row shows non-superconducting "normal state" phases. The colors red/blue represent spin up and down. The wavy lines represent wave-like electrons that occupy momentum eigenstates, while the up/down symbols represent spatially localized electrons. The horizontal arrows specify the changes in Hamiltonian couplings that move the ground state between different phases. For the metal and partially-filled stripe, we find that a small proportion of the electrons form itinerant singlet pairs, giving rise to superconductivity. This instability towards pairing is shown by the downward pointing vertical arrows. In the middle we show a cartoon of our optimization strategy. Each optimization is run with several different mean-field initial conditions (empty circles). These optimizations converge to distinct phases (black circles) that are not sensitive to the details of the initialization.
• Figure 2: Stripes and stripy superconductors in the $t-t^\prime$ model at strong coupling. Simulations are done on a $16 \times 16$ lattice at $U=8$ and $\delta = 1/8$. All color plots are clipped at a maximum value shown in the color scale for ease of visualization. a.- b. Charge and spin correlation functions of the filled stripe at $t^\prime=0$ ( a.) and a half-filled stripe at $t^\prime = -0.3$ ( b.). The spin correlation function is plotted on top, while one-dimensional cuts of the charge, $C_c(x,y=8)$, and sublattice-transformed spin $(-1)^{x } C_s(x,y=8)$ correlations are plotted on the bottom. c. Energies of the filled stripe and half-filled stripe on a $256$ site cluster as a function of $t^\prime$. The statistical error bars are smaller than the data points. d.- e. Color plot of pair correlations, $C_p({\bf x})$ at $t^\prime = 0$ ( d.) and $t^\prime = -0.3$ ( e.) f. Pair correlations as a function of scalar displacement $|{\bf x}|$ for $t^\prime = 0$ on a $256$ site cluster (gold), and for $t^\prime = -0.3$ on $128$ (blue) and $256$ (red) site clusters.
• Figure 3: Long range pairing order in the $t^\prime = 0$ Hubbard model at 1/6 doping. Results are shown for $144$ and $288$ site simulation clusters for $U=4$, $\delta=1/6$a. Color plot of d-wave pair correlation functions of the $288$ site lattice. b. Line plot showing pair correlations at $U=4$ as a function of distance on $144$ (blue) and $288$ (red) site simulation clusters. Lines that denote the average pair correlation, ${\bar{C}}_p$, on both system sizes are shown for reference. c.-d. Color plots of momentum space pairing estimator, $p({\bf k})$ on $144$ ( c.) site and $288$ site ( d.) clusters. e-f. Momentum space pairing $p({\bf k})$ along the one-dimensional cuts shown in c.-d. for the $144$ (blue) and $288$ (red) site clusters.
• Figure 4: Correlation functions in the $t^\prime = 0$ Hubbard model for $\boldsymbol{\delta} \bf{= 1/6}$ at different values of $\bf{U}$. Simulations are shown for different values of $U$ on a $144$ site cluster. a. Spin structure factor, $C_s({\bf k})$ for $U=6$b. Spin structure factor at $U=10$c. One-dimensional cut of the spin structure $C_s({\bf k})$ factor along $k_y= \pi$. d. One dimensional cut of the charge structure factor $C_c({\bf k})$ along $k_y = 0$. e. P( k) plotted along the same cut in Fig. \ref{['fig: pure hubbard']}e. f. P( k) plotted along the same cut in Fig. \ref{['fig: pure hubbard']}f.
• Figure 5: Comparison between methods of computing momentum space pairing at $U=4, \delta = 1/6$ on the 144 site cluster. a. Momentum space pairing estimator using dominant eigenmode of $M_{\bf k \bf k'}$, which is computed as $\sqrt{\lambda_{0}} e_0({\bf k})$ (see Eqn. \ref{['eqn: momentum space low rank']}). b. Momentum space pairing, $p({\bf k})$ using the mixed estimator (Eqn \ref{['eqn: momentum space pairing proxy']}) c. Histogram comparing the eigenvalues of $M_{\bf k, \bf k'}$ using two different sample sizes.