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arXiv:2609.22618

Probing $d$-wave pairing in the $t$-$t'$-$U$ Hubbard model with tensor-backflow wave functions

Xiao Liang

Published Sep 18, 2026• 0 citations
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Abstract

Building on the recently developed tensor-backflow method, which accurately describes energies and stripe states in the Fermi-Hubbard model, we investigate $d$-wave pairing in the two-dimensional $t$-$t'$-$U$ Hubbard model. For the $8\times8$ lattice, we improve the wave function using symmetry projections followed by one Lanczos step, whereas for the $12\times 12$ and $16\times 16$ lattices, we apply one Lanczos step without symmetry projection. We focus primarily on the parameter regime with filling $n=0.875$, on-site repulsion $U=8$ and next-nearest-neighbor hopping $t'=-0.2t$, a regime in which enhanced $d$-wave pairing has been reported. The resulting energies are competitive with those obtained using state-of-the-art neural quantum states. For example, on a $16\times 16$ lattice with periodic boundary conditions, the tensor-backflow wave function after one Lanczos step without explicit symmetry enforcement achieves a relative energy difference of $4.4\times 10^{-3}$ from the symmetry-preserving neural quantum state result. From the real space pair correlations, for $t'=-0.2t$, we observe stronger $d$-wave pair correlations together with the characteristic relative sign between horizontal and vertical bond-pair components. Compared with $t'=0$, finite negative $t'$ increases the spectral weight of the corresponding pair-density matrix, which indicates the enhanced overall $d$-wave pair fluctuations. The leading eigenvalue is enhanced on the $8\times8$ lattice for both with and without symmetry projections. For larger lattices such as $12\times 12$ and $16\times 16$, the leading pair-density matrix spectral weight is distributed among several nearly degenerate eigenmodes rather than being concentrated in a single dominant mode. This indicates enhanced and more broadly distributed $d$-wave pair fluctuations, but no single eigenvalue exhibits the extensive scaling required for ODLRO.

DOI
10.48550/arxiv.2609.22618

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The two-dimensional Hubbard model was originally proposed as the minimal model of uniform ddd-wave superconductivity in the cuprate high-temperature superconductors, but recent calculations with relevant parameters in a realistic doping regime have often found other ground states. Further, uncertainty remains over such results due to the intrinsic bias of different computational methods. Here we address this uncertainty through state-of-the-art neural tensor network and neural backflow state simulations, enabled by large-scale GPU computation, that together reach an unprecedented variational accuracy on lattices with up to 1024 sites. At 1/81/81/8 doping in the pure Hubbard model, as well as with next-nearest-neighbor hopping relevant to the hole-doped cuprates, we reveal a near degeneracy among a low-energy uniform ddd-wave state, superconducting stripes, and non-superconducting stripe states on the milli-ttt energy scale. Our work rebalances the narrative that the Hubbard model supports a single dominant order in relevant parameter regimes towards a picture of extreme competition between stripes and superconductivity.
Superconductivity in the ttt-t′t't′ Hubbard Model from Symmetry-Preserving Neural-Network Quantum States
Despite its fundamental importance in the theory of strongly correlated electrons, the nature of the ground state of the two-dimensional doped Hubbard model remains intensely debated. Variational approaches provide a powerful route to this problem, but their conclusions can depend sensitively on the chosen wave-function parameterization, the mean-field initialization, or the pinning fields used to guide the optimization, as well as on boundary conditions. This can favor one type of symmetry breaking over another, making it difficult to distinguish the genuine interplay of intertwined or competing orders from biases induced by the variational parameterization. Here, we introduce the Symmetry-Preserving Backflow Pairing (SBP) ansatz, a neural-network wave function that respects translational symmetry by construction and thereby avoids these broken-symmetry minima. The SBP ansatz reaches state-of-the-art variational energies for the ttt-t′t't′ Hubbard model on lattices up to 24×2424\times2424×24 with 504504504 electrons, below those of competing pure stripe solutions. By extrapolating to the thermodynamic limit, we find robust evidence for ddd-wave superconducting order, resolving a long-standing question about the 1/81/81/8-doped model at t′/t=−0.2t'/t=-0.2t′/t=−0.2 and U/t=8.0U/t=8.0U/t=8.0. Built on general principles of symmetry and locality, the SBP wave function provides a broadly applicable variational representation for challenging interacting fermionic systems.
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.
The quantum many-body problem is an important topic in condensed matter physics. To efficiently solve the problem, several methods have been developed to improve the representation ability of wave functions. For the Fermi-Hubbard model under periodic boundary conditions, current state-of-the-art methods are neural network backflows and the hidden fermion Slater determinant. The backflow correction is an efficient way to improve the Slater determinant of free particles. In this work we propose a tensor representation of the backflow-corrected wave function; we show that for the spinless $t\text{\ensuremath{-}}V$ model, the energy precision is competitive or even lower than current state-of-the-art fermionic tensor network methods. For models with spin, we further improve the representation ability by considering backflows on fictitious particles with different spins, thus naturally introducing nonzero backflow corrections when the orbital and the particle have opposite spins. We benchmark our method on molecules in the STO-3G basis and the Fermi-Hubbard model with periodic and cylindrical boundary conditions. We show that the tensor representation of backflow corrections achieves competitive or even lower-energy results than current state-of-the-art neural network methods.
We examine the influence of hole doping on superconducting pairing symmetries in the two-dimensional Hubbard model extended by next-nearest-neighbor hopping, nearest-neighbor Coulomb interaction, correlated hopping, exchange interaction and pair-hopping processes. Using Lanczos exact diagonalization supported by projector quantum Monte Carlo method, we calculate superconducting correlation functions corresponding to s-wave, extended s-wave, and d-wave pairing over a broad range of electron concentrations and interaction parameters. We find that hole doping acts as an efficient control parameter of superconducting pairing symmetry, giving rise to robust crossovers between the dominant pairing correlations. While extended s-wave pairing dominates throughout most of the overdoped regime, reducing the hole concentration progressively enhances d-wave correlations over large regions of the ground-state diagram. The critical doping associated with these crossovers depends strongly on the underlying interaction mechanism. In particular, correlated hopping and next-nearest-neighbor hopping substantially modify the location of the pairing-symmetry boundaries, whereas the local Hubbard interaction exhibits a comparatively weaker effect. In contrast to all other interaction channels, a finite nearest-neighbor Coulomb interaction additionally stabilizes conventional s-wave state over a sizeable region of parameter space. Our results demonstrate that superconducting pairing symmetry in the extended Hubbard model is governed by a subtle interplay between carrier concentration and nonlocal electronic interactions.