Probing d-wave pairing in the t-t'-U Hubbard model with tensor-backflow wave functions
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 8×8 lattice, we improve the wave function using symmetry projections followed by one Lanczos step, whereas for the 12× 12 and 16× 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× 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× 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 8×8 lattice for both with and without symmetry projections. For larger lattices such as 12× 12 and 16× 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.
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BibTeX
@misc{liang2026probing,
title = {{Probing $d$-wave pairing in the $t$-$t'$-$U$ Hubbard model with tensor-backflow wave functions}},
author = {Xiao Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22618},
eprint = {2609.22618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22618},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probing d-wave pairing in the t-t'-U Hubbard model with tensor-backflow wave functions
AU - Xiao Liang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22618
UR - https://arxiv.org/abs/2609.22618
AB - 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 8×8 lattice, we improve the wave function using symmetry projections followed by one Lanczos step, whereas for the 12× 12 and 16× 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× 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× 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 8×8 lattice for both with and without symmetry projections. For larger lattices such as 12× 12 and 16× 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.
ER -