AI 中文总结
该研究提出对称性保持回流配对近似，基于其构造的神经网络波函数避免对称性破缺极小值，在t-t'哈伯德模型中获最优变分能量，证实d波超导序，解决了1/8掺杂模型的长期问题。
AI 中文摘要
尽管二维掺杂哈伯德模型在强关联电子理论中具有根本重要性，但其基态的本质仍存在激烈争论。变分方法为该问题提供了有力途径，但其结论可能敏感依赖于所选波函数参数化、平均场初始化、用于指导优化的钉扎场以及边界条件。这可能会偏向某一类对称性破缺而非另一类，使得难以区分交织或竞争序的真实相互作用与变分参数化引入的偏差。在此，我们提出对称性保持回流配对（Symmetry-Preserving Backflow Pairing，SBP）近似，这是一种神经网络波函数，其构造上具有平移对称性，从而避免了这些对称性破缺的极小值。SBP近似在晶格尺寸达24×24、电子数为504的t-t'哈伯德模型中达到了最先进的变分能量，低于竞争的纯条纹解的能量。通过外推至热力学极限，我们发现了d波超导序的有力证据，解决了t'/t=-0.2、U/t=8.0下1/8掺杂模型的长期问题。SBP波函数基于对称性和局域性的一般原理，为具有挑战性的相互作用费米子系统提供了广泛适用的变分表示。
英文摘要
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 $t$-$t'$ Hubbard model on lattices up to $24\times24$ with $504$ electrons, below those of competing pure stripe solutions. By extrapolating to the thermodynamic limit, we find robust evidence for $d$-wave superconducting order, resolving a long-standing question about the $1/8$-doped model at $t'/t=-0.2$ and $U/t=8.0$. Built on general principles of symmetry and locality, the SBP wave function provides a broadly applicable variational representation for challenging interacting fermionic systems.
Comments8 pages, 7 figures, 1 table
