用张量回流波函数探测$t$-$t'$-$U$ Hubbard模型中的$d$波配对

中文摘要 AI 辅助

基于最近发展的张量回流方法（该方法能准确描述Fermi-Hubbard模型中的能量和条纹态），我们研究了二维$t$-$t'$-$U$ Hubbard模型中的$d$波配对。对于$8\ imes8$晶格，我们通过对称性投影后接一次Lanczos步骤来改进波函数，而对于$12\ imes 12$和$16\ imes 16$晶格，我们应用一次Lanczos步骤而不进行对称性投影。我们主要关注填充$n=0.875$、在位排斥$U=8$和次近邻跳跃$t'=-0.2t$的参数区域，该区域此前被报道存在增强的$d$波配对。所得能量与使用最先进的神经量子态获得的能量相当。例如，在具有周期性边界条件的$16\ imes 16$晶格上，不进行显式对称性强制的张量回流波函数在一次Lanczos步骤后，与保持对称性的神经量子态结果相比，相对能量差为$4.4\ imes 10^{-3}$。从实空间对关联来看，对于$t'=-0.2t$，我们观察到更强的$d$波对关联，以及水平和垂直键对分量之间特征性的相对符号。与$t'=0$相比，有限的负$t'$增加了相应配对密度矩阵的谱权重，这表明整体$d$波对涨落增强。在$8\ imes8$晶格上，无论是否进行对称性投影，主导特征值都得到增强。对于更大的晶格如$12\ imes 12$和$16\ imes 16$，主导配对密度矩阵谱权重分布在几个近似简并的本征模上，而不是集中在一个单一主导模上。这表明$d$波对涨落增强且分布更广，但没有单个特征值表现出ODLRO所需的广延标度。

英文摘要

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.

发表机构

• William & Mary（威廉与玛丽学院）

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