On several one-dimensional (1D) and 2D nonbipartite lattices, we study both free and Hubbard interacting lattice fermions when some magnetic fluxes are threaded or gauge fields coupled. First, we focus on finding out the optimal flux which minimizes the energy of fermions at specific fillings. For spin-$1/2$ fermions at half-filling on a ring lattice consisting of odd-numbered sites, the optimal flux turns out to be $\pm{\pi}/2$. We prove this conclusion for Hubbard interacting fermions utilizing a generalized reflection positivity technique, which can lead to further applications on 2D nonbipartite lattices such as triangular and Kagome. At half-filling the optimal flux patterns on the triangular and Kagome lattice are ascertained to be $\pm[{\pi}/2,{\pi}/2]$, $\pm[{\pi}/2,{\pi}/2,0]$, respectively (see the meaning of these notations in the main text). We also find that chirality emerges in these optimal flux states. Then, we verify these exact conclusions and further study some other fillings with the numerical exact diagonalization method. It is found that when it deviates from half-filling, Hubbard interactions can alter the optimal flux patterns on these lattices. Moreover, numerically observed emergent flux singularities driven by strong Hubbard interactions in the ground states, both in 1D and 2D, are discussed and interpreted as some kind of non-Fermi liquid feature.

中文翻译：

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非二分晶格的费米-哈伯德模型：通量问题和紧急手性

在几个一维 (1D) 和 2D 非二分晶格上，我们研究了当一些磁通量被穿线或规范场耦合时自由和哈伯德相互作用的晶格费米子。首先，我们专注于找出在特定填充物处最小化费米子能量的最佳通量。对于由奇数位点组成的环晶格上半填充的自旋 $1/2$ 费米子，最佳通量结果是 $\pm{\pi}/2$。我们利用广义反射正性技术证明了哈伯德相互作用费米子的这一结论，这可以导致在二维非二分晶格（如三角形和 Kagome）上的进一步应用。在半填充时，三角形和 Kagome 晶格上的最佳通量模式确定为 $\pm[{\pi}/2,{\pi}/2]$, $\pm[{\pi}/2,{ \pi}/2,0]$, 分别（参见正文中这些符号的含义）。我们还发现手性出现在这些最佳通量状态中。然后，我们验证了这些精确结论，并用数值精确对角化方法进一步研究了其他一些填充。发现当它偏离半填充时，哈伯德相互作用可以改变这些晶格上的最佳通量模式。此外，在数值上观察到的由基态中强哈伯德相互作用驱动的涌现通量奇点，在 1D 和 2D 中，都被讨论并解释为某种非费米液体特征。哈伯德相互作用可以改变这些晶格上的最佳通量模式。此外，在数值上观察到的由基态中强哈伯德相互作用驱动的涌现通量奇点，在 1D 和 2D 中，都被讨论并解释为某种非费米液体特征。哈伯德相互作用可以改变这些晶格上的最佳通量模式。此外，在数值上观察到的由基态中强哈伯德相互作用驱动的涌现通量奇点，在 1D 和 2D 中，被讨论并解释为某种非费米液体特征。