四分之一填充吸引哈伯德模型中的有限尺寸效应与相互作用驱动的交叉：精确对角化、DMRG与机器学习分析
Finite-size effects and interaction-driven crossovers in quarter-filled attractive Hubbard model: Exact diagonalization, DMRG and machine-learning analysis
AI总结：
本研究采用ED、DMRG及PCA、UMAP等机器学习方法，探究有限宽度圆柱晶格上四分之一填充吸引哈伯德模型的相互作用驱动交叉，明确其与热力学极限BCS-BEC交叉的一致性及特征的有限尺寸稳健性。
AI中文摘要：
我们采用精确对角化（ED）、密度矩阵重整化群（DMRG）及无监督机器学习技术，研究有限宽度圆柱晶格上的四分之一填充吸引哈伯德模型。对基态能量、局域可观测量及关联函数的分析显示，存在由弱关联费米子向紧密结合单重态对主导区域的连续相互作用驱动交叉。该交叉源于动能驱动的费米子巡游性与相互作用驱动的格点对形成之间的竞争，表现出与热力学极限下BCS-BEC交叉一致的行为。空穴结合能计算为对形成提供了直接能量证据：在吸引相互作用整个范围内，两空穴结合能保持为负，而三空穴结合仅在足够强的吸引作用下出现，且呈现显著的有限尺寸依赖性。为获得关联图景的无偏表征，我们对实空间关联矩阵应用主成分分析（PCA）与均匀流形近似与投影（UMAP）。PCA揭示了关联方差的系统再分布，UMAP则识别出弱配对与强配对区域间的清晰分离。两种机器学习方法均独立确定了从传统可观测量推断出的同一交叉区域，同时为多体关联的潜在重组提供了与序参量无关的表征。对配对结构因子及主导PCA方差比的有限尺寸标度分析表明，这些特征随系统尺寸增大仍保持稳健。
英文摘要：
We investigate the quarter-filled attractive Hubbard model on finite-width cylindrical lattices using exact diagonalization (ED), density-matrix renormalization group (DMRG) and unsupervised machine-learning-based techniques. Analysis of the ground-state energetics, local observables and correlation functions reveals a continuous interaction-driven crossover from weakly correlated fermions to a regime dominated by tightly bound singlet pairs. This crossover originates from the competition between kinetic-energy-driven fermionic itinerancy and interaction-driven onsite pair formation and exhibits behavior consistent with the BCS--BEC crossover in the thermodynamic limit. Hole-binding-energy calculations provide direct energetic evidence for pair formation: the two-hole binding energy remains negative throughout the attractive regime whereas three-hole binding emerges only at sufficiently strong attraction and exhibits pronounced finite-size dependence. To obtain an unbiased characterization of the correlation landscape, we apply principal component analysis (PCA) and uniform manifold approximation and projection (UMAP) to the real-space correlation matrices. PCA reveals a systematic redistribution of correlation variance whereas UMAP identifies a clear separation between weak- and strong-pairing regimes. Both machine-learning-based approaches independently identify the same crossover region inferred from conventional observables while providing an order-parameter-independent characterization of the underlying reorganization of many-body correlations. Finite-size scaling analyses of the pairing structure factor and the leading PCA variance ratio demonstrate that these signatures remain robust with increasing system size.
