Principal Component Analysis of Competing Correlations in Quarter-Filled Hubbard Models

Abstract

We present a data-driven analysis of correlation hierarchies in the quarter-filled simple and extended Hubbard models by applying principal component analysis (PCA) to exact-diagonalization (ED) data on 3x4 and 4x4 cylindrical clusters. While the non-interacting limit (U=0) provides a finite-size reference, increasing on-site repulsion U induces localization and reorganizes the low-energy spectrum. For the extended model, we examine moderate (U=4) and strong (U=10) coupling regimes, where conventional structure factors reveal familiar crossovers among charge, spin and local-pairing correlations. PCA of the corresponding correlation matrices captures these crossovers directly from the data, without assuming predefined order parameters by identifying charge-dominated, spin-dominated and pairing-dominated regimes through variance condensation into leading components. This establishes PCA as a transparent, model-agnostic framework for uncovering the hierarchy and competition of correlation channels in finite Hubbard lattice clusters, providing a bridge between exact diagonalization and modern machine-learning diagnostics in strongly correlated systems.

Figures (20)

• Figure 1: Schematic of (a) $3\times4$ and (b) $4\times4$ clusters with periodic boundary conditions (PBC) along $x$ and open boundary conditions (OBC) along $y$. Black dots denote lattice sites, and red dashed lines represent wrap-around bonds due to PBC.
• Figure 2: Ground-state energy per site $E_0/M$ as a function of the on-site interaction $U$ for quarter-filled $3\times4$ (red) and $4\times4$ (blue) Hubbard clusters. The inset shows the excitation gap $\Delta E = E_1 - E_0$, revealing contrasting behaviors: a monotonically opening gap on $3\times4$ versus a collapse on $4\times4$ at large $U$.
• Figure 3: Single-particle molecular-orbital (MO) energy spectra of the non-interacting ($U=0$) quarter-filled Hubbard model on $3\times4$ and $4\times4$ cylindrical lattices. Degeneracy at the Fermi level for $3\times4$ yields a vanishing gap, while a finite highest occupied MO-lowest unoccupied MO spacing appears for $4\times4$.
• Figure 4: Average double occupancy $\bar{d}$ (circles, left axis) and squared local moment $\bar{m}$ (squares, right axis) as functions of $U$ for quarter-filled $3\times4$ (red) and $4\times4$ (blue) clusters. Increasing $U$ suppresses $\bar{d}$ and enhances $\bar{m}$, reflecting progressive electron localization.
• Figure 5: Charge and spin structure factors $S_D(\pi,\pi)$ and $S_L(\pi,\pi)$ versus $U$ for quarter-filled $3\times4$ and $4\times4$ Hubbard clusters. Increasing $U$ suppresses charge modulations but strengthens antiferromagnetic spin correlations.