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

TL;DR

This paper demonstrates how principal component analysis (PCA) applied to exact-diagonalization data can reveal the hierarchy and competition of correlation channels in quarter-filled Hubbard models, offering a model-agnostic diagnostic tool.

Contribution

It introduces PCA as a transparent, data-driven method to identify dominant correlation regimes in finite Hubbard clusters without relying on predefined order parameters.

Findings

PCA captures charge, spin, and pairing crossovers directly from correlation data.

The method identifies regimes dominated by charge, spin, or pairing correlations.

PCA provides a bridge between exact diagonalization and machine-learning diagnostics.

Abstract

We present an unsupervised learning 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 clusters, providing a bridge between exact diagonalization and modern machine-learning diagnostics in strongly correlated systems.