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arXiv:2511.12551 · Physica Scripta

Principal component analysis of competing correlations in quarter-filled Hubbard models

Md Fahad Equbal, Syed R. Hassan, M. A. H. Ahsan

Published May 15, 2026• 0 citations
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Abstract

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 data on 3 × 4 and 4 × 4 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.

DOI
10.1088/1402-4896/ae6ea0

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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 data on 3 × 4 and 4 × 4 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.
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.
We use determinant quantum Monte Carlo (DQMC), in combination with the principal component analysis (PCA) approach to unsupervised learning, to extract information about phase transitions in several of the most fundamental Hamiltonians describing strongly correlated materials. We first explore the zero-temperature antiferromagnet to singlet transition in the periodic Anderson model, the Mott insulating transition in the Hubbard model on a honeycomb lattice, and the magnetic transition in the 1/6-filled Lieb lattice. We then discuss the prospects for learning finite temperature superconducting transitions in the attractive Hubbard model, for which there is no sign problem. Finally, we investigate finite temperature charge density wave (CDW) transitions in the Holstein model, where the electrons are coupled to phonon degrees of freedom, and carry out a finite size scaling analysis to determine Tc{T}_{c}Tc​. We examine the different behaviors associated with Hubbard-Stratonovich auxiliary field configurations on both the entire space-time lattice and on a single imaginary time slice, or other quantities, such as equal-time Green's and pair-pair correlation functions.
We investigate the binding of holes and the emergence of competing spin-charge order in the simple and extended Hubbard model using exact diagonalization on the 3x4 cylindrical lattice. For the simple Hubbard model (V=0), we find weakly bound hole pairing mediated by magnetic correlations at intermediate repulsive U, without any evidence of phase separation. Introducing nearest-neighbor interaction V reveals a rich phase diagram: attractive V drives multi-hole clustering and phase separation with localized magnetic quenching, while repulsive V stabilizes charge-density-wave (CDW) order that coexists with bound hole pairs within a modulated magnetic background. At strong coupling (U=10), the competition sharpens, with attractive V overcoming on-site repulsion to form magnetically quenched clusters and repulsive V producing robust CDW order that constrains pairing. Real-space analysis of spin and charge correlations provides microscopic evidence of distinct binding mechanisms -- phase separation versus correlation-mediated pairing -- depending on the sign and strength of intersite interaction V . Our results establish a comprehensive picture of how nonlocal Coulomb interactions reshape the landscape of hole-binding and collective order in correlated electron systems.
We apply various unsupervised machine learning methods for phase classification to investigate the finite-temperature phase diagram of the spinless Falicov-Kimball model in two dimensions. Using only particle occupation snapshots from Monte Carlo simulations as input, each technique, including a straightforward classification based on principal component analysis (PCA), successfully identifies the phase boundary between ordered and disordered phases, independent of the type of phase transition. Remarkably, these techniques also distinguish between the weakly localized and Anderson-localized regimes within the disordered phase, accurately identifying their crossover, which is a challenging task for standard methods. Among the machine learning approaches used, PCA based analysis outperforms more complex methods, such as neural network predictors and autoencoders. These results underscore the effectiveness of simple unsupervised techniques in examining phase transitions and electron localization in complex correlated systems.
We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow the exploration of different phases and symmetry-breaking, but they can distinguish phase-transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the "charge" correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the "autoencoder method," and we demonstrate that it too can be trained to capture phase transitions and critical points.
Abstract Machine learning techniques have been widely used in the study of strongly correlated systems in recent years. Here, we review some applications to classical and quantum many-body systems and present results from an unsupervised machine learning technique, the principal component analysis, employed to identify the finite-temperature phase transition of the three-dimensional Fermi-Hubbard model to the antiferromagnetically ordered state. We find that this linear method can capture the phase transition as well as other more complicated and nonlinear counterparts.