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 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.
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 study the equilibrium and nonequilibrium formation of four-particle complexes in a balanced two-species Bose-Hubbard model with repulsive intraspecies and attractive interspecies interactions. Using exact diagonalization, we characterize the transition from weakly to strongly correlated dimer and tetramer states along the one- to two-dimensional crossover in coupled-chain geometries by combining local correlation signatures with global diagnostics such as the binding energy and interspecies entanglement entropy. We show that transverse connectivity between chains qualitatively reshapes the phase diagram, substantially enlarging the tetramer region and, in particular, stabilizing weakly bound tetramers when compared to the one-dimensional chains. By tuning the interchain hopping, we identify a transition from a degenerate manifold of spatially separated dimers to a localized tetramer ground state, driven by the lifting of one-dimensional configurational degeneracies and an associated kinetic-energy gain. By extending our analysis to larger complexes of six and eight particles, we reveal that these binding mechanisms are robust and suggest a general pathway for the formation of higher-order complexes. Finally, we demonstrate interaction and geometric quench protocols to dynamically prepare these complexes with high fidelity. Our results provide a microscopic framework for engineering and probing few-body bosonic bound states in tunable lattice geometries.
Confined ultracold atoms in optical lattices provide a versatile platform for simulating lattice models of strongly correlated quantum systems, where pairing phenomena and superfluid phases can be explored under controlled conditions. While the crossover between the Bardeen-Cooper-Schrieffer (BCS) phase and the Bose-Einstein condensation (BEC) is well understood in homogeneous systems, spatial confinement breaks translational symmetry and reshapes correlation patterns, making the BCS-BEC identification in trapped geometries challenging and allowing unconventional phases to emerge with no direct analog in homogeneous systems. Here we present a characterization of the BCS-BEC crossover in harmonically confined one-dimensional Fermi-Hubbard chains. Our analysis combines Density Matrix Renormalization Group (DMRG) simulations and entanglement-based diagnostics with effective models describing the formation of tightly bound fermion pairs. This combined approach enables a detailed understanding of how the interplay between interactions and confinement reshapes the crossover, leading to insulating regions coexisting with persistent superfluid correlations. Within this framework, we further introduce conditioned correlation functions whose power-law decay allows a clear distinction between BCS-like and BEC-like regimes. The consistency between the effective descriptions and the numerical DMRG results yields a unified picture of the crossover in harmonically confined geometries.
We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength U/t=2∼12 (where t is the nearest-neighbor hopping amplitude) and fermion density n=0.25∼1.0. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.
The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy E=T+W is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength ∣U∣/t, onsite potential $\ensuremath{\varepsilon}/t$, and band filling n=Ne/Na. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.
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.