Spin-resolved Mott crossover and entanglement in the half-filled Hubbard model

Abstract

We study interaction-driven crossover from a correlated metallic state to Mott-insulating state on the half-filled 3x3 square cluster using exact diagonalization in the S=1/2 and S=3/2 sectors. Using complementary diagnostics such as order parameters, principal component analysis (PCA) of correlation matrices and quantum geometry, we obtain a unified, data-driven characterization of the crossover. All three diagnostics consistently identify a broad crossover centered in the weak-to-intermediate coupling regime U~2-6; charge gaps open rapidly, double occupancy suppresses, local moments form, entanglement is significantly reduced, PCA concentrates variance into a few dominant modes and the distance matrices reveals rapid wavefunction reorganization. Our multi-pronged approach establishes the Mott crossover as a smooth but well defined reorganization of electronic correlations with pronounced spin-sector dependence in finite systems.

Figures (6)

• Figure 1: Spin-resolved charge gaps as a function of $U$ on the $3\times 3$ square cluster. The low-spin gap $\Delta_c^{low}=E(10,0)+E(8,0)-2E(9,\frac{1}{2})$ and the high-spin gap $\Delta_c^{high}=E(10,1)+E(8,1)-2E(9,\frac{3}{2})$ both increase monotonically with $U$, reflecting the suppression of charge fluctuations. The low-spin sector remains consistently more gapped across the full range of interactions. Inset: The gap difference $\delta\Delta_c$=$\Delta_c^{low}$-$\Delta_c^{high}$ highlights a pronounced maximum at intermediate $U$, indicating a non-trivial spin-dependent crossover in the charge-excited sector.
• Figure 2: Average local moment $\bar{m}$ (main panel) and average double occupancy $\bar{d}$ (inset) as a function of $U$ for the half-filled $3\times 3$ square cluster in the $S=1/2$ and $S=3/2$ sectors. The rapid decrease of $\bar{d}$ at weak-intermediate $U$ and the monotonic growth of $\bar{m}$ signal the loss of on-site charge fluctuations and the concomitant formation of robust local moments.
• Figure 3: von-Neumann entanglement entropy $S_A$ as a function of the interaction strength $U$ for the half-filled $3\times 3$ square cluster in the $S=1/2$ and $S=3/2$ spin sectors. Both sectors exhibit large entanglement at $U=0$, indicative of strong inter-sublattice charge fluctuations; $S_A$ decreases monotonically with $U$, reflecting the suppression of quantum correlations and the buildup of local moments.
• Figure 4: Principal component analysis (PCA) of (a,c) charge correlation functions and (b,d) spin correlation functions, in $S=1/2$ and $S=3/2$ sectors, respectively for the half-filled band of $3\times 3$ square cluster. Panels (a-i), (b-i), (c-i) and (d-i) show the projection of the correlation data onto the space of the first two principal components ($p_1$, $p_2$) for $U=0,2,4,6,8$ and 10 with colors representing the on-site interaction $U$. Panels (a-ii), (b-ii), (c-ii) and (d-ii) display the explained variance-ratio $\tilde{\lambda}_k$ of the first six principal components as functions of the component index k.
• Figure 5: Spatial distribution of the leading PCA eigenvectors (PC1) for charge ($D_{ij}$) and spin ($L_{ij}$) correlation matrices in the $S=1/2$ and $S=3/2$ sectors of the half-filled $3\times 3$ Hubbard cluster. Each $3\times 3$ panel shows the normalized eigenvector amplitudes across lattice sites for increasing $U=0$-10. Charge eigenvectors evolve from inhomogeneous metallic patterns to stabilized checkerboard textures, while spin eigenvectors develop pronounced staggered antiferromagnetic character, especially in the $S=1/2$ sector, signaling the real-space onset of Mott localization.