A structural measure of patterns called C0 was almost proportionally related to the entanglement entropy of half the system — a calculation of the link between two parts of a quantum system — in an arXiv preprint. As C0 changed with the model’s interaction strength, its derivative showed extrema at U/t = 3.50, 4.75 and 5.65, the same values as the reported mean-field phase boundaries.
The result is an association within a calculation, not evidence that structural complexity causes a phase change. The study analyzed spin-resolved density snapshots for an SU(3) Fermi-Hubbard model on a square lattice at one-third filling, with ground states generated by a self-consistent Hartree mean-field calculation that retained the lowest-energy converged solution.
The broad signal and the finer detail
The researchers computed bipartite entanglement entropy from a correlation matrix and compared it with structural-complexity measures applied through rectangular coarse-graining windows — small viewing blocks used to compare patterns at different scales.
C0 was unchanged across the tested 2 × 2, 3 × 2, 3 × 4 and 3 × 3 windows. The first-step dissimilarity measure, D0, closely followed C0 and supplied its dominant contribution, but it depended on window geometry in the antiferromagnetic phases and not in the metallic phase.
The higher-order measure C1 provided a window-dependent fingerprint of magnetic order. Its change with interaction strength had extrema at phase-boundary locations, although the sign across a boundary depended on the window. C1 rose in the tooth phase for the 2 × 2, 3 × 2 and 3 × 4 windows but vanished for 3 × 3; in stripe order, D0 captured the pattern while C1 vanished for 3 × n windows.
In a comparison with classical antiferromagnetic configurations, C1 remained at zero while D0 and C0 stayed constant across the antiferromagnetic phases. Quantum mean-field snapshots broke those constraints.
Built from simulated snapshots
Snapshots at fixed U/t came from many initial conditions and were post-selected with an energy cutoff relative to the minimum energy in the dataset. At least 50 post-selected images were used to tile a region of space for the structural analysis.
What remains untested
The study does not test the measures on experimental quantum-gas-microscope snapshots or provide an exact many-body validation. Its evidence is limited to the modeled SU(3), one-third-filling, square-lattice mean-field setting, with the reported boundaries benchmarked against that mean-field phase diagram.
Filtering affected the finer signal more than the broad one: stricter energy cutoffs lowered C1 and noise but did not change C0, and C1 became sensitive to the cutoff for U/t above 4.75. The total number of converged solutions and snapshots at each interaction strength was not reported, and no formal uncertainty intervals were supplied.
Further tests would need to examine other SU(N) models, fillings, lattice geometries and temperatures, and determine whether C1 can identify magnetic order without phase labels being known in advance.
Paper data and sources
Original title: Structural complexity of an SU(3) Fermi Hubbard model
Authors: Jiani Fu, Zewen Zhang, Eduardo Ibarra-García-Padilla
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text