Ground-state analysis of the Falicov-Kimball model on complete graphs
Bibliographic record
Abstract
The ground-state nature of the FalicovKimball model of mobile electrons and fixed nuclei on complete graphs is investigated. We give a pedagogic derivation of the eigenvalue problem and present a complete account of the ground-state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U [Formula: see text] [Formula: see text]. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half-filled band for U < 0. The model in consideration was proposed and partially solved by Farkasovsky for finite graphs and repulsive on-site interaction U > 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model on bipartite complete graphs and examine how sharp the KennedyLieb variational estimate is as compared with the exact ground state. PACS Nos.: 71.2-b, 02.10Gd
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".