Spin and pseudospin towers of the Hubbard model on a bipartite lattice
Bibliographic record
Abstract
In 1989, Lieb proved two theorems about the Hubbard model. One showed that the ground state of the attractive model was a spin singlet state (S = 0) was unique and was positive definite. The other showed that the ground state of the repulsive model on a bipartite lattice at half-filling has a total spin given by [Formula: see text](N[Formula: see text]NB)/2[Formula: see text], corresponding to the difference of the number of lattice sites on the two sublattices divided by two. In the mid to late 1990s, Shen extended these proofs to show that the pseudospin of the attractive model was minimal until the electron number equaled 2NA where it became fixed at J = [Formula: see text](N[Formula: see text]NB)/2[Formula: see text] until the filling became 2NB, where it became minimal again. In addition, Shen showed that a spin tower exists for the spin eigenstates for the half-filled case on a bipartite lattice. The spin tower says the minimal energy state with spin S is higher in energy than the minimal energy state with spin S − 1 until we reach the ground-state spin given above. One long-standing conjecture about this model remains, namely, does the attractive model have such a spin tower for all fillings, which would then imply that the repulsive model has minimal pseudopsin in its ground state. While we do not prove this last conjecture, we provide a quick review of this previous work, provide a constructive proof of the pseudospin of the attractive model ground state, and describe the challenges by proving the remaining open conjecture.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".