Pairing susceptibility in the weakly interacting multilayer Hubbard model evaluated by direct perturbative expansion
Bibliographic record
Abstract
We present a systematic study of the interaction, doping, and layer dependence of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing susceptibility of the Hubbard model for a stacked 2D square lattice. We perform a multi-index perturbative expansion up to fourth order to obtain coefficients in powers of the Hubbard $U$, the interlayer $V$, and the pair-hopping $J$ interactions. We evaluate the vertex diagrams that contribute to the pairing susceptibility for $\ensuremath{\ell}=2,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}4$ layered models in the $U\text{\ensuremath{-}}V\text{\ensuremath{-}}J$ interaction space. This provides unprecedented access to the pairing amplitudes, allowing us to identify the processes that enhance or reduce pairing. We distinguish pairing within the diagonal channel, ${P}_{d}^{\ensuremath{\parallel}}$, and off-diagonal channel, ${P}_{d}^{\ensuremath{\perp}}$, and find that, in the absence of $J$, the qualitative behavior of the layered system is equivalent to the single-layer model. In the presence of $J$, we show that pairing is enhanced sublinearly with increasing $\ensuremath{\ell}$ and is primarily mediated by the ${P}_{d}^{\ensuremath{\perp}}$ component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized $\ensuremath{\ell}$-dependent equation for ${P}_{d}^{\ensuremath{\perp}}$ to speculate pairing beyond $\ensuremath{\ell}=4$.
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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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".