Eliashberg theory in the weak-coupling limit: Results on the real frequency axis
Bibliographic record
Abstract
We formulate and solve the Eliashberg equations on the imaginary frequency axis at temperatures below ${T}_{c}$ in the weak-coupling limit. We find an excellent scaling at all temperatures, for a fixed coupling strength, and the normalized order parameter exhibits a BCS-like temperature dependence. The hybrid real-imaginary axis equations are also solved to obtain numerically exact analytic continuations from the imaginary frequency axis to the real frequency axis. This provides a determination of the gap edge, which, in the weak-coupling limit, is identical to the order parameter from the imaginary axis. The analytical result for the zero-temperature gap edge deviates from the BCS result by a factor of $1/\sqrt{e}$, which was also obtained for the transition temperature ${T}_{c}$. We show that the normalized gap function on both the real and imaginary frequency axes, for an electron-phonon Einstein spectrum $(\ensuremath{\delta}$ function) of a fixed strength, is a universal function of frequency, independent of temperature. The $1/\sqrt{e}$ correction is a result of this nontrivial frequency dependence in the gap function. This modification, in the gap edge and in ${T}_{c}$, serves to preserve various dimensionless ratios to their BCS values.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
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 teacher head, 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".