Temperature-filling phase diagram of the two-dimensional Holstein model in the thermodynamic limit by self-consistent Migdal approximation
Bibliographic record
Abstract
We study the temperature-filling phase diagram of the single-band Holstein model in two dimensions using the self-consistent Migdal approximation, where both the electron and phonon self-energies are treated on an equal footing. By employing an efficient numerical algorithm utilizing fast Fourier transforms to evaluate momentum and Matsubara frequency summations, we determine the charge-density-wave (CDW) and superconducting transition temperatures in the thermodynamic limit using lattice sizes that are sufficient to eliminate significant finite-size effects present at lower temperatures. We obtain the temperature-filling phase diagrams for a range of coupling strengths and phonon frequencies for the model defined on a square lattice with and without next-nearest-neighbor hopping. We find the appearance of a superconducting dome with a critical temperature that decreases before reaching the ${\mathbf{q}}_{\text{max}}=(\ensuremath{\pi},\ensuremath{\pi})$ CDW phase boundary. For very low phonon frequencies, we also find an incommensurate CDW phase with the ordering vector ${\mathbf{q}}_{\text{max}}\ensuremath{\approx}(\ensuremath{\pi},\ensuremath{\pi})$ appearing between the commensurate CDW and superconducting phases. Our numerical implementation can be easily extended to treat momentum-dependent electron-phonon coupling, as well as dispersive phonon branches, and has been made available to the public.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".