Normal state specific heat in the cuprate superconductors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>La</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>La</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> near the critical point of the pseudogap phase
Bibliographic record
Abstract
The specific heat $C$ of the cuprate superconductors ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ and ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2\ensuremath{-}x\ensuremath{-}y}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ was measured at low temperatures (down to 0.5 K) for dopings $p$ close to ${p}^{★}$, the critical doping for the onset of the pseudogap phase. A magnetic field up to 35 T was applied to suppress superconductivity, giving direct access to the normal state at low temperatures, and enabling a determination of ${C}_{\mathrm{e}}$, the electronic contribution to the normal-state specific heat at $T\ensuremath{\rightarrow}0$. In ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ at $x=p=0.22$, 0.24 and 0.25, ${C}_{\mathrm{e}}/T=15\phantom{\rule{0.28em}{0ex}}\mathrm{to}\phantom{\rule{0.28em}{0ex}}16\phantom{\rule{0.28em}{0ex}}\mathrm{mJ}\phantom{\rule{0.16em}{0ex}}{\mathrm{mol}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}2}$ at $T=2\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, values that are twice as large as those measured at higher doping ($p>0.3$) and lower doping ($p<0.15$). This confirms the presence of a broad peak in the doping dependence of ${C}_{\mathrm{e}}$ at ${p}^{★}\ensuremath{\simeq}0.19$ as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as $T\ensuremath{\rightarrow}0$ such that ${C}_{\mathrm{e}}/T\ensuremath{\sim}B\phantom{\rule{0.16em}{0ex}}ln({T}_{0}/T)$. The peak versus $p$ and the logarithmic dependence versus $T$ are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2\ensuremath{-}x\ensuremath{-}y}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$, we again find that ${C}_{\mathrm{e}}/T\ensuremath{\sim}B\phantom{\rule{0.16em}{0ex}}ln({T}_{0}/T)$ at $p\ensuremath{\simeq}{p}^{★}$, strong evidence that this $ln(1/T)$ dependence of the electronic specific heat---first discovered in the cuprates ${\mathrm{La}}_{1.8\ensuremath{-}x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ and ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$---is a universal property of the pseudogap critical point.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 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".