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 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 | 0.014 | 0.012 |
| Meta-epidemiology (narrow) | 0.007 | 0.015 |
| Meta-epidemiology (broad) | 0.003 | 0.017 |
| Bibliometrics | 0.004 | 0.011 |
| Science and technology studies | 0.012 | 0.016 |
| Scholarly communication | 0.012 | 0.012 |
| Open science | 0.018 | 0.016 |
| Research integrity | 0.014 | 0.015 |
| Insufficient payload (model declined to judge) | 0.879 | 0.013 |
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; both teacher heads agree on what is shown here.
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".