Estimates of the dynamic structure factor for the finite temperature electron liquid via analytic continuation of path integral Monte Carlo data
Bibliographic record
Abstract
Understanding the dynamic properties of uniform electron gas (UEG) is important for numerous applications ranging from semiconductor physics to exotic warm dense matter. In this work, we apply the maximum entropy method (MEM), as implemented by Thomas Chuna , , to path integral Monte Carlo (PIMC) results for the imaginary-time correlation function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>F</a:mi> <a:mo>(</a:mo> <a:mi>q</a:mi> <a:mo>,</a:mo> <a:mi>τ</a:mi> <a:mo>)</a:mo> </a:mrow> </a:math> to estimate the dynamic structure factor <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>S</b:mi> <b:mo>(</b:mo> <b:mi>q</b:mi> <b:mo>,</b:mo> <b:mi>ω</b:mi> <b:mo>)</b:mo> </b:mrow> </b:math> over an unprecedented range of densities at the electronic Fermi temperature. To conduct the MEM, we propose to construct the Bayesian prior <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>μ</c:mi> </c:math> from the PIMC data. Constructing the static approximation leads to a drastic improvement in <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>S</d:mi> <d:mo>(</d:mo> <d:mi>q</d:mi> <d:mo>,</d:mo> <d:mi>ω</d:mi> <d:mo>)</d:mo> </d:mrow> </d:math> estimate over using the simpler random phase approximation (RPA) as the Bayesian prior. We present results for the strongly coupled electron liquid regime with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msub> <e:mi>r</e:mi> <e:mi>s</e:mi> </e:msub> <e:mo>=</e:mo> <e:mn>50</e:mn> <e:mo>,</e:mo> <e:mo>⋯</e:mo> <e:mo>,</e:mo> <e:mn>200</e:mn> </e:mrow> </e:math> , which reveal a pronounced roton-type feature and an incipient double peak structure in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>S</f:mi> <f:mo>(</f:mo> <f:mi>q</f:mi> <f:mo>,</f:mo> <f:mi>ω</f:mi> <f:mo>)</f:mo> </f:mrow> </f:math> for intermediate wave numbers at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:msub> <g:mi>r</g:mi> <g:mi>s</g:mi> </g:msub> <g:mo>=</g:mo> <g:mn>200</g:mn> </g:mrow> </g:math> . We also find that our dynamic structure factors satisfy known sum rules, even though these sum rules are not enforced explicitly. To verify our results, we show that our MEM estimates converge to the RPA limit at higher densities and weaker coupling <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi>r</h:mi> <h:mi>s</h:mi> </h:msub> <h:mo>=</h:mo> <h:mn>2</h:mn> </h:mrow> </h:math> and 5. Further, we compare with two different existing results at intermediate density and coupling strength <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:msub> <i:mi>r</i:mi> <i:mi>s</i:mi> </i:msub> <i:mo>=</i:mo> <i:mn>10</i:mn> </i:mrow> </i:math> and 20, and we find good agreement with more conservative estimates. Combining all of our results for <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>r</j:mi> <j:mi>s</j:mi> </j:msub> <j:mo>=</j:mo> <j:mn>2</j:mn> <j:mo>,</j:mo> <j:mn>5</j:mn> <j:mo>,</j:mo> <j:mn>10</j:mn> <j:mo>,</j:mo> <j:mn>20</j:mn> <j:mo>,</j:mo> <j:mn>50</j:mn> <j:mo>,</j:mo> <j:mn>100</j:mn> <j:mo>,</j:mo> <j:mn>200</j:mn> </j:mrow> </j:math> , we present estimates of a dispersion relation that show a continuous deepening of its minimum value at higher coupling. An advantage of our setup is that it is not specific to the UEG, thereby opening up new avenues to study the dynamics of real warm dense matter systems based on cutting-edge PIMC simulations in future works.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".