Estimates of the dynamic structure factor for the finite temperature electron liquid via analytic continuation of path integral Monte Carlo data

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.