Equation of state of Bose gases beyond the universal regime

The equation of state of dilute Bose gases, in which the energy only depends on the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>s</a:mi></a:math>-wave scattering length, is rather unknown beyond the universal limit. We have carried out a set of diffusion Monte Carlo calculations up to gas parameters of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>2</b:mn></b:mrow></b:msup></b:math> to explore how the departure from the universality emerges. Using different model potentials, we calculate the energies of the gas in an exact way, within some statistical noise, and report the results as a function of the three relevant scattering parameters: the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>s</c:mi></c:math>-wave scattering length <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>a</d:mi><d:mn>0</d:mn></d:msub></d:math>, the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>s</e:mi></e:math>-wave effective range <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>r</f:mi><f:mn>0</f:mn></f:msub></f:math>, and the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>p</g:mi></g:math>-wave scattering length <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>a</h:mi><h:mn>1</h:mn></h:msub></h:math>. If the effective range is not large, we observe universality in terms of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>a</i:mi><i:mn>0</i:mn></i:msub></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mi>r</j:mi><j:mn>0</j:mn></j:msub></j:math> up to gas parameters of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>2</k:mn></k:mrow></k:msup></k:math>. If <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msub><l:mi>r</l:mi><l:mn>0</l:mn></l:msub></l:math> grows, the regime of universality in these two parameters is reduced and effects of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>a</m:mi><m:mn>1</m:mn></m:msub></m:math> start to be observed. In the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mo>(</n:mo><n:msub><n:mi>a</n:mi><n:mn>0</n:mn></n:msub><n:mo>,</n:mo><n:msub><n:mi>r</n:mi><n:mn>0</n:mn></n:msub><n:mo>)</n:mo></n:mrow></n:math> universal regime we propose an analytical law that reproduces fairly well the exact energies. Published by the American Physical Society 2024