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arXiv:1912.06654

Application of machine learning in Bose-Einstein condensation critical-temperature analyses of path-integral Monte Carlo simulations

Adith Ramamurti

Published Dec 14, 2019• 0 citations
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Abstract

We detail the use of simple machine learning algorithms to determine the critical Bose-Einstein condensation (BEC) critical temperature $T_\text{c}$ from ensembles of paths created by path-integral Monte Carlo (PIMC) simulations. We quickly overview critical temperature analysis methods from literature, and then compare the results of simple machine learning algorithm analyses with these prior-published methods for one-component Coulomb Bose gases and liquid $^4$He, showing good agreement.

DOI
10.48550/arxiv.1912.06654

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We detail the use of simple machine learning algorithms to determine the critical Bose-Einstein condensation (BEC) critical temperature TcT_\text{c}Tc​ from ensembles of paths created by path-integral Monte Carlo (PIMC) simulations. We quickly overview critical temperature analysis methods from literature, and then compare the results of simple machine learning algorithm analyses with these prior-published methods for one-component Coulomb Bose gases and liquid 4^44He, showing good agreement.
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Bose-Einstein condensation has been experimentally found to take place in finite trapped systems when one of the confining frequencies is increased until the gas becomes effectively two-dimensional (2D). We confirm the plausibility of this result by performing path-integral Monte Carlo (PIMC) simulations of trapped Bose gases of increasing anisotropy and comparing them to the predictions of finite-temperature many-body theory. PIMC simulations provide an essentially exact description of these systems; they yield the density profile directly and provide two different estimates for the condensate fraction. For the ideal gas, we find that the PIMC column density of the squeezed gas corresponds quite accurately to that of the exact analytic solution and, moreover, is well mimicked by the density of a 2D gas at the same temperature; the two estimates for the condensate fraction bracket the exact result. For the interacting case, we find 2D Hartree-Fock solutions whose density profiles coincide quite well with the PIMC column densities and whose predictions for the condensate fraction are again bracketed by the PIMC estimates.
We determine the critical temperature of hard-sphere Bosons in a cubic box by path-integral Monte Carlo simulations and finite-size scaling. We obtain an enhancement compared to the ideal gas at low densities in the form of a power law in density with exponent 1/31/31/3. At high densities the result for liquid helium, namely a lower critical temperature than in the non-interacting case, is recovered. We give a microscopic explanation for the observed behavior.
We determine the critical temperature of a 3D homogeneous system of hard-sphere bosons by path-integral Monte Carlo simulations and finite-size scaling. In the low density limit, we find that the critical temperature is increased by the repulsive interactions, as $\ensuremath{\Delta}{T}_{C}{/T}_{0}\ensuremath{\sim}({\mathrm{na}}^{3}{)}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.34\ifmmode\pm\else\textpm\fi{}0.03$. At high densities the result for liquid helium, namely, a lower critical temperature than in the noninteracting case, is recovered. We give a microscopic explanation for the observed behavior.