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arXiv:2205.09341 · Journal of the Physical Society of Japan

Comment on “Universal and Non-Universal Correction Terms of Bose Gases in Dilute Region: A Quantum Monte Carlo Study”

Adam Rançon

Published May 17, 2022• 2 citations
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Abstract

We comment on J. Phys. Soc. Jpn. 91, 024001 (2022) by Masaki-Kato et al. and show that their results are in very good agreement with the semi-classical spin-wave approximation of Bose-Hubbard model in the hardcore limit. This allows for recovering the analytical formulae for both the s-wave scattering length and the Lee-Huang-Yang correction.

DOI
10.7566/jpsj.91.066001

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We comment on J. Phys. Soc. Jpn. 91, 024001 (2022) by Masaki-Kato et al. and show that their results are in very good agreement with the semi-classical spin-wave approximation of Bose-Hubbard model in the hardcore limit. This allows for recovering the analytical formulae for both the s-wave scattering length and the Lee-Huang-Yang correction.
We study dilute gases of interacting bosons at zero-temperature in the region where the system is characterized only by the s-wave scattering length. We carry out quantum Monte Carlo simulation of the Bose Hubbard model and a continuous-space hard-core model. Fitting the extended Lee-Huang-Yang formula to the Monte Carlo results establishes the detailed mapping from the lattice model to the continuous field theory characterized only by the s-wave scattering length. Our estimate of the intrinsic s-wave scattering length asa_sas​ of the Bose-Hubbard model is as/al=0.316(2)a_s/a_l = 0.316(2)as​/al​=0.316(2) where ala_lal​ is the lattice constant. It turned out that inclusion of the universal second correction term of O(nlog⁡n)O(n \log n)O(nlogn) makes the fitting worse while it is restored by inclusion of the non-universal third correction term, of which the existence was predicted analytically.
The equation of state of dilute Bose gases, in which the energy only depends on the s -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 10 − 2 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 s -wave scattering length a 0 , the s -wave effective range r 0 , and the p -wave scattering length a 1 . If the effective range is not large, we observe universality in terms of a 0 and r 0 up to gas parameters of 10 − 2 . If r 0 grows, the regime of universality in these two parameters is reduced and effects of a 1 start to be observed. In the ( a 0 , r 0 ) universal regime we propose an analytical law that reproduces fairly well the exact energies. Published by the American Physical Society 2024
We present a novel approach to the Bogoliubov theory of dilute Bose gases, allowing for an elementary derivation of the celebrated Lee-Huang-Yang formula in the Gross-Pitaevskii regime. Furthermore, we identify the low lying excitation spectrum beyond the Gross-Pitaevskii scaling, extending a recent result (Brennecke et al. in Rev Math Phys 34, 2022) to significantly more singular scaling regimes. Finally, we provide an upper bound on the ground state energy in the Gross-Pitaevskii regime that captures the correct expected order of magnitude beyond the Lee-Huang-Yang formula.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.