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arXiv:2112.10301 · Journal of the Physical Society of Japan

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

Akiko Masaki-Kato, Yuichi Motoyama, Naoki Kawashima

Published Jan 18, 2022• 5 citations
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Abstract

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 $a_s$ of the Bose-Hubbard model is $a_s/a_l = 0.316(2)$ where $a_l$ is the lattice constant. It turned out that inclusion of the universal second correction term of $O(n \log n)$ makes the fitting worse while it is restored by inclusion of the non-universal third correction term, of which the existence was predicted analytically.

DOI
10.7566/jpsj.91.024001

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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.
We have studied dilute Bose-Bose mixtures of atoms with attractive interspecies and repulsive intraspecies interactions using quantum Monte Carlo methods at T=0T=0T=0. Using a number of models for interactions, we determine the range of validity of the universal equation of state of the symmetric liquid mixture as a function of two parameters: the sss-wave scattering length and the effective range of the interaction potential. It is shown that the Lee-Huang-Yang correction is sufficient only for extremely dilute liquids with the additional restriction that the range of the potential is small enough. Based on the quantum Monte Carlo equation of state we develop a density functional which goes beyond the Lee-Huang-Yang term and use it together with the local density approximation to determine density profiles of realistic self-bound drops.
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
Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function ${\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))$ once the mass mmm and the sss-wave scattering length ad{a}_{d}ad​ of the bosons are known (ddd is the space dimension). Here $\ensuremath{\mu}$ and TTT denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ is a function of $m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}$. We compute the scaling function F2{\mathcal{F}}_{2}F2​ using a nonperturbative renormalization-group approach and find that both the $\ensuremath{\mu}/{k}_{B}T$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.
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.
For a dilute system of non-relativistic bosons interacting through a positive L1L^1L1 potential vvv with scattering length aaa we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1+12815πρa3+o(ρa3 ))e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))e(ρ)≥4πaρ2(1+15π​128​ρa3​+o(ρa3​)), thereby proving the Lee-Huang-Yang formula for the energy density.