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arXiv:0910.0720 · Physical Review A

Ultracold atoms at unitarity within quantum Monte Carlo methods

Andrew J. Morris, Pablo López Ríos, Richard J. Needs

Published Mar 23, 2010• 33 citations
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Abstract

Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. Our study differs from earlier ones mainly in that we have constructed more accurate trial wave functions and used a larger system size, we have studied the dependence of the energy on the particle density and well width, and we have achieved much smaller statistical error bars. The correct value of the universal ratio of the energy of the interacting to that of the noninteracting gas, $\ensuremath{\xi}$, is still a matter of debate. We find DMC values of $\ensuremath{\xi}$ of $0.4244(1)$ with 66 particles and $0.4339(1)$ with 128 particles. The spherically averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, which suggests that our results for these quantities are very accurate. We find, however, some differences between the VMC and DMC results for the two-body density matrices and condensate fractions, which indicates that these quantities are more sensitive to the quality of the trial wave function. Our best estimate of the condensate fraction of 0.51 is smaller than the values from earlier quantum Monte Carlo calculations.

DOI
10.1103/physreva.81.033619

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Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. Our study differs from earlier ones mainly in that we have constructed more accurate trial wave functions and used a larger system size, we have studied the dependence of the energy on the particle density and well width, and we have achieved much smaller statistical error bars. The correct value of the universal ratio of the energy of the interacting to that of the noninteracting gas, \ensuremathξ\ensuremath{\xi}\ensuremathξ, is still a matter of debate. We find DMC values of \ensuremathξ\ensuremath{\xi}\ensuremathξ of 0.4244(1)0.4244(1)0.4244(1) with 66 particles and 0.4339(1)0.4339(1)0.4339(1) with 128 particles. The spherically averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, which suggests that our results for these quantities are very accurate. We find, however, some differences between the VMC and DMC results for the two-body density matrices and condensate fractions, which indicates that these quantities are more sensitive to the quality of the trial wave function. Our best estimate of the condensate fraction of 0.51 is smaller than the values from earlier quantum Monte Carlo calculations.
We calculate the ground-state properties of an unpolarized two-component Fermi gas with the aid of the diffusion quantum Monte Carlo (DMC) methods. Using an extrapolation to the zero effective range of the attractive two-particle interaction, we find E/EfreeE/{E}_{\mathrm{free}}E/Efree​ in the unitary limit to be 0.212(2), 0.407(2), 0.409(3), and 0.398(3) for 4, 14, 38, and 66 atoms, respectively. Our calculations indicate that the dependence of the total energy on the effective range of the interaction Reff{R}_{\mathrm{eff}}Reff​ is sizable and the extrapolation to Reff=0{R}_{\mathrm{eff}}=0Reff​=0 is therefore important for reaching the true unitary limit. To test the quality of nodal surfaces and to estimate the impact of the fixed-node approximation, we perform released-node DMC calculations for 4 and 14 atoms. Analysis of the released-node and the fixed-node results suggests that the main sources of the fixed-node errors are long-range correlations, which are difficult to sample in the released-node approaches due to the fast growth of the bosonic noise. Besides energies, we evaluate the two-body density matrix and the condensate fraction. We find that the condensate fraction for the 66-atom system converges to 0.56(1) after the extrapolation to the zero interaction range.
We present a study of spin-unpolarized and spin-polarized two-dimensional uniform electron liquids using variational and diffusion quantum Monte Carlo (VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions. Ground-state VMC and DMC energies are obtained in the density range 1 ≤ r s ≤ 40 . Single-particle and many-body finite-size errors are corrected using canonical-ensemble twist-averaged boundary conditions and extrapolation of twist-averaged energies to the thermodynamic limit of infinite system size. System-size-dependent errors in Slater-Jastrow-backflow DMC energies caused by partially converged VMC energy minimization calculations are discussed. We find that, for 1 ≤ r s ≤ 5 , optimizing the backflow function at each twist lowers the twist-averaged DMC energy at finite system size. However, nonsystematic system-size-dependent effects remain in the DMC energies, which can be partially removed by extrapolation from multiple finite system sizes to infinite system size. The DMC energies in the thermodynamic limit are used to parametrize a local spin density approximation correlation functional for inhomogeneous electron systems. Our zero-temperature phase diagram shows a single transition from a paramagnetic fluid to a hexagonal Wigner crystal at r s = 35 ( 1 ) , with no region of stability for a ferromagnetic fluid. Published by the American Physical Society 2024
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
Variational and diffusion Monte Carlo (VMC and DMC) are widely used techniques in the Monte Carlo study of many-body quantum systems. Through efficient use of ran- dom sampling to solve many-dimensional integrals, they can be used to reliably solve the Schrödinger equation. Within the wider ecosystems of electronic structure calculation, quantum chemistry, and quantum simulation, quantum Monte Carlo (QMC) methods are used in tandem with density functional theory (DFT) and other computational methods to deliver insight into a vast array of condensed matter systems. Of particular interest to this work are systems in the intermediate correlation regime, where methods such as simple band theory and the Hubbard model break down. This thesis proposes and tests an explicit three-body backflow term to improve the accuracy of VMC and DMC calculations by incorporating higher-order correlation in order to reduce fixed-node er- ror in DMC, finding statistically significant improvements in ground state energies and improved variance in a water molecule. Then, QMC is used to calculate accurate charge densities for weak and intermediately correlated materials, to test commonly used dens- ity functional approximations in a controlled environment, providing insight into where these approximations perform well, and where they fail. Finally, DMC is used to generate ground state energy data for a dipolar Fermi gas of dressed Rydberg atoms interacting in the intermediate correlation regime. This data is parametrised in terms of interaction strength, for use in generating exchange-correlation functionals, to enable the DFT study of inhomogeneous systems of dressed Rydberg atoms.
We investigate the zero-temperature properties of a diluted homogeneous Bose gas made of NNN particles interacting via a two-body square-well potential by performing Monte Carlo simulations. We tune the interaction strength to achieve arbitrary positive values of the scattering length and compute by Monte Carlo quadrature the energy per particle E/NE/NE/N and the condensate fraction N0/N{N}_{0}/NN0​/N of this system by using a Jastrow ansatz for the many-body wave function, which avoids the formation of the self-bound ground state and describes instead a (metastable) gaseous state with uniform density. In the unitarity limit, where the scattering length diverges while the range of the interatomic potential is much smaller than the average distance between atoms, we find a finite energy per particle [$E/N=0.70\phantom{\rule{4pt}{0ex}}{\ensuremath{\hbar}}^{2}{(6{\ensuremath{\pi}}^{2}n)}^{2/3}/2m$, with nnn the number density] and a quite large condensate fraction (N0/N=0.83{N}_{0}/N=0.83N0​/N=0.83).
Variational and diffusion quantum Monte Carlo (VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions are used to study the spin-polarized three-dimensional uniform electron fluid. We report ground state VMC and DMC energies in the density range $0.5\ensuremath{\le}{r}_{\text{s}}\ensuremath{\le}20$. Finite-size errors are corrected using canonical-ensemble twist-averaged boundary conditions and extrapolation of the twist-averaged energy per particle calculated at three system sizes (N=113,259,and387)(N=113, 259, \mathrm{and} 387)(N=113,259,and387) to the thermodynamic limit of infinite system size. The DMC energies in the thermodynamic limit are used to parametrize a local spin density approximation correlation function for inhomogeneous electron systems.