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arXiv:1812.06521 · Physical Review A

Are smooth pseudopotentials a good choice for representing short-range interactions?

Péter Jeszenszki, Ali Alavi, Joachim Brand

Published Mar 11, 2019• 6 citations
arXivHTMLPDF

Abstract

When seeking a numerical representation of a quantum-mechanical multiparticle problem it is tempting to replace a singular short-range interaction by a smooth finite-range pseudopotential. Finite basis set expansions, e.g., in Fock space, are then guaranteed to converge exponentially. The need to faithfully represent the artificial length scale of the pseudopotential, however, places a costly burden on the basis set. Here we discuss scaling relations for the required size of the basis set and demonstrate the basis set convergence on the example of a two-dimensional system of few fermions with short-range $s$-wave interactions in a harmonic trapping potential. In particular we show that the number of harmonic oscillator basis functions needed to reach a regime of exponential convergence for a Gaussian pseudopotential scales with the fourth power of the pseudopotential length scale, which can be improved to quadratic scaling when the basis functions are rescaled appropriately. Numerical examples for three fermions with up to a few hundred single-particle basis functions are presented and implications for the feasibility of accurate numerical multiparticle simulations of interacting ultracold-atom systems are discussed.

DOI
10.1103/physreva.99.033608

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When seeking a numerical representation of a quantum-mechanical multiparticle problem it is tempting to replace a singular short-range interaction by a smooth finite-range pseudopotential. Finite basis set expansions, e.g., in Fock space, are then guaranteed to converge exponentially. The need to faithfully represent the artificial length scale of the pseudopotential, however, places a costly burden on the basis set. Here we discuss scaling relations for the required size of the basis set and demonstrate the basis set convergence on the example of a two-dimensional system of few fermions with short-range sss-wave interactions in a harmonic trapping potential. In particular we show that the number of harmonic oscillator basis functions needed to reach a regime of exponential convergence for a Gaussian pseudopotential scales with the fourth power of the pseudopotential length scale, which can be improved to quadratic scaling when the basis functions are rescaled appropriately. Numerical examples for three fermions with up to a few hundred single-particle basis functions are presented and implications for the feasibility of accurate numerical multiparticle simulations of interacting ultracold-atom systems are discussed.
Representing a strongly interacting multiparticle wave function in a finite product basis leads to errors. Simple rescaling of the contact interaction can preserve the low-lying energy spectrum and long-wavelength structure of wave functions in one-dimensional systems and thus correct for the basis set truncation error. The analytic form of the rescaling is found for a two-particle system where the rescaling is exact. A detailed comparison between finite Hilbert space calculations and exact results for up to five particles show that rescaling can significantly improve the accuracy of numerical calculations in various external potentials. In addition to ground-state energies, the low-lying excitation spectrum, density profile, and correlation functions are studied. The results give a promising outlook for numerical simulations of trapped ultracold atoms.
We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body interactions, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the no-core shell model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by other methods (in the four-body case). We also present results for finite scattering lengths and nonzero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.
We propose a discrete-space representation of a one-dimensional zero-range odd-parity pseudopotential. The proposed representation is validated by applying it to the analytically solvable case of two fermions in a harmonic trap and successfully recovering the exact energy spectrum and eigenfunctions. Furthermore, we use the square-well and modified Pöschl--Teller potentials as finite-range representations of the odd-parity interaction and study their convergence to the contact interaction when the range tends to zero. Finally, we perform natural orbital analysis and compute the eigenvalues of the one-body density matrix for different particle numbers, examining their dependence on the one-dimensional scattering length and identifying distinct physical regimes.
The contact interaction is often used in modeling ultracold atomic gases, although it leads to pathological behavior arising from the divergence of the many-body wave function when two particles coalesce. This makes it difficult to use this model interaction in quantum Monte Carlo and other popular numerical methods. Researchers therefore model the contact interaction with pseudopotentials, such as the square well potential, whose scattering properties deviate markedly from those of the contact potential. In this article, we propose a family of pseudopotentials that reproduce the scattering phase shifts of the contact interaction up to a hundred times more accurately than the square well potential. Moreover, the pseudopotentials are smooth, resulting in significant improvements in efficiency when used in numerical calculations.
We compute the ground-state energy of two atoms in a one-dimensional geometry of a harmonic optical trap. We obtain a dependence of the energy on a one-dimensional scattering length, which corre-sponds to various strengths of the interaction potential V int (x) = V 0 exp {−2cx 2}. The calculation is performed by numerical and analytical methods. For the analytical method we choose the oscillator representation method (OR), which has been successfully applied to computations of bound states of various few-body systems. The main results of this paper are (1) a numerical investigation of the validity range of the previously used pseudopotential method and (2) an investigation of the validity range of the OR for the potential V(x) = V conf (x) + V int (x) = x 2/2 + V 0 exp {−2cx 2}.
We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.