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arXiv:1108.2687 · Physical Review A

Separable effective interaction for the trapped Fermi gas: The BEC-BCS crossover

C. N. Gilbreth, Y. Alhassid

Published Mar 15, 2012• 7 citations
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Abstract

We extend a recently introduced separable interaction for the unitary trapped Fermi gas to all values of the scattering length. We derive closed expressions for the interaction matrix elements and the two-particle eigenvectors. At unitarity we analytically demonstrate the convergence of this interaction to the zero-range two-body pseudopotential for $s$-wave scattering. We apply this effective interaction to the three- and four-particle systems along the BEC-BCS crossover and find that their low-lying energies exhibit convergence in the regularization parameter that is much faster than for the conventional renormalized contact interaction. We find similar convergence properties of the three-particle free energy at unitarity.

DOI
10.1103/physreva.85.033621

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We extend a recently introduced separable interaction for the unitary trapped Fermi gas to all values of the scattering length. We derive closed expressions for the interaction matrix elements and the two-particle eigenvectors. At unitarity we analytically demonstrate the convergence of this interaction to the zero-range two-body pseudopotential for sss-wave scattering. We apply this effective interaction to the three- and four-particle systems along the BEC-BCS crossover and find that their low-lying energies exhibit convergence in the regularization parameter that is much faster than for the conventional renormalized contact interaction. We find similar convergence properties of the three-particle free energy at unitarity.
We determine the energetically lowest-lying states in the BEC-BCS crossover regime of sss-wave interacting two-component Fermi gases under harmonic confinement by solving the many-body Schr\"odinger equation using two distinct approaches. Essentially exact basis set expansion techniques are applied to determine the energy spectrum of systems with N=4N=4N=4 fermions. Fixed-node diffusion Monte Carlo methods are applied to systems with up to N=20N=20N=20 fermions, and a discussion of different guiding functions used in the Monte Carlo approach to impose the proper symmetry of the fermionic system is presented. The energies are calculated as a function of the sss-wave scattering length as{a}_{s}as​ for N=2−−20N=2--20N=2−−20 fermions and different mass ratios $\ensuremath{\kappa}$ of the two species. On the Bose-Einstein condensate (BEC) and BCS sides, our energies agree with analytically determined first-order correction terms. We extract the scattering length and the effective range of the dimer-dimer system up to $\ensuremath{\kappa}=20$. Our energies for the strongly interacting trapped system in the unitarity regime show essentially no shell structure, and are well described by a simple expression whose functional form can be derived using the local density approximation, with one or two parameters. The universal parameter $\ensuremath{\xi}$ for the trapped system for various $\ensuremath{\kappa}$ is determined, and comparisons with results for the homogeneous system are presented.
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We apply the configuration-interaction method to calculate the spectra of two-component Fermi systems in a harmonic trap, studying the convergence of energies at the unitary interaction limit. We find that for a fixed regularization of the two-body interaction the convergence is exponential or better in the truncation parameter of the many-body space. However, the conventional regularization is found to have poor convergence in the regularization parameter, with an error that scales as a low negative power of this parameter. We propose a new regularization of the two-body interaction that produces exponential convergence for systems of three and four particles. We estimate the ground-state energy of the four-particle system to be (5.045 +/- 0.003) variant Planck's constant over 2 pi omega.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
We study a trapped system of fermions with a zero-range two-body interaction using the shell-model Monte Carlo method, providing ab initio results for the low particle number limit where mean-field theory is not applicable. We present results for the NNN-body energies as function of interaction strength, particle number, and temperature. The subtle question of renormalization in a finite model space is addressed and the convergence of our method and its applicability across the BCS-BEC crossover is discussed. Our findings indicate that very good quantitative results can be obtained on the BCS side, whereas at unitarity and in the BEC regime the convergence is less clear. Comparison to N=2N=2N=2 analytics at zero and finite temperature, and to other calculations in the literature for N>2N>2N>2 show very good agreement.
In this thesis,we investigate the Unitary Fermi gas and its thermodynamic properties. We look into the problem using the Lowest Order Constrained Variational method,using the Jastrow correlation function and determine the thermodynamic quantities of interest around unitarity.From this variational calculation we calculate quantities like energy per particle and chemical potential and study their variation as a function of the s-wave scattering length. We find there are two energy branches which suggests that the way to approach unitarity is not unique.We compare our results with those found from toy model for unitary fermi gas,a single particle in a spherical box interacting with a fixed scatterer at the center of the cavity interactions being modelled by the fermi pseudo potential.We find a surprising qualitative agreement between the results obtained. We also study the BCS-BEC crossover in three dimension using mean field analysis, and evaluate quantities like gap parameter and chemical potential across the whole range of the crossover.