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arXiv:cond-mat/0405045 · Physical Review A

Simple variational approach for an interacting Fermi trapped gas

Rocío Jáuregui, R. Paredes, G. Toledo Sánchez

Published Jan 1, 2004• 2 citations
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Abstract

Quantum states of a two-component Fermi trapped gas are described by introducing an effective trap frequency, determined via variational techniques. Closed expressions for the contribution of a contact interaction potential to the total energy and the pairing interaction are derived. They are valid for both few and large number of particles, given the discrete nature of the formulation, and therefore richer than the continuous expressions, which are perfectly matched. Pairing energies within a shell are explicitly evaluated and its allowed values at a given energy level delimited. We show the importance of the interaction over the trap energy as the number of particles N grows and the temperature decreases. At zero temperature we find a polynomial dependence of the interaction energy on the Fermi energy, whose dominant term at large N corresponds to the mean-field approximation result. In addition, the role of the strength of an attractive potential on the total energy is exhibited.

DOI
10.1103/physreva.69.013606

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Quantum states of a two-component Fermi trapped gas are described by introducing an effective trap frequency, determined via variational techniques. Closed expressions for the contribution of a contact interaction potential to the total energy and the pairing interaction are derived. They are valid for both few and large number of particles, given the discrete nature of the formulation, and therefore richer than the continuous expressions, which are perfectly matched. Pairing energies within a shell are explicitly evaluated and its allowed values at a given energy level delimited. We show the importance of the interaction over the trap energy as the number of particles N grows and the temperature decreases. At zero temperature we find a polynomial dependence of the interaction energy on the Fermi energy, whose dominant term at large N corresponds to the mean-field approximation result. In addition, the role of the strength of an attractive potential on the total energy is exhibited.
We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.
The structure and stability of dilute degenerate Fermi gases trapped in an external potential is discussed with special emphasis on the influence of s- and p-wave interactions. In a first step an effective contact interaction for all partial waves is derived, which reproduces the energy spectrum of the full potential within a mean-field model space. Using the s- and p-wave part the energy density of the multicomponent Fermi gas is calculated in Thomas-Fermi approximation. On this basis the stability of the one- and two-component Fermi gas against mean-field induced collapse is investigated. Explicit stability conditions in terms of density and total particle number are given. For the single-component system attractive p-wave interactions limit the density of the gas. In the two-component case a subtle competition of s- and p-wave interactions occurs and gives rise to a rich variety of phenomena. A repulsive p-wave part, for example, can stabilize a two-component system that would otherwise collapse due to an attractive s-wave interaction. It is concluded that the p-wave interaction may have important influence on the structure of degenerate Fermi gases and should not be discarded from the outset.
The energy density of an interacting Fermi gas in a harmonic potential is derived within localdensity approximation framework,in which the gas is assumed at each point to be locally homogenous. Based on the derived expressions,the thermodynamic stability conditions of trapped gas at zero temperature are studied,and the analytical expressions of stability conditions for d=l,2,3 dimensions are given. Furthermore,the stability phase diagrams are plotted for different dimension.It is shown that it is always stable when the system has repulsive interactions or has no interactions in condition of certain density of particle number,no mater what the dimension is.However,the system with attractive interactions is stable only under the condition of weak interaction strength.For d=1 and d = 3 cases,the critical values of interaction strength have close relationship with particle number density,but as for d= 2 case,the corresponding critical value does not depend on the particle number density explicitly.
We provide an efficient form to express the action of a many-body Hamiltonian of harmonically trapped interacting Fermi particles on wavefunctions built from paired states. The expression is suitable to numerically determine the ground state energy, regardless of the form of the two-body interaction. It takes advantage of the knowledge of the two-particle problem and the inherent properties of the matrix form of the many-body wavefunction. As an example, we evaluate the properties of a system composed of a balanced mixture of two families of fermions confined in a harmonic trap interacting through a short-range exponential potential. Numerical results for N ⩽ 10 and N = 35, 56, 84 and 165 particles of each family are reported. In the strong interacting regime corresponding to an infinite s-wave scattering length, our results give an upper bound to the Bertsch parameter for harmonically trapped systems ( E / E IFG ) 2 = 1 + β ⩽ 0.376 ± 0.008 with E the total energy and E IFG the energy for the analogous ideal Fermi gas. The influence of the harmonic trap and the interaction potential is exhibited in one and two-body correlation functions.
Abstract. We develop a consistent formalism in order to explore the effects of density and spin fluctuations on the quasi-particle properties and on the pairing critical temperature of a trapped Fermi gas on the attractive side of a Feshbach resonance. We first analyze the quasi-particle properties of a gas due to interactions far from resonance (effective mass and lifetime, quasi-particle strength and effective interaction) for the two cases of a spherically symmetric harmonic trap and of a spherically symmetric infinite potential well. We then explore the effect of each of these quantities on Tc and point out the important role played by the discrete level structure. PACS numbers: 03.75.Ss, 21.10.-k, 21.60.JzQuasi-particle properties of trapped Fermi gases 2 1.
We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers N=2−−22N=2--22N=2−−22 using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ETF{E}_{\mathit{TF}}ETF​ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.