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arXiv:cond-mat/9806333 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Effect of quantum group invariance on trapped Fermi gases

Marcelo R. Ubriaco

Published Oct 1, 1998• 6 citations
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Abstract

We study the properties of a thermodynamic system having the symmetry of a quantum group and interacting with a harmonic potential. We calculate the dependence of the chemical potential, heat capacity, and spatial distribution of the gas on the quantum group parameter $q$ and the number of spatial dimensions $D.$ In addition, we consider a fourth-order interaction in the quantum group fields $\ensuremath{\Psi}$ and calculate the ground state energy up to first order.

DOI
10.1103/physreve.58.4191

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We study the properties of a thermodynamic system having the symmetry of a quantum group and interacting with a harmonic potential. We calculate the dependence of the chemical potential, heat capacity, and spatial distribution of the gas on the quantum group parameter qqq and the number of spatial dimensions D.D.D. In addition, we consider a fourth-order interaction in the quantum group fields \ensuremathΨ\ensuremath{\Psi}\ensuremathΨ and calculate the ground state energy up to first order.
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.
A system of a finite number of harmonically trapped fermions in one dimension, in the presence of a static magnetic field, is studied within the framework of the static fluctuation approximation, for different repulsive and attractive potential strengths. Specifically, the thermodynamic properties of the system (the chemical potential, total energy, heat capacity, and entropy), as well as its magnetic properties (the magnetization and susceptibility), are calculated. It is observed that the system remains in an ordered phase for a small number of particles N, even at high temperatures T. Disorder sets in for large N, even at low T. The effect of the potential strength on the heat capacity is particularly tangible in the region bordering the quantum and classical regimes. The effect of the temperature (representing disorder) is directly opposite to that of the magnetic field (representing order), as expected on basic physical grounds. These features are consistent with experimental results.
In view of ongoing experiments to trap ultracold spin-polarized 6Li,{}^{6}\mathrm{Li},6Li, we study various properties of an interacting Fermi gas in a harmonic trap, taking the discrete nature of the unperturbed harmonic trap levels into account exactly. As 6Li{}^{6}\mathrm{Li}6Li has a rather large and negative scattering length, we focus on the effects of the attractive atom-atom interaction on several thermodynamic properties, and on the momentum and density distributions. The dependence of the chemical potential, the specific heat, and the density and momentum distributions on the number of particles in the trap is obtained. We also calculate the energy of the gas. Comparison is made with results of a semiclassical calculation and with the properties of a noninteracting gas. We find that the effect of the interactions is rather large for realistic trap frequencies. Hence it is important to include these interactions in any quantitative predictions relevant for experiments.