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Canadian Journal of Physics

Harmonically trapped one-dimensional Fermi gas using the static fluctuation approximation

H.A. Al-Khzon, H. B. Ghassib, Mohamed K. Al-Sugheir

Published Sep 14, 2015• 7 citations
Publisher

Abstract

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.

DOI
10.1139/cjp-2015-0315

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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.
The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An energy-entropy argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.
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 thermodynamic and magnetic properties of a spin-mixture of 40K atoms in a 1D harmonic trap, in the presence of an external magnetic field H, are studied using the static fluctuation approximation (SFA). The two-body interaction used is the contact potential. The thermodynamic properties include the chemical potential, energy per particle, specific heat capacity, and entropy per particle. The magnetic properties comprise the magnetization per particle and susceptibility per particle. A phase transition is observed for the repulsive interactions around 0.35TF and 0.5TF in the channels {–9/2, –7/2} and {–9/2, –5/2}, respectively. A peak appears in the entropy and susceptibility at effective scattering length |a1d| ≈ 0.16 μm. For the attractive interactions, three different behaviors depending on H are found in the low-temperature limit. The first behavior is when H < Hc, where Hc is the critical magnetic field, and typical peaks emerge. The second behavior is at H = Hc, and a magnetic phase transition occurs. The third behavior is when H > Hc and the system becomes a partially polarized phase. Our results are consistent with the reported results obtained by the local density approximation, thermal Bethe ansatz method, mean-field approximation, and quantum transfer matrix method.
Feynman-type series for grand potential , number of particles, internal energy, magnetization and susceptibility are derived for systems of noninteracting quantum particles in an external magnetic field. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in the presence of a homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single-particle states. N -convergence to universal curves for chemical potential and energy is presented. At low temperatures magnetization of the Fermi gas exhibits a pronounced quantum behaviour.
The thermodynamic properties of 6Li atomic gas system, with imbalanced spin populations trapped in one-dimension, were systematically investigated using the Static Fluctuation Approximation. The two-body interaction used is an attractive contact potential. The effects of gas parameter [Formula: see text] and spin polarization [Formula: see text], on the thermodynamic properties and effective magnetic field were investigated. We observed a decrease in [Formula: see text] and an enhancement in [Formula: see text] and [Formula: see text] with increasing [Formula: see text]. At strong interaction and at [Formula: see text], the behavior of entropy with [Formula: see text] indicated two different phases. At small spin polarization [Formula: see text], the system could be in Fulde–Ferrell–Larkin Ovchinnikov (FFLO) state, while above [Formula: see text], the system might be in normal state. In addition, we found a clear decrease in both [Formula: see text] and [Formula: see text] and an enhancement in [Formula: see text] with the increase of the interaction strength. Our results are consistent with the reported results obtained by the mean-field Bogoliubov–de Gennes method, the Bardeen–Cooper–Schrieffer (BCS) approximation and Nozieres–Schmitt–Rink (NSR) theory.