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arXiv:cond-mat/0610308 · Physical Review B

Exact results for a charged harmonically trapped quantum gas at arbitrary temperature and magnetic field strength

Patrick Shea, Brandon P. van Zyl

Published Nov 29, 2006• 9 citations
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Abstract

An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. Our approach is nonpertabative, and is therefore exact for any temperature and magnetic field strength. We also present a useful factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on the system, and also permits the extension of our analysis to other dimensions, and/or anisotropic confinement.

DOI
10.1103/physrevb.74.205334

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An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. Our approach is nonpertabative, and is therefore exact for any temperature and magnetic field strength. We also present a useful factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on the system, and also permits the extension of our analysis to other dimensions, and/or anisotropic confinement.
Closed form analytical expressions are obtained for the Wigner transform of the Bloch density matrix and for the Wigner phase-space density of a two-dimensional harmonically trapped charged quantum gas in a uniform magnetic field of arbitrary strength, at zero and nonzero temperatures. An exact analytic expression is also obtained for the autocorrelation function. The strong magnetic field case, where only few Landau levels are occupied, is also examined, and useful approximate expressions for the spatial and momentum densities are given.
We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a 'macroscopic' number of particles, and finite-temperatures.
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We investigate the thermodynamic properties of an ideal charged Bose gas confined in an anisotropic harmonic potential and a constant magnetic field. Using an accurate density of states, we calculate analytically the thermodynamic potential and consequently various intriguing thermodynamic properties, including the Bose–Einstein transition temperature, the specific heat, magnetization, and the corrections to these quantities due to the finite number of particles are also given explicitly. In contrast to the infinite number of particles scenarios, we show that those thermodynamic properties, particularly the Bose–Einstein transition temperature depends upon the strength of the magnetic field due to the finiteness of the particle numbers, and the collective effects of a finite number of particles become larger when the particle number decreases. Moreover, the magnetization varies with the temperature due to the finiteness of the particle number while it keeps invariant in the thermodynamic limit N → ∞ .