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Journal of Mathematical Physics

Magnetic Properties of Charged Ideal Quantum Gases in n Dimensions

Robert Mccredie May

Published Oct 1, 1965• 38 citations
Publisher

Abstract

We consider the mathematical model of an ideal gas of charged bosons or fermions in an n-dimensional space, treating n as a continuous variable. The investigation shows the extent to which the magnetic behavior depends on the dimensionality of the system. In particular, the charged Bose gas in a homogeneous magnetic field does not condense unless n > 4, in contrast to the field-free gas which condenses for n > 2: however so long as n > 2 and T < Tc this system completely expels homogeneous magnetic fields weaker than a certain critical field, Hc. (Tc is the field free transition temperature.) This field expulsion comes explicitly from the condensed ground-state bosons for n > 4; but it is still present, and Hc has the same form, for 4 > n > 2 where there is no condensation.

DOI
10.1063/1.1704682

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We consider the mathematical model of an ideal gas of charged bosons or fermions in an n-dimensional space, treating n as a continuous variable. The investigation shows the extent to which the magnetic behavior depends on the dimensionality of the system. In particular, the charged Bose gas in a homogeneous magnetic field does not condense unless n > 4, in contrast to the field-free gas which condenses for n > 2: however so long as n > 2 and T < Tc this system completely expels homogeneous magnetic fields weaker than a certain critical field, Hc. (Tc is the field free transition temperature.) This field expulsion comes explicitly from the condensed ground-state bosons for n > 4; but it is still present, and Hc has the same form, for 4 > n > 2 where there is no condensation.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.
The relationship between the two different ordering procedures taking place in an ideal charged relativistic Bose gas in the presence of a uniform external magnetic field B is investigated in an arbitrary number of spatial dimensions d. In particular the analytical form of the charge-density condensate qc{q}_{c}qc​ and the magnetization density \ensuremath{\rho} are calculated as a function of B and the inverse temperature \ensuremath{\beta} for relativistic bosons without spin. In the small-field limit (B\ensuremath{\rightarrow}0) this system exhibits the complete restoration of the symmetry of the ideal relativistic Bose gas for all d at any finite temperature.
A charged Bose gas under confinements ensures a phase transition, in contrast with the homogeneous system in an external magnetic field. The present investigation concerns charged Bose gases trapped in a two-dimensional potential as well as neutral Bose gases rotating around the z axis. It is found that the systems exhibit quite different magnetic properties. A diamagnetization is determined in the charged system while the rotating system shows up an uncommon magnetic response. However, both models possess finite magnetic susceptibilities at vanishing field or at stopping rotation.
The magnetic properties of charged vector boson gas are studied in the very weak, and very strong (near critical value) external magnetic field limits. When the density of the vector boson gas is low, or when the external field is strong, no true Bose–Einstein condensation occurs, though significant amount of bosons will accumulate in the ground state. The gas is ferromagnetic in nature at low temperature. However, Bose–Einstein condensation of vector bosons (scalar bosons as well) is likely to occur in the presence of a uniform weak magnetic field when the gas density is sufficiently high. A transitional density depending on the magnetic field seems to exist below which the vector boson gas changes its property with respect to the Bose–Einstein condensation in a uniform magnetic field.
It is well-known that the 2-dimensional Bose gas does not condense, and that no transition occurs in its thermodynamic behavior. Despite this, we find that in a magnetic field a 2-dimensional gas of charged, ideal bosons has a nearly sharp transition temperature, below which it displays an "imperfect" Meissner-Ochsenfeld effect, practically indistinguishable from the perfect London one. This fact is relevant to the application of the Bose gas model of superconductivity to thin films.The main interest of the model is, however, that it exhibits an (essentially perfect) Meissner-Ochsenfeld effect without any irregularity in CV(T){C}_{V}(T)CV​(T).
The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited as a first step to tackle the more complex case of a molecular condensate, where several degrees of freedom have to be taken into account. \nIn the charged bose gas, the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to (1/2) NkB and a gradual condensation at lower temperatures. \nIn the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs.