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Physical Review A

Condensation of a charged relativistic Bose gas in the presence of a magnetic field

David E. Miller, P. S. Ray

Published Mar 1, 1986• 10 citations
Publisher

Abstract

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 ${q}_{c}$ 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.

DOI
10.1103/physreva.33.1990

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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.
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 statistical mechanics of a system of non-relativistic charged particles in a constant magnetic field is discussed. The spatial dimension DDD is arbitrary with D≥3D\geq 3D≥3 assumed. Calculations are presented from first principles using the effective action method. For D≥5D\geq 5D≥5 the system has a phase transition with a Bose condensate. We show how the effective action method method deals with in a very natural way with the condensate and study it's rôle in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in DDD spatial dimensions behaves like the free Bose gas in (D−2)(D-2)(D−2) spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any non-zero value of the magnetic field, we show how the specific heat starts to resemble the result for the free gas as the magnetic field is reduced. A number of analytical approximations for the magnetization and specific heat are given and compared with numerical results. In this way we are able to study in precise detail how the B→0B\to 0B→0 limit of the magnetized gas is achieved.
Abstract We study the effects of a uniform and constant magnetic field on the Bose–Einstein condensation (BEC) of relativistic charged scalar bosons. The condensation of magnetized charged bosons is usually discussed in the weak or strong field regimes separately, and in each case, the properties predicted for the condensate are different. Here, we develop a low‐temperature analysis suitable for any field that allows us to observe how the gas behaves for intermediate values of the magnetic field. We find that BEC may occur at finite magnetic fields, despite the gas becoming effectively one‐dimensional due to Landau quantization of the momentum perpendicular to the magnetic axis.
Bose-Einstein condensation of charged scalar and vector particles may actually occur in presence of a constant homogeneous magnetic field, but there is no critical temperature at which condensation starts. The condensate is described by the statistical distribution. The Meissner effect is possible in the scalar, but not in the vector field case, which exhibits a ferromagnetic behavior. Permanent address: Grupo de Fisica Teórica, ICIMAF, Academia de Ciencias de Cuba, Calle E No. 309,Vedado, La Habana 4, Cuba. It was pointed out long ago by Schafroth [1] that for a non-relativistic boson gas, Bose-Einstein condensation (BEC) does not take place in the presence of a constant magnetic field. The problem was studied afterwards by May [2], who investigated the condensation in the non-relativistic case for an arbitrary dimension d and showed that it occurs for d ≥ 5, and later by Daicic et al [3], [4], who extended the considerations made by [2] to the relativistic high temperature case and found new magnetization properties. More recently Toms [5] has proved that BEC in presence of a constant magnetic field does not occur in any number of spatial dimensions, and Elmfors et al [6] consider that a in the 3d
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.
The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited. 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 1/2Nk(B) and a gradual condensation at lower temperatures.