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Teubner-Texte zur Physik

Bose-Einstein Condensation under External Conditions

Klaus Kirsten

Published Jan 1, 1996• 4 citations
Publisher

Abstract

We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation can be given if we adopt generalized ζ-functions to define the quantum theory. We show that the crucial feature governing Bose-Einstein condensation is the dimension q associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. In either case Bose-Einstein condensation can only occur if q ≥ 3 [1]. Several examples, some of them not completely solved before, may be treated very easily using our simple criterion [2]. These examples are charged free gases confined in finite volumes as torus, boxes, spheres, cylinders (see also [3, 4, 5, 6]) and in the presence of general constant external magnetic fields ([7, 8, 9, 10]).

DOI
10.1007/978-3-663-01204-7_29

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We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation can be given if we adopt generalized ζ-functions to define the quantum theory. We show that the crucial feature governing Bose-Einstein condensation is the dimension q associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. In either case Bose-Einstein condensation can only occur if q ≥ 3 [1]. Several examples, some of them not completely solved before, may be treated very easily using our simple criterion [2]. These examples are charged free gases confined in finite volumes as torus, boxes, spheres, cylinders (see also [3, 4, 5, 6]) and in the presence of general constant external magnetic fields ([7, 8, 9, 10]).
We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation can be given if we adopt generalized \\zeta-functions to define the quantum theory. We show that the crucial feature governing Bose-Einstein condensation is the dimension q associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. In either case Bose-Einstein condensation can only occur if q\\ge3.
We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation is given. We show that condensation can occur only if q> 3q>~3q> 3, where qqq is the dimension associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. Furthermore we show that the criterion for the appearance of the Meissner-Ochsenfeld effect is closely connected with that for the appearance of Bose-Einstein condensation.
We discuss the occurrence of Bose-Einstein condensation in systems of noninteracting charged particles in three and one dimensions and in presence of an external magnetic field. In the one dimensional, as well as in the magnetic field cases, although not a critical temperature, a characteristic temperature can be found, corresponding to the case in which the ground state density becomes a macroscopic fraction of the total density. The case of relativistic charged scalar and vector particles is studied. The results give support to the existence of superconductivity in extremely strong magnetic fields, and lead to the prediction of superconductive-ferromagnetic behavior in the vector field case, which might be of interest in condensed matter as well as in cosmology. Some features of the magnetization in the early universe are conjectured.
We discuss the occurrence of Bose-Einstein condensation in systems of noninteracting charged particles in three in one dimensions and in presence of an external magnetic field. In the one dimensional, as well as in the magnetic field cases, although not a critical temperature, a characteristic temperature can be found, corresponding to the case in which the ground state density becomes a macroscopic fraction of the total density. The case of relativistic charged scalar and vector particles is studied. The results obtainedgive support to the existence of superconductivity in extremely strong magnetic fields, and leads to the prediction of superconductive-ferromagnetic behavior in the vector field case, which might be of interest in condensed matter as well as in cosmology. Some features of the magnetization in the early universe are conjectured.
We consider the problem of Bose-Einstein condensation for a system of nonrelativistic spin-0 bosons in a space of arbitrary dimension D. A general static homogeneous magnetic field is imposed. The effective action approach and \ensuremath{\zeta}-function regularization are used. If D=2\ensuremath{\delta} or 2\ensuremath{\delta}+1, a constant magnetic field is characterized by \ensuremath{\delta} independent components. If p\ensuremath{\le}\ensuremath{\delta} of these components are nonzero, the condition for Bose-Einstein condensation to occur is D-2p\ensuremath{\ge}3. This means that if D=2\ensuremath{\delta}, then Bose-Einstein condensation never occurs for p=\ensuremath{\delta}-1 or \ensuremath{\delta}. If D=2\ensuremath{\delta}+1, Bose-Einstein condensation never occurs for p=\ensuremath{\delta}. For D-2p\ensuremath{\ge}3, Bose-Einstein condensation does occur, and we show how it may be interpreted as symmetry breaking to give a ground state which is not constant.
Three examples of spatial Bose-Einstein condensations in which the particles macroscopically occupy the lowest localized state of an inhomogeneous external potential are analyzed. The three cases are (a) a small potential well in a large box which causes a spectrum with a gap, (b) a harmonic oscillator potential, and (c) randomly sized trapping potentials caused by ’’impurities.’’ All three cases are two dimensional so that no Bose condensation occurs without the inhomogeneous potentials. An attempt is made to keep the treatments as mathematically simple as possible. A review of much of the literature of spatial Bose condensations is provided. The special problem of the form of the thermodynamic limit in an inhomogeneous potential is discussed for case (b). Numerical examples applying to monolayer 4He adsorbed on a surface are treated.