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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Bose-Einstein condensation in a general static homogeneous magnetic fieldinebreak and the effective action: The nonrelativistic ideal gas

David J. Toms

Published Feb 15, 1995• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.51.1886

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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.
We present a short review of how the effective action formalism, well known in relativistic quantum field theory, can be used to discuss Bose-Einstein condensation of non-relativistic gases.This method lends itself very naturally to an interpretation of Bose-Einstein condensation in terms of symmetry breaking.It also allows for the definition of a very elegant regularization technique involving generalized ␨ -functions.We show how this method can be used to recover the well known results for the free boson gas, as well as the charged boson gas in a constant magnetic field.A general criterion for interpreting Bose-Einstein condensation in terms of a phase transition with symmetry breaking is given.Finally we present an analysis of Bose-Einstein condensation in a harmonic oscillator confining potential trap, and show how the results of this simple model are in excellent agreement with experiment.
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.
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
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]).
The statistical mechanics of a system of nonrelativistic charged particles in a constant magnetic field is discussed. The spatial dimension D is arbitrary, with D>or=3 assumed. Calculations are presented from first principles using the effective action method. For D>or=5 the system has a phase transition with a Bose condensate. We show how the effective action method deals in a very natural way with the condensate, and study its role in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in D spatial dimensions behaves like the free Bose gas in D-2 spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any nonzero 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-->0 limit of the magnetized gas is achieved.