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Journal of Research of the National Institute of Standards and Technology

Effective action approach to Bose-Einstein condensation of ideal gases

Klaus Kirsten, David J. Toms

Published Jul 1, 1996• 11 citations
Publisher

Abstract

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.

DOI
10.6028/jres.101.049

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•Jun 16, 2013•0 citations
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.
A simple pedagogical introduction to the effective action method of quantum field theory is given at a level suitable for beginning postgraduate students. It is shown how to obtain the effective potential at zero temperature from a regularized zero-point energy. The results are applicable to curved as well as to flat space. The generalization to finite temperatures is also given. It is shown how to obtain high temperature expansions of the thermodynamic potential for the neutral free Bose gas and the charged Bose gas in both the relativistic and nonrelativistic limits. The results are obtained for an arbitrary spatial dimension and in curved space. Results are also obtained for the self-interacting relativistic gas in three spatial dimensions. A detailed discussion of how the formalism may be applied to study Bose-Einstein condensation is given. The interpretation of Bose-Einstein condensation as symmetry breaking is discussed. Application is given to the study of charged bosons, both relativistic and nonrelativistic, in a constant magnetic field. The Meissner effect is obtained for the nonrelativistic gas in three spatial dimensions. The final application is to the study of nonrelativistic bosons in a harmonic oscillator confing potential trap. A number of analytical approaches to this are discussed.
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.
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.
This chapter focuses on the applications involving finite temperature theories. Technically, the grand canonical partition function is easiest to handle and in general it is the one used for the analysis of the thermodynamical properties. In fact, for properties like the energy and the specific heat of the gas this is justified because in the thermodynamical limit the different ensembles are known to give the same answer. The chapter uses the grand canonical ensemble to obtain properties like critical temperature, energy and specific heat. It summarizes the grand canonical description of ideal Bose gases trapped by magnetic fields and describes the formulation of Bose-Einstein condensation as a symmetry-breaking phenomenon.
We rigorously revisit a textbook model used to figure out the Bose-Einstein condensation (BEC) phenomenon created by dilute cold alkali atoms gases in a magnetic-optical trap. It consists of a d-dimensional (d=1,2,3) ideal non-relativistic spin-0 Bose gas confined in a box and trapped in an isotropic harmonic potential. Throughout we review and clarify a series of methods involved in the derivation of the thermodynamics in the grand-canonical situation. To make the derivation consistent with the usual rules of the statistical mechanics, we assign through our open-trap limit approach the role of canonical parameter to a rescaled number of particles (instead of an effective density involving the pulsation of the trap). Within this approach, we formulate an Einstein-like and Penrose-Onsager-like criterion of BEC and show their equivalence. Afterwards, we focus on the spatial localization of the condensate/thermal gas. When dealing with the reduced density matrix, our method is similar to the loop path approach.