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RENORMALIZATION GROUP ANALYSIS OF A CONFINED, INTERACTING BOSE GAS

G. Alber, G. Metikas

Published Jan 1, 2001• 3 citations
OpenAlex

Abstract

The renormalization group is not only a powerful method for describing universal properties of phase transitions but it is also useful for evaluating non-universal thermodynamic properties beyond meanfield theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group changing thus the The recent experimental realization of Bose-Einstein condensation has renewed the interest in the study of trapped, dilute, weakly interacting Bose gases. Despite the long history of this subject there still exist unresolved

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The renormalization group is not only a powerful method for describing universal properties of phase transitions but it is also useful for evaluating non-universal thermodynamic properties beyond meanfield theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group changing thus the The recent experimental realization of Bose-Einstein condensation has renewed the interest in the study of trapped, dilute, weakly interacting Bose gases. Despite the long history of this subject there still exist unresolved
A renormalization-group method is developed with which thermodynamic properties of a weakly interacting, finite-size Bose gas can be investigated. Thereby effects originating from a confining potential are taken into account by periodic boundary conditions and by treating the resulting discrete energy levels of the confined degrees of freedom properly. The resulting density of states modifies the flow equations of the renormalization group in momentum space. It is shown that as soon as the characteristic length of confinement becomes comparable to the thermal wavelength of a weakly interacting and trapped Bose gas its thermodynamic properties are changed significantly. This is exemplified by investigating characteristic bunching properties of the interacting Bose gas which manifest themselves in the second-order coherence factor.
We review some recent theoretical work on the phase transition of interacting Bose gases in the presence of external trapping potentials. A general framework for the study of such questions is presented which is based on the application of perturbative momentum-shell renormalization group methods to the trapped gas in the uncondensed phase. After giving an overview of this approach, we first establish its validity by comparing to previous results for homogeneous and harmonically trapped gases. Using this theoretical framework, we then examine various aspects of how external potentials influence the physics of condensation. (i) By studying the case of general power-law potentials and complemented by arguments from variational perturbation theory, it is quantitatively worked out how a growing inhomogeneity of the trapping potential diminishes nonperturbative effects at the transition. (ii) It is shown how by superimposing a weak periodic potentials on the homogeneous system, the characteristic nonperturbative momentum scale of critical interacting Bose gases can be probed. (iii) For a gas in a random potential, it is studied how condensation is affected by the combined influence of disorder effects and particle interactions.
We apply perturbative renormalization-group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilson's energy-shell renormalization and the $\ensuremath{\epsilon}$ expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our approach to examine the transition temperature of the harmonically trapped Bose gas as a function of the scattering length. The results are compared to previous studies of the problem.
We study the condensation of a homogeneous interacting Bose gas in the presence of a superimposed weak periodic potential. In contrast to, e.g., a harmonic trap, the periodic potential does not reduce the significance of long-wavelength fluctuations and thus influences the phase transition in a more profound way. We present general thermodynamic descriptions of the system above condensation in terms of mean-field as well as renormalization-group theory. Our approaches are based on describing the interplay between the effects of the interactions and the potential by means of a renormalization of the potential amplitude. These theories are then used to study the critical chemical potential and particle density and their dependence on the properties of the applied potential. A main result of our investigations concerns the characteristic nonperturbative momentum scale kc{k}_{c}kc​ of the critical interacting Bose gas. It is shown that the existence of this momentum scale becomes macroscopically manifest in the behavior of the critical thermodynamic properties.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
Due to the observation of Bose-Einstein condensation in dilute atomic 87^{87}87Rb and 23^{23}23Na vapors last year, there is currently a great interest in the properties of a degenerate Bose gas. One aspect that is not yet understood is the effect of the interatomic interactions on the critical temperature of the phase transition. We present here a renormalization group study of the weakly-interacting Bose gas and predict the critical temperature at which Bose-Einstein condensation occurs as a function of the (positive) scattering length aaa of the interatomic potential.