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arXiv:cond-mat/0307387 · Journal of Physics B Atomic Molecular and Optical Physics

The expansion in the symmetry-broken phase of an interacting Bose gas at finite temperature

Georgios Metikas, O. Zobay, G. Alber

Published Nov 4, 2003• 8 citations
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Abstract

We discuss the application of the momentum-shell renormalization group method to the interacting homogeneous Bose gas in the symmetric and symmetry-broken phases. It is demonstrated that recently discussed discrepancies are artefacts due to not taking proper care of infrared divergencies appearing at finite temperature. If these divergencies are taken into account and treated properly by means of the ε expansion, the resulting renormalization group equations and the corresponding universal properties are identical in the symmetric and symmetry-broken phases.

DOI
10.1088/0953-4075/36/22/017

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We discuss the application of the momentum-shell renormalization group method to the interacting homogeneous Bose gas in the symmetric and symmetry-broken phases. It is demonstrated that recently discussed discrepancies are artefacts due to not taking proper care of infrared divergencies appearing at finite temperature. If these divergencies are taken into account and treated properly by means of the ε expansion, the resulting renormalization group equations and the corresponding universal properties are identical in the symmetric and symmetry-broken phases.
We exploit the renormalization-group approach to establish the exact infrared behavior of an interacting Bose system at zero temperature. The local-gauge symmetry in the broken-symmetry phase is implemented through the associated Ward identities, which reduce the number of independent running couplings to a single one. For this coupling the \ensuremath{\epsilon} expansion can be controlled to all orders in \ensuremath{\epsilon} $(=3\ensuremath{-}d).$ For spatial dimensions 1<d< 31<d<~31<d< 3 the Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of the longitudinal correlation function. The Bogoliubov linear spectrum, however, is found to be independent of the critical behavior of this correlation function, being exactly constrained by Ward identities. The new fixed point properly gives a finite value of the coupling among transverse fluctuations, but due to virtual intermediate longitudinal fluctuations, the effective coupling affecting the transverse correlation function flows to zero. As a result, no transverse anomalous dimension is present. This treatment allows us to recover known results for the quantum Bose gas in the context of a unifying framework and also to reveal the nontrivial skeleton structure of its perturbation theory.
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.
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 investigate the phase transition of interacting Bose gases in general power-law traps in the thermodynamic limit. Using energy-shell renormalization and the $\ensuremath{\epsilon}$ expansion, we evaluate the partition function for the uncondensed gas phase within a renormalization-group framework. This approach allows a unified description of homogeneous as well as inhomogeneous and anisotropic systems. Results for the critical temperature are compared to mean-field theory as well as to a local-density approximation based on renormalization-group theory for the homogeneous Bose gas. This comparison indicates the consistency of our approach. We also make suggestions for an optimized trap design in experiments that measure the transition temperature.
The momentum-shell renormalization group (RG) is used to study the condensation of interacting Bose gases without and with disorder. First of all, for the homogeneous disorder-free Bose gas the interaction-induced shifts in the critical temperature and chemical potential are determined up to second order in the scattering length. The approach does not make use of dimensional reduction and is thus independent of previous derivations. Secondly, the RG is used together with the replica method to study the interacting Bose gas with delta-correlated disorder. The flow equations are derived and found to reduce, in the high-temperature limit, to the RG equations of the classical Landau-Ginzburg model with random-exchange defects. The random fixed point is used to calculate the condensation temperature under the combined influence of particle interactions and disorder.