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arXiv:0903.1647 · Physical Review A

Stability of low-dimensional multicomponent dilute Bose gases

Alexei K. Kolezhuk

Published Jan 4, 2010• 16 citations
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Abstract

I show that in low dimensions the interactions in dilute Bose mixtures are strongly renormalized, which leads to a considerable change of stability conditions compared to the mean-field results valid in the high-density regime. Estimates are given for the two-component Bose-Hubbard model and for the $^{87}\mathrm{Rb}--^{41}\mathrm{K}$ mixture.

DOI
10.1103/physreva.81.013601

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I show that in low dimensions the interactions in dilute Bose mixtures are strongly renormalized, which leads to a considerable change of stability conditions compared to the mean-field results valid in the high-density regime. Estimates are given for the two-component Bose-Hubbard model and for the 87Rb−−41K^{87}\mathrm{Rb}--^{41}\mathrm{K}87Rb−−41K mixture.
We study the stability of the dilute Bose-Fermi and Bose-Bose mixtures with repulsive interactions in one and two dimensions in terms of the renormalization group. For the Bose-Fermi mixture, we show that the uniform mixture is stable against de-mixing in the dilute limit. For the Bose-Bose mixture, we give the stability conditions in the dilute limit. As a byproduct, we also calculate the critical temperature for the superfluid phase of the two-dimensional Bose-Fermi mixture in the extremely dilute limit.
We discuss the mean-field theories obtained from the leading order in a large-NNN approximation for one- and two-component dilute Bose gases. For a one-component Bose gas, this approximation has the following properties: the Bose-Einstein condensation (BEC) phase transition is second order but the critical temperature Tc{T}_{c}Tc​ is not shifted from the noninteracting gas value T0{T}_{0}T0​. The spectrum of excitations in the BEC phase resembles the Bogoliubov dispersion with the usual coupling constant replaced by the running coupling constant which depends on both temperature and momentum. We then study two-component Bose gases with both inter- and intraspecies interactions and focus on the stability of the mixture state above Tc{T}_{c}Tc​. Our mean-field approximation predicts an instability from the mixture state to a phase-separated state when the ratio of the interspecies interaction strength to the intraspecies interaction strength (assuming equal strength for both species) exceeds a critical value. At high temperature this is a structural transition and the global translational symmetry is broken. Our work complements previous studies on the instability of the mixture phase in the presence of BEC.
We propose a new method of constructing the effective theory for a multicomponent low-dimensional dilute Bose gas. The method is based on obtaining and solving renormalization group equations for all many-body interactions of density-density type in the one-loop approximation. In contrast to the standard approach based on two-body interactions, our method does not rely on the introduction of density-dependent infrared cutoff, and is able to reproduce exactly the leading term in the energy density of the one-dimensional dilute Bose gas, and to obtain the next term with the error of less than 2%.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
This colloquium surveys a number of theoretical problems and open questions in the field of two-dimensional dilute Bose gases with weak repulsive interactions. In contrast to three dimensions, in two dimensions the formation of long-range order is prohibited by the Bogoliubov-Hohenberg theorem, and Bose-Einstein condensation is not expected to occur. Nevertheless, experimental indications supporting the formation of a condensate in low-dimensional systems have recently been obtained. This unexpected behavior appears to be due to the nonuniformity introduced into a system by the external trapping potential. Theoretical predictions, made for homogeneous systems, require therefore careful reexamination. A number of popular theoretical treatments of the dilute weakly interacting Bose gas are presented and their regions of applicability are discussed. The possibility of Bose-Einstein condensation in a two-dimensional gas, the validity of the perturbative ttt-matrix approximation, and the diluteness condition are issues also discussed in detail.