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Physical Review Letters

Bose-Einstein Phase Transition in an Interacting System

Marshall Luban, Warren D. Grobman

Published Jul 25, 1966• 22 citations
Publisher

Abstract

It is shown that for both the Hartree-Fock and Bogoliubov models of interacting bosons associated with the disappearance of the Bose-Einstein condensation at a temperature ${T}_{c}$, the specific heat ${C}_{V}$ has a square root singularity, ${C}_{V}\ensuremath{\sim}A{({T}_{c}\ensuremath{-}T)}^{\ensuremath{-}\frac{1}{2}}$, and the superfluid density ${\ensuremath{\rho}}_{S}$ is discontinuous, with ${\ensuremath{\rho}}_{S}(T)\ensuremath{-}{\ensuremath{\rho}}_{S}({T}_{c}\ensuremath{-}0)\ensuremath{\sim}B{({T}_{c}\ensuremath{-}T)}^{\frac{1}{2}}$ for temperatures $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}0$. Except near ${T}_{c}={T}_{\ensuremath{\lambda}}$ the theoretical and experimental results for ${\ensuremath{\rho}}_{S}$ are in good agreement.

DOI
10.1103/physrevlett.17.182

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We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.
We investigate Bose-Einstein condensation for ultracold bosonic atoms in two-dimensional systems. The functional renormalization group for the average action allows us to follow the effective interactions from molecular scales (microphysics) to the characteristic extension of the probe lll (macrophysics). In two dimensions the scale dependence of the dimensionless interaction strength $\ensuremath{\lambda}$ is logarithmic. Furthermore, for large lll the frequency dependence of the inverse propagator becomes quadratic. We find an upper bound for $\ensuremath{\lambda}$, and for large $\ensuremath{\lambda}$ substantial deviations from the Bogoliubov results for the condensate depletion, the dispersion relation and the sound velocity. The melting of the condensate above the critical temperature Tc{T}_{c}Tc​ is associated to a phase transition of the Kosterlitz-Thouless type. The critical temperature in units of the density Tc∕n{T}_{c}∕nTc​∕n vanishes for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ logarithmically.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length aaa which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density nnn, and find for the critical temperature Tc{T}_{c}Tc​ a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0T=0T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons NNN, and that the additional trace over NNN that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature TTT goes past the critical transition temperature TcT_cTc​ is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons NNN is taken to be a constant that does not depend on TTT. Consequences on the physics of interacting bosons are briefly discussed.