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Physical review. B, Condensed matter

Goldstone phonons in a Bose-condensed gas at finite temperature: One-loop approximation

Stephen H. Payne, Adam Griffin

Published Dec 1, 1985• 26 citations
Publisher

Abstract

Using a field-theoretic formalism in terms of regular (proper, irreducible) quantities, we give a detailed analysis of the long-wavelength collisionless excitations of a weakly interacting dilute Bose-condensed gas in the intermediate-temperature region. Earlier work by Sz\'epfalusy and Kondor has shown that use of the free-Bose-gas approximation for the regular quantities leads to a phonon with a velocity equal to [${n}_{0}$(T)${V}_{0}$/m${]}^{1/2}$, where ${n}_{0}$(T) is the temperature-dependent condensate of a free Bose gas. In this paper the phonon velocity is worked out with use of the one-loop diagrams for the regular quantities. The result is [${n}_{S}$(T)${V}_{0}$/m${]}^{1/2}$[1-f(T)], where ${n}_{S}$(T) is the Landau superfluid density defined in terms of the Bogoliubov excitations and f(T) is a relatively small correction. Numerical calculations are given for spin-polarized ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ with density n=5\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{3}$. .AE

DOI
10.1103/physrevb.32.7199

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Using a field-theoretic formalism in terms of regular (proper, irreducible) quantities, we give a detailed analysis of the long-wavelength collisionless excitations of a weakly interacting dilute Bose-condensed gas in the intermediate-temperature region. Earlier work by Sz'epfalusy and Kondor has shown that use of the free-Bose-gas approximation for the regular quantities leads to a phonon with a velocity equal to [n0{n}_{0}n0​(T)V0{V}_{0}V0​/m]1/2{]}^{1/2}]1/2, where n0{n}_{0}n0​(T) is the temperature-dependent condensate of a free Bose gas. In this paper the phonon velocity is worked out with use of the one-loop diagrams for the regular quantities. The result is [nS{n}_{S}nS​(T)V0{V}_{0}V0​/m]1/2{]}^{1/2}]1/2[1-f(T)], where nS{n}_{S}nS​(T) is the Landau superfluid density defined in terms of the Bogoliubov excitations and f(T) is a relatively small correction. Numerical calculations are given for spin-polarized H\ensuremath↓{\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}H\ensuremath↓​ with density n=5\ifmmode\times\else\texttimes\fi{}1018{10}^{18}1018 cm3{\mathrm{cm}}^{3}cm3. .AE
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature Tc{T}_{c}Tc​, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature ${T}^{\ensuremath{\star}}>{T}_{c}$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\ensuremath{\langle}A\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\varphi}\ensuremath{\rangle}$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\rangle}$. In the region between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.
Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate.It is based on the coupled evolution equations of the parameters describing the system.These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate.The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation.They were analyzed both analytically and numerically.The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases.We have shown that at low temperature the dependence of the obtained quantities on temperature significantly differs from those obtained by other authors in the semi-phenomenological approaches.Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated.
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.