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Physical Review A

Order-Parameter Fluctuations in a Weakly Interacting Bose Gas near the Superfluid Transition

Harvey Gould, Victor K. Wong

Published Aug 1, 1971• 5 citations
Publisher

Abstract

The contribution of order-parameter fluctuations to the static and dynamic properties of a weakly interacting Bose gas near the Bose-Einstein condensation temperature ${T}_{c}$ is investigated via a novel approach. Self-consistent approximations for the propagator of order-parameter fluctuations $G$ are generated in successive temperature ranges closer and closer to ${T}_{c}$. Mean-field theory corresponds to assuming that the correlation of fluctuations ${G}_{2}$ is given by ${{G}_{2}}^{(1)}=GG$. This first approximation is valid in the range ${\ensuremath{\tau}}_{1}<\ensuremath{\tau}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}<1$, where at $\ensuremath{\tau}={\ensuremath{\tau}}_{1}$ the neglected terms are comparable to those included. In this paper we consider the second approximation ${{G}_{2}}^{(2)}=GGVGG$, which involves the first-order interaction between the fluctuations. Two independent criteria for the range of validity ${\ensuremath{\tau}}_{2}<\ensuremath{\tau}<{\ensuremath{\tau}}_{1}$ of the second approximation are established. It is conjectured that the qualitative results of the present model calculation are applicable to liquid helium in the presently accessible temperature range. The predicted temperature dependences of the specific heat, superfluid density, and fourth sound are in agreement with the scaling hypothesis and available experiments in helium. The condensate density is predicted to vanish linearly with $\ensuremath{\tau}$. It is shown that the contribution of order-parameter fluctuations to the thermal conductivity diverges as ${\ensuremath{\tau}}^{\frac{\ensuremath{-}1}{3}}$ above ${T}_{c}$, and it is suggested that the absence of logarithmic factors is due to the inconsistent treatment of entropy fluctuations.

DOI
10.1103/physreva.4.719

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The contribution of order-parameter fluctuations to the static and dynamic properties of a weakly interacting Bose gas near the Bose-Einstein condensation temperature Tc{T}_{c}Tc​ is investigated via a novel approach. Self-consistent approximations for the propagator of order-parameter fluctuations GGG are generated in successive temperature ranges closer and closer to Tc{T}_{c}Tc​. Mean-field theory corresponds to assuming that the correlation of fluctuations G2{G}_{2}G2​ is given by G2(1)=GG{{G}_{2}}^{(1)}=GGG2​(1)=GG. This first approximation is valid in the range \ensuremathτ1<\ensuremathτ=(T\ensuremath−Tc)Tc<1{\ensuremath{\tau}}_{1}<\ensuremath{\tau}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}<1\ensuremathτ1​<\ensuremathτ=Tc​(T\ensuremath−Tc​)​<1, where at \ensuremathτ=\ensuremathτ1\ensuremath{\tau}={\ensuremath{\tau}}_{1}\ensuremathτ=\ensuremathτ1​ the neglected terms are comparable to those included. In this paper we consider the second approximation G2(2)=GGVGG{{G}_{2}}^{(2)}=GGVGGG2​(2)=GGVGG, which involves the first-order interaction between the fluctuations. Two independent criteria for the range of validity \ensuremathτ2<\ensuremathτ<\ensuremathτ1{\ensuremath{\tau}}_{2}<\ensuremath{\tau}<{\ensuremath{\tau}}_{1}\ensuremathτ2​<\ensuremathτ<\ensuremathτ1​ of the second approximation are established. It is conjectured that the qualitative results of the present model calculation are applicable to liquid helium in the presently accessible temperature range. The predicted temperature dependences of the specific heat, superfluid density, and fourth sound are in agreement with the scaling hypothesis and available experiments in helium. The condensate density is predicted to vanish linearly with \ensuremathτ\ensuremath{\tau}\ensuremathτ. It is shown that the contribution of order-parameter fluctuations to the thermal conductivity diverges as \ensuremathτ\ensuremath−13{\ensuremath{\tau}}^{\frac{\ensuremath{-}1}{3}}\ensuremathτ3\ensuremath−1​ above Tc{T}_{c}Tc​, and it is suggested that the absence of logarithmic factors is due to the inconsistent treatment of entropy fluctuations.
The thermal fluctuations of the order parameter affect the transport of Bose gas above the superfluid transition. Using the time-dependent Ginzburg-Landau theory in the classical region, we calculate the fluctuation corrections to the coefficients of thermal conductivity and shear viscosity. Above the transition temperature T c , the fluctuation contribution to the thermal conductivity diverges as ( T − T c ) -2+ d /2 for dimensions d . The viscosity drops near T c since the fluctuation contributes a correction which varies as − ( T − T c ) −2+ d /2 . The calculations are conducted under the condition that the fluctuation corrections are small compared with the normal part of the transport coefficients.
We compare many-body theories describing fluctuation corrections to the mean-field theory in a weakly interacting Bose-condensed gas. Using a generalized random-phase approximation, we include both density fluctuations and fluctuations in the particle-particle scattering channel in a consistent manner. We also separately examine effects of the fluctuations within the framework of the random-phase approximation. Effects of fluctuations in the particle-particle scattering channel are also separately examined by using the many-body TTT-matrix approximation. We assess these approximations with respect to the transition temperature Tc{T}_{\mathrm{c}}Tc​, the order of phase transition, as well as the so-called Nepomnyashchii-Nepomnyashchii identity, which states the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. Since the construction of a consistent theory for interacting bosons which satisfies various required conditions is a long-standing problem in cold atom physics, our results would be useful for this important challenge.
We consider a three-dimensional weakly interacting Bose gas in the fluctuation region (and its vicinity) of the normal-fluid--superfluid phase transition point. We establish relations between basic thermodynamic functions: density $n(T,\ensuremath{\mu})$, superfluid density ${n}_{s}(T,\ensuremath{\mu})$, and condensate density ${n}_{\text{cnd}}(T,\ensuremath{\mu})$. Being universal for all weakly interacting ${\ensuremath{\mid}\ensuremath{\psi}\ensuremath{\mid}}^{4}$ systems, these relations are obtained from Monte Carlo simulations of the classical ${\ensuremath{\mid}\ensuremath{\psi}\ensuremath{\mid}}^{4}$ model on a lattice. Comparing with the mean-field results yields a quantitative estimate of the fluctuation region size. Away from the fluctuation region, on the superfluid side, all the data perfectly agree with the predictions of the quasicondensate mean-field theory. This demonstrates that the only effect of the leading above-the-mean-field corrections in the condensate-based treatments is to replace the condensate density with the quasicondensate one in all local thermodynamic relations. Surprisingly, we find that a significant fraction of the density profile of a loosely trapped atomic gas might correspond to the fluctuation region.
We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.
We consider a three-dimensional weakly interacting Bose gas in the fluctuation region (and its vicinity) of the normal-superfluid phase transition point. We establish relations between basic thermodynamic functions: density, n(T,μ)n(T,μ)n(T,μ), superfluid density ns(T,μ)n_s(T,μ)ns​(T,μ), and condensate density, ncnd(T,μ)n_{\rm cnd} (T,μ)ncnd​(T,μ). Being universal for all weakly interacting ∣ψ∣4|ψ|^4∣ψ∣4 systems, these relations are obtained from Monte Carlo simulations of the classical ∣ψ∣4|ψ|^4∣ψ∣4 model on a lattice. Comparing with the mean-field results yields a quantitative estimate of the fluctuation region size. Away from the fluctuation region, on the superfluid side, all the data perfectly agree with the predictions of the quasicondensate mean field theory.--This demonstrates that the only effect of the leading above-the-mean-field corrections in the condensate based treatments is to replace the condensate density with the quasicondensate one in all local thermodynamic relations. Surprisingly, we find that a significant fraction of the density profile of a loosely trapped atomic gas might correspond to the fluctuation region.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.