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arXiv:cond-mat/9906344 · Physical Review A

Anomalous condensate fluctuations in strongly interacting superfluids

F. Meier, W. Zwerger

Published Dec 1, 1999• 31 citations
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Abstract

We show that the condensate occupation of a superfluid Bose liquid quite generally exhibits anomalously large fluctuations at finite temperatures $T\ensuremath{\ne}0.$ In three dimensions, the variance $〈\ensuremath{\delta}{N}_{0}^{2}〉$ of the number ${N}_{0}$ of particles in the condensate scales nonlinearly with volume V like ${T}^{2}{V}^{4/3}$ at low T, generalizing the result obtained by Giorgini, Pitaevskii, and Stringari for a weakly interacting Bose gas. In two dimensions there is only a quasicondensate whose fluctuations are of the same order as the mean value.

DOI
10.1103/physreva.60.5133

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We show that the condensate occupation of a superfluid Bose liquid quite generally exhibits anomalously large fluctuations at finite temperatures T\ensuremath≠0.T\ensuremath{\ne}0.T\ensuremath=0. In three dimensions, the variance 〈\ensuremathδN02〉〈\ensuremath{\delta}{N}_{0}^{2}〉〈\ensuremathδN02​〉 of the number N0{N}_{0}N0​ of particles in the condensate scales nonlinearly with volume V like T2V4/3{T}^{2}{V}^{4/3}T2V4/3 at low T, generalizing the result obtained by Giorgini, Pitaevskii, and Stringari for a weakly interacting Bose gas. In two dimensions there is only a quasicondensate whose fluctuations are of the same order as the mean value.
We find that the fluctuations of the condensate in a weakly interacting Bose gas confined in a box of volume VVV follow the law $〈\ensuremath{\delta}{N}_{0}^{2}〉\ensuremath{\sim}{V}^{4/3}$. This anomalous behavior arises from the occurrence of infrared divergencies due to phonon excitations and holds also for strongly correlated Bose superfluids. The analysis is extended to an interacting Bose gas confined in a harmonic trap where the fluctuations are found to exhibit a similar anomaly.
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 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.
Fluctuations of the number of condensed atoms in a finite-size, weakly interacting Bose gas confined in a box potential are investigated for temperatures up to the critical region. The canonical partition functions are evaluated using a recursive scheme for smaller systems, and a saddle-point approximation for larger samples that allows to treat realistic size systems containing up to N ∼10 5 particles. We point out the importance of particle-number constraint and interactions between out of condensate atoms for the statistics near the critical region. For sufficiently large systems, the crossover from the anomalous to normal scaling of the fluctuations is observed. The excitations are described in a self-consistent way within the Bogoliubov-Popov approximation, and the interactions between thermal atoms are described by means of the Hartree-Fock method.
We investigate the behavior of the anomalous correlation function in two-dimensional Bose gas. In the local case, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities are also considered. These effects can modify, in particular, the chemical potential, the ground-state energy, the depletion, and the superfluid fraction. Our predictions are in good agreement with recent analytical and numerical calculations. We show also that the anomalous density presents a significant importance compared to the noncondensed one at zero temperature. The single-particle anomalous correlation function is expressed in two-dimensional homogenous Bose gases by using the density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous manner the true condensate at zero temperature, while it does not exist at finite temperature.
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 ${\ensuremath{\tau}}_{1}<\ensuremath{\tau}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}<1$, where at $\ensuremath{\tau}={\ensuremath{\tau}}_{1}$ the neglected terms are comparable to those included. In this paper we consider the second approximation 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 ${\ensuremath{\tau}}_{2}<\ensuremath{\tau}<{\ensuremath{\tau}}_{1}$ 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 Tc{T}_{c}Tc​, and it is suggested that the absence of logarithmic factors is due to the inconsistent treatment of entropy fluctuations.