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arXiv:cond-mat/9902097 · Physical review. B, Condensed matter

Critical temperature of the superfluid transition in Bose liquids

Sergey M. Apenko

Published Aug 1, 1999• 49 citations
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Abstract

A phenomenological criterion for the superfluid transition is proposed, which is similar to the Lindemann criterion for the crystal melting. Then we derive a formula for the critical temperature, relating ${T}_{\ensuremath{\lambda}}$ to the mean kinetic energy per particle above the transition. The suppression of the critical temperature in a sufficiently dense liquid is described as a result of the quantum decoherence phenomenon. The theory can account for the observed dependence of ${T}_{\ensuremath{\lambda}}$ on density in liquid helium and results in an estimate ${T}_{\ensuremath{\lambda}}\ensuremath{\sim}1.1$ K for molecular hydrogen.

DOI
10.1103/physrevb.60.3052

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A phenomenological criterion for the superfluid transition is proposed, which is similar to the Lindemann criterion for the crystal melting. Then we derive a formula for the critical temperature, relating T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​ to the mean kinetic energy per particle above the transition. The suppression of the critical temperature in a sufficiently dense liquid is described as a result of the quantum decoherence phenomenon. The theory can account for the observed dependence of T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​ on density in liquid helium and results in an estimate T\ensuremathλ\ensuremath∼1.1{T}_{\ensuremath{\lambda}}\ensuremath{\sim}1.1T\ensuremathλ​\ensuremath∼1.1 K for molecular hydrogen.
We further analyze the recently proposed criterion, according to which a superfluid transition occurs whenever the r.m.s. displacement of a particle from its initial position in imaginary time reaches a certain fraction of the interparticle distance. The critical temperature ${T}_{\ensuremath{\lambda}}$ is expressed in terms of the single particle mobility. Within the Drude approximation for the mobility ${T}_{\ensuremath{\lambda}}$ is determined by the friction coefficient (or by the viscosity for dense liquids). To check the validity of the criterion the resulting formula is applied to nonideal Bose gas, to liquid helium and to ${}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}$ mixtures. A reasonable qualitative agreement with available experiments is obtained in all cases. In the case of the Bose gas the present approach results also in an upper bound on the peak value of ${T}_{\ensuremath{\lambda}}{/T}_{\ensuremath{\lambda}}^{0},$ where ${T}_{\ensuremath{\lambda}}^{0}$ is the critical temperature for the ideal gas.
We present a microscopic theory for superfluidity in an interacting many-particle Bose system (such as liquid 4^44He). We show that, similar to superconductivity in superconductors, superfluidity in a Bose system arises from pairing of particles of opposite momenta. We show the existence of an energy gap in single-particle excitation spectrum in the superfluid state and the existence of a specific heat jump at the superfluid transition. We derive an expression for superfluid particle density nsn_sns​ as a function of temperature TTT and superfluid velocity vs{\bf v}_svs​. We show that superfluid-state free energy density FFF is an increasing function of vsv_svs​ (i.e., ∂F/∂vs>0\partial F/\partial v_s > 0∂F/∂vs​>0), which indicates that a superfluid has a tendency to remain motionless (this result qualitatively explains the Hess-Fairbank effect, which is analogous to the Meissner effect in superconductors). We further speculate the existence of the equation ${\bf j}=-Λ\nabla\times \text{\boldmath ωωω}$, where j=nsvs{\bf j} = n_s{\bf v}_sj=ns​vs​ is the superfluid current density, $\text{\boldmath ωωω}=\nabla\times {\bf v}_s$ the superfluid vorticity, and ΛΛΛ a positive constant (with the help of this equation, the Hess-Fairbank effect can be quantitatively described).
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.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
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.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.