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Physical review. B, Condensed matter

Quantum theory of superfluidity and its onset in Bose films

William Frederick Saam

Published Feb 1, 1981• 5 citations
Publisher

Abstract

A theory of superfluidity in two-dimensional Bose systems is presented. In contrast to the classical theory of Nelson and Kosterlitz, the excitation (rotons, ripplons, etc.) are treated quantum mechanically from the outset, resulting in an explicitly temperature-dependent vortex pair coupling proportional to the total mass density minus the normal mass density of the excitations. An application to thick $^{4}\mathrm{He}$ films is presented, and the weakly interacting Bose system is discussed explicitly.

DOI
10.1103/physrevb.23.1485

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Low Temperature Physics•Feb 1, 1999•14 citations
A theory of superfluidity in two-dimensional Bose systems is presented. In contrast to the classical theory of Nelson and Kosterlitz, the excitation (rotons, ripplons, etc.) are treated quantum mechanically from the outset, resulting in an explicitly temperature-dependent vortex pair coupling proportional to the total mass density minus the normal mass density of the excitations. An application to thick 4He^{4}\mathrm{He}4He films is presented, and the weakly interacting Bose system is discussed explicitly.
In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems. Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955). In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state. Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.
Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. Although this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.
We derive a fully quantum-mechanical equation of motion for a vortex in a 2-dimensional Bose superfluid in the temperature regime where the normal fluid density ${\ensuremath{\rho}}_{n}(T)$ is small. The coupling between the vortex ``zero mode'' and the quasiparticles has no term linear in the quasiparticle variables---the lowest-order coupling is quadratic. We find that as a function of the dimensionless frequency $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}=\ensuremath{\hbar}\ensuremath{\Omega}/{k}_{B}T$, the standard Hall-Vinen-Iordanskii equations are valid when $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\ll}1$ (the ``classical regime''), but elsewhere, the equations of motion become highly retarded, with significant experimental implications when $\stackrel{\texttildelow{}}{\ensuremath{\Omega}}\ensuremath{\gtrsim}1$.
Abstract Chapter 2 describes the basic knowledge of Bose-condensed systems in order to study the quantum hydrodynamics for both the superfluid helium and the atomic-gas BECs in the following sections. Starting from the field theoretical formulation, we will first give a self-contained explanation of the derivation of the most fundamental GP equation for weakly interacting Bose gases at zero temperature. We will also highlight the theoretical treatment for incorporating the concept of elementary excitations to describe low-energy excitations of Bose systems, the definition of normal and superfluid components, and eventually the celebrated model of two-fluid hydrodynamics. Finally, the microscopic structure and dynamics of topological excitations obtained from the GP equation are discussed by exemplifying solitons, vortex lines, and vortex rings.
We have derived the nonlinear Schrodinger equation generalizing the Gross-Pitaevskii (GP) equa tion for dilute Bose gas. The derivation is based on the Hartree-Fock time-dependent mean-field theory with an arbitrary intermolecular interaction potential. It is shown that obtained nonlinear Schrodinger equation with appropriate redefinition of coefficients can be used for description of dense Bose fluid at low temperatures. We also present the Schrodinger type equation describing the superfluid component of helium in two fluid hydrodynamics. This approach leads to quantum correction for superfluid component of velocity in two fluid hydrodynamics. We also have derived the nonlinear Schrodinger equation for superfluid He4 film at low temperatures. It is shown that this Schrodinger type equation for superfluid He4 film leads to phonon-roton dispersion relation for elementary excitation at low temperatures.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.