This book is about a certain class of superfluids, namely the high temperature superconductors. Perhaps the simplest of the superfluids is liquid He4. Since the very early days of quantum mechanics it has been known that the helium atom He4 is a boson; it is built of an even number of particles (two electrons, two neutrons and two protons), so that a wave function ^(ri,F2) must be symmetrical in FI — r? where FI, r-z are the electron's coordinates. This has the result that half the rotational states of the molecule H62 are absent and that the scattering of alpha particles by helium for low energies deviates from the Rutherford scattering law. It also has the remarkable result that this is not so for the light isotope He3, which shows that the number of neutrons in the nucleus affects properties depending otherwise only on the electrons and nuclear charge.
Dilute mixtures of
He3 in liquid
He4 are regarded as a set of
He3 atoms intereacting with elementary excitations of the
He4. The coupling between the
He3 and the
He4 is chosen phenomenologically and it is not weak but, for low temperatures, its main effects may be eliminated by means of two canonical transformations. The equations which determine the transformations describe the motion of a
He4 excitation in the presence of one or two
He3 atoms and, in the latter case, they resemble Faddeev's equations in the three-body problem. At low temperatures, there are few
He4 excitations and many of the properties of the mixture are determined by the effective
He3 Hamiltonian produced by the transformation. In this respect, the system behaves like a Fermi liquid. The
He3-
He3 potential is essentially momentum-independent and consists of the Van der Waals interaction together with an effectively repulsive potential induced by the
He4. It is suggested that a Fermi-liquid analysis is the best way of obtaining information about this potential, and the possible existence of a fermion superfluid phase transition is discussed from that point of view.
A rigorous microscopic analysis of a weakly interacting Bose gas at zero temperature shows that the elementary excitation spectrum $\ensuremath{\omega}(k)$ is not an analytic function of the wave vector
k in the long-wavelength limit but has the form $k[{c}_{0}+{c}_{2}{k}^{2}+{c}_{L4}{k}^{4}\mathrm{ln}(\frac{1}{k})+\ensuremath{\cdots}]$. It is pointed out that the structure factor
S(k) is not an analytic function of
k and that the specific heat
Cv(T) at low temperatures
T has the form ${C}_{v}={T}^{3}[{C}_{0}+{C}_{2}{T}^{2}+{C}_{L4}{T}^{4}\mathrm{ln}(\frac{1}{T})+\ensuremath{\cdots}]$. The extension of these results to liquid
He4 is discussed.
The authors present for the first time an effective interaction between two
4He atoms in a bath of liquid
3He at temperatures below 0.1 K. This interaction turns out to be, in its long-range limit, an oscillatory function of the
4He interatomic separation, the oscillations arising essentially from the sharp discontinuity in the momentum distribution at the Fermi surface. In addition, an expression is formulated for the oscillations in the density of the
3He bath induced by one
4He impurity, as well as a modified Friedel sum rule, which relates the relative amount of matter displaced by this impurity to the phase shifts pertaining to the impurity-
3He quasiparticle scattering at the Fermi surface. Finally, the dependence of the results on temperature and on the
3He superfluidity gap is discussed.
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.