Modern trends in Superconductivity and Superfluidity Selected Chapters. M.Yu. Kagan P.L. Kapitza Institute for Physical Problems [email protected] In preparation for Springer-Verlag 1 Chapter 1. Hydrodynamics of rotating superfluids with quantized vortices. 1.1. The foundation of Landau theory for superfluid hydrodynamics. 1.1.1. The essence of the hydrodynamics. Description of the Goldstone modes. 1.1.2. Landau scheme of the conservation laws. Euler equation. 1.1.3. Sound waves in classical liquid. Damping of sound waves. 1.1.4. Rotational fluid. Vorticity conservation. Inertial mode. 1.1.5. Two-velocity hydrodynamics for superfluid helium. vn and vs , ρn and ρs . 1.1.6. First and second sound modes in superfluid liquid. 1.1.7. Gross-Pitaevskii equation for dilute Bose-gas. Connection between superfluid hydrodynamics and microscopic theory at T = 0. 1.2. Hydrodynamics of rotating superfluids. 1.2.1. Andronikashvilli experiments in rotating helium. 1.2.2. Feynman-Onsager quantized vortices. Critical angular velocities ΩC1 and ΩC2 . 1.2.3. Vortex lattice. Nonlinear elasticity theory. Vorticity conservation law. 1.2.4. Hydrodynamics of slow rotations. Hall-Vinen friction coefficients β and β′ . 1.2.5. Linearization of the elasticity theory. Connection between vs and u in linearized theory. 1.2.6. Collective modes of the lattice. Tkachenko waves and Lord Kelvin waves. Melting of the vortex lattice. 1.3. Hydrodynamics of fast rotations. 1.3.1. The foundation of the hydrodynamics of fast rotations. The role of umklapp processes. 1.3.2. The system of nonlinear equations for hydrodynamics of fast rotations. 1.3.3. Linearized system of equations of fast rotations. The spectrum and damping of second sound mode. 1.4. Opposite case of a single bended vortex line for extremely slow rotations (Ω ~ ΩC1 ). 1.4.1. Stabilization of the bending oscillations by rotation. 1.4.2. Visualization of the vortex lattice in rotating superfluid. Packard experiments. 1.4.3. Contribution to normal density and specific heat from bended vortex lines. 1.5. Experimental situation and discussion. How to achieve the limit of the fast rotations at not very high frequencies in He II – 3He mixtures and in superfluid 3He-B. Reference list to Chapter 1. 2 While many forthcoming Chapters of the present manuscript deal with the microscopical models for the description of Quantum Gasses, Fluids and Solids, as well as strongly correlated electron systems, the first four Chapters are devoted to purely phenomenological (macroscopic) description of hydrodynamic phenomena in anisotropic and inhomogeneous superfluids and superconductors. The presentation in these Chapters is based on Landau ideas. Let us stress that Landau theory of hydrodynamics [1.1] based on the conservation laws in the differential form (together with Quantum Mechanics [1.2] and Statistical Physics [1.3]) is one of the masterpieces of Landau-Lifshitz course of the Theoretical Physics. While the essence of this theory is well known (serves as a mothermilk) to Russian researchers from the first steps of their scientific career, in the West many young physicists, specializing on macroscopic models, do not have a deep background in this subject. A good style meanwhile is to know fluently both microscopics and phenomenology from our point of view. That is why from the pedagogical reasons we decided to start the present Chapter with brief description (in Section 1.1) of the Landau theory for hydrodynamics in classical [1.1] and superfluid [1.1, 1.4] liquids. We present Landau scheme of the conservation laws for the description of slowly varying in space and time Goldstone (gapless) collective modes. We will discuss Landau [1.1] (or Landau-Tissa [1.1, 1.5], as often referred to in the West) two-velocity superfluid hydrodynamics for 4He [1.1, 1.4, 1.5] which contains normal and superfluid velocities vn and vs , as well as normal and superfluid densities ρn and ρs, thus describing both the normal and superfluid motion in helium [1.6, 1.28]. We will derive the spectrum and damping of first and second sound in superfluid helium (in He-II). We will get ω = cIk and ω = cIIk (the spectrum is linear for both waves) and compare the velocities cI and cII of these sound waves. We will stress that while in first sound total mass-current j=ρss v + ρ nn v ≠ 0 , in the second sound wave j = 0 but the relative velocity W= vn − v s ≠ 0 . In Section 1.2 we proceed to the hydrodynamics of rotating superfluids with a large number of quantized vortices. We start this Section with famous Andronikashvilli experiments (on non-classical moment of inertia in rotating He-II) [1.7] and discuss Feynmann-Onsager quantization of the vortex lines in superfluid helium [1.8, 1.9, 1.29, 1.30]. We introduce the notion of first and second critical angular velocities of rotation ΩC1 and ΩC2 in helium and stress their correspondence with first and second critical magnetic fields HC1 and HC2 in type – II superconductors [1.10, 1.26, 1.27]. Than we will present the scheme of macroscopic averaging [1.11] for a large number of vortices and construct nonlinear elasticity theory for a 2D (triangular) vortex lattice in helium [1.12]. The linearization of this theory yields well-known Tkachenko modes [1.13] (which describe longitudinal and transverse oscillations of the vortex lattice), as well as Lord Kelvin (Thomson) bending oscillations of the vortex lines [1.14]. We include dissipation in the system and discuss Hall-Vinen friction coefficients β and β` [1.15, 1.24, 1.25] for the scattering of normal component on the vortex lattice. In the end of Section 1.2 we will construct the complete system of equations (which describe hydrodynamics of slow rotations [1.12, 1.22]) based on the Landau scheme of the conservation laws. We will discuss briefly another elegant method to derive a system of hydrodynamic equations based on Poisson brackets [1.19] and emphasize, nevertheless, the advantages of Landau method especially in nonlinear regime. In the next Section (Section 1.3) we consider the different limit for hydrodynamics of fast rotating superfluid. In this case due to umklapp processes [1.10, 1.21] the normal excitations are bound to vortex lattice, and hence the normal and superfluid velocities perpendicular to the vortex lines coincide vn⊥= v s ⊥ . In the same time in the direction parallel to the vortex lines due to translational invariance vn|| ≠ vs|| . Thus we have one-velocity motion in the plane of the vortex lattice v⊥= j ⊥ / ρ and two-velocity motion parallel to the vortex lines. Hence we are in a strongly anisotropic situation where we have a crystal in the plane of the vortex lattice and a 3 standard superfluid in the direction perpendicular to the lattice and parallel to the vortex lines. We will construct the full system of hydrodynamic equations for a regime of rapid rotations and analyze the spectrum and damping of collective excitations (first of all of a second sound mode) in this regime. We will get that the spectrum is linear and sound wave can freely propagate only along the vortex lines where W= vn − v s ≠ 0 , while the spectrum is overdamped in the 2 2 2 perpendicular direction. More specifically for k⊥ = 0 the spectrum reads ω = c2 k z , while for kz = 2 iκ⊥ k ⊥ 0: ω = − [1.12], where κ⊥ is heat conductivity in the direction perpendicular to the vortex C p lines, Cp is a specific heat and kz, k⊥are the components of the k -vector along and perpendicular to the vortex lines, respectively. In the Section 1.4 we consider an opposite case of a single bended line. Here, as shown k 2 1 by Lord Kelvin, the spectrum for the bending oscillations is almost quadratic ω = z ln 2m dk z [1.10, 1.14] (d is a normal vortex core, which in the case of superfluid 4He is of the order of the interatomic distance). Naive considerations show that such quasi 1D system (as a bended vortex line) should be completely destroyed by the thermal fluctuations and experience in analogy with the biophysical systems a phase-transition to the state of a globula [1.10]. We show, however, that in fact [1.12] the bending oscillations correspond to rotation: u u ≠ 0 , where u and u are respectively a local displacement of the bended vortex from a nondeformed position and its time derivative (local velocity). Thus the quanta of the bending oscillations in fact have the rotation moment (- ħ) (diamagnetic situation) and hence the gap ħΩ appears in their spectrum. This gap stabilizes the fluctuations of this 1D system providing a finite ratio u2/ R 2 << 1 of a mean square displacement u 2 to the radius squared R2 of the rotating vessel with helium. It is a reason why a regular (triangular) vortex lattice can be directly visualized (the photograph of the lattice with small vortex displacements can be obtained) in the experiments of Packard [1.10]. We conclude the Chapter with the discussion in Section 1.5 how we can realize the hydrodynamics of fast rotations at not very high frequencies Ω << ΩC2 (note that the second 4 critical angular velocity ΩC2 is very high in superfluid He). We propose to use for that regime 4He-3He mixtures, where 4He is superfluid and has a large number of vortices. In this case 2 3 according to the Bernulli law [1.1] p+ρ vs = const all the He impurities will be driven by the gradient of the pressure ( ∇p ) to the vortex core and organize inside the core the quasi 1D normal component with a free motion only along the vortex lines.