The Superfluid Phase s of Helium 3

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1 The Superfluid Phase s of Helium 3 DIETER VOLLHARD T Rheinisch-Westfälische Technische Hochschule Aachen, Federal Republic of German y PETER WÖLFL E Universität Karlsruhe Federal Republic of Germany

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3 PREFACE LIST OF FREQUENTLY USED SYMBOLS v xvi 1 INTRODUCTION The helium liquids Early history of superfluid 3He Elementary discussion of superfluid 3He The internal structure of Cooper pairs Broken symmetry and the order parameter Orientational effects Textures Superfluid mass currents in 3He-A Dynamic properties Relation to other fields Reviews and introductory articles on superfluid 3He General reviews Reviews of specific topics Discussions in books on related subjects Introductory articles 24 2 THEORY OF NORMAL FERMI LIQUIDS The quasiparticle concept Thermodynamic properties Entropy and specific heat Spin susceptibility Compressibility Transport properties Hydrodynamic equations Kinetic equation Quasiparticle lifetime Local equilibrium The linearized collision integral Calculation of the transport coefficients 40

4 2.4 Collective modes Theories beyond the Fermi-liquid model Microscopic theories Polarization potential theory Extended Fermi-liquid theory Lattice-gas model Phenomenological approaches Pair interaction 57 Further reading PAIR CORRELATIONS IN THE WEAK-COUPLING LIMIT Cooper instability Generalized pairing theory Generalized BCS wave function Diagonalization of the mean-field Hamiltonian Single-particle excitations Singlet versus triplet pairing Pairing theory at finite temperature Effective Hamiltonian Gap equation Critical temperature Free energy Thermodynamic properties of model states p-wave pairing ; the BW and ABM states Gap parameter Specific heat Normal-fluid density Spin susceptibility Expectation value of two-particle quantities 90 Further reading 92 4 BASIC EXPERIMENTAL PROPERTIES Experimental techniques for attaining ultralow temperatures Thermodynamic properties Phase diagram Specific heat Magnetic properties Sound propagation Hydrodynamic properties and superfluidity Normal-fluid density and viscosity Persistent currents Rotating superfluid 3He 109 Further reading 111

5 5 PAIR CORRELATIONS BEYOND WEAK COUPLING General Ginzburg-Landau expansion of the free energy Phenomenological classification of model states Unitary states Relative stability of model states Spin fluctuations and the stability of the ABM state Effects of a magnetic field 129 Further reading BROKEN SYMMETRIES AND MACROSCOPIC ORDER Broken symmetries and group theory The symmetry group describing superfluid 3He A two-dimensional model of superfluid 3He Remaining symmetries of the 3He-A, -A, and -B orde r parameters Symmetry and order-parameter structure Continuous symmetries Discrete symmetries Symmetry and stationary points of a free energy Symmetry reduction due to a magnetic field or spin - orbit coupling Symmetry classification of an exactly solvable case : d-wave pairing Broken symmetries in high-energy physics Orientation of the order parameter by internal residua l interactions and external fields Dipole interaction Orientation induced by a magnetic field Orientation induced by an electric field Effect of superflow Orientational effects in the B phase due to magneti c and electric fields and superflow Surface energies and boundary conditions 185 Further reading SUPERFLOW AND TEXTURES Superfluidity Gradient free energy The A phase The B phase Healing lengths 200

6 7.3 Supercurrents The A phase Finite normal density at T = 0 in the A phase Quantization of circulation Topological investigation of defects Linear defects Topological stability Topological properties of B-phase vortices Topological properties of A-phase vortices at smal l distances Topological properties of A-phase vortices at large distances Symmetry classification of vortices Axisymmetric vortices Energetics of vortices Vortices in the A phase Vortices in the B phase Magnetic properties of vortices in the B phase Rotating superfluid 3He Isotropic superfluid Superfluid 3He Point defects The B phase The A phase Planar defects Topological classification Planar solitons in the A phase Planar solitons in the B phase Surface-induced textures He-B in a slab He-B in a cylinder He-A in a slab He-A in a cylinder He-A in a sphere Stability of superflow and related textural transitions in 3He-A Stability of superflow in the bulk liquid The helical instability of the uniform 1 texture Textures and superflow in the presence of boundaries The effect of superflow on domain walls Flow-induced dynamical textures Pair-breaking critical currents The B phase The A phase Dissipation of superflow Superfluid He The B phase The A phase 33 1 Further reading 337

7 8 SPIN DYNAMICS Derivation of the equations of motion Nuclear magnetic resonance under linear spatiall y homogeneous conditions Limit of zero magnetic field Effect of the magnetic field on NMR Nonlinear NMR phenomena in uniform textures The Leggett equations for the A and B phases Spin dynamics in zero magnetic field Nonlinear spin dynamics in the A phase Nonlinear spin dynamics in the B phase Texture-induced magnetic resonance phenomena Tilted uniform textures NMR shifts induced by superflow General theory of texture-induced NMR shifts in the A phase Dynamics of solitons in the A phase NMR signature of solitons in the A phase General theory of texture-induced NMR shifts in the B phase NMR in rotating superfluid 'He The A phase The B phase Spin-relaxation phenomena 394 Further reading HYDRODYNAMIC THEORY General principles Thermodynamic identities Standard procedure for deriving hydrodynamic equations Equations of motion for the symmetry variables Hydrodynamic equations for the B phase The thermodynamic identity and equilibrium conditions Hydrodynamic equations for the symmetry variables Hydrodynamic equations for the conserved quantities Hydrodynamic equations for the A phase The concept of an intrinsic angular momentum Equations of motion for the symmetry variables Thermodynamic identities and equilibrium conditions Entropy production Derivation of the hydrodynamic currents Further consequences of an intrinsic orbital angula r momentum 427

8 9.4 Hydrodynamics for finite magnetic field The A phase in a magnetic field The B phase in a magnetic field 43 2 Further reading TRANSPORT PROPERTIES Kinetic equations Matrix kinetic equation Conserved quantities and conservation laws Kinetic equation for Bogoliubov quasiparticles Gauge transformation of the kinetic equation Gauge-invariant densities and currents Collision integral Matrix operator for binary collision processes Conservation properties of the collision integral Collision integral for Bogoliubov quasiparticles Relaxation-time approximation Bogoliubov-quasiparticle relaxation rate Transport coefficients Transport coefficients of the B phase Intrinsic spin relaxation Transport coefficients of the A phase Flow in restricted geometries Slip correction to hydrodynamics Andreev reflection Poiseuille flow Vibrating-wire experiments Sound propagation and other flow problems Superfluidity in 3He films Ion mobility 496 Further reading COLLECTIVE MODES Hydrodynamic modes Sound modes Spin-wave modes Other hydrodynamic modes Symmetry classification of order-parameter modes in the collisionless regime The B phase The A phase Time-dependent mean-field theory in the collisionless regime Collective modes in the B phase Collective modes in the A phase Effect of residual interactions 533

9 Observability of collective modes Collisionless sound Phenomenological model Sound propagation in the B phase Sound propagation in the A phase Nonlinear effects of sound propagation Transverse sound and spin waves 54 6 Further reading AMPLIFICATION OF WEAK INTERACTION EFFECTS DUE TO MACROSCOPIC QUANTUM COHERENCE Properties of the BCS pair wave function The permanent orbital magnetic moment of the A phase The permanent electric dipole moment of the B phase 55 5 Further reading 560 REFERENCES 56 1 AUTHOR INDEX 595 SUBJECT INDEX 607

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