Collective Excitations in Unconventional Superconductors and Superfluids

The following sections are included:

• Preface

• Contents

• Introduction

The following sections are included:

• Functional Integrals in Statistical Physics

• Functional Integrals and Diagram Techniques for Bose–particles

• Functional Integrals and Diagram Techniques for Fermi–particles

• Method of Successive Integration over “Fast” and “Slow” Variables

The following sections are included:

• Effective Action Functional of the Superfluid Fermi–gas

• Bose–spectrum of Superfluid Fermi–gas

• Fermi–gas with Coulomb Interaction

The following sections are included:

• Sound Propagation in Superfluid ³He

• Sound Propagation in Conventional Superconductors
  • Attenuation in the normal state
  • Attenuation in the superconducting state
  • Attenuation in the superconducting state in a magnetic field
  • Velocity at the superconducting transition

The following sections are included:

• Introduction: Fermi–systems with Nontrivial Pairing

• Properties of Superfluid Phases in ³He

The following sections are included:

• The Path Integral Approach

• Kinetic–equation Method

The following sections are included:

• The Quadratic Form of Action Functional

• The Collective Mode Frequencies

• Dispersion Corrections to the Collective Mode Spectrum
  • Dispersion laws for rsq– and sq–modes
  • Dispersion induced splitting of the rsq– and sq–mode

• The Pair–breaking–mode Dispersion Low

• Collective Mode Spectrum Calculated by the Kinetic Equation Method

• Fermi–Liquid Corrections

• Textural Effects on the Squashing Modes

• Coupling of Order–Parameter Collective Modes to Ultrasound

The following sections are included:

• A–Phase of ³He

• The Collective Mode Spectrum in the Absence of Magnetic Fields

• The Latent Symmetry, Additional Goldstone Modes, W–Bosons

• The Linear Zeeman Effect for Clapping– and Pair–breaking Modes
  • The equations for the collective mode spectrum in an arbitrary magnetic field and at arbitrary collective mode momenta
  • The collective mode spectrum for small magnetic fields and zero collective mode momenta (linear Zeeman effect for clapping– and pair–breaking–modes)

• Kinetic Equation Results on Collective Modes in A–Phase
  • Sound and the order parameter collective modes
  • Orbital waves and sound

• Textural Effects in A–Phase

The following sections are included:

• Introduction

• Mermin–Star's Phase Diagram Analysis

• Axial Phase
  • Ginzburg–Landau model
  • The second variation of free energy

• Conclusion

The following sections are included:

• Stability of Goldstone–Modes and Their Dispersion Laws

• Stability of Goldstone–Modes in the B–Phase

• Stability of Goldstone–Modes in the Axial A–Phase

• Stability of Goldstone Modes in the Planar 2D–Phase

The following sections are included:

• The Influence of the Dipole Interaction on the Collective Excitations

• The Influence of the Magnetic Field on the Collective Excitations

• Conclusion

The following sections are included:

• The Energy Spectrum and Hydrodynamics of ⁴He in a Strong Electric Field (Macroscopic Approach)

• Superfluid Bose–Systems in the Electric Field Microscopic Approach)

• The Effective Action Functional for the Superfluid ³He in the Electric Field

• The Influence of the Electric Field on the Bose–Spectrum in the B–Phase

• The Influence of the Electric Field on the Bose–Spectrum in the A–Phase

The following sections are included:

• The External Perturbations and the Order Parameter Distortions

• The Collective Mode Spectrum Under the Order Parameter Distortion
  • Dipole interaction
  • Magnetic fields
  • Electric fields
  • Superfluid flow
  • Rotational effects (vortices and gyromagnetism)

• Sound Experiments at the Absorption Edge

• Subdominant f–Wave Pairing Interactions in Superfluid ³He

The following sections are included:

• A Doublet Splitting of the Squashing Mode in Superfluid ³He–B

• The Method of Superfluid Velocity Measurement in ³He–B

The following sections are included:

• Introduction

• Transverse Sound Experiments

• Possible New Phases Near the Boundary

• Different Branches of Squashing Mode

• Deformed B–Phase

• Conclusion

The following sections are included:

• The Planar 2D–Phase of Superfluid ³He

• Collective Modes in ³He–2D at Zero Momenta of Excitations

The following sections are included:

• Introduction

• Axial Phase
  • The model of superfluid ³He
  • The equations for the collective mode spectrum in an arbitrary magnetic field and at arbitrary collective mode momenta
  • The dispersion corrections to collective mode spectrum in ³He–A

• Planar Phase
  • Stability of 2D–phase
  • The equations for collective mode spectrum in ³He–2D
  • The equations for collective mode spectrum in ³He–2D with dispersion corrections
  • The collective mode spectrum in ³He–2D with dispersion corrections

• Conclusion

The following sections are included:

• Calculation of the Collective Mode Spectrum

• Conclusion

The following sections are included:

• Calculation of the Collective Mode Spectrum

• Conclusion

The following sections are included:

• Phase Transitions in Two–Dimensional Systems

• Two–Dimensional Superfluidity

• Quantum Vortices

• One–Dimensional Systems

• Superfluidity in Fermi–Films. Singlet Pairing

• Triplet Pairing. Thick Films

• Model of ³He–Film

• Superfluid Phases of a Two–Dimensional Superfluid ³He

• Bose–Spectrum of the a–Phase

• Bose–Spectrum of the b–Phase

• The Two–Dimensional Superfluidity Must Exist!

• New Possibility for the Search of 2D–Superfluidity in ³He–Films

The following sections are included:

• Superfluidity of ³He, Dissolved in ⁴He

• The Case of s– Pairing in ³He. Effective Action Functional of the ³He–⁴He Solutions

• Bose–Spectrum of the ³He–⁴He Solution

• The Case of p–pairing. The Effective Action Functional of the ³He–⁴He Solution

• Bose–Spectrum of a Solution of the Type ³He–B–⁴He

• Bose–Spectrum of a System of the Type ³He–A–⁴He

• Bose–Spectrum of Films of the Types ³He–a–⁴He and ³He–b–⁴He

• Conclusion

The following sections are included:

• Introduction

• Decoupling of First and Second Sound in Pure Superfluids

• Sounds Coupling in Impure Superfluids
  • Superfluids with different impurities, ³He–⁴He mixtures
  • Sounds coupling in superfluid He in aerogel

• Slow Pressure (Density) Oscillations, Fast Temperature (Entropy) Oscillations

• Fast Mode Frequency Shift at T_C(T_λ)

• Difference in Nature of First and Second Sound in Impure Superfluids

• Sound Conversion Phenomena
  • Conservation Laws in Sound Conversion
  • Sound Conversion in Pure Superfluids
  • Sound Conversion in ³He–⁴He Mixtures

• Sound Conversion Experiments in Pure Superfluids

• Some Possible New Sound Experiments in Impure Superfluids

• Coupling of Two Slow Modes in Superfluid ³He–⁴He Mixture in Aerogel

• Nonlinear Hydrodynamic Equations for Superfluid Helium in Aerogel

• Putterman's Type Equations

• Conclusion

In this Chapter we will develop an approach to the microscopic theory of periodic structures in the framework of the functional integration method following to the book by Popov¹. This approach was suggested by Kapitonov and Popov² and was developed by Andrianov, Kapitonov and Popov^3,4…

As it is well known, the superconductivity phenomenon exists due to attractive interaction between electrons in the narrow energetic layer near the Fermi surface. The effective interaction of electrons consists of two parts: the Coulomb repulsive potential and some attractive potential arising from the phonon exchange. In this Chapter we will derive this effective interaction potential in the framework of the functional integral method following to the book by Popov¹. Here the procedure of the previous section will be slightly changed. At the first stage we integrate over the electron fields with momenta outside the narrow layer near the Fermi surface. Then we integrate over the electric potential field and also over ion trajectories in the phase space. So we come down to the effective interaction potential between electrons near the Fermi–surface…

The following sections are included:

• Models of p– and d–Pairing

• p–Pairing

• d–Pairing

The following sections are included:

• The Discovery of HTSC

• Physical Properties of HTSC
  • Some experimental data

The following sections are included:

• Introduction
  • Superconductivity and Broken Symmetry
  • The symmetry group

• Symmetry Classification of HTSC States
  • Square lattice
  • Tetragonal lattice
  • The orthorhombic lattice
  • Electron–hole symmetry

• Singlet States
  • The gap functions
  • Mixing of states of different irreducible representations
  • Orthorhombicity and twins
  • Multilayer structures

• Pairing Symmetry and Pairing Interactions
  • Two scenarios for d–wave pairing
  • Tests of the pairing interaction
  • Influence of electron–phonon interaction on d_x²−y²–pairing

• Experimental Symmetry Probes
  • Josephson effects
  • Magnetic induction of d_x²−y²+id_xy order in HTSC
  • Transition splitting, spontaneous strain and magnetism
  • Critical phenomena and Gaussian fluctuations
  • Collective modes
  • Exotic vortices
  • Probes of the gap function
  • Distinction of a scalar from a tensor order parameter

• Experimental Evidence for d_x²−y²–Pairing
  • “Clean samples”
  • Impurities

• Irradiation Studies

• List of Abridgements for Chapters XXV and XXVI

The following sections are included:

• Introduction

• Bulk HTSC Under d–Pairing

The following sections are included:

• The Mixture of Two d–Wave States

• Equations for Collective Modes Spectrum in a Mixed d–Wave State of Unconventional Superconductors
  • Model for the mixed state
  • Equations for collective modes spectrum in a mixed d–wave state at arbitrary admixture of d_xy state
  • Equations for collective modes spectrum in a mixed d–wave state at an equal admixtures of d_x²−y² and d_xy states
  • d_x²−y² – state of high temperature superconductors with a small admixture of d_xy–state

• Conclusion

The following sections are included:

• Introduction

• Bulk p–Wave Superconductivity

The following sections are included:

• Two–Dimensional Models of p– and d–Pairing in USC

• p–Pairing
  • Two–dimensional p–wave superconducting states
  • The collective mode spectrum

• Two–Dimensional d–Wave Superconductivity
  • 2D–model of d–pairing in CuO₂ planes of HTSC
  • The collective mode spectrum
  • Lattice symmetry and collective mode spectrum

The following sections are included:

• Physical Properties of Heavy–Fermion Superconductors

• Bulk Heavy–Fermion Superconductors Under d–Pairing

• Conclusion

The following sections are included:

• Relativistic Analogs of ³He

The following sections are included:

• References

• About Authors