Collective Classical And Quantum Fields: In Plasmas, Superconductors, Superfluid 3he, And Liquid Crystals

by Hagen Kleinert

Overview

This is an introductory book dealing with collective phenomena in many-body systems. A gas of bosons or fermions can show oscillations of various types of density. These are described by different combinations of field variables. Especially delicate is the competition of these variables. In superfluid 3He, for example, the atoms can be attracted to each other by molecular forces, whereas they are repelled from each other at short distance due to a hardcore repulsion. The attraction gives rise to Cooper pairs, and the repulsion is overcome by paramagnon oscillations. The combination is what finally led to the discovery of superfluidity in 3He. In general, the competition between various channels can most efficiently be studied by means of a classical version of the Hubbard-Stratonovich transformation.A gas of electrons is controlled by the interplay of plasma oscillations and pair formation. In a system of rod- or disc-like molecules, liquid crystals are observed with directional orientations that behave in unusual five-fold or seven-fold symmetry patterns. The existence of such a symmetry was postulated in 1975 by the author and K Maki. An aluminium material of this type was later manufactured by Dan Shechtman which won him the 2014 Nobel prize. The last chapter presents some solvable models, one of which was the first to illustrate the existence of broken supersymmetry in nuclei.

Product Details

ISBN-13:	9789813223936
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	02/09/2018
Pages:	424
Product dimensions:	6.00(w) x 9.00(h) x 0.94(d)

About the Author

Hagen Kleinert is professor of physics at the Freie Universität Berlin, Germany. As a visiting scientist, he has spent extended periods of time at CERN, the European Organization for Nuclear Research in Geneva; at Caltech in Pasadena; at the Universities of California in Berkeley, Santa Barbara, and San Diego; at the Los Alamos National Laboratory in New Mexico; and at Princeton University, New Jersey. He has made numerous contributions to the understanding of particle physics, mathematical physics, condensed matter physics, chemical physics, and nuclear physics. His two-volume book Gauge Fields in Condensed Matter (World Scientific, 1989) develops a new quantum field-theory of phase transitions on the basis of disorder fields. Such fields have meanwhile become a powerful tool supplementing the order filed introduced by the Landan scholls. Another book on Path Integrals appeared with World Scientific in 1990 (5^th edition, 2009). It is so far the most comprehensive text on this subject. He also published with World Scientific in 2001, in collaboration with V. Schulte-Frohlinde, a thorough review book on the field-theoretic renormalization group approach to critical phenomena. The title is Critical Properties of φ⁴-Theories. The divergent perturbation expansions are evaluated with exponentially fast convergence using a new technique called VPT. This evolved from a paper by the author with Richard Feynman. The list of books by the author continues with Multivalued Filed in Condensed Matter, Electromagnetism, and Gravitation (World Scientific, 2008). There he shows that many phenomena in physics, in particular all those commonly explained by vortices and by gauge theories, can be understood as consequences of a Riemann-sheet nature of fields. His latest book, Particles and Quantum Fields (World Scientific, 2016), explains the presently known elementary particles and studies their interactions using the modern techniques of quantum field theory.

Table of Contents

Preface vii

1 Functional Integral Techniques 1

1.1 Nonrelativistic Fields 2

1.1.1 Quantization of Free Fields 2

1.1.2 Fluctuating Free Fields 4

1.1.3 Interactions 8

1.1.4 Normal Products 10

1.1.5 Functional Formulation 14

1.1.6 Equivalence of Functional and Operator Methods 15

1.1.7 Grand-Canonical Ensembles at Zero Temperature 16

1.2 Relativistic Fields 22

1.2.1 Lorentz and Poincaré Invariance 22

1.2.2 Relativistic Free Scalar Fields 27

1.2.3 Electromaginetic Fields 31

1.2.4 Relativistic Free Fermi Fields 34

1.2.5 Perturbation Theory of Relativistic Fields 37

Notes and References 39

2 Plasma Oscillations 41

2.1 General Formalism 41

2.2 Physical Consequences 45

2.2.1 Zero Temperature 46

2.2.2 Short-Range Potential 47

Appendix 2A Fluctuations around the Plasmon 48

Notes and References 49

3 Superconductors 50

3.1 General Formulation 52

3.2 Local Interaction and Ginzburg-Landau Equations 59

3.2.1 Inclusion of Electromagnetic Fields into the Pair Field Theory 69

3.3 Far below the Critical Temperature 72

3.3.1 The Gap 73

3.3.2 The Free Pair Field 77

3.4 From BCS to Strong-Coupling Superconductivity 91

3.5 Strong-Coupling Calculation of the Pair Field 92

3.6 From BCS Superconductivity near T_c to the onset of pseudogap behavior 100

3.7 Phase Fluctuations in Two Dimensions and Kosterlitz-Thouless Transition 105

3.8 Phase Fluctuations in Three Dimensions 111

3.9 Collective Classical Fields 112

3.9.1 Superconducting Electrons 115

3.10 Strong-Coupling Limit of Pair Formation 117

3.11 Composite Bosons 122

3.12 Composite Fermions 127

3.13 Conclusion and Remarks 129

Appendix 3A Auxiliary Strong-Coupling Calculations 131

Appendix 3B Propagator of the Bilocal Pair Field 133

Appendix 3C Fluctuations Around the Composite Filed 135

Notes and References 138

4 Superfluid ³He 145

4.1 Interatomic Potential 145

4.2 Phase Diagram 147

4.3 Preparation of Functional Integral 149

4.3.1 Action of the System 149

4.3.2 Dipole Interaction 149

4.3.3 Euclidean Action 150

4.3.4 From Particles to Quasiparticles 151

4.3.5 Approximate Quasiparticle Action 152

4.3.6 Effective Interaction 155

4.3.7 Pairing Interaction 158

4.4 Transformation from Fundamental to Collective Fields 159

4.5 General Properties of a Collective Action 164

4.6 Comparison with O(3)-Symmetric Linear σ-Model 169

4.7 Hydrodynamic Propertics Close to T_c 170

4.8 Bending the Superfluid ³He-A 178

4.8.1 Monopoles 179

4.8.2 Line Singularities 182

4.8.3 Solitons 184

4.8.4 Localized Lumps 187

4.8.5 Use of Topology in the A-Phase 188

4.8.6 Topology in the B-Phase 190

4.9 Hydrodynamic Properties at Al Temperatures T ≤ T_c 193

4.9.1 Derivation of Gap Equation 194

4.9.2 Ground State Properties 199

4.9.3 Bending Energies 208

4.9.4 Fermi-Liquid Corrections 218

4.10 Large Currents and Magnetic Fields in the Ginzburg-Landau Regime 227

4.10.1 B-Phase 228

4.10.2 A-Phase 239

4.10.3 Critical Current in Other Phases for T ∼ T_c 240

4.11 Is ³He-A a Superfluid? 248

4.11.1 Magnetic Field and Transition between A- and B-Phases 272

4.12 Large Currents at Any Temperature T ≤ T_c 274

4.12.1 Energy at Nonzero Velocities 274

4.12.2 Gap Equations 275

4.12.3 Superfluid Densities and Currents 283

4.12.4 Critical Currents 285

4.12.5 Ground State Energy at Large Velocities 289

4.12.6 Fermi Liquid Corrections 289

4.13 Collective Modes in the Presence of Current at all Temperatures T ≤ T_c 292

4.13.1 Quadratic Fluctuations 292

4.13.2 Time-Dependent Fluctuations at Infinite Wavelength 295

4.13.3 Normal Modes 298

4.13.4 Simple Limiting Results at Zero Gap Deformation 301

4.13.5 Static Stability 303

4.14 Fluctuation Coefficients 304

4.15 Stability of Superflow in the B-Phase under Small Fluctuations for T ∼ T_c 307

Appendix 4A Hydrodynamic Coefficients for T ≈ T_c 312

Appendix 4B Hyrdodynamic Coefficients for All T ≤ T_c 315

Appendix 4C Generalized Ginzburg-Landau Energy 319

Notes and References 319

5 Liquid Crystals 323

5.1 Maier-Saupe Model and Generalizations 324

5.1.1 General Properties 324

5.1.2 Landau Expansion 326

5.1.3 Tensor Form of Landau-de Gennes Expansion 327

5.2 Landau-de Gennes Description of Nematic Phase 328

5.3 Bending Energy 336

5.4 Light Scattering 338

5.5 Interfacial Tension between Nematic and Isotropic Phases 347

5.6 Cholesteric Liquid Crystals 351

5.6.1 Small Fluctuations above T₁ 354

5.6.2 Some Experimental Facts 355

5.6.3 Mean-Field Description of Cholesteric Phase 357

5.7 Other Phases 362

Appendix 5A Biaxial Maier-Saupe Model 365

Notes and References 368

6 Exactly Solvable Field-Theoretic Models 371

6.1 Pet Model in Zero Plus One Time Dimensions 371

6.1.1 The Generalized BCS Model in a Degenerate Shell 379

6.1.2 The Hilbert Space of the Generalized BCS Model 390

6.2 Thirring Model in 1+1 Dimensions 393

6.3 Supersymmetry in Nuclear Physics 397

Notes and References 397

Index 399