Spin Glasses and Complexity

Preface xi

Introduction: Why Spin Glasses? 1

- 1. Order, Symmetry, and the Organization of Matter 15

- 1.1 The Symmetry of Physical Laws 17

- 1.2 The Hamiltonian 23

- 1.3 Broken Symmetry 26

- 1.4 The Order Parameter 31

- 1.5 Phases of Matter 35

- 1.6 Phase Transitions 39

- 1.7 Summary: The Unity of Condensed Matter Physics 41

2. Glasses and Quenchied Disorder 43

- 2.1 Equilibrium and Non Equilibrium 43

- 2.2 The Glass Transition 45

- 2.3 Localization 49



3. Magnetic Systems 51

- 3.1 Spin 51

- 3.2 Magnetism in Solids 53

- 3.3 The Paramagnetic Phase 55

- 3.4 Magnetization 55

- 3.5 The Ferromagnetic Phase and Magnetic Susceptibility 57

- 3.6 The Antiferromagnetic Phase 59

- 3.7 Broken Symmetry and the Heisenberg Hamiltonian 59



4. Spin Glasses: General Features 63

- 4.1 Dilute Magnetic Alloys and the Kondo Effect 64

- 4.2 A New State of Matter? 65

- 4.3 Nonequilibrium and Dynamical Behavior 71

- 4.4 Mechanisms Underlying Spin Glass Behavior 74

- 4.5 The Edwards-Anderson Hamiltonian 78

- 4.6 Frustration 81

- 4.7 Dimensionality and Phase Transitions 83

- 4.8 Broken Symmetry and the Edwards-Anderson Order Parameter 85

- 4.9 Energy Landscapes and Metastability 86



5. The Infinite-Range Spin Glass 90

- 5.1 Mean Field Theory 90

- 5.2 The Sherrington-Kirkpatrick Hamiltonian 92

- 5.3 A Problem Arises 93

- 5.4 The Remedy 95

- 5.5 Thermodynamic States 97

- 5.6 The Meaning of Replica Symmetry Breaking 98

- 5.7 The Big Picture 109



6. Applications to Other Fields 112

- 6.1 Computational Time Complexity and Combinatorial Optimization 113

- 6.2 Neural Networks and Neural Computation 129

- 6.3 Protein Folding and Conformational Dynamics 144

- 6.4 Short Takes 168



7. Short-Range Spin Glasses: Some Basic Questions 175

- 7.1 Ground States 177

- 7.2 Pure States 188

- 7.3 Scenarios for the Spin Glass Phase of the EA Model 193

- 7.4 The Replica Symmetry Breaking and Droplet/Scaling Scenarios 194

- 7.5 The Parisi Overlap Distribution 197

- 7.6 Self-Averaging and Non-Self-Averaging 199

- 7.7 Ruling Out the Standard RSB Scenario 201

- 7.8 Chaotic Size Dependence and Metastates 203

- 7.9 A New RSB Scenario 206

- 7.10 Two More (Relatively) New Scenarios 211

- 7.11 Why Should the SK Model Behave Differently from the EA Model? 214

- 7.12 Summary: Where Do We Stand? 216



8. Are Spin Glasses Complex Systems? 218

- 8.1 Three Foundational Papers 219

- 8.2 Spin Glasses as a Bridge to Somewhere 227

- 8.3 Modern Viewpoints on Complexity 228

- 8.4 Spin Glasses: Old, New, and Quasi-Complexity 233

Notes 239

Glossary 265

Bibliography 285

Index 309