Table of Contents

Preface vii

List of Figures xv

List of Tables xxii

The Big Picture 1

1 Modern Physics at a Glance 3

Geometry and Phase Transitions, in General 15

2 Phase Transitions and Critical Phenomena 17

2.1 Introduction 17

2.1.1 Evolution of the Universe: Decoupling of the Four Fundamental Forces 18

2.1.2 Three States of Water 19

2.1.3 Spins and Magnetism 21

2.2 Modern Classification of Phase Transitions 23

2.3 First-Order Phase Transitions: Solid-Liquid Transition 24

2.4 Second-Order Phase Transitions: Scaling and Universality 25

2.5 Renormalization Group 26

2.5.1 Kadanoff Picture: Coarse-Graining of Spin Blocks 26

2.5.2 General Formulation 28

2.5.3 Critical Exponents 31

2.5.4 Origin of Universality Class 32

2.5.5 Wilsonian Picture: Momentum-Space Renormalization Group 33

2.6 Mathematical Miscellanies: Semi-Group Structure and Fixed-Point Theorems 34

2.6.1 Semi-groups 34

2.6.2 Miscellany on Fixed-Points 35

2.7 Conclusion 35

3 Overview of Density-Functional Theory 38

3.1 Introduction 38

3.2 Electronic Density-Functional Theory 38

3.3 Classical Density-Functional Theory 42

3.4 Conclusion 46

4 Survey of Solid Geometry and Topology 49

4.1 Introduction 49

4.2 Lattice Symmetry Groups 50

4.3 Two-Dimensional Space Groups 53

4.3.1 Hermann-Mauguin Crystallographic Notation 55

4.3.2 Orbifold notation 57

4.3.3 Why Are There Exactly 17 Wallpaper Groups? 78

4.3.4 Other Aspects of Topology in Physics 84

4.4 Three-Dimensional Point Groups 85

4.4.1 Face-centered Cubic (FCC) Lattices 85

4.4.2 Body-Centered Cubic (BCC) Lattices 88

4.4.3 A15 Lattices 89

4.5 Conceptual Framework of the Foam Model 90

4.6 TheKelvin Problem and the Kepler Conjecture 92

4.7 Conclusion 97

Geometry and Phase Transitions, in Colloidal Crystals 101

5 Lattice Free Energy via the Foam Model 103

5.1 Introduction 103

5.2 Bulk Free Energy 104

5.3 Interfacial Free Energy 109

5.3.1 Charged Colloidal Crystals 109

5.3.2 Fuzzy Colloidal Crystals 111

5.4 Conclusion 112

6 Phases of Charged Colloidal Crystals 115

6.1 Introduction 115

6.2 Phase Transitions of Charged Colloids 117

6.3 Foam Analogy and Charged Colloids 119

6.4 Conclusion 120

7 Elasticity of Colloidal Crystals 122

7.1 Introduction 122

7.2 Foam Analogy and Cubic Elastic Constants 124

7.3 Elasticity of Charged Colloidal Crystals 129

7.4 Elasticity of Fuzzy Colloids 137

7.5 Conclusion 143

Geometry and Phase Transitions, in Topologically Constrained Polymers 145

8 Topologically-Constrained Polymers in Theta Solution 147

8.1 Introduction 147

8.2 O(N)-Symmetric φ6-Theory 148

8.3 Chern-Simons Theory and Writhe 154

8.4 One-Loop Scaling of Closed Polymers 159

8.5 Two-Loop Results 163

8.6 Conclusion 170

Summary 175

9 Final Thoughts 177

Index 179