courses.physics.ucsd.edu...Contents 1 Broken Symmetry 1 1.1 Introduction . . . . . . . . . . . . . ....

Lecture Notes on Condensed Matter Physics

(A Work in Progress)

Daniel ArovasDepartment of Physics

University of California, San Diego

July 7, 2020

Contents

1 Broken Symmetry 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The mother of all Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Classical and Quantum Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Correspondence between quantum and classical statistical mechanics . . . . . . . 6

1.3 Phases of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Beyond the Landau paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Landau Theory of Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Quartic free energy with Ising symmetry . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Cubic terms in Landau theory : first order transitions . . . . . . . . . . . . . . . . 14

1.4.3 Magnetization dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.4 Sixth order Landau theory : tricritical point . . . . . . . . . . . . . . . . . . . . . . 18

1.4.5 Hysteresis for the sextic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.6 Weak crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5 Four Vignettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.1 Lower critical dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5.2 Random systems : Imry-Ma argument . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5.3 Hohenberg-Mermin-Wagner theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5.4 Goldstone’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Appendix : The Foldy-Wouthuysen Transformation . . . . . . . . . . . . . . . . . . . . . . 37

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ii CONTENTS

1.6.1 The Dirac Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.6.2 Emergence of the spin-orbit and Zeeman interaction terms . . . . . . . . . . . . . 38

1.7 Appendix : Ideal Bose Gas Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.8 Appendix : Asymptotic Series in a Zero-Dimensional Field theory . . . . . . . . . . . . . 42

1.9 Appendix : Derivation of Ginzburg-Landau Functional . . . . . . . . . . . . . . . . . . . . 50

1.9.1 Discrete symmetry : Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.9.2 Continuous symmetry : O(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2 Crystal Math 55

2.1 Classification of Crystalline Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.2 Miller indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.1.3 Crystallographic restriction theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.1.4 Enumeration of two and three-dimensional Bravais lattices . . . . . . . . . . . . . 60

2.1.5 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.1.6 Trigonal crystal system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.1.7 Point groups, space groups and site groups . . . . . . . . . . . . . . . . . . . . . . 67

2.2 More on Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.2.1 Standard notation for point group operations . . . . . . . . . . . . . . . . . . . . . 69

2.2.2 Proper point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.2.3 Commuting operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.2.4 Improper point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.2.5 The ten two-dimensional point groups . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.2.6 The achiral tetrahedral group, Td . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.2.7 Tetrahedral vs. octahedral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.2.8 The 32 crystallographic point groups . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.2.9 Hermann-Mauguin (international) notation . . . . . . . . . . . . . . . . . . . . . . 80

2.2.10 Double groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.2.11 The three amigos : D4 , C4v , D2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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2.3 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.3.1 Space group elements and their properties . . . . . . . . . . . . . . . . . . . . . . . 91

2.3.2 Factor groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.3.3 How to make a symmorphic space group . . . . . . . . . . . . . . . . . . . . . . . 97

2.3.4 Nonsymmorphic space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.3.5 Translations and their representations . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.3.6 Space group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.4 Fourier Space Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.4.1 Space group symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.4.2 Extinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.4.3 Sticky bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3 Deformations of Crystals 111

3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.1.1 Stress and strain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.1.2 Elasticity and symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.2 Phonons in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2.1 One-dimensional chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2.2 General theory of lattice vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2.3 Translation and rotation invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.2.4 Phonons in an fcc lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.2.5 Phonons in the hcp structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.2.6 Phonon density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.2.7 Einstein and Debye models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.2.8 Phenomenological theory of melting . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.2.9 Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.2.10 Elasticity theory redux : Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . 133

3.2.11 Elasticity theory in cases with bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.2.12 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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3.2.13 Evaluation of Sll′(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.2.14 Dynamic structure factor for Bravais lattices . . . . . . . . . . . . . . . . . . . . . . 141

3.2.15 Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.2.16 The Mössbauer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4 Electronic Band Structure of Crystals 145

4.1 Energy Bands in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.1.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.1.2 Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.1.3 V = 0 : empty lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.1.4 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.1.5 Solvable model : one-dimensional Dirac comb . . . . . . . . . . . . . . . . . . . . 150

4.1.6 Diamond lattice bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.2 Metals and Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.2.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.2.2 Fermi statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.2.3 Metals and insulators at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.3 Calculation of Energy Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.3.1 The tight binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.3.2 Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.3.3 Tight binding redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.3.4 Interlude on Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.3.5 Examples of tight binding dispersions . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.3.6 Bloch’s theorem, again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.4 Ab initio Calculations of Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.4.1 Orthogonalized plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.4.2 The pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.5 Semiclassical Dynamics of Bloch Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.5.1 Adiabatic evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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4.5.2 Violation of Liouville’s theorem and its resolution . . . . . . . . . . . . . . . . . . 178

4.5.3 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.6 Topological Band Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.6.1 SSH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5 Metals 185

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.2 T = 0 and the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.2.1 Definition of the Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.2.2 Fermi surface vs. Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.2.3 Spin-split Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5.3 Quantum Thermodynamics of the Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . 191

5.3.1 Fermi distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.3.2 Sommerfeld expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.3.3 Chemical potential shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

5.3.4 Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.4 Effects of External Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.4.1 Magnetic susceptibility and Pauli paramagnetism . . . . . . . . . . . . . . . . . . 195

5.4.2 Landau diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.4.3 de Haas-van Alphen oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.4.4 de Haas-von Alphen effect for anisotropic Fermi surfaces . . . . . . . . . . . . . . 204

5.5 Simple Theory of Electron Transport in Metals . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.5.1 Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.5.2 Magnetoresistance and magnetoconductance . . . . . . . . . . . . . . . . . . . . . 209

5.5.3 Hall effect in high fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.5.4 Cyclotron resonance in semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 211

5.5.5 Magnetoresistance in a two band model . . . . . . . . . . . . . . . . . . . . . . . . 212

5.5.6 Optical reflectivity of metals and semiconductors . . . . . . . . . . . . . . . . . . . 213

5.5.7 Theory for Bloch wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

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5.6 Boltzmann Equation in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.6.1 Semiclassical Dynamics and Distribution Functions . . . . . . . . . . . . . . . . . 217

5.6.2 Local Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.7 Conductivity of Normal Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.7.1 Relaxation Time Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.7.2 Optical Conductivity and the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . 225

5.8 Calculation of the Scattering Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

5.8.1 Potential Scattering and Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . 227

5.8.2 Screening and the Transport Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . 229

5.9 Boltzmann Equation for Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.9.1 Properties of Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.10 Magnetoresistance and Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.10.1 Boltzmann Theory for ραβ(ω,B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.10.2 Hall Effect in High Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

5.11 Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.11.1 Boltzmann Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.11.2 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

5.11.3 Calculation of Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 242

5.11.4 Onsager Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5.12 Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.12.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.12.2 Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.12.3 Boltzmann Equation for Electron-Phonon Scattering . . . . . . . . . . . . . . . . . 248

6 Semiconductors and Insulators 253

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

6.1.1 Band gaps and transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

6.1.2 Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

6.1.3 Optical absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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6.1.4 Direct versus indirect gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6.1.5 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6.1.6 Effective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

6.2 Number of Carriers in Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 260

6.2.1 Intrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.2.2 Extrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

6.3 Donors and Acceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

6.3.1 Impurity charges in a semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . 264

6.3.2 Donor and acceptor quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . 265

6.3.3 Chemical potential versus temperature in doped semiconductors . . . . . . . . . . 267

6.4 Inhomogeneous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

6.4.1 Modeling the p -n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.4.2 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6.4.3 MOSFETs and heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6.4.4 Heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6.5 Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.5.1 Maxwell’s equations in polarizable media . . . . . . . . . . . . . . . . . . . . . . . 279

6.5.2 Clausius-Mossotti relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

6.5.3 Theory of atomic polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

6.5.4 Electromagnetic waves in a polar crystal . . . . . . . . . . . . . . . . . . . . . . . . 284

7 Mesoscopia 287

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

7.2 The Landauer Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

7.2.1 Example: Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.3 Multichannel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.3.1 Transfer Matrices: The Pichard Formula . . . . . . . . . . . . . . . . . . . . . . . . 295

7.3.2 Discussion of the Pichard Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.3.3 Two Quantum Resistors in Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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7.3.4 Two Quantum Resistors in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

7.4 Universal Conductance Fluctuations in Dirty Metals . . . . . . . . . . . . . . . . . . . . . 309

7.4.1 Weak Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.5 Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

7.5.1 Characterization of Localized and Extended States . . . . . . . . . . . . . . . . . . 316

7.5.2 Numerical Studies of the Localization Transition . . . . . . . . . . . . . . . . . . . 317

7.5.3 Scaling Theory of Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

7.5.4 Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

8 Hartree-Fock and Density Functional Theories 325

8.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.1.1 Basis states and creation/annihilation operators . . . . . . . . . . . . . . . . . . . 325

8.1.2 The second quantized Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

8.2 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

8.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.3.1 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

8.3.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

8.4 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

8.4.1 Linear response theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

8.4.2 Static screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.4.3 Approximate forms for the polarization function . . . . . . . . . . . . . . . . . . . 343

9 Linear Response of Quantum Systems 345

9.1 Response and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

9.1.1 Forced damped oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

9.1.2 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

9.2 Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

9.2.1 Quantum mechanical response functions . . . . . . . . . . . . . . . . . . . . . . . . 349

9.2.2 Spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

CONTENTS ix

9.2.3 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

9.2.4 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

9.2.5 Continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

9.3 A Spin in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

9.3.1 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

9.4 Density Response and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

9.4.1 Sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

9.4.2 Dynamic Structure Factor for the Electron Gas . . . . . . . . . . . . . . . . . . . . 362

9.5 Charged Systems: Screening and Dielectric Response . . . . . . . . . . . . . . . . . . . . . 367

9.5.1 Definition of the charge response functions . . . . . . . . . . . . . . . . . . . . . . 367

9.5.2 Static screening: Thomas-Fermi approximation . . . . . . . . . . . . . . . . . . . . 368

9.5.3 High frequency behavior of ǫ(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

9.5.4 Random phase approximation (RPA) . . . . . . . . . . . . . . . . . . . . . . . . . . 371

9.5.5 Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

9.5.6 Jellium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

9.6 Electromagnetic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

9.6.1 Gauge invariance and charge conservation . . . . . . . . . . . . . . . . . . . . . . . 377

9.6.2 A sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

9.6.3 Longitudinal and transverse response . . . . . . . . . . . . . . . . . . . . . . . . . 378

9.6.4 Neutral systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

9.6.5 The Meissner effect and superfluid density . . . . . . . . . . . . . . . . . . . . . . 380

10 Landau Fermi Liquid Theory 383

10.1 Normal 3He Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

10.2 Fermi Liquid Theory : Statics and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 385

10.2.1 Adiabatic continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

10.2.2 First law of thermodynamics for Fermi liquids . . . . . . . . . . . . . . . . . . . . 388

10.2.3 Low temperature equilibrium properties . . . . . . . . . . . . . . . . . . . . . . . . 391

10.2.4 Thermodynamic stability at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

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10.3 Collective Dynamics of the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

10.3.1 Landau-Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

10.3.2 Zero sound : free FS oscillations in the collisionless limit . . . . . . . . . . . . . . . 401

10.4 Dynamic Response of the Fermi Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

11 Phenomenological Theories of Superconductivity 405

11.1 Basic Phenomenology of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

11.2 Thermodynamics of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

11.3 London Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

11.4 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

11.4.1 Landau theory for superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

11.4.2 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

11.4.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

11.4.4 Critical current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

11.4.5 Ginzburg criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

11.4.6 Domain wall solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

11.4.7 Scaled Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

11.5 Applications of Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

11.5.1 Domain wall energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

11.5.2 Thin type-I films : critical field strength . . . . . . . . . . . . . . . . . . . . . . . . . 425

11.5.3 Critical current of a wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

11.5.4 Magnetic properties of type-II superconductors . . . . . . . . . . . . . . . . . . . . 430

11.5.5 Lower critical field of a type-II superconductor . . . . . . . . . . . . . . . . . . . . 432

11.5.6 Abrikosov vortex lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

12 BCS Theory of Superconductivity 439

12.1 Binding and Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

12.2 Cooper’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

12.3 Effective attraction due to phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

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12.3.1 Electron-phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

12.3.2 Effective interaction between electrons . . . . . . . . . . . . . . . . . . . . . . . . . 445

12.4 Reduced BCS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

12.5 Solution of the mean field Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

12.6 Self-consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

12.6.1 Solution at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

12.6.2 Condensation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

12.7 Coherence factors and quasiparticle energies . . . . . . . . . . . . . . . . . . . . . . . . . . 453

12.8 Number and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

12.9 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

12.9.1 Isotope effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

12.9.2 Landau free energy of a superconductor . . . . . . . . . . . . . . . . . . . . . . . . 456

12.10 Paramagnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

12.11 Finite Momentum Condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

12.11.1 Gap equation for finite momentum condensate . . . . . . . . . . . . . . . . . . . . 461

12.11.2 Supercurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

12.12 Effect of repulsive interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

12.13 Appendix I : General Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 465

12.14 Appendix II : Superconducting Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 467

13 Applications of BCS Theory 471

13.1 Quantum XY Model for Granular Superconductors . . . . . . . . . . . . . . . . . . . . . . 471

13.1.1 No disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

13.1.2 Self-consistent harmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . 473

13.1.3 Calculation of the Cooper pair hopping amplitude . . . . . . . . . . . . . . . . . . 475

13.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

13.2.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

13.2.2 The single particle tunneling current IN . . . . . . . . . . . . . . . . . . . . . . . . . 479

13.2.3 The Josephson pair tunneling current IJ . . . . . . . . . . . . . . . . . . . . . . . . 485

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13.3 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13.3.1 Two grain junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13.3.2 Effect of in-plane magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

13.3.3 Two-point quantum interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

13.3.4 RCSJ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

13.4 Ultrasonic Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

13.5 Nuclear Magnetic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

13.6 General Theory of BCS Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

13.6.1 Case I and case II probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

13.6.2 Electromagnetic absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

13.7 Electromagnetic Response of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 514

14 Magnetism 519

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

14.1.1 Absence of orbital magnetism within classical physics . . . . . . . . . . . . . . . . 521

14.2 Basic Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

14.2.1 Single electron Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

14.2.2 The Darwin term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

14.2.3 Many electron Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

14.3 The Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

14.3.1 Aufbau principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

14.3.2 Splitting of configurations: Hund’s rules . . . . . . . . . . . . . . . . . . . . . . . . 525

14.3.3 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

14.3.4 Crystal field splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

14.4 Magnetic Susceptibility of Atomic and Ionic Systems . . . . . . . . . . . . . . . . . . . . . 530

14.4.1 Filled shells: Larmor diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

14.4.2 Partially filled shells: van Vleck paramagnetism . . . . . . . . . . . . . . . . . . . 532

14.5 Moment Formation in Interacting Itinerant Systems . . . . . . . . . . . . . . . . . . . . . . 536

14.5.1 The Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

CONTENTS xiii

14.5.2 Stoner mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

14.5.3 Antiferromagnetic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

14.5.4 Mean field phase diagram of the Hubbard model . . . . . . . . . . . . . . . . . . . 541

14.6 Interaction of Local Moments: the Heisenberg Model . . . . . . . . . . . . . . . . . . . . . 544

14.6.1 Ferromagnetic exchange of orthogonal orbitals . . . . . . . . . . . . . . . . . . . . 544

14.6.2 Heitler-London theory of the H2 molecule . . . . . . . . . . . . . . . . . . . . . . . 546

14.6.3 Failure of Heitler-London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

14.6.4 Herring’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

14.7 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

14.7.1 Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

14.7.2 Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

14.7.3 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

14.7.4 Variational probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

14.8 Magnetic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

14.8.1 Mean field theory of anisotropic magnetic systems . . . . . . . . . . . . . . . . . . 559

14.8.2 Quasi-1D chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

14.9 Spin Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

14.9.1 Ferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562

14.9.2 Static Correlations in the Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . 563

14.9.3 Antiferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

14.9.4 Specific Heat due to Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

14.10 Appendix : Generalized Spin Wave Theory for SU(2) Heisenberg Hamiltonians . . . . . . 569

14.10.1 General form of Heisenberg Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 569

14.10.2 Planar spiral phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

14.10.3 Sublattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

14.10.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

14.11 Appendix: The Foldy-Wouthuysen Transformation . . . . . . . . . . . . . . . . . . . . . . 576

14.11.1 Derivation of the spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . 577

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15 Spins, Coherent States, Path Integrals, and Applications 581

15.1 The Coherent State Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

15.1.1 Feynman path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

15.1.2 Coherent state path integral for the ‘Heisenberg-Weyl’ hroup . . . . . . . . . . . . 583

15.2 Coherent States for Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

15.2.1 Coherent state wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

15.2.2 Valence bond states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

15.2.3 Derivation of spin path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

15.2.4 Gauge field and geometric phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

15.2.5 Semiclassical dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

15.3 Other Useful Representations of the Spin Path Integral . . . . . . . . . . . . . . . . . . . . 596

15.3.1 Stereographic representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

15.3.2 Recovery of spin wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

15.4 Quantum Tunneling of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

15.4.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

15.4.2 Instantons and tunnel splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

15.4.3 Garg’s calculation (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

15.5 Haldane’s Mapping to the Nonlinear Sigma Model . . . . . . . . . . . . . . . . . . . . . . 601

15.5.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

15.5.2 Geometric phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

15.5.3 Emergence of the nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . . . 603

15.5.4 Continuum limit of the geometric phase: d = 1 . . . . . . . . . . . . . . . . . . . . 605

15.5.5 The geometric phase in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 606

15.6 Large-N Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

15.6.1 1/N expansion for an integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

15.6.2 Large-N theory of the nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . 608

15.6.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

16 Notes on Line Graphs 615

CONTENTS xv

16.1 Line graphs: Kagomé and Checkerboard Lattices . . . . . . . . . . . . . . . . . . . . . . . 615

16.1.1 Kagomé lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

16.1.2 Checkerboard lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

16.1.3 Square-octagon lattice line graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

16.2 Pyrochlore Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

16.2.1 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

16.2.2 The FCC lattice Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

16.3 Depleted Pyrochlores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

16.3.1 Pyrochlore with 16 element basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

16.3.2 Depleted pyrochlore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

16.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

17 Quadratic Hamiltonians 633

17.1 Bosonic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

17.1.1 Bogoliubov equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

17.1.2 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

17.1.3 A final note on the boson problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

17.2 Fermionic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

17.2.1 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

17.3 Majorana Fermion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

17.3.1 Majorana chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

17.4 Jordan-Wigner Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

17.4.1 Anisotropic XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

17.4.2 Majorana representation of the JW transformation . . . . . . . . . . . . . . . . . . 646

xvi CONTENTS

Chapter 1

Broken Symmetry

1.1 Introduction

What is the world made of? To a philosopher, this question lies at the intersection of ontology (the‘study’ of being and existence) and mereology (the ‘study’ of parts and wholes). To a physicist, theanswer very much depends on whom you ask, because ultimately it is a matter of energy scales. To acondensed matter physicist, the world consists of electrons, nuclei, and photons. That’s pretty much it1.The characteristic energy scales in condensed matter typically range from milli-electron volts (meV) toelectron volts (eV)2. By contrast, the protons and neutrons which constitute nuclei typically have bindingenergies on the order of MeV. The nucleons themselves consist of quarks and gluons. Quarks acquiretheir masses from coupling to the Higgs field, but some 99% of the nucleon mass is due to gluons andvirtual quark-antiquark pairs, i.e. to the physics of QCD binding3.

1.1.1 The mother of all Hamiltonians

The Hamiltonian for a single electron in the presence of a static external potential V (r) and electromag-netic vector potentialA(r) is given by

H =π2

2m+ V (r) +

e~

2mcσ ·H + ~

4m2c2σ ·∇V × π + ~

2

8m2c2∇

2V +(π2)2

8m3c2+ . . . , (1.1)

where π = p+ ecA. Where did this come from? From the Dirac equation,

i~∂Ψ

∂t=

(mc2 + V cσ · πcσ · π −mc2 + V

)Ψ = EΨ . (1.2)

1Sometimes experimentalists use muons or even positrons to probe their samples, but the samples themselves consist of elec-trons and nuclei.

2Sometimes higher energy probes on the order of keV are used, for example in photoemission spectroscopy.3So the next time you overhear someone holding forth about how the Higgs field (or, worse still, the Higgs boson) gives massto everything in the universe, you can tell them that they are full of shit.

1

2 CHAPTER 1. BROKEN SYMMETRY

Figure 1.1: What is the world made of? It depends on whom you ask.

The wavefunction Ψ is a four-component Dirac spinor. Sincemc2 is the largest term for our applications,the upper two components of Ψ are essentially the positive energy components. However, the DiracHamiltonian mixes the upper two and lower two components of Ψ. One can ‘unmix’ them by making acanonical transformation,

H −→ Ĥ ′ ≡ eiS Ĥ e−iS , (1.3)

where S is Hermitian, to render Ĥ ′ block diagonal. WithE = mc2+ε, the effective Hamiltonian is givenby (14.3). This is known as the Foldy-Wouthuysen transformation, the details of which may be foundin many standard books on relativistic quantum mechanics and quantum field theory (e.g. Bjorken andDrell, Itzykson and Zuber, etc.) and are recited in §14.11 below. Note that the Dirac equation leadsto g = 2. If we go beyond “tree level” and allow for radiative corrections within QED, we obtain aperturbative expansion,

g = 2

{1 +

α

2π+O(α2)

}, (1.4)

where α = e2/~c ≈ 1/137 is the fine structure constant4.

4Note that with µn = e~/2mpc for the nuclear magneton, gp = 2.793 and gn = −1.913. These results immediately suggest thatthere is composite structure to the nucleons, i.e. quarks.

1.1. INTRODUCTION 3

Figure 1.2: What is the world made of? Another point of view.

There are two terms in (14.3) which involve the electron’s spin:

Zeeman interaction : ĤZ =e~

2mcσ ·H

Spin-orbit interaction : ĤSO =~

4m2c2σ ·∇V ×

(p+ ecA

).

(1.5)

The numerical value for µB is

µB =e~

2mc= 5.788 × 10−9 eV/G

µB/kB = 6.717 × 10−5 K/G .(1.6)

So on the scale of electron volts, laboratory scale fields (H


For Coulomb interactions, va(r) = −Zae2/r. The ion cores interact as well, with a potential uab(r) =ZaZbe

2/r. Putting it all together, we have5

H =∑

i

π2i2m− e~

2mcH ·

∑

i

σi −∑

i

1.2. CLASSICAL AND QUANTUM STATISTICAL PHYSICS 5

Suppose we have an idealized system described by a model Hamiltonian H which is simpler than Hand where we might hope to make some progress. The thermodynamic properties of the system arecalculable from the Helmholtz free energy F = −kBT lnTr exp(−H/kBT ) or one of its Legendre trans-forms, such as the grand potential7 Ω = −kBT lnTr exp(−K/kBT ), where K = H − µN . We write theHamiltonian as

H = H0 −∑

α

hαQα , (1.9)

where Qα is a Hermitian operator and hα is a conjugate field for each index α. Then the free energy F isa function of the fields, i.e. F = F

({hα}

)and we have that the thermodynamic average of Qα is

〈Qα〉= − ∂F

∂hα. (1.10)

One can now define the susceptibility

χαβ =∂〈Qα〉∂hβ

= − ∂2F

∂hα ∂hβ. (1.11)

Dynamical responses are defined for Hamiltonians of the form

H(t) = H0 −∑

α

hα(t)Qα . (1.12)

Time-dependent first order perturbation theory then yields

〈Qα(t)

〉=

∞∫

−∞

dt′ χαβ(t− t′)hβ(t′) +O(h2) (1.13)

with

χαβ(t− t′) =i

~

〈[Qα(t), Qβ(t

′)]〉

Θ(t− t′) . (1.14)

Note that the response is causal, i.e. the value of〈Qα(t)

〉depends only on the values of

{hβ(t

′)}

for t′ < t.The spectral representation of the response function χαβ(t− t′) is defined via its Fourier transform,

χ̂αβ(ω + iǫ) ≡i

~

∞∫

0

dt〈[Qα(t), Qβ(0)]

〉eiωt e−ǫt (1.15)

=1

~Z

∑

m,n

e−βE0m

{〈m∣∣Qβ

∣∣n〉 〈n∣∣Qα

∣∣m〉

ω + (E0n − E0m)/~+ iǫ−〈m∣∣Qα

∣∣n〉 〈n∣∣Qβ

∣∣m〉

ω − (E0n −E0m)/~+ iǫ

},

where Z = Tr exp(−H0/kBT ) and where H0 |n〉 = E0n |n〉.

Finally, the indices α and β may be appended by spatial coordinates in the case of local operatorsQα(r),in which case we write

H(t) = H0 −∑

α

∫ddr hα(r, t)Qα(r) . (1.16)

7Also called the Landau free energy.


For example, Qα(r) could be taken to be the local density n(r) or a component Sα(r) of the local spin

density. One then defines the response function

χαβ(r, t | r; , t′) ≡δ〈Qα(r, t)

〉

δhβ(r′, t′)

. (1.17)

1.2.1 Correspondence between quantum and classical statistical mechanics

In classical statistical physics, in systems where spontaneous symmetry breaking results in an orderedphase in which the local order parameter field takes on a finite average, i.e. 〈φ(r)〉 6= 0, the connectedcorrelation function at large distances in the vicinity of the transition behaves as (for r fixed and T → Tc)

C(r, T ) =〈φ(r)φ(0)

〉− 〈φ(r)〉〈φ(0)〉 ∼ Ar2−d e−r/ξ , (1.18)

where ξ(T ) ∼ |T − Tc|−ν is the spatial correlation length and ν the correlation length exponent8. Prettymuch all classical phase transitions are described by a continuum field theory of sorts, with a free energyfunctional

F[φ(r)

]=

∫ddx f(φ,∇φ) . (1.19)

Here φ(x) = {φ1, . . . , φN} is an N -component real field, and by ∇φ we mean the set of dN first deriva-tives ∂φa/∂xi with a ∈ {1, . . . , N} and i ∈ {1, . . . , d}. The partition function is given by the functionalintegral

Z =

∫Dφ(r) e−F [φ(r)] . (1.20)

For the Ising model, one generally has N = 1 and one may take

fIsing(φ,∇φ) =12(∇φ)

2 + 12aφ2 + 14bφ

4 , (1.21)

i.e. a scalar φ4 theory in d dimensions.

For bosonic quantum systems, the path integral formulation results in the following prescription. Thepartition function is once again given by a functional integral,

Z =

∫Dφ(r, τ) e−SE[φ(r,τ)] , (1.22)

but now the field φ(r, τ) is dependent both on spatial coordinates r as well as a ”Euclidean time” coor-dinate τ ∈

[0, ~/kBT

], where T is the temperature. SE[φ] is the Euclidean action,

SE[φ(r, t)] =

~β∫

0

dτ

∫ddx LE(φ,∇φ, ∂τφ) , (1.23)

whereLE(φ,∇φ, ∂τφ) is the Euclidean Lagrangian density. Note that spacetime consists of a ’slice’ of finitetemporal width ~/kBT . If we define the coordinate x

0 ≡ cτ , where c is a (possibly arbitrary) constant8Precisely at T = Tc, the correlation function behaves as

〈φ(r)φ(0)

〉∼ r−(d−2+η), where η is the so-called anomalous dimension.

1.2. CLASSICAL AND QUANTUM STATISTICAL PHYSICS 7

with dimensions of velocity, then we see that the action is given by a (d+1)-dimensional integral over aslab of finite widthLτ = cτ/kBT in the x

0 dimension. Hence the oft-heard maxim, quantum mechanics in d(space) dimensions is equivalent to classical statistical mechanics in (d+1) dimensions. While this is essentiallytrue, it may be the case, as we emphasize below, that the resulting (d + 1)-dimensional classical modelderived from a given quantum theory will be anisotropic in that the x0 direction is special, both due toits finite extent for T > 0 systems and also due to the form of the derivative terms in the Lagrangiandensity. Indeed, nonrelativistic quantum theories generally result in such anisotropic classical models,although an emergent low-energy relativistic symmetry is sometimes present for particular systems,such as quantum antiferromagnets or acoustic phonons, or perhaps in the vicinity of a quantum criticalpoint, such as near the tips of the ’Mott lobes’ of the Bose Hubbard model.

For bosonic systems, φ(r, τ) may be taken to be an N -component real field, or a complexified versionthereof9. As an example of a Euclidean Lagrangian density, consider the case

LE(φ,∇φ, ∂τφ) =1

2

(∂φ

∂τ

)2+ 12K(∇φ)

2 + V (φ) . (1.24)

This model may possess various discrete or continuous symmetries. For example, if we let g ∈ O(N)be an arbitrary rotation in the internal space of the field φ, and if V (gφ) = V (φ) for all such g, thenthe model is said to possess an O(N) symmetry (which may be broken - see below. Another importantsymmetry of the model is the O(d + 1) symmetry resulting from mixing space and time coordinates(with appropriate rescaling so as to render them of the same dimension). The real time correlations ofthis model will then exhibit a Lorentz symmetry. But not all QFTs have such a relativistic symmetry.Consider the case N = 2, for example, with

LE(φ,∇φ, ∂τφ) = i φ1 ∂τ φ2 − i φ2 ∂τ φ1 + 12K(∇φ)2 + V (φ2) . (1.25)

Here φ = (φ1, φ2). The ’kinetic’ term is only linear in time derivatives, while the spatial derivative termsappear quadratically. This model has no relativistic symmetry, but it does have a global O(2) symmetry.

For fermionic systems, one also has a path integral, but it is over anticommuting Grassmann fieldsψσ(r, t) and their conjugates ψ̄σ(r, τ), which we shall discuss in due time

10. An example might be

LE(ψσ, ψ̄σ ,∇ψσ,∇ψ̄σ, ∂τψσ, ∂τ ψ̄σ) = ψ̄σ(~ ∂τ − µ−

~2

2m∇

2)ψσ + Uψ̄↑ψ̄↓ψ↓ψ↑ (1.26)

Note that the Laplacian term may be spatially integrated by parts to yield a term proportional to ∇ψ̄σ ·∇ψσ. This LE corresponds to the Hubbard model.

As we approach a quantum phase transition, which occurs at temperature T = 0 and a critical valueg = gc of some coupling, the spatial correlation length ξ(g) grows as ξ(g) ∼ |g − gc|−ν . For systems withrelativistic invariance, the correlation length in the imaginary time direction ξτ (g) behaves in the sameway, i.e. with the same correlation length exponent. But in general, the temporal exponent is differentand may be written as ντ = zν, where z is the dynamic critical exponent. Thus, ξτ (g) ∼ |g − gc|−zν .As g → gc, both ξ(g) and ξτ (g) will diverge. But at finite temperature, the width of the time slice isfinite, given by ~/kBT . Thus, when g is sufficiently close to gc such that ξτ (g) ≫ Lτ = ~c/kBT , where9For example (φ1, φ2) → φ ≡ φ1 + iφ2.10Real time, rest assured.


Figure 1.3: Spatial and temporal correlations in the vicinity of a quantum phase transition. (The param-eter c carries dimensions of velocity and may be arbitrary.)

c is an arbitrary measure of velocity, the fields will be ’locked’ in the imaginary time direction. In thiscase, the expression for the partition function reverts to the classical result. What this tells us is thatquantum mechanics is irrelevant at any finite temperature T , i.e. QM does not change the critical exponentsat any finite temperature phase transitions. It should be emphasized that the above argument presumesthe existence of a local order parameter field φ(r) whose correlations asymptotically decay on the scaleof the correlation length ξ(g, T ). In the case of topological phases, where there is no such local orderparameter, this argument may not apply.

1.3 Phases of matter

Condensed matter physicists are not interested in electrons, nuclei, and photons in isolation. Ratherit is the collective properties of these interacting constituents which leads to the remarkably rich phe-nomenology of condensed matter. This richness is manifested by the bewildering array of phases ofmatter that can be conjured. Until relatively recently, phases of matter were described based on the Lan-dau paradigm, in which an ordered phase is described by one or more nonzero order parameters, each ofwhich describes a broken global symmetry.

1.3.1 Spontaneous symmetry breaking

In quantum mechanics, the eigenstates of a Hamiltonian H0 which commutes with all the generatorsof a symmetry group G may be classified according to the representations of that group. Typically thisentails the appearance of degeneracies in the eigenspectrum, with degenerate states transforming intoeach other under the group operations. Adding a perturbation V to the Hamiltonian which breaks Gdown to a subgroup H will accordingly split these degeneracies, and the new multiplets of H = H0 + Vare characterized by representations of the lower symmetry group H.

1.3. PHASES OF MATTER 9

Figure 1.4: Left: Phase diagram of H2O in the (T, p) plane. Right: Low temperature (T ≈ 150mK)longitudinal resistivity ρxx and Hall resistivity ρxy as a function of applied magnetic field in a two-dimensional electron gas system (GaAs/AlGaAs heterostructure), from R. Willett et al., Phys. Rev. Lett.59, 1776 (1987). Each dip in ρxx and concomitant plateau in ρxy corresponds to a distinct phase of matter.

In quantum field theory, or in the thermodynamic limit of a classical system, as a consequence of theinfinite number of degrees of freedom, symmetries may be spontaneously broken. This means that evenif the Hamiltonian H (or action S) for the field theory is invariant under a group G of symmetry trans-formations, the ground state or thermodynamic density matrix may not be invariant under the fullsymmetry group G. The presence or absence of spontaneous symmetry breaking (SSB), and its detailedmanifestations, will in general depend on the couplings, or the temperature in the case of quantum sta-tistical mechanics. SSB is usually associated with the presence of a local order parameter which transformsnontrivially under some group operations, and whose whose quantum statistical average vanishes in afully symmetric phase, but takes nonzero values in symmetry-broken phase11. The parade example isthe Ising model, H = −∑i 1 and T < Tc , the system actuallypicks the left or the right well, so that 〈φ(r)〉 6= 0. Another example is the spontaneously broken O(2)invariance of superfluids, where the boson annihilation operator ψ(r) develops a spontaneous average〈ψ(r)

〉=√n0 e

iθ , where n0 is the condensate density and θ the condensate phase.

Truth be told, the above description is a bit of a swindle. In the ferromagnetic (Jij > 0) Ising model, for

11While SSB is generally associated with the existence of a phase transitions, not all phase transitions involve SSB. Exceptionsinclude the Kosterlitz-Thouless transition, and also those topological phases which have no local order parameter.


example, at T = 0, there are still two ground states, |⇑ 〉 ≡ |↑↑↑ · · · 〉 and |⇓ 〉 ≡ |↓↓↓ · · · 〉 . The (ergodic)zero temperature density matrix is ρ0 =

12 |⇑ 〉〈 ⇑| + 12 |⇓ 〉〈 ⇓| , and therefore 〈σi〉 = Tr

(ρ0 σi

)= 0. The

order parameter apparently has vanished. WTF?! There are at least two compelling ways to resolve thisseeming conundrum:

(a) First, rather than defining the order parameter of the Ising model, for example, to be the ex-pected value m = 〈σi〉 of the local spin12, consider instead the behavior of the correlation functionCij = 〈σi σj〉 in the limit dij = |Ri − Rj | → ∞ . In a disordered phase, there is no correla-tion between infinitely far separated spins, hence limdij→∞Cij = 0 . In the ordered phase, this isno longer true, and we define the spontaneous magnetization m from the long distance correlator:m2 ≡ limdij→∞〈σi σj〉 . In this formulatio, SSB is associated with the emergence of long-ranged orderin the correlators of operators which transform nontrivially under the symmetry group.

(ii) Second, we could impose an external field which explicitly breaks the symmetry, such as a ZeemantermH ′ = −h∑i σi in the Ising model. We now compute the magnetization (per site)m(T, h, V ) =〈σi〉 as a function of temperature T , the external field h, and the volume V of our system. The orderparameter m(T ) in zero field is then defined as

m(T ) = limh→0

limV→∞

m(T, h, V ) . (1.27)

The order of limits here is crucially important. The thermodynamic limit V → ∞ is taken first,which means that the energy difference between |⇑ 〉 and |⇓ 〉, being proportional to hV , diverges,thus infinitely suppressing the |⇓ 〉 state if h > 0 (and the |⇑ 〉 state if h < 0). The magnitude of theorder parameter will be independent on the way in which we take h→ 0, but its sign will dependon whether h→ 0+ or h→ 0−, with sgn(m) = sgn(h). Physically, the direction in which a systemorders can be decided by the presence of small stray fields or impurities. An illustration of howthis works in the case of ideal Bose gas condensation is provided in the appendix §1.7 below.

Note that in both formulations, SSB is necessarily associated with the existence of a local operator Oiwhich is identified as the order parameter field. In (i) the correlations

〈OiOj

〉exhibit long-ranged order

in the symmetry-broken phase. In (ii) Oi is the operator to which the external field hi couples.

For a ferromagnet, the order parameter is the magnetization density,m, and the broken symmetry is thegroup of rotations O(3) or SU(2), or possibly Z2. Under a group operation g ∈ G, the order parametertransforms as m → gm. For a crystal or charge density wave, the order parameters are the Fouriercomponents ̺G of the density at a series of wavevectors which comprise the reciprocal lattice of thestructure. The broken symmetry is that of continuous translation, i.e. the group Rd under addition, andunder the group operation ta corresponding to translation by a ∈ Rd, we have ̺G → ta ̺G = ̺G eiG·a.A given order parameter φ will in general depend on temperature T , pressure p, applied magnetic fieldH , various coupling parameters which enter the Hamiltonian {gi}, etc. A multidimensional plot whichlabels the various phases of a system as a function of all these parameters is known as a phase diagram.Points, lines, or surfaces separating different phases are the loci of phase transitions, where the free energyand the order parameters of the system exhibit singularities. At a first order transition, certain propertieschange discontinuously. The canonical example of a first order transition is the freezing or boiling of

12We assume translational invariance, which means 〈σi〉 is independent of the site index i.

1.3. PHASES OF MATTER 11

liquid water to solid ice or gaseous vapor. These transitions involve a discontinuous change in thedensity as T or p is varied so as to cross a phase boundary13. At a second order transition, one or moreorder parameters vanish as a phase boundary is approached. Within the Landau paradigm, a nonzeroorder parameter is associated with the spontaneous breaking of a symmetry, such as spin rotation (in amagnet), space translation (in a crystal), etc. Furthermore, a second order transition between phases Aand B is possible only if the symmetry groups GA of the A phase and GB of the B phase, both of whichmust be subgroups of the symmetry group G of the Hamiltonian, satisfy GA ⊂ GB or GB ⊂ GA. A phasetransition which takes place at T = 0 as a function of other parameters (p, H , etc.) is called a quantumphase transition.

1.3.2 Beyond the Landau paradigm

The classical O(2) model in d = 2 is precluded from achieving long-ranged order and spontaneous O(2)symmetry breaking at any finite temperature by the Hohenberg-Mermin-Wagner theorem, which weshall discuss below in §1.5.3. Nevertheless, the model does exhibit a second order phase transition,known as the Berezinskii-Kosterlitz-Thouless (BKT) transition14. The energy density in the continuummodel is written as ε(r) = 12ρs(∇Θ)

2, where ρs is the stiffness parameter and Θ(r) is a planar angle, with Θand Θ+2πn identified for all integers n. The model features point defects which are known as vortices,about which the winding number

∮C dr·∇Θ along a small loop C enclosing the vortex is 2πq, where q ∈ Z

is the vorticity. Integrating the energy density for a single q = 1 vortex with Θ(r) = tan−1(y/x) yieldsE = πρs ln(R/a), whereR is the system radius and a an ultraviolet cutoff, which is naturally imposed inany lattice-based model. Since a vortex can be in any location, its entropy is S = kB ln(R

2/a2), and we seethat free vortices should proliferate when F = E−TS = (πρs−2kBT ) ln(R/a) < 0, i.e. T > Tc = πρs/2kB .This very crude derivation, which neglects interactions between the vortices, is essentially correct. Thephase transition is associated with an unbinding of vortex-antivortex pairs. In the confined phase, whereT < Tc , the correlation function C(r, T ) = 〈eiΘ(r) e−iΘ(0)〉 decays as a power law, and there is no long-ranged order associated with a spontaneous breaking of O(2). In the plasma phase, where T > Tc , freevortices and antivortices (i.e. defects with qi < 0) are present, and C(r, T ) decays exponentially with afinite correlation length ξ. We will discuss the BKT transition and its analysis via the renormalizationgroup later in this course.

Over the past 25 years or so, a new paradigm has emerged for certain phases of matter which haveno order parameter in the conventional sense. These are called topological phases, and are exemplifiedby the phases of the quantum Hall effect. Typically topological phases exhibit a bulk excitation gap,meaning it requires a finite amount of energy in order to excite these systems in their bulk, and in thisrespect, they are akin to insulators. However, at the edges of finite systems, there exist gapless edgestates which may carry current. The structure of the edge states is intimately related to the nature ofthe bulk phase, which despite being a condensate of sorts, has no local order parameter. Topologicalphases are of intense current interest, both theoretically and experimentally, but they remain somewhatexotic, as compared with, say, metals. Yet the metallic phase, which is an example of a Landau Fermi

13The phase boundary between liquid and vapor terminates in a critical point, and it is therefore possible to continuously evolvefrom liquid to vapor without ever encountering a singularity. Thus, from the point of view of order parameters and brokensymmetry, there is no essential distinction between liquid and vapor phases.

14V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971); J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid State Phys. 6, 1181 (1973).


liquid, is difficult to characterize in terms of a conventional order parameter15. Another violation of theLandau paradigm which has been revealed in recent years is the (at this point purely theoretical) notionof deconfined quantum criticality. A deconfined quantum critical point violated the Landau requirementthat the symmetry groups on either side of the transition have a subgroup relation.

Experimental probes

We will initially study two important phases of condensed matter in the solid state: metals and insu-lators16. Later we shall expand our horizons and consider magnets, superconductors, quantum Hallphases, and spin liquids. These various phases are revealed through various experimental probes, in-cluding thermodynamic measurements such as specific heat and magnetic susceptibility, transport measure-ments such as electrical resistivity and thermal conductivity, and spectroscopies such as Auger, photoe-mission and scanning probe measurements, and optical properties as measured in reflectivity, Kerr effect,dichroism, etc. The theoretical understanding of such probes typically involves the computation of cor-relation (or, equivalently, response) functions within the framework of quantum statistical physics.

When falsifiable theoretical models and their solutions are in harmonious agreement with experiment,and predict new observable phenomena, we can feel confident that we are indeed beginning to under-stand what the world is made of.

1.4 Landau Theory of Phase Transitions

Landau’s theory of phase transitions is based on an expansion of the free energy of a thermodynamicsystem in terms of an order parameter, which is nonzero in an ordered phase and zero in a disorderedphase. For example, the magnetizationM of a ferromagnet in zero external field but at finite temperature

typically vanishes for temperatures T > Tc, where Tc is the critical temperature, also called the Curietemperature in a ferromagnet. A low order expansion in powers of the order parameter is appropriatesufficiently close to the phase transition, i.e. at temperatures such that the order parameter, if nonzero,is still small.

1.4.1 Quartic free energy with Ising symmetry

The simplest example is the quartic free energy,

f(m,h = 0, θ) = f0 +12am

2 + 14bm4 , (1.28)

where f0 = f0(θ), a = a(θ), and b = b(θ). Here, θ is a dimensionless measure of the temperature. If forexample the local exchange energy in the ferromagnet is J , then we might define θ = kBT/zJ , where

15The order parameter of a Fermi liquid is usually taken to be the quasiparticle weight Z, which, in isotropic systems, character-izes the discontinuous drop in the momentum occupation function n(k) as one crosses the Fermi surface at k = kF.

16As we shall see, semiconductors form a sub-class of insulators.

1.4. LANDAU THEORY OF PHASE TRANSITIONS 13

Figure 1.5: Phase diagram for the quartic Landau free energy f = f0 +12am

2 + 14bm4 − hm, with b > 0.

There is a first order line at h = 0 extending from a = −∞ and terminating in a critical point at a = 0.For |h| < h∗(a) (dashed red line) there are three solutions to the mean field equation, corresponding toone global minimum, one local minimum, and one local maximum. Insets show behavior of the freeenergy f(m).

z is the lattice coordination number. Let us assume b > 0, which is necessary if the free energy is to bebounded from below17. The equation of state ,

∂f

∂m= 0 = am+ bm3 , (1.29)

has three solutions in the complex m plane: (i) m = 0, (ii) m =√−a/b , and (iii) m = −

√−a/b . The

latter two solutions lie along the (physical) real axis if a < 0. We assume that there exists a uniquetemperature θc where a(θc) = 0. Minimizing f , we find

θ < θc : f(θ) = f0 −a2

4bθ > θc : f(θ) = f0 .

(1.30)

The free energy is continuous at θc since a(θc) = 0. The specific heat, however, is discontinuous acrossthe transition, with

c(θ+c)− c(θ−c)= −θc

∂2

∂θ2

∣∣∣∣θ=θc

(a2

4b

)= −θc

[a′(θc)

]2

2b(θc). (1.31)

17It is always the case that f is bounded from below, on physical grounds. Were b negative, we’d have to consider higher orderterms in the Landau expansion.


The presence of a magnetic field h breaks the Z2 symmetry of m→ −m. The free energy becomes

f(m,h, θ) = f0 +12am

2 + 14bm4 − hm , (1.32)

and the mean field equation isbm3 + am− h = 0 . (1.33)

This is a cubic equation for m with real coefficients, and as such it can either have three real solutionsor one real solution and two complex solutions related by complex conjugation. Clearly we must havea < 0 in order to have three real roots, since bm3 + am is monotonically increasing otherwise. Theboundary between these two classes of solution sets occurs when two roots coincide, which meansf ′′(m) = 0 as well as f ′(m) = 0. Simultaneously solving these two equations, we find

h∗(a) = ± 233/2

(−a)3/2b1/2

, (1.34)

or, equivalently,

a∗(h) = − 322/3

b1/3 |h|2/3. (1.35)

If, for fixed h, we have a < a∗(h), then there will be three real solutions to the mean field equationf ′(m) = 0, one of which is a global minimum (the one for which m · h > 0). For a > a∗(h) there is onlya single global minimum, at which m also has the same sign as h. If we solve the mean field equationperturbatively in h/a, we find

m(a, h) =h

a− ba4h3 +O(h5) (a > 0)

= ±|a|1/2

b1/2+

h

2 |a| ±3 b1/2

8 |a|5/2 h2 +O(h3) (a < 0) .

(1.36)

1.4.2 Cubic terms in Landau theory : first order transitions

Next, consider a free energy with a cubic term,

f = f0 +12am

2 − 13ym3 + 14bm4 , (1.37)

with b > 0 for stability. Without loss of generality, we may assume y > 0 (else sendm→ −m). Note thatwe no longer have m → −m (i.e. Z2) symmetry. The cubic term favors positive m. What is the phasediagram in the (a, y) plane?

Extremizing the free energy with respect to m, we obtain

∂f

∂m= 0 = am− ym2 + bm3 . (1.38)

This cubic equation factorizes into a linear and quadratic piece, and hence may be solved simply. Thethree solutions are m = 0 and

m = m± ≡y

2b±√( y

2b

)2− ab. (1.39)


Figure 1.6: Behavior of the quartic free energy f(m) = 12am2 − 13ym3 + 14bm4. A: y2 < 4ab ; B: 4ab <

y2 < 92ab ; C and D: y2 > 92ab. The thick black line denotes a line of first order transitions, where the

order parameter is discontinuous across the transition.

We now see that for y2 < 4ab there is only one real solution, at m = 0, while for y2 > 4ab there are threereal solutions. Which solution has lowest free energy? To find out, we compare the energy f(0) withf(m+)

18. Thus, we setf(m) = f(0) =⇒ 12am2 − 13ym3 + 14bm4 = 0 , (1.40)

and we now have two quadratic equations to solve simultaneously:

0 = a− ym+ bm2

0 = 12a− 13ym+ 14bm2 = 0 .(1.41)

Eliminating the quadratic term givesm = 3a/y. Finally, substitutingm = m+ gives us a relation betweena, b, and y:

y2 = 92 ab . (1.42)

Thus, we have the following:

a >y2

4b: 1 real root m = 0

y2

4b> a >

2y2

9b: 3 real roots; minimum at m = 0

2y2

9b> a : 3 real roots; minimum at m =

y

2b+

√( y2b

)2− ab

(1.43)

18We needn’t waste our time considering the m = m− solution, since the cubic term prefers positive m.


The solutionm = 0 lies at a local minimum of the free energy for a > 0 and at a local maximum for a < 0.

Over the range y2

4b > a >2y2

9b , then, there is a global minimum at m = 0, a local minimum at m = m+,

and a local maximum at m = m−, with m+ > m− > 0. For2y2

9b > a > 0, there is a local minimum ata = 0, a global minimum at m = m+, and a local maximum at m = m−, again with m+ > m− > 0.For a < 0, there is a local maximum at m = 0, a local minimum at m = m−, and a global minimum atm = m+, with m+ > 0 > m−. See fig. 1.6.

With y = 0, we have a second order transition at a = 0. With y 6= 0, there is a discontinuous (first order)transition at ac = 2y

2/9b > 0 and mc = 2y/3b . This occurs before a reaches the value a = 0 where thecurvature at m = 0 turns negative. If we write a = α(T − T0), then the expected second order transitionat T = T0 is preempted by a first order transition at Tc = T0 + 2y

2/9αb.

1.4.3 Magnetization dynamics

Suppose we now impose some dynamics on the system, of the simple relaxational type

∂m

∂t= −Γ ∂f

∂m, (1.44)

where Γ is a phenomenological kinetic coefficient. Assuming y > 0 and b > 0, it is convenient toadimensionalize by writing

m ≡ yb· u , a ≡ y

2

b· r , t ≡ b

Γy2· s . (1.45)

Then we obtain∂u

∂s= −∂ϕ

∂u, (1.46)

where the dimensionless free energy function is

ϕ(u) = 12ru2 − 13u3 + 14u4 . (1.47)

We see that there is a single control parameter, r. The fixed points of the dynamics are then the stationarypoints of ϕ(u), where ϕ′(u) = 0, with

ϕ′(u) = u (r − u+ u2) . (1.48)

The solutions to ϕ′(u) = 0 are then given by

u∗ = 0 , u∗ = 12 ±√

14 − r . (1.49)

For r > 14 there is one fixed point at u = 0, which is attractive under the dynamics u̇ = −ϕ′(u) sinceϕ′′(0) = r. At r = 14 there occurs a saddle-node bifurcation and a pair of fixed points is generated, onestable and one unstable. As we see from fig. 1.5, the interior fixed point is always unstable and the twoexterior fixed points are always stable. At r = 0 there is a transcritical bifurcation where two fixed pointsof opposite stability collide and bounce off one another (metaphorically speaking).


Figure 1.7: Fixed points for ϕ(u) = 12ru2 − 13u3 + 14u4 and flow under the dynamics u̇ = −ϕ′(u).

Solid curves represent stable fixed points and dashed curves unstable fixed points. Magenta arrowsshow behavior under slowly increasing control parameter r and dark blue arrows show behavior underslowly decreasing r. For u > 0 there is a hysteresis loop. The thick black curve shows the equilibriumthermodynamic value of u(r), i.e. that value which minimizes the free energy ϕ(u). There is a first orderphase transition at r = 29 , where the thermodynamic value of u jumps from u = 0 to u =

23 .

At the saddle-node bifurcation, r = 14 and u =12 , and we find ϕ(u =

12 ; r =

14) =

1192 , which is positive.

Thus, the thermodynamic state of the system remains at u = 0 until the value of ϕ(u+) crosses zero. Thisoccurs when ϕ(u) = 0 and ϕ′(u) = 0, the simultaneous solution of which yields r = 29 and u =

23 .

Suppose we slowly ramp the control parameter r up and down as a function of the dimensionless times. Under the dynamics of eqn. 1.46, u(s) flows to the first stable fixed point encountered – this is alwaysthe case for a dynamical system with a one-dimensional phase space. Then as r is further varied, ufollows the position of whatever locally stable fixed point it initially encountered. Thus, u

(r(s)

)evolves

smoothly until a bifurcation is encountered. The situation is depicted by the arrows in fig. 1.7. Theequilibrium thermodynamic value for u(r) is discontinuous; there is a first order phase transition atr = 29 , as we’ve already seen. As r is increased, u(r) follows a trajectory indicated by the magentaarrows. For an negative initial value of u, the evolution as a function of r will be reversible. However,if u(0) is initially positive, then the system exhibits hysteresis, as shown. Starting with a large positivevalue of r, u(s) quickly evolves to u = 0+, which means a positive infinitesimal value. Then as r isdecreased, the system remains at u = 0+ even through the first order transition, because u = 0 is anattractive fixed point. However, once r begins to go negative, the u = 0 fixed point becomes repulsive,

and u(s) quickly flows to the stable fixed point u+ =12 +

√14 − r. Further decreasing r, the system

remains on this branch. If r is later increased, then u(s) remains on the upper branch past r = 0, until


the u+ fixed point annihilates with the unstable fixed point at u− =12 −

√14 − r, at which time u(s)

quickly flows down to u = 0+ again.

1.4.4 Sixth order Landau theory : tricritical point

Finally, consider a model with Z2 symmetry, with the Landau free energy

f = f0 +12am

2 + 14bm4 + 16cm

6 , (1.50)

with c > 0 for stability. We seek the phase diagram in the (a, b) plane. Extremizing f with respect to m,we obtain

∂f

∂m= 0 = m (a+ bm2 + cm4) , (1.51)

which is a quintic with five solutions over the complex m plane. One solution is obviously m = 0. Theother four are

m = ±

√√√√− b2c±√(

b

2c

)2− ac. (1.52)

For each ± symbol in the above equation, there are two options, hence four roots in all.

If a > 0 and b > 0, then four of the roots are imaginary and there is a unique minimum at m = 0.

For a < 0, there are only three solutions to f ′(m) = 0 for real m, since the − choice for the ± sign underthe radical leads to imaginary roots. One of the solutions is m = 0. The other two are

m = ±

√

− b2c

+

√( b2c

)2− ac. (1.53)

The most interesting situation is a > 0 and b < 0. If a > 0 and b < −2√ac, all five roots are real. Theremust be three minima, separated by two local maxima. Clearly if m∗ is a solution, then so is −m∗. Thus,the only question is whether the outer minima are of lower energy than the minimum at m = 0. Weassess this by demanding f(m∗) = f(0), where m∗ is the position of the largest root (i.e. the rightmostminimum). This gives a second quadratic equation,

0 = 12a+14bm

2 + 16cm4 , (1.54)

which together with equation 1.51 gives

b = − 4√3

√ac . (1.55)

Thus, we have the following, for fixed a > 0:

b > −2√ac : 1 real root m = 0−2√ac > b > − 4√

3

√ac : 5 real roots; minimum at m = 0 (1.56)

− 4√3

√ac > b : 5 real roots; minima at m = ±

√

− b2c

+

√( b2c

)2− ac

The point (a, b) = (0, 0), which lies at the confluence of a first order line and a second order line, isknown as a tricritical point.


Figure 1.8: Behavior of the sextic free energy f(m) = 12am2+ 14bm

4+ 16cm6. A: a > 0 and b > 0 ; B: a < 0

and b > 0 ; C: a < 0 and b < 0 ; D: a > 0 and b < − 4√3

√ac ; E: a > 0 and − 4√

3

√ac < b < −2√ac ; F: a > 0

and −2√ac < b < 0. The thick dashed line is a line of second order transitions, which meets the thicksolid line of first order transitions at the tricritical point, (a, b) = (0, 0).

The case of barium titanate

The compound BaTiO3 is a ferroelectric in which, at sufficiently low temperatures, a spontaneous electricpolarization density P develops. The Landau free energy density may be written as

f(P, ε) = f0 +12aP

2 + 14bP4 + 16cP

6 − dEP + εP 2 + ε2

2k, (1.57)

whereE is the electric field and ε is a component of the the strain field19. Note that the coupling betweenstrain and polarization is linear in the former and quadratic in the latter, which is a consequence ofsymmetry associated with the displacive transition. In other materials such as KH2PO4, where thesymmetry is lower, the coupling is of the form εP .

19Recall the strain tensor in a solid is given by εij =12

( ∂ui∂xj

+∂uj∂xi

), where u(r) is the local displacement field.


Ti4+

Ba2+

O2−

Figure 1.9: High temperature cubic perovskite crystal structure of BaTiO3. Ba2+ sites are in green, Ti4+

in blue, and O2− in red. The yellow arrow shows the direction in which the Ti4+ ion moves as thematerial is cooled below Tc within the displacive model. Image credit: Wikipedia.

Setting ∂f/∂ε = 0 we obtain ε = −kdP 2, resulting in the the effective free energy density

feff(P ) = f0 +12aP

2 + (14b− 12kd2)P 4 + 16cP 6 − EP . (1.58)

If b > 0, and dk2 > 12b, the second order transition is driven to become first order due to the couplingto strain. However, it is oftentimes possible under experimental conditions to ensure that the strain isalways zero, for example in the case of a thin epitaxial film whose lattice constants are perfectly matchedto a substrate. In this case, stresses develop which constrain the strain to be ε = 0, and in the absence ofan electric field E the transition is second order.

1.4.5 Hysteresis for the sextic potential

Once again, we consider the dissipative dynamics ṁ = −Γ f ′(m). We adimensionalize by writing

m ≡√|b|c· u , a ≡ b

2

c· r , t ≡ c

Γ b2· s . (1.59)

Then we obtain once again the dimensionless equation

∂u

∂s= −∂ϕ

∂u, (1.60)

whereϕ(u) = 12ru

2 ± 14u4 + 16u6 . (1.61)In the above equation, the coefficient of the quartic term is positive if b > 0 and negative if b < 0. Thatis, the coefficient is sgn(b). When b > 0 we can ignore the sextic term for sufficiently small u, and werecover the quartic free energy studied earlier. There is then a second order transition at r = 0.


Figure 1.10: Free energy ϕ(u) = 12ru2 − 14u4 + 16u6 for several different values of the control parameter

r.

New and interesting behavior occurs for b > 0. The fixed points of the dynamics are obtained by settingϕ′(u) = 0. We have

ϕ(u) = 12ru2 − 14u4 + 16u6

ϕ′(u) = u (r − u2 + u4) .(1.62)

Thus, the equation ϕ′(u) = 0 factorizes into a linear factor u and a quartic factor u4 − u2 + r which isquadratic in u2. Thus, we can easily obtain the roots:

r < 0 : u∗ = 0 , u∗ = ±√

12 +

√14 − r

0 < r < 14 : u∗ = 0 , u∗ = ±

√12 +

√14 − r , u∗ = ±

√12 −

√14 − r

r > 14 : u∗ = 0 .

(1.63)

In fig. 1.11, we plot the fixed points and the hysteresis loops for this system. At r = 14 , there are twosymmetrically located saddle-node bifurcations at u = ± 1√

2. We find ϕ(u = ± 1√

2, r = 14) =

148 , which is

positive, indicating that the stable fixed point u∗ = 0 remains the thermodynamic minimum for the freeenergy ϕ(u) as r is decreased through r = 14 . Setting ϕ(u) = 0 and ϕ

′(u) = 0 simultaneously, we obtain

r = 316 and u = ±√32 . The thermodynamic value for u therefore jumps discontinuously from u = 0 to

u = ±√32 (either branch) at r =

316 ; this is a first order transition.

Under the dissipative dynamics considered here, the system exhibits hysteresis, as indicated in the fig-ure, where the arrows show the evolution of u(s) for very slowly varying r(s). When the control param-


Figure 1.11: Fixed points ϕ′(u∗) = 0 for the sextic potential ϕ(u) = 12ru2− 14u4+ 16u6, and corresponding

dynamical flow (arrows) under u̇ = −ϕ′(u). Solid curves show stable fixed points and dashed curvesshow unstable fixed points. The thick solid black and solid grey curves indicate the equilibrium thermo-dynamic values for u; note the overall u→ −u symmetry. Within the region r ∈ [0, 14 ] the dynamics areirreversible and the system exhibits the phenomenon of hysteresis. There is a first order phase transitionat r = 316 .

eter r is large and positive, the flow is toward the sole fixed point at u∗ = 0. At r = 14 , two simultaneoussaddle-node bifurcations take place at u∗ = ± 1√

2; the outer branch is stable and the inner branch un-

stable in both cases. At r = 0 there is a subcritical pitchfork bifurcation, and the fixed point at u∗ = 0becomes unstable.

Suppose one starts off with r ≫ 14 with some value u > 0. The flow u̇ = −ϕ′(u) then rapidly resultsin u → 0+. This is the ‘high temperature phase’ in which there is no magnetization. Now let r increaseslowly, using s as t

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