Contents

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Preface xvii

1 Overview 11.1 Condensed matter physics 11.2 An example - H2O 3

1 Gaseous and liquid states 32 The liquid-gas phase transition 43 Spatial correlations in the liquid state 54 Ice - crystallized water 85 Broken symmetry and rigidity 106 Dislocations - topological defects 127 Universality of the water example 138 Fluctuations and spatial dimension 159 Overview of book 16

1.3 Energies and potentials 171 Energy scales 172 Van der Waals attraction 183 Molecular hydrogen - the Heitler-London approach 204 Hard-sphere repulsion 225 Exchange interaction and magnetism 246 The hydrogen molecule, molecular orbitals, and bands

in metals 25Bibliography 28References 28

2 Structure and scattering 292.1 Elementary scattering theory - Bragg's law 292.2 Photons, neutrons, or electrons 332.3 The density Operator and its correlation functions 342.4 Liquids and gases 38

1 Hard-sphere liquids 402.5 Crystalline solids 43

1 Unit cells and the direct lattice 432 The reciprocal lattice 45

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3 Periodic functions 464 Bragg scattering 47

2.6 Symmetry and crystal structure 491 Two-dimensional Bravais lattices 502 Three-dimensional Bravais lattices 533 Close packed structures 564 Space groups 57

2.7 Liquid crystals 581 Isotropic, nematic and cholesteric phases 582 Smectics-A and -C 613 Hexatic phases 654 Discotic phases 685 Lyotropic liquid crystals and microemulsions 68

2.8 One- and two-dimensional order in three-dimensionalmaterials 71

2.9 Incommensurate structures 772.10 Quasicrystals 822.11 Magnetic order 852.12 Random isotropic fractals 90Appendix 2A Fourier transforms 97

1 One dimension 972 d dimensions 993 Transforms an a lattice 100

Bibliography 101References 102Problems 103

3 Thermodynamics and statistical mechanics 1083.1 Thermodynamics of homogeneous fluids 108

1 The first law of thermodynamics 1092 The second law of thermodynamics 1113 The third law of thermodynamics 1114 Thermodynamic potentials 1125 Stability criteria 1136 Homogeneous functions 1157 Equations of state 116

3.2 Statistical mechanics: phase space and ensembles 1173.3 The ideal gas 1223.4 Spatial correlations in classical systems 1233.5 Ordered systems 1273.6 Symmetry, order parameters, and models 132

1 Discrete symmetries 1352 Continuous symmetries 137

Contents ix

3 Models 139Appendix 3A Functional derivatives 140Bibliography 142References 142Problems 142

4 Mean-field theory 1444.1 Bragg-Williams theory 1464.2 Landau theory 1514.3 The Ising and n-vector models 152

1 The nonlocal susceptibility and the correlation length 1542 O„ symmetry 1563 Some mean-field transitions 157

4.4 The liquid-gas transition 1591 The critical point and the critical isochore 1622 The coexistence curve 165

4.5 The first-order nematic-to-isotropic transition 1684.6 Multicritical points 172

1 Tricritical points 1732 Metamagnets and FeC12 1753 Hei - He4 mixtures and the Blume-Emery-Griffiths

model 1794 Bicritical and tetracritical points 1815 Lifshitz points 184

4.7 The liquid-solid transition 1881 Are all crystals BCC? 1892 Criterion for freezing 1923 Improvements of the theory 1924 Changes in density 1945 Density functional theory 195

4.8 Variational mean-field theory 1981 Two inequalities 1982 The mean-field approximation 2003 The s-state Potts model 2014 The O„ classical Heisenberg model 2025 Debye-Hückel theory 204

Bibliography 208References 209Problems 209

5 Field theories, critical phenomena, and therenormalization group 213

5.1 Breakdown of mean-field theory 214

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1 Mean-field transitions revisited 2165.2 Construction of a field theory 217

1 Coarse graining 2172 Lattice field theories and their continuum limit 2193 Gaussian integrals 2214 Mean-field theory from functional integrals 2235 Breakdown of mean-field theory revisited 225

5.3 The seif-consistent field approximation 2261 The n-vector model in the limit n --> co 229

5.4 Critical exponents, universality, and scaling 2301 Exponents and scaling relations 2302 Scaled equation of state 2343 Multicritical points 2354 Amplitude ratios 2365 Theoretical calculations of critical exponents and

amplitude ratios 2375.5 The Kadanoff construction 2375.6 The one-dimensional Ising model 242

1 Exact solution 2422 Decimation and renormalization 245

5.7 The Migdal-Kadanoff procedure 2481 The Ising model an a hypercubic lattice 2482 General properties of recursion relations 2523 The Potts lattice gas and krypton an graphite 253

5.8 Momentum shell renormalization group 2561 Thinning of degrees of freedom and rescaling 2562 Correlation functions 2603 The Gaussian model 2614 The e-expansion 2635 n-vector model with cubic anisotropy 2676 Quadratic anisotropy 2697 Crossover 2708 Dangerous irrelevant variables 2739 The utility of the e-expansion 275

Appendix 5A The Hubbard-Stratonovich transformation 276Appendix 5B Diagrammatic perturbation theory 277Bibliography 283References 283Problems 283

6 Generalized elasticity 2886.1 The xy-model 289

1 The elastic free energy 289

Contents xi

2 Boundary conditions and external fields 2903 The Josephson scaling relation 2924 Fluctuations 2935 Long-range order, quasi-long-range order, and disorder 2956 Resistance of a conducting medium 297

6.2 O„ symmetry and nematic liquid crystals 2981 n-vector elastic energy 2982 The Frank free energy of nematic liquid crystals 2983 Cells with non-uniform n 3004 The Freedericksz transition 3025 The twisted nematic display 3046 Fluctuations and light scattering 306

6.3 Smectic liquid crystals 3081 The elastic free energy 3092 Fluctuations 3123 Nonlinearities 3144 The nematic-to-smectic-A transition 315

6.4 Elasticity of solids : strain and elastic energy 3161 The strain tensor 3162 The elastic free energy 3183 Isotropic and cubic solids 3194 Fluctuations 3215 Mercury chain salts - one-dimensional crystals 3226 Xenon an graphite - a two-dimensional crystal 3247 Vacancies and interstitials 3258 Bond-angle order and rotational and translational

elasticity 3289 Elastic constants from density functional theory 329

6.5 Lagrangian elasticity 3301 Classical theory of elasticity 3302 Elasticity of classical harmonie lattices 332

6.6 Elasticity of solids : the stress tensor 3341 The Lagrangian stress tensor 3342 Stress-strain relations 3373 The Eulerian stress tensor 338

6.7 The nonlinear sigma model 341Bibliography 347References 347Problems 347

7 Dynamics : correlation and response 3537.1 Dynamic correlation and response functions 354

1 Correlation functions 354

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2 Response functions 355

7.2 The harmonic oscillator 3591 The undamped oscillator 3592 The damped oscillator 3603 The response function 3624 Dissipation 365

7.3 Elastic waves and phonons 3661 Sound waves in an elastic continuum 3662 Acoustic phonons in a harmonic lattice 367

7.4 Diffusion 3691 Fick's law 3692 The Green function and dynamic response 3703 The response function 3714 External potentials and the Einstein relation 3735 Brownian motion 3756 Cooperative diffusion versus seif-diffusion 3767 Master equation for diffusion an a lattice 378

7.5 Langevin theory 3811 Random forces and thermal equilibrium 3812 Correlation functions for diffusion 3833 Short-time behavior 3854 Fluctuation-dissipation theorem for the harmonic

oscillator 3875 The Fokker-Planck and Smoluchowski equations 388

7.6 Formal properties of response functions 3901 Response to external fields 3902 Symmetry properties of response functions 3923 Dissipation 3944 Spectral representations of x,,,

j395

5 The fluctuation-dissipation theorem 3976 Sum rules and moment expansions 398

7.7 Inelastic scattering 3991 Scattering geometry and partial cross-sections 3992 Fermi golden rule and neutron scattering 4003 The Fermi pseudopotential 4024 Coherent and incoherent scattering 4045 Cross-sections and correlation functions 4056 Neutron scattering from crystals 4067 Magnetic scattering 4078 How neutron scattering experiments are actually done 4089 Scattering of charged particles and photons 410

Bibliography 411References 411

Contents

Problems

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411

8 Hydrodynamics 4178.1 Conserved and broken-symmetry variables 4178.2 A tutorial example - rigid rotors an a lattice 419

1 Description of the model 4202 The disordered phase 4213 The ordered phase 4264 Excitations from the classical ground state 4305 The Goldstone theorem 4326 Kubo formulae 4327 Summary 433

8.3 Spin systems 4341 Spin dynamics 4342 Generalized Heisenberg models 4353 The planar magnet 4364 The isotropic antiferromagnet 4385 Isotropic ferromagnets 439

8.4 Hydrodynamics of simple fluids 4401 Conservation laws 4412 Thermodynamics with mass motion 4433 The entropy production equation 4444 Dissipationless hydrodynamics 4455 Dissipation 4466 The Navier-Stokes equations 4487 Hydrodynamic modes 4498 Light scattering 4529 Two-component fluids 453

8.5 Liquid crystals, crystalline solids, and superfluid helium 4541 Nematic liquid crystals 4542 Smectic-A liquid crystals 4563 Crystalline solids 4594 Superfluid helium 460

8.6 Stochastic models and dynamic critical phenomena 4641 Critical slowing down and the conventional theory 4642 Dissipative dynamics 4663 Dynamic scaling 4694 Poisson bracket terms 4725 Models with Poisson brackets 4756 Mode-mode coupling 477

8.7 Nucleation and spinodal decomposition 4791 Nucleation with a nonconserved order parameter 4802 Symmetric unstable quench with model A dynamics 483

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3 Conserved order parameters and spinodaldecomposition 484

491BibliographyReferences 491

Problems 492

9

Topological defects 4959.1 Characterization of topological defects 495

1 Vortex pairs 4992 Order parameters with more than two components 4993 Order parameter spaces and homotopy 501

9.2 Examples of topological defects 506

1 Vortices in xy-models 506

2 Dislocations in smectic liquid crystals 507

3 Periodic solids 512

4 Volterra construction 515

5 Hexagonal and close-packed lattices 515

6 Disclinations in crystals 5177 Strength of crystals 5188 Crystal growth 522

9 Grain boundaries 522

10 Nematic and hexatic liquid crystals 5249.3 Energies of vortices and dislocations 526

1 Simple calculation of xy-vortex energies 5262 Analogy with magnetism 530

3 Energies of dislocations in crystals 5314 Dislocations in smectic liquid crystals 536

9.4 Vortex unbinding and the Kosterlitz-Thouless transition 5421 Vortices and the spin-wave stiffness 542

2 Vortex unbinding in two dimensions - the Kosterlitz-Thouless transition 544

3 Superfluid helium films 5519.5 Dislocation mediated melting 555

1 Effects of a substrate 5582 Experiments and numerical simulation 559

9.6 The twist-grain-boundary phase 5611 Structure of the TGB phase 561

2 The thermodynamic critical field 5643 The lower critical field 565

4 The upper critical field 5665 X-ray scattering 5686 Analogy with superconductivity 571

Appendix 9A Notes an the Kosterlitz-Thouless transition 573

Contents xv

1 Integration of the KT recursion relations 5732 Longitudinal and transverse response 5753 The spin correlation function 577

Appendix 9B Duality and the Villain model 5781 Potts models 5792 The xy-, Villain, and lattice Coulomb-gas models 582

Bibliography 584References 584Problems 585

10 Walls, kinks and solitons 59010.1 Some simple examples 59110.2 Domain walls in mean-field theory 595

1 The 04 kink 5972 The sine-Gordon soliton 5993 Dynamics 599

10.3 The Frenkel-Kontorowa model 6011 Introduction 6012 Discommensurations 6023 Devi1's staircases and the FK phase diagram 6034 The continuum approximation 6055 Nature of solutions 6086 The minimum energy solution 6107 Repulsive interaction between discommensurations 6138 X-ray diffraction 6139 Compressional elastic constants 61410 Phasons 61511 Pinned phasons 61712 Extension to two dimensions 618

10.4 Fluctuating walls 6201 Differential geometry and the total surface area 6202 Curvature 6233 Energy of a surface 6254 Fluctuations in the harmonie approximation 6265 Nonlinearities and renormalization in fluid membranes 6296 Polymerized membranes 630

10.5 Arrays of fluctuating walls 6351 Fluctuating walls and steric entropy 6352 Honeycomb lattice of walls 6383 Elasticity of sterically stabilized phases 6384 Dislocations and the CI transition 640

10.6 Roughening and faceting 6431 The solid-an-solid and discrete Gaussian models 643

xvi Contents

2 The roughening transition 6463 Faceting 648

Bibliography 655References 656Problems 656

Glossary 662Index 685