Beyond the Crystalline State

1. Introduction.- 2. Variety in Structures.- 2.1 Crystals.- 2.2 Incommensurate and Long-Period Structures.- 2.3 Quasicrystals.- 2.4 Liquid Crystals.- 2.5 Glass.- 2.6 Systems with Quasi Long-Range Order.- 2.7 Overview.- 3. Order Out of Disorder.- 3.1 Landau Theory.- 3.2 Conjugate Field.- 3.3 Symmetry Breaking: Further Aspects.- 3.4 Goldstone Modes.- 3.5 Generalized Rigidity.- 3.6 Quasi LRO.- 3.7 Overview.- 4. Defects and Topology.- 4.1 Basic Strategy.- 4.2 Some Basic Concepts of Topology.- 4.3 Continuous Groups and Topological Spaces.- 4.4 The First or the Fundamental Homotopy Group and Defects.- 4.5 Some Examples.- 4.6 Stability.- 4.7 Combination of Defects.- 4.8 Other Homotopy Groups.- 4.9 Ordered Media with Broken Translational Symmetry.- 4.10 Summary.- 5. Structures by Projection.- 5.1 Concerning Tilings.- 5.2 Regular Polytopes.- 5.3 Amorphous Structures from Mappings of Polytopes.- 5.4 Line Defects in Amorphous Structures.- 5.5 Disclinations and Frank-Kasper Chains.- 5.6 Mapping from S3 to E3.- 5.7 Defects and Star Mapping.- 5.8 Mapping by Disclination Procedure.- 5.9 Decoration.- 5.10 Defects in the CRN.- 5.11 Amorphous Structures by Projection of Hyperbolic Tilings.- 5.12 Polymers and Polytopes.- 5.13 Quasicrystals by the Projection Method.- 5.14 Generalization.- 5.15 Some Comments on the Projection Method.- 5.16 Miller Indices for Quasicrystals.- 5.17 Diffraction Patterns of Quasicrystals.- 5.18 Incommensurate Crystals.- 5.19 Summary.- 6. Beyond Simple Geometry.- 6.1 Some Basics.- 6.2 Landau Theory and Ordered Atomic Structures.- 6.3 Orientational Ordering.- 6.4 Orientational Order Versus Translational Order.- 6.5 Landau Theory and Amorphous Structures.- 6.6 Landau Theory and Liquid Crystals.- 6.7 Hydrodynamics.- 6.8 Fluctuations and the Landau Theory.- 6.9 Frustration and the Disruption of Order.- 6.10 Defect-Dominated Structures.- 6.11 Overview.- 7. Tilings in One Dimension.- 7.1 Structures and Competing Periodic Potentials.- 7.2 Portrait of the Penrose Chain.- 7.3 Spatial Chaos and Amorphous Structures.- 7.4 Summary.- 8. Ergodicity Breaking.- 8.1 Basic Ideas.- 8.2 Time Scales and Broken Ergodicity.- 8.3 Broken Ergodicity and Symmetry Breaking.- 8.4 The Spin Glass.- 8.5 The Case of Glass.- 8.6 Generalization.- 9. Symmetry Breaking - A Second Look.- 9.1 Orbits and Strata in Crystal Physics.- 9.2 Symmetry Breaking and Strata.- 9.3 Isotropy Subgroups of the Euclidean Group E(3).- 9.4 More About Extensions to E(3).- 9.5 Patterns in Nonequilibrium Systems.- 9.6 Cylindrical Crystallography.- Appendix: Special Topics.- A. Hydrodynamics.- A.1 Hydrodynamic Equations.- A.2 Ordered Media with Continuous Broken Symmetries.- A.2.1 Hydrodynamics of a Solid.- A.3 The Poisson Bracket Method in Hydrodynamics.- A.4 Summary.- B. Curved Space and Parallel Transport of Vectors.- B.1 Parallel Transport of Vectors.- B.2 The Covariant Derivative.- B.3 The Curvature.- B.4 The Torsion.- B.5 Mapping from Curved Space to Flat Space.- D. Some Aspects of Group Theory.- D.1 Group Morphisms.- D.2 Transformation Group, Group Action and Orbits.- E. A Brief Introduction to Homotopy and Lie Groups.- E.1 Topology.- E.2 Elements of Homotopy Theory.- E.2.1 The First Homotopy Group.- E.2.2 Higher Homotopy Groups.- E.3 Continuous Groups and Lie Groups.- F. Local Gauge Invariance and Gauge Theories.- F.1 Internal Connection.- F.2 Gauge Field Theory.- F.3 U(1) Gauge Symmetry.- F.4 Non-Abelian Gauge Groups.- F.5 Gauge Theory of Dislocations and Disclinations.- References.- Author Index.