Frustrated Spin Systems (Third Edition)

"Frustrated spin systems have been first investigated five decades ago. Well-known examples include the Ising model on the antiferromagnetic triangular lattice studied by G H Wannier in 1950 and the Heisenberg helical structure discovered independently by A Yoshimori, J Villainn and T A Kaplan in 1959. However, extensive investigations on frustrated spin systems have really started with the concept of frustration introduced at the same time by G Toulouse and by J Villain in 1977 in the context of spin glasses. The frustration is generated by the competition of different kinds of interaction and/or by the lattice geometry. As a result, in the ground state all bonds are not fully satisfied. In frustrated Ising spin systems, a number of spins behave as free spins. In frustrated vector spin systems, the ground-state configuration is usually non-collinear. The ground state of frustrated spin systems is therefore highly degenerate and new induced symmetries give rise to unexpected behaviors at finite temperatures. Many properties of frustrated systems are still not well understood at present. Theoretically, recent studies shown in this book reveal that established theories, numerical simulations as well as experimental techniques have encountered many difficulties in dealing with frustrated systems. In some sense, frustrated systems provide an excellent testing ground for approximations and theories. Experimentally, more and more frustrated materials are discovered with interesting properties for applications"--Publisher's website CONTENTS Preface of the First Edition Preface of the Second Edition Preface of the Third Edition 1. Frustration — Exactly Solved Frustrated Models 1.1. Frustration: An Introduction 1.1.1. Definition 1.1.2. Non-collinear spin configurations 1.2. Frustrated Ising Spin Systems 1.3. Mapping between Ising Models and Vertex Models 1.3.1. The 16-vertex model 1.3.2. The 32-vertex model 1.3.3. Disorder solutions for 2D Ising models 1.4. Reentrance in Exactly Solved Frustrated Ising Spin Systems 1.4.1. Centered square lattice 1.4.1.1. Phase diagram 1.4.1.2. Nature of ordering and disorder solutions 1.4.2. Kagom ́e lattice 1.4.2.1. Model with NN and NNN interactions 1.4.2.2. Generalized Kagom ́e lattice 1.4.3. Centered honeycomb lattice 1.4.4. Periodically dilute centered square lattices 1.4.4.1. Model with three centers 1.4.4.2. Model with two adjacent centers 1.4.4.3. Model with one center 1.4.5. Random-field aspects of the models 1.5. Evidence of Partial Disorder and Reentrance in Other Frustrated Systems 1.6. Conclusion 1.7. Note Added for the Third Edition Acknowledgments References 2. Properties and Phase Transitions in Frustrated Ising Systems 2.1. Introduction 2.2. Frustrated Lattices 2.2.1. Antiferromagnetic triangular lattice (2D) 2.2.2. Villain lattice (2D) 2.3. Stacked Frustrated Lattices (3D) 2.4. Ising Models on Antiferromagnetic Triangular Lattice:Effect of the Ferromagnetic Next-Nearest-NeighborInteractions 2.4.1. Model S3 2.4.2. Model S2 2.4.3. Model S1 2.5. Thermal Properties of Ising Models on Stacked Antiferromagnetic Triangular Lattice 2.6. Ising Model on Stacked Antiferromagnetic Triangular Lattice with NNN Interaction 2.6.1. Mean-field analysis 2.6.2. Monte Carlo analysis in two dimensions 2.6.3. Monte Carlo analysis in three dimensions 2.6.4. New type of intermediate phase in the generalized six-state clock model 2.7. Ising Model with Large S on Antiferromagnetic Triangular Lattice 2.8. Ising Model with Infinite Spin on Antiferromagnetic Triangular Lattice 2.9. Ising-like Heisenberg Model on Antiferromagnetic Triangular Lattice 2.10. Ising Model with Infinite Spin on Stacked Antiferromagnetic Triangular Lattice 2.11. Phase Diagram in Spin Magnitude vs. Temperature for Ising Models with Spin S on Stacked Antiferromagnetic Triangular Lattice 2.12. Effect of Antiferromagnetic Interaction between Next-Nearest-Neighbor Spins in xy-Plane 2.13. Three-Dimensional Ising Paramagnet 2.14. Antiferromagnets on Corner-Sharing Lattices 2.15. Concluding Remarks Acknowledgments References 3. Renormalization Group Approaches to Frustrated Magnets in d = 3 3.1. Introduction 3.2. The STA Model and Generalization 3.2.1. The lattice model, its continuum limit, and its symmetries 3.2.2. The Heisenberg case 3.2.3. The XY case 3.2.4. Generalization 3.3. Experimental and Numerical Situations 3.3.1. The XY systems 3.3.1.1. The experimental situation 3.3.1.2. The numerical situation 3.3.1.3. Summary 3.3.2. The Heisenberg systems 3.3.2.1. The experimental situation 3.3.2.2. The numerical situation 3.3.2.3. Summary 3.3.3. The N = 6 STA 3.3.4. Summary 3.4. A Brief Chronological Survey of the Theoretical Approaches 3.5. The Perturbative Situation 3.5.1. The Nonlinear Sigma (NLσ) model approach 3.5.2. The Ginzburg–Landau–Wilson model approach 3.5.2.1. The RG flow 3.5.2.2. The three- and five-loop results in d = 4−  3.5.2.3. The improved three- and five-loop results 3.5.2.4. The three-loop results in d = 3 3.5.2.5. The large-N results 3.5.2.6. The six-loop results in d = 3 3.5.3. The six-loop results in d = 3 re-examined 3.5.3.1. Summary 3.6. The Effective Average Action Method 3.6.1. The effective average action equation 3.6.2. Properties 3.6.3. Truncations 3.6.4. Principle of the calculation 3.6.5. The O(N) × O(2) model 3.6.5.1. The flow equations 3.6.6. Tests of the method and first results 3.6.7. The physics in d = 3 according to the NPRG approach 3.6.7.1. The physics in d = 3 just below Nc(d): Scaling with a pseudo-fixed point and minimum of the flow 3.6.7.2. Scaling with or without pseudo-fixed point: The Heisenberg and XY cases 3.6.7.3. The integration of the RG flow 3.6.7.4. The Heisenberg case 3.6.7.5. The XY case 3.6.8. Summary 3.7. Conclusion and Prospects 3.8. Note Added in the Second Edition 3.9. Note Added by the Editor for the Third Edition Acknowledgments References 4. Phase Transitions in Frustrated Vector Spin Systems: Numerical Studies 4.1. Introduction 4.2. Breakdown of Symmetry 4.2.1. Symmetry in the high-temperature region 4.2.2. Breakdown of symmetry for ferromagnetic systems 4.2.3. Breakdown of symmetry for frustrated systems 4.2.3.1. Stacked triangular antiferromagnetic lattices 4.2.3.2. BCT Helimagnets 4.2.3.3. Stacked J1–J2 square lattices 4.2.3.4. The simple cubic J1–J2 lattice 4.2.3.5. J1–J2–J3 lattice 4.2.3.6. Villain lattice and fully frustrated simple cubic lattice 4.2.3.7. Face-centered cubic lattice 4.2.3.8. Hexagonal-close-packed lattice (hcp) 4.2.3.9. Pyrochlores 4.2.3.10. Other lattices 4.2.3.11. STAR lattices 4.2.3.12. Dihedral lattices VN,2 4.2.3.13. Right-handed trihedral lattices V3,3 4.2.3.14. P-hedral lattices VN,P 4.2.3.15. Ising and Potts-VN,1 model 4.2.3.16. Ising and Potts-VN,2 model 4.2.3.17. Landau–Ginzburg model 4.2.3.18. Cubic term in Hamiltonian 4.2.3.19. Summary 4.3. Phase Transitions between Two and Four Dimensions: 2 J1/2) 5.2.3. Non-magnetic region (J2  J1/2) 5.2.3.1. Series expansions 5.2.3.2. Exact diagonalizations 5.2.3.3. Quantum Monte Carlo 5.3. Valence Bond Crystals 5.3.1. Definitions 5.3.2. One-dimensional and quasi- one-dimensional examples (spin-1/2 systems) 5.3.3. Valence Bond Solids 5.3.4. Two-dimensional examples of VBC 5.3.4.1. Without spontaneous lattice symmetry breaking 5.3.4.2. With spontaneous lattice symmetry breaking 5.3.5. Methods 5.3.6. Summary of the properties of VBC phases 5.4. Large-N Methods 5.4.1. Bond variables 5.4.2. SU(N) 5.4.3. Sp(N) 5.4.3.1. Gauge invariance 5.4.3.2. Mean-field (N = ∞ limit) 5.4.3.3. Fluctuations about the mean-field solution 5.4.3.4. Topological effects — instantons and spontaneous dimerization 5.4.3.5. Deconfined phases 5.5. Quantum Dimer Models 5.5.1. Hamiltonian 5.5.2. Relation with spin-1/2 models 5.5.3. Square lattice 5.5.3.1. Transition graphs and topological sectors 5.5.3.2. Staggered VBC for V/J > 1 5.5.3.3. Columnar crystal for V