Quantum Magnetism (Lecture Notes in Physics, 645)

✍ Scribed by Ulrich Schollwöck (editor), Johannes Richter (editor), Damian J.J. Farnell (editor), Raymond F. Bishop (editor)

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 488
• Edition: 2004
• Category: Library

✦ Synopsis

The investigation of magnetic systems where quantum effects play a dominant role has become a very active branch of solid-state-physics research in its own right. The first three chapters of the Quantum Magnetism survey conceptual problems and provide insights into the classes of systems considered, namely one-dimensional, two-dimensional and molecular magnets. The following chapters introduce the methods used in the field of quantum magnetism, including spin wave analysis, exact diagonalization, quantum field theory, coupled cluster methods and the Betheansatz. The book closes with a chapter on quantum phase transitions and a contribution that puts the wealth of phenomena into the context of experimental solid-state physics. Closing a gap in the literature, this volume is intended both as an introductory text at postgraduate level and as a modern, comprehensive reference for researchers in the field.

✦ Table of Contents

front-matter
Chapter 1
1.1 Introduction
1.2 S = 1 Heisenberg Chain
1.2.1 Ferromagnetic Phase
1.2.2 N´eel Phase
1.2.3 XY Phase
1.2.4 The Isotropic Heisenberg Antiferromagnet and Its Vicinity
1.2.5 The Dynamical Structure Factor of the XXZ Chain
1.2.6 Modi.ed S=1/2 Chains
1.2.7 The XXZ Chain in an External Magnetic Field
1.2.8 E.ects of 3D Coupling
1.3 Spin Chains with S > 1/2
1.3.1 S = 1 Haldane Chain
1.3.2 Integer vs Half-Odd-Integer S
1.3.3 The AKLT Model and Valence Bond Solid States
1.3.4 Spin Chains with Alternating and Frustrated Exchange
1.3.5 Frustrated Chains with Anisotropy: Quantum Chiral Phases
1.4 S = 1 2 Heisenberg Ladders
1.4.1 Quantum Phases of Two-Leg S = 1/2 Ladders
1.4.2 Matrix Product Representation for the Two Leg S = 1/2 Ladder
1.4.3 Matrix Product States: General Formulation
1.4.4 Excitations in Two-Leg S=1/2 Ladders
1.4.5 Multileg Ladders
1.5 Modi.ed Spin Chains and Ladders
1.5.1 S = 1 Ladders with Four-Spin Interaction
1.5.2 S = 1 Bilinear-Biquadratic Chain
1.5.3 Mixed Spin Chains: Ferrimagnet
1.6 Gapped 1D Systems in High Magnetic Field
1.6.1 The Critical Phase and Gapped (Plateau) Phase
1.6.2 Magnetization Cusp Singularities
1.6.3 Response Functions in the High-Field Phase
Chapter 2
2.1 Introduction
2.2 Archimedean Lattices
2.2.1 Characteristics and Geometry
2.2.2 Relationships Between the Lattices
2.3 Criteria for N´eel Like Order
2.3.1 Order Parameter
2.3.2 Mechanism of Symmetry Breaking – The Pisa Tower of Quasi-degenerate Joint States (QDJS)
2.3.3 Finite-Size Scaling
2.4 Magnetic Ground-State Ordering for the Spin Half HAFM on the Archimedean Lattices
2.4.1 Semi-classical N´eel Ordering on Bipartite Lattices
2.4.2 Semi-classical LRO on Frustrated Lattices
2.4.3 Absence of Semi-classical LRO on Frustrated Lattices – The Kagom´e (T8) and the Star (T9) Lattices
2.4.4 Summary and Comparison
2.5 Quantum Phase Transitions in 2D HAFM – The CaVO J - J Model and the Shastry-Sutherland Model
2.6 Magnetization Process
2.6.1 Square Lattice
2.6.2 Triangular Lattice
2.6.3 Kagom´e Lattice
2.6.4 Independent Magnons and Macroscopic Magnetization Jumps
2.6.5 Shastry-Sutherland Model Versus SrCu2(BO3)2
Chapter 3
3.1 Introduction
3.2 Substances
3.3 Experimental Work
3.3.1 Experimental Methods
3.3.2 Phenomena of Current Interest
3.4 Theoretical Techniques and Results
3.4.1 Hamiltonian
3.4.2 Evaluating the Spectrum
3.4.3 Properties of Spectra
3.5 Dynamics
3.5.1 Tunneling
3.5.2 Relaxation Dynamics
Chapter 4
4.1 Introduction
4.2 Dyson–Maleev Formalism
4.2.1 Classical Reference State
4.2.2 Boson Hamiltonian
4.2.3 Quasiparticle Representation
4.3 Spin Wave Analysis of Quasi-1D Ferrimagnets
4.3.1 Linear Spin Wave Approximations
4.3.2 Spin Wave Interactions
4.3.3 Comparison with Numerical Results
4.4 Applications to 2D Heisenberg Antiferromagnets
4.4.1 Square-Lattice Antiferromagnet
4.4.2 Triangular-Lattice Antiferromagnet
4.5 Modi.ed Spin Wave Theories
4.5.1 Square-Lattice Antiferromagnet at Finite T
4.5.2 Applications to Finite-Size Systems
4.6 Concluding Remarks
Chapter 5
5.1 Introduction
5.2 Lanczos Algorithm
5.2.1 Algorithm
5.2.2 Space Group Symmetries
5.2.3 Construction of the Hilbert Space
5.2.4 Construction of the Hamiltonian Matrix
5.3 Examples of Translationally Invariant Spin Gapped Systems
5.3.1 Application to the 2D J1 - J2 Model
5.3.2 Application to Spin-Peierls Chains
5.4 Lanczos Algorithm for Non-uniform Systems: Application to Doped SP Chains
5.4.1 Doped Coupled Frustrated Spin-1 2 Chain with Four-Spin Exchange
5.4.2 Con.nement
5.4.3 E.ective Interaction
5.5 Conclusion
Chapter 6
6.1 Introduction
6.2 Path Integral for Spin Systems
6.3 E.ective Action for Antiferromagnetic Spins Chains
6.4 The Hamiltonian Approach
6.5 The Non-linear Sigma Model and Haldane’s Conjecture
6.6 Antiferromagnetic Spin Ladders
6.7 Chains with Alternating Bonds
6.8 The Two-Dimensional Heisenberg Antiferromagnet
6.9 Bosonization of 1D Systems
6.9.1 XXZ Chain in a Magnetic Field: Bosonization and Luttinger Liquid Description
6.9.2 Thermodynamics and Correlations
6.9.3 SU(2) Point via Non-abelian Bosonization
6.9.4 Modi.cations of the XXZ Chain
6.9.5 RG Analysis of the Scalar Field Perturbed by Vertex Operators
6.9.6 Charge Degrees of Freedom: Hubbard and t - J Models
6.9.7 Two-Leg Heisenberg Ladder
6.9.8 Higher Spin Chains: Non-abelian Bosonization
6.9.9 N-Leg Ladders in a Magnetic Field: Gap for Non-zero Magnetization
Appendix: The Scalar Boson in 2D, a c = 1 Conformal Field Theory
Chapter 7
7.1 Introduction
7.2 The CCM Formalism
7.3 The XXZ Model
7.3.1 The CCM Applied to the XXZ Model Using a z-Aligned N´eel Model State
7.3.2 The LSUB2 Approximation for the Spin-Half, Linear-Chain XXZ Model
7.3.3 The SUB2 Approximation for the Spin-Half, Linear-Chain XXZ Model
7.3.4 CCM Results for the Spin-Half Square-Lattice XXZ Model Using a z-Aligned Model State
7.3.5 CCM Results for the Spin-Half Square-Lattice XXZ Model Using a Planar Model State
7.3.6 Quantum Criticality of the Antiferromagnetic Phase Transition for the Spin-Half Square-Lattice XXZ Model
7.3.7 CCM Prediction of the Nodal Surface of the Spin-Half Square-Lattice Heisenberg Model
7.4 The J–J Model: A Square-Lattice Model with Competing Nearest-Neighbour Bonds
7.5 An Interpolating Kagom´e/Triangle Model
7.5.1 CCM Treatment of the Interpolating Kagom´e/Triangle Model
7.5.2 CCM Results for the Ground-State Properties
7.5.3 Evaluation of the Perturbation Series Using CCM
7.6 The J1–J2 Ferrimagnet
7.7 Conclusion
Chapter 8
8.1 Introduction
8.2 Integrable Exchange Hamiltonians
8.3 Lattice Path Integral and Quantum Transfer Matrix
8.4 Bethe Ansatz Equations for the Spin-1/2 XXZ Chain
8.5 Manipulation of the Bethe Ansatz Equations
8.5.1 Derivation of Non-linear Integral Equations
8.5.2 Integral Expressions for the Eigenvalue
8.6 Numerical Results for Thermodynamical Quantities
8.7 Thermal Transport
8.8 Summary
Chapter 9
9.1 Introduction
9.2 Coupled Dimer Antiferromagnet
9.2.1 Phases and Their Excitations
9.2.2 Bond Operators and Quantum Field Theory
9.2.3 Quantum Criticality
9.3 In.uence of an Applied Magnetic Field
9.3.1 Weak Fields
9.3.2 Strong Fields
9.4 Square Lattice Antiferromagnet
9.4.1 Paramagnetic Phase
4.1.1 S Even Integer
4.1.2 S Half-Odd-Integer
4.1.3 S Odd Integer
9.4.2 Critical Theory
4.2.1 Lattice Model at N = 1
4.2.2 Easy Plane Model at N = 2
9.5 Triangular Lattice Antiferromagnet
9.6 Conclusions
Chapter 10
10.1 Introduction and General Remarks
10.1.1 Antiferromagnetic Correlations and Superconductivity
10.1.2 Quantum Criticality and the Low Energy Spectrum of Quantum Spin Systems
10.1.3 Frustration
10.1.4 Orbital Related E.ects
10.2 Interplay of Structural and Electronic Properties
10.2.1 Concepts Based on Structure and Chemistry
10.2.2 Angle Dependence of Superexchange
10.3 Copper-Oxygen Coordinations
10.3.1 3D Dimerized Systems and E.ects in Large Magnetic Fields
10.3.2 2D Dimerized Systems
10.3.3 Dimerized Spin Chains
10.3.4 Triangular Lattices and Tetrahedra
10.4 Vanadium-Oxygen Coordinations
10.4.1 MV2O5 and Related Compounds with Charge Ordering dInstabilities



10.4.5 Depleted Lattices – Playing with Valences (V4+/V5+)
10.4.6 J1-J2 Model on a Square Lattice
10.4.7 Exotic Topologies
10.5 Titanium-Oxygen Coordinations
10.5.1 The Pyroxene NaTiSi2O6
10.5.2 The Bilayer System TiOCl
10.5.3 The Pyrochlore MgTi2O4
10.6 Conclusion
back-matter