1 Introduction ................................................. 1
References ................................................... 5
2 Spin Models .................................................. 7
2.1 Spin Angular Momentum ................................... 7
2.2 Coupled Spins .......................................... 10
2.3 Two Interacting Spin-1/2's ............................. 11
2.4 Commutators and Quantum Numbers ........................ 14
2.5 Physical Picture ....................................... 16
2.6 Infinite Arrays of Spins ............................... 16
2.7 ID Heisenberg Chain with S = 1/2 and Nearest-Neighbour
Interaction ............................................ 18
References .................................................. 19
3 Quantum Treatment of the Spin-1/2 Chain ..................... 21
3.1 General Remarks ........................................ 21
3.2 Aligned State .......................................... 22
3.3 Single Deviation States ................................ 23
3.4 Two Deviation States ................................... 27
3.4.1 Form of the States .............................. 33
3.5 Three Deviation States ................................. 36
3.5.1 Bethe Ansatz for SZ/T= N/2 - 3 .................. 36
3.6 States with an Arbitrary Number of Deviations .......... 37
Reference ................................................... 38
4 The Antiferromagnetic Ground State .......................... 39
4.1 The Fundamental Integral Equation ...................... 39
4.2 Solution of the Fundamental Integral Equation .......... 43
4.3 The Ground State Energy ................................ 45
References .................................................. 47
5 Antiferromagnetic Spin Waves ................................ 49
5.1 The Basic Formalism .................................... 49
5.2 Magnetic Field Behaviour ............................... 57
References .................................................. 59
6 The XY Model ................................................ 61
6.1 Introduction ........................................... 61
6.2 Change from Spin Operators to Fermion Operators ........ 62
6.3 Fourier Transform ...................................... 68
6.4 Quasiparticle Operators ................................ 69
6.5 Quasiparticle Energies ................................. 71
6.6 Ground State Energy of the XY-Model .................... 73
References .................................................. 75
7 Spin-Wave Theory ............................................ 77
7.1 Introduction ........................................... 77
7.2 Ferromagnetic Spin-Wave Theory ......................... 80
7.3 Antiferromagnetic Spin-Wave Theory ..................... 82
References .................................................. 88
8 Numerical Finite-Size Calculations .......................... 89
8.1 Introduction ........................................... 89
8.2 A Simple Example ....................................... 90
8.3 Results in 1D .......................................... 92
8.4 Results in 2D .......................................... 95
References .................................................. 96
9 Other Approximate Methods ................................... 99
9.1 Introduction ........................................... 99
9.2 Variational Method ..................................... 99
9.3 Variational Monte Carlo Method ........................ 101
9.4 The Green Function Monte Carlo Method ................. 104
9.5 Perturbation Theory ................................... 105
References ................................................. 108
10 The Coupled Cluster Method ................................. 109
10.1 Introduction .......................................... 109
10.2 The CCM Formalism ..................................... 110
10.3 The XXZ-Model ......................................... 115
10.3.1 The LSUB2 Approximation for the Spin-Half,
Square-Lattice XXZ-Model for the z-Aligned
Model State .................................... 117
10.3.2 The SUB2 Approximation for the Spin-Half,
Square-Lattice XXZ-Model of the z-Aligned
Model State .................................... 122
10.3.3 High-Order CCM Calculations Using a
Computational Approach ......................... 125
10.3.4 Excitation Spectrum of the Spin-Half Square-
Lattice XXZ-Model for the z-Aligned Model
State .......................................... 127
10.4 The Lattice Magnetisation ............................. 129
References ................................................. 134
11 Quantum Magnetism .......................................... 135
11.1 Introduction .......................................... 135
11.2 One-Dimensional Models ................................ 136
11.2.1 The Spin-Half J1-J2 Model on the Linear
Chain .......................................... 136
11.2.2 The s = 1 Heisenberg Model on the Linear
Chain .......................................... 138
11.2.3 The s = 1 Heisenberg-Biquadratic Model on the
Linear Chain ................................... 139
11.3 The s = 1/2 Heisenberg Model for Archimedean
Lattices .............................................. 140
11.4 Spin Plateaux ......................................... 145
11.5 The Spin-Half J1-J2 Model on the Square Lattice ....... 146
11.6 The Shastry-Sutherland Antiferromagnet ................ 147
11.7 Conclusions .......................................... 149
References ................................................. 150
Index ......................................................... 153
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