Theory of Magnetism

The following sections are included:

• Dedication
• Foreword
• Acknowledgments
• Contents

The following sections are included:

• Introduction
• Paramagnetism of a free electron gas
• Paramagnetism of a system of free atoms
• Diamagnetism of many-electron atoms
• Free electron gas in a strong magnetic field: Landau diamagnetism
  • Landau's levels
  • Degeneracy of Landau's levels
  • Quantization of electron orbit
  • Diamagnetic susceptibility
• De Haas-van Alphen effect
• Conclusion
• Problems

The following sections are included:

• Introduction
• Antisymmetric wave function
• Hartree-Fock equation
  • Single-electron terms
  • Interaction terms
• Results for an interacting electron gas
  • Direct-interaction energy and exchange-interaction energy
  • Effective mass
  • Total energy of an interacting electron gas
  • Paramagnetic-ferromagnetic transition
• Conclusion
• Problems

We show in this chapter the origin of exchange interactions between spins in crystalline solids. One of the most popular models for spin-spin interactions is the Heisenberg model. This model along with other spin models such as the Ising, XY and Potts models have been largely used to study magnetic properties of materials…

This chapter is devoted to the mean-field theory as applied to ferromagnetism and antiferromagnetism. A short treatment of a simple model of ferrimagnetism is also included. The origin of the ferromagnetism and other exchange interactions have been shown in chapter 3. The mean-field theory, sometimes called molecular-field approximation, is the simplest approximation for systems of interacting spins. Though simple, its results bear essential physical properties in high dimensions as will be seen below.

Elementary excitations of a system of interacting spins have a wave nature as in many other systems such as collective vibrations of coupled atoms in crystals (phonons), waves of charge density in plasmas, etc. In spin systems, the collective excitations are called spin waves or magnons when they are quantized. Spin waves propagate in magnetically ordered systems in the way phonons do in crystalline solids. At finite temperatures, as long as the magnetic order exists, i.e. T < T_c, spin waves are the only physical process that determines low-T magnetic properties of the system…

The Green's function is a very general method in quantum field theory. It can be used for various problems in many areas such as condensed matter, nuclear physics and elementary-particle physics [50]. The general formulation is rather complicated, abstract, not suitable for a quick application. For our purpose, we present in this chapter a simplified version which, from the beginning, aims at an application to systems of interacting spins. This formulation does not need much of basic knowledge in quantum field theory. A basic level in quantum mechanics is enough to understand and to master the method…

The following sections are included:

• Introduction
  • Order parameter
  • Order of the phase transition
  • Correlation function — Correlation length
  • Critical exponents
  • Universality class
• Landau-Ginzburg theory
  • Mean-field critical exponents
  • Correlation function
  • Corrections to mean-field theory
• Renormalization group
  • Transformation of renormalization group — Fixed point
  • Scaling variables
  • Critical surfaces
  • Critical exponents
• Phase transition in some particular systems
  • Exactly solved spin systems
  • Kosterlitz-Thouless transition
• Frustrated spin systems
  • Definition
  • Phase transition
• Conclusion
• Problems

To understand methods of Monte Carlo simulation for the study of phase transitions we need a background taught in statistical physics and a programming language. The first method of simulation has been introduced in the 50's by Metropolis [96]. But simulations become popular only from the early 90's when several new methods of simulation of high precision have been introduced and at the same time powerful computers have become accessible for a large number of researchers [17]. Today, numerical simulations are considered as an investigation method which is as important as theory and experiment in the study of properties of nature. Numerical simulations serve as a complementary tool, and in many cases they are the only way to study very complex systems that theory and experiment cannot investigate properly. Numerical simulations allow us to test the validity of theoretical approximations, to compare quantitatively simulated results with experimental data, and to propose interpretations. In addition, real systems have many parameters but often only a few of them govern their main properties. Theories and experiments cannot test all of them for a simple reason that experimental realizations and theoretical calculations take time and cost. Simulations can help identify relevant mechanisms before realizing experimental setups or constructing models…

The following sections are included:

• Introduction
• Surface physics
• Surface magnetism: Generalities
  • Surface magnon modes
  • Reconstruction of surface magnetic ordering
  • Surface phase transition
• Semi-infinite solids
• Ferromagnetic films with different lattices
  • Method
  • Results
• Antiferromagnetic films
  • Films of simple cubic lattice
  • Films of body-centered cubic lattice
• Frustrated films
  • Model
  • Ground state
  • Green's function method
  • Phase transition and phase diagram of the quantum case
  • Monte Carlo results
• Conclusion
• Problems

The magnetic phase transition is experimentally known to give rise to an anomalous temperature-dependence of the electron resistivity in ferromagnetic crystals. Phenomenological theories based on the interaction between itinerant electron spins and lattice spins have been suggested to explain these observations. In this chapter, we recall these theories and introduce a method of Monte Carlo simulation to study the behavior of the resistivity of the spin current as a function of temperature (T) from low-T ordered phase to high-T paramagnetic phase in ferromagnets and antiferromagnets. Frustrated systems and thin films are also shown. As a demonstration of its efficiency, we show the results obtained for MnTe…

The following sections are included:

• Solutions of problems of chapter 1
• Solutions of problems of chapter 2
• Solutions of problems of chapter 3
• Solutions of problems of chapter 4
• Solutions of problems of chapter 5
• Solutions of problems of chapter 6
• Solutions of problems of chapter 7
• Solutions of problems of chapter 8
• Solutions of problems of chapter 9
• Solutions of problems of chapter 10

The following sections are included:

• Appendix A. Introduction to Statistical Physics
  • A.1 Introduction
  • A.2 Isolated systems: Microcanonical description
    • A.2.1 Fundamental postulate
    • A.2.2 Applications
  • A.3 Systems at constant temperature: Canonical description
    • A.3.1 Applications
  • A.4 Open systems at constant temperature: Grand-canonical description
    • A.4.1 Applications
  • A.5 Fermi-Dirac and Bose-Einstein statistics
  • A.6 Phase space — Density of states
• Appendix B. A Simple Monte Carlo Program for Ising Model

The following sections are included:

• Bibliography
• Index