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When we think of magnetism, the familiar ferromagnet often comes to mind. Yet magnetism hosts far richer phenomena: antiferromagnets with alternating spins, topological textures like skyrmions, and the newly discovered altermagnets. Beyond these ordered states, quantum fluctuations can even prevent spins from settling altogether, giving rise to exotic quantum spin liquids. Magnetism, therefore, is not just a simple alignment of spins, but a vast playground of diverse and fascinating quantum phenomena.
In this lecture, we explore the theoretical foundations of the different forms of magnetism. We examine the quantum mechanical origins of magnetic interactions and provide an overview of the collective magnetic states that can emerge at the macroscopic level. Key mathematical results, such as the Mermin–Wagner theorem and the Lieb–Schultz–Mattis–Hastings theorem, are introduced and discussed in the context of quantum magnets. We then present an introduction to spin-wave theory and its applications, before turning to the theory of quantum spin liquids and the concept of topological order.
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