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Dynamics of Quantum Spin Systems Spins embody intrinsic quantum mechanical degrees of freedom capable of manifesting collective behaviors absent in classical systems. This unique attribute of spin systems has prompted numerous efforts aimed at unveiling novel quantum states of matter, such as topologically ordered spin liquids. Understanding spin dynamics offers a systematic method for uncovering new states of matter, as these states are frequently preceded by anomalous excitations within magnetically ordered states. In this series of three lectures, I will introduce a recently developed generalization of classical and semiclassical approaches to the dynamics of quantum spin systems. A notable advantage of classical approaches is their numerical efficiency, with costs scaling linearly in the number of degrees of freedom, contrasting the exponential cost of solving the exact quantum mechanical problem. Specifically, my aim is to extend the classical limit of quantum spin systems to encompass a broad range of materials whose semiclassical behavior cannot be adequately captured by the traditional largeS limit. This generalization involves employing coherent states of SU(N) instead of the traditional SU(2) coherent states and considering an infinite sequence of (degenerate) irreducible representations of SU(N) labeled by the parameter λ₁. A generalized classical limit is attained by taking the limit λ₁ → ∞. During the first two lectures, I will introduce the generalized classical limits of quantum spin systems and discuss the novel linear and nonlinear dynamics emerging from the generalized classical equations of motion. The emergence of CPN1 skyrmions in realistic quantum spin models will illustrate the novel nonlinear dynamics. Additionally, we will explore how the generalized classical dynamics provides a formal scheme for renormalizing the traditional (largeS) classical limit. The third lecture will be dedicated to the largeN approaches arising from our generalization of the classical limit. As demonstrated in this lecture, a generalized largeN limit is achieved by fixing λ₁ = 1 and sending N (the group) to infinity. We will observe that largeN treatments complement semiclassical methods and become indispensable in the vicinity of "quantum melting points". Zoom Link: https://tennessee.zoom.us/j/4991007715 Host: Christopher Fryer and Enrique Batista (CNLS) 