Spins represent intrinsic quantum mechanical degrees of freedom that can exhibit collective behaviors absent in classical systems. This distinct property of spin systems has motivated extensive efforts to uncover novel quantum states of matter, such as topologically ordered spin liquids. Understanding spin dynamics provides a systematic approach to identifying new states of matter, as these states are often preceded by anomalous excitations within magnetically ordered phases.
I will provide an overview of semi-classical and large-N approaches for modeling quantum spin dynamics. Specifically, I aim to broaden the classical limit of quantum spin systems to include a variety of materials whose semi-classical behavior extends beyond what is captured by the traditional large-S limit. This generalization employs SU(N) coherent states in place of the conventional SU(2) coherent states, yielding a generalized classical limit by considering a sequence of irreducible representations of a given Lie group. This novel approach not only generalizes the equations of motion but also broadens the scope of topological textures that can emerge in magnetic systems. Similarly, a generalized large-N limit is achieved by considering a sequence of Lie groups and the irreducible representations within each. These generalized large-N methods offer valuable insights, especially near quantum critical points where long-range order is continuously suppressed.

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