Phase space geometry of collective spin systems: Scaling and fractality

We examine the scaling of the inverse participation ratio of spin coherent states in the energy basis of three collective spin systems: a bounded harmonic oscillator, the Lipkin-Meshkov-Glick model, and the quantum kicked top. The finite-size quantum probing provides detailed insights into the structure of the phase space, particularly the relationship between fixed points in classical dynamics and their quantum counterparts in collective spin systems. We introduce a finite-size scaling mass exponent that makes it possible to identify conditions under which a power-law behavior emerges, allowing one to assign a fractal dimension to a coherent state. For the quantum kicked top, the fractal dimension of coherent states-when well-defined-exhibits three general behaviors: one related to the presence of fixed points and two associated with regular and chaotic dynamics. The finite-size scaling analysis paves the way toward exploring collective spin systems relevant to quantum technologies within the quantum-classical framework.