Phase space geometry of collective spin systems: Scaling and Fractality

TL;DR

This paper investigates the phase space structure of collective spin systems using finite-size scaling of the inverse participation ratio, revealing fractal dimensions linked to classical dynamics and critical points, with implications for quantum technologies.

Contribution

It introduces a finite-size scaling mass exponent to identify power-law behaviors and assign fractal dimensions to coherent states in collective spin systems.

Findings

Fractal dimensions vary with classical dynamics and critical points.

Three behaviors of fractal dimension in the Quantum Kicked Top.

Finite-size scaling aids in understanding quantum-classical correspondence.

Abstract

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 critical 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 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 critical 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.