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arXiv:2502.15169 · Physical review. E

Phase space geometry of collective spin systems: Scaling and fractality

Miguel Gonzalez, M. A. Bastarrachea-Magnani, Jorge G. Hirsch

Published Jun 6, 2025• 4 citations
arXivHTMLPDF

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 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.

DOI
10.1103/784q-xqmm

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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.
We study spin squeezing, an intrinsically quantum property, in the quantum kicked top model exhibiting chaotic behaviours in the classical limit. We show that spin squeezing can reveal the underlying chaotic and regular structures in phase space given by a sphere with a unit radius, namely, spin squeezing vanishes after a very short time for an initial spin coherent state centred in a chaotic region of the phase space, whereas spin squeezing occurs over a long period of time for the coherent state centred in an elliptic fixed point. We also study the distribution of the mean spin directions when quantum dynamics take places and give a comparison between the dynamics of spin squeezing and quantum entanglement.
Quantum coherence reflects the origin of quantumness and might be capable of extracting the subtle nature of a system. We investigate the ground-state coherence and steered coherence in the Lipkin-Meshkov-Glick model and show that they detect faithfully the quantum phase transitions of this model. Moreover, we carry out scaling analysis on the coherence and steered coherence by means of the continuous unitary transformation method and find that the scaling exponents are uniquely determined by the phase region of this model. These results may provide useful insights into the mechanism underlying quantum criticality in many-body systems.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
The quantum kicked top (QKT) is one of the most widely studied models in quantum chaos, providing a minimal yet powerful framework for exploring the relationship between classical nonlinear dynamics and quantum behavior. Unlike many chaotic systems with infinite-dimensional Hilbert spaces, the QKT possesses a finite-dimensional Hilbert space, making it analytically and numerically controllable while still showing a rich dynamical phenomena. In this chapter, we present a comprehensive introduction to the QKT as a paradigmatic model of quantum chaos. Starting from the classical kicked top, we derive the discrete nonlinear map governing the dynamics on the unit sphere and analyze its phase space structure through fixed points, stability analysis, bifurcations and Lyapunov exponents. We then discuss the role of symmetries, including rotational and time-reversal symmetry, and how their breaking modifies the dynamics. The quantum description is developed using Floquet theory, where the periodically driven spin system is represented by a unitary Floquet operator acting on a (2j+1)(2j+1)(2j+1)-dimensional Hilbert space. Within this framework, signatures of quantum chaos such as spectral statistics, entanglement generation and recurrences are discussed. The model also admits an interpretation as a system of interacting qubits, enabling explicit few-qubit realizations and direct connections with quantum information measures through reduced density matrices and entanglement entropy. By linking classical phase space structures with quantum dynamical indicators, the QKT provides a clear setting to investigate the emergence of chaotic behavior in the semiclassical limit. The chapter, therefore, highlights the quantum kicked top as a bridge between nonlinear classical dynamics, quantum chaos and modern quantum information science.
Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.
We analyze the interplay of chaos, entanglement, and decoherence in a system of qubits whose collective behavior is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement depends on the support of the initial state on regular versus chaotic Floquet eigenstates, whose phase-space distributions are concentrated on the corresponding regular or chaotic eigenstructures. We include the effect of decoherence via a realistic model and show that the signatures of chaos in the entanglement dynamics persist in the presence of decoherence. In addition, the classical chaos affects the decoherence rate itself.