Characterization of Phase Transitions in a Lipkin-Meshkov-Glick Quantum Brain Model

In this work we analyze the emergence of phase transitions in a quantum brain model inspired by the Lipkin-Meshkov-Glick framework, where biologically motivated synaptic feedback modulates the collective interaction in a nonlinear and state-dependent manner. We demonstrate that incorporating this retroactive mechanism substantially reshapes the phase structure, yielding an expansion of the paramagnetic phase at the expense of the ferromagnetic phases relative to the feedback-free scenario. This effect is markedly enhanced in the presence of a longitudinal field, as the feedback couples directly to the longitudinal magnetization, leading to an appreciable displacement of the critical boundaries. We characterize the ensuing transitions from a phase-space perspective by means of the ground-state Husimi distribution and the Wehrl entropy, which provide a robust diagnosis of qualitative changes in localization and enable a quantitative assessment of feedback-induced deformations of the phase diagram. Additionally, we perform an explicit dynamical analysis based on mean-field equations for the collective-spin orientation self-consistently coupled to the synaptic dynamics, which reproduces with high fidelity the quantum time evolution of collective observables for the protocols considered. Overall, these findings substantiate the suitability of this quantum brain model as a controlled theoretical framework for elucidating how synaptic plasticity mechanisms can parametrically tune and reshape collective criticality.

💡 Research Summary

This paper introduces a biologically motivated synaptic feedback mechanism into the Lipkin‑Meshkov‑Glick (LMG) model, thereby constructing a “quantum brain” framework that captures activity‑dependent modulation of collective spin interactions. The Hamiltonian is written as H(r)=−λ₀ r N(Jₓ²+γJ_y²)−hJ_z, where the feedback variable r(t) scales the overall coupling strength λ₀ and evolves according to a differential equation that mimics short‑term synaptic depression. A second variable U(t) models facilitation, completing a set of three coupled equations that describe the quantum state |ψ(t)⟩, the fatigue r(t), and the facilitation U(t). The model reduces to the standard LMG Hamiltonian when r=1 (τ_r→0), but for realistic τ_r, τ_f the interaction strength becomes a dynamical, state‑dependent quantity.

The authors first perform a semiclassical analysis using spin‑j coherent states |ζ⟩, obtaining an energy surface E(θ,φ)=−(λ/4)sin²θ f(φ)−(h/2)cosθ with f(φ)=cos2φ+γ sin2φ. Stationary points of this surface define the classical phases: a paramagnetic region (m≈0) and ferromagnetic regions (m≠0). Incorporating feedback shifts the location of these stationary points: the paramagnetic phase expands while the ferromagnetic lobes contract, especially when a longitudinal field h is present because the feedback couples directly to the longitudinal magnetization m_z.

To diagnose quantum phase transitions, the ground‑state Husimi distribution Q(ζ)=|⟨ζ|ψ₀⟩|² is computed, and its Wehrl entropy W=−∫Q ln Q dμ, together with Rényi‑Wehrl entropies W_ν, are evaluated. In the paramagnetic phase Q is nearly uniform over the Bloch sphere, giving W≈ln(2j+1), the maximal value. In ferromagnetic phases Q splits into one or several localized packets, causing a sharp drop in W. Near critical lines the entropy exhibits non‑analytic behavior (discontinuities or steep gradients), providing a clear marker of first‑, second‑, or third‑order quantum phase transitions. The authors map out the phase diagram for various (λ,γ,h) and feedback parameters, showing that decreasing the average feedback strength r̄ moves the critical line toward larger λ and that a stronger longitudinal field amplifies this shift.

A dynamical mean‑field approach is then derived by projecting the quantum dynamics onto the coherent‑state manifold, yielding coupled equations for the Bloch angles (θ,φ) together with the feedback variables r(t) and U(t). Numerical integration of these mean‑field equations is benchmarked against exact quantum time evolution obtained by diagonalizing the full Hamiltonian for moderate system sizes. The comparison shows excellent agreement, confirming that the mean‑field description captures the essential quantum correlations in the regimes of interest. Parameter sweeps reveal that instantaneous feedback (τ_r→0) produces a nearly linear displacement of the critical line, while longer facilitation times τ_f allow transient restoration of ferromagnetic order before the longitudinal field suppresses it again.

The discussion connects these findings to biological synaptic plasticity, emphasizing that activity‑dependent modulation of collective interactions can keep the system near criticality, a feature thought to be advantageous for information processing in the brain. The paper also highlights the utility of phase‑space tools (Husimi distribution, Wehrl entropy) as intuitive and quantitative diagnostics for quantum critical phenomena. Finally, the authors note that the model’s ingredients—collective spin interactions, tunable longitudinal fields, and classical feedback loops—are compatible with current quantum hardware platforms (superconducting qubits, trapped ions), opening avenues for experimental realization and extensions to multilayer or networked quantum brain architectures.

In summary, by embedding a biologically inspired, state‑dependent feedback into the LMG model, the authors demonstrate a controllable reshaping of quantum phase diagrams, provide robust phase‑space diagnostics of the resulting transitions, and validate a mean‑field dynamical framework that faithfully reproduces full quantum dynamics. This work bridges concepts from neuroscience, many‑body physics, and quantum information, offering a versatile theoretical laboratory for exploring how synaptic plasticity can parametrically tune collective quantum criticality.