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

Abstract

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.

Figures (9)

• Figure 1: Phase diagram of the LMG based brain model for $\tau_r\rightarrow 0$ (that is the standard LMG model), $h=0$ (left), $h=0.5$ (center) and $h=1$ (right). Different colors corresponds to different phases with the code explained in the color bar on the right corresponding to FMX ($m_x\neq 0$), FMY ($m_y\neq0$) and PM ($m_x=m_y=0$ and $m_z\neq 0$))
• Figure 2: Characterization of the quantum phase transitions of the LMG model ($\tau_r=0$) in terms of the Wehrl entropy, clearly depicting strong changes in the phase-space localitation associated to the quantum phase transitions illustrated in Fig. \ref{['figura1']} for $h=0$.
• Figure 3: Characterization of the quantum phase transitions of the LMG model ($\tau_r=0$) in terms of the Wehrl entropy, clearly depicting strong changes in the phase-space localitazion associated to the quantum phase transitions illustrated in Fig. \ref{['figura1']} for $h=0.5$.
• Figure 4: Characterization of the quantum phase transitions of the LMG model ($\tau_r=0$) in terms of the Wehrl entropy, clearly depicting strong changes in the phase-space localitazion associated to the quantum phase transitions illustrated in Fig. \ref{['figura1']} for $h=1$.
• Figure 5: Phase diagram of the LMG model with synaptic feedback (the so-called quantum brain model) for $h=0$ (left), $h=0.5$ (center) and $h=1$ (right). In all situations $\tau_r=10,$$\mathcal{U}=0.5$ and $\tau_f=0.$