Chaos in the holographic matrix models for meson and baryon

TL;DR

The paper analyzes chaos in holographic meson and baryon matrix models derived from the top-down D4/D6/$\overline{D6}$ setup, reducing the dynamics to coupled oscillators with $m_a$ and $g$ determined by holographic data. Classical chaos is quantified with Poincaré sections and Lyapunov exponents, revealing a chaos-onset with energy in the mesonic case, while the baryonic sector remains chaotic across energies; large-$N_c$ perturbation shows chaos weakens and tends to integrability as $N_c\to\infty$. Quantum chaos is studied via microcanonical and thermal OTOCs, showing temperature-driven saturation and an opposite large-$N_c$ scaling between OTOCs and their asymptotics, consistent with a suppression of scrambling at large color number. The results link chaos features to symmetry breaking/restoration and gauge–gravity duality, offering insights into hadronic dynamics and quantum gravity from a tractable holographic matrix-model framework.

Abstract

In recent years, the investigation of chaos has become a bridge connecting gravity theory and quantum field theory, especially within the framework of gauge-gravity duality. In this work, we study holographically the chaos in the matrix models for meson and baryon, which are derived from the $\mathrm{D4}/\mathrm{D6}/\overline{\mathrm{D6}}$ approach as a top-down holographic model for QCD. Since these matrix models can be simplified into coupled oscillator models with special parameters, we analyze the chaos in the resultant coupled oscillators. In the analysis of the classical chaos, we calculate numerically the orbits on the Poincaré section, the Lyapunov exponent as a function of the total energy and derive the large $N_{c}$ behavior analytically, then discuss the possible phase structure both in the mesonic and baryonic matrix models. These analyses suggest that chaos might serve as an order parameter to detect the gauge theory with spontaneous breaking or restoration of symmetry. Besides, in the analysis of the quantum chaos, we demonstrate the numerical calculation of the OTOCs and analytically derive their large $N_{c}$ behavior by using the perturbation method in quantum mechanics. The numerical calculation illustrates there is a critical temperature, as a critical energy scale, that the OTOC begins to saturate, which covers qualitatively the classical analysis of the Lyapunov exponent. And the large $N_{c}$ analytics indicates the OTOCs are suppressed by the growth of $N_{c}$. Overall, the investigation of chaos in this work may be helpful to identify common features shared by the matrix models, hadronic physics, gauge theory, quantum mechanics, and gravity theory.

Figures (13)

• Figure 1: The classical trajectory on the Poincare section of the mesonic matrix model at $x\left(t\right)=0$ with various energy $E=1,15,30,80$. The horizontal axis is $y\left(t\right)$, the vertical axis is $p_{y}=\dot{y}\left(t\right)$.
• Figure 2: The classical trajectory on the Poincare section of the baryonic matrix model at $x\left(t\right)=0$ with various energy $E=1,10$. The horizontal axis is $y\left(t\right)$ or $w\left(t\right)$, the vertical axis is $p_{y}=\dot{y}\left(t\right)$ or $p_{w}=\dot{w}\left(t\right)$.
• Figure 3: Lyapunov coefficient of the mesonic matrix model . Upper: Lyapunov coefficient as a function of time $t$. Lower: Lyapunov coefficient as a function of $E$ and $N_{c}$.
• Figure 4: Lyapunov coefficient of the baryonic matrix model . Upper: Lyapunov coefficient as a function of time $t$. Lower: Lyapunov coefficient as a function of $E$ and $N_{c}$.
• Figure 5: The quantum properties of the mesonic matrix model including the microcanonical and thermal OTOCs, average of the quantum Lyapunov coefficient and eigen energies. Upper: The microcanonical OTOC $c_{n}\left(t\right)$ and thermal OTOC $C_{T}\left(t\right)$ as functions of time $t$ at various temperatures with $N_{c}=10$. Lower: Eigen energies $E_{n}$ as a function of quantum number $n$ and the average of quantum Lyapunov coefficient $L$ as a function of temperature $T$ with various $N_{c}$.