Study of chaos and scrambling in hairy AdS Soliton

TL;DR

This work addresses how chaos and information scrambling emerge in a 5D hairy AdS soliton, a horizonless, confining holographic background. It combines classical probes (geodesics and closed strings), minisuperspace quantisation with RMT diagnostics, and holographic OTOC/shockwave and entanglement wedge techniques to study scrambling. The main findings reveal an IR chaotic window in the string spectrum that flows to UV integrability, with a finite butterfly velocity $v_B$ and a kinematic bound $\lambda_L = 2\pi/\beta$ governing temporal scrambling in the IR; the hair parameter $\alpha$ tunes the strength and extent of chaos and the associated phase transitions. The results link confinement physics, spectral statistics, and operator growth, offering a cohesive framework for chaos in horizonless holographic backgrounds and informing phase-structure analyses in confining gauge theories. Overall, the study demonstrates how scalar hair can dynamically drive transitions between chaotic and integrable regimes and connects these to insulator/superconductor-like phase behavior in the dual theory.

Abstract

In this work, we perform a comprehensive study of the classical and quantum chaos in a candidate five-dimensional hairy AdS soliton. It is a horizonless geometry holographically dual to a confining field theory with finite scalar potential. We probe classical chaos by using particle geodesics and closed classical string. While the former shows no signature of chaos, the latter provides chaotic dynamics of the string using the Lyapunov exponent and the evolution of the Poincaré section. We perform an independent spectral analysis using the tools of the random matrix theory (RMT), namely the level space distributions and the Dyson-Mehta(DM) $Δ_{3}$-statistics. We observe a clear transition from the low energy Wigner-Gaussian Orthogonal Ensemble (GOE) distribution to the high energy Poisson distribution. This signifies a flow from quantum chaos in the infrared to integrability in the ultraviolet. We quantitatively characterize the inherent quantum scrambling in the dual theory by computing the butterfly velocity, the rate of spatial spread of the information scrambling, inside the bulk. We undergo two independent holographic methods -- entanglement wedge reconstruction and derivation of out-of-time-ordered correlators via shockwave analysis. In these methods, we heuristically consider the region near the soliton tip to provide the infrared physics of scrambling in analogy with the near-horizon region of a black hole. We find that the hair parameter controls various scrambling properties. Finally, we make comments on the interplay between insulator/superconductor phase transition in hairy soliton geometry and dynamical transition from integrability to chaos as both of these are affected by the presence of the hair parameter.

Figures (10)

• Figure 1: Time dependence of the Rindler momentum $p_{\rho}(t)$ at the parameters $\alpha=-0.1,\, x_{s}\approx 2.50,\, \lambda= 1,\, x_{0}\approx 2.30$. We set $l=10,\, m=1,\, E=5$.
• Figure 2: Poincaré sections for hairy AdS soliton for different energies at parameters: $n=1\,,\, l=10\,,\, \lambda=0.15\,,\, k=0.12\,,\,\alpha=-0.1$ with inital conditions $x(0)=1.2$ , $s(0)=0$ , $p_{x}(0)=0$.
• Figure 3: Lyapunov exponents at low and high energy values $E=0.22$ (left) and $E=2.0$(right), respectively. The other parameter values are the same as those used in Fig \ref{['fig:hairy-ads-soliton-poincare']}.
• Figure 4: Typically $f(x)=0$ admits two roots: one lying at small $x$ and the other at large $x$ at parameter values $\alpha=-1.5,\;\;l=10.2, \;\; \lambda=0.1$.
• Figure 5: For the case $\alpha = 0$ with $V_{\max} = 400$ having energy cutoffs $E^2 < 200$ and $E^2 < 400$, the upper panel presents the spectral rigidity and Dyson–Mehta statistics. The dashed blue and dotted red curves correspond to the Poisson and Wigner GOE predictions, respectively. The results show consistency with the Poisson distribution across both the low- and high-energy regimes. In the lower panel, the level-spacing distribution in the AdS soliton is depicted. Here too, for both low and high energies, the spectrum exhibits level repulsion while maintaining an overall behavior that remains close to the Poisson distribution.