Exponential growth of thermal out-of-time-order correlator (OTOC) is an indicator of a possible gravity dual, and a simple toy quantum model showing the growth is being looked for. We consider a system of two harmonic oscillators coupled nonlinearly with each other, and numerically observe that the thermal OTOC grows exponentially in time. The system is well-known to be classically chaotic, and is a reduction of Yang-Mills-Higgs theory. The exponential growth is certified because the growth exponent (quantum Lyapunov exponent) of the thermal OTOC is well matched with the classical Lyapunov exponent, including their energy/temperature dependence. Even in the presence of the exponential growth in the OTOC, the energy level spacings are not sufficient to judge a Wigner distribution, hence the OTOC is a better indicator of quantum chaos.

We propose a deep learning method to build an AdS/QCD model from the data of hadron spectra. A major problem of generic AdS/QCD models is that a large ambiguity is allowed for the bulk gravity metric with which QCD observables are holographically calculated. We adopt the experimentally measured spectra of ρ and a 2 mesons as training data, and perform a supervised machine learning which determines concretely a bulk metric and a dilaton profile of an AdS/QCD model. Our deep learning (DL) architecture is based on the AdS/DL correspondence [K. Hashimoto, S. Sugishita, A. Tanaka, and A. Tomiya, Phys. Rev. D 98, 046019 (2018)] where the deep neural network is identified with the emergent bulk spacetime.

It is challenging to quantify chaos of QCD, because non-perturbative QCD accompanies non-local observables. By using holography, we find that QCD strings at large Nc and strong coupling limit exhibit chaos, and measure their Lyapunov exponent at zero temperature. A pair of a quark and an antiquark separated by Lq in the large Nc QCD is dual to a Nambu-Goto string hanging from the spatial boundary of the D4-soliton geometry. We numerically solve the motion of the string after putting a pulse force on its boundaries. The chaos is observed for the amplitude of the force larger than a certain lower bound. The bound increases as Lq grows, and its dependence is well approximated by a hypothesis that the chaos originates in the endpoints of the QCD string.

Measuring chaos of QCD-like theories is a challenge for formulating a novel characterization of quantum gauge theories. We define a chaos phase diagram of QCD allowing us to locate chaos in the parameter space of energy of homogeneous meson condensates and the QCD parameters such as pion/quark mass. We draw the chaos phase diagrams obtained in two ways: first, by using a linear sigma model, varying parameters of the potential, and second, by using the D4/D6 holographic QCD, varying the number of colors N c and the 't Hooft coupling constant λ. A scaling law drastically simplifies our analyses, and we discovered that the chaos originates in the maximum of the potential, and larger N c or larger λ diminishes the chaos.