Four-point function in the IOP matrix model

TL;DR

The paper studies chaos diagnostics in tractable large-N matrix oscillator models (IP and IOP) by computing planar four-point functions and analyzing out-of-time-order correlators. It shows that, despite ladder resummations and analytic control, both models fail to exhibit exponential OTO growth in the planar limit, indicating a lack of chaos under these assumptions. The results suggest that simple single-flavor, weakly interacting setups with heavy fundamentals do not reproduce black-hole-like chaotic dynamics, and point to required ingredients (e.g., enhanced self-interactions or multiple flavors) for chaos in holographic analogs.

Abstract

The IOP model is a quantum mechanical system of a large-$N$ matrix oscillator and a fundamental oscillator, coupled through a quartic interaction. It was introduced previously as a toy model of the gauge dual of an AdS black hole, and captures a key property that at infinite $N$ the two-point function decays to zero on long time scales. Motivated by recent work on quantum chaos, we sum all planar Feynman diagrams contributing to the four-point function. We find that the IOP model does not satisfy the more refined criteria of exponential growth of the out-of-time-order four-point function.

Figures (10)

• Figure 1: The basic graphical unit of the Hamiltonian (\ref{['eq:Liu']}) studied in Liu.
• Figure 2: The basic graphical unit of the IP model (\ref{['HIP']}) studied in IP. It is like the diagram in Fig. \ref{['fig:FLd']}, but cut in half. A single line is a fundamental, a double line is an adjoint.
• Figure 3: The basic graphical unit of the SYK model (\ref{['SYKH']}). The solid lines are fermions $\chi_i$, the dotted line is the coupling $J_{j k l m}$.
• Figure 4: The dashed lines indicate $J_{j k l m}$, while the sold lines are the fermions $\chi_i$. Treating $J_{j k l m}$ as a quantum field, the quantum corrections to the two-point function are suppressed by $1/N^3$.
• Figure 5: The Schwinger-Dyson equation for the propagator $G(\omega)$ in the IP model, in the planar limit. Arrows point from creation operators toward annihilation operators. A single line denotes the free propagator $G_0(\omega)$, a line with a shaded box is the dressed propagator $G(\omega)$, and a double line is the adjoint propagator $K(\omega)$. Iterating generates a sequence of nested rainbow diagrams.