Many-body chaos at weak coupling (1512.07687v2)

Abstract: The strength of chaos in large $N$ quantum systems can be quantified using $\lambda_L$, the rate of growth of certain out-of-time-order four point functions. We calculate $\lambda_L$ to leading order in a weakly coupled matrix $\Phi^4$ theory by numerically diagonalizing a ladder kernel. The computation reduces to an essentially classical problem.

Many-body Chaos at Weak Coupling

The paper "Many-body Chaos at Weak Coupling" by Douglas Stanford presents a quantitative paper of chaos in large $N$ quantum systems using $\lambda_L$, the Lyapunov exponent that measures the rate of growth of certain out-of-time-order correlators (OTOCs). The focus is on calculating $\lambda_L$ to leading order in a weakly coupled matrix $\Phi^4$ theory at finite temperature.

The paper leverages a classical analogy, transforming the computation into a classical problem by using a ladder kernel that is numerically diagonalized. The central aim is to evaluate $\lambda_L$ for a single Hermitian matrix field $\Phi_{ab}$ in four-dimensional spacetime, governed by a Lagrangian density with a $\Phi^4$ interaction term.

The analysis begins with the introduction of chaos indicators through non-time-ordered four-point functions, specifically the square of a commutator $c(t)$, which characterizes the butterfly effect. This is described using a thermal expectation framework where the function $c(t)$ is expected to demonstrate a period of exponential growth. $\lambda_L$ provides a quantitative measure of this chaos, bounded by $\lambda_L \le 2\pi/\beta$.

In the context of a weakly coupled quantum field theory, the paper introduces the 't Hooft coupling $\lambda = g^2 N$ with the goal of computing $\lambda_L$ to leading order in this coupling. The approach involves evaluating thermal Feynman diagrams for an index-averaged squared commutator. Notably, the paper adapts methods from high energy scattering theory, drawing parallels to BFKL analysis.

A crucial aspect of the computation involves restricting to planar diagrams and focusing on the fastest-growing time contributions at each perturbative order. This leads to summing only ladder diagrams, facilitating a numerical eigenvalue computation to extract $\lambda_L$. Through this process, the paper reports a significant numerical result: for small bare mass $m$, $\lambda_L$ is found to be proportional to $\lambda^2/m\beta^2$, highlighting the dominance of low-energy quantum collisions.

This paper suggests several theoretical and practical implications:

1. Theoretical Insights: The findings are linked to many-body dynamics in quantum systems, drawing parallels to classical epidemic models where chaos spreads through particle collisions. This perspective enhances our understanding of non-equilibrium phenomena in quantum chromatography and field theory.
2. Thermal and Mass Sensitivity: The numerical result indicating a $1/m$ scaling for $\lambda_L$ suggests an IR divergence, emphasizing that chaos at weak coupling is influenced significantly by low energy quanta rather than thermal-scale modes.
3. Quantum Field Comparison: The paper contrasts quantum field theory chaos with holographic models, where similar ladder diagrams yield exact results, saturating known bounds. It posits that weakly coupled gauge theories on hyperbolic space exhibit $\lambda_L \propto \lambda$, yet suggests that finite temperature flat-space scenarios lack this property due to the nature of cubic vertex interactions.
4. Potential Extensions: The paper opens pathways for further exploration across different quantum field theories, particularly focusing on the spatial growth of operator commutators and their diffusive or ballistic propagation characteristics.

Future developments could involve exploring these dynamics within more complex field theories, studying the saturation of chaos indicators in lattice simulations, or applying insights to condensed matter physics where similar quantum chaotic phenomena may manifest.

Overall, Stanford's paper builds upon the groundwork laid by previous studies on quantum chaos and extends it into a weak coupling regime, reinforcing the significance of perturbative methodologies in calculating Lyapunov exponents and characterizing chaos in quantum systems.