Scrambling in Yang-Mills

TL;DR

This paper models the large N Yang-Mills theory as a graph-based Hamiltonian to analyze how quickly information scrambles and equilibrates, confirming fast scrambling behavior at weak coupling.

Contribution

It introduces a graph-based Hamiltonian framework for large N Yang-Mills theory to study scrambling and equilibration dynamics.

Findings

Scrambling time aligns with the fast scrambling conjecture.

System exhibits equilibration with relaxation time proportional to p/λ.

Typical graph structure determines the dynamics and scrambling behavior.

Abstract

Acting on operators with a bare dimension $\Delta\sim N^2$ the dilatation operator of $U(N)$ ${\cal N}=4$ super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees of freedom are associated with the vertices of the graph while edges correspond to terms in the Hamiltonian. The graph has $p\sim N$ vertices. Using this Hamiltonian, we study scrambling and equilibration in the large $N$ Yang-Mills theory. We characterize the typical graph and thus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in a time consistent with the fast scrambling conjecture. Further, the system exhibits a notion of equilibration with a relaxation time, at weak coupling, given by $t\sim{p\over\lambda}$ with $\lambda$ the 't Hooft coupling.