Many-body chaos at weak coupling

TL;DR

This work computes the chaotic growth rate $λ_L$ for a weakly coupled large-$N$ matrix φ^4 theory at finite temperature by summing ladder diagrams and solving a resulting integral equation for an on-shell momentum distribution. The dominant contribution arises from a ladder series whose largest eigenvalue sets $λ_L$, found numerically to scale as $λ_L frac{~}{~} λ^2$ with a pronounced infrared sensitivity $~1/m$ for small mass, and enhanced scaling $λ^{3/2}/β$ when the tree-level mass vanishes due to thermal mass effects. The analysis clarifies the relationship between quantum chaos indicators and classical collision dynamics, and situates the weakly coupled results alongside BFKL-like analyses and strong-coupling (Kitaev) results, highlighting both universal features and theory-specific differences. The study also outlines corrections in coupling and $1/N$ that could modify the growth and saturation behavior of $C(t)$, suggesting rich avenues for future work across weakly coupled theories.

Abstract

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

Figures (7)

• Figure 1: The squared commutator (\ref{['tocompute']}) can be expanded to four terms represented by the path integral contours shown. The vertical segment (ends should be identified) represents the imaginary-time circle, and the horizontal folds implement the real time evolution to produce $\Phi_{ab}(t)$. The two folds are separated by half of the thermal circle.
• Figure 2: When one interaction vertex is integrated over one of the folds, we get a self-energy correction for one of the retarded propagators. The seven other terms on the LHS differ in whether the vertex is on the top piece or bottom piece of the fold, and in the arrangement of the external operators. The sum (with signs) turns all horizontal lines into retarded propagators.
• Figure 3: When both vertices in the $O(\lambda^2)$ correction are integrated over the same fold, we get the two self-energy diagrams shown in (a). When the vertices are integrated over different folds, we get additional self-energy corrections and then, finally, the one-rung diagram shown at right in (b). For this diagram, we emphasize which propagators are retarded and which are Wightman.
• Figure 4: The sum over indices in $C(t)$ is equivalent to contracting the external operators in the four point function with semicircle caps. The one rung diagram then has three index structures that each contribute $16N^4$, where $16 = 4\cdot 4$ is a combinatoric factor that arises from the possibility of "rotating" each of the vertices in the plane of the diagram. The total factor is $48N^4$; dividing by $N^4$ to turn the sum into an average, we get 48.
• Figure 5: Ladder diagrams for $f(\omega,k)$ are shown. In these diagrams, diagonal lines are dressed retarded propagators, and the loops are Wightman correlators $\tilde{G}$. Frequency $\omega$ flows into the diagram from the left corner, which has an implied sum over momenta. The shaded blob represents the full $f$. It satisfies the recursion relation shown on the bottom line. At large $t$, the zero-rung term on the right hand side can be ignored, so we have a homogeneous equation that states that the large-time behavior of the sum is unchanged if we add one extra rung.