Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory

We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form $⟨W(t)VW(t)V⟩$. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large $c$ Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\sim t_{*}-(\beta /2\pi )\log {\beta }^2{E}_w{E}_v$, where $t_{*}$ is the fast scrambling time $(\beta /2\pi )\log c$ and $E_w,E_v$ are the energy scales of the $W,V$ operators.

Figure 1

Left: The spacetime arrangement of the $W$ and $V$ operators. Right: Their locations after the conformal mapping, viewed in the Rindler patch on the boundary of ${\mathrm{AdS}}_3$ (gray) covered by $x,t$. The union of the gray and yellow regions is the Poincaré patch covered by $z,\overline{z}$.

Figure 2

The paths taken by the cross ratio $z$ during the continuations corresponding to (from left to right) $⟨WVWV⟩$, $⟨WWVV⟩$, and $⟨WVVW⟩$. Only in the first case does the path pass around the branch point at $z=1$.