Two-dimensional conformal field theory and the butterfly effect

TL;DR

The paper analyzes chaos in two-dimensional conformal field theories by studying out-of-time-order four-point functions, focusing on the large-$c$ Virasoro identity block. By analytically continuing Euclidean correlators to the second sheet, the authors show that the Virasoro identity block reproduces bulk AdS$_3$ shock-wave results, yielding a scrambling-time behavior $t_* = (\beta/2\pi) \log c$ with a delay set by the operator energies $E_w$ and $E_v$. The key result is a concrete expression for the one-sided OTOC decay after scrambling: $\frac{\langle W(t+i\epsilon_1)V(i\epsilon_3) W(t+i\epsilon_2)V(i\epsilon_4)\rangle_\beta}{\langle W(i\epsilon_1) W(i\epsilon_2)\rangle_\beta \langle V(i\epsilon_3) V(i\epsilon_4)\rangle_\beta} \approx \left(1 + \frac{24\pi i h_w}{\epsilon_{12}^* \epsilon_{34}} e^{\frac{2\pi}{\beta}(t - t_* - x)}\right)^{-2 h_v}$. The discussion addresses limitations (stringy corrections, non-identity blocks) and suggests extensions to multi-point correlators and tensor-network perspectives. Overall, the work provides a field-theoretic route to fast scrambling in 2d CFTs that aligns with holographic gravity results and clarifies the roles of identity-block contributions and high-spin exchanges in chaotic dynamics.

Abstract

We study chaotic dynamics in two-dimensional conformal field theory through out-of-time order thermal correlators of the form $\langle W(t)VW(t)V\rangle$. We reproduce bulk calculations similar to those of [1], by studying the large $c$ Virasoro identity block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\sim t_* - \fracβ{2π}\log β^2E_w E_v$, where $t_*$ is the scrambling time $\fracβ{2π}\log c$, and $E_w,E_v$ are the energy scales of the $W,V$ operators.

Figures (2)

• 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 AdS${}_3$ (grey) covered by $x,t$. The union of the grey and yellow regions are the Poincare patch covered by $z,\bar{z}$.
• Figure 2: The paths taken by the cross ratio $z$ during the continuations corresponding to (from left to right) $\langle WVWV\rangle$, $\langle WWVV\rangle$, and $\langle WVVW\rangle$. Only in the first case does the path pass around the branch point at $z = 1$. Our choice of $t >x>0$ breaks the symmetry between $z$ and $\bar{z}$, and the corresponding figures for $\bar{z}$ are very boring: the $\bar{z}$ coordinate never circles one.