Real Time Dynamics in Low Point Correlators in 2d BCFT

TL;DR

The paper investigates whether boundary effects in 2d BCFT can encode higher-point chaotic dynamics in lower-point correlators. It develops analytic control using the Virasoro identity block in large-c CFTs and the doubling trick to study BCFT three-point and two-point structures, identifying scrambling-type time dependence and persistent pole-skipping in BCFT contexts. The results show that 3-point BCFT configurations can reproduce scrambling behavior with a scrambling time t* = (beta/(2*pi)) log c and that pole-skipping persists for certain BCFT 2-point setups, linking to maximal Lyapunov exponents. Together, these findings suggest higher-point chaotic information may be encoded in lower-point BCFT correlators via analytic structure and boundary conditions.

Abstract

In this article, we demonstrate how a 3-point correlation function can capture the out-of-time-ordered features of a higher point correlation function, in the context of a conformal field theory (CFT) with a boundary, in two dimensions. Our general analyses of the analytic structures are independent of the details of the CFT and the operators, however, to demonstrate a Lyapunov growth we focus on the Virasoro identity block in large-c CFT's. Motivated by this, we also show that the phenomenon of pole-skipping is present in a 2-point correlation function in a two-dimensional CFT with a boundary. This pole-skipping is related, by an analytic continuation, to the maximal Lyapunov exponent for maximally chaotic systems. Our results hint that, the dynamical content of higher point correlation functions, in certain cases, may be encrypted within low-point correlation functions, and analytic properties thereof.

Figures (7)

• Figure 1: The simplest configuration in which a $2$-point correlator becomes a $4$-point correlator, in the presence of a boundary. The horizontal line, on which ${\rm Im}(z) =0$, denotes the boundary; ${\cal O}_1$ and ${\cal O}_2$ are two operators in the upper half plane (UHP). These operators have corresponding mirror images $\tilde{{\cal O}_1}$ and $\tilde{{\cal O}_2}$, located appropriately in the lower half plane. This particular $2$-point correlator, however, cannot be written as a $4$-point OTOC.
• Figure 2: The non-trivial $3$-point correlator which becomes a $4$-point correlator, in the presence of a boundary. The horizontal line, as before, denotes the boundary; ${\cal O}_1$ and ${\cal O}_2$ are two operators inserted on the boundary, and $\phi$ is an operator inserted in the bulk. The latter one has a corresponding mirror image $\tilde{\phi}$, located appropriately in the lower half plane. This particular $3$-point correlator in the BCFT can be written as a non-trivial $4$-point OTOC in the plane. Here, $z_1$ and $z_2$ both lie on the ${\rm Im}(z) =0$ line.
• Figure 3: $\mathbb{I}$ is the Virasoro identity block coming purely from the vacuum sector. Contributions from other primaries(and their descendants) belong to the sum over blocks $g_{\Delta,l}(z,\bar{z})$.
• Figure 4: On the right is shown the analytic continuation for the TOC cases while the diagram on th left shows the analytic continuation in the OTOC case.
• Figure 5: The simplest $2$-point correlator in a BCFT, which yields a non-trivial pole-skipping structure. The $2$-point correlator, with this structure, becomes a $3$-point correlator.