Phases of scrambling in eigenstates

TL;DR

This work analyzes scrambling in heavy eigenstates of holographic 2D CFTs using the monodromy method to relate heavy-light correlators to thermal blocks. It proves an ETH-like statement: for $H$ above the BTZ threshold, light operators in a heavy eigenstate behave thermally with an emergent temperature $\beta_H$, yielding maximal OTOC scrambling with $\lambda_L = 2\pi/\beta_H$; below threshold, the scrambling is non-ergodic and oscillatory due to conical-defect backgrounds. The authors develop a coordinate-transformation framework (w-plane) that isolates heavy contributions and relates $z$-plane blocks to $w$-plane blocks, with the identity block providing the thermal behavior and non-identity blocks requiring extra integration constants. They provide explicit calculations for HHLL configurations, extract the chaos exponent from Regge-limit OTOCs, and discuss the implications for bulk/boundary duality and non-ergodic phases in holographic CFTs.

Abstract

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent $2πT_{\rm eff}$. Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations.

Figures (2)

• Figure 1: Application of our main result \ref{['eq.ourResult']} to the butterfly effect ('scrambling') in heavy states, i.e. an out-of-time-order "HHLLLL" correlation function. As indicated in purple, the effect of the heavy insertions can be interpreted as dressing the propagator, $s^{-1}$, of the leading mode, the "scramblon" Kitaev-talks:2015, exchanged between the light operators on the second sheet by the exponential factor $s^{-1} = e^{-i\alpha t}$. In the ergodic phase, $\alpha = i |\alpha|$, this leads to maximal scrambling, while in the non-ergodic phase, $\alpha = |\alpha|$, this results in an oscillatory OTOC. The bulk interpretation of the scramblon picture is a single graviton exchange in the heavy background. Full details of the calculation summarized here can be found in section \ref{['sec.Lyapunov']} below.
• Figure 2: We will treat the $\mathcal{Q}$ operators as parametrically lighter than the $O_H$. We can then impose that the $O_H$ fuse to the identity operator, which, to leading order results in $\tilde{T}_{\rm cl}(w\rightarrow\infty)=O(w^{-4})$.