Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks

TL;DR

This work analyzes the time evolution of Renyi entanglement entropy for locally excited states in 2D CFTs with large central charge, focusing on how the log-time growth coefficient $B(n,h_O)$ depends on the replica number $n$ and the excitation dimension $h_O$. Using Zamolodchikov's recursion to numerically study vacuum conformal blocks in the replicated theory ($c = n c_{CFT}$) and applying HHLL approximations in heavy-light regimes, the authors uncover a three-region phase structure separated by the value $h_O=c/32$, including a new universal formula $\,\Delta S_A^{(n)}\simeq \frac{n c}{24(n-1)} \log t$ for region (i) with $n\ge2$. They show that the vacuum block coefficients $c_n$ exhibit polynomial growth in region (i) but exponential growth in regions (ii) and (iii), with the asymptotics governed by $A$ and $\alpha$ tied to HHLL data; these results yield region-wise expressions for $B(n,h_O)$ and reveal how entanglement growth interpolates between known HHLL and small-$h_O$ limits. The findings connect entanglement dynamics to chaotic features via OTOC behavior and provide a nonperturbative, numerically supported picture of Renyi entropy evolution in holographic CFTs.

Abstract

We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of $\log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov's recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$, drastically changes when the dimensions of external primary states reach the value $c/32$. In particular, when $h_O\geq c/32$ and $n\geq 2$, we find a new universal formula $ΔS^{(n)}_A\simeq \frac{nc}{24(n-1)}\log t$. Our numerical results also confirm existing analytical results using the HHLL approximation.

Figures (8)

• Figure 1: We summarized the behaviors of the logarithmic growth of Renyi entanglement entropy $\Delta S^{(n)}_A(t)$. The green and red regions can be well approximated by the HHLL conformal block approximation.
• Figure 2: The positions of operators in the replica computation (\ref{['eq:defREE']}).
• Figure 3: The time evolution of cross ratio $z$.
• Figure 4: The sign behaviors of coefficients $c_n$. Actually, we plotted sign$(c_n)\cdot \log|c_n|$ against $n$ for $n\leq 600$. The left plot is for $(h_A,h_B)={c\over 24}(0.7,0.1)$, which is in the region (iii) and $c_n$ is always positive. The right one is for $(h_A,h_B)={c\over 24}(0.8,0.1)$, which is in the region (ii) and has alternating signs.
• Figure 5: The sketch of behaviors of $c_n$ for various values of $(h_A,h_B)$.