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Out-of-Time-Ordered Correlators in $(T^2)^n/\mathbb{Z}_n$

Pawel Caputa, Yuya Kusuki, Tadashi Takayanagi, Kento Watanabe

TL;DR

The work analyzes out-of-time-ordered correlators (OTOCs) with twist operators in the cyclic orbifold CFT $(T^2)^n/\mathbb{Z}_n$ to understand information scrambling in 2d CFTs. By expressing the four-point function through the function $F_n(z,\bar{z})$ built from the Riemann-Siegel theta function and performing the OTO continuation, the authors distinguish rational and irrational compactification parameters $\eta=R^2$. For rational $\eta= p/p'$, late-time OTOCs approach a monodromy-determined constant tied to modular data, with explicit parity-dependent behavior in the $n=2$ case, while for irrational $\eta$ the OTOCs decay polynomially in time; higher $n$ generalizes these results with the decay becoming steeper as $n$ grows. The results connect to Renyi entropy growth and suggest a nuanced classification of 2d CFTs bridging rational, irrational, and holographic-like chaotic behavior, and they point to rich future directions in large-$n$ limits and symmetric orbifolds.

Abstract

In this note we continue analysing the non-equilibrium dynamics in the $(T^2)^n/\mathbb{Z}_n$ orbifold conformal field theory. We compute the out-of-time-ordered four-point correlators with twist operators. For rational $η\ (=p/q)$ which is the square of the compactification radius, we find that the correlators approach non-trivial constants at late time. For $n=2$ they are expressed in terms of the modular matrices and for higher $n$ orbifolds are functions of $pq$ and $n$. For irrational $η$, we find a new polynomial decay of the correlators that is a signature of an intermediate regime between rational and chaotic models.

Out-of-Time-Ordered Correlators in $(T^2)^n/\mathbb{Z}_n$

TL;DR

The work analyzes out-of-time-ordered correlators (OTOCs) with twist operators in the cyclic orbifold CFT to understand information scrambling in 2d CFTs. By expressing the four-point function through the function built from the Riemann-Siegel theta function and performing the OTO continuation, the authors distinguish rational and irrational compactification parameters . For rational , late-time OTOCs approach a monodromy-determined constant tied to modular data, with explicit parity-dependent behavior in the case, while for irrational the OTOCs decay polynomially in time; higher generalizes these results with the decay becoming steeper as grows. The results connect to Renyi entropy growth and suggest a nuanced classification of 2d CFTs bridging rational, irrational, and holographic-like chaotic behavior, and they point to rich future directions in large- limits and symmetric orbifolds.

Abstract

In this note we continue analysing the non-equilibrium dynamics in the orbifold conformal field theory. We compute the out-of-time-ordered four-point correlators with twist operators. For rational which is the square of the compactification radius, we find that the correlators approach non-trivial constants at late time. For they are expressed in terms of the modular matrices and for higher orbifolds are functions of and . For irrational , we find a new polynomial decay of the correlators that is a signature of an intermediate regime between rational and chaotic models.

Paper Structure

This paper contains 10 sections, 67 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. OTOC in $(T^2)^n/\mathbb{Z}_n$
  3. The $n=2$ orbifold
  4. Rational $\eta=p/p'$
  5. RCFT approach
  6. Direct approach
  7. Irrational $\eta\neq p/p'$
  8. Higher $n$ orbifolds
  9. Conclusions
  10. Cross-ratios

Figures (3)

  • Figure 1: The plots of the absolute value $| F_{2} (t) |$ for even $\eta= 2, 4, 6, 8$ (left) and odd $\eta = 1, 3, 5, 7$(right). We set $|t-1|<10^{-8}, x=1, \beta =0.1, \epsilon_1 = \epsilon_2/6 = \epsilon_3/4 = \epsilon_4/8 = 10^{-10}$.
  • Figure 2: Left : Full numerical late time plots of $\mathrm{Im} \ F_{2} (t)$ for $\eta = \sqrt{2}$ (blue) and plots of the analytically approximated formula (orange).
  • Figure 3: Ordering of the twist operators in the OTOC. dashed lines denote the "trajectory" of the two operators with the progress of time.