Fractional Statistics and the Butterfly Effect

TL;DR

The paper connects the late-time behavior of out-of-time-ordered correlators in (1+1) rational conformal field theories to fractional statistics in (2+1) topologically ordered states through bulk-boundary correspondence. It shows that the surviving late-time value of the OTOC is governed by universal RCFT data, specifically the monodromy scalar expressible in terms of the modular S-matrix, and illustrates this with Ising, compact boson, and SU(2) WZW models. A bulk interpretation via anyon braiding explains the link between boundary chaos and bulk fractional statistics, while averaging over random operators ties the chaos measure to topological entanglement entropy. The work also discusses experimental pathways for observing these effects and suggests broader implications for integrable systems and future research directions.

Abstract

Fractional statistics and quantum chaos are both phenomena associated with the non-local storage of quantum information. In this article, we point out a connection between the butterfly effect in (1+1)-dimensional rational conformal field theories and fractional statistics in (2+1)-dimensional topologically ordered states. This connection comes from the characterization of the butterfly effect by the out-of-time-order-correlator proposed recently. We show that the late-time behavior of such correlators is determined by universal properties of the rational conformal field theory such as the modular S-matrix and conformal spins. Using the bulk-boundary correspondence between rational conformal field theories and (2+1)-dimensional topologically ordered states, we show that the late time behavior of out-of-time-order-correlators is intrinsically connected with fractional statistics in the topological order. We also propose a quantitative measure of chaos in a rational conformal field theory, which turns out to be determined by the topological entanglement entropy of the corresponding topological order.

Figures (10)

• Figure 1: Illustration of the thermal expectation value of OTOC as an inner product of states $|x\rangle$ and $|y\rangle$. The operator $V$ and $W$ act on a pure state $|\beta \rangle$ (the grey disk) which is a purification of the thermal state. We can imagine $W(t)$ as a small perturbation in late time $t\gg \beta$: if $W={\rm Identity}$ is trivial, then state $|x\rangle=|y\rangle$; if $W$ is a non-trivial perturbation and the system is "chaotic", we expect the "butterfly" $W$ causes a big difference on states $|x\rangle$ and $|y\rangle$. Therefore, we can use the inner product $\langle y | x \rangle$ to quantify the butterfly effect.
• Figure 2: (a) Winding $\eta(t)$ in the moduli space: $\eta(t)\in \mathbb{C}-\lbrace0,1,+\infty \rbrace$ winds around $z=1$ clockwisely and induces a linear transformation on the space of conformal blocks $V_{\overline{a}a\overline{b}b}\simeq V_{b}^{a\overline{a}b}$, which defines the the monodromy matrix. (b) The diagrammatic representation of the monodromy matrix $\widetilde{M}= \widetilde{M}[a,b]$. For fixed $a,b$, $\widetilde{M}[a,b]$ acts by braiding the two lines $\overline{a},b$. See appendix \ref{['appendix: notations and conventions']} for detailed conventions of the diagrammatics.
• Figure 3: The distributions of $|r\left[V_i,V_j\right]|$ in $\operatorname{SU}(2)$ WZW models.
• Figure 4: The bulk-boundary correspondence. (a) At a fixed time $t^*$, the CFT state at two boundaries belong to sector $\mathcal{H}_a\otimes \mathcal{H}_{\overline{a}}$; (b) The spacetime picture for a pair of anyons $a,\overline{a}$ created and passed through the boundary at fixed time $t^*$. The state was in identity sector before $t^*$: $\psi(t<t^*)\in \mathcal{H}_1 \otimes \overline{\mathcal{H}}_1$, and shifted to sector $\mathcal{H}_a \otimes \overline{\mathcal{H}}_{\overline{a}}$ after time $t^*$.
• Figure 5: OTOC is mapped to a time-ordered four-point function on the two boundaries of a strip, by shifting along the light cones (see equation (\ref{['eqn:OTO after shifting']}).