Fractional Statistics and the Butterfly Effect

TL;DR

This paper explores the deep connection between quantum chaos, fractional statistics, and topological order, revealing how late-time correlator behavior in conformal field theories relates to topological properties and chaos measures.

Contribution

It establishes a link between the butterfly effect in rational conformal field theories and fractional statistics in topologically ordered states, introducing a chaos measure tied to topological entanglement entropy.

Findings

Late-time correlator behavior is governed by the modular S-matrix and conformal spins.

The butterfly effect correlator decay is connected to fractional statistics in topological states.

A new chaos measure is proposed, related to topological entanglement entropy.

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.