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.
Highlights
We are interested in the size of the operator.) Recently, a generalization of such quantities have been studied in many-body systems, where operators x and p are replaced by generic many-body operators. [2,3,4]
Our result shows that when time t goes to ∞, of-time-ordered correlators (OTOCs) in (1 + 1)-dimensional rational conformal field theories (RCFTs) are intrinsically related to the fraction statistics [14, 15] in (2 + 1)-dimensional topological order
We studied OTOC in the context of RCFTs, and relate its behavior to the universal algebraic data of RCFT, such as the monodromy matrix and the modular S-matrix
Summary
Before going to the detailed discussions, we first declare some general definitions for OTOC. Previous studies [2, 3, 7, 18] of such correlation function focused on the early time β < t < tscr, i.e., between the dissipation time and the scrambling time, and found interesting “Lyapunov behavior” for those systems that can be holographically described by Einstein gravity. For a generic system that does not have a large separation between dissipation time td ∼ β and scrambling time tscr, the “Lyapunov behavior” is not welldefined. We denote the inner product x|x = V †W †(t)W (t)V β by g(t). In the regime t β, such a four point function generically factorizes to g(t) ∼ V †V β W †W β, which represents a normalization for operators W and V. We will study the late time behavior of f (t) in the context of RCFTs
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
