Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory

Abstract

We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form hWðtÞVWðtÞVi. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large c Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of ∼t à − ðβ=2πÞ log β 2 E w E v , where t à is the fast scrambling time ðβ=2πÞ log c and E w ; E v are the energy scales of the W; V operators. Introduction.-In classical mechanics, the butterfly effect is a vivid diagnostic of chaos: small perturbations grow rapidly to affect the entire system. In quantum mechanics, studies of chaos have often focused on less direct measures, such as the statistics of level spacings in the energy spectrum [1,2], or the local properties of energy eigenvectors [3][4][5]. In this Letter, following Refs. [6-9] we will focus on a notion of quantum chaos that is closely related to the classical butterfly effect.Consider a pair of rather general Hermitian operators V and W in a quantum mechanical system. If the system is strongly chaotic, we expect perturbations by V to affect later measurements of W, for almost any choice of operators V and W [10]. In other words, we expect the commutator ½V; WðtÞ to become large. A useful diagnostic is the expectation value of the square of the commutator [11]:

Chaos inAdS2Holography

“…In the setting of two-dimensional CFT, Roberts and Stanford [31] have shown that large central charge, a large gap for higher-spin Virasoro primary operators, and a "low" density of light states is enough to guarantee a maximal Lyapunov exponent λ L = 2πT . Their argument would go through more or less unaltered for a CQM invariant under a Virasoro symmetry with large central charge, assuming that such a thing existed.…”

Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

“…A particularly interesting type of four-point function is the out-of-time-order correlation function (OTOC) F (t) = χ j (t)χ k (0)χ j (t)χ k (0) , which has been proposed as a measure of chaos in quantum systems [15][16][17][18][19][20] (see refs. [21][22][23][24] for experimental proposals for measuring OTOC).…”

Worldline approach to semi-classical conformal blocks

“…evolution of entanglement entropy in excited states created by local operators [13][14][15]; the issue of universality in 2d conformal field theories [16,17]; entanglement entropy in higher spin theories [18]; and the emergence of chaos in thermal systems [19].…”