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arXiv:1907.08763 · Journal of High Energy Physics

Real time dynamics from low point correlators in 2d BCFT

Suchetan Das, Bobby Ezhuthachan, Arnab Kundu

Published Dec 1, 2019• 51 citations
arXivHTMLPDF

Abstract

Abstract In this article, we demonstrate how a 3-point correlation function can capture the out-of-time-ordered features of a higher point correlation function, in the context of a conformal field theory (CFT) with a boundary, in two dimensions. Our general analyses of the analytic structures are independent of the details of the CFT and the operators, however, to demonstrate a Lyapunov growth we focus on the Virasoro identity block in large-c CFT’s. Motivated by this, we also show that the phenomenon of pole-skipping is present in a 2-point correlation function in a two-dimensional CFT with a boundary. This pole-skipping is related, by an analytic continuation, to the maximal Lyapunov exponent for maximally chaotic systems. Our results hint that, the dynamical content of higher point correlation functions, in certain cases, may be encrypted within low-point correlation functions, and analytic properties thereof.

DOI
10.1007/jhep12(2019)141

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Abstract In this article, we demonstrate how a 3-point correlation function can capture the out-of-time-ordered features of a higher point correlation function, in the context of a conformal field theory (CFT) with a boundary, in two dimensions. Our general analyses of the analytic structures are independent of the details of the CFT and the operators, however, to demonstrate a Lyapunov growth we focus on the Virasoro identity block in large-c CFT’s. Motivated by this, we also show that the phenomenon of pole-skipping is present in a 2-point correlation function in a two-dimensional CFT with a boundary. This pole-skipping is related, by an analytic continuation, to the maximal Lyapunov exponent for maximally chaotic systems. Our results hint that, the dynamical content of higher point correlation functions, in certain cases, may be encrypted within low-point correlation functions, and analytic properties thereof.
In this article, we demonstrate how a 3-point correlation function can capture the out-of-time-ordered features of a higher point correlation function, in the context of a conformal field theory (CFT) with a boundary, in two dimensions. Our general analyses of the analytic structures are independent of the details of the CFT and the operators, however, to demonstrate a Lyapunov growth we focus on the Virasoro identity block. Motivated by this, we also show that the phenomenon of pole-skipping is present in a 2-point correlation function in a two-dimensional CFT with a boundary. This pole-skipping is related, by an analytic continuation, to the maximal Lyapunov exponent for maximally chaotic systems. Our results hint that, the dynamical content of higher point correlation functions, in certain cases, may be encrypted within low-point correlation functions, and analytic properties thereof.
A bstract Motivated by the recent connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators (OTOCs), we study the pole structure of thermal two-point functions in d -dimensional conformal field theories (CFTs) in hyperbolic space. We derive the pole-skipping points of two-point functions of scalar and vector fields by three methods (one field theoretic and two holographic methods) and confirm that they agree. We show that the leading pole-skipping point of two point functions is related with the late time behavior of conformal blocks and shadow conformal blocks in four-point OTOCs.
Abstract We compute Out-of-Time-Order correlators (OTOCs) for conformal field theories (CFTs) subjected to either continuous or discrete periodic drive protocols. This is achieved by an appropriate analytic continuation of the stroboscopic time. After detailing the general structure, we perform explicit calculations in large-c CFTs where we find that OTOCs display an exponential, an oscillatory and a power-law behaviour in the heating phase, the non-heating phase and on the phase boundary, respectively. In contrast to this, for the Ising CFT representing an integrable model, OTOCs never display such exponential growth. This observation hints towards how OTOCs can demarcate between integrable and chaotic CFT models subjected to a periodic drive. We further explore properties of the light-cone which is characterized by the corresponding butterfly velocity as well as the Lyapunov exponent. Interestingly, as a consequence of the spatial inhomogeneity introduced by the drive, the butterfly velocity, in these systems, has an explicit dependence on the initial location of the operators. We chart out the dependence of the Lyapunov exponent and the butterfly velocities on the frequency and amplitude of the drive for both protocols and discuss the fixed point structure which differentiates such driven CFTs from their undriven counterparts.
Abstract We study the out-of-time-ordered correlator (OTOC) in a zero temperature 2d large-c CFT under evolution by a Liouvillian composed of the Virasoro generators. A bound was conjectured in [1] on the growth of the OTOC set by the Krylov complexity which is a measure of operator growth. The latter grows as an exponential of time with exponent 2α, which sets an upper bound on the Lyapunov exponent, ΛL≤ 2α. We find that for a two dimensional zero temperature CFT, the OTOC decays exponentially with a Lyapunov exponent which saturates this bound. We show that these Virasoro generators form the modular Hamiltonian of the CFT with half space traced out. Therefore, evolution by this modular Hamiltonian gives rise to thermal dynamics in a zero temperature CFT. Leveraging the thermal dynamics of the system, we derive this bound in a zero temperature CFT using the analyticity and boundedness properties of the OTOC.
Abstract We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.
We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form ⟨W(t)VW(t)V⟩⟨W(t)VW(t)V⟩⟨W(t)VW(t)V⟩. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large ccc Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\ensuremath{\sim}{t}_{*}\ensuremath{-}(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}{\ensuremath{\beta}}^{2}{E}_{w}{E}_{v}$, where t∗{t}_{*}t∗​ is the fast scrambling time $(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}c$ and Ew,Ev{E}_{w},{E}_{v}Ew​,Ev​ are the energy scales of the W,VW,VW,V operators.