Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2006.00974 · Journal of High Energy Physics

Pole-skipping of scalar and vector fields in hyperbolic space: conformal blocks and holography

Yongjun Ahn, Viktor Jahnke, Hyun-Sik Jeong, Keun-Young Kim, Kyung-Sun Lee, Mitsuhiro Nishida

Published Sep 1, 2020• 50 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep09(2020)111

Related papers

Scrambling in hyperbolic black holes: shock waves and pole-skipping

Yongjun Ahn, Viktor Jahnke, Hyun-Sik Jeong +1 more

A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result $$ {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} $$ v B T = 1 2 π ℓ = 1 d − 1 and the planar result $$ {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} $$ v B T ≫ 1 ℓ = d 2 d − 1 .

arXiv:1907.08030•Journal of High Energy Physics•Oct 1, 2019•71 citations

Real Time Dynamics in Low Point Correlators

Suchetan Das, Arnab Kundu, Bobby Ezhuthachan

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.

•Jul 20, 2019•3 citations

Real time dynamics from low point correlators in 2d BCFT

Suchetan Das, Bobby Ezhuthachan, Arnab Kundu

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.

arXiv:1907.08763•Journal of High Energy Physics•Dec 1, 2019•51 citations

Quantum chaos and pole skipping in two-dimensional conformal perturbation theory

Curtis T. Asplund, Sebastian Fischetti, Alexandra Miller +1 more

A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.

arXiv:2509.18540•Journal of High Energy Physics•Sep 3, 2026•0 citations

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

Daniel A. Roberts, Douglas Stanford

We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form $⟨W(t)VW(t)V⟩$. 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 $\ensuremath{\sim}{t}_{*}\ensuremath{-}(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}{\ensuremath{\beta}}^{2}{E}_{w}{E}_{v}$, where ${t}_{*}$ is the fast scrambling time $(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}c$ and ${E}_{w},{E}_{v}$ are the energy scales of the $W,V$ operators.

arXiv:1412.5123•Physical Review Letters•Sep 22, 2015•466 citations

Thermal one-point functions and their partial wave decomposition

Ilija Burić, Francesco Russo, Volker Schomerus +1 more

A bstract In this work we address partial wave decompositions of thermal one-point functions in conformal field theories on S 1 × S d− 1 . With the help of Casimir differential equations we develop efficient algorithms to compute the relevant conformal blocks for an external field of arbitrary spin and with any spin exchange along the thermal circle, at least in three dimensions. This is achieved by identifying solutions to the Casimir equations with a special class of spherical functions in the harmonic analysis of the conformal group. The resulting blocks are then applied to study the decomposition of one-point functions of the scalar ϕ 2 and the stress tensor T for a three-dimensional free scalar field ϕ . We are able to read off averaged OPE coefficients into exchanged fields of high weight and spin for a complete set of tensor structures. We also extract an asymptotic behaviour of conformal blocks and use it to analyse the density of heavy-heavy-light OPE coefficients for spinning operators, comparing it with semi-classical predictions, such as the dimensions of operators at large charge.

arXiv:2408.02747•Journal of High Energy Physics•Dec 3, 2024•8 citations
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.
A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result vB(T=12πℓ)=1d−1 {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} vB​(T=2πℓ1​)=d−11​ v B T = 1 2 π ℓ = 1 d − 1 and the planar result vB(T≫1ℓ)=d2(d−1) {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} vB​(T≫ℓ1​)=2(d−1)d​​ v B T ≫ 1 ℓ = d 2 d − 1 .
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.
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.
A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.
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.
A bstract In this work we address partial wave decompositions of thermal one-point functions in conformal field theories on S 1 × S d− 1 . With the help of Casimir differential equations we develop efficient algorithms to compute the relevant conformal blocks for an external field of arbitrary spin and with any spin exchange along the thermal circle, at least in three dimensions. This is achieved by identifying solutions to the Casimir equations with a special class of spherical functions in the harmonic analysis of the conformal group. The resulting blocks are then applied to study the decomposition of one-point functions of the scalar ϕ 2 and the stress tensor T for a three-dimensional free scalar field ϕ . We are able to read off averaged OPE coefficients into exchanged fields of high weight and spin for a complete set of tensor structures. We also extract an asymptotic behaviour of conformal blocks and use it to analyse the density of heavy-heavy-light OPE coefficients for spinning operators, comparing it with semi-classical predictions, such as the dimensions of operators at large charge.