# Pole skipping The original work on pole skipping found an interesting connection to [[0008 Quantum chaos|quantum chaos]]. At a special point $\omega=i \lambda, k=i \lambda / v_{B}$ in the complex frequency and momentum plane, the *energy density two point function* exhibits the following behaviour: $G_{T^{00} T^{00}}^{R}(\omega, k)=\frac{b(\omega, k)}{a(\omega, k)},$where $\lim _{(\omega, k) \rightarrow \mathcal{P}_{c}} a(\omega, k)=\lim _{(\omega, k) \rightarrow \mathcal{P}_{c}} b(\omega, k)=0.$This means that a would-be pole is skipped -- hence the name. However, pole-skipping happens at other points and for other Green's functions. Some of them are called lower-half-plane pole-skipping (see below for references), but they are not fundamentally different. Holographically, the [[0325 Quasi-normal modes|quasi-normal]] spectra of the dual gravitational backgrounds give the location (in momentum space) of the skipped poles of the retarded correlators in the gauge theory, supplying important information about the theory’s quasiparticle spectra and transport (kinetic) coefficients. The universality of this behaviour can be directly studied by studying the perturbation to the Einstein equation. ## Refs - original - [[2017#Grozdanov, Schalm, Scopelliti]] (numerical) - [[2018#Blake, Lee, Liu]] (EFT) - [[2018#Blake, Davison, Grozdanov, Liu]] - general understanding - classification - [[2020#Ahn, Jahnke, Jeong, Kim, Lee, Nishida]] - regularity at horizon - [[2019#Blake, Davison, Vegh]] - [[NatsuumeOkamura201905a]][](https://arxiv.org/abs/1905.12014) - relation between [[0482 Out-of-time-order correlator|OTOC]] and pole-skipping - [[2020#Kim, Lee, Nishida]] - connection to symmetry - [[2022#Wang, Wang]]: pole skipping from boost symmetry for arbitrary integer spins - [[2023#Ning, Wang, Wang]]: pole skipping from boost symmetry for arbitrary half-integer spins - [[2024#Knysh, Liu, Pinzani-Fokeeva]]: relation of pole skipping to horizon symmetry - relation to the normal mode spectrum - [[2023#Natsuume, Okamura (Jul)]] (in [[0567 AdS soliton|AdS soliton]]) - [[0026 Bulk reconstruction|bulk reconstruction]] (in a broad sense) - [[2023#Grozdanov, Lemut, Pedraza]] - relation to univalence - [[Grozdanov2020]] - relation to duality constraints - [[2024#Grozdanov, Vrbica]] - lower half plane - [[2019#Blake, Davison, Vegh]] and [[2019#Grozdanov, Kovtun, Starinets, Tadic (Apr, Long)]] - [[2020#Abbasi, Kaminski]]: pattern of lower half plane coming from constraints - special backgrounds - [[Sil2020]][](https://arxiv.org/pdf/2012.07710.pdf): anisotropic plasma - [[YuanGe2020]][](https://arxiv.org/pdf/2012.15396.pdf): Lifshitz, AdS2, Rindler - [[YuanGe2021]][](https://arxiv.org/pdf/2110.08074.pdf): acoustic BHs - [[JansenPantelidou2020]][](https://arxiv.org/pdf/2007.14418.pdf): charged fluids - [[AbbasiTahery2020]][](https://arxiv.org/pdf/2007.10024.pdf): finite chemical potential - [[2023#Grozdanov, Vrbica]]: massive BH with flat, spherical and hyperbolic horizons - relation to convergence of the gradient expansion - [[2019#Grozdanov, Kovtun, Starinets, Tadic (Apr, Short)]] - [[2019#Grozdanov, Kovtun, Starinets, Tadic (Apr, Long)]] - away from maximal [[0008 Quantum chaos|chaos]] - [[2020#Choi, Mezei, Sarosi]] - rotation - [[2021#Blake, Davison]]: Kerr-AdS (analytically continue the discrete expansion coefficients, i.e., take $k$ to be continuous even though the perturbation would not generally be regular) - [[2020#Natsuume, Okamura]]: lower half plane poke skipping for rotating BTZ and its extremal limit - [[2020#Liu, Raju]][](https://arxiv.org/abs/2005.08508): rotating BTZ in TMG - [[2022#Amano, Blake, Cartwright, Kaminski, Thompson]]: 5D Myers-Perry BH - [[2023#Jeong, Ji, Kim]]: rotating BTZ and general spin - spherical - [[2023#Grozdanov, Vrbica]]: sec. 5 for spherical black holes - higher derivative or modified gravity - [[2018#Grozdanov]]: GB and $Riem^4$ corrections to the upper half plane skipped pole - [[2019#Wu]]: corrections to the lower half plane poles - [[NatsuumeOkamura201909]]: correction to lower-half plane poles with 4-derivative gravity and $RFF$ correction - [[2020#Liu, Raju]]: rotating BTZ in TMG - [[AbbasiTabatabaei2019]][](https://arxiv.org/pdf/1910.13696.pdf): Einstein-Maxwell-CS; [[0482 Out-of-time-order correlator|OTOC]] and pole-skipping match - [[BaishyaNayek2023]][](https://arxiv.org/pdf/2301.03984.pdf): GB coupled to $\phi^p$ for some scalar field $\phi$ and integer $p$ - string theory - [[MahishSil2022]][](https://arxiv.org/pdf/2202.05865.pdf) - [[AmrahiAsadiTaghinavaz2023]][](https://arxiv.org/abs/2305.00298): 1RCBH -- solution to a consistent truncation of [[0332 Supergravity|sugra]] - other geometry - [[AhnJahnkeJeongKim2019]][](https://arxiv.org/abs/1907.08030): hyperbolic BH - relation to scattering - [[2021#Natsuume, Okamura]] - relation to quantum critical point - [[2023#Abbasi, Landsteiner]] - relation to IR fixed point - [[2023#Jeong]] - higher bosonic spin - [[2016#Perlmutter]]: higher spin in [[0117 Shockwave|shockwave]] calculation - [[2018#Haehl, Rozali]]: spin-3 (CFT) - [[2021#Kim, Lee, Nishida]]: considered several higher spin fields - [[2022#Wang, Wang]]: arbitrary integer spin - fermions - have $\omega$ at half-integer $i2\pi T$ - pointed out in [[2019#Blake, Davison, Vegh]] using results in [[2009#Iqbal, Liu]] - [[2019#Ceplak, Ramdial, Vegh]]: spin-1/2 - [[2021#Ceplak, Vegh]]: spin-3/2 - [[2023#Ning, Wang, Wang]]: general fermionic fields - [[2023#Baishya, Chakrabarti, Maity]]: effect of scalar condensation - matter fields (bosonic) - [[NatsuumeOkamura201905b]][](https://arxiv.org/pdf/1905.12015.pdf): bulk scalar field, the bulk Maxwell vector and scalar modes, and the shear mode of gravitational perturbations - [[AhnJahnkeJeongKimLeeNishida202006]][](https://arxiv.org/abs/2006.00974) - [[2021#Jeong, Kim, Sun]]: Einstein-Maxwell-Dilaon-axion model with the matter fields (including finite charge and magnetic fields) - [[2022#Wang, Pan]]: generalised models of Maxwell theory with form fields - zero temperature - [[2020#Natsuume, Okamura]] - non-integer frequencies - [[2020#Choi, Mezei, Sarosi]] - [[2020#Ahn, Jahnke, Jeong, Kim, Lee, Nishida]] (Type III) - non-black hole background - [[2023#Natsuume, Okamura (Jun)]] - de Sitter - [[2024#Yuan, Ge, Kim]] - non-holographic - maximal - [[2018#Haehl, Rozali]] - [[HaehlReevesRozali2019]] - [[2019#Das, Ezhuthachan, Kundu]]: [[0181 AdS-BCFT]][](https://arxiv.org/pdf/1907.08763.pdf) - non-maximal - [[2019#Mezei, Sarosi]] - [[2020#Choi, Mezei, Sarosi]] - [[Ramirez2020]][](https://arxiv.org/abs/2009.00500) ## Relation to [[0482 Out-of-time-order correlator|OTOC]] The OTOC looks like $\langle V(t, \vec{x}) W(0) V(t, \vec{x}) W(0)\rangle_{\beta_{0}}=1-\epsilon e^{\lambda\left(t-|\vec{x}| / v_{B}\right)}+\cdots$which $\sim 1-\epsilon e^{-i \omega t+i k|\vec{x}|}+\cdots$for $\omega=i\lambda$ and $k=i\lambda/v_B$. This was first pointed out in [[2018#Blake, Davison, Grozdanov, Liu]] for planar black holes in Einstein gravity and in [[2022#Wang, Wang]] for planar black holes in general [[0006 Higher-derivative gravity|higher-derivative gravity]]. A non-planar generalisation was given in [[2022#Amano, Blake, Cartwright, Kaminski, Thompson]]. More generally, for maximally chaotic theories, pole-skipping always happens at the right $\lambda_L$ and $v_B$ defined via [[0482 Out-of-time-order correlator|OTOC]]; for (non-holographic) non-maximally chaotic theories, this is not necessarily true but still subtly related. See [[2020#Choi, Mezei, Sarosi]]. ## Relation to regularity of solutions It was noticed in [[1999#Horowitz, Hubeny]] that at special values of the frequency, the outgoing mode is regular at the horizon, but it was argued there that it cannot happen for the theory considered there. Then in [[2000#van den Brink]] the existence of the special frequency was found and studied. Recently in [[2019#Blake, Davison, Vegh]], the relation to pole-skipping was discussed, which we summarise now. At special values of $\omega_{n}=-i 2 \pi T n$, both branches satisfy the ingoing condition (regular at horizon), but log terms can be used to destroy one of them for generic values of $k$; however at special values of $k$, the log terms are *not* present; these are values at which pole-skipping happens. Some of them depend on $\delta \omega /\delta k$ as one goes away from the point, and they are called Type I, according to [[2020#Ahn, Jahnke, Jeong, Kim, Lee, Nishida]]. If they depend on higher order quantities like $(\delta k)^2$, but not $\delta \omega /\delta k$, they are classified as Type II.