0167 Butterfly velocity - otherworldlines

# Butterfly velocity In relativistic theories, there is an upper bound on the speed of propagation, that is (of course) the speed of light. For non-relativistic theories, is there a speed limit? While there might not be strict causality, there is a bound for how large the commutator can be: the [[0322 Lieb-Robinson bound|Lieb-Robinson bound]]. The **butterfly velocity** is a low energy effective [[0322 Lieb-Robinson bound|Lieb-Robinson bound]] (as explained in [[2016#Roberts, Swingle]]), and is state dependent. It is a characteristic velocity of [[0008 Quantum chaos|chaotic]] propagation. Define $C(t)=-\left\langle[W(t), V(0)]^{2}\right\rangle.$For large-$N$ gauge theories with $O(N^2)$ degrees of freedom per site: $C(x, t)=\frac{K}{N^{2}} e^{\lambda_{L}\left(t-x / v_{B}\right)}+O\left(N^{-4}\right),$where $\lambda_L$ is the [[0466 Lyapunov exponent|Lyapunov exponent]] (conjectured to be bounded by $\lambda_{L} \leq 2 \pi / \beta$ by [[2015#Maldacena, Shenker, Stanford]]) and $v_B$ is the butterfly velocity. We can take this as a definition of $v_B$. For [[0001 AdS-CFT|holographic]] theories, it can be computed using four independent methods (see below) at leading order in $G_N$ (i.e., the classical limit). ## Bounds - [[2016#Mezei]] - from footnotes 13 of [[2018#Jahnke (Review)]]: The above bound can also be violated by [[0006 Higher-derivative gravity|HDG]] corrections, but $v_B$ remains bounded by the speed of light as long as causality is respected. For instance, in 4-dimensional Gauss-Bonnet (GB) gravity, the butterfly velocity surpasses the speed of light for $\Lambda_{GB} <-3/4$, but causality requires $λ_{GB} >-0.19$ - typos: should be 5 bulk dimensions; two parameters have difference conventions $\Lambda_{GB}=\frac{\lambda_{GB}}{(D-3)(D-4)}$ where $D$ is total dimension so it should be: the butterfly velocity surpasses the speed of light for $\Lambda_{GB} <-3/4=-0.75$, but causality requires $\Lambda_{GB}=2\lambda_{GB} >-2*0.19=-0.38$ ## Holographic calculations There are four methods of computing the butterfly velocity in holography: 1. [[0117 Shockwave|shockwave]] 2. [[0007 RT surface|entanglement surface]] 3. [[0179 Pole skipping|pole-skipping]] 4. [[0433 Membrane theory of entanglement dynamics|membrane theory]] The value of butterfly velocity depends on the theory: - Einstein gravity - [[2014#Roberts, Stanford, Susskind]] using method 1 - [[0425 Gauss-Bonnet gravity|GB gravity]]: - [[2014#Roberts, Stanford, Susskind]] using the method 1 - general four-derivative theories: - [[2016#Mezei, Stanford]]: methods 1 and 2 agree - [[2019#Mezei, Virrueta (Dec)]]: methods 1 and 4 agree - Weyl cubed: - [[2018#Grozdanov]]: methods 1 and 3 agree - $f$(Riemann): - [[2022#Dong, Wang, Weng, Wu]]: methods 1 and 2 agree - most general [[0006 Higher-derivative gravity|HDG]]: - [[2022#Wang, Wang]]: methods 1 and 3 are equivalent - [[2025#Chua, Hartman, Weng]]: all three are equivalent ## Alternative (non-holographic) computations - using a real-time local correlator, $\langle\mathcal{O}(\mathbf{x}, t) \mathcal{O}(0,0)\rangle$: see e.g. [[2022#Nahum, Roy, Vijay, Zhou]] ## Related topics and refs - [[0179 Pole skipping]] - [[0434 Diffusivity]] - [[0008 Quantum chaos]] - [[0466 Lyapunov exponent]] - some subtleties about the butterfly calculation: [[2016#Mezei]] - generalisation beyond the thermal case: [[2017#Qi, Yang]] ![[A0004 Butterfly on muddy road.png]]