# Lyapunov exponent The (quantum) Lyapunov characterises how [[0008 Quantum chaos|chaotic]] a quantum system is. It can be defined via [[0482 Out-of-time-order correlator|OTOC]] as: $ \langle V(t) W(0) V(t) W(0)\rangle \sim 1-e^{\lambda_L\left(t-t_*\right)}+\cdots. $ There is also a classical analogue, but we will not discuss it here. The quantum one is more interesting. In fact, there is a [[0474 Chaos bound|bound]] on the value of the (quantum) Lyapunov exponent which has no classical analogue. ## Approaches 1. [[0482 Out-of-time-order correlator|OTOC]] (or holographically the [[0117 Shockwave|shockwave]]): this is the *definition* of the Lyapunov exponent 2. [[0179 Pole skipping|pole-skipping]] (both holographic and non-holographic) 3. from particle motion on BH background (holographic): - see e.g. [[YuChenGao2022]][](https://arxiv.org/pdf/2202.13741.pdf) Einstein-Maxwell-Dilaton-Axion model with angular momentum for the particle 4. non-holographic bound: [[2015#Maldacena, Shenker, Stanford]], and generalised to velocity dependent bound [[2019#Mezei, Sarosi]] (see [[0474 Chaos bound|chaos bound]]) ## VDLE (Velocity dependent Lyapunov exponent) - in general (non-maximally chaotic), $\operatorname{OTOC}(t, x)=1-\frac{\#}{N} \exp \left[\lambda\left(\frac{|x|}{t}\right) t\right]$ where $\lambda(u)$ is called VDLE - define the ordinary Lyapunov exponent as $\lambda_L=\lambda(0)$ (i.e. when nothing is moving) and define butterfly velocity as $\lambda(u_B)=0$, i.e., the boundary of butterfly cone - then, chaos bound translate to $\lambda(u) \leq 2 \pi \beta^{-1}\left(1-|u| / u_{B}\right)$ - $\lambda(u)$ is a competition between Regge and stress tensor - define: $\lambda_{L}^{(T)}=2 \pi / \beta$ and $u_{B}^{(T)}$ are stress-tensor contributions to [[0008 Quantum chaos|chaos]] - below some critical $u_*$, Regge dominates - above $u_*$, stress tensor dominates, and $\lambda(u)=\lambda_{L}^{(T)}\left(1-\frac{|u|}{u_{B}^{(T)}}\right)$ (maximal chaos) - If $u_*=0$, so stress tensor always dominate => always maximal ## Connections with other things - connection to [[0012 Hawking-Page transition|Hawking-Page]]? - [[GuoLuMuWang2022]][](https://arxiv.org/pdf/2205.02122.pdf) - onset of chaos - [[2022#Horowitz, Leung, Queimada, Zhao]] ## Lyapunov exponent for general spins - 4-point function of large-$N$ CFTs in *vacuum* in the Regge limit (thermal OTOC at Rindler time) - [[2012#Costa, Goncalves, Penedones]] - [[2007#Cornalba]] - [[2018#Kravchuk, Simmons-Duffin]] - calculations in *vacuum* AdS - [[2006#Cornalba, Costa, Penedones, Schiappa (a)]] - [[2006#Cornalba, Costa, Penedones, Schiappa (b)]] - [[2007#Cornalba, Costa, Penedones]] - [[2016#Perlmutter]] - $\mathcal{N}=2$ SUSY JT: [[DelgadoForste2022]][](https://arxiv.org/pdf/2209.15456.pdf) ## Examples - [[2023#Kalloor, Sharon]] ## Lyapunov in large-$p$ SYK - $\lambda_L=\frac{2\pi v}{\beta}$ where $\frac{\pi v}{\beta\mathcal{J}}=\cos(\pi v/2)$ - low temperature: $\beta \mathcal{J}\gg 1$, $v\to1$, maximal chaos - high temperature: $\lambda_{L}\to2\mathcal{J}$ ## Generalisations - for *dissipative* quantum chaos - [[2024#Garcia-Garcia, Verbaarschot, Zheng]]