# Quantum chaos **Chaos** is a familiar concept to most of us: when life is not going as expected, it is chaotic. In physics, chaos describes a similar "unexpectedness", or more precisely the "unpredictability" of certain systems. In a chaotic physical system, a small change in the initial state can cause large deviations in the outcome. In quantum physics, in general, a system with complicated Hamiltonian or in contact with thermal bath is chaotic. A useful quantity that characterises chaos is the (squared) commutator between two generic operators: $C(t)=\left\langle[V(t), W(0)]^\dagger[V(t), W(0)]\right\rangle,$where $\langle\cdot\rangle=Z^{-1} \operatorname{tr}\left[e^{-\beta H} \cdot\right]$ denotes the expectation value. This is related to the [[0482 Out-of-time-order correlator|OTOC]] by $C(t)=2-2 \operatorname{Re}\left\langle V(0)^{\dagger} W\left(t\right)^{\dagger} V( 0) W\left(t\right)\right\rangle_\beta.$ Another way to define or characterise quantum chaos is through the statistics of energy levels. The level statistics of a quantum system approaches either the Wigner-Dyson or the Poisson distribution when the energy levels are dense (semiclassical limit in the case of few-body systems or thermodynamic limit in the case of many-body systems). This definition does not require the system to have a classical limit. ## Refs - reviews - [[2018#Jahnke (Review)]] holographic chaos - [[UllmoTomsovic2014]] [](http://www.lptms.u-psud.fr/membres/ullmo/Articles/eolss-ullmo-tomsovic.pdf) - [[2015#D'Alessio, Kafri, Polkovnikov, Rigol (Review)]] - shock wave as bulk dual - [[2013#Shenker, Stanford (Jun)]] - extensions see refs 17-35 in [[2018#Huang]] - localised [[2014#Roberts, Stanford, Susskind]] - stringy corrections [[2014#Shenker, Stanford]] - no longer maximally chaotic - relation to Lieb-Robinson bound [[2016#Roberts, Swingle]] - also calculates butterfly velocity - relation between $v_E$ and $v_B$ etc - [[2016#Mezei]] - [[2016#Mezei, Stanford]] - higher-spin - [[2016#Perlmutter]] - [[NarayanYoon2019]][](https://arxiv.org/pdf/1903.08761.pdf): 3d higher spin gravity - scrambling - [[2008#Sekino, Susskind]] - [[2023#Dowling, Kos, Modi]]: scrambling v.s. cahos - chaos effective theory - introduced in [[2018#Blake, Lee, Liu]] - aspects of chaos in 2d CFT - [[2023#Haehl, Reeves, Rozali]] ## Approaches to quantum chaos 1. semi-classical chaos - when the classical limit of chaotic - first development by [[LarkinOvchinnikov1969]] 2. random matrix theory - compare the spectrum of energies to that of random matrices ## Special topics - chaos in quantum channels - famous paper [[HosurQiRobertsYoshida2015]] - [[0320 Chaos-protected locality]] ## Dissipative quantum chaos - definition using Lyapunov exponent - [[2024#Garcia-Garcia, Verbaarschot, Zheng]] ## Maximally chaotic systems saturate the bound on Lyapunov - why? - hydrodynamic origin [[2021#Blake, Liu]] [](https://arxiv.org/abs/2102.11294) - its nature is fundamentally different to non-maximally chaotic system ([[2021#Blake, Liu]]) ## Holographic calculations - original [[2013#Shenker, Stanford (Jun)]] - dual to rotating BTZ - [[CrapsKhetrapalRabideau2021]][](https://arxiv.org/pdf/2107.13874.pdf) - disruption to [[0300 Mutual information]] - [[2014#Leichenauer]] ## Chaotic system with charge (symmetry) - [[2020#Liu]] - talks about [[0187 Global symmetries in QG]] and [[0177 Weak gravity conjecture]] using information theory by studying charged chaotic systems - [[0326 Charged BH in holography]] ## In string theory - [[GrossRosenhaus2021]][](https://arxiv.org/abs/2103.15301) - AdS$_2$ string - [[2022#Giombi, Komatsu, Offertaler]] - [[2023#Bianchi, Firrotta, Sonnenschein, Weissman]] ## Direct field theory calculations - 2d CFT: [[2014#Roberts, Stanford]] - by studying large $c$ Virasoro identity block - weakly coupled systems - [[2015#Maldacena, Shenker, Stanford]] - weakly coupled $\Phi^4$ theory by numerically diagonalising ladder kernel - [[PlamadealaFradlin2018]][](https://arxiv.org/abs/1802.07268) - quantum Lifshitz model - random unitary models - [[2017#Nahum, Vijay, Haah]] - [[KhemaniVishwanathHuse2017]][](https://arxiv.org/abs/1710.09835) - [[RakovszkyPollmannVonKeyserlingk2017]][](https://arxiv.org/abs/1710.09827) - spin chains - [[LuitzBarLev2017]] - [[HeylPollmanDora2018]] - [[LinMotrunich2018]] - [[XuSwingle2018]] - [[BohrdtMendlEndresKnap2016]] - zero temperature and quench [](https://arxiv.org/pdf/2109.02132.pdf) - [[0482 Out-of-time-order correlator|OTOC]] calculation in SYK - [[GuKitaevZhang2021]][](https://arxiv.org/pdf/2111.12007.pdf) ## Quantum information diagnostic tools of chaos - [[0199 Relative entropy]] - [[NakagawaSarosiUgajin2018]][](https://arxiv.org/abs/1805.01051) - [[0204 Quantum complexity]] - [[Magan2018]][](https://arxiv.org/abs/1805.05839) and [[MansooriQaemmaqami2017]][](https://arxiv.org/abs/1711.09749) - [[0052 Pseudo-entropy]] - [[2024#He, Lau, Zhao]] ## OTOC v.s. time-ordered - see [[CotlerSchusterMohseni2022]][](https://arxiv.org/pdf/2208.02256.pdf) - [[0482 Out-of-time-order correlator|OTOC]] contains information that would be much harder to obtain using only time-ordered correlators ## Weak chaos vs strong chaos The "quality" of chaos can be characterised by Thouless energy which is the scale of energy level repulsion, i.e., two levels separated by $E_{\text{Thouless}}$ still has (effective) level repulsion. This can be manifested in the [[0062 Spectral form factor|SFF]]. The linear growth at large $t$ starts much earlier for strong chaos. ## BPS - eigenvalue chaos: [[0579 Random matrix theory|RMT]] statistics of energy levels; does not apply to BPS states - eigenvector chaos: eigenstates resemble random vectors (wrt simple operators) ([[0040 Eigenstate thermalisation hypothesis|ETH]]) applies to BPS states with small modifications ## Related - [[0167 Butterfly velocity]] - [[0474 Chaos bound]] ![[A0009 White swirls.jpeg]]