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Home , Quantum chaos

Summer study course on many-body chaos Pablo Poggi Center for and Control (CQuIC) University of New Mexico

Session 1: Overview of classical and Hamiltonian chaos Wednesday June 2nd 2021 Background picture: Tent Rocks, New Mexico (USA) This course 2/17

• Format: self-study. Not a class. Not a seminar. • Participants read materials about the topics they have chosen, and then use what they learned to introduce the topic and lead a discussion about it. • Goals: learn about the fundamentals and get a taste of new developments in a fast-moving field. Accumulate material for future reference. • Start at ∼ 4 pm MDT – we go until 5.15 pm MDT including discussion • Schedule and materials available online at http://www.unm.edu/~ppoggi/ • Presentations will be recorded each week • UNM people can access recordings via Stream using their NetID • People outside of UNM can access a shared folder with the recordings via this link (password required) Quantum chaos: why? 3/17

Chaos in classical dynamical systems - deterministic and Extreme sensitivity of 1 trajectories to changes in initial conditions Quantum No trajectories, evolve quasi-periodically in time. No obvious characterization of chaos systems 1980’s: G. Casati, M. Berry, O. Correspondence: generic properties of quantum systems whose Bohigas, M. Giannoni, C. Schmit ‘Quantum Chaology’ 2,3 classical counterpart is chaotic (level statistics, eigenstate properties, semiclassical analysis, connection to theory) Complete characterization of quantum chaos? • Dynamics: connection to and thermalization in statistical mechanics 5 • Include systems without a well-defined . Quantum integrability. 4, 9, 10 • Properties of quantum features (i.e. entanglement) in generic (nonintegrable) quantum many body systems

Hard to simulate classically (in general) Dynamics of interacting quantum systems out of equilibrium Quantum computers to study quantum chaos! Quantum Sensitivity of quantum states to external perturbations Reliable quantum information processing 6,7 information

Dynamics of quantum information and generation of randomness Scrambling and random circuits 8 Today’s menu 4/17

푝 ℘ ℘ ℘ ℘ 1. and Hamiltonian formalism ℘ ℘ 푞 2. Integrability in classical systems and KAM theorem

3. Area-preserving maps and transition to chaos 풛∗

4. Lyapunov exponents 푊− 푊+

5. Ergodicity and

Today’s presentation is mostly based on “Nonlinear dynamics and quantum chaos: an Introduction” – by S. Wimberger Hamiltonian systems 5/17

• Lagrangian formalism “configuration space” (degrees of • Hamiltonian formalism “” freedom)

Or, in fancy form: e.g. Particle in 1D Equations of Symplectic matrix

푝2 Pendulum 퐻 휃, 푝 = 휃 − 푚푔퐿 푐표푠 휃 휃 2푚 Autonomous and non autonomous systems 퐸1 • 퐻(푞, 푝) time-independent ⇒ 퐻 푞 푡 , 푝 푡 = 퐻 푞 0 , 푝 0 = 퐸 휃 • 퐻(푞, 푝, 푡) = 퐸(푡) explicitly time-dependent 퐸2 푝휃 • Mappable to an autonomous system in an extended phase space 퐸3 • Can be visualized using surfaces of section (coming in a few slides) … Liouville’s theorem: time-evolution of a Hamiltonian system preserves phase space volume 휃 Canonical transformations 6/17

• I can take my original Hamiltonian and transform coordinates

• Relevant transformations are those which preserve the form of Hamilton’s EOMs:

• Canonical Transformation ℳ is canonical if and only if transformations • Poisson bracket:

• Suppose we have a canonical transformation ℳ such that 퐻′ = 퐻′(푷) (independent of Q)

and

• In this coordinates, the system is easily solvable We look for a transformation such that • ℳ has a generating function 푀(풒, 푷) such Hamilton-Jacobi equation that 푝 = 휕푀/휕푞 and 푄 = 휕푀/휕푃 푘 푘 푘 푘 (one nonlinear partial differential equation) Integrable systems 7/17

• Hamilton-Jacobi equation is hard to solve in general (except 푛 = 1, or separable systems) • In some cases, this can be achieved via identifying constants of motion

Integrability: A Hamiltonian system with 푛 degrees of constants freedom and 푠 = 푛 constants of motion 푐푖 풒, 풑 is called involution integrable independent

• In that case, the canonical transformation gives a solution to the H-J equation with 푰풊 = 풄풊

• Dynamics of an is restricted to a n-dimensional hypersurface in the 2n-dimensional phase space, which can be thought of as torus

Motion is periodic if 흎. 풌 = 0 for some 풌 ∈ ℤ푛

• Property: For 푛 = 1, all autonomous Hamiltonian systems are integrable: the required constant of motion is the energy 퐻(푞, 푝) Quantum integrability – session 4 KAM theorem 8/17

How stable are the ‘regular’ structures (invariant tori) of 푯 ? Let ퟎ • Kolmogorov – Arnold – Moser (KAM) theorem: for small enough 휀, Integrable Non-integrable there exists a torus for 퐻 with angle variables (휃, 퐼) perturbation New torus is ‘near’ the old one, and they transform smoothly

Comment: for a given energy, a tori is fixed, and KAM Sketch of proof in Section 3.7.4 of “Nonlinear dynamics and quantum chaos: an Introduction” by assumes is nonresonant (motion is not periodic) S. Wimberger

Consequence: regular structures persist under the presence of a small (non-integrable) perturbation. In this regime, the system is ‘quasi-integrable’ (constant of motion are only approximate)

KAM regime – quasi-integrability Chaos Breakdown ofw integrability and perturbation transition to chaos strength

complete integrability Poincaré maps and transition to chaos 9/17

• We want to study the transition between regular and chaotic motion in simple cases: small 푛 (degrees of freedom) • Recall all autonomous systems with 푛 = 1 are integrable – we need to add time-dependence

Surface of section (SOS): a tool for visualizing trajectories in low dimensional systems and to identify differences between regular and chaotic motion • n=1, non conservative and periodic 풒 • n=2, conservative

Poincaré Map (℘): Gives the evolution in the S.O.S. 퐳퐧+ퟏ = ℘ 풛풏 푝 ℘ • Discrete • is unique (each initial condition has its own trajectory) ℘ • Inherits many properties from the continuous dynamical system. ℘ ℘ • Area preserving map (as Liouville’s theorem) ℘ ℘

• If the system shows a periodic for an initial condition 푧0 ∈ 푆푂푆, then 풌 푞 the map has a fixed point of some order 푘 at 풛ퟎ: ℘ 풛ퟎ = 풛ퟎ Fixed points, stability and instability 10/17

Fixed points: ℘ 풛∗ = 풛∗ for 푧∗ ∈ 푆푂푆 푛 = 1 ∗ ∗ ∗ 휕℘푞 휕℘푞 ℘ 풛 + 훿풛 = 풛 + ℳ 풛 . 훿퐳 휕℘ 휕푞 휕푝 Near a fixed point, dynamics is given by the tangent map ℳ 풛 = = ∗ 휕풛 휕℘푝 휕℘푝 Eigenvalues of ℳ 풛 : 휕푞 휕푝 solutions of 푥2 − 푇푟 ℳ 푥 + det ℳ = 0 So, Tr(ℳ 풛∗ ) determines type of motion near fixed point: = 1

Tr ℳ < 2 → 푥 = 푒±푖훽 Tr ℳ > 2 → 푥 = 푒±휆 Very unstable under ± ± Separatrix: 푊 and 푊 coincide Elliptic fixed point Hyperbolic fixed point − + perturbations!

rotation

separatrix 풛∗ 풛∗ 푝휃

libration 푊− 푊+

Tr ℳ = 2 → 푥± = ±1 unstable and stable Parabolic fixed point manifolds 휃 pendulum Perturbed motion: all hell breaking loose 11/17

We introduce a nonintegrable perturbation:

푊− and 푊+ coincide (separatrix) Fixed 퐼

푊− and 푊+ intersect (homoclinic points) Variable 퐼 (broken integrability)

• A homoclinic point (HP) is a point in the SOS which belongs to both 푊− and 푊+

• Since 푊± are ‘invariant curves’ ℘ 푊± = 푊±, then if 푧 is a HP ⇒ ℘(푧) is a HP. So: there is an infinite sequence of them, bunched up together

• Points in the either 푊± which are not HP’s, also evolve according to ℘, area preserving map

℘ Complicated motion, highly unstable with large … Homoclinic deviations with respect to points initial conditions! – onset of Chaos Fixed point Homoclinic tangle and stochastic layer 12/17

Homoclinic tangle

From S. Wimberger, Nonlinear dynamics and quantum chaos: an Introduction From R. Ramirez-Ros, Physica D: Nonlinear Phenomena, 210 149-179 (2005) Stochastic layer: complex motion around the (former) separatrix, makes the SOS trajectories look like randomly distributed points 휖 = 0 휖 > 0 Driven pendulum

A. Chernikov, R. Zagdeev and G. Zaslavsky, “Chaos: how regular can it be?” Today 41, 11, 27 (1988) Transition to chaos in a kicked top 13/17

Phase space variables are the components of the angular 2 2 2 2 momentum 푱 = (퐽푥, 퐽푦, 퐽푧). Also, 푱 = 퐽푥 + 퐽푦 + 퐽푧 is conserved perturbation Phase space dimension = 2 ⇒ 풏 = ퟏ

F. Haake, M. Kus and R. Scharf, Classical and quantum chaos for a kicked top. Z. Phys. B – Cond. Mat. 65, 381-395 (1987) Lyapunov exponents 14/17

Chaos exponential separation of nearby phase space trajectories

• Dynamics near a fixed point ℘ 풛∗ + 훿풛 = 풛∗ + ℳ 풛∗ . 훿퐳 푒휆 0 • Hyperbolic FP: ℳ 풛∗ = (in normal coordinates) 0 푒−휆 ∗ 푛휆 −푛휆 • After 푛 time steps: ℘푛 = 풛 + 푒 훿풛풖 + 푒 훿풛풔

Lyapunov exponent (흀)

Kicked top More generally: 1 | ℘ 풛+훿풛 −℘ 풛 | • Maximal Λ 풛 = lim lim log 푛 푛 푛→∞ 훿풛→0 푛 | 훿풛 | • For a generic 훿풛, Λ 풛 = max. eigenvalue of ℳ 풛

• If I take a set of {훿풛풊}, I get a ‘Lyapunov spectrum’ {휆푖}

• In a globally chaotic regime, Λ 풛 is mostly independent of initial condition

OTOCs and scrambling – Sessions 6 and 7 From M. Muñoz et al PRLPhys. Rev. Lett. 124, 110503 (2020) Ergodicity and mixing 15/17

푡 Some notation: 푇: Dynamical system 푇: 퐺 × Ω → Ω 푇 휔 ∈ Ω 퐺 ∼ time Ω ∼ phase space 휈: Measure such that 휈(Ω) = 1, invariant under 푇: 휈 Tt A = 휈(A)

Ergodicity A dynamical system is ergodic if all 푻-invariant sets (푻풕 푨 = 푨) are such that either 흂 퐀 = ퟏ 풐풓 ퟎ i.e, there cannot be an invariant set which is not a fixed point, or the whole space

Orbits fill the whole Phase space average = time-average: where available phase space

Note that: Ergodicity Chaos Integrable systems can lead to ergodicity in the available phase space (torus)

Mixing A dynamical system is (strongly) mixing if for any two sets 푨, 푩 ⊆ 훀,

푡 푇푡(퐴) 푇 (퐴) Mixing ⇒ Ergodicity Chaos ⇒ Mixing Not mixing Mixing 퐴 퐴 Ω Ω Thermalization in closed quantum systems – Session 5 Ergodic hierarchy and correlation functions 16/17

Take a correlation function with functions in Ω and

Ergodic hierarchy Ergodic Weak Mixing Mixing ⊃ … 푪(풕) = ퟎ ⊃ |푪 풕 | = ퟎ 푪(풕) → ퟎ ⊃

• Analyzing the behavior of correlation functions is From P. Claeys and A. Lamacraft, “Ergodic and useful for systems with many d.o.f.s where is hard Nonergodic Dual-Unitary to characterize chaos ‘globally’ (Lyapunovs, etc) Quantum Circuits with Arbitrary Local Dimension” PRL 126 100603 (2021) Summary 17/17

• Integrable systems are characterized by conserved quantities which allow for the dynamics to be solved using action- angle variables. Motion lies on invariant tori • Upon addition of a nonintegrable perturbation, tori persist (KAM theorem) for a while. After that, proliferation of instabilities leads to complex motion, and hypersensitivity to initial conditions • Global chaotic behavior can be characterized at short times via Lyapunov exponents, and for long times via mixing and ergodicity

Next week: Introduction to quantum chaos. Some good reads to get some background: • F. Haake – ‘Quantum signatures of chaos’ – Chapter 1 • M. Berry – ‘Chaos and the semiclassical limit of ’ (link) • D. Poulin – ‘A rough guide to quantum chaos’ (link)

References • Main reference: Nonlinear dynamics and quantum chaos: an Introduction – by Sandro Wimberger • S. Aravinda et al, From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy, :2101.04580. • A. Chernikov, R. Zagdeev and G. Zaslavsky, Chaos: how regular can it be?. Physics Today 41, 11, 27 (1988) • Notes from the course “Chaos and Quantum Chaos 2021” at TU Dresden, taught by Roland Ketzmerick. Some materials publicly available in English at https://tu-dresden.de/mn/physik/itp/cp/studium/lehrveranstaltungen/chaos-and-quantum-chaos-2021?set_language=en Extra stuff Autonomous and non autonomous systems • 퐻(푞, 푝) time-independent ⇒ 퐻 푞 푡 , 푝 푡 = 퐻 푞 0 , 푝 0 = 퐸 • 퐻(푞, 푝, 푡) = 퐸(푡) explicitly time-dependent New EOMs 푝0 Can be mapped to an 푑푞′ 휕ℋ 푑푝′ 휕ℋ 푝′ = −퐸, 푝 ℋ 푝′, 푞′ = 퐻 푞, 푝, 푡 − 퐸(푡) autonomous system = , = − ′ in an extended phase 푞′ = (푡, 푞) 푑휏 휕푝′ 푑휏 휕푞 space New Hamiltonian 푞0 This leads to

푑푞0 휕ℋ 푑푡 휕ℋ • = ↔ = − = 1 → 휏 = 푡 + 푐표푛푠푡. This variable becomes ‘trivial’ 휕퐻 휕퐻 푑휏 휕푝0 푑휏 휕퐸 Also 푞ሶ = , 푝ሶ = − 휕푝 휕푞 푑푝 휕ℋ 푑퐸 휕ℋ “one and a half • 0 = − ↔ − = − One extra independent variable for the old variables 푑휏 휕푞0 푑푡 휕푡 degrees of freedom”

Phase space variables are the components of the angular 2 2 2 2 momentum 푱 = (퐽푥, 퐽푦, 퐽푧). Also, 푱 = 퐽푥 + 퐽푦 + 퐽푧 is conserved perturbation Phase space dimension = 2 ⇒ 풏 = ퟏ Existence / separability of H-J equations and integrability

From: ‘Integrable vs non-integrable systems’ on Physics.StackExchange. See also: ‘Constants of motion vs integrals of motion vs first integrals’ on Physics.StackExchange