summer study course on many-body quantum chaos
TRANSCRIPT
Summer study course on many-body quantum chaos
Session 1: Overview of classical mechanics and Hamiltonian chaosWednesday June 2nd 2021
Pablo PoggiCenter for Quantum Information and Control (CQuIC)University of New Mexico
Background picture: Tent Rocks, New Mexico (USA)
2/17This course
β’ Format: self-study. Not a class. Not a seminar.
β’ 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.
β’ Goals: learn about the fundamentals and get a taste of new developments in a fast-moving field. Accumulate material for future reference.
β’ Start at βΌ 4 pm MDT β we go until 5.15 pm MDT including discussion
β’ Schedule and materials available online at http://www.unm.edu/~ppoggi/
β’ Presentations will be recorded each week
β’ UNM people can access recordings via Stream using their NetID
β’ People outside of UNM can accessa shared folder with therecordings via this link (password required)
3/17Quantum chaos: why?
Chaos in classical dynamical systems - deterministic randomness and butterfly effect Extreme sensitivity of trajectories to changes in
initial conditionsQuantum systems
No trajectories, observables evolve quasi-periodically in time. No obvious characterization of chaos
Correspondence: generic properties of quantum systems whose classical counterpart is chaotic (level statistics, eigenstate properties, semiclassical analysis, connection to random matrix theory)
1980βs: G. Casati, M. Berry, O. Bohigas, M. Giannoni, C. Schmit
βQuantum Chaologyβ
Complete characterization of quantum chaos?
β’ Dynamics: connection to ergodicity and thermalization in statistical mechanics
β’ Include systems without a well-defined classical limit. Quantum integrability.
β’ Properties of quantum features (i.e. entanglement) in generic (nonintegrable) quantum many body systems
1
2,3
5
4, 9, 10
Quantum information
Dynamics of interacting quantum systems out of equilibriumHard to simulate classically (in general)
Quantum computers to study quantum chaos!
Sensitivity of quantum states to external perturbations Reliable quantum information processing
Quantum metrology
Dynamics of quantum information and generation of randomness Scrambling and random circuits
6,7
8
4/17Todayβs menu
1. Classical mechanics and Hamiltonian formalism
2. Integrability in classical systems and KAM theorem
3. Area-preserving maps and transition to chaos
4. Lyapunov exponents
5. Ergodicity and mixing
Todayβs presentation is mostly based on βNonlinear dynamics and quantum chaos: an Introductionβ β by S. Wimberger
π
π
β
β
β
β ββ
πβ
π+πβ
5/17Hamiltonian systems
β’ Lagrangian formalism
β’ Hamiltonian formalism
βconfiguration spaceβ
βphase spaceβ(degrees of freedom)
Equations of motion
Or, in fancy form:
Symplectic matrix
Autonomous and non autonomous systems
β’ π»(π, π) time-independent β π» π π‘ , π π‘ = π» π 0 , π 0 = πΈ
β’ π»(π, π, π‘) = πΈ(π‘) explicitly time-dependent
β’ Mappable to an autonomous system in an extended phase space
β’ Can be visualized using surfaces of section (coming in a few slides)
π
e.g. Particle in 1D
Pendulum π» π, ππ =ππ2
2πβπππΏ πππ π
π
ππ
πΈ1
πΈ2
πΈ3
β¦
Liouvilleβs theorem: time-evolution of a Hamiltonian system preserves phase space volume
6/17Canonical transformations
β’ I can take my original Hamiltonian and transform coordinates
β’ Suppose we have a canonical transformation β³ such that π»β² = π»β²(π·) (independent of Q)
and
β’ In this coordinates, the system is easily solvable
β’ β³ has a generating function π(π,π·) such that ππ = ππ/πππ and ππ = ππ/πππ
We look for a transformation such that
Hamilton-Jacobi equation(one nonlinear partial differential equation)
β’ Relevant transformations are those which preserve the form of Hamiltonβs EOMs:
Canonical transformations
β’ Transformation β³ is canonical if and only if
β’ Poisson bracket:
7/17Integrable systems
β’ Hamilton-Jacobi equation is hard to solve in general (except π = 1, or separable systems)
β’ In some cases, this can be achieved via identifying constants of motion
Quantum integrability β session 4
Integrability: A Hamiltonian system with π degrees of freedom and π = π constants of motion ππ π, π is called
integrable
constants
involution
independent
β’ In that case, the canonical transformation gives a solution to the H-J equation with π°π = ππ
β’ Property: For π = 1, all autonomous Hamiltonian systems are integrable: the required constant of motion is the energy π»(π, π)
β’ Dynamics of an integrable system is restricted to a n-dimensional hypersurface in the 2n-dimensional phase space, which can be thought of as torus
Motion is periodic if π. π = 0 for some π β β€π
8/17KAM theorem
How stable are the βregularβ structures (invariant tori) of π―π?
Consequence: regular structures persist under the presence of a small (non-integrable) perturbation. In this regime, the system is βquasi-integrableβ (constant of motion are only approximate)
Let
Integrable(π, πΌ)
Non-integrable perturbation
β’ Kolmogorov β Arnold β Moser (KAM) theorem: for small enough ν, there exists a torus for π» with action angle variables
New torus is βnearβ the old one, and they transform smoothly
Comment: for a given energy, a tori is fixed, and KAM assumes is nonresonant (motion is not periodic)
Chaosperturbation
strength
complete integrability
KAM regime βquasi-integrability
wBreakdown of integrability and transition to chaos
Sketch of proof in Section 3.7.4 of βNonlinear dynamics and quantum chaos: an Introductionβ by S. Wimberger
9/17PoincarΓ© maps and transition to chaos
PoincarΓ© Map (β): Gives the evolution in the S.O.S. π³π§+π = β ππ
β’ We want to study the transition between regular and chaotic motion in simple cases: small π (degrees of freedom)
β’ Recall all autonomous systems with π = 1 are integrable β we need to add time-dependence
β’ Discrete dynamical system
β’ Time evolution is unique (each initial condition has its own trajectory)
β’ Inherits many properties from the continuous dynamical system.
β’ Area preserving map (as Liouvilleβs theorem)
β’ If the system shows a periodic orbit for an initial condition π§0 β πππ, then
the map has a fixed point of some order π at ππ: βπ ππ = ππ
π
π
β
β
β
β β
β
Surface of section (SOS): a tool for visualizing trajectories in low dimensional systems and to identify differences between regular and chaotic motion
β’ n=1, non conservative and periodic
β’ n=2, conservativeπ
10/17
So, Tr(β³ πβ ) determines type of motion near fixed point:
Fixed points, stability and instability
Fixed points: β πβ = πβ for π§β β πππ
Near a fixed point, dynamics is given by the tangent map β³ π =πβ
ππ=
πβπ
ππ
πβπ
ππ
πβπ
ππ
πβπ
ππ
π = 1
β πβ + πΏπ = πβ +β³ πβ . πΏπ³
= 1
Eigenvalues of β³ πβ :
solutions of π₯2 β ππ β³ π₯ + det β³ = 0
Tr β³ < 2 β π₯Β± = πΒ±ππ½
Elliptic fixed point
Tr β³ = 2 β π₯Β± = Β±1Parabolic fixed point
πβ
Tr β³ > 2 β π₯Β± = πΒ±π
Hyperbolic fixed point
unstable and stablemanifolds
πβ
π+πβ
Very unstable under perturbations!
Separatrix: πβ and π+ coincide
pendulumπ
ππ
rotation
libration
separatrix
11/17Perturbed motion: all hell breaking looseWe introduce a nonintegrable perturbation:
πβ and π+ coincide (separatrix) Fixed πΌ
πβ and π+ intersect (homoclinic points) Variable πΌ (broken integrability)
β’ A homoclinic point (HP) is a point in the SOS which belongs to both πβ and π+
β’ Since πΒ± are βinvariant curvesβ β πΒ± = πΒ±, then if π§ is a HP β β(π§) is a HP. So: there is an infinite sequence of them, bunched up together
β’ Points in the either πΒ± which are not HPβs, also evolve according to β, area preserving map
Complicated motion, highly unstable with large
deviations with respect to initial conditions! β onset of
Chaos
β
Fixed point
Homoclinic points
β¦
12/17Homoclinic tangle and stochastic layer
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
Driven pendulumπ = 0 π > 0
A. Chernikov, R. Zagdeev and G. Zaslavsky, βChaos: how regular can it be?β Physics Today 41, 11, 27 (1988)
13/17Transition to chaos in a kicked topPhase space variables are the components of the angular
momentum π± = (π½π₯ , π½π¦ , π½π§). Also, π±2 = π½π₯2 + π½π¦
2 + π½π§2 is
conserved
Phase space dimension = 2 β π = πperturbation
F. Haake, M. Kus and R. Scharf, Classical and quantum chaos for a kicked top. Z. Phys. B β Cond. Mat. 65, 381-395 (1987)
14/17Lyapunov exponents
exponential separation of nearby phase space trajectoriesChaos
β’ Dynamics near a fixed point β πβ + πΏπ = πβ +β³ πβ . πΏπ³
β’ Hyperbolic FP: β³ πβ = ππ 00 πβπ
(in normal coordinates)
β’ After π time steps: βπ = πβ + ππππΏππ + πβπππΏππ
Lyapunov exponent (π)
More generally:
β’ Maximal Lyapunov exponent Ξ π = limπββ
limπΏπβ0
1
πlog
| βπ π+πΏπ ββπ π |
| πΏπ |
β’ For a generic πΏπ, Ξ π = max. eigenvalue of β³ π
β’ If I take a set of {πΏππ}, I get a βLyapunov spectrumβ {ππ}
β’ In a globally chaotic regime, Ξ π is mostly independent of initial condition
OTOCs and scrambling β Sessions 6 and 7 From M. MuΓ±oz et al PRLPhys. Rev. Lett. 124, 110503 (2020)
Kicked top
15/17Ergodicity and mixing
Some notation: π: Dynamical system π: πΊ Γ Ξ© β Ξ© πΊ βΌ time Ξ© βΌ phase space ππ‘ π β Ξ©
π: Measure such that π(Ξ©) = 1, invariant under π: π Tt A = π(A)
Ergodicity A dynamical system is ergodic if all π»-invariant sets (π»π π¨ = π¨) are such that either π π = π ππ π
i.e, there cannot be an invariant set which is not a fixed point, or the whole space
Phase space average = time-average: where Orbits fill the whole 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 π¨,π© β π,
π΄
ππ‘(π΄)
Ξ©
ππ‘(π΄)
Ξ©π΄
Not mixing MixingMixing β Ergodicity Chaos β Mixing
Thermalization in closed quantum systems β Session 5
16/17Ergodic hierarchy and correlation functions
Take a correlation function
with functions in Ξ©
and
Ergodic hierarchy
Ergodic Weak Mixing Mixing
πͺ(π) = π |πͺ π | = π πͺ(π) β πβ β β β¦
β’ Analyzing the behavior of correlation functions is useful for systems with many d.o.f.s where is hard to characterize chaos βgloballyβ (Lyapunovs, etc)
From P. Claeys and A. Lamacraft, βErgodic and Nonergodic Dual-Unitary Quantum Circuits with Arbitrary Local Hilbert Space Dimensionβ PRL 126 100603 (2021)
17/17Summary
β’ Integrable systems are characterized by conserved quantities which allow for the dynamics to be solved using action-angle variables. Motion lies on invariant tori
β’ 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
β’ Global chaotic behavior can be characterized at short times via Lyapunov exponents, and for long times via mixing and ergodicity
References
β’ Main reference: Nonlinear dynamics and quantum chaos: an Introduction β by Sandro Wimberger
β’ S. Aravinda et al, From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy, arxiv:2101.04580.
β’ A. Chernikov, R. Zagdeev and G. Zaslavsky, Chaos: how regular can it be?. Physics Today 41, 11, 27 (1988)
β’ Notes from the course βChaos and Quantum Chaos 2021β 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
Next week: Introduction to quantum chaos. Some good reads to get some background:
β’ F. Haake β βQuantum signatures of chaosβ β Chapter 1
β’ M. Berry β βChaos and the semiclassical limit of quantum mechanicsβ (link)
β’ D. Poulin β βA rough guide to quantum chaosβ (link)
Extra stuff
Autonomous and non autonomous systems
β’ π»(π, π) time-independent β π» π π‘ , π π‘ = π» π 0 , π 0 = πΈ
β’ π»(π, π, π‘) = πΈ(π‘) explicitly time-dependent
πβ² = βπΈ, ππβ² = (π‘, π)
Can be mapped to an autonomous system in an extended phase
space
β πβ², πβ² = π» π, π, π‘ β πΈ(π‘)ππβ²
ππ=
πβ
ππβ²,ππβ²
ππ= β
πβ
ππβ²
π0
π0
New Hamiltonian
New EOMs
This leads to
β’ππ0
ππ=
πβ
ππ0β
ππ‘
ππ= β
πβ
ππΈ= 1 β π = π‘ + ππππ π‘.
β’ππ0
ππ= β
πβ
ππ0β β
ππΈ
ππ‘= β
πβ
ππ‘
This variable becomes βtrivialβ
One extra independent variableβone and a half
degrees of freedomβ
Also αΆπ =ππ»
ππ, αΆπ = β
ππ»
ππ
for the old variables
Phase space variables are the components of the angular momentum π± = (π½π₯ , π½π¦ , π½π§). Also, π±2 = π½π₯
2 + π½π¦2 + π½π§
2 is
conserved
Phase space dimension = 2 β π = πperturbation
Existence / separability of H-J equations and integrability
From: βIntegrable vs non-integrable systemsβ on Physics.StackExchange. Seealso: βConstants of motion vs integrals of
motion vs first integralsβ onPhysics.StackExchange