PHYC 500.001 Adv Sem: “Many-body quantum chaos”
Course coordinator: Pablo M. Poggi (ppoggi@unm.edu)
Course duration: June 2nd
to August 4th 2021 (there will be ten sessions in total). All
sessions will be held online via Zoom.
Format: This course is a survey of topics broadly related to
chaos in quantum systems, with emphasis in its manifestations in many-body
quantum systems. The course is designed
to be a ‘self-study’ program. In the first two sessions, we will introduce some
basic aspects of quantum chaos and give an overview of the topics that will be
treated later on. Reading material will be assigned every week, including
lecture notes and research articles. In the remaining sessions, course
participants will present sections from the reading material and facilitate
group discussions. Participants are encouraged to also propose other topics of
their interest.
Prerequisites: This course assumes familiarity with quantum
mechanics at the level of Physics 521-522.
Assignments: All participants are required to read the materials
assigned for each session and
participate in group discussions. Participants should
be willing to present assigned material in later
sessions.
Syllabus and
program: Sessions 1 and 2 will be given by course
coordinator. Sessions 3 through 10 will be based presentations given by course
participants.
Sessions 1 and 2
(June 2nd and 9th): Introduction and general aspects of
quantum chaos
·
Notions from classical mechanics:
Chaos and integrability, mixing, ergodicity
· Aspects of quantum chaos in systems with few degrees of
freedom
o
Correlation in energy spectra.
Level spacing statistics.
o
Eigenstate structure and relation
to random vectors
o
Dynamics: Loschmidt
echo and spectral form factors
·
Chaos in many-body quantum
systems – overview of the course
References: [Haake] [Wimberger] [Gubin2012]
Session 3 (June 16th): Random matrix theory and chaos.
Gaussian and circular random matrix ensembles. Random eigenvectors
and Porter Thomas distribution. Random matrix ensembles for many-body systems.
Reference: [Haake]. See also: [QChaosSantos]
Session 4 (June 23rd): Integrability in quantum systems
Notions of integrability in quantum systems. Integrability in
many-body systems. Interacting vs non interacting models. Free fermions, Bethe
ansatz and Richardson-Gaudin models.
References:
[Caux2011] – [VieiraCostello2017]
Session 5 (June 30th): Thermalization in closed quantum
systems
Equilibration, thermalization, and connection to random matrix
aspects of quantum chaos. Eigenstate thermalization hypothesis (ETH).
Constrained equilibrium and generalized Gibbs ensembles.
Reference:
[D’Alessio2016]. See also: [Borgonovi2016], [Rigol2008], [Kaufman2016], [QChaosNation]
Session 6 (July 7th): Scrambling and
out-of-time-order correlators I
Information spreading and scrambling in many body systems.
Operator spreading and operator entanglement.
References:
[Hosur2016], [SwingleLecNotes]. See also [Swingle2016], [Zhuang2019], [Google2020]
Session 7 (July
14th): Scrambling and out-of-time-order correlators II
Semiclassical theory of OTOCs and connection to Lyapunov
exponents. Scrambling vs relaxation. Out-of-time-ordered and time-ordered
correlation functions.
References: [Zhuang2019],
[Swingle2016], [Rozenbaum2017], [Roberts2017]
Session 8 (July 21st): Random unitary
evolution and quantum complexity
Haar randomness. Unitary t-designs and frame potentials. Random
quantum circuits.
Main reference:
[Roberts2017]. See also [Harrow2009], [Brandao2012].
Session 9 (July 26th):
Deviations from chaos and ETH: Many body localization and quantum many body
scars.
References: [Nandkishore2015]
[QChaosBorhdt] [QChaosBurnell]
Session 10 (July 28th):
Solvable chaotic systems (dual unitary circuits)
References: [Claeys2021] [Bertini
2020] [QChaosKos] [Aravinda2021]
Other topics
(didn’t make the cut!)
·
Quantum information scrambling in
black holes [Hayden2007] [Maldacena2015]
·
Solvable chaotic systems (dual
unitary circuits)
·
Sachdev-Ye-Kitaev
(SYK) model and quantum chaos
·
Deviations from ETH: many body
localization / quantum many body scars
·
Chaos in open quantum systems
(random Liouvillians)
·
Ergodicity and dynamical
typicality
·
Chaos and information (KS
entropy)
·
Quantum echoes and
irreversibility
·
Adiabatic gauge potentials and
sensitivity of eigenstates to perturbations
References
[Aravinda2021] Aravinda et al, From dual-unitary to quantum Bernoulli
circuits: Role of the entangling power in constructing a quantum ergodic
hierarchy. Arxiv: 2021.04580 (link)
[Bertini
2020] Bertini and Piroli, Scrambling
in random unitary circuits: exact results. Phys. Rev. B 2020 (link)
[Borgonovi2016] Borgonovi, Izrailev, Santos, Zelevinsky, Quantum chaos and thermalization in isolated
systems of interacting particles. Phys. Rep. 2016 (link)
[Brandao2012]
Hunter-Jones, Local Random Quantum Circuits are Approximate
Polynomial-Designs, Comm. Math. Phys. 2012 (link)
[Caux2011] Caux
and Mossel, Remarks on the notion of quantum Integrability, J. Stat.
Mech. 2011 (link)
[Claeys2021] Claeys
and Lamacraft, Ergodic
and Nonergodic Dual-Unitary Quantum Circuits with Arbitrary Local Hilbert Space
Dimension, Phys. Rev. Lett. 2021 (link)
[Cotler2017] Cotler,
Hunter-Hones, Liu and Yoshida, Chaos, complexity, and random matrices, HEP 2017
(link)
[D’Alessio2016] D’Alessio, Kafri, Polkovnikov and Rigol. From
quantum chaos and eigenstate thermalization to statistical mechanics and
thermodynamics. Adv. Phys. 2016 (link)
[Google2020] Google team and
others, Information Scrambling in Computationally Complex Quantum Circuits,
arxiv 2021 (link)
[Gubin2012] Gubin
and Santos, Quantum chaos: an introduction via chains of interacting spins
˝. Am. J. Phys. 2012 (link)
[Haake] F. Haake, Quantum Signatures of Chaos. Springer, Boston MA.
[Harrow2009]
Harrow and Low, Random Quantum Circuits are Approximate 2-designs, Comm.
Math. Phys. 2009 (link).
[Hayden2007] Hayden and Preskill, Black Holes as Mirrors, JHEP 2007 (link)
[Hosur2016] Hosur, Qi, Roberts
and Yoshida, Chaos in quantum channels, JHEP 2016 (link)
[Kaufman2016] Kaufman et al, Quantum
thermalization through entanglement in an isolated many-body system.
Science 2016 (link)
[Maldacena2015] Maldacena, Shenker, Stanford, A bound on chaos, arxiv 2015 (link)
[Nandkishore2015] Nandkishore
and Huse, Many-body localization and
thermalization in quantum statistical mechanics. Annu.
Rev. Condens. Matter Phys. 2015 (link)
[PolkovnikovLecNotes]
A. Polkovnikov, Lecture notes on Quantum Ergodicity. http://physics.bu.edu/~asp28/teaching/PY_747.pdf
[QChaosBohrdt]
A. Bohrdt, Probing dynamics in quantum simulators,
https://www.youtube.com/watch?v=yyZOi1BVPZI
[QChaosBurnell]
F. Burnell, Exact models for many-body quantum scars, https://www.youtube.com/watch?v=IBpMcRd4T54
[QChaosKos]
P. Kos, Solvable chaotic many-body quantum systems, https://www.youtube.com/watch?v=jL12q5F8XMw
[QChaosNation] C. Nation, Thermalization
dynamics and the emergence of Brownian motion in chaotic quantum systems: https://www.youtube.com/watch?v=Rr_1oLfS8nw
[QChaosSantos] L. Santos, Indicators of
many-body quantum chaos: https://www.youtube.com/watch?v=h1-xFUJ_T_s&t=2950s
[Rigol2008] Rigol,
Dunjko and Olshanii, Thermalization
and its mechanism for generic isolated quantum systems. Nature 2008 (link)
[Roberts2017] Roberts and
Yoshida, Chaos and complexity by design. JHEP 2017 (link)
[Rozenbaum2017] Rozenbaum, Ganeshan and Galitsky,
Lyapunov Exponent and Out-of-Time-Ordered Correlator’s Growth Rate in a Chaotic
System, PRL 2017 (link)
[Swingle2016] Swingle, Bentsen, Schleier-Smith and Hayden, Measuring the scrambling of
quantum information, PRA 2016 (link)
[SwingleLecNotes]
B. Swingle. Lecture notes on Quantum information scrambling (link)
[VieiraCostelloCourse]
P. Vieira and K. Costello. Introduction to quantum integrable systems (link)
[Wimberger] S. Wimberger, Nonlinear dynamics and quantum chaos. Springer.
[Zhuang2019] Zhuang, Schuster,
Yoshida and Yao, Scrambling and complexity in phase space. Phys. Rev. A
2019 (link)