Quantum Matter: Concepts and Models
PhD course (7.5/15 p)

Lecturer and examiner
Henrik Johannesson, Soliden 3007, phone: 0768 237042, e-mail: henrik.johannesson@physics.gu.se

Guest lecturers
Abolfazl Bayat (UESTC), Hans-Peter Eckle (Ulm), Ulf Gran (Chalmers),
Hans Hansson (Stockholm), Mariana Malard (Brasilia)



About the course

All matter is quantum matter. While in the everyday world quantum effects are erased by information loss into a thermal
environment, quantum matter reveals its true character in the new worlds that can can now be created in laboratories.
These worlds exhibit properties that seem strange to us - many electrons cooperate to produce fractional electron charges,
electromagnetic response is quantized in units of fundamental constants of Nature, and charge can move freely on the
surface of an insulator. The study of quantum matter brings in ideas and techniques from condensed matter physics,
the physics of cold atoms, photonics, and quantum information science, making it a rapidly developing field of relevance
for fundamental physics as well as for applications in future quantum technologies.

In this PhD level course, offered in reading period 2 and 3 of this academic year (2019-20), we will study some of the
problems that make up this fascinating field of research. The focus will be on the central concepts, models, and analytical
methods that form its theoretical basis. You are expected to come with a firm grounding in quantum mechanics, statistical
physics, and condensed matter physics. Some familiarity with the language of quantum field theory and many-body theory
will be helpful, but is not necessary.

The course consists of four independent modules - Topological Quantum Matter, Quantum Systems Out of Equilibrium,
Strongly Correlated Quantum Matter, and Quantum Matter Meets Quantum Information. You can choose to sign up for
all four modules (or any two modules of your choice) for which you will earn 15 (7.5) points, based on attendance of lectures,
solutions of homework problems, and a small project at the end of the course.

Examination
Homework problems.

Schedule
Tuesdays and Thursdays, Nov 5 - Dec 19, Jan 14 - Feb 25, March 10 - 17, 10:00 - 11:45, in F-N6115




Lectures


Part I: Topological Quantum Matter


Nov 5 Introductory lecture. [slides from the lecture]

Some general information about the course; "What is quantum matter (really)?"; Topological quantum matter: some background;
Classifying phases by broken symmetries... and by topology!; How it all started: the integer quantum Hall effect; 3 min crash course
in topology; Gauss-Bonnet, Chern, Berry and all that... ; Cross fertilization between topology and physics; What's to come...



Nov 7 Berry phase. [lecture notes]

Discretized picture; Parallel transport; Berry connection; "Small" and "large" gauge transformations; Spin-1/2 particle in a magnetic field;
Berry curvature and Berry flux

Reading material
B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013), chapt. 2
D. Vanderbilt, Berry Phases in Electronic Structure Theory (Cambridge University Press, 2018), chapt. 3
T. D. Stanescu, Topological Quantum Matter and & Quantum Computation (CRC Press, 2013), Sec. I (The Geometric Phase)
Di Xiao, Ming-Che Chang, and Qian Niu, Rev. Mod. Phys. 82, 1959 (2010)
K. Durstberger, Geometric Phases in Quantum Theory
Y. Ben-Aryeh, Berry and Pancharatnam Topological Phases of Atomic and Optical Systems



Nov 12 Chern theorem. Adiabatic evolution. [lecture notes]

Proof of the Chern theorem; Interpreting Chern numbers on a torus; Adiabatic evolution

Reading material
Chern theorem
B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013), sec. 3.6
D. Vanderbilt, Berry Phases in Electronic Structure Theory (Cambridge University Press, 2018), chapt. 3
adiabatic evolution
M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes
M. V. Berry, Some geometric phases
general reference on topology for physicists
M. Nakahara, Geometry, Topology and Physics (IOP Publishing, 2003)



Nov 14 "Berryology" of electron structure. [lecture notes]

Review of electron structure theory; Berry phases and fluxes on the Brillouin zone; Wannier states

Reading material
S. M. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge University Press, 2019), Secs. 7.3, 7.4, 13.4



Nov 19 "Modern theory of electric polarization". SSH model. [lecture notes]

Modern theory of electric polarization. Introduction to the SSH model

Reading material
electric polarization
R. Resta and D. Vanderbilt, in Physics of Ferroelectrics: A Modern Perspective, eds. K. M. Rabe et al. (Springer, 2007).
SSH model
J. K. Asboth, L. Oroszlany, and A. Palyi, A Short Course on Topological Insulators, chapter 1
N. Batra and G. Sheet, Understanding Basic Concepts of Topological Insulators Through Su-Schrieffer-Heeger (SSH) Model



Nov 21 Symmetry-protected topological phases. [lecture notes] [slides]

Chiral, particle-hole, and time-reversal symmetries. Periodic table of symmetry-protected topological phases

Reading material
ten-fold way
C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of Topological Quantum Matter with Symmetries
general introductions to topological insulators
B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013)
T. D. Stanescu, Topological Quantum Matter and & Quantum Computation (CRC Press, 2013)
S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matter (Springer, 2012)
C. L. Kane, Topological Band Theory and the Z_2 Index
R. Shankar, Topological Insulators - A Review



Nov 26 Boundary states, Dirac theory, Jackiw-Rebbi, and all that... [lecture notes] [slides]

Protection of boundary states by chiral symmetry; 1D Dirac Hamiltonian from SSH; Charge fractionalization

Reading material
Dirac theory approach
S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matter, chapt. 1 (Springer, 2012)
C. L. Kane, Topological Band Theory and the Z_2 Index, sec. 3.3
charge fractionalization
R. Jackiv, Fractional Charge from Topology in Polyacetylene and Graphene , sec. 3.3



Dec 3 Integer quantum Hall effect: basics [lecture notes] [slides: intro] [slides: Kubo]

Basics on IQHE, Hall conductivity from the Kubo formula

Reading material
general
M. O. Goerbig, Quantum Hall Effects
D. Tong, Lectures on the Quantum Hall Effect , 1. The Basics & 2. The Integer Quantum Hall Effect
quantum Hall effect in graphene
M. O. Goerbig, The Quantum Hall Effect in Graphene - A Theoretical Perspective



Dec 5 Hall conductance and Chern numbers; Haldane model [lecture notes] [slides: spectral flow] [slides: Haldane model]

Hall conductance from Kubo on a torus, spectral flow, Chern insulators

Reading material
E. Fradkin, Field Theories of Condensed Matter Physics, (Cambridge University Press, 2013), secs. 12.3-12.7; 16.1-16.3



Dec 10 Quantum spin Hall effect; basics of superconductivity [lecture notes] [slides]

Quantum spin Hall effect; Z_2 index; superconductivity: from s-wave to p-wave

Reading material
quantum spin Hall effect
J. Maciejko et al., The Quantum Spin Hall Effect
Z_2 index
J. E. Moore and L. Balents, Topological invariants of time-reversal-invariant band structure
topological superconductivity
B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013), chapt. 16
M. Sato and Y. Ando, Topological superconductors: a review



Dec 12 Majorana physics [lecture notes] [slides]

Kitaev chain; Majorana zero modes; non-Abelian statistics, quantum gates from braiding Majoranas

Reading material
Kitaev chain and Majorana zero modes
J. Alicea, New directions in the pursuit of Majorana fermions in solid state physics
J. Sau, Kitaev chain and bulk-edge correspondence
Y. Oreg, From Kitaev model to an experiment
Braiding Majoranas
M. Leijnse and K. Flensberg Introduction to topological superconductivity and Majorana fermions
Bernard van Heck, Braiding of Majoranas
Annica Black-Schaffer, Topological Superconductors and Majorana Fermions




Part II: Quantum Systems out of Equilibrium


Dec 17 Equilibration and thermalization [lecture notes] [slides]

Closed quantum many-body systems out of equilibrium: old problems, new directions!;
equilibration and thermalization; eigenstate thermalization hypothesis

Reading material
overview
J. Eisert, M. Friesdorf, and C. Gogolin Quantum many-body systems out of equilibrium
equilibration and thermalization
M. Rigol, V. Dunjko, and M. Olshani, Thermalization and its mechanism for generic isolated quantum systems
J. M. Deutsch, Eigenstate thermalization hypothesis
L. D'Alessio et al., From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
random matrix theory
T. Guhr, Random Matrix Theory in Physics
G. Giacomo, M. Novaes, and P. Vivo, Introduction to Random Matrices - Theory and Practice



Dec 19 Periodically driven quantum systems [lecture notes] [slides]

Floquet theory

Reading material
A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven systems from a Floquet-space perspective,
Sec. 1 and 2, App. A
M. Bukov, L. D'Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems, Sec. 1 and 2



Feb 4 Floquet topological engineering [slides 1] [slides 2]

Illustrating Floquet theory: Periodically driven SSH model; High-frequency drives and the Magnus expansion;
Topological phases from periodic drives

Reading material
S. Blanes et al., A pedagogical approach to the Magnus expansion
M. S. Rudner and N. H. Lindner, Floquet topological insulators: from band structure engineering to novel quantum phenomena



Feb 11 & 19 Dynamical quantum phase transitions [lecture notes 11] [lecture notes 19] [slides 11] [slides 19]

Briefing on cold atom physics; Quantum phase transitions: equilibrium and nonequilibrium;
Nonanalyticities and Fisher zeros; Future challenges...

Reading material
cold atoms in optical lattices
I. Bloch, Ultracold atoms in optical lattices
D.-W. Zhang et al., Topological quantum matter with cold atoms
phase transitions: thermal, quantum, and dissipative
M. Vojta, Thermal and Quantum Phase Transitions
M. Kessler et al., Dissipative Phase Transition in a Central Spin System
dynamical quantum phase transitions
M. Heyl, Dynamical quantum phase transitions: a review




Part III: Strongly Correlated Quantum Matter


Jan 14 Background and motivation [slides]

Conventional approach to interacting electrons: DFT, Diagrammatic many-body theory, Landau Fermi liquid theory.
Breakdown of Fermi liquid theory: Strongly correlated electron systems.

Reading material
Philosophical musings
R. B. Laughlin and D. Pines,The Theory of Everything
Density Functional Theory
K. Capelle, A Bird's-Eye View of Density Functional Theory
Many-body theory of the interacting electron gas, Landau Fermi liquid theory
P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015), Chapt. 1, 6, 7
Non-Fermi liquids
C. M. Varma, Z. Nussinov, and W. van Saarloos, Singular Fermi Liquids



Jan 16 Mariana Malard: Introduction to the renormalization group [slides]

Renormalization group, basic concepts and formulas

Reading material
M. Malard, Sine-Gordon Model - Renormalization Group Solutions and Applications



Jan 21 Mariana Malard: More about RG; bosonization [slides]

Renormalization group procedures: Green's functions and averaging; Interacting electrons in 1D, bosonization



Jan 23 Mariana Malard: sine-Gordon model [slides]

sine-Gordon model from bosonization



Jan 28 Mariana Malard: Kosteritz-Thouless phase transition [slides]

Kosterlitz-Thouless phase transition from perturbative RG



Jan 30 Mariana Malard: Spin lattices and spin waves [slides]

Heisenberg model; 1D antiferromagnets; "Haldane conjecture"; 3D ferromagnets and spinwave theory



Feb 6 Ulf Gran: String theory methods in condensed matter physics [slides]

General motivation; Holographic duality; Graphene



Feb 13 Hans Hansson: Fractional quantum Hall effect and topological order [lecture notes 1] [lecture notes 2]

Characteristic properties of topologically ordered states; Thin torus limit of quantum Hall states;
Fractional statistics; Chern-Simons theory; Edge states

Reading material
T. H. Hansson and T. K. Kvorning, Effectice Field Theories for Topological States of Matter



Feb 18 Hans-Peter Eckle: Integrable models of quantum matter [lecture notes]

1D Hubbard and Heisenberg models

Reading material
Hans-Peter Eckle, Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz (Oxford University Press, 2019)



Feb 20 Hans-Peter Eckle: Bethe Ansatz [lecture notes]

Integrability; Algebraic Bethe Ansatz

Reading material
Hans-Peter Eckle, Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz (Oxford University Press, 2019)




Part IV: Quantum matter meets quantum information


Feb 25 Abolfazl Bayat: Quantum entanglement measures [slides]

Basic concepts, Quantum operations; Pure state entanglement; Mixed state entanglement

Reading material
M. B. Plenio and S. Virmani, An introduction to entanglement measures
J. Gray, L. Banchi, A. Bayat, and S. Bose, Machine Learning Assisted Many-Body Entanglement Measurement



Feb 26 Abolfazl Bayat: Special lecture on the many-body localization transition [slides]

Anderson localization; Many-body localization; Scaling near the MBL transition

Reading material
F. Alet and N. Laflorencie, Many-body localization: an introduction and selected topics
J. Gray, S. Bose, and A. Bayat, Many-body Localization Transition: Schmidt Gap, Entanglement Length & Scaling



Feb 27 Abolfazl Bayat: Quantum simulations of quantum many-body ground states [slides]

Quantum simulators: digital and analog; Quantum circuits; Certification problem; Variational quantum eigensolver;
Quantum field theory in the lab; Quantum impurity entanglement

Reading material
I. M. Georgescu, S. Ashhab, and F. Nori, Quantum Simulation
U. Farooq et al., Adiabatic many-body state preparation and information transfer in quantum dot arrays
A. Bayat et al., Certification of spin-based quantum simulators



Homework problems

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