**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.

Homework problems.

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

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...

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

Berry curvature and Berry flux

B. A. Bernevig,

D. Vanderbilt,

T. D. Stanescu,

Di Xiao, Ming-Che Chang, and Qian Niu, Rev. Mod. Phys.

K. Durstberger,

Y. Ben-Aryeh,

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

B. A. Bernevig,

D. Vanderbilt,

M. V. Berry,

M. V. Berry,

M. Nakahara,

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

S. M. Girvin and K. Yang,

Modern theory of electric polarization. Introduction to the SSH model

R. Resta and D. Vanderbilt, in

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

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

C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of Topological Quantum Matter with Symmetries

B. A. Bernevig,

T. D. Stanescu,

S.-Q. Shen,

C. L. Kane, Topological Band Theory and the Z_2 Index

R. Shankar, Topological Insulators - A Review

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

S.-Q. Shen,

C. L. Kane, Topological Band Theory and the Z_2 Index, sec. 3.3

R. Jackiv, Fractional Charge from Topology in Polyacetylene and Graphene , sec. 3.3

Basics on IQHE, Hall conductivity from the Kubo formula

M. O. Goerbig, Quantum Hall Effects

D. Tong, Lectures on the Quantum Hall Effect , 1. The Basics & 2. The Integer Quantum Hall Effect

M. O. Goerbig, The Quantum Hall Effect in Graphene - A Theoretical Perspective

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

E. Fradkin,

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

J. Maciejko et al., The Quantum Spin Hall Effect

J. E. Moore and L. Balents, Topological invariants of time-reversal-invariant band structure

B. A. Bernevig,

M. Sato and Y. Ando, Topological superconductors: a review

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

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

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

Closed quantum many-body systems out of equilibrium: old problems, new directions!;

equilibration and thermalization; eigenstate thermalization hypothesis

J. Eisert, M. Friesdorf, and C. Gogolin Quantum many-body systems out of equilibrium

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

T. Guhr, Random Matrix Theory in Physics

G. Giacomo, M. Novaes, and P. Vivo, Introduction to Random Matrices - Theory and Practice

Floquet theory

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

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

Topological phases from periodic drives

S. Blanes

M. S. Rudner and N. H. Lindner, Floquet topological insulators: from band structure engineering to novel quantum phenomena

Briefing on cold atom physics; Quantum phase transitions: equilibrium and nonequilibrium;

Nonanalyticities and Fisher zeros; Future challenges...

I. Bloch, Ultracold atoms in optical lattices

D.-W. Zhang

M. Vojta, Thermal and Quantum Phase Transitions

M. Kessler

M. Heyl, Dynamical quantum phase transitions: a review

Conventional approach to interacting electrons: DFT, Diagrammatic many-body theory, Landau Fermi liquid theory.

Breakdown of Fermi liquid theory: Strongly correlated electron systems.

R. B. Laughlin and D. Pines,The Theory of Everything

K. Capelle, A Bird's-Eye View of Density Functional Theory

P. Coleman,

C. M. Varma, Z. Nussinov, and W. van Saarloos, Singular Fermi Liquids

Renormalization group, basic concepts and formulas

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

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

sine-Gordon model from bosonization

Kosterlitz-Thouless phase transition from perturbative RG

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

General motivation; Holographic duality; Graphene

Characteristic properties of topologically ordered states; Thin torus limit of quantum Hall states;

Fractional statistics; Chern-Simons theory; Edge states

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

1D Hubbard and Heisenberg models

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

Integrability; Algebraic Bethe Ansatz

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

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

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

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

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

Quantum simulators: digital and analog; Quantum circuits; Certification problem; Variational quantum eigensolver;

Quantum field theory in the lab; Quantum impurity entanglement

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