Topological Quantum Matter
7.5 ECTS, fall 2014

Lecturer and examiner
Henrik Johannesson, Soliden 3007, phone: 786 9164, e-mail: henrik.johannesson@physics.gu.se



About the course

This is a course intended for PhD students and second year Masters students with a strong theory background. The first part of the course, with 10 double hour lectures in the period 12/9 - 3/10, 30/10 - 31/10, will provide for the very basics of the subject, essentially covering chapters 1-6, 9, and 10 in the book Topological Insulators: Dirac Equation in Condensed Matters by Shun-Qing Shen. The aim is to introduce language and concepts necessary to study the research literature in the subject. The lectures will be properly cushioned! Although the focus of the course will be on symmetry-protected topological matter (topological insulators and topological superconductors), topologically ordered matter (phases with long-range quantum entanglement) will also be discussed.

The second part of the course will comprise five guest lectures in the period 3/11 - 12/12, with discussion of some more advanced material and applications.



Course material
Scanned lecture notes will be made available online. Additional texts will be distributed in class or linked to the course home page.


Examination
Homework problems and a project based on one of the guest lectures.


Schedule
Friday 12/9: 13:15 - 15:00 in F7107 (introductory lecture)
Wednesdays 17/9, 24/9,1/10: 13:15 - 15:00 in N6115
Fridays 19/9, 26/9, 3/10: 13:15 - 15:00 in N6115
Thursday 30/10: 15:15 - 17:00 in Faraday
Friday 31/10: 13:15 - 15:00 in Faraday
One additional double-hour lecture to be scheduled later.

Guest lectures: to be announced.



Suggested reading

Lecture 1 and 2: Introduction (mostly about topological insulators). Dirac equation. Jackiw-Rebbi solution. Quadratic correction to the Dirac equation. Skyrmions. Zero-energy boundary solutions. The slides from my introductory lecture are available here, with my lecture notes here. My notes follow the first chapter in Shun-Qing Shen's book rather closely. Before going through my notes (or your own notes from the lectures!), you may wish to read the popular-science style introduction to topological insulators by Charlie Kane and Joel Moore. For those of you who wish some more background reading already at this stage of the course, I can recommend the review on the quantum spin Hall effect by Macijeko, Hughes, and Zhang. (For the purpose of this course: A "quantum spin Hall system" is just another name for a 2D topological insulator.) If you feel that you need to refresh yourself about basics and notation of Dirac theory, I suggest that you work through p 25-31 of this quick- and-dirty review. The two-component spinor formalism that can be used in 1D and 2D is easier to handle, and you will get to practice on it in several of the homework problems!

Lecture 3: Helical edge states in 2D. Surface states in 3D. Lattice version of Dirac theory with a quadratic term. My lecture notes are here. For a quick reminder about some basics about lattices, reciprocal space, Brillouin zones, and all that jazz, see this. If you need a referesher also of second quantization, have a look at my appendix C in Kinaret and Johannesson.

Lecture 4: Bound states at the ends of a 1D lattice. Topological invariants: Warming up with Gauss-Bonnet. Berry phase. Lecture notes. A thorough discussion of the adiabatic theorem (that we exploited to derive the Berry phase) can be found in Messiah's book on quantum mechanics.

Lecture 5: Berry connection and Berry curvature. Electromagnetic analogue: Aharonov-Bohm effect. Illustration: Spin in a magnetic field. Magnetic monopoles. Lecture notes. For a nice text on the Aharonov-Bohm effect, and geometric phases in general, see Chapter 4 of Curt Wittig's Lecture Notes on Quantum Chemistry. For those of you who want some more mathematical flesh on Berry phases, the review by Ben-Aryeh is quite accessible, with a good list of reference.

Lecture 6: Calculating the Berry curvature. Intro to topological band theory. Su-Schrieffer-Heeger model. Lecture notes.

Lecture 7: Topological phase transitions. Symmetry protection. Kitaev's model for a 1D spinless p-wave superconductor. Lecture notes.


Next lecture: Thursday October 30 in Faraday, 15:15 - 17:00. During the recess you may wish to do some reading on your own (besides working on the homwework problems!). The two standard review articles on topological insulators and topological superconductors (symmetry protected topological matter) are X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011) and M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). With the background gained from the first weeks of the course, you should be able to profitably read selected parts of these excellent reviews. For a really nice lecture by Charlie Kane on topological band theory, click here. A colloquium by Shoucheng Zhang given at MIT in 2009 is also highly recommended.


Lecture 8: Majorana zero modes. Lecture notes. For some additional reading, please see the first three sections in the review by Leijnse and Flensberg. More material, but still quite accessible, can be found in the review by Alicea. On a lighter note, have a look at Frank Wilczek's Majorana returns and the recent MIT Technology Review on quantum computing using Majorana zero modes.



Homework problems

HW set 1

HW set 2

HW set 3

HW set 4