# Topological Insulators Lecture Notes Homework Market

#### Instructor: Kai Sun

**Email:** sunkai@umich.edu**Phone:** 734-764-0730 **Office: ** 2245 Randall **Homepage:**Kai Sun's Homepage

#### Time and Place:

**Time:** 2:30-4pm Tuesday and Thursday**Place: ** 335 West Hall**Office Hour:** Wednesday 3:00-4:00pm, 2245 Randall.

#### Announcement:

Office hour on 12/12/2012 is changed to 2:00-3:00pm

No lectures in the week of 10/15/2012-10/19/2012

For the week of 10/15/2012-10/19/2012, the office hour is on Friday 10/19/2012 at 1-2pm

#### Course Description:

This course reviews the developments in modern condensed matter physics, as well as their connections to and impacts on other branches of physics. The course presents the physical pictures for each of the topics, along with relevant experimental and theoretical techniques. The course contains three parts: (1) Topological states of matter, including topological insulators and superconductors, fractional topological insulators, fractional and nonabelian statistics, topological field theory, (2) strongly correlated systems, including conventional and unconventional superconductors, Mott insulators, 1D Luttinger liquid and other nonFermi liquids, and (3) frustrated spin systems, including spin liquids and lattice gauge theory.

#### Prerrequisite:

Basic knowledge on quantum mechanics and statistical physics are required. An introductory solid state physics course (undergraduate or 500 level) are strongly recommended. I will provide necessary introduction for relevant background on (1) band structure and (2) second quantization, but the students are expected to be familiar with the basic concepts for at least one of the two.

Quantum field theory is NOT required and will be covered as part of the course, which will be constructed from a condensed matter point of view.

#### Lecture notes:

Chapter 1: Band structure theory (9/4/2012)

Chapter 2: Topological insulator part I (phenomena): what is a topological insulator (9/6/2012)

Chapter 3: Topological insulator part II (band structure theory)

Complex phase of a wave-function and the central charge: The fact that phase is unimportant is very important. (9/11/2012)

Berry connection, Berry curvature and Hall conductivity: can we have E and B fields in k-spaces? (9/13/2012)

Topological index, the Chern number and quantized Hall conductivity: why topological insulators are topological? (9/18/2012)

Second quantization and tight binding models: simplified model to study band structures (9/20/2012)

Tight binding models part II: an example with two bands (9/25/2012)

Tight binding models part III: a topologically nontrivial example (9/27/2012)

The model of Haldane: Dirac points and Chern insulators. (10/2/2012 and 10/4/2012)

Interactive figures: the Honeycomb lattice model and the model of Haldane (requires Wolfram Mathematica or Wolfram CDFPlayer. Mathematica is available on most department-owned computers/laptops and CDFPlayer is a free software offered by Wolfram).

Symmetry properties and Berry flux: why Dirac points are so stable? (10/9/2012)

Chiral edge states: what happens on the edge of a Chern insulator? (10/11/2012)

Interactive figure: the edge states in the model of Haldane

Chapter 4: Topological insulator part III (gauge structure theory)

Chern-Simon's gauge theory and quantized Hall conductivity: how to describe an insulator? (10/11/2012)

Chern-Simon's gauge theory part II: the interplay between charge and magnetic flux (10/23/2012)

Chapter 5: Time-reversal invariant topological insualtors

Quantum spin Hall insulators Can we have a time-reversal invariant topological insulator? (10/25/2012)

Time-reversal symmetry, the theorem of Kramers and 3D topological insulator Can we have a topological insulator in 3D? (10/30/2012)

Chapter 6: Superconductors

Superfuild and Bose–Einstein condensate (11/1/2012)

the BCS theory (11/6/2012 and 11/8/2012)

Gauge theory of superconductivity (11/13/2012)

High temperature superconductivity (11/15/2012)

Chapter 7: Frustrated magnets and 1D quantum systems (11/20/2012)

Chapter 8: Quantum field theory and Green's functions (11/22/2012)

Background: Why time-ordering?(11/27/2012)

Green's functions: Boundary conditions and connections between different Green's functions (11/29/2012)

Interacting systems and Feynman diagram: (12/4/2012 and 12/6/2012)

Chapter 9: Quantum phase transition and quantum criticality (12/11/2012)

#### Homeworks:

Problem set #1 (Due 10/25/2012).

Problem set #2 (Due 12/14/2012).

#### Course Plan:

Band structures: metals, insulators and beyond (graphene, Dirac Fermions and topological semi-metals)

Band structure revisit: position operator and topological insulators

Superconductivity: BCS theory, gauge symmetry and topological superconductors

1D Luttinger Liquids: beyond Fermi liquid and Fermions=Bosons

2D fractional quantum Hall: beyond Fermions and Bosons (anyons, fractional charge, fractional and nonabelian statistics)

Quantum field theory and Feynman diagram on a nutshell

Fermi liquid theory from quantum field theory

Quantum phase transition and quantum criticality: concept and examples

Spin liquids

Phonons

#### Homework: (1-2 homework set per month)

#### Term Paper: (due 12/17/2012)

There will not be a midterm exam but there will be a final term paper. A subject matter can be any topic related to many-body physics. It does NOT need to be restricted to condensed matter physics or topics that we discussed in class. Interesting developments in other areas will also be welcomed, as long as the subject is relevant to many-body physics.

I will also provide some suggested topics for your consideration, but you are NOT required to choose from this list. I strongly encourage all the students to discuss with me about the topic you plan to work on and all the students need to send the topic they choose to me by email by Oct. 26. If you need extra time to decide the subject, please let me know.

The term paper must explain (1) what is the motivation for this study, i.e. why the subject is interesting, (2) what has been done, e.g., theoretical model, calculations and necessary approximations, experimental techniques and signatures, and (3) what are the conclusions. Unnecessary technique details are NOT required, because the main purpose of the term paper is to demonstrate your understanding on the broad topic. if technique details are too involved and disrupt the natural logical flow of the paper, they can be put in an Appendix. The term paper must be submitted electronically in PDF format. The paper should be at least seven (7) pages long (including title page), double spaced pages in 10pt. font. The title page must include the title, your name and an abstract. References should be listed at the end of the paper.

Please be aware that (1) any citations and quotations need to be clearly indicated, and (2) term papers from other courses cannot be used.

#### References:

- Xiao-Gang Wen, Quantum Field Theory of Many-body Systems, Oxford University Press.
- Eduardo Fradkin, Field Theories of Condensed Matter Systems, Addison Wesley, Redwood City.
- Subir Sachdev, Quantum Phase Transitions, Cambridge University Press.
- John Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics.
- Leo P. Kadanoff and Gordon Baym, Quantum Statistical Mechanics, Advanced Books Classics.

A remarkable discovery in recent years is that there exist various kinds of topological insulators and superconductors characterized by a periodic table according to the system symmetry and dimensionality. To physically realize these peculiar phases and study their properties, a critical step is to construct experimentally relevant Hamiltonians that support these topological phases. We propose a general and systematic method based on the quaternion algebra to construct the tight-binding Hamiltonians for all the three-dimensional topological phases in the periodic table characterized by arbitrary integer topological invariants, which include the spin-singlet and the spin-triplet topological superconductors, the Hopf, and the chiral topological insulators as particular examples. For each class, we calculate the corresponding topological invariants through both geometric analysis and numerical simulation.

- Received 5 November 2013
- Revised 6 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.075126

©2014 American Physical Society

## Comments