#### Instructor: Kai Sun

**Email:**
sunkai@umich.edu

**Phone:** 734-764-0730

**Office: ** 2245
Randall **Homepage:** Kai Sun's Homepage

#### Time and Place:

**Time:** 1:00-2:30pm Tuesday and Thursday

**Place: **335 WH

**Office Hour:**
2:00-3:00pm Wendesday, 2245 Randall.

#### Announcement:

(Sep. 3, 2013) There will be no lecture on Sep 10 and Sep 12, due to an international conference. We will have two makeup lectures. Please participate in the doodle poll posted on the ctools page of our course to select your preferred time for the first makeup lecture (under announcement).

#### 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 materials covered in this course include Green’s functions and Feynman diagrams, weakly-correlated electronic systems and the Fermi liquid theory of Landau, (quantum and classical) phase transitions and spontaneous symmetry breaking, the quantum Hall effect and topological insulators, strongly-correlated electronic systems and non-Fermi liquids, etc.

#### Prerrequisite:

One introductory level solid state physics course is prerequired (e.g. Physics 520 or 463)

Quantum field theory is NOT required and will be covered as part of the course.

#### Textbook:

No required textbook. Lecture notes will be avaible online.

#### Lecture notes:

Chapter 1: Quantum field theory and Green's function. [Reference: Leo P. Kadanoff and Gordon Baym, Quantum Statistical Mechanics]

[Sep. 3: page 1-5]; [Sep. 5: page 6-10]; [Sep. 6: page 11-14]; [Sep 17: page 15-17] [Sep 19: page 18-22]

Why do we use second quantization?

Why do we use time-ordered products?

Why do we use Green's functions?

What are Feynman diagrams and Dyson's equations?

Chapter 2: Fermi liquid theory. [Reference: Gerald D. Mahanm, Many-Particle Phyiscs.]

Why can we treat electrons in a metal as free fermions? The rigirious answer comes from the quantum field theory.

Chapter 3: Topological insulators Part I: Phenomena

Chapter 4: Topological insulators Part II: Phase and topology

Chapter 5: Topological insulators Part III: examples and tight-binding models

Interactive figure: The model of Haldane

Interactive figure: Edge States

Chapter 6: Topological insulators Part IV: gauge theory

Chapter 7: Time-reversal invariant topological insulators (will not discuss in class)

Chapter 8: Phase transition

Interactive figure: Phonon modes

Chapter 9: Superconductivity

Chapter 10: Magnetism in one-dimension

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

Problem set #1 (Due 10/3/2013).

Problem set #2 (Due 10/17/2013).

#### Course Plan:

**Quantum field theory and Feynman diagram:** Quantum field theory is a powerful theoretical tool and is widely used in the investigation of various quantum systems. Although the name may sound a bit scary, the quantum field theory is in fact a natural extension of quantum mechanics. In this part, we will show that to utilize quantum mechanics to describe a system composed by indistinguishable particles, quantum field theory is a very natural choice. It arises naturally when we try to solve the equations of motion in the Heisenberg picture.

**Fermi liquid theory:** In Solid State Physics I, interactions between electrons are ignored. This free-electron approximation is NOT just a simple approximation, but is supported by a deep reason. In this section, to practice the quantum field theory that we just learned, we will use the quantum field theory to prove that the free-electron approximation is indeed a valid approximation in metals. The goal in this part is to fully understand the Fermi liquid theory of Landau.

**Band structure theory beyond metals and insulators:** In this part, we revisit the band structure theory that we have learned in solid state physics I. We have already known that the band structure theory tells us that there are in general two types of solids: insulators and metals. Here, we study more exotic situations, such as graphene, Dirac Fermions and topological semi-metals.

**Topological insulator:** In this part, we investigate the topology of the ground state wavefunction and its implications.

**Phase transitions and spontaneous symmetry breaking.**

**Superconductivity:** The BCS theory of superconductivity has been covered in Solid State Physics I. Here, we revisit this topic from a different point of the view, focusing on the gauge symmetry breaking, which is the fundamental reason of superconductivity.

**1D Luttinger Liquids:** In 3D, the spin–statistics theorem tells us that there are only two types of quantum particles: bosons with integer spins or fermions with half-integer spins. In 1D or 2D systems, this law doesn’t need to be obeyed and thus more exotic states of matter can be discovered. In this part, we show that in 1D there is no clear distinction between bosons and fermions. In fact, we can use a fermionic theory to describe some systems made by bosons (and vice versa).

**Fractional quantum Hall effect: **In this part, we continue the discussion about how to go beyond the spin–statistics theorem. In 2D fractional quantum Hall systems, the quasi-particle excitations are anyons, which carry fractional charge of an electron and fractional statistics.

#### Course work and grading:

Your class grade will be based on problem sets (50%), final presentation (15%) and term paper (35%).

#### Term Paper: (due date to be determined)

The 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.

You need to send the topic to me by email (or in person) by Nov. 7 for approval, and I strongly encourage each of you to discuss with me about the topic in person. 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.

#### Final Presentation: (date to be determined)

Each of you will have 8 mins to present your term paper in front of the class. You will have two addition minutes to answer questions raised by the audience.

#### References:

- B. Andrei Bernevig with Taylor L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press (2013).
- John Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge University Press (1996).
- P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, (2000).
- Eduardo Fradkin, Field Theories of Condensed Matter Systems (2nd edition), Cambridge University Press, (2013).
- Leo P. Kadanoff and Gordon Baym, Quantum Statistical Mechanics (Advanced Books Classics), Westview Press (1994).
- Gerald D. Mahanm, Many-Particle Phyiscs, Plenum, New York (1990).
- Philip Phillips, Advanced Solid State Physics (2nd Edition), Cambridge University Press (2012).
- Subir Sachdev, Quantum Phase Transitions (2nd Edition), Cambridge University Press (2011).
- Xiao-Gang Wen, Quantum Field Theory of Many-body Systems, Oxford University Press (2007).