Quantum Matter:
Coherence and Correlations
FUF010 (7.5 ECTS)

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



About the course

All matter is quantum matter. In the everyday world quantum effects are masked by many disturbances but in the new worlds that have been created in the laboratories in the last few decades quantum matter reveals its true character. These worlds exhibit properties that seem strange to us - charge can move freely on the surface of an insulator, many electrons conspire to produce fractional charges, and the charge and spin of an electron may be ripped apart and move independently of each other, just to mention a few of the consequences of quantum coherence and correlations.

In this course we will study the transition between the ordinary and quantum worlds in terms of coherent quantum phenomena and correlation effects in these novel systems. We will focus on systems of interacting fermions (e.g. electrons in nanoscale devices), but will also discuss bosonic systems, including "cold atoms". A number of theoretical techniques will be presented, useful for studying quantum coherence and correlation effects.

The course is recommended for students in the MSc program in Applied Physics with focus on theory and/or nanophysics. It is also well suited for students following the program in Fundamental Physics, and for students in the MC2 Nanoscale Science and Technology program with an interest in the theoretical foundation of nanoscience.



Course material
J. Kinaret and H. Johannesson, Quantum Matter: Coherence and Correlations,
supplemented by free course ware and excerpts from review articles.


Examination
Homework problems. For grade 4 or 5: a supplementary oral exam.


Schedule
please click here




Suggested reading


Week 1: Quantum Hall effect

My lectures this first week followed Secs. 2 and 3.1 in Mark Goerbig, Quantum Hall Effects rather closely. Goerbig's review contains a lot of information. Use your lecture notes as a guide to what to focus on! Also, read the introductory Sec. 1!

For some (by now) classical monographs on the quantum Hall effects, see Refs. [1]-[5] in Goerbig. Another excellent and authoritative review, also available online is Steve Girvin, The Quantum Hall Effect: Novel Excitations and Broken Symmetries. Secs. 1.1 - 1.4 cover essentially the same ground as we did in the lectures. For a (very) condensed picture of the integer quantum Hall effect (the one we studied this week), you may wish to have a look at Sec 1.2.2. in Kinaret & Johannesson.

Material handed out at the introductory lecture



Week 2: Topological quantum matter

My two lectures this week attempted to show you the basic ideas that go into the explanation of the quantization of the Hall conductance via topological arguments. For an excellent and easy-to-read introduction to the subject, see A Topological Look at the Quantum Hall Effect. More material, also nicely presented, can be found In Joel Moore's Berkeley lectures, part 1 and part 2.

Those of you interested in venturing into the details of how one actually measures the Hall conductance are encouraged to read Sec 3.3 in Mark Goerbig's review. Laughlin's original gauge argument from 1981 can be found here, and is recounted in the book Condensed Matter Field Theory by Altland and Simons.

For an introduction to geometric phases in quantum mechanics, see the Masters thesis by Durstberger.

A hot topic today in the study of topological quantum matter are the recently discovered topological insulators, of which the quantum spin Hall insulators are the simplest examples. More on that Monday next week! If you are curious and want to know a bit more about this fascinating subject, see Chapter 2 of Anders Ström's licentiate thesis which also contains an introduction to some useful concepts in topology for physicists, as well as suggestions for further reading.

Slides from my Monday Lecture 15/4 (course week 3!)



Week 3 and 4: Graphene

For reviews on graphene I recommend the reviews that you can download from the webpage of the mesoscopic physics group at the University of Manchester. The Nature Materials and Science reviews have a lot of information about the experimental side. They also have a few popular science articles about graphene. You can also check out the wikipedia entry on graphene and the scientific background from the Nobel Foundation.

For a thorough overview of the theoretical work and the basic physics of graphene I recommend the Reviews of Modern Physics article by Castro Neto et al.. You can find a discussion of the tight-binding approach and how to get the Dirac equation in Sec. II. There they also discuss the transmission through a barrier (Sec. II.B.1). The calculation of the conductivity at the Dirac point using the Landauer formula is discussed in Sec. III.I around Eq. (179). There's also references to the original papers. If you are at home and cannot access the published articles you can almost always find them by searching the arXiv. For an exposition of the anomalous quantum Hall effect in graphene, see the corresponding sections in Mark Goerbig's review Here are two chapters from the thesis of Johan Nilsson (guest lecturer in the course last year!) about the tight-binding approach in graphene (and bilayer graphene, for those of you interested!): chapter 2 and chapter 3.



Week 5 and 6: Cold atoms

For a quick "warm-up" (!), tune in to Introduction to Cold Atom Physics.

As I discussed in my lecture, the booming field of cold atoms has been made possible by breakthroughs in cooling and trapping techniques (Nobel Prize in Physics 1997), and secondly, from a clever use of the Feshbach resonance for controlling interactions between atoms. For an early review of the first item by one of the 1997 Nobel Prize winners and his collaborator, see Cooling and Trapping Atoms. For an easy-going piece on the Feshbach resonance, see Frank Wilzcek on Herman Feshbach. For some more substance, the term paper by former MIT-student Sara Campbell is quite good but you may need to refresh yourself about some basics of scattering theory in quantum mechanics to appreciate it... Pick up your 4th year QM book, use your lecture notes as a guide, and then return to the very informative text by Campbell. [Please note: the scattering phase exp(i delta_l) is missing on the right-hand side of Eq. (6), and there is a missing minus sign in one of the exponents in Eq. (55).]

For an excellent review of cold atom physics in optical lattices, see the Physics Nature Review by Immanuel Bloch from 2005. The same author has also recently written about quantum simulations using cold atoms, see page 47f in the Quantum Frontiers issue of Physics World, March 2013. For an easy-to-read little piece on how ultracold atoms trapped in crystals of light may solve "one of the most famous problems in theoretical physics", have a look at The strong-correlations puzzle.

For an authoritative review of cold Fermi gases as of 2008: arXiv:0801.2500.

Slides from my Cold Atom lectures



Week 6 and 7: QCD in a quantum dot: Kondo physics

As I tried to hint at in my Wednesday lectures this week, quantum impurity physics has become a paradigm for a wide variety of physical systems which involve strong correlations/interactions. In many cases models of quantum impurities can be solved exactly by using sophisticated tools from modern theoretical physics (Bethe Ansatz, conformal field theory,...) and the results can be compared to data from high-precision experiments. An unusual and very privileged situation in physics research! To get some more feeling for the importance of quantum impurity physics, in particular its most pristine realization - the Kondo effect - try to read the first two sections in Andy Schoefield's review on Non-Fermi liquids. You may also want to have a look at Piers Coleman's "Many-Body Physics: Unfinished Revolution". Simply focus on the stuff you find accessible and interesting and skip the rest! With this warm-up you are ready to tackle the two main texts for this course week: "Revival of the Kondo Effect" by Kouwenhoven and Glazman, and pages 71 - 88 in Philip Phillips excellent book "Advanced Solid State Physics" which cover the very basics of Kondo physics. My Monday morning lecture (next week, week 7!) will follow this text rather closely, and, with your lecture notes as a guide, the reading of these pages (as well as the corresponding homework problem) should come easy.

For those of you curious to learn more, I can suggest the wikipedia and scholarpedia articles on the Kondo effect and references therein, as well as the excellent review by Pustilnik and Glazman on Kondo effect in quantum dots.




NOT INCLUDED THIS YEAR ''Splitting an electron'': the Luttinger liquid

The theory of Luttinger liquids is an important piece of modern physics. In a way, it is a theorist's "dream come true". The availability of sophisticated mathematical and conceptual tools in 1D quantum physics (bosonization being one of them!) makes possible well controlled, or even exact, analytical results for a fully interacting many-particle system. And these results can be tested against high-precision experiments in mesoscopic and nanoscale physics! Most importantly, the physics that comes out challenges our conventional wisdom about how a many-electron system should behave (contrast "spin-charge separation" of a Luttinger liquid to the mundane picture of quasi-particles in a Fermi liquid a la Landau!). As such, it opens a door to a different universe in quantum physics.

Suggested reading: Introductory part of sec 2.2 + secs. 2.2.4, 3.1.1, 3.1.2 in Kinaret&Johannesson. Read pp 59-65 carefully, the rest you may read at sight.

For supplementary reading, see Sebastian Eggert's review at http://arxiv.org./pdf/0708.0003 (close to Kinaret&Johannesson in spirit, approach, and notation). Other, somewhat more advanced texts include Kurt Schönhammer's review at http://arxiv.org/pdf/cond-mat/0305035, and David Senechal's introduction to bosonization at http://arxiv.org/pdf/cond-mat/9908262 (the latter which requires some grounding in quantum field theory). For the "real thing", you should go to Thierry Gimarachi's book, Quantum Physics in One Dimension (Oxford University Press, 2003).

For a nice review of the experimental situation (as of 2010), see Electron liquids and solids in one dimension by V. V. Deshpande et al.







Homework problems

No. 1

No. 2

No. 3

No. 4

No. 5