| Instructor: | Olexei Motrunich
Office: Sloan Annex 127 Phone: (626) 395-8894 Email: motrunch |
|---|---|
| Class Meets: | Tue, Th 9:00 - 10:30, Downs 107 |
| Office Hours: | Tue 5:00-6:00
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| Teaching Assistant: | Peter Brooks
Office: 239 Annenberg Email: pbrooks@caltech.edu Office hours: Wed 3:00-4:00 |
| References for the course: |
M. Kardar, "Statistical Physics of Fields".
MIT lectures
J. B. Kogut, "An introduction to lattice gauge theory and spin systems" Rev. Mod. Phys. 51, 659 (1979). Rev. Mod. Phys. 51, 659 (1979) S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, "Continuous quantum phase transitions", Rev. Mod. Phys. 69, 315 (1997). N. Goldenfeld, "Lectures on Phase Transitions and the Renormalization Group". J. Cardy, "Scaling and Renormalization in Statistical Physics". I. Herbut, "A Modern Approach to Critical Phenomena". S.-k. Ma, "Modern Theory of Critical Phenomena". K. Huang, "Statistical Mechanics". X.-G. Wen, "Quantum field theory of many-body systems." |
| Homework and Grading: |
There will be a weekly (or biweekly) homework assignment
anounced in class (and via email), due one week later. At the end of the term, each student will make a presentation (30-minute lecture or term paper) on a topic related to the course. Grades will be based on the homework (65%) and presentation (35%) |
Course Description:
This is an advanced course in Statistical Mechanics focusing
on qualitative aspects in prototypical lattice models of magnets,
superfluids, and gauge field theories
(Kogut's RMP article provides good examples of topics).
Topics (tentative):
Lattice models; phases and phase transitions; quantum-classical
mapping
Spin waves. Quasi-long-range order in the 2D XY model.
Vortices in the 2D XY model and Kosterlitz-Thouless transition.
2D melting.
Dualities and thinking in terms of topological defects
(2D Ising; 3D Ising and Ising gauge theory; 3D XY model;
compact electrodynamics)
Prerequisites:
Basic statistical mechanics; renormalization group; quantum mechanics;
some field theory is helpful but not required.
Phy127c 2009 course plan
More links to Statstical Physics courses on the web
Mike Cross' Phy127c 2005 lectures
LECTURES:
| Lecture 1: | Lattice models. Overview of the course.
Suggested reading: Sections I,II from Kogut's RMP; Lectures 13,14 from Kardar Lectures . Scratch lecture notes: Overview of lattice models and Monte Carlo . Historical reading: Interesting interview with A. Polyakov recalling pre-Wilson days when condensed matter and field theory appeared as separate universes; and Wilson's Nobel lecture . |
|---|---|
| Lecture 2: | Quantum-classical mapping (Euclidean path integrals);
quantum Ising model and mapping to classical Ising model.
Suggested reading: Sections III,IVA from Kogut's RMP (it may be easier to read these "backwards" from Hamiltonians to path integrals). Section II of S. L. Sondhi et al, Rev. Mod. Phys. 69, 315 (1997). has a very clear discussion of the path integral for the quantum XY model. For still different examples of path integrals and their applications, you can read lectures 16-18 of Mike Cross' lecture notes Lecture notes: Euclidean path integrals for quantum Ising and XY models . |
| Lecture 3: | Quantum-classical mapping - connection with Feynman path integrals
in quantum mechanics. Classical Ising model: pictures of the phases and
series expansions: low-temperature series.
Suggested reading: Kardar Chapter 7 or Lectures 15,16 from Kardar Lectures . Sections III,IV from Kogut's RMP Lecture notes: Ising series expansions (roughly following Kardar's lectures) |
| Lecture 4: | High-temperature series for the Ising model.
Duality in the 2d classical Ising model.
Started Quantum Ising model: pictures of the phases (ground states and
excitations) -- will cover next time.
Suggested reading: Sections III,IV from Kogut's RMP Lecture notes: Quantum Ising phases |
| Lecture 5: | Quantum Ising model: pictures of the phases (ground states and
excitations). Self-duality in the 1+1D Ising.
Suggested reading: Sections III,IV from Kogut's RMP Lecture notes: see Lecture 4. |
| Lecture 6: | Classical Ising gauge theory.
Elitzur's theorem.
Suggested reading: Section V from Kogut's RMP Lecture notes: Classical Ising gauge theory |
| Lecture 7: | Classical Ising gauge theory continued.
High-temperature and low-temperature expansions and duality to global
Ising model in 3D. Wegner-Wilson-Polyakov loop.
Suggested reading: Same as lecture 6. |
| Lecture 8: | Quantum Ising gauge theory and its Euclidean path intgral.
Wilson-Polyakov loops and confinement of charges.
Simple pictures of the phases and excitations.
Suggested reading: Sections V-VI from Kogut's RMP Lecture notes: Quantum Ising gauge theory | Lecture 9: | Quantum Ising gauge theory: Simple pictures of the phases and excitations.
Suggested reading: Same as lecture 8. |
| Lecture 10: | XY model at low temperatures -- spin waves
Suggested reading: Kardar Chapter 3.1-3.3 (Lectures 3, 21 from Kardar Lectures ); sections VII A,B from Kogut's RMP Lecture notes: Spin wave treatment in the classical XY model ; Addendum: spin waves in the quantum rotor model. |
| Lecture 11: | Vortices in the 2d XY model -- 2d Coulomb gas
Suggested reading: Kardar Chapter 8.2-8.3; Lectures 21, 22 from Kardar Lectures ; section VII from Kogut's RMP Lecture notes: Coulomb gas description of vortices in the 2d XY model |
| Lecture 12: | Kosterlitz-Thouless RG for the 2d Coulomb gas; KT transition
Suggested reading: Kardar Chapter 8.2-8.3; Lecture 22 from Kardar Lectures ; section VII from Kogut's RMP Original very clear paper by Kosterlitz: J. Phys. C: Solid State Phys. 7 1046 (1974). Lecture notes: KT RG and KT transition |
| Lecture 13: | Analysis of KT equations.
Started other representations of the XY model.
Suggested reading: Same as Lecture 12. Lecture notes: Representations (variants) of the XY model |
| Lecture 14: | Representations (variants) of the XY model: Villain; current loops;
Coulomb gas; Sine-Gordon
Suggested reading: See Lecture 13. Also Kogut's RMP Sec. VII; Herbut Chapter 6 Lecture notes: See lecture 13. Analysis of Sine-Gordon model |
| Lecture 15: | Sine-Gordon model
Suggested reading: Kogut's RMP Sec. VII; Herbut Chapter 6; Wen Chapter 3 Lecture notes: Analysis of Sine-Gordon model |
| Lecture 16: | Sine-Gordon model and Luttinger liquids
Suggested reading: Wen Chapter 3 Lecture notes: Luttinger liquids |
| Lecture 17: | 3D XY in terms of vortices. Duality to Higgs model.
Review of Landau-Ginzburg theory of charged superconductors
Suggested reading: Herbut Ch. 7 and Ch. 3. For a very nice concise review of Landau-Ginzburg theory, see Lecture 14 in Mike Cross's lectures Lecture notes: 3D XY duality |
| Lecture 18: | Landau-Ginzburg theory of charged superconductors: flux quantization
and Abrikosov-Nielsen vortices. Discussion of 3D XY <-> Higgs model
duality. Wrap-up
Suggested reading: Same as Lecture 17 |