| Instructor: | Olexei Motrunich
Office: Sloan Annex 127 Phone: (626) 395-8894 Email: motrunch |
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| Class Meets: | Tue, Th 9:00 - 10:30, Downs 107 |
| Office Hours: | Tue 5:00-6:00
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| Teaching Assistant: | Hsin-Hua Lai
Office: Sloan Annex 121 Email: hsinhua Office hours: Wed 4:00-5:00 |
| References for the course: |
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). M. Kardar, "Statistical Physics of Fields". MIT lectures 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.
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 . |
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| Lecture 2: | Quantum-classical mapping (Euclidean path integrals);
quantum Ising and XY models.
Suggested reading: Sections III,IVA from Kogut's RMP (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 quantum XY model. Lecture notes: Euclidean path integrals for quantum Ising and XY models . |
| Lecture 3: | Finish quantum-classical mapping.
Classical Ising model: pictures of the phases and series expansions.
Suggested reading: 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: | 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: Quantum Ising phases |
| Lecture 5: | Classical Ising gauge theory.
Elitzur's theorem. Wegner loop correlator.
Suggested reading: Section V from Kogut's RMP Lecture notes: Classical Ising gauge theory |
| Lecture 6: | Wegner-Wilson-Polyakov loop and confinement/deconfinement
of charges. Quantum Ising gauge theory (path integral
and started discussion of phases in the quantum setting).
Suggested reading: Section V from Kogut's RMP Lecture notes: Quantum Ising gauge theory |
| Lecture 7: | Quantum Ising gauge theory: picture of the phases and
excitations. Duality with global Ising model in 2+1d.
Started XY models.
Suggested reading: Sections V-VI from Kogut's RMP Lecture notes: same as Lecture 6 |
| Lecture 8: | XY model at low temperatures -- spin waves
Suggested reading: 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 9: | Vortices in the 2d XY model -- 2d Coulomb gas
Suggested reading: 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 10: | Kosterlitz-Thouless RG for the 2d Coulomb gas; KT transition
Suggested reading: 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 11: | Analysis of KT equations.
Started other representations of the XY model.
Suggested reading: Same as Lecture 10. Lecture notes: Representations (variants) of the XY model |
| Lecture 12: | Analysis of KT equations.
Representations (variants) of the XY model: Villain; current loops; Coulomb gas;
Sine-Gordon
Suggested reading: See Lecture 11. Also Kogut's RMP Sec. VII; Herbut Chapter 6 Lecture notes: See lecture 11. |
| Lecture 13: | Sine-Gordon model
Suggested reading: Kogut's RMP Sec. VII; Herbut Chapter 6; Wen Chapter 3 Lecture notes: Analysis of Sine-Gordon model |
| Lecture 14: | Sine-Gordon model and Luttinger liquids
Suggested reading: Wen Chapter 3 Lecture notes: Luttinger liquids |
| Lecture 15: | Bosons at half-filling in 1+1D. Start 3D XY duality
Suggested reading: Same as lecture 14; Herbut Ch. 7 (and Ch. 3). Lecture notes: 3D XY duality |
| Lecture 16: | 3D XY in terms of vortices. 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: same as lecture 15 |
| Lecture 17: | Finish review of Landau-Ginzburg theory of charged superconductors:
flux quantization and Abrikosov-Nielsen vortices.
Finish 3D XY <-> Higgs model duality.
Suggested reading: Same as Lecture 16 Lecture notes: Same as lecture 15 |
| Lecture 18: | Compact Electrodynamics: monopoles; confinement in 2+1D;
phases in 3+1D. Wrap-up
Suggested reading: Polyakov Ch. 4; Herbut Ch. 7.4 Lecture notes: Compact QED |