Advanced Statistical Physics (P817), Spring 2024

Instructor: Romain Vasseur, Associate Professor
office: Hasbrouck 405A
email: rvasseur[at]umass[dot]edu
office hours: by appointment and email, or visit my office

Lectures: MoWe 2.30-3.45pm, Has Lab 136

Suggested reading materials:

• Scaling and Renormalization in Statistical Physics, John Cardy,  Cambridge University Press
• Statistical Physics of Fields, Kardar, Cambridge University Press
• Lectures on Phase Transitions and the Renormalization Group, Nigel Goldenfeld,  Frontiers in Physics Series (Vol. 85), Westview Press
• Principles of Condensed Matter Physics, P.M. Chaikin & T.C. Lubensky, Cambridge University Press
• Phase transitions and Renormalization Group, Jean Zinn-Justin, Oxford Graduate Texts

Website: https://websites.umass.edu/rvasseur/teaching/. Course materials will also be made available on Slack.

Grading:
The course grade will be based on the problem sets (70%) and a take-home final (30%).

Topics:

• Introduction: Critical phenomena and simple models. Mean-field theory.
• Ginzburg-Landau theory: fluctuations, Ginzburg criterion, upper/lower critical dimensions, Goldstone modes.
• The Renormalization Group
• The perturbative RG:
  • Operator product expansions
  • epsilon expansion
  • O(N) model and Large N limit
• XY model: Topological defects, Coulomb gas and Berezinskii-Kosterlitz–Thouless transition
• Advanced topics (time permitting):
  • Nonlinear sigma model
  • Random Systems
  • (Brief) Introduction to conformal field theory

Lecture Notes:

1. Critical Phenomena
2. Ginzburg-Landau theory & Fluctuations
3. The Renormalization Group
4. Perturbative RG
5. KT transition
6. Nonlinear sigma model
7. Disordered systems
8. CFT

Problem Sets: posted on Slack.

Schedule:

• 2/5: Intro to critical phenomena, Chapter 1: pages 1-3
• 2/7: Models & Mean-field theory, Remarks about SSB. Chapter 1: pages 4-7
• 2/12: Remarks about SSB, Landau-Ginzburg free energy: Chapter 1: page 8, Chapter 2: pages 1-3. 
• 2/14: Saddle point approximation, mean-field exponents and correlation functions. Chapter 2: pages 4-6.
• 2/21: Correlation functions, Ginzburg criterion, upper critical dimension and Gaussian fluctuations. Chapter 2: pages 7-9 + appendix page 17. 
• 2/22: functional Gaussian integrals, Gaussian fluctuations and specific heat. Chapter 2: pages 9-10 + appendix pages 15-16 on Gaussian integrals. 
• 2/26: Lower critical dimension, Peierls argument, Goldstone modes. Chapter 2: pages 11-13.
• 2/28: Mermin-Wagner theorem, RG solution of the 1d Ising model. Chapter 2: pages 13-14. Chapter 3: pages 1-2.
• 3/4: Block spins, general RG theory, universality. Chapter 3: pages 3-5.
• 3/6: Universality, scaling and critical exponents, Chapter 3: pages 6-8.
• 3/11: Correlation length scaling, irrelevant variables and corrections to scaling, finite-size scaling. Chapter 3: pages 8-9.
• 3/25: Scaling operators and correlation functions. Start real space RG for 2d Ising. Chapter 3: pages 9-11.
• 3/27: Real space RG for the 2D Ising model. Power counting in the phi^4 theory. Chapter 3: Pages 12-14.
• 3/29:  Operator product expansions, perturbative RG. Chapter 4: Pages 1-4.
• 4/1: phi^4 theory: Gaussian fixed point, Wick’s theorem, normal order, OPEs. Pages 4-7.
• 4/3: Wilson-Fisher fixed point and epsilon expansion. Irrelevant variables. Pages 8-10.
• 4/8: O(N) model, dangerously irrelevant variables and phi^4 theory for d>4. Chapter 4: pages 10-11, page 15.
• 4/10: Large N limit of the O(N) model. Chapter 4: pages 12-14.
• 4/15:  XY Model: Monte Carlo Simulations, High T and low T expansions. Chapter 5: pages 1-2.  
• 4/17:  Quasi-long range order, topological defects, free energy argument, dipoles and vortices. Chapter 5: pages 3-5.
• 4/22: Coulomb gas mapping and duality to sine-Gordon field theory. Chapter 5: pages 6-8.
• 4/24:  Stiffness renormalization, KT RG equations. Chapter 5: pages 9-11.
• 4/29: RG analysis of the KT transition, scaling. Chapter 5: pages 11-15.
• 5/1:  Disordered systems: quenched randomness, replica trick.  Harris criterion. Chapter 7.