Advanced Statistical Physics (P817), Spring 2022
Instructor: Romain Vasseur, Associate Professor
office: Hasbrouck 405A
office hours: by appointment and email, or visit my office
Lectures: MoWe 4.00-5.15pm, Has Add 104B
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://blogs.umass.edu/rvasseur/teaching/. Course materials will also be made available on Slack.
The course grade will be based on the problem sets (60%), on a take-home final (20%), and a final term paper (20%).
- 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
- Critical Phenomena
- Ginzburg-Landau theory & Fluctuations
- The Renormalization Group
- Perturbative RG
- KT transition
- Nonlinear sigma model
- Disordered systems
Solutions will be made available on Slack, or upon request by email.
- 1/26: Intro to critical phenomena, Chapter 1: pages 1-3
- 1/31: Models & Mean-field theory, Chapter 1: pages 4-7
- 2/2: Remarks about SSB, Landau-Ginzburg free energy: Chapter 1: page 8, Chapter 2: pages 1-3.
- 2/7: Saddle point approximation, mean-field exponents and correlation functions. Chapter 2: pages 4-6.
- 2/9: Correlation functions, Ginzburg criterion, upper critical dimension and Gaussian fluctuations. Chapter 2: pages 7-9 + appendix page 17.
- 2/14: functional Gaussian integrals, Gaussian fluctuations and specific heat. Chapter 2: pages 9-10 + appendix pages 15-16 on Gaussian integrals.
- 2/16: Lower critical dimension, Peierls argument, Goldstone modes. Chapter 2: pages 11-13.
- 2/22: Mermin-Wagner theorem, RG solution of the 1d Ising model. Chapter 2: pages 13-14. Chapter 3: pages 1-2.
- 2/23: Block spins, general RG theory, universality. Chapter 3: pages 3-5.
- 2/28: Scaling and critical exponents, Chapter 3: pages 6-8.
- 3/2: Irrelevant variables and corrections to scaling, finite-size scaling, scaling operators and correlation functions. Chapter 3: pages 9-11.
- 3/7: Real space RG for the 2D Ising model. Power counting in the phi^4 theory. Chapter 3: Pages 12-14.
- 3/9: Operator product expansions, perturbative RG. Chapter 4: Pages 1-4.
- 3/21: End of derivation of the perturbative RG equations. phi^4 theory: Gaussian fixed point, Wick’s theorem, normal order. Pages 4-7.
- 3/23: OPEs for the Gaussian fixed point, Wilson-Fisher fixed point and epsilon expansion. Pages 7-9.
- 3/28: Irrelevant operators, O(N) model, dangerously irrelevant variables and phi^4 theory for d>4. Chapter 4: pages 9-11, page 15.
- 3/30: XY Model: Monte Carlo Simulations, High T and low T expansions. Chapter 5: pages 1-2.
- 4/4: Quasi-long range order, topological defects, free energy argument, dipoles and vortices. Chapter 5: pages 3-5.
- 4/6: Coulomb gas mapping and duality to sine-Gordon field theory. Chapter 5: pages 6-8.
- 4/11: Stiffness renormalization, KT RG equations. Chapter 5: pages 9-11.
- 4/13: RG analysis of the KT transition, scaling. Chapter 5: pages 11-15.
- 4/20: Roughening transitions, Large N limit of the O(N) model. Chapter 4: pages 12-14.
- 4/25: Nonlinear sigma model, beta function, asymptotic freedom. Chapter 6: pages 1-3.
- 4/27: d+2 epsilon expansion. Disordered systems: quenched randomness, replica trick. Chapter 6: page 4. Chapter 7: pages 1-2.
- 5/2: Harris criterion. A few words about conformal invariance. Chapter 7: page 3. Chapter 8.
Suggestions for final project:
- Disordered systems: replica trick, Harris criterion, Imry-Ma argument
- RG in high-energy physics vs statistical mechanics
- Quantum criticality: quantum to classical mapping, example: transverse field Ising chain
- Exact solution of the 2D Ising model using Majorana fermions (possibility to collaborate with “quantum criticality”)
- Percolation: mapping onto Potts model, critical behavior
- Polymers: limit n–> 0 of the O(n) model, critical behavior
- Surface critical behavior (Cardy)
- Topological terms in non-linear sigma models
- Spin glasses
- Momentum-shell RG (compare to OPE approach developed in class)
- Critical dynamics: reaction diffusion models (Cardy)
- Monte Carlo simulations and finite-size scaling: write a MC code for the 2D Ising model and extract critical exponents
- Dynamic of growing surfaces, KPZ equation (Kardar)
- Integrable statistical mechanics models
- Conformal Field Theory (Finite size scaling)
- CFT (global conformal invariance, correlation functions + overview in 2d)
- Neural networks: Hopfield model
- Scaling in non-equilibrium systems (Goldenfeld)
- Lee-Yang theorem
- Caldeira-Leggett model (qubit in dissipative environment)