Advanced Statistical Physics (P817), Spring 2020
Instructor: Romain Vasseur, Assistant Professor
office: Hasbrouck 405A
office hours: by appointment and email, or visit my office
Lectures: MW 4.00-5.15pm, room: Has 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://blogs.umass.edu/rvasseur/teaching/advanced-statistical-physics-p817-spring-2020. Course materials will also be made available on Moodle.
The course grade will be based 65% on the homework and 35% on a final term paper.
- 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
- HW1_MeanField due Feb 7 by 5pm
- HW2_GinzburgLandau due Feb 28 by 5pm
- HW3_Scaling due March 23 by 5pm
- HW4_perturbative_RG due April 10
- HW5_KT due May 1
Solutions available on Moodle, or upon request by email.
- 1/22: Intro to critical phenomena, Chapter 1: pages 1-4
- 1/27: Mean-field theory, Chapter 1: pages 5-8
- 1/29: Landau-Ginzburg theory and saddle-point, Chapter 2: pages 1-4
- 2/3: Mean-field critical exponents and correlation functions, Chapter 2: pages 5-7 + appendix page 17
- 2/5: Ginzburg criterion and functional Gaussian integrals, Chapter 2: pages 8-9 + appendix pages 15-16 on Gaussian integrals
- 2/10: Fluctuations about the saddle point, upper/lower critical dimension, Peierls’ argument. Chapter 2: pages 9-12.
- 2/12: Goldstone modes and Mermin-Wagner theorem, Chapter 2: pages 12-14.
- 2/18: Block spins, RG transformations, Chapter 3: pages 1-3.
- 2/19: RG flows, scaling variables, universality and scaling, Chapter 3: page 4-7.
- 2/24: Scaling and critical exponents, corrections to scaling and finite size scaling, Chapter 3: pages 7-9.
- 3/2: Scaling operators and correlation functions, start real space RG for Ising, Chapter 3: pages 10-12.
- 3/3: Real space RG for the 2D Ising model, dimensional analysis of LG theory. Perturbative RG: general idea. Chapter 3: pages 12-14. Chapter 4: page 1.
- 3/9: OPEs and 1-loop perturbative RG equations. Chapter 4: pages 2-4.
- 3/11: Gaussian fixed points, Normal order and gaussian OPEs. Chapter 4: pages 5-7.
- 3/12: large N limit of the O(N) model. Chapter 4: pages 12-14.
- SPRING BREAK. Summary latex Notes
- 3/23: Gaussian OPEs and Wilson-Fisher fixed point. Chapter 4: pages 7-9.
- 3/25: Epsilon expansion, irrelevant operators and O(N) model. Chapter 4: pages 9-11.
- 3/30: Dangerously irrelevant variables and phi^4 theory for d>4. XY Model: Monte Carlo Simulations, High T expansion. Chapter 4: page 15. Chapter 5: pages 1-2.
- 4/1: XY Model: Low T expansion, quasi-long range order, vortices and free energy argument. Chapter 5: pages 2-4.
- 4/6: Dipoles and Coulomb gas mapping. Chapter 5: pages 5-7.
- 4/8: Sine-Gordon theory and vortex operators. Chapter 5: pages 7-9.
- 4/13: Stiffness renormalization. Chapter 5: pages 9-11.
- 4/15: RG analysis of the KT transition, scaling. Chapter 5: pages 11-15.
- 4/22: Non-linear sigma model, beta function. Chapter 6: pages 1-3.
- 4/27: O(N) model in d=2+epsilon. Disordered systems and replica trick. Chapter 6: page 4. Chapter 7: pages 1-2.
- 4/29: Harris criterion. A few words about conformal invariance. Chapter 7: page 3.
- iPad notes (online lectures): link
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)