Advanced Statistical Physics (P817), Spring 2020

Advanced Statistical Physics (P817), Spring 2020

Instructor: Romain Vasseur, Assistant Professor
office: Hasbrouck 405A
email: rvasseur[at]umass[dot]edu
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://websites.umass.edu/rvasseur/teaching/advanced-statistical-physics-p817-spring-2020. Course materials will also be made available on Moodle.

Grading:
The course grade will be based 65% on the homework and 35% on a final term paper.

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:

Solutions available on Moodle, or upon request by email.

Schedule:

  • 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)

Recent Posts