Advanced Statistical Physics (P817), Spring 2022

Advanced Statistical Physics (P817), Spring 2022

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 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://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 (60%), on a take-home final (20%), and a final term paper (20%).

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 will be made available on Slack, or upon request by email.

Schedule:

  • 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)

Recent Posts