PHYS 7240 - Advanced Statistical Mechanics

  • Lecture 1 - Introduction and overview
    • Review of statistical mechanics
    • Eigenstate Thermalization Hypothesis (ETH)
    • Ensembles
    • Reduced density matrix
    • Entanglement entropy
    • Course overview
  • Lecture 2 - Magnetism: exact solutions and mean-field theory
    • Background on magnetism
    • Langevin paramagnetism
    • Spin exchange
    • Magnetism: Heisenberg model and crystalline anisotropies
    • Exact solution of 1d Ising model via transfer matrix
    • Exact solution of 2d Ising model via 1d quantum transverse field Ising model, Jordan-Wigner transformation to p-wave superconductor, Bogoluibov-de Gennes, Majorana fermions
    • Weiss and Landau mean-field theory (d=∞ limit)
    • Exact solution of N → ∞ of O(N) model
  • Lecture 3 - Landau theory of phase transitions and its breakdown
    • Landau theory of phase transitions
    • Nonuniform boundary conditions: soliton
    • Ising PM-FM transition
    • Liquid-gas transition
    • Normal-to-superfluid transition
    • Goldstone modes and G/H counting
    • Tricritical point
    • Liquid Crystals:
      • Isotropic-Nematic transition
      • Nematic-Smectic-A transition (de Gennes model)
      • Smectic anaomalous elasticity (Grinstein and Pelcovits)
      • Cholesterics - chiral nematics and their smectic elasticity
    • Critical states of soft matter
    • Liquid-to-crystal transition: freezing
    • Breakdown of Landau theory
  • Lecture 4 - Field theory primer
    • Gaussian integrals
    • Functional integrals, correlation functions
    • Perturbative expansion
    • Breakdown of perturbation theory and Ginzburg criterion
  • Lecture 5 - Renormalization group
    • Introduction, motivation and scaling theory
    • General philosophy and structure of RG
    • Real-space RG and application to 1d Ising model
    • Physical observables by matching analysis
    • Momentum-shell RG and e-expansion Φ4 -theory
    • RG flows and fixed points
    • "Dangerously" irrelevant operators
    • O(N) model
    • Nonlinear σ-model via e = d - 2 expansion
  • Lecture 6 - Stability of ordered phases: Goldstone modes and topological defects
    • Stability of phases and their Goldstone modes
    • Weakly interacting superfluids via Bogoluibov theory and phase-density coherent state path integrals
    • XY-model
    • Hohenberg-Mermin-Wagner-Coleman theorems
    • Topological defects: vortices, dislocations, disclinations, solitons
    • Kosterliz-Thouless transition
    • Duality, Coulomb gas, and sine-Gordon model
    • Dualities: Kramer-Wannier, 2+1 TFIM to Z2 Gauge theory, E-B, classical 2d XY to Coulomb gas to sine-Gordon, classical 3c XY to superconductor (U(1) gauge theory), quantum 2+1d bosons to Abelian-Higgs vortex gauge theory
    • Commensurate-Incommensurate Pokrovsky-Talapov transition.
    • Roughening transition
    • Nonlinear O(3) sigma-model and its disordering in d=2+eps dimensions
    • Large N expansion of the O(N) sigma model
    • Critical states of soft matter
  • Lecture 7 - Random heterogeneity: systems with quenched disorder
    • Quenched versus annealed disorder
    • Methods: replica trick and correlators
    • Disorder near a continuous phase transitions: random field, random bond, Harris and Imry-Wortis criteria
    • Physical systems:
      • charge-density wave
      • vortex lattice
      • polymerized membranes
      • liquid crystals in aerogel
      • liquid crystal with a "dirty" substrate
    • 2d random field finite temperature RG (Cardy-Ostlund)
    • Zero temperature functional RG
  • Lecture 8 - Superconductivity: Fluctuations, dissipation and phase transitions
  • Lecture 9 - Equilibrium and Non-equilibrium Hydrodynamics
    • Brownian motion (Einstein, Smoluchowski, Langevin theories)
    • Fokker-Planck equation