Critical Phenomena and the Renormalisation Group

Ray Jones

  • Introduction: phenomenology, order parameters, critical temperatures and exponents.
  • Magnetism as a paradigm: Ising and Heisenberg models and arguments about the possibility of a phase transition for d=1 and d=2. Spin waves. n and d.
  • Fluctuations: linear response theory, fluctuation dissipation theorem, correlation functions and \chi (q).
  • Classical mean field theory applied to the Ising model and calculation of critical exponents: Ornstein -Zernicke theory and the role of the correlation length \xi (T). Failings of the theory and the Ginzburg criterion- the special role of d=4.
  • Scaling: the static scaling hypothesis, exponent (in)equalities, Kadanoff block scaling.
  • Introduction to the Wilson renormalisation group: role of fixed points and scale invariance, the correlation length exponent \nu and the linearised transformation.
  • Approximate real space renormalisation group analysis of the Ising model when d=2.
  • Universality.
  • Percolation theory: an introduction to the ideas; calculation of p* and \nu by real space renormalisation group methods.
  • Introduction to reciprocal space renormalisation group methods.
  • The Kosterlitz-Thouless transition.