Phase transitions and critical phenomena

In this lecture we will establish and develop a framework in which critical phenomena associated with classical and quantum phase transitions can be described. The important concept of universality in statistical mechanics provides the basement for the general framework and leads to fundamental tools such as scaling theory and the renormalisation group. These phenomenological approaches allow the description of large scale behaviour in quantum as well as classical field theories and are at the interface between condensed matter and high energy physics. The crucial role of symmetry and topology will be emphasised along the way.

Course description

After a short introduction to critical phenomena which will touch upon the concept of phase transitions, order parameters, response functions and universality, we will review the Ginzburg-Landau theory, in particular mean field theory, critical exponents, symmetry breaking, Goldstone modes, critical dimensions, fluctuations and correlation functions, as well as the Ginzburg criterion. The scaling theory then deals with self-similarity, the scaling hypothesis, Kadanoff’s heuristic renormalisation group (RG), the Gaussian model, fixed points and critical exponent identities. After establishing Wilson’s momentum space RG and the concept of relevant, irrelevant and marginal parameters we will introduce the epsilon-expansion. Finally, we will also discuss topological phase transitions by means of the non-linear sigma-model and the XY-model, and introduce the concepts of algebraic order, topological defects, confinement and the Kosterlitz-Thouless phase transition.

Assistant

Literature

There are many good textbooks covering parts of the lecture. Some of these can be found in the library under the signature OKF.

  • N.D. Goldenfeld, Lectures on phase transitions
  • S.K. Ma, Modern theory of critical phenomena [OKF 116]
  • J.J. Binney, N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The theory of critical phenomena (An introduction o the renormalization group) [OKF 116]
  • J. Cardy, Scaling and renormalization in statistical physics [OKF 205]
  • K.G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55 (1983) 583
  • C. Domb and M. Green, Phase transitions and critical phenomena, Vol. 1-9 [OKF114]
  • K.G. Wilson and J. Kogut, The renormalization group and the epsilon expansion, Physics Reports 12 (1974) 75