Phase Transitions and Critical Phenomena
The course is held in the first half of the spring semester (27.02 - 27.03) 2018. There will be five meetings on Tuesdays (27.02 - 27.03) in room 119 and three meetings on Thursdays (01.03 - 15.03) in room B1. The meetings take place from 14-16.
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.
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, if time permits, 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.
Gain familiarity with the concept of universality at phase transitions,
develop a framework to describe critical phenomena, derive tools for
quantitative descriptions, understand the role of symmetry and topology