# Phase Transitions and Critical Phenomena

## Course description

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.

Outline:

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.

Learning outcomes:

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