Phase transitions and critical phenomena

Objectives

Phase transitions are a pervasive concept in physics. Most of the real world as we see it is the result of a phase transition, in which a symmetry is spontaneously broken by the collective behavior of interacting particles/fields. The concept of symmetry, and spontaneous breaking thereof, allows us to classify different collective phases, and to understand the existence of phase transitions separating them. Landau’s mean-field theory captures the essence of phase classification, as well as most of the phenomenology of the phase transitions; yet an accurate description of transitions requires to treat fluctuations and correlations accurately, a seemingly impossible task. This task has required the introduction of a whole new approach, namely renormalization, allowing to unveil how a simple organizing principle — universality — emerges from the complexity of phase transitions. This course will discuss/contrast the microscopic approach and field-theory approach to phase transitions, and their application to different areas of physics, including classical and quantum systems.

Course outline

1. Introduction: importance of phase transitions across the physical spectrum, thermodynamics of phase transitions, symmetry and symmetry breaking.

2. Phenomenological approach to phase transitions: Ginzburg-Landau theory. First- and second-order transitions, multi-critical points.

3. Scaling concept and phenomenological renormalization.

4. Real-space renormalization: rise and fall.

5. Momentum-space renormalization: transitions close to the upper critical dimension.

6. Phase transitions in low dimensions; the Kosterlitz-Thouless transition.

7. Dynamics of phase transitions.

8. Quantum phase transitions.

Textbooks

Many excellent books exist on the subject of phase transitions and critical phenomena. Here is the list of a few which may be useful as a guide during the course:

J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman The Theory of Critical Phenomena - An Introduction to the Renormalization Group Clarendon Press, Oxford, 1992

N. Goldenfeld Lectures on Phase Transitions and the Renormalization Group Addison & Wesley, 1992

J. Cardy Scaling and Renormalization in Physics Cambridge University Press, 1996

P. Chaikin and T. Lubensky Principles of Condensed Matter Physics Cambridge University Press, 2000

Pre-requisites

Mandatory: statistical physics (L3), quantum mechanics (L3)

Recommended: phase transitions (M1), non-equilibrium statistical physics (M2), quantum field theory (M2)

Exam

Written exam