LEARNING OUTCOMES

Verify and stimulate the basic knowledge on statistical physics. To take on recent arguments in the simplest possible context so to stimulate interest for the recent deveopments of statistical mechanics.

AIMS AND LEARNING OUTCOMES

Specifically, the learning outcomes of the course are:

• Introduction of the concepts of symmetries and scales and their use in building effective field theories of fields.
• Familiarize with the concepts of critical phenomena and phase transitions and their description in terms of scale invariant effective field theories of fields.
• Introduction of the path integral method and the renormalization group in quantum field theory.
• Introduction of the concepts of spontaneous symmetry breaking and Goldstone bosons and their relations with phase transitions.
• Learning how to describe relevant physical phenomena in terms of effective field theories.

PREREQUISITES

The attendance of the mandatory courses of the first semester of the Master in Physics, and in the particular the course "Theoretical Physics", is desirable in order to understand the arguments which will be discussed during the course.

TEACHING METHODS

Blackboard lectures and exercise sessions.

SYLLABUS/CONTENT

1. The Ising model:
   •  Description in terms of spins
   • Mean field approximation: from spins to fields
2. The Landau approach to phase transitions:
   • Continuous phase transitions
   • First order phase transitions
   • The concept of "Universality"
3. The Ginzburg-Landau theory:
   • applications to the ferromagnetic and superconducting phase transitions
4. The path integral:
   • defining the thermodynamical quantities using the path integral
   • The Gaussian path integral
   • Correlation length and critical dimension
   • Analogies with quantum field theory
5. The renormalization group:
   • Scale transformations and critical exponents
   • The Gaussian fixed point
   • Relevant, marginal and irrelevant perturbations at the fixed point
   • Interactions and renormalization group: Beta functions and Feynman diagrams
6. Continuous symmetries
   • Continuous symmetries and phase transitions
   • Spontaneous symmetry breaking and Goldstone bosons
   • O(N) models
   • Sigma models
   • The Kosterlitz-Thouless phase transition
7. Effective field theory and the Fermi surface
   • The renormalization group applied to the Fermi liquid theory and the superconductive instability
8. Introduction to conformal field theories and their application to the description of critical points

Where possible the symbolic calculus program Mathematica will be used to illustrate applications of the techniques explained during the course.

RECOMMENDED READING/BIBLIOGRAPHY

• Nigel Goldenfeld, Phase Transitions and the Renormalization Group
• Mehran Kardar, Statistical Physics of Fields
• John Cardy, Scaling and Renormalisation in Statistical Physics
• Chaikin and Lubensky, Principles of Condensed Matter Physics
• Shankar, Quantum Field Theory and Condensed Matter
• Alexey Polyakov, Gauge Fields and Strings