Statistical Mechanics

A.Y. 2019/2020
Overall hours
Learning objectives
The main learning objective of the course is to provide a modern mathematical introduction to classical and quantistic statistical mechamics.
Expected learning outcomes
At the end of the course, students will be able to study, by means of modern mathematical tools, the main phenomena described by classical and quantistic statistical mechanics.
Course syllabus and organization

Unique edition

Lesson period
Second semester
Course syllabus
Spin models.
Exact Solution of the Ising model in one dimension; Transfer matrix method and Multipolygon expansion. Absence of Phase transition.

The infinite range Ising model; exact solution and mean field critical exponents.

Grassmann algebra and Grassman Integrals.

The Diimer model in two dimensions and Kasteleyn solution. Height function and dimer correlations.

The exact solution of the 2-dimensional nearest-neighbor
Ising model:
Multipolygon expansion, dimer mapping and Grassmann integral representation.
Derivation of the free energy in the thermodynamic limit and existence of phase transition.
Ising model and Dirac fermions.

The concept of universality. The next-to-nearest neighbor Ising model and its representation in terms of a non Gaussian Grassmann Integral.

Feynman graphs representation of Grassmann integrals and Infrared Divergences.

Introduction to the Renormalization Group; multiscale expansion, Weinberg Theorem,
Localization operators, overlapping divergences and clusters. Superrenormalizability and
of the next-to-nearest neighbor Ising model.
Prerequisites for admission
Basic knowledge of mathematics and physics
Teaching methods
The lectures are traditional and the frequence is suggested.
1)V. Mastropietro: Non perturbative renormalziaton. World Scientific

2) C. Thomson Mathematical Statistical Mechanics. Princeton University Press

3)C. Itzykson, J. Drouffe Statistical Field Theory. Cambridge University Press

4)Notes on
Assessement methods and criteria
The final exam is oral and it consists in the presentation of arguments explained in the course and in the solution of exercises similar to ones seen in the class (for instance perturbative computations at lowest order or power counting),
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor: Mastropietro Vieri
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