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 universality of the next-to-nearest neighbor Ising model.
Prerequisites for admission
Basic knowledge of mathematics and physics
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
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),