The course gives a theoretical introduction to the problem of phase transitions in statistical mechanics and to Renormalziation Group. In particular It is considered the Ising model, the Onsager exact solution, the non gaussian fermion and Boson functional integrals, the symmetry breaking, the critical phenomena, the Wilsonian Renormalization Group , the notion of universality, the vertex and dimer models, the phi4 model in several dimensions.
Expected learning outcomes
At the end the student will know the concepts of Phase Transitions and Renormalization Group, and he will be able for instance 1)to compute the correlations and the free energy of the Ising model in one dimension 2)To prove the phase transition in the infinite range Ising model 3)To compute the critical temperature in the Ising model in 2 dimensions and some critical exponents using Onsager solution 4)To compute correlations in the dimer model 5) To write the Feynman graph expansion in certain models 6) To compute the scaling dimensions in the RG sense and to say if they are relevant irrelevant or marginal 7)To compute the perturbative corrections to the critical temperature in models like next to nearest neighbor Ising model.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
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 4)Notes on http://users.mat.unimi.it/users/mastropietro/dispms3.pdf
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)