Statistical Mechanics
A.Y. 2018/2019
Learning objectives
Introduction to the theory of critical phenomena and phase transitions
in statistical mechanics, from the 2D Ising model exact solution to Renormalization Group.
in statistical mechanics, from the 2D Ising model exact solution to Renormalization Group.
Expected learning outcomes
Basic knowledge of Mathematical Physics and Probability.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Lesson period
Second semester
Course syllabus
Aim: Introduction to the theory of critical phenomena and phase transition
using exact solutions, quantum field theory and renormalization group methods
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.
using exact solutions, quantum field theory and renormalization group methods
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.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor:
Mastropietro Vieri