With this course, students will learn some fundamental notions of the physics of critical phenomena in many-body systems, such as scale invariance, the renormalization group and the universality of a physical theory. By means of the exercises proposed and solved in the class, they will become familiar with mathematical tools of daily use in the most advanced areas of research in theoretical and statistical physics.

Elements of probability theory and complex analysis, Fourier analysis, basic knowledge of statistical mechanics, and phase transitions. Basic knowledge of stochastic processes.

0. Introductory material (~3h): - self-similarity, scale invariance, fractal geometry, dimensional analysis and scaling. 1. Percolation theory (~5h): - phenomenology and mathematical representation of the percolation transition, exact and approximate results on low-dimensional regular lattices, recursive solution on Bethe lattices, calculation of mean-field critical exponents. 2. Landau theory (~6 h): - review of mean-field theories, phenomenological description of phase transitions, functional integrals, mean-field theory as a saddle-point approximation, mean-field critical exponents. 3. Breakdown of mean-field theory (~8 h): - validity of the Landau approximation, fluctuations, and Ginzburg criterion, upper critical dimension; - Gaussian model, exact calculations of critical exponents; - lower-critical dimension and topological excitations, discrete and continuous symmetry breaking (Goldstone modes and domain walls). 4. Scaling hypothesis and the renormalization group (~8 h): - scaling form of the mean-field equations, scaling beyond mean-field - the Renormalization Group transformation - Wilson’s momentum-shell renormalization procedure, fixed-points, and critical exponents for the Gaussian model. 5. Perturbative Momentum-shell Renormalization Group (~15h): - coarse-graining and integration over fast modes, perturbative expansion of the action, Feynman diagrams, - renormalization of the action and evaluation of momentum-shell integrals, derivation of RG flow equations, epsilon-expansion - Gaussian and Wilson-Fisher fixed points in the phi^4 scalar theory, computation of non-mean-field values of critical exponents - renormalization of the O(n) model, other examples. 6. Field Theory and Non-equilibrium Statistical Physics (~15h). Selected topics among: - equilibrium dynamics of a field, linear response and fluctuation-dissipation theorem; - response functional formalism, non-equilibrium critical dynamics, interface growth, Edwards-Wilkinson, and Kardar-Parisi-Zhang universality classes; - Peliti-Doi formalism, reaction-diffusion systems, and absorbing phase transitions; - slow relaxation in the dynamics of the p-spin spherical model, mode-coupling equations, ergodicity breaking; - quantum phase transitions using path-integral formalism

The course is characterized by classroom-taught lectures mostly delivered using the blackboard (by means of a graphic tablet and slides in case of lectures delivered online). Exercises will be tackled and solved during the classes or proposed as homework, then solved in the class few lectures later.

- Mehran Kardar, Statistical Physics of Fields, Cambridge University Press, 2007 - John Cardy, Scaling and Renormalisation in Statistical Physics, Cambridge University Press, 1996 - Nigel Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, CRC Press, 2018 - Giorgio Parisi, Statistical Field Theory, Addison-Wesley, 1988. - complete lecture notes will be provided by the teacher

Lecture slides; Lecture notes; Exercises; Exercise with solutions ;

You can take this exam before attending the course

Exam: Compulsory oral exam;

The understanding of the topics covered in the course and the student's ability to apply the notions learned to the resolution of exercises and problems arising in the physics of complex systems will be assessed through an oral exam. The oral exam is organized into two parts. In the first part, the student is asked to solve a problem that requires the application of theoretical ideas and mathematical techniques learned during the course (i.e. statistical mechanics and field-theoretic formulation, scaling relations, perturbative expansion, diagrammatics, and renormalization group transformations). Examples of similar exercises will be provided (and solved) in the classes. After reading the text of the problem, the student will be given a short time (about 15-20 minutes) to review the lecture notes / personal notes and an additional 30-45 min to sketch the development and solution of the problem. Then the oral exam will start with the student discussing the resolution of the problem on the blackboard. The rest of the oral exam will consist of three or four broad theoretical questions on the main topics of the lectures. The typical total duration of the oral exam varies between 2 and 2.5 hours.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.