Mandatory course for the Master in Physics of Complex Systems (national track), 2nd year, 1st term. The course aims to provide an introduction to statistical field theory and its application to the study of phase transitions and critical phenomena in complex systems. The formulation of interacting many-body systems in terms of continuous fields makes possible to go beyond mean-field approximations (Landau theory) to evaluate the effects of spatial fluctuations and correlations and their influence on the collective behaviour of systems near a phase transition. To this purpose, the course will provide an introduction to the mathematical tools (functional integrals, perturbative expansions, Feynman diagrams) and physical concepts (scale invariance, renormalization) that are at the basis of modern statistical physics. The revolutionary idea of ​Wilson’s renormalization group will be introduced and discussed by means of examples. The same methods will be also applied to the study of collective non-equilibrium phenomena.

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, basic knowledge of statistical mechanics and phase transitions.

1. Landau theory (~10 h): - phenomenological description of phase transitions, saddle-point approximation and mean-field theory, critical exponents; - continuous and discrete symmetry breaking (Goldstone modes and domain walls), lower-critical dimensions; - Gaussian integrals, role of fluctuations, Ginzburg criterion, upper critical dimensions. 2. From Scaling invariance to Renormalization (~10h): - scaling theory and invariance, coarse-graining process, the effective Hamiltonian, functional derivatives, Green functions; - Gaussian model, Wilson’s momentum-shell renormalization procedure, fixed-points and critical exponents for the Gaussian model. 3. Perturbative Renormalization Group (~20h): - perturbation theory in coordinate and momentum space, Wick’s theorem, Feynman diagrams, vertex functions and Legendre transform, loop expansion in the phi4 theory, 1/N expansion in the O(N) theory; - power counting and divergences, renormalization of mass, field and couplings, dimensional regularization, minimal subtraction scheme, RG flow equations, fixed points for the phi4 and O(n) theories, critical exponents; - basics of functional (non-perturbative) renormalization group methods. 4. Field Theory and Non-equilibrium Statistical Physics (~20h). Selected topics among: - equilibrium dynamics of a field, linear response and fluctuation-dissipation theorem; - Martin-Siggia-Rose 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.

The course is characterized by frontal lectures mostly delivered using the blackboard. Plots and computational results will be also shown by means of computer presentations prepared by the teacher and then made available to the students. Homework exercises will be proposed, 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. - additional material provided by the teacher

Modalità di esame: Prova scritta (in aula); Prova orale facoltativa;

Exam: Written test; Optional oral exam;

... The understanding of the subjects covered in the course and the ability of the student to apply the learned concepts to the resolution of exercises and problems in the physics of complex systems will be evaluated through a written exam. The exam will focus on the entire program of the course and it will consist of two parts: 1) a set of multiple-choice questions, that will act as a threshold to assess minimal theoretical knowledge, 2) resolution of a multistep guided problem requiring the application of scaling ideas and to carry out calculations (perturbative expansion and renormalization). Students obtaining a grade of 27/30 or higher are allowed (upon request by the student) to do a short oral exam (30-45 min) to possibly improve their grade. This exam will focus on the entire program of the course and it will consist of an initial question on a topic selected by the student plus two other theoretical questions.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.