Course Unit Page

Academic Year 2019/2020

Learning outcomes

At the end of the course the student will acquire some knowledge about the physics of strongly interacting many body systems. More specifically he/she will be able to use both exact and approximation methods for the study of phase transitions and critical phenomena. These techniques will be seen in action by analysing some paradigmatic models, both in the classical and in the quantum setting.

Course contents


  • Review on classical phase transitions: theory and examples
  • Critical exponents and universality classes
  • Spontaneous Symmetry Breaking (SSB); Goldstone and Mermin-Wagner theorems
  • The Landau-Ginzburg paradigm
  • Magnetic transitions
  • Superconductivity


  • Introduction to Quantum Phase Transitions
  • Exactly solvable models in 1D
    1. the XX and the XY models
    2. the Bethe Ansatz solution: the Lieb-Liniger model, remarks on integrability
  • BKT transitions and vortices: The x-y-model and generalized elasticity, Topological defects and vortices
  • Topological phase transitions: the SSH model, the Kitaev chain and p-wave topological superconductors in 2d, Berry phase and topological Chern insulators, the classification of topological phases


A detailed syllabus with references can be found in IOL.

E. Ercolessi, G. Morandi, F. Napoli - Statistical mechanics: an intermediate course, II edition

S. Sacdhev, Quantum phase transitions

F. Franchini, An introduction to integrable techniques for one dimensional quantum systems, arXiv:1609.02100

B. Sutherland, Beautiful models

P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics

J.K. Asboth, L. Oroszany, A. Palyi, A short course on topological insulators

B.A. Bernevig, Topological Insulators and Topological Superconductors

B.A. Bernevig, T. Neupert, Topological superconductors and category theory, arXiv:1506.05805;

L. Fidkowski, A. Kitaev, Topological phases of fermions in one dimensions, arXiv:1008.4138

Teaching methods

The course consists of about 48 hours of class lectures, given by the teachers at the blackboard and/or with slides.

Assessment methods

Oral exam: students are asked to prepare a seminar on some topic of interest, which has to be agreed with the teachers.

A list of possible topics can be found in IOL.

Students should demonstrate to be familiar and have a good understanding of the different sujects.

They will be asked to both present an introduction to the main general topics and to prove more specific results, making connections among the different parts of the syllabus.

The organization of the presentation and a rigorous scientific language will be also considered for the formulation of the final grade.

The “cum laude” honor is granted to students who demonstrate a personal and critical rethinking of the subject.

According to the general rules of the University, students will be allowed to reject the grade only once, but they can withdraw at any time during the exam.

Teaching tools

Notes and slides, available to download from the university repository

Office hours

See the website of Elisa Ercolessi

See the website of Cristian Degli Esposti Boschi