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QUANTUM PHASES OF MATTER

Academic Year 2022/2023 - Teacher: DARIO GAETANO ZAPPALA'

Expected Learning Outcomes

The course aims to provide the basic elements of the physics of strongly interacting many body systems at  very low temperature and of the physics of  second order phase transitions. Various examples of specific models are analyzed, in order to give a first introduction to the classification of phases of matter under the influence of strong quantum fluctuations.

  • Knowledge and understanding.  Critical understanding of the main phenomena that identify the properties of strongly interacting many body systems at low temperature. Capability of exploiting  those skills acquired during the course. Knowledge of the basic mathematical tools and procedures, as well as of the scientific method, currently used in research in physics. 

  • Applying knowledge and understanding. Capability of computing observables related to subject of the course, also including the required approximations or approximation schemes.  Capability of  characterising the main relevant features of a phenomenon, including order of magnitude and degree of approximation. Skills  in the use of  numerical computation with the help of electronic devices.  Skills in formulating a personal interpretation or assessment of a specific problem concerning  the course.

  • Communication skills. Communication skills shown during the course and at the final examination. Students are invited to intervene and actively participate to classes.

  • Learning skills. Ability in searching for specific issues through  specialised literature, in order to clarify or improve the understanding of specific issues encountered during the study of the subject.

Course Structure

Teaching consists of frontal lectures, with calculations displayed on blackboard or 

on projected slides. Only in case of emergency (coronavirus or other), teaching could be switched to  remote or mixed form.

Required Prerequisites

Basics of statistical mechanics, of many body theory, of second quantisation and of quantum field theory.


Attendance of Lessons

Attending classes of the course is, in general, mandatory (see "Regolamento Didattico del Corso di Studi"). Exceptions to this rule must be singularly considered and assessed.

Detailed Course Content

1Path Integral and its application in quantum mechanics, statistical mechanics, quantum field theory.
2Classical phase transitions. Singularities and order of the transition. Symmetry, symmetry breaking and order paramer. Ginzburg Landau Theory.
3Dimensional scaling. Relation among critical exponents. Wilson Renormalization Group and determination of critical exponents. Epsilon expansion. Connection with the renormalization of Quantum Field Theory
4Mermin-Wagner theorem and no ferromagnetic ordered phase in two dimensions. 
5Quantum Phase Transitions. Relation between d quantum, and d+1 classical phase transitions
6Examples of quantum-classical dimensional crossover: one and two dimensional Ising model. Transfer matrix formalism. Quantum Rotor model.
7Examples of Quantum Phase Transitions. The Bose-Hubbard model and physical realizations.
8Transverse Ising Model in one-dimension: ground state, quantum critical point, duality argument, exact solution by Jordan-Wigner transformation.
9Effects of quantum criticality at finite temperature. Thermal crossover and quantum critical region. 
10Goldstone theorem.
11Topological Kosterlitz-Thouless phase transition.

Textbook Information


  1. R. Feynmann, "Statistical Mechanics: A Set Of Lectures", (Frontiers in Physics) CRC press, 1972.

  2. S. Sachdev, “Quantum Phase Transitions” (Cambridge University press 2011).


  3. X.G. Wen, “Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons”, (Oxford University press 2007).

  4. G. Mussardo, "Il modello di Ising. Introduzione alla teoria dei campi e delle transizioni di fase", Boringheri 2010

  5. Lecture notes.

Course Planning

 SubjectsText References
1Path Integral (4h)1) and 5) of the text list.
2Classical phase transitions. Symmetry, symmetry breaking. Ginzburg Landau Theory.(4h)2) and 5) of the text list.
3Dimensional scaling. Critical exponents. Wilson Renormalization Group. Epsilon expansion. Connection with the renormalization of Quantum Field Theory. (6h)5) of the text list.
4Mermin-Wagner theorem.(2h)5) of the text list.
5Quantum Phase Transitions. Relation between d quantum, and d+1 classical phase transitions.(4h)2) and 5) of the text list.
6Quantum-classical dimensional crossover: 1 & 2 dimensional Ising model. Transfer matrix formalism. Quantum Rotor model. (4h)2) and 5) of the text list.
7Quantum Phase Transitions. The Bose-Hubbard model and physical realizations. (4h)2) and 5) of the text list.
8Transverse Ising Model in 1 dimension: ground state, quantum critical point, duality argument, exact solution by Jordan-Wigner transformation.(4h)3), 4)) and 5) of the text list.
9Effects of quantum criticality at finite temperature. Thermal crossover and quantum critical region. (2h)3), 4) and 5) of the text list.
10Goldstone Theorem. (4h)5) of the text list.
11Topological Kosterlitz-Thouless phase transition. (4h)5) of the text list.

Learning Assessment

Learning Assessment Procedures

The assessment  of the level of knowledge achieved by the students, besides the active participation to classes, is essentially based on the final examination, which consists of an oral interview on three issues where,  usually, the first one involves relatively complex calculations. 
On personal basis, the student is allowed to present a particular subject, chosen beforehand together with the professor, in any case only to improve the  particular grade  achieved in the standard examination.

Examples of frequently asked questions and / or exercises

The following list of issues represents a plain example of the kind of questions raised during the examination and, by no means, represents an exhaustive list of questions.

Computation of Wilson-fisher equations. 
Determination of relevant/irrelevant directions of the Renormalization Group. 
Computation of the crititical exponents. 
Ising model in 1 or 2 dimensions. 
Quantum critical point structure at low temperature.