QUANTUM PHASES OF MATTER
Academic Year 2026/2027 - Teacher: MARCO RUGGIERIExpected Learning Outcomes
The course aims to provide the basic elements of the physics of strongly correlated systems at low temperatures and of the physics of second-order phase transitions. Various examples of specific models are analyzed in order to provide a first introduction to the classification of phases of matter dominated by strong quantum fluctuations.
▪ Knowledge and understanding. Critical understanding of the main phenomena characterizing the low-temperature properties of strongly interacting quantum many-body systems. Ability to synthesize the knowledge and skills acquired during the course. Knowledge of mathematical and computational tools, mastery of the scientific method and of the procedures commonly used in Physics research.
▪ Applying knowledge and understanding. Ability to calculate observables, also by means of the necessary approximations, within the topics covered in the course. Ability to identify the essential elements of a phenomenon, in terms of orders of magnitude and the required level of approximation. Ability to use analytical and numerical mathematical tools and information technologies, including software development. Ability to make autonomous judgments. Ability to formulate and discuss personal interpretations of physical phenomena.
▪ Communication skills. Communication skills within the context of the course and during the final examination. Students will be encouraged to actively participate during classroom lectures.
▪ Learning skills. Ability to access specialized literature. Ability to use databases and bibliographic and scientific resources to extract information and insights useful for better framing and developing one's own study and research work.
Course Structure
The course consists of lectures delivered at the blackboard or through slide presentations.
Only in the event of unforeseen emergency circumstances, the course may be delivered in a "blended" or "distance learning" mode.
Required Prerequisites
Fundamentals of statistical mechanics, fundamentals of many-body physics, second quantization, and fundamentals of quantum field theory.
Attendance of Lessons
Detailed Course Content
- Path integral and its applications in quantum mechanics, statistical mechanics, and quantum field theory.
- Classical phase transitions. Singularity and order of the transition. Symmetry, symmetry breaking, and order parameter. Ginzburg-Landau theory.
- Dimensional scaling. Relations among critical exponents. Wilson's Renormalization Group and determination of critical exponents. Epsilon expansion. Connection with renormalization in quantum field theory.
- Mermin-Wagner theorem and absence of an ordered phase in two dimensions.
- Quantum phase transitions. Relation between a quantum phase transition in d dimensions and a classical phase transition in d+1dimensions.
- Examples of classical-quantum dimensional crossover: Ising model in 1 and 2 dimensions. Transfer matrix formalism. Quantum Rotor Model.
- Examples of quantum phase transitions. Bose-Hubbard model and physical realizations.
- Transverse Ising model in one dimension: ground state, quantum critical point, duality arguments, exact solution through Jordan-Wigner transformations.
- Effects of quantum criticality at finite temperature. Thermal crossover and quantum critical region.
- Goldstone theorem.
- Kosterlitz-Thouless topological phase transition.
Textbook Information
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Path integral and applications (4 hours) | |
| 2 | Classical phase transitions, symmetry, spontaneous symmetry breaking, order parameter, Ginzburg–Landau theory (6 hours) | |
| 3 | Scaling, critical exponents, Wilson's Renormalization Group, epsilon expansion, connection with QFT (10 hours) | |
| 4 | Mermin–Wagner theorem (2 hours) | |
| 5 | Quantum phase transitions and d↔d+1d \leftrightarrow d+1 mapping (4 hours) | |
| 6 | Classical–quantum crossover: 1D/2D Ising model, transfer matrix, quantum rotor model (5 hours) | |
| 7 | Bose–Hubbard model and physical realizations (4 hours) | |
| 8 | 1D transverse Ising model: ground state, quantum critical point, duality, Jordan–Wigner transformations (6 hours) | |
| 9 | Quantum criticality at finite temperature, thermal crossover, quantum critical region (4 hours) | |
| 10 | Goldstone theorem (2 hours) | |
| 11 | Kosterlitz–Thouless transition (3 hours) |
Learning Assessment
Learning Assessment Procedures
Examples of frequently asked questions and / or exercises
The questions listed below are provided merely as examples and do not in any way constitute an exhaustive list of possible examination questions.
- Derivation of the Wilson-Fisher equations.
- Determination of the relevant/irrelevant directions of the Renormalization Group.
- Calculation of critical exponents.
- Properties of the Ising Model in 1 or 2 dimensions.
- Structure of a quantum critical point at low temperature.