QUANTUM PHASES OF MATTER
Academic Year 2023/2024 - Teacher: Luigi AMICOExpected 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
Required Prerequisites
Basics of statistical mechanics, of many body theory, of second
Attendance of Lessons
Detailed Course Content
Path Integral and its application in quantum mechanics, statistical mechanics, quantum field theory. | ||
2 | Classical phase transitions. Singularities and order of the transition. Symmetry, symmetry breaking and order paramer. Ginzburg Landau Theory. | |
3 | Dimensional scaling. Relation among critical exponents. Wilson Renormalization Group and determination of critical exponents. Epsilon expansion. Connection with the renormalization of Quantum Field Theory | |
4 | Mermin-Wagner theorem and no ferromagnetic ordered phase in two dimensions. | |
5 | Quantum Phase Transitions. Relation between d quantum, and d+1 classical phase transitions | |
6 | Examples of quantum-classical dimensional crossover: one and two dimensional Ising model. Transfer matrix formalism. Quantum Rotor model. | |
7 | Examples of Quantum Phase Transitions. The Bose-Hubbard model and physical realizations. | |
8 | Transverse Ising Model in one-dimension: ground state, quantum critical point, duality argument, exact solution by Jordan-Wigner transformation. | |
9 | Effects of quantum criticality at finite temperature. Thermal crossover and quantum critical region. | |
10 | Goldstone theorem. | |
11 | Topological Kosterlitz-Thouless phase transition. |