LEARNING OUTCOMES

The lectures aim at providing the tools for a theoretical understanding of the effects of topology in condensed matter physics. The main aim is to address topological phase transitions in fermionic and spin systems.

AIMS AND LEARNING OUTCOMES

By the end of the lectures, the students will be able to:
Understand the effects of topology on condensed matter systems,
Know the basics about conductance quantization in quantum Hall systems, Thouless pumping, Jackiw-Rebbi model, and Majorana fermions,
interpret the experimental results about the topological phases of matter

PREREQUISITES

• FISICA DELLA MATERIA 2 61844
• FISICA TEORICA 61842
• FISICA NUCLEARE, DELLE PARTICELLE E ASTROFISICA 2 61847

TEACHING METHODS

blackboard lectures (online if necessary) where the concepts are explained and, if necerrary, derived. The slides will be used to better describe the relevant experiments.

SYLLABUS/CONTENT

In the first part, the phase transitions are discussed, with particular reference to second order phase transitions. The aspects that will be addressed are

the phenomenology,

the general picture, as emerging from the Ginzburg-Landau theory, 

Spontaneous symmetry breaking and Goldstone modes,

The exact solution of the one-dimensional Ising model,

The approximate solution of the two-dimensional Ising model, based on the quantum-classical correspondence.

In the second part the following topics will be discussed:

the Hubbadr model,

the Heisenberg model as a limit of the Hubbard model,

Ferromagnetic magnons,

Antiferromagnetic magnons,

The xy model and the quantum phase transitions,

The Kitaev chain and the topological phase transitions,

The Jackiw-Rebbi model,

Thouless pumping,

The quantization ofthe Hall conductance.

RECOMMENDED READING/BIBLIOGRAPHY

SUGGESTED BOOKS * H. Bruus, K. Flensberg, "Many-body Quantum Theory in Condensed Matter Physics" Oxford University Press (2004). * G.F. Giuliani, G. Vignale. "Quantum theory of the electron liquid"". Cambridge University Press (2005). * B. A. Bernevig, T. Hughes, "Topological insulators and topological superconductors". Princeton University Press (2013).