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This unit offers an advanced introduction to modern quantum Condensed Matter Physics, in particular the physics of topological and strongly correlated quantum systems.  

This unit aims to provide students with a comprehensive understanding of the theoretical framework underlying modern condensed matter physics, where quantum statistics, topology, and strong correlations play a central role. After introducing the second-quantized formalism for bosons and fermions, the course develops the conceptual and mathematical tools necessary to analyse quantum Hall systems, superconductivity, as well as topological and strongly-correlated phases of matter. Students will develop the ability to connect abstract theoretical constructs to experimentally observable phenomena and to use these ideas to access current research in condensed matter physics. 

SYLLABUS  (S1, 33 lectures – includes example classes) 

1.          Introduction to Bose and Fermi operators (2 lecture): 1D spin chains and mapping to bosonic and fermionic problems. Jordan-Wigner and Holstein-Primakoff transformations. 

2.          Symmetry and topology (8 lectures): recap of the SSH model, winding number and its connection to polarization. Chern physics: Haldane and Kane-Mele models, Berry phase and connection to the quantum Hall effects. Symmetries and topological classification. 

3.          Quantum Hall and edge transport (8 lectures): anomalous Hall effect, integer quantum Hall effect, TKNN formula. Edge states and Landauer-Buttiker formalism. Fractional quantum Hall effect, Laughlin argument in the quantum Hall effect, composite fermions. Fractional Chern insulators. 

4.          Competing orders and superconductivity (7 lectures): phenomenology and BCS theory, topological superconductivity, the Kitaev chain.  

5.          Emergent phenomena in frustrated lattices (7 lectures): frustration in 2D and 3D lattices, quantum spin liquids, Kitaev model, entanglement, toric code. 

6.          Revision (1 lecture) 

 Example classes: W4 (2h), W7 (2h), W11 (2h) 

ILO 1 - Apply advanced theoretical frameworks to model and interpret the behaviour of quantum many-body systems. 

ILO 2 - Apply concepts from topology, quantum mechanics, and condensed matter theory to derive and interpret the physical consequences of models describing superconductivity, quantum Hall effects, and topological insulators. 

ILO 3 - Solve complex quantitative and conceptual problems involving topological phases and strongly correlated systems.

Assessment methods

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