MESOSCOPIC AND TOPOLOGICAL SYSTEMS

Academic Year 2026/2027 - Teacher: FRANCESCO MARIA DIMITRI PELLEGRINO

Expected Learning Outcomes

The course aims to provide the fundamental concepts of mesoscopic physics, with particular emphasis on electronic transport, fluctuations, and single-electron effects, while also introducing the electronic properties of graphene and the basic concepts underlying topological systems.

Knowledge and understanding. By the end of the course, students will have acquired knowledge of the main theoretical approaches used to describe electronic transport in mesoscopic systems, both in the semiclassical and quantum regimes, as well as of fluctuation phenomena and single-electron effects. They will also have acquired knowledge of the electronic structure of graphene and of the fundamental concepts underlying the description of one- and two-dimensional topological systems.

Applying knowledge and understanding. By the end of the course, students will be able to identify the main physical regimes relevant to transport in mesoscopic systems and to select the appropriate theoretical approaches and approximations for their description. They will also be able to apply the formalisms introduced during the course to simple problems in mesoscopic transport and to use the fundamental concepts required to characterize the electronic properties of graphene and topological systems.

Course Structure

The course is delivered through lectures. The theoretical concepts and the corresponding formalisms are accompanied by the discussion of examples, applications, and experimental platforms, highlighting the connection between the models introduced and experimentally observed phenomena. During the lectures, the formal steps and derivations required for the understanding of the models are developed in detail, so that individual study is mainly devoted to consolidating and reworking the concepts discussed in class. Particular attention is paid to the different physical regimes, the approximations adopted, and the limits of validity of the models.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above to ensure consistency with the syllabus.

Required Prerequisites

Essential knowledge: harmonic oscillator, angular momentum and Pauli theory of spin, approximation methods in quantum mechanics; quantum statistics.

Important knowledge: free electrons, electrons in crystals, phonons.

Useful knowledge: scattering theory; kinetic theory.

Attendance of Lessons

Attendance is not mandatory, but it is strongly recommended. During the lectures, the theoretical formalism is developed in detail, together with the discussion of less intuitive conceptual aspects. Participation in class allows students to interact directly with the instructor and to clarify both the more complex formal steps and those concepts that require deeper reflection, facilitating their understanding through discussion and exchange during the lectures.

Detailed Course Content

Semiclassical theory: Semiclassical Boltzmann equation, Relaxation time approximation, Elastic scattering, Diffusive limit, Inelastic scattering, Thermoelectric effects.

Scattering approach to quantum transport: Scattering region, leads, and reservoirs, Scattering matrix, Conductance from scattering, Resonant tunneling, Introduction to localization.

Fluctuations and correlations: Definition and main characteristics of noise, Scattering approach to noise, Boltzmann-Langevin approach, Introduction to the effect of noise on quantum dynamics.

Single-electron effects: Charging energy, Tunnel Hamiltonian and tunneling rates, Master equation, Cotunnelling.

Graphene: Electron structure of monolayer graphene, electrical doping, Landau levels in monolayer graphene.

Topological materials in one and two dimensions: SSH model, Berry phase, Chern number, Current operator and particle pumping, Chern insulators (QWZ model), 2-dimensional time-reversal invariant topological insulators, Electrical conduction of edge states. Majorana fermions.

Textbook Information

[1] T. T. Heikkilä, The Physics of Nanoelectronics: Transport and Fluctuation Phenomena at Low Temperatures, Oxford Master Series in Physics (2013).

[2] M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge University Press (2009).

[3] J.K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions, Springer (2016).

[4] S. M. Girvin, K. Yang, Modern Condensed Matter Physics, Cambridge University Press (2019).

[5] Online course on topology in condensed matter, https://topocondmat.org

[6] M.O. Goerbig, Online course on Introduction to Quantum Mesoscopic Transport and Topological Matter, https://cnrs.hal.science/hal-04248649/

Course Planning

	Subjects	Text References
1	Semiclassical theory (5h)	[1] Chapt. 2
2	Scattering approach to quantum transport (8h)	[1] Chapt. 3
3	Fluctuations and correlations (6h)	[1] Chapt. 6
4	Single electron effects (4h)	[1] Chapt. 7
5	SSH model (3h)	[3] Chapt. 1
6	Berry phase, polarization, and Chern number (3h)	[3] Chapt. 2-3
7	Current Operator and Particle Pumping (2h)	[3] Chapt. 4-5
8	Chern insulators (2h)	[3] Chapt. 6
9	Time-Reversal Symmetric Two-Dimensional Topological Insulators (2h)	[3] Chapt. 8
10	Electrical Conduction of Edge States (2h)	[3] Chapt. 10
11	Graphene (4h)	[2] Chapt. 1-2
12	Majorana Fermions (1h)	[5]

Learning Assessment

Learning Assessment Procedures

Oral examination on a topic chosen by the student, to be presented with due attention to formal details, and on two additional topics selected by the lecturer. The examination is intended to assess both the student’s knowledge and understanding of the course contents and their ability to use the formalisms introduced, as well as to identify the relevant physical regimes and the approximations appropriate for their description.

For assessment, the following aspects are taken into account: the relevance of the answers to the questions asked, the level of detail and depth of the contents presented, the ability to establish connections with other topics included in the course, the ability to provide examples, the correct use of formal tools, the appropriate use of scientific language, and clarity of exposition.

Learning assessment may also be carried out online, should circumstances require it.

To ensure equal opportunities and in compliance with current legislation, interested students may request an individual meeting in order to arrange any compensatory and/or dispensatory measures, in accordance with the learning objectives and their specific needs. Students may also contact the CInAP representative of their Department.

Examples of frequently asked questions and / or exercises

The questions listed below do not constitute an exhaustive list, but represent some examples of possible examination questions.

“What physical quantities should be compared to distinguish the different transport regimes, and what form do the corresponding transport equations take?”

“What are the main peculiarities of the Landau levels in monolayer graphene?”

“How can the topological properties of the SSH model be characterized?”

Degree Course in

Physics