Nanophysics specialization



Biophysics specialization



Period: Second        ECTS: 4

Course contents

Topic 1 Introduction to path integrals

Feynman path integral in quantum mechanics. Functional integrals in quantum field theory. Fermion and boson propagators. Perturbation theory: Feynman diagrams and Dyson equation.

Topic 2 Applications of the functional integral to condensed matter

RPA approximation to the electron gas. Fermi liquid theory. Broken symmetry and mean field theory.

Topic 3 Criticality and the renormalization group

Phase transitions and criticality. Introduction to the renormalization group in quantum field theory. Applications of the renormalization group to simple problems.

Topic 4 The Hubbard model and bosonization

The 1D Hubbard model with the renormalization group. Bosonization. Luttinger liquid and spin-charge separation.

Topic 5 Diagrammatic approach to out-of-equilibrium systems

The Keldysh formalism. Diagrammatic expansions in out-of-equilibrium systems. The triangular representation.

Topic 6 Applications to electronic transport

Electronic transport in nano structures. Stationary and time-dependent transport.  Analysis of fluctuations: full counting statistics.

Topic 7 Out-of-equilibrium systems with broken symmetry

Superconductivity. Nambu representation. Andreev reflection, Andreev states and multiple Andreev reflection. Basic concepts about topological superconductors.

Topic 8 Out-of-equilibrium systems of interacting electrons

Transport in quantum dots.  Coulomb blockade and Kondo effect.


  • J. W. Negele, and H. Orland, “Quantum Many-Particle Systems”, Frontiers in Physics (1997).
  • A. Altland and B. Simons, “Condensed Matter Field Theory”, 2nd Edition, Cambridge (2010).
  • P. Coleman, “Introduction to Many-Body Physics”, Cambridge (2015).
  • A.A. Abrikosov, L.P. Gorkov y I.E. Dzyaloshinski: “Methods of Quantum Field Theory in Statistical Physics” (Dover Books).
  • A. M. Tsvelik: “Quantum Field Theory in Condensed Matter Physics” (Cambridge, 1996).
  • A. Kamenev: “Field Theory of Non-Equilibrium Systems” (Cambridge 2011).
  • H. Bruus, K. Flensberg, “Many-Body Quantum Theory in Condensed Matter Physics: An Introduction” (Oxford Graduate Texts 2004).


Coordinator  Jaime Merino

More info on the course official guide (Guía docente)