Description and contextualization of the subject

Esta asignatura se divide en dos partes. La primera parte abarca teoría de la materia condensada y estadística cuántica, introduciendo conceptos y técnicas fundamentales. Se comienza con el formalismo de segunda cuantización y las estadísticas de fermiones y bosones, y se introducen conceptos como las funciones de Green y los diagramas de Feynman. También se consideran las interacciones electrón-electrón mediante la aproximación de Hartree-Fock y los líquidos de Fermi. La segunda parte se centra en la fenomenología de la superconductividad, abordando los materiales superconductores, el efecto Meissner y las diferencias entre superconductores de tipo I y tipo II. Se introducen teorías como la de London y Ginzburg-Landau, junto con la teoría BCS, discutiendo los pares de Cooper y el efecto Josephson.

Learning outcomes of the subject

Knowledge or content:

RCO1. Demonstrate the ability to explain the fundamental principles of the quantum world, both at a basic and technical level.

RCO2. Have a basic knowledge of the relevant literature in quantum mechanics and be capable of effectively reading and understanding research articles.

RCO3. Be able to initiate the development of original ideas and applications within the context of quantum physics research.

RCO4. Possess the capacity for independent research, synthesis, and be able to present in a clear and structured way complex issues related to the various areas of quantum mechanics addressed in this Master¿s program.

RCO5. Under supervision, demonstrate the ability to write and defend original work that meets the quality standards required for publication in high-impact indexed journals.

RCO9. Know the basic literature and demonstrate the ability to solve standard problems in the field of Quantum Statistical Physics.

RCO11. Know the basic literature and demonstrate the ability to solve standard problems in the field of Condensed Matter Physics.



Competencies:

RC1. Possess and understand knowledge that provides a basis or opportunity for developing and/or applying original ideas, often in a research context.

RC2. Apply acquired knowledge and problem-solving skills in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their field of study.

RC3. Demonstrate the ability to integrate knowledge and address the complexity of formulating judgments based on incomplete or limited information, including reflection on social and ethical responsibilities linked to the application of their knowledge and judgments.

RC4. Communicate conclusions, as well as the underlying knowledge and rationale, clearly and unambiguously to both specialized and non-specialized audiences.

RC5. Possess learning skills that enable continued study in a largely self-directed or autonomous manner.



Abilities or skills:



RHE1. Demonstrate proficiency in using tools for bibliographic resource searches.

RHE2. Exhibit critical capacity to read research articles and incorporate their findings into one¿s own work.

RHE3. Write and present original work in one of the official languages and in English.

RHE4. Communicate scientific concepts and results clearly and effectively to both specialized and non-specialized audiences, through presentations and publications.

RHE5. Demonstrate the ability for autonomous learning and staying current with scientific and technological advances.





RHT1. Understand and apply the fundamental principles of quantum mechanics to analyze and solve problems in basic research in quantum science.

RHT3. Effectively integrate into a fundamental or applied research project involving quantum aspects, and solve problems in multidisciplinary environments.

RHT4. Evaluate and select appropriate tools and techniques for research in fundamental physics.

Temary

PART I

Preliminaries: introduction to second quantization. Fermion and boson statistics.

Green¿s function approach: time evolution operator. Gell-Mann-Low¿s theorem. Green¿s function (GF) definition. Non-interacting one-fermion GF. Linear response. Wick¿s theorem. Feynman diagrams.

Green¿s function at finite temperature: GF in the Matsubara frequency domain. Analytic continuation and retarded/advanced GF. Techniques: equation of motion and Matsubara summations.

Electron-electron interactions: Hartree-Fock approximation. Self-energy. Dyson equation. Fermi liquids. Free-electron polarizability. RPA.

Impurities: Potential scattering. Friedel¿s oscillations. Anderson impurity model.

Strongly correlated systems: Hubbard¿s Hamiltonian and approximations based on the GF.

Magnetic response: Stoner criterion. Magnetic instability in RPA.





PART II

Phenomenology of superconductivity: Superconducting materials - absence of low-energy excitations. Isotope effect. The Meissner-Ochsenfeld effect. Perfect diamagnetism. Type I and type II superconductivity. London theory, flux quantization and Ginzburg-Landau equations.

Electrons in metals: Non-interacting Fermi gas. Recap second quantization for fermions; distribution function for non-interacting Fermi gas. Electron-phonon interaction; repulsive and attractive electron-electron interaction.

The BCS theory of superconductivity: Mean-field Hamiltonian, Cooper pairs, the BCS wave function. Energy gap and quasiparticle states.

The critical temperature. Electron tunneling between normal and superconducting metals.

The Josephson effect: Cooper-pair tunneling between superconductors.

Inhomogeneous superconductors: Bogolubov-deGennes equations; Andreev reflection; Andreev bound states.

Bibliography

Basic bibliography

S. Doniach and E. Sondheimer, Green's Functions For Solid State Physicists, Imperial College Press, 1998.

Gerald D. Mahan, Many-Particle Physics (3rd Edition). Springer Science 2000.

Henrik Bruus, Many-body quantum theory in condensed matter physics: an introduction. Oxford University Press, 2004.

Ottfried Madelung, Introduction to Solid-State Theory, Springer, 2012.

A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, 1993.



P.G. de Gennes, Superconductivity of Metals and Alloys, Benjamin 1966.

M. Tinkham, Introduction to Superconductivity, 2nd Edition, McGraw-Hill 1996.

N. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College Publishing 1976.

In-depth bibliography

A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems, Dover Publications, 2003.

Patrick Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific, 1999.

Robert M. White, Quantum Theory of Magnetism, Springer, 2007.