# Condensed matter from a many-electron point of view

## Frequency

Only this year

## Summary

The course will introduce basic concepts for the description of electrons in condensed matter from a microscopic many-body approach, and relate results obtained by quantum field theoretical methods to properties of the many-electron wave functions underlying quantum Monte Carlo calculations.

## Content

Description of the behavior of electrons in matter constitutes a big part of condensed matter and solid state theory.

Non-relativistic quantum mechanics, Pauli principle, and Coulomb interaction are enough to formulate a microscopic theory, and ab-initio approaches based on approximate solutions to the many-electron problem have been well established in material science.

However, most of practical descriptions are based on mappings to effectively non-interacting electrons, approximately including Coulomb interactions either phenomenologically (mean-field like) or adding them perturbatively, sometimes both. Despite their enourmous success

in describing and predicting material properties compared to experiment, theoretical justification is less obvious.

In this course, I will briefly introduce numerical and analytical methods which aim to solve the fulll underlying microscopic (electronic) Hamiltonian, or provide a systematic analysis based on quantum field theoretical methods. I will then discuss from a many-body perspective how the common basic description of electrons in matter emerges in terms of non-interacting quasi-particles in normal Fermi liquids or electrons and holes interacting via screened Coulomb interaction in semiconductors.

Preliminary outline

- Numerical approaches: Quantum Monte Carlo methods
- Analytical methods: Green'€™s functions, perturbation theory, diagrams
- Many-Fermion wave functions (electron gas)
- From finite to infinite matter: thermodynamic limit considerations
- Born-Oppenheimer approximation, nuclear motion, phonons, band-structure
- Insulator and semiconductors: do electrons interact via bare or screened Coulomb potential?
- Normal Fermi liquids, Luttinger'€™s theorem
- ...

## Keywords

interacting electrons, Fermi liquid theory, electron gas, diagramatic and quantum Monte Carlo methods

## Learning Prerequisites

## Recommended courses

quantum mechanics, statistical physics

## Learning Outcomes

By the end of the course, the student must be able to:

- analytical and numerical methods for interacting electrons in normal Fermi liquids (metal) and semiconductors/insulators

## Dans les plans d'études

**Forme de l'examen:**Oral (session libre)**Matière examinée:**Condensed matter from a many-electron point of view**Cours:**12 Heure(s)**Exercices:**2 Heure(s)**TP:**2 Heure(s)**Type:**optionnel