Methods for strongly correlated quantum many-body systems

Quantum many body problems take a central role in many different fields in physics, such as condensed matter physics, particle physics or nuclear physics. Often the complexity of the system leads to unsurmountable difficulties in solving the many-body equations, and approximate effective solutions have to be found. In a few outstanding cases, finding such effective solution is still not solved, and a number of the outstanding problems in theoretical physics are exactly of that kind: the strong-coupling regime of quantum chromodynamics is one example, the mechanism of high-temperature superconductivity is another.

In this course, we explore the basic structure of the quantum many-body problem that appears in its many incarnations across different fields in physics. This is done by treating simple toy models such as the Ising model: we show how perturbation theory fails in solving these types of models, discuss what are quantum phase transitions, show how ideas from the renormalization group allow us to define “effective descriptions” for these models, the central role of entanglement, how quasiparticles arise, etc. Then we discuss exciting many-body phenomena phenomena in condensed matter physics (e.g., high-Tc superconductivity), high-energy field theory (e.g., confinement) and nuclear physics (e.g., nuclear deformations).

In a parallel part of this course, we introduce different modern methods for solving the quantum many-body problem numerically. Here we start from the numerical renormalization group, which exemplifies the idea that a successful numerical method should somehow capture the relevant degrees of freedom in a given many-body system. Then we go on to explore how this idea functions in other numerical methods such as Hartree-Fock theory, density functional theory, the Quantum Monte Carlo method, tensor networks, and neural networks. We show how these methods are used to approach the phenomena that were discussed before. Here we want to show that numerical methods are not to be thought of as black boxes, but are a modern way to gain physical understanding of a quantum many-body system.

Topics :

    1) Introduction to QMB physics, including second quantization formalism
    2) Hartree-Fock (mean field) approximations and DFT approach
    3) Application to nuclear physics : Many-body physics for nuclei, spontaneous symmetry breaking in nuclei, Hartree-Fock-Bogoliubov theory
    4) Numerical renormalization group
    5) Quantum Ising model : phase transitions, symmetry breaking, quasiparticles
    6) Quantum Monte Carlo methods
    7) More advanced models : Heisenberg model (continuous symmetries and fractionalization), Hubbard model (superfluidity and high-Tc superconductivity), Schwinger model (lattice gauge theory)
    8) Entanglement and tensor networks

In this course, we will combine :
- ex-cathedra theory classes
- hands-on exercise sessions
- individual student projects

 

Références, bibliographie et lectures recommandées

- Alexander Altland and Ben Simons – Condensed matter field theory
- Ring and Schuck – The nuclear many-body problem
- Assa Auerbach – Interacting electrons and quantum magnetism
- John Cardy – Scaling and renormalization in statistical physics
- Eduardo Fradkin – Field theories of condensed matter physics
- Subir Sachdev – Quantum phase transitions
- Steven Simon – The Oxford solid state basics
- Kenneth Wilson – The renormalization group: Critical phenomena and the Kondo problem
- Tao Xiang – Density matrix and tensor network renormalization
 

Contacts

Laurens Vanderstraeten
laurens.vanderstraeten@ulb.be

Wouter Ryssens
wouter.ryssens@ulb.be

Campus

Plaine

Méthode(s) d'évaluation

• Projet
• Examen oral

Projet

Examen oral

• Examen à livre ouvert

• Question ouverte à développement long
• Question ouverte à réponse courte

Langue(s) d'évaluation

• anglais
• (éventuellement français )