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Numerical methods for quantum systems
The goal of this course is to introduce the main numerical methods used to solve the quantum many-body problem, namely the many-body Schrödinger equation, which is at the heart of quantum mechanics. Its exact solution being known only for a very limited number of cases, mastering and developing approximate, albeit accurate, numerical tools for solving this problem is of fundamental importance. Indeed, their cutting-edge use allows us to understand, model, and predict phenomena occurring in systems that go from optical lattices to condensed matter. For instance, the development of quantum technologies, as well as of electronic, magnetic, and topological devices, heavily rely upon the numerical solutions provided by these methods.
Syllabus
In this course, we will follow a teaching/learning approach that combines detailed explanations of the underlying formalisms to solve the quantum many-body problem and the introduction to their actual implementations in working computer codes. The lectures (cours magistraux) on the most theoretical part will be followed by the implementation of the corresponding algorithms into codes that will be written and run during hands-on sessions (travaux pratiques). In the last part of the course, the codes previously written will be applied to more challenging problems, with the aim at describing the most recent cold atoms experiments or condensed matter systems. These applications will be carried out in teams of two-three students each.
The lectures will cover the density functional theory (DFT), dynamical mean field theory (DMFT), exact diagonalization (ED), matrix product states (MPS), and quantum Monte Carlo (QMC) methods.
Plan of the course:
- Introduction (4h)
- DFT (4h) + hands-on (4h)
- ED (4h) + hands-on (4h)
- MPS (4h) + hands-on (4h)
- DMFT(4h) + hands-on (4h)
- QMC (4h) + hands-on (4h)
- Projects on quantum systems (8h)
Prerequisites
Examination
Homework (30%), written project report (35%), oral presentation (35%)