Monday 12:00 - 13:30, Wednesday 8:15 - 9:45, seminar room: 0.03 (new theory building)
The lecture starts on Wednesday, 14.10.2026, 8:15.

For more information, please check the ILIAS-page.

Contents:

This lecture gives an introduction to numerical methods for the investigation of classical and quantum many-particle systems. Already on the classical level (see Sec. 1), the interactions between the particles give rise to non-trivial behaviour which cannot be (in most cases) derived using analytical methods. Sections 2-4 deal with quantum spin models (mainly the Heisenberg model) and the diagonalization of small clusters is introduced as the simplest numerical approach (Sec. 2.2). Various physical properties can be studied within this framework (Sec. 3), but the extension to larger systems (more than ~15 spins) requires more advanced methods (Sec. 4). Finally, fermionic many-particle systems - with the Hubbard model as the most important example - are introduced in Sec. 5.

Module description of the primary area of specialization `Solid State Theory/Computational Physics'

Table of Contents:

1. classical many-particle systems
   1.1 cellular automata
   - rule N, Game of Life, ASEP, Nagel-Schreckenberg model
   1.2 statistical physics of classical many-particle systems
   - Metropolis algorithm for the classical Ising model
   1.3 Newtonian dynamics of classical many-particle systems
   
2. quantum-mechanical spin models - introduction
   2.1 quantum spin models
   - Heisenberg model, Kitaev model
   2.2 diagonalization of small clusters
   - Hamilton matrix
3. quantum-mechanical spin models - calculation of physical properties
   3.1 spin correlations
   - T=0 and T>0
   3.2 entanglement
   - reduced density operator, entanglement entropy, Schmidt decomposition
4. numerical methods
   4.1 exact diagonalization (Lanczos)
   4.2 iterative diagonalization
   - application to the 1d Heisenberg model, truncation, relation to rg methods (see Sec. 7)
   4.3 quantum Monte Carlo (QMC)
   - application to the 1d Heisenberg model, Suzuki-Trotter decomposition, world lines
5. fermionic many-particle systems
   5.1 second quantization
   5.2 fermionic models
   - Hubbard model, single-impurity Anderson model
   5.3 diagonalization of small clusters