Monday 14:00 - 15:30, Wednesday 8:00 - 9:30, seminar room: 0.03 (new theory building)

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. The focus is on models of strongly correlated electron systems (Hubbard model, single-impurity Anderson model) and quantum spin models (Heisenberg model, Kitaev model). The physical phenomena (Mott transitions, Kondo physics, spin liquid physics, etc.) these models are supposed to describe, are quite often out of the reach of analytical techniques - this triggered the development of very powerful numerical approaches, see Secs. 4 and 7 in the table of contents. The lecture also includes a brief introduction to basic theoretical concepts, such as Green functions, continued fraction expansions, reduced density matrices, and entanglement measures.

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