# Lecture: Computational Many-Body Physics

summer term 2022

Lectures will be held in presence on Wednesday 8:00 - 9:30.
- Seminar room: 0.03 (new theory building)

On Monday 14:00 - 15:30, there will be:
- Tutorials in presence every two weeks (dates see below); seminar room: 0.01 (new theory building).
- There will also be a separate online group (via Zoom, same date and time).
- Online-lectures (in the remaining slots).

The online lectures will be mainly used to discuss organizational issues, the exercise sheets and questions related to the topics of the lecture.

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'
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
6. Green functions for fermionic systems
6.1 basic definitions
6.2 equations of motion and continued fractions
7. renormalization group methods
7.1 general concepts
7.2 numerical renormalization group (NRG)
7.3 density-matrix renormalization group (DMRG)

Tutorials:

The tutorials will be held on Mondays, 14:00 - 15:30
Dates: 25.04 - 09.05 - 23.05 - 13.06 - 27.06 - 11.07

Tutors: Vahideh Eshaghian, Ana-Luiza Ferrari

Solutions can be submitted in groups of up to three students (via ILIAS).
Requirements for the admission to the module exam:

To obtain the signature on the form Anmeldung zur Modulprüfung im Masterstudiengang, at least 50% of the points from the exercises are required. This applies to the overall score, not to the score for the individual sheets.

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