# Lecture: Computational Many-Body Physics

**summer term 2019**

Monday 14:00 - 15:30, Wednesday 8:15 - 9:45

Seminarraum 0.01 TP (new theory building)

Seminarraum 0.01 TP (new theory building)

This lecture gives an introduction to numerical methods
for the investigation of classical (Sec. 1) and quantum (Secs. 2-8)
many-particle
systems. The focus in 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 8 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,
entanglement measures.

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

**Contents:**(preliminary)

- 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 classical many-particle systems - quantum-mechanical spin models - introduction

2.1 Heisenberg models

2.2 diagonalization of small clusters

- Hamilton matrix - quantum-mechanical spin models - calculation of physical properties

3.1 spin correlations

-*T=0*and*T>0*

3.2 thermodynamics

3.3 entanglement

- reduced density operator, entanglement entropy, Schmidt decomposition - numerical methods

4.1 exact diagonalization (Lanczos)

4.2 iterative diagonalization

- application to the 1d Heisenberg model, truncation, relation to rg methods (see Sec. 8)

4.3 quantum Monte Carlo (QMC)

- application to the 1d Heisenberg model, Suzuki-Trotter decomposition, world lines - fermionic many-particle systems

5.1 second quantization

5.2 diagonalization of small clusters - overview: models of quantum many-particle systems

6.1 Heisenberg model

6.2 Kitaev model

6.3 Hubbard model

6.4 single-impurity Anderson and Kondo model

6.5 periodic Anderson model - Green functions for fermionic systems

7.1 basic definitions

7.2 equations of motion and continued fractions - renormalization
group methods

8.1 general concepts

8.2 numerical renormalization group (NRG)

8.3 density-matrix renormalization group (DMRG)

The script of the previous lecture, given in the summer term 2017, is available here. This script covers most of the material presented in the current lecture, although in a different order.

**Tutorials:**

Mondays, 14:00 - 15:30, Seminarraum 0.01 TP (new theory building)

Dates: April 15, May 6, May 20

Tutor: Nico Gneist

The exercise sheets contain both analytical and programming exercises. We recommend to use the Julia programming language. Please send the solutions via e-mail to Nico Gneist (gneist@thp...).

Exercises:

**Requirements for the admission to the module exam:**

For the signature on the form "Anmeldung zur Modulprüfung im Masterstudiengang", each student has to select one topic from the list of projects and present the results in a talk of about 20 minutes. The talks will be scheduled for the end of the summer term.

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