** Numerische Methoden für Vielteilchensysteme**

** Tuesday 14-16, Thursday 9-11, Seminaraum 5. **

This course
is intended for students from the 5^{th} semester onwards. It covers a series of modern topics in solid
state physics with emphasis placed on numerical methods. For example, the Hubbard model is the generic
model required to understand the physics of transition metal oxides, a class of
organic materials, as well as the physics of optical lattices. Impurity models, such as the Anderson model,
are the key to the understanding of heavy fermion
materials, the Kondo effect in quantum
dots, as well as the physics of
transition metal oxides in the limit where the coordination number is very
large. Here is a tentative outline of
the course.

1. The

a.
The
central limit theorem

b.
Generating
Markov chains

c.
The
one dimensional Ising model. Exact solution and

2. Introduction to models of correlated
electron systems

a.
Hubbard
model

b.
The
Heisenberg model

c.
The

d.
The
Kondo model.

3. Simulations of quantum spin models

a.
Trotter
decomposition.

b.
The
world line path integral formulation of the Hesenberg
model.

c.
Local
and loop updates.

d.
Stochastic
series expansions.

e.
Aspects
of the physics of quantum antiferromagnets.

4. Auxiliary field methods.

a.
The
Hirsch-Fye impurity algorithm.

b.
The
Kondo resonance and the Kondo effect in quantum dots.

c.
Application
of the Hirsch-Fye algorithm
to dynamical mean-field theories .

5. Exact diagonalization
methods. Ground state static and dynamical properties.