# Thesis projects

Please contact Ralf Bulla for a full list of topics suitable for bachelor and master thesis projects.Here is a preliminary list:

## The two-impurity Kondo problem in spin systems

Here we consider two impurity spins (S_{1} and
S_{2}) weakly coupled
to two different sites of a cluster of spin-1/2 particles. The cluster
is modeled by a Heisenberg or Kitaev model, or combinations
of both. The aim of this project is the numerical calculation of
the spin-spin correlations between the two impurity spins and
its dependence on the model parameters. Of particular interest
are the conditions under which the cluster mediates a long-range
correlation between the impurities.

## Block-tridiagonalization of multi-impurity Anderson models

The multi-impurity Anderson model contains *M* impurity sites
coupled to different sites of a fermionic tight-binding
model. This is illustrated in the figure for *M=2* for both a
chain (a) and a square lattice (b). Using a method called
block-tridiagonalization, these models can be mapped onto a model
in which the *M* impurities are attached to *M*
coupled chains, as shown in part (c) of the figure.
The first part of the project is to develop a computer code
which performs the block-tridiagonalization for an arbitrary
number of impurities coupled to a square lattice. The second part
deals with the systematic investigation of the disordered case: How
does the randomness introduced for the original two-dimensional
model influence the parameters of the coupled chains?

## Classification of Kitaev clusters

The Kitaev model is a quantum mechanical spin model with a very
specific coupling between the spins on different sites of a lattice:
each spin (on site i, characterized by Pauli matrices
σ_{i}^{α})
is coupled to three neighbouring sites (on sites j), with a coupling
of the form
σ_{i}^{α}σ_{j}^{α}
(α = x,y,z).
This project is focussed on the Kitaev model on small clusters with
N=4,6,8,... sites and the aim is to classify these Kitaev clusters
according to symmetries, conserved quantities, the spectrum of
eigenenergies, etc. The corresponding SchrÃ¶dinger equation can be
solved either numerically (by setting up Hamilton matrices) or by
using a Majorana fermion representation of the spin operators.

## Entanglement entropy in the random Heisenberg model

Basically all eigenstates of many-body Hamiltonians show
some degree of entanglement - in other words, product states are
rare exceptions. Now consider the one-dimensional Heisenberg model
and a division of the system in two parts (A and B), as shown
in Fig. a). The entanglement entropy *S _{e}* measures
the degree of entanglement between the subsystems A and B and its

*L*-dependence (

*L*is the number of sites in A) is shown in Fig. b) for the groundstate of the antiferromagnetic case. The question to be studied in this project is how the entanglement properties change in a

*random*Heisenberg model, with the randomness introduced, for example, by setting the nearest-neighbour couplings

*J*as random numbers in some interval

_{n}*J*<

_{min}*J*<

_{n}*J*. Averaging over many sets of {

_{max}*J*} gives a distribution

_{n}*P(S*.

_{e})## The two-channel Kondo problem in spin systems

Consider a one-dimensional Heisenberg model in which an "impurity"
spin is coupled to either one or two spin chains, see Figs. a) and
b), respectively. The case of a single chain (with a chain usually
referred to as channel or bath) gives rise to the well known Kondo
effect, characterized by the screening of the impurity spin by the
bath. In contrast, the two-channel version of the model is
characterized by an *overscreening* of the impurity spin, with
physical properties which differ qualitatively from the
single-channel case. The aim of this project is to identify these
differences via a numerical solution of the SchrÃ¶dinger equation for
small clusters. The characteristics of the one- and two-channel
cases should be visible, for example, in the spectrum of
eigenenergies, the spin-spin correlation and the entanglement
between impurity and bath.

## Iterative diagonalization of the Heisenberg chain

## Kondo physics and percolation