# Thesis projects

Here is a list of topics suitable for bachelor and master thesis projects:## 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.

## 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 Schrödinger equation can be
solved either numerically (by setting up Hamilton matrices) or by
using a Majorana fermion representation of the spin operators.

## Kondo physics and percolation

## Engineering of harmonic chains

Consider a (semi-infinite) harmonic chain with the first body of
the chain displaced at time t=0, q_{o}(t=0)>0, and all other
bodies at rest (at their respective equilibrium positions). The
parameters of the chain, i.e. the values of the masses m_{i}
and spring constants k_{i}, determine the precise form of the
displacement q_{o}(t). The aim of this project is to
calculate the chain parameters m_{i} and k_{i} for a
given q_{o}(t), this means to engineer a harmonic chain to
produce a desired q_{o}(t). On a technical level this can be
achieved by a Laplace tranform of q_{o}(t), which results in
a function Q_{o}(s), and a subsequent continued fraction
decomposition of Q_{o}(s). This procedure clearly cannot work
for any given q_{o}(t), so one of the questions to look at is
which constraints can be assigned to q_{o}(t) such that it can
be realized as the displacement of a harmonic chain.

## Zero-point entropy of quantum impurity systems

The thermodynamic entropy of a quantum system in the limit of
temperature to zero is given by S(T to 0) = ln(d_{g}),
with d_{g} the degeneracy of the ground state. This
degeneracy can only have integer values, d_{g}=1,2,3,...
There are, however, quantum impurity systems for which d_{g}
seems to acquire non-integer values: as an example, the two-channel
Kondo model has a zero-point entropy of S_{imp} = 0.5 ln(2),
corresponding to d_{g}=sqrt(2). The idea of this project is
to derive this anomalous value of S_{imp} from the
specific fixed point structure of the two-channel
Kondo model.