# 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.

## Iterative diagonalization of the Heisenberg chain

The Schrödinger equation for a cluster of interacting quantum particles can be solved by diagonalization of the corresponding Hamilton matrix, provided the numerical diagonalization of this matrix does not exceed the limits of computing time and memory. In this exact (or full) diagonalization approach, a single matrix is set up for the whole system. In the iterative diagonalization approach, however, one site of the system is added in each iteration - the enlarged cluster is then diagonalized using the information that has been kept from the previous iteration. Combined with a suitable truncation scheme, the iterative diagonalization allows to treat much larger system sizes. The aim of this project is to set up such an iterative diagonalization for the one-dimensional Heisenberg model. Various truncation schemes can be used here, based on the spectrum of eigenenergies or the entanglement spectrum.

## 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.

## Three-body periodic orbits

Consider three bodies (with equal masses *m _{i} = m*)
moving in a plane and interacting via the gravitational
forces between the bodies. Starting from random initial
values for the coordinates and velocities typically leads
to chaotic (non-periodic) orbits. The search for stable and
periodic orbits of the three-body problem started already
in the 18th century and new families of such orbits
have been found only very recently
(see C. Moore,
Phys. Rev. Lett.

**70**, 3675 (1993); M. Šuvakov and V. Dmitrašinović, Phys. Rev. Lett. 110, 114301 (2013)). To investigate these orbits numerically, a computer program has to be set up which solves the coupled set of differential equations (using, for example, the Runge-Kutta method). The aim of this project is to reproduce the periodic orbits as described in the literature and to study their stability with respect to perturbations. Possible extensions are the search for periodic orbits in i) the three-dimensional case and ii) for more than three bodies.