# 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.

## 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.

## 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})## 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

## Time dependence in quantum spin models

Consider the ground state |psi_g> of a quantum spin model,
such as the one-dimensional anti-ferromagnetic Heisenberg
model. At time t=0, one of the spins (with index 1) is
flipped: |psi(t=0)> = S^{+}_{1}|psi_g>. The resulting
state is, in general, not an eigenstate of the Hamiltonian,
and therefore might show a fairly complicated development
in time. The aim of this project is the numerical calculation
of |psi(t)> for spin models on small clusters, either via full
diagonalization of the Hamilton matrix, or via discretization
of the time-dependent Schrödinger equation.