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 (S1 and S2) 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 σiα) is coupled to three neighbouring sites (on sites j), with a coupling of the form σiασjα (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 Se 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 Jn as random numbers in some interval Jmin < Jn < Jmax. Averaging over many sets of {Jn} gives a distribution P(Se).

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 Schrödinger 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