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.

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).

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.