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 (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.

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, qo(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 mi and spring constants ki, determine the precise form of the displacement qo(t). The aim of this project is to calculate the chain parameters mi and ki for a given qo(t), this means to engineer a harmonic chain to produce a desired qo(t). On a technical level this can be achieved by a Laplace tranform of qo(t), which results in a function Qo(s), and a subsequent continued fraction decomposition of Qo(s). This procedure clearly cannot work for any given qo(t), so one of the questions to look at is which constraints can be assigned to qo(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(dg), with dg the degeneracy of the ground state. This degeneracy can only have integer values, dg=1,2,3,... There are, however, quantum impurity systems for which dg seems to acquire non-integer values: as an example, the two-channel Kondo model has a zero-point entropy of Simp = 0.5 ln(2), corresponding to dg=sqrt(2). The idea of this project is to derive this anomalous value of Simp from the specific fixed point structure of the two-channel Kondo model.