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.

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, 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.

Three-body periodic orbits

Consider three bodies (with equal masses mi = 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.