Beschreibung:
Abstract: In this thesis we present a first study of the multifractal properties of the ground state of the one-dimensional Bose-Hubbard model in Fock space using two different bases, Fock and momentum basis. In the beginning we give a brief overview of the properties of the Bose-Hubbard model, introduce the concept of multifractality, and discuss exact diagonalisation and quantum Monte Carlo, the numerical methods we employ throughout the present work. Then we treat the limits of vanishing and infinite bosonic interaction strengths analytically, and their vicinities by using perturbation theory. We find that the limit of vanishing interaction exhibits non-trivial multifractality in the Fock basis, and that the generalised fractal dimensions exhibit (dominantly) logarithmic finite-size corrections in this limit. Our observations for other values of the interaction strength suggest that this form of the corrections might be equally valid. In order to get access to the multifractal properties at arbitrary values of the interaction strength, we use exact diagonalisation (for system sizes up to L = 10 at unit filling) and quantum Monte Carlo simulations (which enable us to reach L = 30 in certain cases, corresponding to a Hilbert space of size N = 6*10^16, and therefore to one of the largest ever considered for a multifractal analysis). While we see that the generalised fractal dimensions in the Fock basis decrease monotonously when the interaction strength is increased, we observe an inverted behaviour in the momentum basis, i.e. a monotonous increase with increasing interaction strength. Furthermore, our results suggest the existence of non-trivial multifractality in the ground state for a large range of interaction values. Finally, we investigate whether it is possible to characterise the superfluid to Mott insulator transition with the help of multifractality. We find that an analysis of the generalised fractal dimensions for different densities exposes qualitatively the transition. We furthermore consider different ideas, originating from our results in combination with other approaches used in the literature, to quantitatively characterise the transition. While the investigated methods carry great potential, we confirmed that a quantitatively accurate location of the transition using these techniques, if possible, requires access to even larger system sizes