The focus of this thesis is in the area of Abelian and Non-Abelian gauge fields that can be efficiently simulated in ultracold atomic systems. These exotic fields and their impact on fermionic particles is studied in the context of two-dimensional systems mainly in optical lattices, that also are available with current experimental techniques. Behaviour of a particle under the influence of such non-Abelian gauge field is contrasted with the standard case of homogeneous magnetic field. Its spectrum and the transport properties such as quantum Hall effect are investigated. The conditions for the energy bands to form a Hofstadter-butterfly-like gaps in an non-Abelian field are given. We show that as long as the Wilson loop for the field is constant, its non-Abelian character does not destroy the big gaps and hence, allows for the integer quantum Hall effect (IQHE). A family of new butterfly spectra is found and the modified IQHE is calculated with the use of Chern numbers. Further, the spectrum of the system is studied in detail and it is demonstrated that it can exhibit anomalies i.e. Dirac cones. The elementary excitations of a system with such spectrum are massless fermions traveling with a modified speed of light similarly to the Majorana fermions in the graphene described by the Dirac equation. We further show that in the case of synthetic non-Abelian gauge field these cones can be squeezed and the speed of light then depends on the direction. Under the conditions of such squeezing the interactions are considered and the first steps towards the analysis of the Fracitonal quantum Hall effect (FQHE) in the presence of non-Abelian field are cone. The matrix elements of the interaction matrix are analytically calculated.


Monday October 3, 2011, 10:00.
Blue Lecture Room

Thesis Co-Advisors: Prof. Maciej Lewenstein, Prof. Jakub Zakrzewski, Dr. Pietro Massignan