Licentiate thesis defense: Frustration in a bipartite lattice

In this Licentiate thesis we consider a system of interacting bosons on a two dimensional lattice, a model that recently was realized experimentally using ultracold atoms trapped in a periodic potential — an optical lattice. The spectrum of single particles in periodic potentials comprises energy bands. The corresponding states of the bands have an orbital structure, like in atoms or molecules. By making the potential bipartite such that every second well becomes shallow and every second becomes deep it is possible to hybridize so called s- and p-bands. On the shallow site the s-orbital state will be the relevant one, while it on the deep sites will be two p-orbital states. Thereby, the lattice is called an s-p lattice. Turning to a many-body picture we derive a Bose-Hubbard model, here the two main ingredients will be the tunneling of atoms from one site to its neighbours and the onsite interaction between the atoms. The ground state can be either an insulating state with a fixed number of atoms per lattice site or an superfluid phase where the atoms are completely delocalized in the lattice. A feature of the s-p-model that derives from its bipartite nature; is that in many scenarios the sorbital atoms freeze out from the picture and an effective models for the p-orbital atoms can be obtained. This model contains tunneling between the nearest neighboring sites as well as between next-nearest neighbouring sites. Depending on the sign of the tunneling amplitudes, the two terms compete and may give rise to frustration, i.e. the system does not know how to order. We explicitly demonstrate such frustration in the s-p-lattice and discuss some prospects if this frustration can lead to an exotic state of matter which is called quantum spin liquids.”