The ordered phases of magnetic materials are well understood within the framework of the mean-field approaches. The excitations above such an ordered state can have nontrivial topological properties, like a finite Chern number and edge states. Furthermore, competing interactions in a quantum spin system can lead to entangled states which are beyond the reach of the traditional mean-field description. In such systems, the order is often lost — they form the wide class of spin liquids, where the ground states can be characterized by topological invariants.

 

 In this Ph.D. project, we will search for the ground states and excitations in spin liquids and other entangled systems. What type of states can we describe with low dimensional tensor networks, how can we extend the AKLT type wave-functions to higher dimensional SU(2) spin models of localized electrons and SU(N) models of ultracold atoms? Can we excite non-abelian excitations? Can we construct magnetic systems where the second Chern number plays a role? This is a theoretical study, requiring the application of analytical and numerical methods.