Research project Topological phases and their transitions
The last decade has seen the advance of topological phases of matter, such as topological insulators and topological superconductors. Topological phases of matter are characterized by topological invariants, describing the non-local properties of such phases. For systems of non-interacting fermions, the possible topological phases have been classified.
One of the most striking results is that a topological one-dimensional superconducting phase, can be realized in strongly spin-orbit coupled nano-wires, proximity coupled to an ordinary superconductor. Such a system exhibits Majorana bound states at its edges, or, in other words, half a fermionic mode at each edge. Several recent experiments have detected signatures, which could very well originate from such Majorana boundstates.
In this projekt, several important issues concerning topological phases will be studied in detail. We will study the properties of the Majorana bound states from a theoretical perspective, by studying the bound states appearing different types of Josephson junctions. In addition, we design and analyze microscopic models exhibiting Majorna bound states, which allows to study the effects of interactions between the bound sates, as well as the the effects of a varying number of bound states. These studies are of prime importance to unravel the delicate properties of the Majorana bound states, which is necessary if one aims to use these bound states as topologically protected quantum bits in a quantum computation setting.
Of equal importance is to understand the possible phase transitions from a topological phase to a normal phase, and phase transitions between different topological phases. We aim to study these topological phase transitions by means of designing microscopic models, that allow for the study of the phase transition. The key advance here is to take both the gapped bulk as well as the gapless edges of the topological phases into account simultaneously. Detailed knowledge of the inter play between the bulk and the edge of the system will provide crucial information on the nature of the phase transitions between different topological phases.
The field studying topological phases has greatly benefitted from the classification of topological phases of non- interacting fermionic systems. A big challenge is to make progress in classifying which topological phases are possible in the presence of interactions. It is clear that it is not possible to give a full answer to this big question, but important progress is within reach, by making use of the powerful correspondence between the bulk and the boundary of topological phases, and by constraining oneself to two-dimensional topological phases. In this way, one can study one-dimensional gapless critical systems. For such systems, there exist many techniques one can employ, ranging from powerful numerical methods to constructing exactly solvable models.
The knowledge gained in the proposed projects will lead to an enhanced understanding of topological phases of matter, which is crucial both if one tries to exploit the properties of topological phases that already have been discovered, as well as in guiding the experimental search for knew topological phases of matter.
