Critical behaviour of collective excitations as a probe for exotic quantum phases - Bound states in a bilayer of Heisenberg models on the kagome lattice

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The investigation of exotic phases with topological quantum order represents one of the most interesting and fascinating topics in modern physics. Such gapped quantum phases are highly entangled states where elementary properties like the number of ground states depends on the genus of the topology. Furthermore, such phases have highly exotic elementary excitations, so-called anyons, having a particle statistic different from fermions and bosons. Topologically ordered quantum systems are therefore of fundamental interest for basic science. Furthermore, such phases are the relevant objects for topological quantum computation. Here quantum information is stored and processed in the topological properties of these systems such that one has a topological protection against local decoherence which is the central advantage of this concept compared to other approaches in quantum information.It is therefore of big relevance to identify realistic models having topological quantum order. To this end geometrical frustration is likely a key knob in order to destabilize other conventionally ordered states of matter. The holy gral in the research on frustated quantum systems is the Heisenberg model on the two-dimensional Kagome lattice. After decades of intensive research recent numerical results indicates a topologically ordered gapped spin liquid ground state. It is the central idea of this project to study two kagome planes which are coupled with an unfrustrated Heisenberg coupling in the transverse direction. The fundamental question asked in this project is whether one can understand the nature and the properties of the exotic topological order in the Kagome Heisenberg model by studying the breakdown of the trivial phase at large transverse couplings where one finds dominantly singlets on the transverse dimers. One then might expect that the phase transition can be described as the condensation of two-particle bound states with total spin zero. We therefore would like to calculate single and two-particle excitations energies as high-order series expansions out of the valence bond crystal phase present at large transverse couplings. This should allow to gain an improved understanding of exotic phases due to the investigation of the critical breakdown of conventional phases on a very general level.