Topological properties of a quantum state of matter describe global signatures which are invariant under weak local perturbations of the system. Such scenarios constitute a major branch of contemporary condensed matter physics and exhibit remarkable phenomena such as quantized edge channels of a bulk-incompressible phase as well as fractionalized quantum numbers and statistics of quasiparticles. In the TOPOLECTRICS ERC starting grant research plan, we investigate topological quantum phases which result from electronic interactions. The key objective is to provide a rigorous link between bare electronic models and low energy effective models hosting emergent topological quantum phases. This task is approached from two perspectives. First, in a top-down approach, renormalization group schemes are developed and employed for weakly and strongly interacting electrons to derive effective models from realistic bare electronic scenarios. We investigate how Fermi liquids can host topological phases such as Chern insulators, topological insulators, and topological superconductors as well as how Mott phases of frustrated magnets can drive the system into a spin liquid regime. The central goal is to identify the crucial parameters which control the stabilization of such phases and to provide a direct feedback for experimental setups. Second, in a bottom-up approach, entanglement spectrum analysis along with reverse Hamiltonian model building is applied to topologically ordered quantum states such as fractional quantum Hall states, spin liquids, spin chain states, as well as fractional Chern and fractional topological insulators. The aim is to develop effective models hosting such topological quantum states and to reconnect them with bare electronic models as well as to investigate key experimental signatures. Ultimately, the goal is to fuse both approaches to develop a microscopically substantiated procedure to identify electronic topological quantum phases in nature.
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