Periodic Reporting for period 4 - WASCOSYS (Wavefunctions for strongly correlated systems) Reporting period: 2018-11-01 to 2020-06-30 Summary of the context and overall objectives of the project "Complex quantum systems which exhibit strong correlations are at the heart of many exciting phenomena and open problems in physics and beyond. For instance, in the fractional quantum Hall effect (FHQE), the electrons ""fractionalize"", i.e. break up into several parts, and this gives rise to novel robust ways to do ultra-precise measurements or to build a quantum computer. As another example, the study of complex molecules, such as for drug development, crucially relies on our ability to simulate complex correlated quantum systems. And fundamental questions about the constituents of matter, such as strong confinement of quarks, yet again have complex and strongly correlated quantum systems at their heart. Gaining a better understanding of these systems is thus important both from a fundamental and a technological point of view.Yet, the same reason which makes these systems exhibit such a rich behavior also makes them extremely hard to understand: The presence of complex quantum correlations, termed ""entanglement"". Entanglement, on the other hand, has always been a central topic of study in Quantum Information Theory, where a wide range of tools for its understanding and manipulation has been developed. Thus, it is suggestive to apply quantum information concepts to enhance our understanding of complex quantum systems.The goal of the project WASCOSYS - ""Wavefunctions for strongly correlated systems"" - has been to apply tools from Quantum Information Theory to build a framework for the study of complex quantum systems, based on their entanglement structure. The central tool are so-called tensor networks: They provide the natural language to describe those systems based on the structure of their entanglement, and allow to reconcile the local structure imposed by physical interactions with the globally emergent entanglement pattern which gives rise to exotic phases. Throughout the project, our team has developed a diverse range of new tools to study and understand the physics of strongly correlated systems, and we have applied these tools to study a wide range of problems of interest in the field of unconventional quantum mechanical materials. Our results span a wide range from fundamental mathematical questions to realistic physical applications: We have established a systematic mathematical framework to fully classify all types of physics which exotic quantum materials can display, we have developed simulation techniques to provide an in-depth analysis of their properties, and we have applied all those to the wide range of physical materials where we expect to see unconventional physics driven by quantum effects. The results of the project will, in combination with experimental findings and quantum simulations, ultimately lead to new applications and materials based on strongly correlated quantum matter." Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In the project WASCOSYS, we have developed a comprehensive framework for the description of strongly correlated quantum systems based on their entanglement. To this end, we have explored several frontiers. First, we have derived the mathematical formalism for a full classification of the different types of global entanglement patterns in two dimensions and for a wide range of symmetries, which allows us to assess a diverse range of different exotic states of matter which can appear in complex quantum systems. A particular focus has been on the way how the bulk properties of a system are related to the behavior seen at its boundary, which is a key feature of quantum systems exhibiting complex entanglement patterns, and how the boundary can be used to classify their physics. Second, we have built a comprehensive framework for the understanding of so-called topological phases and transitions between them, using the combination of a quantum information driven entanglement-based description and a mechanism known as anyon condensation. This set of tools allows to study topological phases and phase transitions at a level of detail which is inaccessible with established approaches. It thus provides us with the means of probing hitherto unobservable properties of these systems.Third, we have developed a range of techniques which allow us to generalize the quantum information based description of many-body states with tensor networks to new regimes. This includes question as diverse as so-called chiral spin liquids, systems in the continuum, systems with various symmetries, or the construction of hierarchies of increasingly complex models.Finally, we have combined all these tools to perform a broad in-depth study of a wide range of interesting systems which are candidates for exotic physical phenomena. In particular, we have studied under which conditions quantum spin systems exhibit topological spin liquid behavior, as well as chiral topological order. We have also systematically explored the way in which these sought-for phases can be stabilized and thus realized in physical scenarios.The results have been disseminated to the scientific community and beyond in over 100 presentations at conferences, workshops, and summer schools, as well popular talks and articles. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The project WASCOSYS had led to several results which go beyond the state of the art.First, it has led to the development of a range of new tools to probe exotic so-called topological phases of matter, and in particular topological spin liquids. This has been achieved by using that the quantum-information based description gives direct access to quantum correlations (entanglement) and thus allows to measure otherwise inaccessible properties. We have developed a wide range of tools to this end, which will in addition prove useful in future work.Next, WASCOSYS has given rise to the development of a broad and powerful mathematical framework to classify the entanglement structure of these systems. This has several ramifications: First, it allowed us to classify complex quantum phases and the way in which they interplay with symmetries beyond what has been known hitherto. Second, it provides the right framework for the correct and unbiased simulation of these systems. And third, it provides a range of novel powerful mathematical tools to further push the boundaries in the mathematical classification of these systems.Lastly, the project WASCOSYS has provided us with a range of ways to construct wavefunctions for the study of complex quantum systems, such as for chiral spin liquids, or the study of topological spin liquids in different setups. The results which we derived in the project using these tools allowed us to assess the properties of such chiral and non-chiral spin liquids beyond the state of the art, but moreover they provide the right language and a powerful set of tools to further explore a wide range of systems in the future.