For higher-order topological insulators, we developed a mathematical framework that allows to predict which materials have this property. Based on our prediction we found that elementary bismuth is an iconic representative of this new phase of matter. Together with experimental groups, we collected evidence that our prediction is actually true. From this starting point, we developed the concept of higher-order topology in many further directions. Among others, we found criteria under which two-dimensional crystals (like graphene) have corner charges (the lower dimensional version of the conducting hinge modes). Furthermore, we studied higher-order topological superconductors, which have corner or hinge modes that are called Majorana fermions and may be relevant for building quantum computers. These developments are flanked by many other subprojects and methodological progress. Most prominently, we work on a code package called Wannier Berri, which allows to calculate response functions of real materials, based on band structure calculations, with extremely high precision. We tested the code on several occasions in collaboration with experimental groups or when we predict new topological quantum materials.
For the numerical investigation of strongly interacting phases of matter, we have mostly focused on code development and benchmarking of the method of neural network quantum states. In that way, PARATOP team members contributed to NetKet, the most prevalent software package in this filed. The results are very promising. We tested the new method on 1D systems, and on the 2D J1-J2 Heisenberg model on the square lattice, which is a iconic model of frustrated magnetism. We found that ground state energy and low-lying excitation energies with new records in numerical precision. Even more challenging was the study of a three-dimensional quantum magnet, the Heisenberg model on the pyrochlore lattice. We established the existence of symmetry broken states in this model. More broadly, we studied various applications of machine learning technology to condensed matter physics.
Finally, in close collaboration with experimental groups, we were able to study unconventional orders in a new class of materials, so-called Kagome-metals. The name derives from a lattice motive, the Kagome structure, which provides a background for studying the interplay of topology, electronic correlations and strong frustration.
Dissemination of the project results was achieved through many talks of PARATOP team members at renown conferences (e.g. three invited talks at APS march meetings), public code packages including lectures and tutorials (NetKet, Wannier Berri), and towards an interest public via popular science and review articles.
