Topological states of matter from first principles

Topological states of matter received increasing attention in recent years due to their fascinating properties. These materials that are otherwise insulating show conducting states at their surfaces or edges, where these conducting states are protected topologically and therefore robust against disorder or other small perturbations. Around five years ago, the first materials that could show these new state of matter were proposed, and soon afterward already found in experiments. This success triggered a whole new field of solid state research. However, all those first realizations of topological states have been found in materials where electrons can be described as non-interacting. The driving force for the occurrence of topological states is the spin-orbit coupling, and since the impact of this effect grows with the fourth power of the atomic number, materials consistent of heavy elements like bismuth are predestined for showing topological insulating phases. Theoretical models for these systems are quite straight forward to study, because of the absence of electronic correlations, making a description in single- particle pictures possible. Only recently, also transition metal compounds came into focus of the research. Several compounds based on the 5d element Iridium were proposed to exhibit topological phases. In the case of these oxides, the theoretical description is not as simple as in the materials mentioned above, because a new player enters the game, the Coulomb interaction. Electrons in open d shells are subject to enhanced electronic interactions, being of the same order of magnitude as the spin- orbit coupling. Hence, several energy scales, i.e. interaction, spin-orbit coupling, kinetic energy, and also crystal field effects, have to be taken into account in an appropriate way. The goal of this project is to set up an efficient first principle tool for the calculation of measurable properties of these new materials with predictive power. Question that we address are for instance: How do strong electronic correlations compete or cooperate with the spin-orbit coupling? Under what conditions can correlated matter show topological phases? How do we have to design new materials that show well-defined topological properties, and what new properties arise due to strong correlations? We will investigate several materials by first-principle techniques. Starting from the crystal structure as the only input, we calculate electronic and topological properties, using a combination of electronic structure calculations with modern many-body techniques. These methods, such as the dynamical mean-field theory or the variational cluster approach, allow to treat all energy scales in the compound on the same footing. Moreover, also the interaction parameters will be calculated from first-principles, making this approach truly ab-initio without external parameters. In that way, we will for the first time connect real materials with models for topological insulators. Details of the band structure and also the atomic multiplet structure will be included, which has not been done so far. We will investigate why some iridate materials like Na2IrO3 are insulators and others are not. Another important class of materials will be the pyrochlore iridates. They have been proposed as topological insulators, but theoretical calculations have never tackled the real band structure of the systems, neglecting material specific properties. The rich phase diagram of transition metal oxides, including metallic, insulating, magnetic, and topological phases gives the opportunity for completely new effects, exactly because of the interplay of the different order parameters. This point will be the focus of this project. The theoretical results will be compared with experimental data on topological insulators. A close collaboration with the experimental group of Dr. Erik van Heumen will be set up in order to facilitate this part of the project. In addition, we will use the developed tools in order to predict new properties, or even completely new materials with well-designed properties. This project will thus contribute substantially to the understanding of correlated topological matter.

Strongly correlated electron systems exhibit a variety of interesting effects that are due to the interactions between particles and cannot occur for independent particles. Famous examples include unconventional superconductivity, colossal magnetoresistance, or so-called heavy fermion compounds. In recent years, a special class of materials has come into focus in international research, namely materials that contain heavy atomic nuclei. In these elements, the spin-orbit interactions of the electrons come to the fore and change the properties compared to similar compounds consisting of lighter elements. For example, in iridium oxides, the electron interactions are so strongly enhanced that an insulator is formed. Furthermore, spin-orbit coupling is also responsible for so-called topological properties of materials. These properties are determined by gross global features such as the symmetry class of the materials, and are thus robust against small external perturbations. The theoretical description of such properties and materials is extremely complex, as commonly used methods, both analytical and numerical, reach their limits. In order to be able to make statements about the materials properties despite this, approximations are necessary, but in the optimal case they are small and can be controlled. One of these approximation methods is the dynamic mean field theory, which we also use in this project. In combination with band structure methods, this allows for a very complete calculation of material properties in silico, i.e. completely on the computer with a minimal number of adjustable parameters. This procedure is called ab initio calculations. The aim of this project was to further develop some of these numerical methods, and also to implement new methods, in order to improve the predictions of ab initio calculations, or to apply them in parameter regimes where it was not previously possible. With the new methods, it is possible to perform calculations at the zero temperature, with an energy resolution that was previously unattainable. With this, we were able to investigate why some materials do not show magnetically ordered states, although this would be expected at first glance. Another application have been topological materials, where we could clarify the mutual influence of spin-orbit coupling and electronic correlations. Even if these questions and statements may seem purely academic at first glance, a fundamental understanding of such material properties is important in order to learn from them for future material design. If we want to optimize certain properties, it is very important to understand the starting point very well. This project has made its contribution to this.