Combining the GW Approximation with the Hubbard I Approximation for strongly correlated materials

In this thesis, the GW approximation (GWA, Green's function G times screened interaction W) and the Hubbard I approximation (HIA) are combined in a non-self-consistent one-shot calculation to determine the electronic structure of a one-dimensional strongly-correlated model. The scheme was chosen to incorporate both screening effects through the GWA and strong on-site correlations through the atomic HIA. The resulting self-energies of both methods are summed together and another term is subtracted to correct double-counting. The examined double-counting terms were the local self-energy of the GWA result, the GWA self-energy of the atomic model used in the HIA and lastly the impurity self-energy within the GWA. The self-energy of the atomic model in the HIA was solved analytically. It was found that all three approaches yield non-causal features, which increase with the on-site Coulomb repulsion, in the resulting spectral function. The local GWA self-energy was observed to perform best in terms of causality and computational effort. Changes in the resulting quasi-particle structure in the spectral function showed that screening effects and local correlations were included successfully.In the world of Solid State Physics we are interested in creating and improving approximations to determine the electronic properties of materials. With better knowledge of the properties, we can make electric devices faster, more efficient and more precise. Since there are many different types of materials, sadly there is no such thing as one method to approximate them all. Especially the electronic structure of so-called strongly correlated materials often deviate from the truth for traditional approximations. Strongly correlated materials are called strongly correlated, because the electrons which inhabit the same orbital of an atom in the material, experience very strong repulsion from each other. This produces a lot of interesting phenomena like high-temperature superconductivity, colossal magnetoresistance or conductor-insulator transitions. In fact, this repulsion is so strong compared to influences of the electrons from nearby particles, that the Hubbard I Approximation gets away by simulating a single atom (or more specifically a single unit cell) and mapping the result on the whole solid. Another interesting phenomenon in solids is the screening effect which occurs when an electron leaves the solid. This can be achieved for example by shining light on things. After the electron leaves, a positively charged hole is left at its former place instead. Surrounding electrons are then attracted by this potential and move closer. These surrounding electrons now screen the hole, so that other electrons feel the presence of the hole much less. This gathering of electrons around the hole is referred to as a quasi-particle. An approximation which incorporates this effect thoroughly, is the so-called GW Approximation. In this thesis work, the Hubbard I Approximation and the GW Approximation were combined to include both strong correlation and screening effects for a one-dimensional model. The inclusion of both these effects were indeed observed in the results. Moreover, since the two methods partially cover the same effects, double-counting appears in the combination. Three approaches were tested to mitigate this problem