The description of electronic behavior within solids is a major part of modern Condensed Matter Physics. It is well known that depending on the precise conditions, very diverse
phenomena arise from the interacting electrons in the material. To make predictions, it is therefore crucial to understand the electronic structure in a material and to compute its electronic spectrum. This thesis discusses three different aspects of electronic spectra including their numerical solution, each highlighting a distinct approach.
In a first part, this thesis presents a numerical solution of many-electron spectra on small clusters of IrO6 octahedra. Such clusters are relevant in the field of strongly coupled matter as they give rise to the elementary building blocks of many topological spin systems, localized j = 1/2 moments. Exact diagonalization of the full many-electron interaction Hamiltonian is utilized to compute multi-particle spectra with respective eigenstates. Subsequently, these eigenstates are further used for numerical calculations of resonant inelastic X-ray scattering (RIXS) amplitudes. The numerical approach is versatile enough to be applied to different examples in this thesis, covering single-site RIXS spectra as in Ba2CeIrO6,
materials with local clusters like Ba3InIr2O9 and Ba3Ti3−xIrxO9 and Kitaev materials such as Na2IrO3 and α-RuCl3. In particular, interference effects in the RIXS amplitudes are shown to play a crucial role of determining the nature of delocalized eigenstates in these materials.
In a second part, supersymmetry is used to link the spectra of electronic lattice models with bosonic counterparts. To this endeavor, an exact lattice construction is introduced,
underlying the supersymmetric identification and providing a visual representation of the supersymmetric matching. As a first instance of the supersymmetric map, it will be shown
that models of complex fermions and models of complex bosons are supersymmetrically related if they reside on the two sublattices of a bipartite lattice. Another similar identification is introduced for Majorana fermions on a bipartite lattice which can be related to real boson models on one of the sublattices, allowing for the explicit construction of related mechanical models. As examples of this classical construction, the Kitaev model and a second
order topological insulator with floppy corner modes are discussed. In both examples, the supersymmetrically related mechanical model is shown to exhibit the same spectral
properties as its quantum mechanical analogue and even inherit topologically protected localized corner modes.
In a third part, the electronic spectra of general Moiré materials are investigated at the example of twisted bilayer graphene. This part demonstrates that statistical principles
are best suited to describe the vast number of bands originating from the large Moiré unit cells. The statistical description reveals a localization mechanism in momentum space which is investigated and described. The mechanism does not only apply to all parts of the spectrum in twisted bilayer graphene but is also believed to apply to generic Moiré materials. Moreover, exceptions from this general mechanism in twisted bilayer graphene are discussed in detail which turn out to be described by harmonic oscillator states
