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This thesis examines the electronic properties of two materials that promise the realization and observation of novel exotic quantum phenomena. For this purpose, angle-resolved photoemission forms the experimental basis for the investigation of the electronic properties. Furthermore, the magnetic order is investigated utilizing X-ray dichroism measurements.
First, the bulk and surface electronic structure of epitaxially grown HgTe in its three-dimensional topological insulator phase is investigated. In this study, synchrotron radiation is used to address the three-dimensional band structure and orbital composition of the bulk states by employing photon-energy-dependent and polarization-dependent measurements, respectively. In addition, the topological surface state is examined on in situ grown samples using a laboratory photon source. The resulting data provide a means to experimentally localize the bulk band inversion in momentum space and to evidence the momentum-dependent change in the orbital character of the inverted bulk states.
Furthermore, a rather new series of van der Waals compounds, (MnBi\(_2\)Te\(_4\))(Bi\(_2\)Te\(_3\))\(_n\), is investigated. First, the magnetic properties of the first two members of the series, MnBi\(_2\)Te\(_4\) and MnBi\(_4\)Te\(_7\), are studied via X-ray absorption-based techniques. The topological surface state on the two terminations of MnBi\(_4\)Te\(_7\) is analyzed using circular dichroic, photon-energy-dependent, and spin-resolved photoemission. The topological state on the (MnBi\(_2\)Te\(_4\))-layer termination shows a free-standing Dirac cone with its Dirac point located in the bulk band gap. In contrast, on the (Bi\(_2\)Te\(_3\))-layer termination the surface state hybridizes with the bulk valences states, forming a spectral weight gap, and exhibits a Dirac point that is buried within the bulk continuum. Lastly, the lack of unambiguous evidence in the literature showing a temperature-dependent mass gap opening in these magnetic topological insulators is discussed through MnBi\(_2\)Te\(_4\).
This thesis contains two major parts: The first part introduces the reader into three independent concepts of treating strongly correlated many body physics. These are, on the analytical side the SO(5)-theory (Chap.3), which poses the general frame. On the numerical side these are the Stochastic Series Expansion (SSE) (Chap.1) and the Contractor Renormalization Group (CORE) approach (Chap. 2}). The central idea of this thesis was to combine these above concepts, in order to achieve a better understanding of the high-T_c superconductors (HTSC). The results obtained by this combination can be found in the second major part of this thesis (chapters 4 and 5). The main idea of this thesis, i.e., to combine the SO(5)-theory with the capabilities of bosonic Quantum-Monte Carlo simulations and those of the CORE approach, has been proven to be a very successful Ansatz. Two different approaches, one based on symmetry and one on renormalization-group arguments, motivate an effective bosonic Hamiltonian. In a subsequent step the effective Hamiltonian has been simulated efficiently using the SSE. The results reproduce salient experiments on high-T_c superconductors. In addition, it has been shown that the model can be extended to capture also charge ordering. These results also form a profound basis for further studies, for example one could address the open question of SO(5)-symmetry restoration at a multicritical point in the extended pSO(5) model, where longer ranged interactions are included.