The localized/delocalized duality of 5f electrons plays an important role in understanding the complex physics of actinides. Band-structure calculations based on the ad hoc assumption that 5f electrons are simultaneously localized and delocalized explained the observed dHvA experiments very well. This ad hoc assumption also gives the correct equilibrium volume for delta-Pu. Experimentally, the duality of 5f electrons is observed by inelastic neutron scattering experiments, or by soft X-ray angle-resolved photoelectron spectroscopy. It is worth recalling that the origin of partial localization in the 3d and 5f systems is quite different. In compounds with 3d electrons, the large crystalline electric field set up by the surrounding environment of transition metal ions plays a major role. On the other hand, in 5f systems, the Hund's rule correlations play the key role whilst the crystalline electric field is less important. In this thesis we have studied the effect of intra-atomic correlations on anisotropies in hopping matrix elements of different 5f orbitals. For that purpose, we used the effective model that includes on-site interactions that are responsible for Hund's rules and effective hopping terms that result from the hybridization of different 5f orbitals with the environment. Two different approximations, namely, rotationally invariant slave-boson mean-field (RISBMF) and infinite time-evolving block decimation (iTEBD), have been used to investigate the ground-state properties of the Hamiltonian. We have demonstrated that Hund's rule correlations enhance strongly anisotropies in hopping matrix elements. For a certain range of 5f bandwidth parameters this effect may result in a complete suppression of hopping processes for some of 5f orbitals, i.e., the system is in a partially localized phase. Within the RISBMF method, we calculated the ground-state properties and the phase diagram of the system. The suppression of hopping processes in some of 5f orbitals due to Hund's rule correlations can be seen through orbital-dependent quasiparticle weights. In a mean-field theory, a quasiparticle weight of zero for an orbital means a complete suppression of hopping processes in this orbital. Thus, quasiparticle weights and occupation numbers were used to classify partially localized phases. In the calculated phase diagram we obtain four partially localized phases that can be separated into two different sets. In the first set electrons in two orbitals are localized. In the second, electrons in one orbital are localized. The difference between the two sets is not simply the number of localized orbitals but the mechanism for the partial localization. For the first set, the Hund's rule mechanism applies: only those 5f electrons that enable the remaining ones to form a Hund's rule state will delocalize. This mechanism requires to have at least two localized orbitals, therefore it is definitely not applicable to those phases with only one localized orbital. For the second set, a situation similar to a single-band Mott-Hubbard transition applies. The direct on-site Coulomb interaction between jz and -jz electrons plays the key role for understanding the partial localization transition. In order to assess the validity of the RISBMF results we have used the iTEBD method to calculate the ground-state properties of a 1D system. Qualitatively, the two approaches agree with each other. However, we found an area where the RISBMF yields an artificial ground-state. Note that the mean-field method is worst for a 1D system. Therefore one shoud not judge from it the quality of the RISBMF method for the more general case.

Since the genesis of condensed matter physics, strongly correlated fermionic systems have shown a variety of fascinating properties and remain a vital topic in the field. Such systems arise through electronic interaction, and despite decades of intensive research, no holistic approach to solving this problem has been found. During that time, physicists have compiled a wealth of individual experimental and theoretical results, which together give an invaluable insight into these materials, and, in some instances, can explain correlated phenomena. However, there are several systems that stubbornly refuse to fall completely in line with current theoretical descriptions, among them the high-\( T_c{}\) cuprates and heavy fermion compounds. Although the two material classes have been around for the better part of the last 50 years, large portions of their respective phase diagram are still under intensive debate. Recent experiments in several electron-doped cuprates compounds, e.g. neodymium cerium copper oxide (Nd\(_{2x}\)Ce\(_x\)CuO\(_4\)), reveal a charge ordering about an antiferromagnetic ground state. So far, it has not been conclusively clarified how this intertwining of charge and spin polarization comes about and how it can be reconciled with a rigorous theoretical description. The heavy-fermion semimetals, on the other hand, have enjoyed renewed scientific interest with the discovery of topological Kondo insulators, a new material class offering a unique interface of topology, symmetry breaking, and correlated phenomena. In this context, samarium hexaboride (SmB\(_6\)) has emerged as a prototypical system, which may feature a topological ground state. In this thesis, we present a spin rotational invariant auxiliary particle approach to investigate the propensities of interacting electrons towards forming new states of order. In particular, we study the onset of spin and charge order in high-\( T_c{}\) cuprate systems and Kondo lattices, as well as the interplay of magnetism and topology. To that end, we use a sophisticated mean-field approximation of bosonic auxiliary particles augmented by a stability analysis of the saddle point via Gaussian fluctuations. The latter enables the derivation of dynamic susceptibilities, which describe the response of the system under external fields and offer a direct comparison to experiments. Both the mean-field and fluctuation formalisms require a numerical tool that is capable of extremizing the saddle point equations, on the one hand, and reliably solving a loop integral of the susceptibility-type, on the other. A full, from scratch derivation of the formalism tailored towards a software implementation, is provided and pedagogically reviewed. The auxiliary particle method allows for a rigorous description of incommensurate magnetic order and compares well to other established numerical and analytical techniques. Within our analysis, we employ the two-dimensional one-band Hubbard as well as the periodic Anderson model as minimal Hamiltonians for the high-\( T_c{}\) cuprates and Kondo systems, respectively. For the former, we observe a regime of intertwined charge- and spin-order in the electron-doped regime, which matches recent experimental observations in the cuprate material Nd\(_{2x}\)Ce\(_x\)CuO\(_4\). Furthermore, we localize the emergence of a Kondo regime in the periodic Anderson model and establish the magnetic phase diagram of the two-band model for topological Kondo insulators. The emerging antiferromagnetic ground state can be characterized by its topological properties and shows, for a non-trivial phase, topologically protected hinge modes. / Stark korrelierte Fermionen in einem Festkörper-Kristallgitter weisen eine Vielzahl faszinierender kollektiver Eigenschaften auf und stellen damit eines der konzeptionell reichhaltigsten Themenkomplexe auf dem Gebiet der Physik der kondensierten Materie da. Die dazu nötigen Mechanismen lassen sich auf die elektronische Coulomb-Wechsel-wirkung zurückführen und sind trotz jahrzehntelanger intensiver Forschung bis heute nicht geschlossen gelöst worden. Vielmehr wurden - Stück für Stück - experimentelle und theoretische Einzelergebnisse zusammen getragen, die nicht nur einen tiefen Einblick in diese Materialien geben, sondern in einigen Fällen sogar korrelierte Phänomene erklären können. Allerdings gibt es durchaus Strukturen, die sich hartnäckig weigern, mit den bisherigen theoretischen Beschreibungen vollständig übereinzustimmen, darunter die Kuprat-Hochtemperatursupraleiter und die Schwer-Fermionenverbindungen. Obwohl diese beiden Materialklassen seit etwa 50 Jahren erforscht werden, sind große Teile ihrer jeweiligen Phasendiagramme noch nicht abschließend entschlüsselt. Experimente an mehreren elektronendotierten Kuprat-verbindungen, z. B. Neodym-Cerium-Kupferoxid (Nd\(_{2x}\)Ce\(_x\)CuO\(_4\)), zeigen unter anderem eine Ladungsdichtewelle, die auf einem antiferromagnetischen Grundzustand beruht. Bislang ist nicht abschließend geklärt, wie diese Verschränkung von Ladungs- und Spinpolarisation zustande kommt und wie sie mit einer strengen theoretischen Beschreibung in Einklang zu bringen ist. Schwer-Fermionen Halbmetalle erleben mit der Entdeckung der topologischen Kondo-Isolatoren eine Renaissance und bieten eine einzigartige Schnittstelle zwischen Topologie, Symmetriebrechung und korrelierten Phänomenen. Der wahrscheinlich vielversprechendste Kandidat dieser neuen Materialklasse ist Samariumhexaborid (SmB\(_6\)). In dieser Arbeit nutzen wir einen spinrotationsinvarianten Hilfsteilchenansatz um die Emergenz neuer Ordnungszustände wechselwirkender Elektronen zu untersuchen. Im Besonderen interessiert uns das Zusammenspiel von Spin- und Ladungsdichtewellen in den Hochtemperatur Kupraten und Kondo-systemen, sowie die Interaktion von Magnetismus und Topologie. Dazu verwenden wir eine hoch parametrische Molekular-Feld-Analyse der bosonischen Hilfsteilchen, die anschließend durch eine Stabilitätsanalyse des Sattelpunkts ergänzt wird. Sowohl die Molekular-Feld-Approximation, als auch der Fluktuations-Formalismus erfordern ein numerisches Softwaretool, das in der Lage ist sowohl Sattelpunkt-Gleichungen als auch Loopintegral präzise zu lösen. Wir präsentieren eine pädagogisch aufgearbeitete, von Grund auf entwickelte Herleitung des Formalismus, die auf eine Software-Implementierung zugeschnitten ist. Der Hilfsteilchenansatz erlaubt überdies eine rigorose Beschreibung inkommensurabel magnetischer Ordnungen und reproduziert etablierten numerischen und analytische Ergebnisse in guter Übereinstimmung. Für unsere Analyse verwenden wir sowohl das zweidimensionale Einband-Hubbard- als auch das periodische Anderson-Modell als minimalen Hamitonian für die Hochtemperatur-Kuprate bzw. Kondo-Systeme. Im Falle der Kuprate finden wir eine Phase, die durch eine kombinierte Ladungs- und Spinordnung im elektronendotierten Parameterbereich gekennzeichnet ist und überdies gut mit experimentellen Beobachtungen im Kupratmaterial Nd\(_{2x}\)Ce\(_x\)CuO\(_4\) übereinstimmt. Des Weitern wird das Auftreten des Kondo-Regimes im periodischen Anderson-Modell untersucht und das magnetische Phasendiagramm des Zwei-Band-Hamiltonians eines topologischen Kondo-Isolators kartiert. Der antiferromagnetische Grundzustand kann durch eine topologische Invariante charakterisiert werden und zeigt für eine nicht-triviale Phase eindimensionale topologisch geschützte Kantenmoden.

The localized/delocalized duality of 5f electrons plays an important role in understanding the complex physics of actinides. Band-structure calculations based on the ad hoc assumption that 5f electrons are simultaneously localized and delocalized explained the observed dHvA experiments very well. This ad hoc assumption also gives the correct equilibrium volume for delta-Pu. Experimentally, the duality of 5f electrons is observed by inelastic neutron scattering experiments, or by soft X-ray angle-resolved photoelectron spectroscopy. It is worth recalling that the origin of partial localization in the 3d and 5f systems is quite different. In compounds with 3d electrons, the large crystalline electric field set up by the surrounding environment of transition metal ions plays a major role. On the other hand, in 5f systems, the Hund's rule correlations play the key role whilst the crystalline electric field is less important. In this thesis we have studied the effect of intra-atomic correlations on anisotropies in hopping matrix elements of different 5f orbitals. For that purpose, we used the effective model that includes on-site interactions that are responsible for Hund's rules and effective hopping terms that result from the hybridization of different 5f orbitals with the environment. Two different approximations, namely, rotationally invariant slave-boson mean-field (RISBMF) and infinite time-evolving block decimation (iTEBD), have been used to investigate the ground-state properties of the Hamiltonian. We have demonstrated that Hund's rule correlations enhance strongly anisotropies in hopping matrix elements. For a certain range of 5f bandwidth parameters this effect may result in a complete suppression of hopping processes for some of 5f orbitals, i.e., the system is in a partially localized phase. Within the RISBMF method, we calculated the ground-state properties and the phase diagram of the system. The suppression of hopping processes in some of 5f orbitals due to Hund's rule correlations can be seen through orbital-dependent quasiparticle weights. In a mean-field theory, a quasiparticle weight of zero for an orbital means a complete suppression of hopping processes in this orbital. Thus, quasiparticle weights and occupation numbers were used to classify partially localized phases. In the calculated phase diagram we obtain four partially localized phases that can be separated into two different sets. In the first set electrons in two orbitals are localized. In the second, electrons in one orbital are localized. The difference between the two sets is not simply the number of localized orbitals but the mechanism for the partial localization. For the first set, the Hund's rule mechanism applies: only those 5f electrons that enable the remaining ones to form a Hund's rule state will delocalize. This mechanism requires to have at least two localized orbitals, therefore it is definitely not applicable to those phases with only one localized orbital. For the second set, a situation similar to a single-band Mott-Hubbard transition applies. The direct on-site Coulomb interaction between jz and -jz electrons plays the key role for understanding the partial localization transition. In order to assess the validity of the RISBMF results we have used the iTEBD method to calculate the ground-state properties of a 1D system. Qualitatively, the two approaches agree with each other. However, we found an area where the RISBMF yields an artificial ground-state. Note that the mean-field method is worst for a 1D system. Therefore one shoud not judge from it the quality of the RISBMF method for the more general case.

In this thesis we employ various mean-field approaches to study the shortrange interaction effects in topological insulators. We start with the Kane-Mele model on the decorated honeycomb lattice and study the stability of topological insulator phase against different perturbations. We establish an adiabatic connection between a noninteracting topological insulator and a strongly interacting spin liquid in its Majorana fermion representation. We use the Hartree-Fock mean-field approach, slave-rotor approach and slave-boson approach to study correlation effects related to topological insulators. With the spontaneous symmetry breaking mechanism, we can have an interaction driven topological insulator with extended Hubbard models on the kagome lattice and decorated honeycomb lattice. For the interplay among spin-orbit coupling, distortion and correlation effect in transition metal oxides, we use the slave-rotor mean-field approach to study its phase transition. We identify regimes where a strong topological Mott insulator and a weak topological insulator reside due to the strong Coulomb interaction and distortion. This is relevant to experiments with the transition metal oxides as they hold promise to realize topological insulators. To study the doping effects and a possible spin liquid in Kane-Mele-Hubbard model on the honeycomb lattice, we employ the slave-boson mean-field approach which is appropriate for the intermediate interaction strength. We compare our results with those obtained from other methods. / text