The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.

In the investigation of correlated electron systems which are characterized by strong spin-orbit coupling, one of the central challenges is the description of the complex interplay of different microscopic energy scales and the elucidation of its influence on the formation of exotic electronic phases like complex ordering phenomena and superconductivity. In the present thesis, exemplary three case studies of iridium-based compounds are presented, in which the effects of such an interplay have been investigated employing state-of-the-art synchrotron-based techniques. The particular focus is set on experimental possibilities to influence this equilibrium utilizing external parameters. In the first study, magnetic excitations are investigated in iridate double perovskites, which exhibit a nonmagnetic ground state. Upon increasing the influence of kinetic contributions, the potential condensation of these excitations is predicted to drive a novel kind of magnetic transitions, called ’excitonic magnetism’. A comprehensive investigation of the dynamics of these excitations via resonant inelastic x-ray scattering allows for an estimation of the relevant energy scales. These results indeed reveal that the influence of kinetic contributions is too small to drive such a transition under ambient conditions. Therefore the influence of excitonic magnetism on the macroscopic properties of the investigated compounds can be excluded. In the second case, the development of a new experimental setup is presented, facilitating the investigation of complex ordering phenomena at low temperatures as a function of pressure via resonant elastic x-ray scattering. This setup has been developed and implemented as part of this work in strong collaboration with the staff of the beamline P09 at the synchrotron PETRAIII (DESY). The functionality of this setup has been illustrated by measurements of the resonant magnetic x-ray scattering in the spin-orbit coupled Mott-insulator Sr 2IrO4. Since the magnetic ground state and magnetic order in iridates result from a complex interplay of different microscopic energy scales, these systems are particularly susceptible to external influences like hydrostatic pressure. In the third case, structural phase transitions are investigated in the iridium-based dichalcogenide IrTe2. Despite the macroscopic itinerant properties of IrTe2, the phase transitions are characterized by the formation of strongly localized states. These transitions have been investigated in the course of this work using single crystal x-ray diffraction experiments as a function of hydrostatic pressure and temperature. The presented experimental data show that these strongly localized states are stabilized with increasing pressure, which is observed as an increased density of Ir-Ir dimer bonds.