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This thesis reports results of studies conducted on the equilibration of two systems and consists of two parts: the first part deals with the optimisation of deep neural networks, whereas the second part with the decay of non-equilibrium states in strongly Rashba-coupled systems at low temperature. Deep learning is a conceptually simple, highly effective, and widely used tool, yet there remains insufficient understanding for why it works. The optimisation of deep neural networks with common algorithms such as stochastic gradient descent performs unexpectedly well given the complexity of the underlying high-dimensional non-convex minimisation problem. The first part of this thesis therefore looks at the optimisation procedure from the perspective of statistical physics. This allows us to interpret the loss function landscape of deep neural networks as the counterpart of the potential energy landscape in molecular systems and the optimisation of the network as its equilibration dynamics. Using landscape exploration tools developed in theoretical chemistry, we resolve the structure of the loss function landscape, from which we can draw conclusions for the relaxational dynamics of typical optimisers and, consequently, for deep learning. The second part investigates how a non-equilibrium charge-carrier chirality distribution in a clean, strongly Rashba-coupled system at low temperatures decays over time. We first motivate this analysis based on experimental studies of transport properties in Rashba materials at low temperatures and subject to external magnetic fields. We investigate whether chirality imbalances could serve as the source for those experimental observations and develop a framework that models the behaviour of such a system. We then proceed with a more general theoretical study of the equilibration mechanisms of chirality in low-temperature strongly Rashba-coupled systems and compute the relaxation timescales of those mechanisms.

Many-body quantum mechanics is the fundamental theory behind many areas of modern science, such as condensed-matter physics, nuclear physics, and quantum chemistry. It is also notoriously hard: the classical picture of particles with well-defined positions and velocities is replaced by an intricate interference pattern between all their possible trajectories, captured by the quantum wave function. The exponentially large information content of wave functions makes direct simulation of large, strongly interacting quantum systems impossible, and necessitates strategies to manage the complexity in an analytically or computationally tractable manner. The bulk of this thesis explores two such strategies in the context of quantum spin liquids. In these materials, competition between incompatible interactions results in robust, massive entanglement, down to zero temperature. Such ground states give rise to a range of exotic behaviour, such as topological order and fractionalised excitations: understanding these is a central challenge in the physics of strongly correlated materials. Several quantum-spin-liquid phases are underpinned by strict local conservation laws, which give rise to lattice gauge theories with exotic quasiparticle excitations, such as emergent photons or magnetic monopoles. We developed a systematic approach, based on a large-S bosonisation formalism, to extract gauge-theoretic descriptions from such constrained Hamiltonians automatically, and thus make them amenable to the powerful techniques of quantum field theory. The same field theories also allow us to simulate quantum-spin-liquid systems semiclassically, that is, to replace spin-1/2 operators with classical ``compass needles,'' removing the computational complexity of quantum entanglement without losing the physical behaviour. We demonstrated this approach on quantum spin ice, a paradigmatic and experimentally relevant model of quantum spin liquids and, by simulating it on unprecedented large system sizes, obtained novel insights about its quasiparticles. The success of neural networks in a range of machine-learning problems makes them a natural candidate for representing highly entangled quantum states, allowing in principle an accurate simulation of large, challenging quantum systems with modest computational resources. However, most current approaches using such neural quantum states suffer from the infamous Monte-Carlo sign problem, making deep neural networks unable to learn ground states in antiferromagnetic and fermionic systems. We studied the possible origins of this sign problem and proposed a neural-network ansatz that is able to overcome the sign problem for unfrustrated antiferromagnets. In addition to this main theme, I have been part of an experiment--theory collaboration on understanding the unique magnetoresistance properties of the classical frustrated magnet Ho2Ir2O7. We discovered a mechanism by which antiferromagnetic domains can be coupled to an external magnetic field via an intercalated spin-ice system: such control is highly desirable for spintronics applications. We have also identified scattering channels through which magnetic monopoles give rise to a significant contribution to the resistivity of Ho2Ir2O7: this allows us to directly measure their density in a simple and flexible experiment. Finally, I report studies on the localisation properties of quasicrystals; namely, the discovery of thermodynamic universality not described by the usual power laws in one-dimensional quasiperiodic models, and of a two-dimensional quasicrystal in which localised and partially extended states coexist without a mobility edge.

Lightwave electronics is based on the idea of using the instantaneous field of strong optical waves to drive and sculpt electronic states on ultrafast time scales. By utilizing the oscillation cycle of light, electronic currents can be switched thousand times faster than traditional electronics to enable petahertz electronics, properties of electronic states in solids become accessible for quantum information applications, and access to correlated many-body states in solids is provided. In this Thesis, I present a comprehensive many-body quantum theory based on the quantum-dynamic cluster expansion approach to quantitatively describe lightwave-driven many-body excitations in quantum materials. The theory is applied to analyze lightwave excitations in quantitative theory-experiment comparisons which led to the development of an ecosystem of new lightwave-based techniques to characterize quantum materials and probe quantum correlations in situ. The theory is also extended to describe spatially inhomogeneous excitations and nanostructures. First, the theoretical background of the quantum-dynamic cluster expansion is summarized and the theory to describe lightwave excitations in two-dimensional materials derived. I describe how the resulting semiconductor Bloch equations can be combined with ab initio density-functional theory computations and solved numerically on the full two-dimensional Brillouin zone for a predictive description of the excitation dynamics of realistic materials. By analyzing the emission dynamics of lightwave-driven coherent excitons in quantum materials, we discovered emergent interference patterns in momentum space, crystal-momentum combs, which precisely locate the emission of harmonic sidebands in momentum space and connect their intensities to the electronic band structure. I apply this connection to reconstruct the band structure from harmonic sideband emission, light that is emitted from a semiconductor in a nonlinear process after excitation with two colors, and demonstrate the approach for two-dimensional tungsten diselenide in a comprehensive theory-experiment comparison. Harmonic sidebands in solids are generated from lightwave-driven electron-hole recollisions and information about the electronic dynamics is encoded in the emitted light. In a quasiparticle-collider approach, we utilize extremely nonperturbative effects to detect millielectronvolt correlations with attosecond precision, beating the Heisenberg uncertainty limit by two orders of magnitude. We find that strong excitonic correlations in monolayers of tungsten diselenide create a distinct delay in the attoclocking setup compared to a bulk sample with weaker correlations. In a detailed theory-experiment analysis, I demonstrate direct clocking of quasiparticle correlations with attosecond precision which I connect to the dynamics of electron-hole pairs in phase-space using a Wigner-function representation. The valley-degree of freedom, emergent in some two-dimensional quantum materials, can be switched coherently on ultrafast time scales using lightwave excitations with potential applications in solid-state based classical and quantum-information processing. I apply the developed theory to predict conditions for efficient coherent valleytronic switches and propose how sequencing of multiple switches with petahertz clock rates could be realized and detected experimentally. Finally, I extend the semiconductor Bloch equations to describe spatial semiconductor and nanostructure excitation by using a Wigner-function based formulation of the single and two-particle correlations for an efficient and intuitive representation in phase space. I find that the spatial dynamics of coherent excitations exhibits a quantum character while electrons, holes, and incoherent excitons show a classical behavior in their spatial kinetics. These properties are identified in a careful theoretical analysis and compared to experimental investigations of the coherent and incoherent dynamics of spatially local excitations.

Angle-resloved photoemission spectroscopy (ARPES) is a powerful technique to study the electronic structure in solids. Its unique ability of resolving the energy and momentum information of electrons inside a solid provides an essential tool in measuring the electronic structure of solids. ARPES has made great contributions in the understanding of correlated system such as high-Tc superconductors and ruthenates. The Metal-insulator transition is a fundamental problem in condensed matter physics. The calcium substituted strontium ruthenate, Ca2-xSrxRuO4, provides a good platform to study the metal-insulator transition in multi-orbital systems. This system has a complex phase diagram that evolves from a p-wave superconductor to a Mott insulator. One of important projects of this thesis focuses on Ca2-xSrxRuO4 The growing evidence for coexistence of itinerant electrons and local moments in transition metals with nearly degenerate d orbitals suggests that one or more electron orbitals undergo a Mott transition while the others remain itinerant. We have observed a novel orbital selective Mott transition (OSMT) in Ca1.8Sr0.2RuO4 by ARPES. While we observed two sets of dispersing bands and Fermi surfaces (FSs) associated with the doubly-degenerate dyz and dzx orbitals, the Fermi surface associated with the dxy orbital which has a wider bandwidth is missing as a consequence of selective Mott localization. Our theoretical calculations have demonstrated that this unusual OSMT is mainly driven by the combined effects of inter-orbital carrier transfer, superlattice potentials and orbital degeneracy, whereas the bandwidth difference plays a less important role. Another important project of this thesis focuses on the recently discovered iron-pnictides superconductors. The idea of inter-FS scattering associated with the near-nesting condition has been proposed to explain the superconductivity in the pnictides. The near-nesting condition varies upon the carrier doping which shifts the chemical potential. We have performed a systematic photoemission study of the chemical potential shift as a function of doping in a pnictide system based on BaFe2As2. The experimentally determined chemical potential shift is consistent with the prediction of a rigid band shift picture by the renormalized first-principle band calculations. This leads to an electron-hole asymmetry (EHA) due to different Fermi velocities for different FS sheets, which can be calculated from the Lindhard function of susceptibility. This built-in EHA from the band structure, which is fully consistent with the experimental phase diagram, strongly supports that inter-FS scattering over the near-nesting Fermi surfaces plays a vital role in the superconductivity of the iron pnictides.