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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.