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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.

Weak attractive interactions in a spin-balanced fermion gas are known to induce a Cooper pairing instability where the fermion pairs have zero net momentum, leading to BCS superconductivity. Extensions for spin-imbalanced systems include breached superfluids, Fulde-Ferrell-Larkin-Ovchinnikov theory, and pair density wave theory. We propose a further extension of a superconducting state, dubbed a communal pairing state, whose underlying components are superpositions of Cooper pairs that share minority-spin fermions. This state includes correlations between all available fermions on both Fermi surfaces and is shown to be energetically favourable to the Fulde-Ferrell-Larkin-Ovchinnikov state. Our numerical quantum Monte Carlo study of finite spin-imbalanced systems provides clear evidence of the existence of such exotic pairing states. In spin-balanced systems, temporal fluctuations of the order parameter promote communal pairing, leading to a widening in momentum space of the superconducting gap and a decrease of the chemical potential.

Condensed matter physics spans molecules, fluids, and solids. Mathematical models have been developed to simulate physical systems, but exact solutions are special leaving the many-body Schr\"odinger equation analytically intractable. Therefore, approximations have to be constructed to make those difficult-to-solve problems tractable. This thesis adapts different levels of approximation, developing an exact formalism for a quantum operator, investigating the origins of intriguing phenomena in solids using first principle methods, and heuristic data-driven machine learning to guide biomedical developments. The quantum Monte Carlo method is used to study molecules where the formalism for force constants matrix is created for the first time using variational and diffusion Monte Carlo, which are the most accurate technique in electronic structure calculation but more resource demanding. We make the atomic relaxation and vibrational frequency calculation possible via the direct approach. The performance has been tested on a diverse set of case studies, with varying symmetries and mass distributions, and shows the established formalism outperforms leading computational methods over an order of magnitude in terms of the accuracy of vibrational frequency relative to experiment. This opens the way for self-contained quantum Monte Carlo simulations from relaxing atomic positions to calculating vibrational frequencies without being limited to the accuracy from density functional theory for structure relaxation. Density functional theory is applied to investigate the large and complex solids of perovskites and heterostructures efficiently. We explore the origins of Raman modes for transient photophysics in layered $2$D perovskites and discover the twisting of organic cations could influence the excitons in the inorganic layer. Understanding the coherent interplay of excitons, spins, and phonons facilitates the design of efficient light emission devices and solar cells. Inspired by an unexpected exchange bias effect in topological insulator superlattices, we introduce magnetic models to investigate the doping effect and predict the interfacial coupling between dopants induces exchange bias. We provide a new pathway for achieving the high-temperature quantum anomalous Hall effect in exchange bias stabilised magnetic systems by interface engineering. Finally, we show machine learning can deliver new insights for systems lacking theoretical models, and where data collection is expensive and difficult. We design a neural network model capable of handling missing data and estimating prediction uncertainties. We apply it to study the efficacy of biomedical stem cell fluids for cartilage repair and identify critical properties, optimal dose, and predicted therapeutic outcomes for personalised therapy. We provide references for scientists and clinicians to design better therapy strategies, and the technology can be adapted for addressing research problems across different subjects.

From the alignment of magnets to the melting of ice, the transition between different phases of matter underpins our exploitation of materials. Both a quantum and a classical phase can undergo an instability into another state. In this thesis, we study the stability of matter in both contexts: topological states and crystalline solids. We start with the stability of fractional quantum Hall states on a lattice, known as fractional Chern insulators. We investigate, using exact diagonalization, fractional Chern insulators in higher Chern bands of the Harper-Hofstadter model, and examine the robustness of their many-body energy gap in the effective continuum limit. We report evidence of stable states in this regime; comment on two cases associated with a bosonic integer quantum Hall effect; and find a modulation of the correlation function in higher Chern bands. We next examine the stability of molecules using variational and diffusion Monte Carlo. By incorporating the matrix of force constants directly into the algorithms, we find that we are able to improve the efficiency and accuracy of atomic relaxation and eigenfrequency calculation. We test the performance on a diverse selection of case studies, with varying symmetries and mass distributions, and show that the proposed formalism outperforms existing restricted Hartree-Fock and density functional theory methods. Finally, we analyze the stability of three-dimensional crystals. We note that for repulsive Coulomb crystals of point nuclei, cubic systems have a zero matrix of force constants at second order. We investigate this by constructing an analytical model in the tight-binding approximation, and present a phase diagram of the most stable crystal structures, as we tune core and valence orbital radii. We reconcile our results with calculations in the nearly free electron regime, as well as current research in condensed matter and plasma physics.

Interacting Fermi gases are one of the chief paradigms of condensed matter physics. They have been studied since the beginning of the development of quantum mechanics, but continue to produce surprises today. Recent experimental developments in the field of ultracold atomic gases, as well as conventional solid state materials, have produced new and exotic forms of Fermi gases, the theoretical understanding of which is still in its infancy. This Thesis aims to provide updated tools and additional insights into some of these systems, through the application of both numerical and analytical techniques. The first Part of this Thesis is concerned with the development of improved numerical tools for the study of interacting Fermi gases. These tools take the form of accurate model potentials for the dipolar and contact interactions, as found in various ultracold atomic gas experiments, and a new form of Jastrow correlation factor that interpolates between the radial symmetry of the inter-electron Coulomb potential at short inter-particle distances, and the symmetry of the numerical simulation cell at large separation. These methods are designed primarily for use in quantum Monte Carlo numerical calculations, and provide high accuracy along with considerable acceleration of simulations. The second Part shifts focus to an analytical analysis of spin-imbalanced Fermi gases with an attractive contact interaction. The spin-imbalanced Fermi gas is shown to be unstable to the formation of multi-particle instabilities, generalisations of a Cooper pair containing more than two fermions, and then a theory of superconductivity is built from these instabilities. This multi-particle superconductivity is shown to be energetically favourable over conventional superconducting phases in spin-imbalanced Fermi gases, and its unusual experimental consequences are discussed.