Free-particle descriptions of topological quantum matter and many-body localisation

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The notion of a free particle is at the heart of theoretical physics. This simple notion allows us to describe a wide variety of systems in nature: for example, we explain atoms as collections of free electrons, and electromagnetic radiation as a set of free harmonic oscillators. But in real world, particles also interact with one another. Interactions have particularly striking effects in quantum systems, where they lead to long-range correlations and quantum entanglement. This makes the theoretical description of quantum many-particle systems very challenging. At the same time, there is growing interest in using interactions as a resource that could revolutionise technology. Modern technology has been based on quantum materials such as semiconductors, superconductors and magnets, which form the building blocks of lasers, transistors, computers, etc. In recent years, there has been a shift towards making such devices more powerful by exploiting the full power of quantum physics, which will be achieved by utilising quantum correlations and entanglement. Thus, at this stage, there is a compelling theoretical and practical need to understand and manipulate the effects of interactions in quantum systems.This proposal will investigate two physical systems of current interest where interactions lead to novel physical phenomena: (1) Topological phases of matter, which include Majorana and parafermion spin chains, spin liquids and fractional quantum Hall states. These phases form as a result of "non-perturbative" effects of interactions, and cannot be described as a collection of free electrons. This gives them unique properties such as robustness under arbitrary, but sufficiently weak, perturbations. Because of this special rigidity, topological quantum matter is being used as a building block of more robust quantum technologies, designed to be resilient to environmental perturbations.(2) Non-equilibrium dynamics and thermalisation in quantum many-particle systems. Typical quantum systems are ergodic: they quickly reach thermal equilibrium because the interactions between their constituent particles quickly erase the memory of the system's initial condition. However, recent work on "many-body localisation" shows that there exist large classes of strongly-disordered, interacting quantum systems which fail to reach thermal equilibrium. These systems are thus non-ergodic, which means that quantum effects in them can persist for unusually long times, thus providing another route of protection for quantum technology.In this proposal we will develop a new approach to describe interaction effects in strongly-correlated phenomena including topological phases of matter and many-body localisation. We will advance the modelling of many-body systems in random environments using state-of-the-art numerical simulations. Our theoretical investigation on the effects of topology and many-body localisation in quantum matter will impact several experiments on cold atoms, trapped ions, defects in solids, etc. Finally, we will explore the possibility of realising phases with topological order in random environments, and propose schemes for quantum information storage and processing with an enhanced stability against thermalisation.