Non-perturbative and stochastic approaches to many-body localization

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A central question in physics concerns the complexity of any given system. What is the nature of quantities characterising the system behaviour, and which characteristics can be safely ignored? How sensitive is the system against external perturbations and how predictable is its dynamics? How many parameters need to be controlled to manipulate the system in a specific way? These questions not only lie at the heart of fundamental fields such as statistical mechanics, but also provide the key to the characterisation of equilibrium phases and nonequilibrium dynamics and pervade practical applications in quantum information, control and sensing.Meeting the urgent need to understand these questions in the quantum setting, the scientific community has recently identified a key paradigm that should hold many of the required answers. This concerns large but finite many-body systems with disorder, low spatial dimensionality and local interactions. These ingredients have been seen to conspire in a persistence of local memory of initial conditions, a fascinating phenomenon known as many-body localisation (MBL). The understanding of this phenomenon is still in its infancy. The theoretical arguments for MBL are thorough, but also in essence qualitative, while quantitative insight to date mainly arrives from numerical investigations of relatively small (in real-world terms) systems. This state of affairs has given rise to a proliferation of characterisations based, among others, on transport, thermalisation, entanglement, dynamics, whose detailed mutual relationships are mostly unclear. These shortcomings become even more pressing as the very first dedicated experiments target yet another set of characteristics, the directly observable consequences in real-world systems.In this project, we take a step back and ask the question: which level of understanding could be accepted to constitute a rather complete description of many-body localisation and delocalisation? Taking a leaf out of the book from non-interacting systems, we argue that this has to involve, as a central piece, a statistical description in quasi-one dimensional settings. We approach this goal from (I.) a rich one-dimensional model system originating from lattice field theory (the Schwinger model, a version of quantum electrodynamics) which is amenable to a detailed treatment and (II.) a stochastic description based on fundamental composition rules of the density matrix (the key object to describe entanglement), allowing comprehensive access to the generic system behaviour. Amongst the observable consequences, we aim to (III.) discriminate between general and system-specific aspects in the MBL phase and of the phase transition, including experimentally testable signatures, and (IV.) extend the considerations to topological phases, where the system displays order due to intricate quantum effects.