Robust many-body Quantum phenomena through Driving and Dissipation

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Schoolchildren learn of three phases of matter: Liquid, solid and gas. This list is glaringly incomplete at low temperatures where quantum mechanical principles more markedly determine material properties, leading to a plethora of new states including superconductors and topological phases, many of which have profound practical applications. The theoretical search for phases has implicitly assumed that all such systems come to thermodynamic equilibrium. This seemingly solid assumption underlies all thermodynamics, of which Einstein once said: "It is the only physical theory of universal content, which I am convinced...will never be overthrown". Two important questions arise: 1) is the assumption valid, i.e., do all systems tend to equilibrium and 2) when they do, how do they get there? "No" is the shocking answer to the first question: some systems do not equilibrate, and a host of new non-equilibrium phases exist beyond the standard thermodynamic paradigm. The realisation was sparked by the discovery of many-body localised (MBL) phases, for which strong disorder prevents the equilibration of, e.g., charge and energy, contrary to the standard assumptions of thermodynamics. As revolutionary as MBL systems are, they do settle in the following sense: left undisturbed, observables settle (are static) at late times. Astonishingly, through a combination of both periodic driving and disorder, one can realise systems that robustly both fail to equilibrate and settle! Many-body localised time crystals (TCs) are a prominent example: for almost any initial state, their local observables eventually oscillate with a period that is an integer multiple (e.g., 2T) of the underlying driving period T, hence the system spontaneously breaks time translation symmetry. A prominent open question is whether the effect can be robust in an open quantum system, i.e., one subject to environmental dissipation. The second question concerns how strongly correlated systems that do settle generate the dissipation that drives them towards a steady state, as well as quantitative questions of how bulk transport coefficients (conductivity, etc.) depend on the underlying microscopic parameters, how systems respond to external dissipation, and what features of transport are universal and robust. This field has a long history, few controlled results, and is of great practical importance: further progress will aid in the search for new materials with specified transport properties, and clarify existing experimental questions (below). Our increased understanding of novel quantum phases revolutionised the material world. This project aims to continue the revolution, extending our understanding of the non-equilibrium frontier, searching for new robust---and hence potentially useful---quantum phenomena. Our primary focus will be on a poorly understood class of many-body systems, namely those subject to both environmental dissipation and periodic driving. We will explore various phenomena in this setting, most prominently aiming to: 1) develop the theory behind a new notion of time crystal stable in the presence of dissipation, 2) formulate a theory of quantum information/entanglement spreading and 3) expand our cachet of solvable models with dissipation. We will 4) use our theoretical results in 2) to develop new numerical techniques able to probe the experimentally relevant late time regimes in many-body systems. Finally, we will 6) transform our theoretical progress into experimental proposals pertaining to, amongst other possibilities: a) the exciting prospect that dissipation---rather than being an impediment---can be used to enhance the stability, and hence practical utility, of time crystals; b) exotic transport in interacting systems, e.g., anomalous spin diffusion in 1D quantum magnets; c) a careful assessment of claimed sightings of time crystals in experiments, particularly in Nitrogen vacancy centre platforms.