Interacting dynamics and exotic phases of quantum lattice systems

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Quantum lattice systems lie at the intersection of many-body condensed matter physics, quantum information theory and, generally, mathematical statistical physics. The overall objective of this project is to explore the analytic structure of these quantum mechanical models that describe condensed matter at the microscopic scale. Physically motivated questions that arose in the context of the quantum Hall effect, recently driven by quantum information theory and supported by experiments in optical lattices, shall be put in a mathematical framework where spectral, dynamical and statistical mechanical questions can be addressed. The project aims at developing algebraic and analytic methods to lay the foundations of a rigorous study of novel exotic states of matter characterized by topological order or many-body localization. Handling the specific difficulties arising in particular from the non-perturbative nature of interactions between an arbitrarily large number of particles requires an extension of our knowledge of the statistical mechanics of infinite quantum lattice systems: dynamical locality, properties of ground states, structure of eigenstates, and effects of disorder. The project is organized in three principal themes. The first builds on very recent results concerning the driven dynamics of quantum lattice systems to obtain a general C*-algebraic adiabatic theory in the infinite volume limit, with improved, exponential error bounds. The second aims at a better understanding of the invariants of topological matter, such as (fractional) quantum Hall systems and topological insulators, in an interacting setting. The third theme is to push the quasi-adiabatic continuation technique beyond the domain of its current and very successful application, in particular to gapless disordered systems. Expected further applications of the program range from spectral stability results to a better understanding of the correlation structure of dynamically localized quantum lattice systems. HQPs recruited specifically for this project will be involved in all stages, with research questions adapted to their level of technical abilities and to the time horizon set by their program.