Waves and Topology in Quantum Materials

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Designing materials with revolutionary properties is a central theme of industrial and applied research. Prototypes include topological insulators (extraordinarily robust one-way communication devices that promise to stabilize quantum computations), invisibility cloaks (textiles that deviate light away from themselves) and atomic-thin batteries (such as paints that absorb and reutilize energy from Wi-Fi or telecommunication waves). This project aims to (a) develop quantitative tools to predict the behavior of quantum particles within such tailored environments; (b) apply these tools to engineer materials with optimized anomalous effects; and (c) train a workforce with a solid background in both quantum science and quantitative analysis. These objectives can aid the design of a new generation of technological devices via an enhanced understanding of quantum phenomena.The focus will be on three themes that carry high promises in technological fields: (i) the asymmetric propagation of energy along edges of topological insulators; (ii) the generation and the effects of pseudo-magnetism; and (iii) the density of states in thin-layer materials. The first topic quantitatively analyzes the characteristics of energy transport between topologically distinct materials: the group velocity of waves, the profile of asymmetric channels, the influence of perturbations. The second topic investigates mathematically Landau levels and quantum Hall effects in custom-made materials that enact artificial magnetic fields of intensities inaccessible to classical experiments. The last topic focuses on twisted bilayer graphene, a revolutionary structure that - for specifically tuned parameters - exhibits high-density state-packing and superconductivity. The mathematical investigation of these phenomena relies on developing and applying diverse and sophisticated analytic tools such as semiclassical calculus for periodic operators, scattering theory and distorted plane waves in Dirac-like systems, multiscale and homeogenization techniques of quasiperiodic structures, and spectral theory of non-selfadjoint operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.