The Topological Inverse Problem For Functional Materials: A New Chapter In The Geometrization Of Physics

The recent classification of all symmetry-indicated topologies in crystals has led to the identification of thousands of Topological Materials (TM).

Yet, the predictive design of functional TM remains a major challenge since little is known about the microscopic origin of topology in matter.This project establishes the first geometric solution to the Topological Inverse Problem (TIP), namely it defines the strict conditions for the microscopic lattice Hamiltonian and the tight-binding parameters for a fixed band structure topology in a given symmetry class.Part A: the geometric solution to the TIP is greatly generalized to a) multi-gap conditions, b) many-band contexts, c) higher dimensional topologies, d) the 10-fold periodic table of topological matter.Part B: the framework is extended to include all principal groups of crystal symmetries in 2D and 3D.Part C: group theory-based lattice Hamiltonians are combined with DFT and wannierized computation of tight-binding parameters, in order to a) greatly refine the assessment of existing TM, and b) establish the tools for the predictive design of functional TM.Part D: new observables are defined from higher structures, as e.g. the non-Abelian braiding of topological charges and the many-band quantum metric.Part E: observation of new topologies in cold atoms and metamaterials.This project constitutes a breakthrough in the fundamental understanding of topology in TM, and it starts a new chapter in the geometrization of physics.