Topological dynamics on patterns of resonators with a focus on group symmetries

• Applicant: Dr. Tom Stoiber
• Subject Area: Mathematics ; Theoretical Condensed Matter Physics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 521291358

An important branch of mathematical solid-state physics in recent years is the classification and bulk-boundary correspondence of topological insulators via K-theory and operator-algebraic methods. While this approach originates from a quantum-mechanical description of electronic matter, it is now understood that it can also be used to classify topological meta-materials, which influence the propagation of classical mechanical, electronmagnetic or acoustical waves. The place of atoms is then taken by larger building blocks, which can be modeled mathematically as resonators whose vibrations are described by linear dynamics. If those dynamics are based on Galilean-invariant physics then the underlying patterns in which one arranges the resonators have a chief influence on the possible types of topological insulators, e.g. quasi-periodic patterns often lead to additional phason degrees of freedom which can be used to effectively model higher-dimensional systems. This project intends to study mathematical aspects surrounding the construction, classification and stability of topological meta-materials. The first is that, in contrast to the original quantum-mechanical models, resonators are often asymmetric and can be rotated arbitrarily with respect to each other other. This leads to important constraints on possible Galilean-invariant dynamics and requires the study of new groupoid-C*-algebras which take these features into account on an operator-algebraic level. The second aspect will be a special case of the first: the study of a construction principle for crystalline topological insulators with a given symmetry group, where one places asymmetric resonators along the orbit under a discrete subgroup of the euclidean group. This results in insulators that can be modeled by elements of group-C*-algebras and therefore enables the use of powerful mathematical methods such as the Baum-Connes conjecture. Those shall be used to solve concrete problems regarding the topological classification, computation of topological invariants and bulk-boundary correspondence. The third aspect to be studied is the stability of crystalline topological insulators under random disorder. While one heuristically expects that at least weak disorder has no effect on the topological classification, it is difficult to formulate a mathematical argument since the usual classification using equivariant K-theory is hard to reconcile with symmetry-violating perturbations.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professor Dr. Emil Prodan