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Applied index theory for quantum and classical systems

Subject Area Mathematics
Theoretical Condensed Matter Physics
Term from 2015 to 2023
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 281672735
 
Final Report Year 2023

Final Report Abstract

Topological insulators are solid state systems of electrons that are insulating in the sense that there are no or at least no conducting quantum states at the Fermi level, but are moreover topological in the sense that one can extract non-trivial invariants from the band theory, often by explicit integral formulas. In the standard situations, these topological invariants are either Chern numbers or winding numbers. The most robust of these invariants are integer-valued invariants that are called strong. Their importance lies in the by now well-known fact that non-triviality of the invariants implies that surfaces of the material harvest edge states that are conducting and not susceptible to Anderson localization. This is the core implication of the bulk-boundary correspondence which can be proved by means of index theory (K-theory, K-homology or KK-theory). Moreover, index theorems connect the topological invariants to indices of Fredholm operators which are naturally associated to the mathematical data of the system. This allows to show that the phenomena described above are stable under perturbations of the system, for example by a small random potential. Other topological invariants are real-valued and called weak, while yet others are Z2-valued and only well-defined in presence of either time-reversal or particle-hole symmetry. All the above theory was mainly developed for electrons in solid state systems, but it is by now well-established in the physics community that all these concepts and results extend to other systems that can be described by similar wave equation, in particular, photonic crystals and metamaterials. Some elements of the theory also apply to semimetals which typically appear as transition points between topological insulators. The main results of this research project provide new techniques to compute the topological invariants by means of the spectral localizer which is a Dirac-like operator containing the Hamiltonian as a topological mass term. This leads to a highly efficient numerical technique to compute the topological invariants - likely the best available at present. The strong integer-valued invariants can be read off as the half-signature of the finite-volume restrictions of the spectral localizer, while the Z2 -invariants are given the sign of the Pfaffian of a skew version of the spectral localizer. Weak invariants are computed as the signature density of the spectral localizer. The low lying spectrum of the spectral localizer also allows to compute invariants of ideal semimetals. Furthermore, the theory of invariants and their index representation was extended to semimetals. For so-called chiral semimetals such as graphene, a bulk-boundary correspondence is proved that connects the weak bulk invariants to the density of surface states by a remarkable identity. To achieve the above goals, various mathematical tools were developed which have their own intrinsic value. Besides several publications in high-ranked journals, two mathematical monographs were written during the second funding period. One develops noncommutative harmonic analysis and its application to index theory. The second monograph provides an in-depth treatment of spectral flow and its importance for index theory.

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