Approximate symmetries and conservation laws in topological insulators and associated Z-invariants - Doll, Nora; Schulz-Baldes, Hermann (Annals of Physics, 2020)
Solid state systems with time reversal symmetry and/or particle–hole symmetry often only have Z2-valued strong invariants for which no general local formula is known. For physically relevant values of the parameters, there may exist approximate symmetries or almost conserved observables, such as the spin in a quantum spin Hall system with small Rashba coupling. It is shown in a general setting how this allows to define robust integer-valued strong invariants stemming from the complex theory, such as the spin Chern numbers, which modulo 2 are equal to the Z2-invariants. Moreover, these integer invariants can be computed using twisted versions of the spectral localizer.
CRIS-Team Scopus Importer:2020-07-10
Keywords: Approximate laws; Spectral localizer; Strong invariants

Orientation flow for skew-adjoint Fredholm operators with odd-dimensional kernel - Doll, Nora (Journal of Functional Analysis, 2024)
The orientation flow of paths of real skew-adjoint Fredholm operators with invertible endpoints was studied by Carey, Phillips and Schulz-Baldes. For paths of real skew-adjoint Fredholm operators with odd-dimensional kernel the orientation flow is defined with respect to a real one-dimensional reference projection. It is homotopy invariant and fulfills a concatenation property. When applied to closed paths it is independent of the reference projection and provides an isomorphism of the fundamental group of the space of real skew-adjoint Fredholm operators with odd-dimensional kernel to Z2. As an example the orientation flow of the magnetic flux inserted in a half-sided Kitaev chain is studied.
CRIS-Team Scopus Importer:2023-10-20
Keywords: Skew-adjoint operators; Spectral flow

Parity as Z2-valued spectral flow - Doll, Nora; Schulz-Baldes, Hermann; Waterstraat, Nils (Bulletin of the London Mathematical Society, 2019)
This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray-Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the Z2-valued spectral flow of an associated path of chiral skew-adjoints. Furthermore, the related notion of Z2-index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non-trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non-linear partial differential equation.
CRIS-Team WoS Importer:2019-09-17

Skew localizer and Z2-flows for real index pairings - Doll, Nora; Schulz-Baldes, Hermann (Advances in Mathematics, 2021)
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are 64 such real index pairings of real K-theory with real K-homology. For 16 of them, the index of the Fredholm operator representing the pairing vanishes, but there is a secondary Z2-valued invariant. The first set of results provides index formulas expressing each of these 16 Z2-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from an unbounded Fredholm module and shows that the Z2-invariant can be computed as the sign of the Pfaffian of the skew localizer and in 8 of the cases as the sign of the determinant of the off-diagonal entry of the skew localize. This is of relevance for the numerical computation of invariants of topological insulator
CRIS-Team Scopus Importer:2021-10-22
Keywords: Index theory; Spectral flow; Z2-invariant

Spectral Flow. A Functional Analytic and Index-Theoretic Approach - Doll, Nora; Schulz-Baldes, Hermann; Waterstraat, Nils (De Gruyter, 2023)
This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory. Presents a self-contained functional analytic treatment of spectral flow Includes a detailed analysis of the homotopy theory of the set of unbounded selfadjoint Fredholm operators Provides an analysis of Bott-Maslov index via spectral flow </li
CRIS-Team Scopus Importer:2025-07-10

Spectral flow: A functional analytic and index-Theoretic approach - Doll, Nora; Schulz-Baldes, Hermann; Waterstraat, Nils (De Gruyter, 2023)
This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.
CRIS-Team Scopus Importer:2023-07-07
Keywords: Indextheorie von Fredholm-Paaren; Oszillationstheorie für Jacobi-Operatoren und Streutheorie; Selbstadjungierte Fredholm-Operatoren; Variationsbifurkationstheorie