Spectral flow for real skew-adjoint Fredholm operators

TL;DR

This paper introduces a $$-valued spectral flow for real skew-adjoint Fredholm operators, capturing eigenfunction orientation changes and linking to topological invariants in quantum models.

Contribution

It provides an analytic, homotopy-invariant definition of $$-spectral flow for real skew-adjoint Fredholm operators, connecting it to topological indices and physical applications.

Findings

Defines a $$-valued spectral flow counting eigenfunction orientation changes.

Proves the spectral flow satisfies concatenation and homotopy invariance.

Links the spectral flow to zero energy states and $$-polarization in Majorana chains.

Abstract

An analytic definition of a $\mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through $0$ along the path. The $\mathbb{Z}_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $\mathbb{Z}_2$-index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the $\mathbb{Z}_2$-polarization in these models.