Topological index formula in physical waves: spectral flow, Chern index and topological contacts

TL;DR

This paper links spectral flow in quantum Hamiltonians to topological invariants like the Chern index, illustrating the Atiyah Singer index formula with applications across physics.

Contribution

It establishes a direct relation between eigenvalue exchange in spectral bands and the Chern index of associated vector bundles, highlighting a topological interpretation.

Findings

Eigenvalue exchange number equals the Chern index.

Topological contact phenomena are distinguished from band exchange.

Applications demonstrated in molecular, plasma, and geophysical physics.

Abstract

We study a family of pseudodifferential operators (quantum Hamiltonians) on $L^{2}(\mathbb{R}^{n};\mathbb{C}^{d})$ whose spectrum exhibits two energy bands exchanging a finite number of eigenvalues. We show that this number coincides with the Chern index of a vector bundle associated to the principal symbol (the classical Hamiltonian). This result provides a simple yet illustrative instance of the Atiyah Singer index formula, with applications in areas such as molecular physics, plasma physics or geophysics. We also discuss the phenomenon of topological contact without exchange between energy bands, a feature that cannot be detected by the Chern index or K theory, but rather reflects subtle torsion effects in the homotopy groups of spheres.