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

TL;DR

The work studies a parametric family of pseudodifferential Hamiltonians with two energy bands and a finite-gap structure, showing that the spectral flow count $\\mathcal{N}$ of eigenvalues exchanged between bands equals the Chern index $\\mathcal{C}$ of the associated symbol bundle, i.e. $\\mathcal{N}=\\mathcal{C}$. It develops a transparent normal-form model in dimension $n=1$ to illustrate the equality $\\mathcal{N}_{E}=\\mathcal{C}_{E}=+1$ and then unifies the result in a general framework where a gap-hypothesis on the symbol $H_{\\mu}$ and its Weyl quantization yield the index formula $\\mathcal{N}_{H}=\\mathcal{C}_{H}$. The authors provide a two-pronged proof strategy: (i) reduce to normal-form models $E^{(n,\\mathcal{C})}$ and verify $\\mathcal{N}_{E}=\\mathcal{C}_{E}$, and (ii) extend to general symbols via deformation arguments and Bott periodicity, complemented by the Fedosov–Hörmander index theorem. They also discuss topological contact without exchange, a phenomenon not detected by Chern/K-theory but tied to torsion in homotopy groups, with implications for when a spectral gap can be opened.

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.

Figures (6)

• Figure 2.1: Energy levels (in $cm^{-1}$) of the $CD_{4}$ molecule (carbon with four deuterium atoms) as a function of the total angular momentum $J\in\mathbb{N}$ (a conserved quantity corresponding to rotational energy). The fine structure of the spectrum reflects the slow rotational motion, while the broad structure corresponds to the faster vibrational dynamics. The spectrum exhibits clusters of energy levels, with some levels crossing or connecting different clusters fred-borisfred-boris01fred-boris02fred-mikael-04boris1.
• Figure 2.2: Spectrum of (\ref{['eq:modele']}).
• Figure 2.3: Chern index $\mathcal{C}$ of a rank one bundle over $S^{2}$, computed by the winding number of the clutching function.
• Figure 2.4: Eigenvalues from (\ref{['eq:val_p']}). We have $\omega^{-}\left(\mu,x,p\right)\leq-\left|\mu\right|$, $\omega^{+}\left(\mu,x,p\right)\geq\left|\mu\right|$. The red domain represents the possible values of $\omega_{-}\left(\mu,x,p\right)$ with $\mu$ fixed and $\left(x,p\right)\in\mathbb{R}^{2}$. Similarly, the blue domain represent $\omega_{-}\left(\mu,x,p\right)$. The degeneracy is at $\left(\mu,x,p\right)=\left(0,0,0\right)$.
• Figure 3.1: Illustration of the assumption (\ref{['eq:gap']}). On figure (a), for parameters $\left(\mu,x,p\right)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ in the green domain, we assume that the spectrum of the hermitian matrix $H_{\mu}\left(x,p\right)$, has $r$ eigenvalues smaller than $-C$ and that the others are greater than $C>0$. Equivalently, on figure (b), the spectrum $\omega$ of $H_{\mu}\left(x,p\right)$ for any $\left(x,p\right)$ is contained in the red domain.

Theorems & Definitions (25)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Definition 3.1
• Definition 3.2