Comment on ``Near-field spin Chern number quantized by real-space topology of optical structures''

TL;DR

This paper clarifies that the so-called 'real-space spin Chern number' is actually the Euler characteristic of a surface, and the claimed new invariant is just a restatement of the Chern-Gauss-Bonnet theorem, not a novel concept.

Contribution

It explains that the 'real-space spin Chern number' is equivalent to the Euler characteristic, emphasizing it is not a new invariant but a known topological property.

Findings

The 'Spin Chern number' equals the Euler characteristic of the surface.

The result is a direct application of the Chern-Gauss-Bonnet theorem.

No new topological invariant is introduced.

Abstract

In the reference Phys. Rev. Lett. 132, 233801 (2024), the authors claim to have introduced a ''real-space spin Chern number'' as well as a ''Spin Berry connection'' and a ''Spin Berry curvature''. The main finding of their letter is the statement that the integral of the ''Spin Berry curvature'' over the surface is equal to the ''Spin Chern number'' which is the Euler characteristic of the surface. What the authors show is that, given a vector field tangent to a surface, there is a connection whose curvature gives the Euler characteristic when it is integrated over the surface. The point of this comment is to explain that no new invariant has been defined and that the result shown is the exact statement of the Chern-Gauss-Bonnet theorem, in the particular case of a surface. Since the ''real-space spin Chern number'' is equal to the Euler characteristic, it is not a new invariant but just another name for the same thing. Moreover, the Euler number characterizes the surface and not the polarization state of the field.