I had a question about Haldane's wormhole interpretation (picture below). I believe he first proposed it in his paper Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property. To my understanding, this is for 3D band structures, where a gapless point initially gives one Dirac string, which then immediately splits to two Dirac points and creates a Berry flux loop.
My question is, let's say we have a 2D model (instead of 3D) - such as gapped graphene or the Haldane model with a K and K' point - with a band touching ONLY at K'. Am I correct in saying that:
- the Hilbert spaces of the two tori (one torus per band) have a wormhole/string there as well (since the gapless point is a source of Berry flux, where the Berry curvature is a dirac delta function)? i.e. this isn't exclusively a 3D interpretation?
- that instead of having a pair of wormholes (Dirac point splitting), we have only one source of Berry curvature flux (i.e. only one Dirac string connecting points of each torus corresponding to the K' point)?
Or, does this picture fail in 2D?