https://phys.org/news/2019-01-topological-quantities.html

The so-called Chern index is another, albeit very abstract, topological quantity. This is a special case in topology. Recently, it was discovered that the Chern index has a density—called the Chern marker. This gives information about the topology of an entire system even by just looking at the local level. The solid line in figure 1 shows the spatial distribution of the Chern marker, where each local point has its own value. (Note that this graph would make no sense for a global property like the number of holes in an object.) The local points do however need to abide by one rule at the global level: the average of all local values should always be zero.

Leiden physicist Marcello Caio, together with a team of scientists from the United Kingdom, has now discovered that if you perturb a system, the local Chern values will distribute by flowing from the boundaries toward the interior of the system. In contrast, global topological properties are robust against perturbations—the number of holes in a donut only changes after a big bite. In figure 1, a perturbation causes the graph to reshape to keep honoring the requirement that its integral should be zero. Caio and his colleagues discovered that this doesn't happen instantaneously, so there must be a flow of the Chern marker through the system. This constitutes topological current, in analogy to a conventional electric current.