# Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to the case of e.g. the electric field of mono- and multipoles in electrodynamics, or the currents circulating an Abrikosov vortex in superconductivity (which is a topological defect where the phase of the underlying order parameter diverges, but the amplitude goes smoothly to zero). In these examples, we are dealing with real topological defects with a diverging vector field, if I understand it correctly. They are non-trivial and "interesting" from a physics perspective.

In the vector field that I'm studying, however, the magnitude of the vector field goes to zero smoothly in the critical points, and the underlying order parameter does not diverge. The critical points still obey the Poincaré-Hopf sum rule, which I have verified for different differential topologies. Are such critical points in general "trivial" (since the magnitude of the vector field is well-defined and goes to zero), or can they still be "interesting" from a physical perspective? I think the critical points I'm referring to might usually be called "zeros" instead, but I'm not well-versed in topology or the vector calculus terminology. Sadly I cannot come up with a well-known example that is an analogue of the vector field I'm studying.

• What is the physical context you are considering? We can't tell you what features of a field may or may not be physically interesting if we don't know what the field is – By Symmetry May 9 at 10:35
• @BySymmetry An example would be a superflow with critical points, but with an underlying phase that never winds 2*pi (i.e. no topological defects in the form of Abrikosov vortices). – fromGiants May 17 at 15:56
• FWIW, this is e.g. used in localization of path integrals, see e.g. here & here. – Qmechanic May 17 at 23:59
• @Qmechanic do you mind elaborating a bit more? How is it used and what is it's significance? – fromGiants May 19 at 9:40