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.