Does Current Density Decay to Zero at Vortex Centers in 2D Systems?

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The discussion centers on the behavior of current density, denoted as ##\vec J##, in two-dimensional systems, specifically at vortex centers. It is established that ##\vec J## decays to zero at the centers of vortices, indicating that the current density is non-zero elsewhere in the material. The participant also infers that the magnetic field associated with these vortices should similarly decay to zero, as the current density approaches zero at the vortex centers. This conclusion is drawn from the derived expression for ##\vec J##, which highlights the unique characteristics of current behavior in these systems.

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Hello, I have derived an expression for ##\vec J## in a particular, unusual physical problem in 2D. The expression is different from ##\vec 0## everwhere in the material except at at least two different points.

Can I conclude that the only way for this to occur is that ##\vec J## itself decays to ##\vec 0## the closer it gets to the center of the vortices? I assume that the current density vanishes exactly at the center of vortices of currents.

Also, what can I conclude about the magnetic field?
 
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Would you be able to share the expression or perhaps have an image?

I would assume that the magnetic field decays to zero since at those vortices the current goes to zero.
 

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