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enter image description here let us consider the two-dimensional configuration shown in Fig. 3.1a. The lengths of the arrows represent the magnitude of φ, while their directions indicate the orientation in the $φ_1 -φ_2$ plane. Regardless of the behavior of the fields in the interior region, it is clear that the fields cannot be continuously deformed to a vacuum solution, such as that shown in Fig. 3.1b.

How can I get the idea of vortex from the above statement?

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It would help if you mention the source from which you borrowed fig 3.1. – user10001 Apr 30 '13 at 19:20
Classical Solutions in Quantum Field Theory Solitons and Instantons in High Energy Physics ERICK J. WEINBERG Columbia University page no : 39 – Unlimited Dreamer Apr 30 '13 at 19:37
Thanks for the nice reference :) – user10001 Apr 30 '13 at 20:54
up vote 4 down vote accepted

It's a bit hard to be sure without seeing the whole text, but it looks like they're discussing the problem of of obtaining finite minimum energy solutions of a gauge/Higgs system. In 3 space dimensions, for example, for the Georgi-Glashow model, $$ \mathcal{L}= \frac{1}{2}Tr(F^{\mu\nu}F_{\mu\nu})+Tr(D_{\mu}\phi D^{\mu}\phi)-\frac{\lambda}{4}(|\phi|^2-v^2)^2 $$ to minimize the energy you want the curvature to vanish at infinity, so the potential becomes pure gauge.

Moreover $\phi^a \phi^a=v^2$ defines a 2 sphere in internal group space. So looking at the behaviour of the Higgs field on the $S^2$ at infinity we have a map $S^2\rightarrow S^2$. Now there is the boring solution where $A_{\mu}=0$ and $\phi$ is a constant at infinity. This is like your picture (b). But there is also the possibility that $\phi$ follows the direction defined by the polar coordinates $\theta, \psi$ on the $S^2$ at infinity. This is a winding number 1 solution and is depicted in (a) - this is 'tHooft's hedgehog configuration. Things like (a) are monopoles.

Now you can do the same thing in two spatial dimensions instead of three. $\phi$ is just a complex number, and the vacuum manifold in internal group space is an $S^1$ this time. If we work out what the gauge potential must do to make the energy of a two dimensional hedgehog finite, it turns out the the $A$ field is pointing in a direction tangential to the $S^1$ at infinity. This is the vortex - you can't extend $A$ back towards the origin without hitting singular behaviour

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Think you got (a) and (b) swapped around. Otherwise great answer. – Michael Brown May 1 '13 at 0:57
@MichaelBrown sorry, yes I did - I'll edit the answer. Thanks! – twistor59 May 1 '13 at 6:17
This is about global vortices in two dimensions. Fields cannot be continuously deformed to a vacuum solution can you focus on this please? – Unlimited Dreamer May 1 '13 at 8:38
@UnlimitedDreamer suppose you try to smoothly deform (a) into (b). You start by adding an upward component to the arrows each of the arrows. You then gradually increase the upward component until they're all pointing up like (b). The LHS ones turn clockwise, the RHS ones anticlockwise. That works fine except for the very bottom downward arrow. Does it turn upward by going clockwise or anticlockwise? This is the singularity. It's typical of a global topological property of a field, in this case the winding number. – twistor59 May 1 '13 at 9:26

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