i'm studying Kosterlitz THouless transition and i have a doubt: what is a vortex anti-vortex configuration?

Is this thing?

enter image description here

or this one

enter image description here

I think that they are quite different !


They are different (but see below $^*$). For a vortex of strength $+1$, if you walk around the defect clockwise (at a safe distance from the core, so that the local spin direction is always well defined) then the spins complete one full turn, also in a clockwise direction. You can see that this is true for both the vortices in your top diagram (which needs a bit of artistic licence, because it looks like it comes from a fluid flow system rather than a spin system $^*$). If, when you look at the right hand vortex, you prefer to take the walk around it in the anticlockwise direction, that's fine, and you will notice that the spins complete a full turn in the same (anticlockwise) direction. So, it is still a strength $+1$ vortex.

In the lower picture, the right one is also a vortex, but the left one is an antivortex, of strength $-1$. Taking a clockwise walk around it, the spins rotate by one full turn in the anticlockwise sense.

It is worth remembering that the hamiltonian for the XY model is invariant to a global rotation of the spins, by the same angle in the same direction. This can change the appearance of snapshots of spin configurations quite dramatically! But it has no effect on the energy, or indeed on the topology of the defects. There are some very nice animations on this blog describing the Kosterlitz-Thouless transition. Look particularly at the section which poses "Puzzle 1" and "Puzzle 2". You'll see an animation showing a left-hand vortex changing smoothly into a right-hand vortex. There's also some interesting stuff in the comments section at the bottom of that page.

$^*$ EDIT. I belatedly realized that the top picture of the OP is most likely the gradient of the angle field, rather than an imperfect representation of the spins themselves. So, the two pictures are most likely different representations of the same configuration of a vortex and an antivortex. In the continuum representation, the angle field around a strength $+1$ vortex at the origin is $\theta^+(x,y)=\tan^{-1}(y/x)$, and the gradient is $\nabla\theta^+=(-y,x)/r^2$, where $r^2=x^2+y^2$. The spins may be represented as unit vectors $(\cos\theta^+,\sin\theta^+) = (x/r,y/r)$. Around a strength $-1$ antivortex at the origin, $\theta^-(x,y)=-\tan^{-1}(y/x)$, $\nabla\theta^-=(y,-x)/r^2$, and $(\cos\theta^-,\sin\theta^-) = (x/r,-y/r)$. Here is the combined gradient field for an antivortex (blue) and vortex (red).

gradient field of antivortex+vortex

This is similar to the top picture in the OP. Now here is the spin vector diagram for the same configuration, with an arbitrary angle $\theta_0=\pi/2$ added throughout.

enter image description here

This is similar to the bottom picture in the OP. The argument about following the rotation of the spins on taking a walk around the defect(s) applies to the spin picture, but clearly the gradient picture is consistent with it.

Calculations and plots created in Maple.

theta := arctan(y, x-.5)-arctan(y, x+.5)+(1/2)*Pi;
f := fieldplot([cos(theta), sin(theta)], x = -1 .. 1, y = -1 .. 1, arrows = medium, axes = none);
p := pointplot([-.5, 0, .5, 0], color = [blue, red], symbol = solidcircle, symbolsize = 30);
display(p, f);
g := gradplot(theta, x = -1 .. 1, y = -1 .. 1, fieldstrength = maximal(1.5), arrows = medium, axes = none);
display(p, g);
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  • $\begingroup$ Thank you. I have another question. On my book i read that the second configuration can be transformed into a ground state in which all the spin are alligned through a continuous rotation. How can i figure out this thing? Because i've tried to rotate the spins in every way but i see always a discontinuity. Thanks $\endgroup$ – MementoMori Sep 28 '18 at 7:08
  • $\begingroup$ Depends what is meant by a "continuous rotation". As explained in the answer, uniform global rotation of the spins in the plane (rotating them all by the same angle) definitely cannot do this. If you want to pursue this, I suggest that you post a new question, giving full details of the claim and a reference to the book in which you found it. $\endgroup$ – user197851 Sep 28 '18 at 7:29
  • $\begingroup$ It is possible that they mean that the spins are rotated, not all by the same angle, but in such a way that the two defects annihilate each other. The generation of a defect pair, and its annihilation, are also shown in one of those animations I referred to. But it's impossible to say for sure if this is what was meant, without more detail. $\endgroup$ – user197851 Sep 28 '18 at 7:36
  • $\begingroup$ I belatedly realized that your first picture was probably the gradient field, and the second picture was the spin configuration: different representations of the same antivortex + vortex pair. I've edited my answer accordingly. This makes more sense. Sorry I was a bit slow there, I only realized this when I read your second question physics.stackexchange.com/questions/431278/… $\endgroup$ – user197851 Sep 29 '18 at 10:44
  • $\begingroup$ OK thanks. But now i have a doubt about the definition of vortex and antivortex. The definition that you have given in the first answer is different from what you have used when you claim that the first picture in a vortex anti vortex configuration, isn't it? $\endgroup$ – MementoMori Sep 29 '18 at 11:21

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