====== Vortices ======

<tabbox Why is it interesting?> 

<tabbox Layman> 

<blockquote>
An example of a vortex is universally encountered by people taking
baths or washing dishes. As the water flows down the drain, it
circulates. We cannot interpolate the circulating velocity field 
all the way to the center of the vortex since it would
have to become multi-valued. Instead the fluid density in the central
region of the vortex vanishes.

<cite>https://arxiv.org/pdf/hep-ph/9710292.pdf</cite>
</blockquote>
  
<tabbox Student> 

<blockquote>
Consider the Lagrangian:
\begin{equation}
L = |\partial _\mu \phi |^2 - {\lambda\over 4}(|\phi |^2 -\eta^2 )^2
\label{glstring}
\end{equation}
where $\phi$ is now taken to be a complex scalar field.
The Lagrangian is invariant under
$$
\phi \rightarrow \phi ' = e^{i\alpha} \phi
$$
and hence the model has a $U(1)$ (global) symmetry. The vacuum
expectation value of $\phi$ is $\eta e^{i\alpha}$ where $\alpha$
can take any value. So the ground state of the model has 
continuous degeneracy. The degeneracy is labelled by the phase
angle $\alpha$ and hence the vacuum manifold is a circle.

Vortices are formed if we consider the model in two spatial dimensions
and let $\alpha$ be such as to wrap around the vacuum manifold. For
example, we could take $\alpha =\theta$, the polar angle.
Then, since the field is single valued everywhere, there
must be at least one point at which $\phi =0$. The field carries
energy at this point since $\phi =0$ is not on the vacuum manifold.
The location of this point may be defined as the location of a vortex.


<cite>https://arxiv.org/pdf/hep-ph/9710292.pdf</cite>
</blockquote>
 
<tabbox Researcher> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Examples> 

--> Example1#

 
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--> Example2:#

 
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<tabbox FAQ> 
  
<tabbox History> 
Lord Kelvin [[http://www.pbs.org/wgbh/nova/blogs/physics/2011/12/beautiful-losers-kelvins-vortex-atoms/|famously tried]] to explain the different atoms as vortices. 

</tabbox>