# Vortices

## Layman

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.

## Student

Consider the Lagrangian: $$L = |\partial _\mu \phi |^2 - {\lambda\over 4}(|\phi |^2 -\eta^2 )^2 \label{glstring}$$ 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.

## Researcher

The motto in this section is: the higher the level of abstraction, the better.

Example1
Example2:

## History

Lord Kelvin famously tried to explain the different atoms as vortices.