# Tag Archives: Vortex

## Just a little thought on Aharonov-Bohm Destruction of Superconductivity

Many experimenters in the past have exploited the similarities between superconductors and superfluids to come up with new ideas. One of the most important of these analogies is the Hess-Fairbank effect in a superfluid (pdf!) and the corresponding Meissner effect in a superconductor. The Hess-Fairbank effect is often taken as the pre-eminent experimental signature of a superfluid.

For those who are unfamiliar with the Hess-Fairbank effect, let me describe it briefly. It was found that if one rotates a cylindrical container of helium slowly (i.e. with angular velocity less than half the critical velocity, $\omega < \omega_c/2 \equiv \hbar/2mR^2$), and one cools below the transition temperature into the superfluid phase towards $T \rightarrow 0$, that the liquid in the container remains at rest despite the rotating walls!

We can write the Hamiltonian for the liquid in the rotating container in the rotating frame (not the lab frame!):

$H'(\textbf{r}',\textbf{p}') = \frac{(\textbf{p}'-m \vec{\omega} \times\textbf{r}')^2}{2m} + V'(\textbf{r}')$

Besides an unimportant centrifugal term which I’ve buried in the definition of $V'(\textbf{r}')$, there is a strong mathematical analogy between the Hamiltonian above and that which describes a superconductor in the presence of a vector potential (in the lab frame):

$H(\textbf{r},\textbf{p}) = \frac{(\textbf{p}-e\textbf{A}(\textbf{r}))^2}{2m} + V(\textbf{r})$

One can see that $e \textbf{A}(\textbf{r})$ corresponds to $m \vec{\omega} \times\textbf{r}$. Furthermore, if we assume a constant magnetic field (as in the Meissner effect), we can write for $\textbf{A} = \frac{1}{2} \textbf{B} \times \textbf{r}$ and the correspondence becomes $\vec{\omega} \leftrightarrow e\textbf{B}/2m$.

Consider now what happens when one rotates the container containing the superfluid quickly (faster than the critical angular velocity). I have blogged previously about this scenario and the elegant experiment showing the proliferation of quantized vortices. In fact, if one rotates the cylinder quickly enough, the superfluidity is destroyed entirely.

Now, let us consider switching to a multiply connected geometry. It is important to note that the kinetic energy terms in the Hamiltonians remain the same if we were to use an annular container for the superfluid and a ring in the superconducting case with an Aharonov-Bohm flux through the center. For the annular container, the superfluidity would still be destroyed if the container were rotated quickly enough. This would lead one to conclude that for a superconducting ring, a strong enough Aharonov-Bohm flux would also destroy the superconductivity. There would not be an actual magnetic field applied to the superconductor (i.e. not the Meissner effect in the usual sense), but the superconductivity would be destroyed through a pseudo-Meissner effect nonetheless.

I have to admit that I have not come across an experimental paper demonstrating this effect, so please share if you know of one, as I’m sure this idea has been around for a while! It is just interesting to think that one could destroy superconductivity without actually ‘touching’ it with any kind of measurable field.

## Quantized Vortices in Superfluid 4He

There are a couple nice old PRLs  from the Packard Group at Berkeley demonstrating the existence of quantized vortices and the vortex pattern in superfluid 4He. See here (pdf!) and here (paywall). Just as a little bit of background for those who are unfamiliar: below 2.17K helium undergoes a liquid to liquid phase transition from a “normal” liquid to a superfluid. The superfluid is characterized by several properties including zero viscosity (similar to electrons in a superconductor), second sound (effectively, temperature waves), and quantized vorticity.

The observation of this latter property was captured vividly in the series of images taken by the Packard Group shown below. To induce vortex formation, the authors rotated the bucket in which the superfluid had been placed. Because of the zero viscosity, the superfluid remained still until a critical velocity was reached where a single vortex formed in the center of the bucket. As the angular velocity was increased, more and more vortices started to form and the authors show the pattern formed by these vortices in the presence of 1-11 vortices.

Interestingly, there are two different stable configurations for which 6 vortices can form as shown in the figure. I happen to know that Richard Feynman, who had done a lot of the prior theoretical work on vortices in superfluid 4He, sent a personal letter to the authors of these papers to thank them for their elegant experiment.