# Tag Archives: Aharonov-Bohm

## 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.

## General Aspects of Topology in Quantum Mechanics

Condensed matter physics has, in the past ten years or so, made a left turn towards studying topological properties of materials. Following the discovery of the Quantum Hall Effect (QHE) in 1980, it took about 25 years to experimentally discover that similar phenomenology could occur in bulk samples in the absence of a magnetic field in topological insulators. In the current issue of Nature Physics, there are three papers demonstrating the existence of a Weyl semimetal in TaAs and NbAs. These states of matter bear a striking similarity to quantum mechanical effects such as the Aharonov-Bohm effect and the Dirac monopole problem.

So what do all of these things have in common? Well, I vaguely addressed this issue in a previous post concerning Berry phases, but I want to elaborate a little more here. First it should be understood that all of these problems take place on some sort of manifold. For instance, the Aharonov-Bohm effect takes place in a plane, the Dirac monopole problem on a 3D sphere and the problems in solid-state physics largely on a torus due to periodic boundary conditions.

Now, what makes all of these problems exhibit a robust topological quantization of some sort is that the Berry connection in these problems cannot adequately be described by a single function over the entire manifold. If one were to attempt to write down a function for the Berry connection, there would necessarily exist a singularity somewhere on the manifold. But because the Berry connection is not an observable, one can just write down two (or more) different functions on different parts (or “neighborhoods”) of the manifold. The price one has to pay is that one has to “patch” the functions together at the boundary of the neighborhoods. Therefore, the existence of the topological quantization in most of the problems described above arise because of a singularity in the Berry connection somewhere on the manifold that cannot be gotten rid of with a gauge transformation.

For instance, for the Aharonov-Bohm effect, the outside of the solenoid and the inside of the solenoid must be described by different functions, or else the “outside function” would be singular at the center of the solenoid.  Qualitatively, one can think of the manifold as a plane with a hole punched in the middle of it. In the case of the Dirac monopole, the magnetic monopole itself is the position of the singularity and there is a hole punched in 3-dimensional space.

There is an excellent discussion on both these problems in Sakurai’s quantum mechanics textbook. I particularly like the approach he takes to the Dirac monopole problem, which he adapted from Wu and Yang’s elegant solution. The explanation of the QHE using similar ideas was developed in this great (but unfortunately quite mathematical) paper by Kohmoto (pdf!). I realize that this post only sketches the main point (with perhaps too much haste), but I hope that it will be illuminating to some.

Update: I have written a guest post for Brian Skinner’s blog Gravity and Levity where I discuss the topics here in a little more detail. You can read the post here if you’re interested.

## Bardeen, CDWs and Macroscopic Quantum Phenomena

There is a well-written 1990 Physics Today article by John Bardeen entitled Superconductivity and Other Macroscopic Quantum Phenomena (pdf!). For those who are unaware, Bardeen was a two-time Nobel Laureate in Physics for inventing the transistor and secondly for the BCS theory of superconductivity.

Later in his career, Bardeen focused on the theory of transport in quasi-1D charge density wave materials. Bardeen was vocal in advocating that the transport in these materials must be understood in a quantum mechanical manner whereas most other physicists working on the problem treated it as a classical one (see True Genius by Daitch and Hoddeson). In the Physics Today article, he describes why he believes that the CDW sliding in these quasi-1D materials must be viewed as a manifestation of a macroscopic quantum phenomenon similar to that in superconductors and superfluids.

While Bardeen seemed to have lost his battle against the mainstream condensed matter physics community on this point upon his death in 1991, some interesting work has taken place since his death that has started to provide evidence for his perspective. In 1997, Monceau and co-workers showed the presence of Aharonov-Bohm-like oscillations in the CDW compound NbSe$_3$, with an oscillation period of, interestingly, $hc/2e$. While his tunneling theory of CDW transport may have been incorrect, his view of CDW transport as a macroscopic quantum phenomenon may yet be vindicated.

A lot of interest in these problems dissipated as scientists shifted to work on the problem of high temperature superconductivity following the discovery of the cuprates in 1986. However, it seems to me that there are still many unresolved issues in these compounds that persist to the present day that were cast aside rather than figured out.

As Paul Valery once said:

A poem is never finished, only abandoned.

The same can aptly be said about scientific problems.

## It’s Not Just a (Berry) Phase

Just a few years after Berry’s breakthrough paper (pdf!) on the phase now named after him, it was recognized by Zak (pdf!) that this effect could play a role in a solid state setting. Zak realized that in the Bloch Hamiltonian, the crystal momentum, k, could be treated as a “parameter” similar to how other “parameters” had been treated in Berry’s original work. More concretely:

$H(\textbf{k}) = e^{-i\textbf{k}\cdot r}He^{i\textbf{k}\cdot r}$

While phenomena that are now considered prototypical Berry phase effects were understood before the Berry paper, it was Berry who unified many concepts under a general framework. For instance, the Aharonov-Bohm effect, the Dirac monopole problem and the Integer Quantum Hall Effect were all understood separately, but the Berry phase concept presented a huge leap forward in consolidating these seemingly disparate phenomena.

It should be mentioned that the three cases mentioned above are all instances where the Berry phase is topological and results in a robust quantization through a topological invariant (known as the Chern number). However, the Berry phase does not have to be topological in nature and can have “geometrical” consequences. Many of these “geometrical” consequences are discussed in a great review article by Xiao, Chang and Niu.

One particular “geometrical” consequence stands out: the semiclassical equations of motion for electrons in a solid, as detailed in textbooks such as Ashcroft and Mermin, are incomplete! The application of an electric field in crystals with either broken time-reversal symmetry or broken inversion symmetry may, under certain circumstances, exhibit a transverse velocity associated with a non-zero Berry curvature (i.e. an application of an electric field in the x-direction will result in electrons travelling in the y-direction!). This experimental paper by NP Ong’s group demonstrates this effect (pdf!), and this paper by Karplus and Luttinger (paywall!) theoretically explains the transverse velocity in ferromagnets, again before Berry’s seminal paper.

I’ll also mention briefly that the seemingly mundane phenomenon of electrical polarization in solids is another case that cannot be explained without Berry-esque concepts, but I’ll leave that topic for another day.

It truly is stunning how many effects can be accounted for under the umbrella of Berry/Zak ideas. I’ll even restrain myself from mentioning topological insulators and Chern insulators (paywall) / the quantum anomalous hall effect (paywall).