If you are, by any chance, following progress in the field of Majorana bound states, then you are for sure super excited about ample Majorana results arriving this Fall. On the other hand, if you just heard about these elusive … Continue reading

If you are, by any chance, following progress in the field of Majorana bound states, then you are for sure super excited about ample Majorana results arriving this Fall. On the other hand, if you just heard about these elusive states recently, it is time for an update. For physicists working in the field, this Fall was perhaps the most exciting time since the first experimental reports from 2012. In the last few weeks there was not only one, but at least three interesting manuscripts reporting new insightful data which may finally provide a definitive experimental verification of the existence of these states in condensed matter systems.

But before I dive into these new results, let me give a brief history on the topic of Majorana states and their experimental observation. The story starts with the young talented physicist Ettore Majorana, who hypothesized back in 1937 the existence of fermionic particles which were their own antiparticles. These hypothetical particles, now called Majorana fermions, were proposed in the context of elementary particle physics, but never observed. Some 60 years later, in the early 2000s, theoretical work emerged showing that Majorana fermionic states can exist as the quasiparticle excitations in certain low-dimensional superconducting systems (not a real particle as originally proposed, but otherwise having the exact same properties). Since then theorists have proposed half a dozen possible ways to realize Majorana modes using readily available materials such as superconductors, semiconductors, magnets, as well as topological insulators (for curious readers, I recommend manuscripts [1, 2, 3] for an overview of the different proposed methods to realize Majorana states in the lab).

The most fascinating thing about Majorana states is that they belong to the class of anyons, which means that they behave neither as bosons nor as fermions upon exchange. For example, if you have two identical fermionic (or bosonic) states and you exchange their positions, the quantum mechanical function describing the two states will acquire a phase factor of -1 (or +1). Anyons, on the other hand, can have an arbitrary phase factor e^{i}^{φ} upon exchange. For this reason, they are considered to be a starting point for topological quantum computation. If you want to learn more about anyons, check out the video below featuring IQIM’s Gil Refael and Jason Alicea.

Back in 2012, a group in Delft (led by Prof. Leo Kouwenhoven) announced the observation of zero-energy states in a nanoscale device consisting of a semiconductor nanowire coupled to a superconductor. These states behaved very similarly to the Majoranas that were previously predicted to occur in this system. The key word here is ‘similar’, since the behavior of these modes was not fully consistent with the theoretical predictions. Namely, the electrical conductance carried through the observed zero energy states was only about ~5% of the expected perfect transmission value for Majoranas. This part of the data was very puzzling, and immediately cast some doubts throughout the community. The physicists were quickly divided into what I will call enthusiasts (believers that these initial results indeed originated from Majorana states) and skeptics (who were pointing out that effects, other than Majoranas, can result in similarly looking zero energy peaks). And thus a great debate started.

In the coming years, experimentalists tried to observe zero energy features in improved devices, track how these features evolve with external parameters, such as gate voltages, length of the wires, etc., or focus on completely different platforms for hosting Majorana states, such as magnetic flux vortices in topological superconductors and magnetic atomic chains placed on a superconducting surface. However, these results were not enough to convince skeptics that the observed states indeed originated from the Majoranas and not some other yet-to-be-discovered phenomenon. And so, the debate continued. With each generation of the experiments some of the alternative proposed scenarios were ruled out, but the final verification was still missing.

Fast forward to the events of this Fall and the exciting recent results. The manuscript I would like to invite you to read was just posted on ArXiv a couple of weeks ago. The main result is the observation of the perfectly quantized 2e^{2}/h conductance at zero energy, the long sought signature of the Majorana states. This quantization implies that in this latest generation of semiconducting-superconducting devices zero-energy states exhibit perfect electron-hole symmetry and thus allow for perfect Andreev reflection. These remarkable results may finally end the debate and convince most of the skeptics out there.

To fully appreciate these results, it is useful to quickly review the physics of Andreev reflection (Fig. 1c-e) that occurs at the interface between a normal region with a superconductor [4]. As the electron (blue) in the normal region enters a superconductor and pulls an additional electron with it to form a Copper pair, an extra hole (red) is left behind (Fig. 1(c)). You can also think about this process as the transmission through two leads, one connecting the superconductor to the electrons and the other to the holes (Fig. 1d). This allows us to view this problem as a transmission through the double barrier that is generally low. In the presence of a Majorana state, however, there is a resonant level at zero energy which is coupled with the same amplitude with both electrons and holes. This in turn results in the resonant Andreev reflection with a perfect quantization of 2e^{2}/h (Fig. 1e). Note that, even in the configuration without Majorana modes, perfect quantization is possible but highly unlikely as it requires very careful tuning of the barrier potential (the authors did show that their quantization is robust against tuning the voltages on the gates, ruling out this possibility).

Going back to the experiments, you may wonder what made this breakthrough possible? It seems to be the combination of various factors, including using epitaxially grown superconductors and more sophisticated fabrication methods. As often happens in experimental physics, this milestone did not come from one ingenious idea, but rather from numerous technical improvements obtained by several generations of hard-working grad students and postdocs.

If you are up for more Majorana reading, you can find two more recent eye-catching manuscripts here and here. Note that the list of interesting recent Majorana papers is a mere selection by the author and not complete by any means. A few months ago, my IQIM colleagues wrote a nice blog entry about topological qubits arriving in 2018. Although this may sound overly optimistic, the recent results suggest that the field is definitely taking off. While there are certainly many challenges to be solved, we may see the next generation of experiments designed to probe control over the Majorana states quite soon. Stay tuned for more!!!!!!

Here’s a cute connection between topological entropy, braids, and the golden ratio. I learned about it in this paper: • Jean-Luc Thiffeault and Matthew D. Finn, Topology, braids, and mixing in fluids. Topological entropy I’ve talked a lot about entropy on this blog, but not much about topological entropy. This is a way to define […]

Here’s a cute connection between topological entropy, braids, and the golden ratio. I learned about it in this paper:

• Jean-Luc Thiffeault and Matthew D. Finn, Topology, braids, and mixing in fluids.

I’ve talked a lot about entropy on this blog, but not much about topological entropy. This is a way to define the entropy of a continuous map from a compact topological space to itself. The idea is that a map that mixes things up a lot should have a lot of entropy. In particular, any map defining a ‘chaotic’ dynamical systems should have positive entropy, while non-chaotic maps maps should have zero entropy.

How can we make this precise? First, cover with finitely many open sets Then take any point in apply the map to it over and over, say times, and report which open set the point lands in each time. You can record this information in a string of symbols. How much information does this string have? The easiest way to define this is to simply count the total number of strings that can be produced this way by choosing different points initially. Then, take the logarithm of this number.

Of course the answer depends on typically growing bigger as increases. So, divide it by and try to take the limit as Or, to be careful, take the lim sup: this could be infinite, but it’s always well-defined. This will tell us how much new information we get, on average, each time we apply the map and report which set our point lands in.

Of course the answer also depends on our choice of open cover So, take the supremum over all finite open covers. This is called the **topological entropy** of

Believe it or not, this is often finite! Even though the log of the number of symbol strings we get will be larger when we use a cover with lots of small sets, when we divide by and take the limit as this dependence often washes out.

Any braid gives a bunch of maps from the disc to itself. So, we define the **entropy of a braid** to be the minimum—or more precisely, the infimum—of the topological entropies of these maps.

How does a braid give a bunch of maps from the disc to itself? Imagine the disc as made of very flexible rubber. Grab it at some finite set of points and then move these points around in the pattern traced out by the braid. When you’re done you get a map from the disc to itself. The map you get is not unique, since the rubber is wiggly and you could have moved the points around in slightly different ways. So, you get a bunch of maps.

I’m being sort of lazy in giving precise details here, since the idea seems so intuitively obvious. But that could be because I’ve spent a lot of time thinking about braids, the braid group, and their relation to maps from the disc to itself!

This picture by Thiffeault and Finn may help explain the idea:

As we keep move points around each other, we keep building up more complicated braids with 4 strands, and keep getting more complicated maps from the disc to itself. In fact, these maps are often chaotic! More precisely: they often have positive entropy.

In this other picture the vertical axis represents time, and we more clearly see the braid traced out as our 4 points move around:

Each horizontal slice depicts a map from the disc (or square: this is topology!) to itself, but we only see their effect on a little rectangle drawn in black.

Okay, now for the punchline!

**Puzzle 1.** Which braid with 3 strands has the highest entropy per generator? What is its entropy per generator?

I should explain: any braid with 3 strands can be written as a product of generators Here switches strands 1 and 2 moving the counterclockwise around each other, does the same for strands 2 and 3, and and do the same but moving the strands clockwise.

For any braid we can write it as a product of generators with as small as possible, and then we can evaluate its entropy divided by This is the right way to compare the entropy of braids, because if a braid gives a chaotic map we expect powers of that braid to have entropy growing linearly with

Now for the answer to the puzzle!

**Answer 1.** A 3-strand braid maximizing the entropy per generator is And the entropy of this braid, per generator, is the logarithm of the golden ratio:

In other words, the entropy of this braid is

All this works regardless of which base we use for our logarithms. But if we use base e, which seems pretty natural, the maximum possible entropy per generator is

Or if you prefer base 2, then each time you stir around a point in the disc with this braid, you’re creating

bits of unknown information.

This fact was proved here:

• D. D’Alessandro, M. Dahleh and I Mezíc, Control of mixing in fluid flow: A maximum entropy approach, *IEEE Transactions on Automatic Control* **44** (1999), 1852–1863.

So, people call this braid the **golden braid**. But since you can use it to generate entropy forever, perhaps it should be called the *eternal* golden braid.

What does it all mean? Well, the 3-strand braid group is called , and I wrote a long story about it:

• John Baez, This Week’s Finds in Mathematical Physics (Week 233).

You’ll see there that has a representation as 2 × 2 matrices:

These matrices are shears, which is connected to how the braids and give maps from the disc to itself that shear points. If we take the golden braid and turn it into a matrix using this representation, we get a matrix for which the magnitude of its largest eigenvalue is the square of the golden ratio! So, the amount of stretching going on is ‘the golden ratio per generator’.

I guess this must be part of the story too:

**Puzzle 2.** Is it true that when we multiply matrices of the form

or their inverses:

the magnitude of the largest eigenvalue of the resulting product can never exceed the th power of the golden ratio?

There’s also a strong connection between braid groups, certain quasiparticles in the plane called Fibonacci anyons, and the golden ratio. But I don’t see the relation between these things and topological entropy! So, there is a mystery here—at least for me.

For more, see:

• Matthew D. Finn and Jean-Luc Thiffeault, Topological optimisation of rod-stirring devices, *SIAM Review* **53** (2011), 723—743.

Abstract.There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of material lines is conveniently given by the topological entropy of the rod motion. We discuss the problem of optimising such rod devices from a topological viewpoint. We express rod motions in terms of generators of the braid group, and assign a cost based on the minimum number of generators needed to write the braid. We show that for one cost function—the topological entropy per generator—the optimal growth rate is the logarithm of the golden ratio. For a more realistic cost function,involving the topological entropy per operation where rods are allowed to move together, the optimal growth rate is the logarithm of the silver ratio, We show how to construct devices that realise this optimal growth, which we callsilver mixers.

Here is the silver ratio:

But now for some reason I feel it’s time to stop!

Today was a low-research day, because [reality]. However, Kate Storey-Fisher (NYU) and I had a great discussion with Josh Ruderman (NYU) about anomalies in the LSS. As my loyal reader knows, we are looking at constructing a statistically valid, safe search for deviations from the cosmological model in the large-scale structure. That search is going to focus towards the overlap (if there is any overlap) between anomalies that are safe to systematic problems with the data (that is, anomalies that can't be mocked by reasonable adjustments to our beliefs about our selection function) and anomalies that live in spaces suggested or predicted by theoretical ideas about non-standard cosmological theories. In particular, we are imagining theories that have the dark sector do interesting things at late times. We didn't make concrete plans in this meeting, except to read down literatures about late decays of the dark matter, dark radiation, and other kinds of dark–dark interactions that could be happening in the current era.