For my November 2017 diary, go here.

Diary — December 2017

John Baez

December 6, 2017

A crystal made of electrons

Electrons repel each other, so they don't usually form crystals. But if you trap a bunch of electrons in a small space, and cool them down a lot, they will try to get as far away from each other as possible — and they can do this by forming a crystal!

This is sometimes called an electron crystal. It's also called a Wigner crystal, because the great physicist Eugene Wigner predicted in 1934 that this would happen.

Only since the late 1980s have we been able to make Wigner crystals in the lab. A crystal can only form if the electron density is low enough. This is due to the uncertainty principle of quantum mechanics, which implies that even at absolute zero, electrons wiggle around — and they do this more when they're densely packed! When the density is low, they settle down and form a crystal.

But when an electron gas is rapidly cooled, sometimes it doesn't manage to form a perfect crystal. It can form a glass! This is called a Coulomb glass.

It's an amazing world we live in, where people can study a glass made of electrons.

We can do other cool stuff, like create electron crystals in 2 dimensions using electrons trapped on a thin film of metal. That's what this picture shows. It's a theoretical picture, but you can trust it, since we understand the laws of physics needed to figure out what electrons do when trapped in a disk. The density here is low enough that the uncertainty principle doesn't play a significant role — so we can visualize the electrons as dots with a well-defined position.

The lines between the dots are just to help you see what's going on. In general, a 2-dimensional electron crystal wants to form a triangular lattice. But a triangular lattice doesn't fit neatly into a disk, so there are defects — places where things go wrong.

Puzzle 1. What is happening at the blue defects?

Puzzle 2. What is happening at the red defects?

Puzzle 3. What can you say about the number of blue defects and the number of red defects? Do these numbers obey some rule?

Puzzle 3 has some very interesting answers: see the comments on my G+ post and my somewhat more detailed blog article on this topic.

To know if a uniform electron gas at zero temperature forms a crystal, you need to work out its so-called Wigner–Seitz radius. This is the average inter-particle spacing measured in units of the Bohr radius. The Bohr radius is the unit of length you can cook up from the electron mass, the electron charge and Planck's constant. (It's also the average distance between the electron and a proton in a hydrogen atom in its lowest energy state.)

Simulations show that a 3-dimensional uniform electron gas crystallizes when the Wigner–Seitz radius is at least 106. In 2 dimensions, it happens when it's at least 31.

The picture above was drawn by Arunas.rv and placed on Wikicommons on a Creative Commons Attribution-Share Alike 3.0 Unported license.

December 9, 2017


In certain crystals you can knock an electron out of its favorite place and leave a hole: a place with a missing electron. Sometimes these holes can move around like particles. And naturally these holes attract electrons, since they are places an electron would want to be.

Since an electron and a hole attract each other, they can orbit each other. An orbiting electron-hole pair is a bit like a hydrogen atom, where an electron orbits a proton.

An orbiting electron-hole pair is called an exciton, because it's really just a special kind of 'excited' electron — an electron with extra energy, not in its lowest energy state where it wants to be.

An exciton usually doesn't last long: the orbiting electron and hole spiral towards each other, the electron finds the hole it's been seeking, and it settles down. Typical lifetimes range from picoseconds to nanoseconds.

But excitons can last long enough to do interesting things. In 1978 the Russian physicist Abrikosov wrote a short and very creative paper in which he raised the possibility that excitons could form a crystal in their own right! He called this new state of matter excitonium.

In fact his reasoning was very simple.

Just as electrons have a mass, so do holes. That sounds odd, since a hole is just a vacant spot where an electron would like to be. But such a hole can move around, and it takes force to accelerate it, so it acts just like it has a mass! The precise mass of a hole depends on the nature of the substance we're dealing with.

Now imagine a substance with very heavy holes.

When a hole is much heavier than an electron, it will stand almost still when an electron orbits it. So, they form an exciton that's very similar to a hydrogen atom, where we have an electron orbiting a much heavier proton.

Hydrogen comes in different forms: gas, liquid, solid... and at extreme pressures, like in the core of Jupiter, hydrogen becomes metallic. So, we should expect that excitons can come in all these different forms too!

We should be able to create an exciton gas... an exciton liquid... an exciton solid.... and under certain circumstances, a metallic crystal of excitons. Abrikosov called this metallic excitonium.

People have been trying to create this stuff for a long time. Some claim to have succeeded. But a new paper claims to have found something else: a Bose-Einstein condensate of excitons:

There's a pretty good simplified explanation at the University of Illinois website: However, the picture here shows domain walls moving through crystallized excitonium — I think that's different than a Bose-Einstein condensate! I'm a bit confused.

I urge you to look at Abrikosov's paper. It's two pages long and beautiful:

He points out that previous authors had the idea of metallic excitonium. Maybe his new idea was that this might be a superconductor — and that this might explain high-temperature superconductivity. The reason for his guess is that metallic hydrogen, too, is widely suspected to be a superconductor.

Later Abrikosov won the Nobel prize for some other ideas about superconductors. I think I should read more of his papers.

Puzzle 1. If a crystal of excitons conducts electricity, what is actually going on? That is, which electrons are moving around, and how?

This is a fun puzzle because an exciton crystal is a kind of abstract crystal created by the motion of electrons in another, ordinary, crystal.

Puzzle 2. Is it possible to create a hole in excitonium? If so, it possible to create an exciton in excitonium? If so, is it possible to create meta-excitonium: an crystal of excitons in excitonium?

December 18, 2017

What's cooler than a superfluid?

When you cool helium enough, it becomes a superfluid. It can then do amazing things like climb out of a cup, as shown here. What could be cooler than that? A supersolid.

In a superfluid, the atoms become exactly the same in every way. They're not even in different places: they're all spread out everywhere. This lets them move in lock step. The viscosity drops to zero. So a superfluid can do things like climb out of a cup thanks to the tiny attraction it feels to the walls of the cup, and the force of gravity pulling down. Each atom is both inside the cup and outside getting pulled down!

This is only possible thanks to quantum mechanics — and only because the most common form of helium is a boson. Every particle in nature is either a boson or fermion. Two particles that are fermions can't be in the same state. But bosons can. And at low temperatures, identical bosons like to be in the exact same state. This is called a Bose–Einstein condensate.

So what's a supersolid?

When you compress liquid helium enough, it becomes a crystal. But as with many crystals, there will be vacancies: places where an atom is missing.

In an ordinary crystal, the vacancies can move around. But in solid helium, the vacancies are bosons, so it's possible that at low temperatures they will form a superfluid! The result is called a 'supersolid'.

In short: a supersolid is a crystal where vacancies form a Bose–Einstein condensate, allowing them to flow through the crystal with no viscosity. It's like a superfluid formed by the absence of particles, moving like ghosts through a crystal!

Here the white circles represent vacancies. Click for more details!

Now for the complicated part. There have been a lot of arguments about whether helium can form a supersolid. The current consensus seems to be that it can't. However, people claim to have made supersolids using other materials. So the idea is still very interesting.

Here's the story, paraphrased from Wikipedia:

While several experiments yielded negative results, in the 1980s, John Goodkind from UCSD discovered the first 'anomaly' in a solid by using ultrasound. Inspired by his observation, Eun-Seong Kim and Moses Chan at Pennsylvania State University saw phenomena which were interpreted as supersolid behavior. Specifically, they observed an unusual decoupling of the solid helium from a container's walls which could not be explained by classical models but which was consistent with a superfluid-like decoupling of a small percentage of the atoms from the rest of the atoms in the container. If such an interpretation is correct, it would signify the discovery of a new quantum phase of matter.

The experiment of Kim and Chan looked for superflow by means of a "torsional oscillator." To achieve this, a turntable is attached tightly to a spring-loaded spindle; then, instead of rotating at constant speed, the turntable is given an initial motion in one direction. The spring causes the table to oscillate similarly to a balance wheel. A toroid filled with solid helium-4 is attached to the table. The rate of oscillation of the turntable and toroid depend on the amount of solid moving with it. If there is frictionless superfluid inside, then the mass moving with the doughnut is less, and the oscillation will occur at a faster rate. In this way, one can measure the amount of superfluid existing at various temperatures. Kim and Chan found that up to about 2% of the material in the doughnut was superfluid. (Recent experiments have increased the percentage to over 20%). Similar experiments in other laboratories have confirmed these results.

In short, without all jargon: if you have a supersolid, you can twist it back and forth and the liquid formed by the vacancies will not turn back and forth, because it can flow right through the crystal.

But then comes the controversy:

A mysterious feature, not in agreement with the old theories, is that the transition continues to occur at high pressures. High-precision measurements of the melting pressure of helium-4 have not resulted in any observation of a phase transition in the solid.

Prior to 2007, many theorists performed calculations indicating that vacancies cannot exist at zero temperature in solid helium-4. While there is some debate, it seems more doubtful that what the experiments observed was the supersolid state. Indeed, further experimentation, including that by Kim and Chan, has also cast some doubt on the existence of a true supersolid. One experiment found that repeated warming followed by slow cooling of the sample causes the effect to disappear. This annealing process removes flaws in the crystal structure.

Furthermore, most samples of helium-4 contain a small amount of helium-3. When some of this helium-3 is removed, the superfluid transition occurs at a lower temperature, which suggests that the superflow is involved with actual fluid moving along imperfections in the crystal rather than a property of the perfect crystal.

In 2009, it was proposed to realize a supersolid in an optical lattice. Starting from a molecular quantum crystal, supersolidity is induced dynamically as an out-of-equilibrium state. While neighboring molecular wave functions overlap, two bosonic species simultaneously exhibit quasicondensation and long-range solid order, which is stabilized by their mass imbalance. This proposal can be realized in present experiments with bosonic mixtures in an optical lattice that features simple on-site interactions.

Experimental and theoretical work continues in hopes of finally settling the question of the existence of a supersolid.

In 2012, Chan repeated his original experiments with a new apparatus that was designed to eliminate any contribution from elasticity of the helium. In this experiment, Chan and his coauthors found no evidence of supersolidity.

Too bad! But...

In 2017, two research groups from ETH Zurich and from MIT reported on the first creation of a supersolid with ultracold quantum gases. The Zurich group placed a Bose–Einstein condensate inside two optical resonators, which enhanced the atomic interactions until they start to spontaneously crystallize and form a solid that maintains the inherent superfluidity of Bose–Einstein condensates. The MIT group exposed a Bose–Einstein condensate in a double-well potential to light beams that created an effective spin-orbit coupling. The interference between the atoms on the two spin-orbit coupled lattice sites gave rise to a density modulation that establishes a stripe phase with supersolid properties.

In short: there's still hope that people can create supersolids, but it will take more experiments to be sure.

December 30, 2017

In math, all sufficiently beautiful entities are connected to all others. Here's another example. A regular octahedron has 12 edges. A regular icosahedron has 12 corners. So there could be a way to draw the icosahedron with its corners on the edges of the octahedron. And yes — there, is!

But as a final twist of the knife, you don't put the corners at the middle of the edges. That wouldn't work. Instead, each of the corners divides each of the edges according to the golden ratio!

Math just had to do that.

I got this beautiful image here:

Puzzle 1. What shape has 12 corners, with one located exactly in the center of each edge of the regular octahedron?

Puzzle 2. Can you make or find an animated gif of that shape morphing into a regular icosahedron as its corners move from the midpoints of the octahedron edges to the points shown here?

Puzzle 3. How many ways are there to create a regular icosahedron whose corners lie on the edges of a given regular octahedron?

Puzzle 4. How many ways are there to create a regular octahedron edges contain the edges of a given regular icosahedron?

The answers can be found in the comments on my G+ post.

For my January 2017 diary, go here.

© 2017 John Baez