\We all know about solids, liquids, and gases from school. Heat up ice, and you get water; heat up water, and you get vapor. We may even have been slightly baffled if we saw this phase diagram with “pressure” added to the mix

I see here a solid phase, a liquid phase, and a gas phase, but what is this “Critical point”? If you tune your temperature and pressure just right you can smoothly cross over from liquid to gas without ever undergoing a phase transition. Without getting into the molecular details, we can think of phases as particular valleys between mountains, and water wants to reach the absolute lowest point. Sometimes there are two valleys, but one is lower, and sometimes there is just one valley.

In fact, this “number of valleys” is why we see this odd behavior. If we sit at 100 degrees C and decrease or increase the pressure, there are two energy minima2—two valleys. At small pressure, the deepest valley is on the gas side, and at large pressure, the deepest valley is on the liquid side. As you then tune pressure across that one-bar point, one valley gets deeper than the other—it’s the true minimum! Yet, to get from one valley to the next, you need some energy to get you over that mountain in between. That’s the phase transition. However, that's not the only option. As the temperature increases, the mountain in between gets smaller and smaller until, at the critical point, it finally disappears, and the two valleys merge.

Without two distinguished valleys, there is no need to scale the mountain and no need for a phase transition. Liquid smoothly and easily becomes gas. At the temperatures above the critical point, you cannot meaningfully distinguish water and gas. OK, so perhaps we only have two phases?

Not quite; look at this more fleshed-out version of the phase diagram:

When ice forms, it adopts a low-energy crystal structure. However, there are numerous crystal structures to choose from. In fact, as you change pressure and temperature, it can completely reorganize how the ice bonds together into a crystal. This leads to over 20 phases of ice, labeled by some of the Roman numerals above.3

Then what are the phases? Solids undergo their own phase transitions—structural phase transitions. Are these not phases of matter? If they are, then we have already exceeded our three phases of matter just within water. But phases go beyond temperature and pressure. They also possess a multitude of interesting properties, particularly at that critical point. We'll cover some of that in detail next week.

Quasicrystals, a beautiful manifestation of something without a strict crystalline symmetry but nonetheless shows order, won a Nobel prize in 2011. A bit more recently, in 2018, a dodecagonal graphene quasicrystal (two sheets of graphene twisted 30 degrees with respect to each other) made its way onto the cover of Science1.

This was just the beginning as well, there have been other instances where experimentalists take two layers of atomically thin materials (like graphene, a single layer of carbon) and obtain patterns that are quasicrytalline. One comes from some Rutgers experimentalists involving two layers of graphene and hexagonal boron nitride2.

I’ve been exploring (with collaborators) how this kind of phenomena could help or impede some interesting effects like superconductivity, but that’s a story for another day.

This phenomena inspired me to make what is known as a Penrose tiling so you can see how twisting two layers of graphene at 30 degrees with respect to each leads to a quasicrystal. (Individually, each is a tiling of “hexagons.”) Here it is partially filled up:

One can tell how this is done: You find the points where two hexagons are on top of each other, put down a point, and connect. There are three shapes: a rhombus, an equilateral triangle, and a square. This can be done along the entire sheet to create an amazing looking pattern. For completeness, we can fill in the rest of the pictured grid to obtain:

The pattern starts to look even more intriguing the further out in the tiling you go. There is much to learn about such physical systems and their quasiperiodic cousins.

The idea came from classical chaos and control, and is pretty simple1. Imagine standing in a chaotic wilderness dotted with a few tiny, perfectly calm mountaintops. Place a boulder on one of those peaks and it can sit there forever—but miss the summit by a hair and it tumbles into the valley below. We want to escape the chaos of avalanches and rock slides, so our task is to sit a boulder up on the top of a mountain.

Now suppose you randomly stop paying attention. Every so often (flip a coin, roll a die), you look up and find the boulder, and nudge it some fraction of the way back up the mountain. This sounds thoughtless—counterintuitive even—but leads us to an interesting realization: There is a critical “nudge-rate” that will get our boulder up the mountain. However, any less often than this value, the boulder is lost to the valleys below. This is a classical phase transition in disguise.

We extended the idea to a quantum-mechanical boulder, and quantum fuzziness changes everything. These types of dynamics have little points of stability classically, but when you look at them quantum mechanically, there is no way to park a boulder at these points. Quantum mechanics has this bad habit of blurring those otherwise stable points.

This is really a battle with quantum uncertainty. If we put our quantum boulder at the top of the hill it is stopped (zero velocity) and we know where it is, but this violates the uncertainty principle! If we know where it is, we should have no idea how fast it’s moving! Even if you get a quantum boulder to the top of the mountain, it will always fall back down.

This alters our whole conception of “control” from before2, but miraculously, there is still a phase transition, but it behaves differently. Whenever we push our quantum boulder up the hill at just the right rate, it still never reaches the top, but it starts to be up there a lot more often than normal. Formally, the probability P(h) that the boulder sits at a height h takes the schematic form

(This looks like it diverges at the summit but that’s the secret sauce of quantum uncertainty, this denominator never actually gets to zero.)

The result is the following plot from the preprint:

The horizontal access is the “nudge-rate” p and the y-axis represents (roughly) how often the boulder is at the top of the hill. Classically, you’d get perfect control for p > 0.5, but quantumly, at p = 0.5, you can see that above that, it’s not at the summit a lot, but we’ve reached a threshold. Even at p = 0.8, you “only” hold the peak about 80% of the time.

So we trade absolute control to something a bit more “uncertain.” We now get to the top more often than not since quantum mechanics tends to get in the way and push us down the side of the mountain again.

Behind the scenes are simulations, analytic calculations, and connections to random walks, turbulence, and even market dynamics, all of which justify the “Universality” in the title. If you’d like the technical story—including how weak measurements implement the quantum mechanical “nudge”—check out our preprint!