Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. [Emphasis added] The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation.

The full post is here:

I highlighted above a key phrase “If it can find a path to the bottom, you get soaked.” What I didn’t say, but should have is that the water was being forced through the pipes, not just dropping down due to gravity. This is a very important point since our phases and phase transition changes dramatically if we just let gravity do the work. In the case of the water being “forced,” it can travel back up pipes if it helps it find its way out and onto your head, but in the case when only gravity is present, it falls down the pipes. To facilitate gravity, we’ll turn the pipes 45 degrees, and if we insert water at a single point on top, it could look like this:

This setup is a different problem called directed percolation. It also has a phase transition, but one that is different in some fundamental ways from regular percolation.

Before we explore its stranger properties, we can ask, “At what likability threshold do you remain dry?” Well, this happens to have a transition chance of 35.53%!2 This system is a lot more generous, keeping you dry even when a majority of people dislike you. This number comes from numerical computations which have been done rather precisely, and we can even compute it ourselves. In fact, you can see this clearly with this plot

Notice that as we make the system bigger and bigger, the chance of getting soaked less than 35.53% increases and above it, it decreases. This is the same kind of hallmark of a phase transition as we saw in our previous case.

We can also look at the water as it flows down the system to see the clusters that make it from top to bottom

There is still a fractal-looking pattern at the transition point. With all of these similarities with the regular percolation problem from the last post, what is different? And why is that plot so long and skinny? If gravity wants to pull you down, is that somehow altering the motion down, making it distinct from the motion left or right?

Well, if you go back to the two plots above, you’ll notice a few things that really make them differ from the percolation plots. In the fine print of the first, I’ve noted that the vertical distance is L^1.58, so for a horizontal size of 40, the vertical size is roughly 340! That is definitely not a square. And in the second plot, there appears to be more vertical distance than horizontal distance. What is special about this 1.58 number3? It turns out, it’s a critical exponent in this problem, a universal aspect of directed percolation, that distinguishes it from regular percolation. We will call it z = 1.58 the dynamical critical exponent since it is revealed as water flows down in time (dynamically). This dynamical exponent z can reveal itself by looking at these “long and skinny” setups, but be masked by the square setup.

Universality and the finite size of our system

One thing we took away in the previous post was that we lose any sense of scale at this type of phase transition4. But whenever we have “only” thousands of pipes, the size of the system provides a scale! This is the main reason why we begin to see smooth curves and not sharp jumps in quantities. If the system of pipes were infinite (and we had infinite time for the water to go down the pipes), the probability you get soaked would be 100% less than the 35.53% likeability and 0% more than 35.53% likeability. For physical systems, the finite size is often not a huge issue since the scale is closer to the 10²³ atoms present in macroscopic systems, and so even things that are technically smooth curves look very sharp.

The problem of size becomes more severe with directed percolation because horizontal and vertical distances start behaving differently thanks to gravity. In this case, if we lay out our nice grid of 10 × 10, 20 × 20, or 30 × 30, we start to notice that the likeability threshold where you stop getting soaked, seems to depend on the size of the system more than before. In actuality it doesn’t, but for these small sizes, you are definitely getting soaked well into the so-called “Dry Phase” we previously labeled. This is seen in the red curves here where each bigger square has a curve underneath the last:

Gravity has done something to the system. Flowing down is different from flowing left or right. In fact, if we flow down by some amount h and over to the right by some distance w, then at the directed percolation transition point

The amount water flows down is related to how far it flows to the right or left by this weird, fractional power of w. This 1.58 is z, our new dynamical critical exponent, which is a universal feature of directed percolation5. It tells us that if we make a system 30 pipes wide, it should extend roughly 30^1.58 ≈ 216 pipes in height to begin picking out the phase transition effectively. The blue curves in the above plot show this and notice how they all converge on one point; that point is the phase transition. It is revealed by small sizes! To understand why, just think about how the curves are changing as we make the system bigger and bigger.

The red curves will still converge to the phase transition, but it takes larger system sizes for it to reveal itself. This is related to the property that at the phase transition there is no longer a sense of scale, but away from the transition, the vertical scale of clusters could be so large that our puny 60-by-60 grid cannot even begin to reveal it. So if we sit at say a likeability of 0.4 in the 60-by-60 grid, we can say that the vertical size of a typical cluster is most likely more than 60.

A different phase transition but connections to new types of physics

This “gravity mode” for our game show we may call “easy mode” since it requires less of the audience to like you, but the implications here are wide. This type of phase transition has been seen in many kinds of local dynamics where there is a preferred configuration or state. These called an absorbing state phase transitions, and they are a property of certain random dynamical systems. Gravity has provided the distinction here, but more generically, causality and time itself provide that direction, leading to dynamics that obey the same universality as directed percolation.

In our everyday experience of the world, things have precise positions, speeds, and outcomes. You throw a baseball—you know where it’s going. But when we zoom in to the world of atoms and particles, things get weird — and the rules change.

2. The Probabilistic Nature (Uncertainty and Superposition)

🗨️ Metaphor:

"Imagine flipping a coin, while it is spinning in mid-air, it spins in mid-air being both at heads and tails at the same time, with the probability of being heads or tails is still 50-50. At this point, if we want to describe the state of this system (the coin), it would be a combination of both heads and tails — until you look, and then you can say whether the coin landed on heads or tails. That’s how particles behave in the quantum world: they exist in a state made of both heads and tails, a superposition of states, until they’re measured.

🎯 Main idea:

• Quantum Particles don’t have exact positions or velocities—just probabilities.

• Measurement collapses the particle’s wavefunction to a definite value.

Let’s look more closely at the idea that Particles behave probabilistically

In classical mechanics, we think of a particle as a tiny object with a definite position and velocity at any time. But in quantum mechanics, particles like electrons that are described by a wavefunction, a mathematical function that tells you the probability of finding the particle in different places. You can think of the particle not as a dot but as a fuzzy cloud, where he denser the cloud in one spot, the more likely the particle is to be found there.

This is why we say: "Particles don't have exact positions or velocities—just probabilities."

🎵 The Wave Nature of Matter

In our everyday life, we see systems that exhibit wave properties. Things like sound waves, water waves (surface waves), waves on a cable (vibrating), or if you live in certain places, you may experience seismic waves. These are all classical physics examples that are described by wave equations, where the disturbance propagates through a medium or field, transferring energy without necessarily transferring matter.

For example, when waves meet (i.e., waves in water), they combine through a process called interference. This can take a few forms:

· Constructive Interference: When the crests (high points) and troughs (low points) of two waves line up, they reinforce each other, creating a larger wave. Think of two ripples on a pond colliding and forming a bigger splash.

· Destructive Interference: When a crest meets a trough, they cancel out to some extent—sometimes completely—resulting in a smaller or flat wave.

This blending of energy is happening constantly in light, sound, water waves, and even quantum systems.

Below in Figure 1, is an example of superpositions of waves. The top image highlights full constructive interference and the bottom image shows destructive interference. You can see that the maximum of the two waves is 1 and its minimum is -1, where 1 and -1 are called the wave's amplitude. For these two points, for complete constructive interference, the superposition of these waves yields 2 (superposition means at each position point you add the two waves together) for the maximum and -2 for the minimum. For complete destructive interference, you can see the waves when at each point you add them together (superposition), completely cancel out (equal 0). This situation is often called completely out-of-phase . Using the same two points as in our constructive interference example, you now see that wave 1 equals 1 and wave 2 equals -1. In fact, for all the points, the two waves are equal but of opposite sign (meaning one is positive, say +1, and the other is -1). The superposition of these two waves produces 0 for all points.

Below in Figure 2, the waves are slightly shifted along the position axis (x-axis). Using our same points as before, you can see that the superposition wave doesn’t quite equal 2 and -2; they are less than 2 and greater than -2 (-2 is less than -1.9, say, meaning -2 does not get more negative, it is heading upwards towards 0). This is because each wave’s maximum and minimum values occur at different points in space, and this is true for the values of the superposition wave at all points in space. Imagine you fix wave 2, and you slowly pull wave 1 to the right (wave 1 could be referred to as phase-shifted relative to wave 2). The superposition wave continues to have positive values and negative values going towards 0. Once the maximums of wave 1 line up with the minimums of wave 2, the superposition wave is 0 for all points. This is the complete destructive interference as we saw in Figure 1. Now, if you continue to pull wave 1 to the right, the superposition wave starts growing, and if you keep pulling to the right, it will reach the complete constructive interference pattern like in Figure 1.

Notice the superposition wave (like the other waves) starts to repeat the pattern. The point where the pattern repeats itself would define the superposition wave’s wavelength 𝛌. Now imagine, if you had lots of waves where some are shifted relative to our wave 1, at some points in position, we will get a maximum amplitude resulting from constructive interference but necessarily complete constructive interference, giving the highest point of a wave (crest of water wave), while for others, we may get destructive interference, leading to the minimum amplitude (trough of a wave) and other intermediate amplitude that help to make-up the entire wave. Hopefully, this simplistic model helps us to understand how waves form and how you can get a big wave from many small waves.

Another feature of waves is that they have a wavelength that describes how far they propagate in space before repeating the same pattern over and over. If you remember what the mathematical functions, sine and cosine, they are waves that repeat in space and have a wavelength. Now the important part is that the momentum, p, of these waves is inversely proportional to their wavelength, that is, p=1/𝛌. So if you have a short wavelength, you have a large momentum, and vice versa.

These waves follow classical equations — disturbances that move through a medium, transferring energy. But in quantum mechanics, the wave isn't a ripple in water or air — it’s a probability wave.

Now comes the key idea: wave-particle duality. Particles act like waves. And waves behave very differently from particles in one crucial way:

A wave that's localized in space (i.e., sharply peaked in position) must be made by combining many different wavelengths. Think of a big wave in the ocean; it is formed by lots of waves coming together to form this big wave. This combining of waves also means you have a wide range of momenta.

Correspondingly, a wave with a defined momentum (i.e., well-defined momentum) must be spread out in space.

For example, let’s look at music and a pure note on a tuning fork (single frequency = defined momentum) lasts long but is hard to pin down in time (spread out). However, a short drumbeat is localized in time (defined position) but contains a spread of frequencies (momentum uncertainty).

For example, let’s look at music and a pure note on a tuning fork (single frequency = defined momentum) lasts long but is hard to pin down in time (spread out). However, a short drumbeat is localized in time (defined position) but contains a spread of frequencies (momentum uncertainty).

This is a fundamental mathematical property of waves called the Fourier transform. A Fourier transform contains both sine and cosine, just as waves, but is a more complicated function that involves complex numbers. The point about the Fourier transform is that you can obtain sine and cosine from it.

3. The Heisenberg Uncertainty Principle: Knowing Less to Understand More

One of the most famous — and misunderstood — ideas in quantum mechanics is the Heisenberg Uncertainty Principle.

It’s often summed up like this: You can’t know both where something is and how fast it’s moving — at the same time — with perfect precision.

At first glance, that sounds like a problem with our measuring tools, as if we just need better microscopes or sensors. But that’s not it.

This principle isn’t about technological limitations — it’s a fundamental property of nature.

What does it mean?

In classical physics, if you know where a car is and how fast it’s going, you can predict exactly where it’ll be a few seconds later. But in the quantum world, if you try to pin down the position of a particle more precisely, you automatically become less certain about its momentum (its speed and direction) — and vice versa.

It’s not because the particle is misbehaving — it’s because particles aren’t like tiny billiard balls. They behave like waves, and waves don’t have sharp edges.

🌀 Wave Metaphor

Think of a musical note. If a sound wave is spread out in time — like a long, steady tone — it has a very precise frequency (pitch). But if it’s a short, sharp “ping,” its frequency becomes less certain. You trade time for pitch.

In the same way, if a particle’s wave is sharply localized in space (you know where it is), the range of its momentum values must broaden. If the wave is spread out (you don’t know exactly where it is), the momentum is better defined.

🔬 So what’s uncertain?

It’s not that the particle is jittering around randomly. Instead:

• Before measurement, a particle’s position and momentum are both described by a range of probabilities.

• The more tightly you narrow one, the more uncertain the other becomes.

The Heisenberg Uncertainty Principle can be written down as,

𝚫p𝚫x ≤ ℏ/2

• 𝚫x is the uncertainty in position

• 𝚫p is the uncertainty in momentum

• ℏ is Planck’s constant (a very small number)

Let’s try to understand this formula a little better. In quantum mechanics, particles like electrons aren’t just little dots — they also act like waves. This means we describe them with wave packets, which are like short-lived ripples or pulses spread out over space.

To make a wave packet that’s narrow in space (so we know roughly where the particle is), we have to combine many different waves (i.e., sine waves) with various wavelengths and frequencies (think back to our above example of waves).

That’s because a single sine wave, for example, stretches out infinitely — it doesn’t give you a clear position. Only by mixing waves with different wavelengths (and therefore different momenta) can we build a localized bump.

So: Precise position → requires many different wavelengths → high momentum uncertainty.

Now reverse it. If we only use one sine wave, it has a very clear wavelength (momentum), but it stretches out forever — the particle could be anywhere.

So: Precise momentum → means the particle is spread out → high position uncertainty.

This trade-off is at the heart of the uncertainty principle:

𝚫p𝚫x ≤ ℏ/2

Here, 𝚫x is the uncertainty in position, 𝚫p is the uncertainty in momentum, and ℏ is a very tiny constant from quantum physics.

The key message: > The more precisely you know where something is, the less precisely you can know how fast it's going — and vice versa.

Imagine building a short splash on a pond with water waves (see Figure-3):

• A small, sharp splash uses many different ripple sizes (frequencies).

• A pure, smooth ripple has just one frequency but spreads out.

That’s the uncertainty principle in action, hiding in the rhythm of waves.

So what that tells us is that as we become more and more certain about the location of a particle (𝚫x is getting smaller and smaller, heading to 0), 𝚫p is getting larger and larger, heading to ∞. This tells us that if we knew x exactly, then we would not know the momentum p of the particle, since the uncertainty 𝚫p is infinite.

The Core Idea:

You can’t precisely know both where something is (position) and how fast it’s going or in what direction (momentum) at the same time. The more accurately you try to measure one, the fuzzier the other becomes.

🧠 Everyday Analogy:

Imagine you're trying to photograph a speeding car at night.

• If you use a fast shutter, you can see exactly where the car is, but the picture will be blurry — you can’t tell how fast it was going.

• If you use a slow shutter, you get a motion blur — which tells you how fast it was moving, but now you don’t know exactly where it was.

That’s the uncertainty principle in action: precision in one area means fuzziness in the other.

Again, this isn’t just a limitation of our instruments — it's a fundamental property of nature. It's like the universe itself has this built-in fuzziness at tiny scales.

This principle also tells us why electrons just don't spiral into the nucleus of an atom.

Because you can’t precisely know both the position and momentum of a particle at the same time.

If an electron got too close to the nucleus, its position would be very well known (i.e., tightly confined in space). According to the uncertainty principle, this would mean its momentum becomes highly uncertain. Because the kinetic energy is directly calculated from the momentum, and since you have large momentum fluctuations, you will have large kinetic energy.

This tells us that confining the electron too tightly costs energy — a lot of energy. That energy cost balances out the attractive pull of the nucleus. The result? The electron occupies a fuzzy “cloud” of most likely locations (remember it is based on probabilities)— what we call an orbital — and it doesn't just fall in.

This quantum balancing act gives rise to stable atoms, the periodic table, chemistry, etc.

Wave-particle duality

Wave-particle duality is one of the most astonishing ideas in modern physics. It says that tiny things—like electrons and light—can behave like particles and waves, depending on how you look at them.

• Waves (like ocean waves, or ripples in a pond, or even sound waves) are spread out, continuous disturbances. They travel, they can interfere with each other (creating bigger or smaller waves), and they bend around corners. You can't point to "one wave" and say it's at a single, precise location.

• Particles (like a baseball, or a tiny pebble) are distinct, localized objects. They have a definite position, mass, and can be tracked as they move from one point to another.

The Classical Difference: In our ordinary experience, something is clearly either a wave or a particle. Never both.

🌍 In the Classical World

In everyday experience:

• Objects are either particles (like baseballs) or waves (like sound or water ripples).

• Particles have defined positions and travel along clear paths.

• Waves are spread out, overlap, and interfere, but they don't "exist" in a single spot.

Think of throwing a rock into a pond—either you're dealing with the rock or the ripples it creates, never both at once.

⚛️ In the Quantum World

The Quantum Twist: Wave-Particle Duality

But when we zoom down to the incredibly tiny, fundamental level of reality – the quantum realm – things get weird. Particles like electrons, and even light itself (which we classically considered a wave), don't always fit neatly into one category. This is wave-particle duality:

• Light, for instance, can behave like a spread-out wave (which is why it can create interference patterns, just like water waves). But it can also act like a stream of tiny, discrete particles called photons (which is how it knocks electrons off a metal surface in the photoelectric effect, acting like tiny billiard balls).

• Similarly, electrons (which we think of as particles making up atoms) can, under certain experimental conditions, exhibit wave-like behavior, creating interference patterns as if they were spread out and passing through multiple places at once. Yet, when we try to pinpoint their location, they act like a localized particle.

This means a single electron, shot toward a double slit, doesn't just go through one slit—it behaves as if it explores all possibilities at once, producing an interference pattern typical of waves.

🤔 So What Does This Mean?

The amazing part is that a quantum entity isn't just sometimes a wave and sometimes a particle. Instead, it possesses both wave-like and particle-like properties simultaneously, and the act of observation or the type of experiment we perform determines which aspects we will observe. You can't observe both characteristics at the same exact time in the same experiment.

This seemingly paradoxical idea is a cornerstone of quantum mechanics and is absolutely essential for understanding how the universe works at its most fundamental level. It underpins all modern technologies from lasers and transistors to medical imaging and the very concept of quantum computing.

The objects aren't just "here or there"—they are probabilistic ripples, until observed.

Wave-particle duality is nature’s way of whispering: “The world is more nuanced than it seems.”

What is the Percolating Phase?

Imagine you’re on a game show called Are You Likeable? (the least-likeable game show). The rules of the game are simple:

1. You stand on the stage and try to win over the audience.

2. Each audience member votes whether they like you or not.

3. But the twist: votes aren’t tallied—they control a system of pipes above your head.

That system of pipes looks something like this2

Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead3. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation4.

But where is the phase transition?

Aside from asking when my game show will be green-lit5, we can ask: When are you most likely to get wet? If the audience is huge, and fewer than 50% of them like you, it’s nearly guaranteed that the water will find a path—you’ll be soaked. But if more than 50% like you, chances are good that you’ll stay dry.

This is a phase transition6: a sudden change in the system’s behavior. For a moderately sized audience, it looks something like this:

Your probability of getting soaked forms a curve that sharpens up with increasing audience size—becoming a near step at 50%. That is known as the percolation threshold.

This is hard to visualize, though; luckily this problem admits some very nice pictures. For instance, here is the problem with a large number of pipes:

If a pipe is blue, water is in it, and all the blue clusters flow down from the top. Notice what happens around 50%: even though spigots are randomly being turned off, the flow from top to bottom is entirely stopped around this value. Something is happening within the system that is allowing water to pass through on one side of the transition and not the other.

A closer look at the transition

To dig deeper, we simulate what happens at this threshold. Each spigot is either open or closed (randomly determined). If we visualize the grid (say 1024×1024 spigots), it looks like visual static: black and white dots with no obvious pattern7:

But now let’s color each connected cluster of open spigots—where water could flow and fill up this section of pipes. Suddenly, structure emerges. Some clusters are small, others large. If one spans from top to bottom, water flows and we’re in the percolating phase. If not, we’re in the non-percolating phase. At the transition (within the static above, we get this for the twelve largest clusters:

At exactly the percolation threshold (50% for the pipes above), there’s no single dominant cluster, but also no clear typical size. Instead, there’s a wide distribution of cluster sizes. The critical state behaves differently than either phase.

Scale Invariance: A Hallmark of Criticality

Let’s zoom out. Suppose we double the grid to 2048×2048.

The largest clusters are definitely larger—the largest here is 400,000 pipes/spigots large, while for the previous 1024×1024 case the largest was 180,000 large—but the pattern still looks… the same. We doubled the size, but if we slightly blur our vision, we cannot distinguish these two plots (even though one is quadruple the area of the other). Look at 512×512—even that looks similar:

You would be hard-pressed to say which one is larger if you blurred your eyes. This problem is apparent even down to 256×256 or 128×128.

This is called scale invariance—there is no characteristic length scale at the phase transition. It’s one of the defining features of what are known as second-order phase transitions.

It also explains why, at the threshold, you have a 50% chance of getting soaked. The largest cluster might span the system, but it might just as well fall short. There’s no guarantee either way.

Fractals in the Flow

These clusters within the above pictures don’t look like regular 2D structures or even 1D lines; they are, in fact, fractals. They aren’t exactly self-similar but they do behave the same at different scales. They fill space with a fractal dimension: not quite 1D, not quite 2D. In two-dimensional percolation, the clusters have a dimension of 91/48 ≈ 1.896—a universal number shared by all systems in this class, regardless of lattice type or other microscopic details.

This is part of the beauty of percolation: It shows us visually the underlying mathematical structure and even reveals some universality of phase transitions.

Why This Matters

Percolation is just one example, but it captures the essence of what physicists mean when they talk about “phases of matter.” It isn’t always about exotic particles or extreme temperatures turning gas into plasma. Sometimes, it’s about whether a liquid can find its way through a series of pipes. It’s about symmetry, structure, and emergence.

You’ve experienced water’s phases: ice, liquid, steam. But nature offers many more—some with no neat label like “solid” or “gas.” The theory of phase transitions explains them. Percolation is a window into that wider world.

In 2018, everything changed when two layers of graphene were twisted to reveal superconductivity! The twist itself is interesting (I briefly discussed it in a previous post), but the key takeaway is that these materials now come with an extra knob for accessing new phases of matter. It’s remarkable. We can first think of these materials like Lego blocks:

Each layer is a different material: mix and match, and you might discover an exotic new phase. This “Lego” idea had already been in the air before 2018, but the physics since then has shown that it’s not just about stacking—we can twist too, creating not just patterns, but new ways for electrons to move.

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We knew these patterns would occur, but we didn’t realize we could make it superconduct. Now we can stack and twist to great effect. Of course, twisted bilayer graphene isn’t about to revolutionize high-speed trains (it goes superconducting at only 4K1), but the way it goes superconducting is eerily reminiscent of higher-temperature superconductors. That means it might help us understand those other materials better.

And once people started twisting, they didn’t stop. We now have twisted multilayers of graphene, transition-metal dichalcogenide (TMD) bilayers2, and more. But it doesn’t end there; you can also apply magnetic fields, electric fields, and pattern the lattice in sophisticated ways. With all that in mind, here’s a short and incomplete survey of some of the exotic phases in these materials:

“Fractional… what now?”

All of these phases are exceptionally hard to understand and model. Some of the best minds in the field are actively working on them. One particularly exciting phase is the fractional Chern insulator, which could be useful for quantum computing.

But even setting aside applications, what’s astonishing is that all of these phenomena come from nothing more than electrons moving on a lattice and experiencing a few fields. Nature seems to treat electrons like Play-Doh, shaping them into wildly different quantum phases.

This is a deep and fundamental question: What can be accomplished using electrons alone?

\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.

First, let me tell you about myself and my expertise. I have a Ph.D. in theoretical condensed matter physics, which is the study of solids, surfaces, interfaces, and liquids. This also includes disordered systems. I have also worked in catalysis and energy storage. I have been doing active research with my first published article in 1987 to the present, which is about 38 years of active work. I have worked at two Department of Energy National Laboratories, Oak Ridge National Laboratory and Pacific Northwest National Laboratory. I am currently a full professor of Physics at Louisiana State University, where I have been for the last twelve years. I currently have 3 funded research projects, one on “Improving Transmon Qubit Performance,” the second is on “Directed Assembly of Metastable States for Harnessing Quantum Effects,” and the third is on “Enabling Formate-Based Hydrogen Storage and Generation via Multimetallic Alloy Catalysts.”

Let me start by saying this: I don’t think science is broken. In fact, science needs new ideas to survive. That’s what keeps it alive. It’s not about guarding the past—it’s about exploring what comes next.

So, What Is Science Anyway?

At its heart, science is just a way of asking questions and trying to find honest answers. There’s a process to it, and it goes something like this:

1. You notice something interesting.

2. You ask a question about it.

3. You come up with a possible explanation—what we call a hypothesis.

4. You test that idea through experiments, models, or observation.

5. You look at the results and ask, “Was I right?”

6. Whether the answer is yes or no, you learn something—and you keep going.

7. Finally, you share what you found, so others can learn from it too.

That’s it. And we repeat this process again and again. If the answer turns out to be wrong, that’s still progress. It tells us what doesn’t work, which is just as important as figuring out what does.

Let’s Pick a Scientific Process Example from our previous blog post on Quantum materials: The Quantum Topological Material Bi₂Se₃

🔍 Observation

Some newly discovered materials—like bismuth selenide (Bi₂Se₃)—conduct electricity on their surfaces while remaining insulating inside. This unusual behavior hints at a new phase of matter.

❓ Question

Why do these materials conduct only on their surfaces? What quantum mechanical principles govern this behavior? Can this property be harnessed for new electronic or quantum computing applications?

💡 Hypothesis

These materials are topological insulators. Their unique surface conductivity arises from strong spin-orbit coupling, which protects surface states from scattering due to defects or impurities. These protected states are a result of the material’s non-trivial topological order.

⚗️ Experiment

Researchers test the hypothesis by:

• Synthesizing high-quality crystals of Bi₂Se₃.

• Measuring surface conductivity via scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) for directly observing the electronic states.

• Applying magnetic fields and introducing defects to see if the surface states remain intact.

• Using transport measurements to compare the bulk and surface contributions to conductivity.

📊 Analysis

• ARPES data shows Dirac-like surface states.

• STM confirms surface conduction pathways even when the bulk is insulating.

• Magnetic fields break time-reversal symmetry, gapping the surface states—validating the topological protection mechanism.

✅ Conclusion

The data confirm that Bi₂Se₃ is a 3D topological insulator. Its conductive surface states are protected by time-reversal symmetry, making it a strong candidate for spintronic devices and fault-tolerant quantum computing.

📢 Communication

Findings are published in Nature Materials and Science, presented at condensed matter physics conferences, and used to inspire further research into topological superconductors, Majorana fermions, and quantum information systems.

🔍 Why It Matters

This discovery has opened up new paths in quantum electronics, where data can flow with minimal energy loss, and in quantum computing, where these materials could enable robust, error-resistant qubits.

Why This Matters

This is how new medications move from lab benches to pharmacy shelves. The scientific process ensures treatments are both safe and effective before reaching the public.

But Is Science Open to New Ideas?

Yes, absolutely—but it’s not always easy. Getting a scientific paper published takes a lot of work. Top-tier journals reject around 80 to 95% of the papers they receive. Even solid mid-tier journals reject more than half. Why? Because these journals want well-supported, clearly written work that pushes knowledge forward. Lastly, here is an interesting statistic, do you know that that roughly 1% of the scientific workforce publishes every single year. That’s a small fraction maintaining a consistent online publishing presence. It is truly difficult getting articles published.

That doesn’t mean new or unusual ideas get shut out. They just have to be backed up with solid evidence—and explained clearly. I’ve seen good ideas get rejected just because the writing was hard to follow, or the data didn’t quite hold up. That’s why I often help younger researchers refine their papers. Heck, I’ve had others help me, too. We’re all learning as we go.

And yes, rejection stings. It really does. But it’s not about ego—it’s about making the work stronger.

Here’s the Bottom Line

Science is hard. But it’s supposed to be. We ask tough questions and hold ourselves to high standards. That’s how we make sure the answers we get are worth something.

Is it perfect? No. But the system is built to self-correct. Bad ideas eventually fall away. Good ones rise to the top—even if it takes a few tries.

Science isn’t just a pile of facts. It’s a conversation—a messy, exciting, and sometimes frustrating conversation about how the world works. And yes, it still works.

So the next time someone says science is too closed off, remind them: science is open—to anyone willing to do the work, ask the hard questions, and follow the evidence wherever it leads.

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!

As we demand more power, faster performance, and smaller devices, silicon is running into serious challenges—slower electron speeds, overheating, and physical limits to how small we can make transistors. Enter: quantum materials.

These are next-generation materials with properties that go far beyond what traditional silicon can offer. They’re not just incremental improvements—they represent a fundamental shift in how we think about electronics, computing, and even national defense.

The Rising Stars of Quantum Materials

Let’s break down a few of the most promising candidates:

• Gallium Nitride (GaN) – GaN has high electron mobility and a wide bandgap, which means it can handle higher voltages and switch faster than silicon. It’s already making waves in 5G technology, electric vehicles, and advanced radar systems.

• Silicon Carbide (SiC) – Known for its durability in extreme environments, SiC is great for high-power electronics. It keeps running in high temperatures while minimizing energy loss, making it ideal for heavy-duty industrial and defense applications.

• Graphene & 2D Materials – Think of materials like molybdenum disulfide (MoS₂) and tungsten diselenide (WSe₂). These ultra-thin, ultra-conductive materials could lead to super-efficient, flexible transistors and wearables.

• Topological Insulators – These are the real game-changers. They allow electricity to flow on their surfaces while remaining insulating inside. Even better? Their surface states are resistant to impurities and defects, making them incredibly reliable. This makes them strong contenders for spintronics (electronics that use electron spin) and quantum computing.

Why Silicon Can’t Keep Up

Here’s the problem: silicon has limitations we can’t engineer our way around anymore.

1. Slower switching speeds – Compared to newer materials like GaN, silicon just isn’t as fast.

2. Overheating – As we pack more transistors into chips, heat buildup becomes a serious issue, requiring costly cooling systems.

3. Miniaturization limits – We’re approaching atomic scales, and quantum effects are interfering with performance.

4. Power inefficiency – Silicon struggles in high-power applications like data centers, defense tech, and electric transportation.

In short, we need something better—and quantum materials are stepping up.

Powering the Future: Quantum, Secure, and Ultra-Precise

One of the most exciting classes of these materials is quantum topological materials. These materials don’t just offer better performance—they behave in fundamentally different ways, thanks to their quantum properties.

• Topological Protection – Their electronic states are robust against defects, making devices more reliable.

• High-speed, low-power performance – Their exotic structures allow electrons to move faster with less energy loss.

• Quantum Stability – In quantum computing, they help build stable qubits—units of quantum information that don’t fall apart under noise, heat, or time.

These properties make them ideal for:

• Quantum Computers – Topological superconductors could unlock fault-tolerant quantum computing by protecting qubits from errors.

• Quantum Sensors – These materials are incredibly sensitive to their environment, opening the door to ultra-precise sensors for medical imaging, navigation, and even detecting gravitational waves.

• National Security & Defense – From secure communications to subsurface detection and stealth-defeating tech, the strategic applications are massive.

• Quantum Metrology – Think of this as the science of ultra-precise measurement. It’s the backbone of GPS, secure transactions, and global communications.

The Big Picture

Quantum materials—and especially topological quantum materials—represent more than a new tool in the electronics toolbox. They’re reshaping what’s possible in technology, science, and security.

With their unique electronic structures, extreme sensitivity, and resistance to defects, these materials could power everything from next-gen smartphones to quantum computers, while enabling breakthroughs in physics and transforming how we interact with the world.

The future of electronics isn’t just smaller and faster—it’s quantum.

It feels tone-deaf to launch into pure science in this Substack when we’ve had one of the largest assaults on science in my lifetime. For instance, just yesterday the administration moved to restrict entry to the US for international students going to Harvard, directly following a broader travel ban affecting our department. At the same time, I don’t want to turn this into a polemic. So, I won’t. Instead, I’ll tell you about what people are feeling. If you want to know more, Doug Natelson’s blog Nanoscale Views (a Rice Professor closely following developments at NSF), reporting from the New York Times, and the reporting in Science magazine are all good sources.

One prominent feeling is anger: fury toward travel bans, the detaining of international researchers, canceled funding, and rhetoric promising more. As stated by our vice president: “Universities in our country are fundamentally corrupt and dedicated to deceit and lies, not to the truth. […] They pursue deceit and lies.”1 This denigration and these attacks create a climate of fear, especially for those who are most vulnerable: the students.

When I decided to pursue a graduate degree in physics, I did so because I wanted to understand how the universe works in the language of physics and mathematics. Graduate school is stressful, and often, the introduction to research can be a sharp transition for people; suddenly, the problems may not have solutions, or they might be way more complicated than you (or your advisor) anticipated. Our country has been known for its research culture and its excellent universities worldwide, and that has attracted, to the US, talent from across the globe. I personally know scientists from all continents which I met while they were attending or working at US institutions. Even before these recent policies, getting a visa was incredibly stressful for international researchers. Now, they are afraid to exercise their First Amendment rights2 and anxious about leaving and re-entering the US. Uncertain visa conditions disrupt their lives, hindering their ability to travel, share results, and fully participate in science.

There is also concern among faculty that the National Science Foundation will no longer be a reliable source of funding if the presidential budget is adopted. Already, divisions are being abolished or restructuring to focus on funding research in artificial intelligence, quantum information science, biotechnology, nuclear energy, or translational science3. Research into materials and high-energy physics could come to a sudden halt. This has many rattled, and we fund our research groups with federal research grants. If we lose that, we lose the ability to fund our students and postdocs4. Even temporary interruptions have this human cost in addition to devastating entire projects.

We hear all the moves against us at universities5, and it hits, creating an atmosphere of anger and fear. I am sad and disappointed as well. Countries are trying to entice away US scientists. France and the EU are already allocating funds and are already hiring US scientists. China directly offered a deal to a Nobel laureate affected by US cuts (he declined). I have heard genuine sentiments from people considering leaving the US. After elections, some might half-jokingly talk about leaving. But this time, it’s different. Researchers fear losing their careers and see moving as a real option to continue innovating. And how can the US compete with any other country if we lose our researchers? We won’t develop the next touch screens and lithium-ion batteries, nor find the next cure for tuberculosis, innovations born at US universities.

I hope those in charge walk us back from the edge and support the center of science in the world: US universities and the people who work here. In the meantime, call your representatives and senators. Tell them clearly: cutting NSF funding directly harms America’s scientific leadership. The American Physical Society’s advocacy webpage can guide you. Pay attention locally, too, as some states (like Indiana) are considering similar harmful measures.

Scientists, professors, graduate students, and researchers are not, in any sense, the enemy. We are part of what makes America great.

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There’s a lot of chaos in the world. No, really—both literally and figuratively. In the news, as well as in my home department at LSU, I hear about uncertainty and chaos with student visas, science funding cuts, and more. I also study chaos in my research and teach physics. This past spring (2025), I taught a course on Quantum Technology. I’ve gathered a lot through all of this, so here I am, on Substack.

My job involves a lot of writing—papers, grants, proposals (for all sorts of things), referee reports, and endless notes. Still, I find myself wanting to practice writing, from technical to conversational. Partly, it’s to build writing confidence and momentum, so that the next big grant doesn’t feel quite so intimidating.

What you’ll find on here is a bit eclectic.

My interests run wide. Even within theoretical physics, my curiosity roams. I’ve studied, among other things, the Casimir effect, topological materials, analog gravity, quantum field theory in curved space, quantum information dynamics, and moiré materials. I’ve got thoughts on all of it, and I plan to share them. Not only that, but other sciences, philosophies, and ideas aren’t safe either.

To help categorize this nonsense, I plan on releasing four types of posts:

• The Observable — The main column with in-depth takes on subjects I (hopefully) know something about.

• Quantum Bites — These will be short explanations of some concept in physics for the general reader. (Yes, I had to restrain myself from calling them Quantum Bytes.)

• The Margins — Where I scribble thoughts on things outside of my domain: current events, philosophy, other sciences.

• Field Notes — Technical dispatches in physics or mathematics, aimed at experts and soon-to-be experts.

Naturally, this will evolve as I go.

The timing will be weekly, with a longer main piece each month.

Every day, I try to write something. If it’s not directly work-related (papers and grants and reports—oh my!), it’s notes to myself or, now, something for this Substack.

That said, I write slowly and thoughtfully, often rewriting until it feels just right. In this way, I expect to write roughly three shorter pieces and one longer piece in a month. What counts as “short”? We’ll figure that out together. As for how much of each type—Margins, Field notes, Bites—well, who knows?

These updates will come on Thursday evening.

But why does Quantum Matter?

I have answered Why Quantum Matters—but not why quantum matters. Simply put, quantum mechanics, so far as we know, is the underlying framework for every physical thing in the universe. It’s passed test after test, even when it upends our intuitions (ever hear of quantum entanglement?).

You can find plenty of quotes about God and dice, spooky action at a distance, and all the interpretation drama. Honestly, I hate1 this PR quantum has accumulated over the last hundred years. Takes like the Copenhagen interpretation have warped how quantum science is perceived, and it doesn’t help when physicists shrug and say “shut up and calculate”2.

I want to unpack the truly bizarre parts and show how, with a little work, they can still make sense. And quantum isn’t just about the “very small”; it shapes technologies we use every day3. To get there, we’ll need to understand how classical physics emerges from all the quantum funk.

We’ve figured out a lot about the world, but there’s still so much more to learn. Sharing insights and participating in this grand human experiment of science and progress matters. I hope to share what I’ve picked up along the way, from the giants who’ve shaped my thinking about physics, and maybe a little about life too. Come along for the ride.

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