What is condensed matter theory, and why is it so hard?

This post is about the general approach to physics that people who work on the theory of condensed matter take. As I’ll explain, it is basically impossible to calculate anything exactly, and so the whole field relies on choosing smart approximations that allow you to make some progress. Exactly what kind of approximations you make depends on what you want to achieve, and I’ll describe some of the common ones below.

But before that, what is ‘condensed matter physics’? Roughly speaking, it refers to anything that is a solid or a liquid (and also some gases) that you can see in the Real World around you. So it’s not stars and galaxies and space exploration, it’s not tiny sub-atomic particles like quarks and higgs bosons like they talk about at CERN, and it’s not what happened in the first fractions of a second after the Big Bang, or what it’s like inside a black hole. But it is about the materials that make the chip inside your phone or power a laser, or about making batteries store energy more efficiently, or finding new catalysts that make industrial chemical production cheaper (okay, so that one crosses into chemistry as well, but the lines are fuzzy!), or its about making superconductivity work at higher temperatures.

Why is it so hard?

Really, what makes condensed matter physics different from many other types of physics is that in many situations, the behaviour of the materials is governed by how many many particles interact with each other. Think about a small piece of metal for instance: You have millions and millions of atoms that form some bonds which give it a solid shape. Then some of the electrons in those atoms disassociate themselves and become a bit like a liquid that can move around inside the metal and conduct electricity or heat, or make the metal magnetic. In a small piece of metal there will be $10^22$ atoms. (That notation means that the number is a one with twenty two zeroes after it. So it’s a lot.) And all of these atoms have an electric field which is felt by all the other atoms so that they all interact with each other. It is, in principle, possible to write down some equations which describe this, but there is no way that anyone can make a solution for these equations and work out exactly how all these atoms and electrons behave. I don’t just mean that it’s very difficult, I mean that it is mathematically proven to be impossible!

Watcha gunna do?

So, that begs the question what can we do? It is easy to connect a bit of metal to a battery and a current meter and see that it can conduct electricity, but how do we describe that theoretically? There are several different approaches to making the approximations needed, so I’ll try to explain them now.

1. Use symmetry. By the magic of mathematics, the equations can often be simplified if you know something about the symmetry of the material you want to investigate. For example, the atoms in many metals sort themselves into a crystal lattice of repeated cubes. Group theory can then be used to reduce the complexity of the equations in a very helpful way. For instance, it might be possible to tell whether a material will conduct electricity or not even at this level of approximation. But this symmetry analysis contains an assumption because in reality materials won’t completely conform to the symmetry. They may have impurities in them, or the crystal structure might have irregularities, for example. So this isn’t a magic bullet. And also this might well not reduce the equations enough that they can be solved, so it is usually just a first step.

2. From this point, it is often possible to make simplifying assumptions so that the mathematically impossible theory becomes something that can be solved. Of course, by doing this you lose quite a lot of detail. It’s like the “spherical cows” analogy. In principle, cows have four legs, a tail, a head, and maybe some udders. But say you wanted to work out how many cows you could safely fit into a field. You don’t need to know any of that detail, so you can think of the cows as being a sphere which consumes a certain amount of hay each day. You can do something similar about the metal: Instead of keeping track of every detail, you can forget that the atoms have an internal structure (spherical atoms!). Or you could assume that the atoms interact with the electrons in a particularly simple way so that you can focus just on the disassociated electrons. Or you could assume that the electrons don’t interact with each other, but only with the atoms. In the jargon of the field, this general approach is called finding an “effective theory”. These theories can often give quite good estimates of not only whether a material will conduct, but how well it will do it.

3. These days, computers are really fast, and they can be used to numerically solve equations that are almost exact. However, computers are not good enough that they can do this for $10^22$ atoms, so if you want to keep quite close to the original equations, they might be able to do fifty or so. Maybe a hundred. In the jargon, these methods are called “ab-initio” (from the beginning) because they do not make any approximations unless they absolutely have to. The fact that you can’t treat too many atoms limits what these methods can be applied to. For instance, they can be quite good for molecules, and crystals where the periodic repetition is not too complicated. But for these situations, you can get a level of detail which is simply impossible in the effective theories. So there’s a trade-off. And computers are getting better all the time so this is one area that will see a lot of progress.
4. The final way that I’ll describe is sort-of the inverse process. Instead of starting from the mathematics which are impossible, you can start from experimental data and try to work backwards towards the theoretical description that gives you the right answer. Sometimes this is used in conjunction with one of the other methods as a way to give you some clues about what assumptions to make.

So, that’s how you do theory in condensed matter. Numbers 2 and 4 are basically my day job, on a good day at least!

Particle-wave duality and the two slit experiment

Particle-wave duality is the concept in quantum mechanics that small objects simultaneously behave a bit like particles and a bit like waves. This comes very naturally from the mathematics, but instead of talking about those boring details, I’m going to describe a famous experiment that proves it.

Diffraction

It’s called the two slit experiment, and I’ve sketched how it works in the picture on the right. Before going into the full details, let’s look at the upper part of the picture. This shows a light wave shining on a barrier with a small slit in it. The thin black lines show the position of the peaks of the wave that describes the traveling light. Some of the light can get through that slit, but in doing so, it changes its form to become a circular wave with the slit at its source. This is called diffraction, and leads to a distinctive pattern when the light hits a screen placed some way behind the barrier. The red line behind the barrier shows the intensity of the light hitting the screen. This demonstrates that light can behave in a wave-like way because if the light was just particles you would not see the diffraction pattern, but there would be a small spot of light on the screen in line with the slit.

Now look at the lower part of the picture. Now the screen has been replaced with a second barrier that has two slits in it. Both of these slits act like the first one: they diffract the light that is coming through. So behind the second barrier, there are now two waves of light, one coming from each slit. These two waves interfere with each other, so that the pattern of light seen on the screen (the red line) looks very different from that made by just one slit. (I did actually calculate what the light should look like before I drew these pictures, so I hope both of the red lines are actually correct!) Interference is the process of these wave adding together to form one single pattern. The value of a light wave at a particular position can be either positive or negative. In the picture, the thin black lines show where the waves are at their maximum – so where they are their most positive. Exactly half-way between a pair of lines they are at their most negative. If the two waves are both positive at a particular position (like exactly at the center of the screen) then they add together to give intense light. But if one is positive and one is negative then they will cancel each other out and leave almost no light.

Electrons

That’s not very controversial. But it starts to get a bit more weird when you repeat the same experiment but using a beam of electrons instead of a beam of light. Electrons are one of the three types of “particle” which make up an atom: The protons and neutrons bind together to form the nucleus, and then electrons “orbit” around it. Until this experiment was done for the first time, most physicists thought that electrons were particles. But the result of the experiment was the same kind of two-slit diffraction pattern that they got when they used light. The electrons that went through each of the slits were interfering with each other just like the light waves did. The only possible conclusion: these electrons were also wave-like.

Then, they pushed the experiment a bit further. They had the same barriers, but instead of using a beam of electrons, they fired them through one at a time. Astonishingly, even though there was only one electron, the result was still a two-slit diffraction pattern. Somehow, the electron was going through both slits and interfering with itself. Conclusion: Electrons are not just wave-like when there are lots of them, they are wave-like on their own!

Now it gets weird

To try and verify this, they modified their apparatus to include detectors at both of the slits so they could tell which slit the electron was going though. Expecting to find a signal from both detectors, they were surprised to find that only one of the detectors sensed an electron going though, and instead of the two-slit diffraction pattern, they now saw a one-slit pattern on the screen. If they did the experiment with the detectors turned off, the two-slit diffraction pattern reappeared. It seemed like asking the electron which slit it had gone through forced it to choose one or the other. But get this: The experimentalists got sneaky. They took the electron detectors away and instead made slits that could be opened and closed very quickly. Starting with both slits open, they fired one electron from the gun. After it had passed the barrier with the two slits, but before it reached the screen, they closed one of the slits. Any guesses as to what pattern was measured on the screen?

They saw a single-slit diffraction pattern! Somehow, the electron knew that one of the slits had been closed after it went through, and behaved like only the other one had been open the whole time. This hints at many deep issues about quantum measurement and (gulp!) the nature of reality itself. But I’ll save that discussion for another time.

This experiment has been repeated with many different objects used instead of the light or electrons. Protons, whole atoms, and buckyballs all show the same behavior, so this is without doubt a general feature in quantum mechanics and not something oddly specific to light and electrons. In fact, once you allow for the possibility of wave-like particles, you start to see the effects of them in many places, including in the behavior of electrons in the materials which make computer chips and all the rest of information technology. So it’s a pretty big deal.

And finally…

One final point of detail which I think is worth pointing out. In the first paragraph, I mentioned that “small objects” are needed to do this experiment. But what does “small” mean in this context? It turns out, this can be written down in a really simple equation. The de Broglie wavelength, referred to by the symbol $\lambda$, is the wavelength associated with the quantum object. It turns out, that to see the wave-like properties, the size of the slits has to be similar to $\lambda$.

The formula is $\lambda = h / mv$. Here, $h$ is just a number that comes from quantum mechanics and can be forgotten about. The $m$ and $v$ are the mass and speed associated with the particle-like properties of the object. So, the heavier the “particle”, the smaller the associated wavelength is. This explains why you don’t see any wave-like effects for people or cars or golf balls. Just to illustrate the kind of size that we talking about, light has a $\lambda$ of half a micron or so. For electrons, it’s a few nanometers, and for buckyballs, it’s a few thousandths of a nanometer.

What is superconductivity?

Most fundamentally, a superconductor is a material which becomes a perfect conductor with no electrical resistance when it gets cold enough. It was first discovered in 1911 when some Dutch experimentalists were playing around with a new way of cooling things down, and one of the things they tried was to measure the electrical resistance of various metals as they got colder and colder. Some metals just kept doing the same things that were expected based on how they behave at higher temperatures. But for others (like mercury) the resistance suddenly dropped to zero when the temperature was lowered to within a few degrees of absolute zero: they became perfect conductors. By perfect, I mean that the amount of energy that was lost as electricity went along the superconducting wire was zero. Nowadays, superconductors are very useful materials and are used in a variety of technologies. For example, they make the coils of the powerful magnets inside an MRI machine or a maglev train, they can allow ultra-precise measurements of magnetic fields in a device called a SQUID (superconducting quantum interference device), and in the future, there is some chance that junctions between different superconductors might be crucial for implementing a quantum computer.

So, how does this work?

Before I try to explain that, there is one crucial bit of terminology that I have to introduce. The types of particles that make up the universe can be classified into two types: One type is called fermions, the other type is called bosons. The big difference between these two types of particles is that for fermions, only one particle can ever be in a particular quantum state at any given time. For bosons, many particles can all be in the same state at the same time. The particles that carry electricity in metals are electrons, and they are a type of fermion. But when two fermions pair up and form a new particle, this new particle is a type of boson. Superconductivity happens when the electrons are able to form these boson pairs, and these pairs then all occupy the lowest possible energy state. In this state, they behave like a big soup of charge which can move without losing energy, and this gives the zero resistance for electrical current which we know as superconductivity.

This leaves a big unanswered question: How do the electrons pair up in the first place? If you remember back to high school, you probably learned that two objects with the same charge will repel each other, but that opposite charges attract. All electrons have negative charge and so should always repel, so how do they stay together close enough to make these pairs? The answer involves the fact that the metal in which the electrons are moving also contains lots of atoms. These atoms are arranged in a regular lattice pattern but they have positive charge because they have lost some of their electrons. (This is where the free electrons that can form the pairs come from.) So, as an electron moves past an atom, there is an attractive force between them, and the atom moves slightly towards the electron. Because electrons are small and light, they can move through the lattice quickly. The atoms are big and heavy so they move slowly and it takes them some time to go back to their original position in the lattice after the electron has gone by. So, as the electron moves through the lattice, it leaves a ripple behind it. A second electron some distance from the first one now feels the effect of this ripple, and because the atoms are positively charged, it is attracted to it. So, the second electron is indirectly attracted to the first, making them move together in a pair.

In the language of quantum mechanics, these ripples of the atoms are called phonons. (The name comes from the fact that these ripples are also what allows sound to travel through solids.) From this point of view, the first electron emits a phonon which is absorbed by the second electron, effectively gluing them together. But why does the metal have to get very cold before this phonon glue can be effective? The reason is that heat in a crystal lattice can also be thought of in terms of phonons. When the metal is warm, there are lots and lots of phonons flying around all over the place and it’s too chaotic for the electrons to feel the influence of just the phonons that were emitted by other electrons. As the metal cools down, the number of temperature phonons reduces, leaving only the ones that came from the other electrons, which allows the glue to work.

Two disclaimers

Two quick disclaimers before I finish.

Number one: I glossed over one inconvenient fact when I described the electrons and atoms interacting with each other. I made it sound like they were small particles moving around like billiard balls. For the atoms, this is a reasonable picture because they pretty much have to stay near their lattice positions. But the electrons are not like that at all. Perhaps you’ve heard of particle-wave duality? In quantum mechanics, small objects like electrons are simultaneously a bit like particles and a bit like waves. That’s true here for the electrons, so they are not little billiard balls but are more wave-like. This makes it more difficult to have a good mental picture of what they’re doing, but the basics of the mechanism are still true.

Secondly, this post has been about the type of superconductivity that occurs in metals. The temperature associated with this kind of superconductivity is quite low – a few degrees above absolute zero. But there are other kinds of superconductivity which can occur at much higher temperatures. (Imaginatively, this is usually called ‘high temperature superconductivity’!) This works in a very different way to what I’ve talked about here. It’s also not very well understood and is and active area of research. Perhaps I’ll write something about that another time.