# What is the holographic correspondence?

One of the hardest things to describe in theoretical physics is what happens when lots of particles interact with each other. Essentially, it is impossible to solve this problem exactly, and so the approaches that are currently used rely on several types of approximation.

What I want to describe is how, maybe, approaches in String Theory might be used to solve some of these really important “hard” problems. There’s no way that I can explain all the details (honestly, I don’t understand them!) but hopefully this will be a picture of how weird, esoteric, and very mathematical concepts can be say something useful about reality.

This approach is generically called “holography” for reasons that will become clear(er) later.

One of the approximate approaches to describing interacting particles that has been used to great effect is called “perturbation theory”. This applies when the interactions between the particles are relatively weak. How it works could be a whole post in itself, but perhaps for now it is enough to say that the existence of perturbation theory makes some problems with weak interactions “easy” in the sense that they can be approximately solved.

Crucially, it turns out that many of the complicated string theories that try to describe how quantum gravity works have interactions between particles which can be treated in perturbation theory.

The point of holography is that it might be possible to discover a dictionary or a way of translating between the “easy” string theory and a “hard” theory with strong interactions. Using this dictionary, it is possible to start from the “hard” theory, translate the calculation into the “easy” gravity analogue, do the calculation, and translate the results back to the “hard” context.

The diagram above is a sketch of how to visualise this process. The “easy” gravity theory exists in a bulk with a certain number of dimensions, whereas the “hard” theory lives in a space which is one dimension smaller, at the edge (or “boundary”). This is where the term holography comes from: The physical theory is a hologram which is projected from the bulk like R2D2’s message from Princess Leia.

Most intriguingly, when the “hard” theory has a temperature above absolute zero (which all physical materials must have) the gravity theory contains a black hole at its centre which has an event horizon.

So, the calculation for the complicated experimental quantity that you are interested in on the boundary can be translated through the bulk to the event horizon of the black hole. There, the properties of the theory on the boundary get converted into the properties of space-time near the black hole. This is what he dictionary does. Perturbation theory can then be used to get an approximate answer in that context. Finally, the answer is moved back through the bulk to the boundary where it can be interpreted in the original context.

Of  course, the technical details of how to actually do this in mathematics is very complicated, but there is one well-understood example of this process.

Quarks are fundamental particles and can be glued together to make protons and neutrons. The particles which do the glueing are called gluons. The gluons and the quarks are strongly interacting and so they fall into the category of “hard” theories. But, there is a well-defined correspondence between a supersymmetric particle theory which lives in eight spatial dimensions and one time dimension (so, nine in total) and an “easy” string theory which lives in ten dimensions. This correspondence has been used to derive results which would otherwise not be possible.

One of the current questions for people who work on holography is whether this is just a fortuitous specific case, or whether these correspondences are more general.

In condensed matter, there are also strongly interacting materials which theorists find very difficult to describe. One really important example is the high temperature superconductor materials.

The question is whether a holographic correspondence can be found for a theory that can make predictions about these materials? To put that another way, is there a higher-dimensional, gravity-like theory which gives a theory for a superconductor as its hologram?

A lot of people are looking at this question at the moment.

There are some encouraging things which have been done already. For example, the materials which go superconducting at low temperature also have weird behaviour at higher temperatures where they don’t superconduct. These properties have been calculated within the gravity theory, and shows some similar features to those seen in experiments.

But there is also a lot that is not known yet. For example, it is very difficult to include effects of the underlying material crystal, or include the existence of the quantum-mechanical spin of the particles. Both of these details will be important to design new materials which sustain superconductivity at even higher temperature.

This is really a field which is still in its infancy, but the underlying idea behind it is intriguing: if the theorists working on it can progress to the point where it can make predictions, it would be very exciting indeed.

# What is high temperature superconductivity?

It was March, 1987. The meeting of the Condensed Matter section of the American Physical Society. It doesn’t sound like much, but this meeting has gone down in history as the “Woodstock of Physics”. Experimental results were shown which proved that superconductivity is possible at a much higher temperature than had ever been thought possible. This result came completely out of left field and captured the imagination of physicists all over the world. It has been a huge area of research ever since.

But why is this a big deal? Superconductors can conduct electricity without any resistance, so it costs no energy and generates no heat. This is different from regular metal wires which get hot and lose energy when electricity passes through them. Imagine having superconducting power lines, or very strong magnets that don’t need to be super-cooled. This would lead to huge energy savings which would be great for the environment and make a lot of technology cost less too.

I guess it makes sense to clarify what “high temperature” means in this context. Most superconductors behave like normal metals at regular temperatures, but if they are cooled far enough (beyond the “critical temperature”, which is usually called Tc) then their properties change and they start to superconduct. Traditional superconducting materials have a Tc in the range of a few Kelvin, so only a few degrees above absolute zero. These new “high temperature” materials have their Tc at up to 120 Kelvin, so substantially warmer, but still pretty cold by everyday standards. (For what it’s worth, 120K is -153°C.)

But, if we could understand how this ‘new’ type of superconductivity works, then maybe we could design materials that superconduct at everyday temperatures and make use of the technological revolution that this would enable.

Unfortunately, the elephant in the room is that, even after thirty years of vigorous research, physicists currently still don’t really understand why and how this high Tc superconductivity happens.

I have written about superconductivity before, but that was the old “low temperature” version. What happens in a superconductor is that electrons pair up into new particles called “Cooper pairs”, and these particles can move through the material without experiencing collisions which slow them down. In the low temperature superconductors, the glue that holds the pairs together is made from vibrations of the crystal structure of the material itself.

But this mechanism of lattice vibrations (phonons) is not what happens in the high temperature version.

To explain the possible mechanisms, it’s important to see the atomic structure of these materials. To the right is a sketch of one high Tc superconductor, called bismuth strontium calcium copper oxide, or BSCCO (pronounced “bisco”) for short. The superconducting electrons are thought to live in the copper oxide (CuO4) layers.

One likely scenario is that instead of the lattice vibrations gluing the Cooper pairs together, it is fluctuations of the spins of the electrons that does it. Of course, electrons can interact with each other because they are electrically charged (and like charges repel each other), but spins can interact too. This interaction can either be attractive or repulsive, strong or weak, depending on the details.

In this case, it is thought that the spins of the electrons in the copper atoms are all pointing in nearly the same direction. But these spins can rotate a bit due to temperature or random motion. When they do this, it changes the interactions with other nearby spins and can create ripples in the spins on the lattice. In an analogy with the phonons that describe ripples in the positions of the atoms, these spin ripples can be described as particles called magnons. It is these that provide the glue: Under the right conditions, they can cause the electrons to be attracted to each other and form the Cooper pairs.

Another possibility comes from the layered structure. If electrons in the CuO4 layers can hop to the strontium or calcium layers, and then hop back again at a different point in space, this could induce correlations between the electrons that would result in superconductivity. (I appreciate that it’s probably far from obvious why this would work, but unfortunately, the explanation is too long and technical for this post.)

In principle, these two different mechanisms should give measurable effects that are slightly different from each other because the symmetry associated with the effective interactions are different. This would allow experimentalists to tell them apart and make the conclusive statement about what is going on. Naturally, these experiments have been done but so far, there is no consensus within the results. Some experiments show symmetry properties that would suggest the magnons are important, others suggest the interlayer hopping is important. Personally, I tend to think that the magnons are more likely to be the reason, but it’s really difficult to know for sure and I could well be wrong.

So, we’re kinda stuck and the puzzle of high Tc superconductivity remains one of condensed matter’s most tantalising and most embarrassing enigmas. We know a lot more than we did thirty years ago, but we are still a very long way from having superconductors that work at everyday temperatures.

# 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.